The Pennsylvania State University
The Graduate School
Department of Geosciences
FAULT-PROPAGATION FOLD KINEMATICS AND DEFORMATION RATES IN THE
NORTH CANTERBURY FOLD AND THRUST BELT, SOUTH ISLAND, NEW
ZEALAND
A Dissertation in
Geosciences
by
David O. S. Oakley
2016 David O. S. Oakley
Submitted in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
May 2017 ii
The dissertation of David O. S. Oakley was reviewed and approved* by the following:
Donald Fisher Professor of Geosciences Dissertation Advisor Chair of Committee
Kevin Furlong Professor of Geosciences
Roman DiBiase Assistant Professor of Geosciences
Derek Elsworth Professor of Energy and Mineral Engineering
Demian Saffer Professor of Geosciences Associate Head for Graduate Programs and Research
*Signatures are on file in the Graduate School
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ABSTRACT
Here we investigate fault-propagation fold kinematics in North Canterbury, New
Zealand, addressing questions of how kinematic model parameters can be constrained and different models distinguished and how marine terrace uplift rates reflect fold kinematics. Kinematic models are powerful tools in the study of fault-related folding, but they are subject to problems of non-uniqueness and uncertainty. The North Canterbury fold and thrust belt provides a location where actively growing basement-involved fault- propagation folds can be studied, where uplifted marine terraces provide critical information on fold growth rates, and where the results of kinematic models can inform understanding of deformation in a seismically active and tectonically complex region. We begin by developing methods to fit trishear kinematic models to data and to estimate model uncertainty using Markov chain Monte Carlo (MCMC) methods. We then use amino acid racemization, infrared stimulated luminescence, and radiocarbon dating to provide new age dates for marine terraces uplifted by folding and faulting in North
Canterbury, where ages were poorly known before. Using the new ages, we calculate uplift rates for the marine terraces, which reveal significant temporal and spatial variations. We use two anticlines along the North Canterbury coast as examples to show that marine terraces can be used to constrain fault-propagation fold kinematic models, both by serving as originally horizontal surfaces to be restored and by facilitating comparison of uplift rates at different structural positions. These approaches allow us to distinguish between trishear and kink-band kinematic models and to constrain the values of trishear parameters, eliminating models that are consistent with the geologic evidence
iv but not the terrace uplift. By incorporating terrace uplift into MCMC simulations, we are also able to provide estimates of fault slip rate and age of folding. Ages are consistent with previous estimates, while fault slip rates are likely somewhat higher than previously thought. Finally, we test models for fault-propagation folding in North Canterbury that incorporate listric faults, we consider the implications of recent earthquake sequences and of the reactivation of inherited normal faults for understanding fault geometry at depth, and we construct a regional cross section to estimate shortening across the North
Canterbury fold and thrust belt. We find that models of rigid basement block rotation on listric faults, although often used to explain basement-involved folding, are not consistent with the style of faulting and folding seen in North Canterbury. Instead, we develop a model combining trishear with simple shear on steep listric faults, which serves to explain the regional characteristics of faulting and folding in North Canterbury. We also compare this model to the simpler fault geometries tested previously and consider the possibility that not all faults in North Canterbury fit the same model. Depth to detachment is poorly constrained by our kinematic models, but a mid-crustal detachment as proposed by previous authors is consistent with our results. Total shortening estimated from our regional cross section is consistent with the low end of estimates from the geodetic shortening rate across the fold belt and the expected age at which folding began.
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TABLE OF CONTENTS
List of Figures ...... vii
List of Tables ...... xiv
Acknowledgements ...... xvi
Chapter 1 Introduction ...... 1
1-1. Fault-Related Fold Kinematics ...... 1 1-2. North Canterbury Fold and Thrust Belt ...... 4 1-3. Overview of Chapters ...... 8 1-4. References ...... 12
Chapter 2 Inverse trishear modeling of bedding dip data using Markov chain Monte Carlo methods ...... 28
Abstract ...... 28 2-1. Introduction ...... 29 2-2. Methods ...... 31 2-3. Results ...... 37 2-4. Discussion ...... 45 2-5. Conclusions ...... 49 2-6. References ...... 51
Chapter 3 Quaternary marine terrace chronology, North Canterbury, New Zealand using amino acid racemization and infrared stimulated luminescence ...... 69
Abstract ...... 69 3-1. Introduction ...... 70 3-2. Background ...... 71 3-3. Methods ...... 77 3-4. Results ...... 80 3-5. Discussion ...... 83 3-6. Conclusions ...... 92 3-7. References ...... 93
Chapter 4 Uplift Rates of Marine Terraces as a Constraint on Fault-Propagation Fold Kinematics: Examples from the Hawkswood and Kate Anticlines, North Canterbury, New Zealand ...... 117
Abstract ...... 117 4-1. Introduction ...... 118 4-2. Background ...... 119 4-3. Methods ...... 123 4-4. Haumuri Bluff Results ...... 130
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4-5 Kate Anticline Results ...... 138 4-6. Discussion ...... 147 4-7. Conclusions ...... 154 4-8. References ...... 155
Chapter 5 Kinematic Modeling of Listric Faulting and Basement-Involved Folding in the North Canterbury Fold and Thrust Belt, South Island, New Zealand ...... 197
Abstract ...... 197 5-1. Introduction ...... 198 5-2. Background ...... 199 5-3. Methods ...... 204 5-4. Results ...... 206 5-5. Discussion ...... 217 5-6. Conclusions ...... 225 5-7. References ...... 226 Appendix A Supplementary Tables for Chapter 3 ...... 259 Appendix B GPS Data From Terrace Profiles ...... 281 Appendix C A New, Mostly Analytic Trishear Solution ...... 289 Appendix D Structural Data ...... 299 Appendix E Haumuri Bluff Kinematic Modeling – Supplementary Figures ...... 313 Appendix F Kate Anticline Seismic Sections ...... 315 Appendix G Kate Anticline Kinematic Modeling – Supplementary Figures ...... 322
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LIST OF FIGURES
Figure 1-1: Examples of fault-related folds, following three different kinematic models: (a) kink-band fault bend fold, (b) kink-band parallel fault-propagation fold, and (c) trishear fault-propagation...... 23
Figure 1-2: The Pacific-Australian plate boundary through New Zealand. MFS: Marlborough Fault System, NCFTB: North Canterbury Fold and Thrust Belt...... 24
Figure 1-3. North Canterbury Fold and Thrust Belt and surrounding structural domains. NCFTB: North Canterbury Fold and Thrust Belt, MFS: Marlborough ...... 26
Figure 1-4: Major structures and geology of the North Canterbury Fold and Thrust Belt. Modified from T. Gardner, unpublished...... 27
Figure 2-1. The geometry of the trishear zone, trishear coordinate system, and model parameters...... 54
Figure 2-2. The synthetic cross section and the model parameters used to create it. The bold line is the profile along which data were taken...... 55
Figure 2-3: Probability density functions for synthetic model parameters generated from a grid search. The dashed lines show the parameter values used to create the original model...... 56
Figure 2-4: Histograms of results from the RAM algorithm. Each histogram is divided into 100 equally spaced bins. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model...... 57
Figure 2-5: (a) Histograms of results from the APT algorithm. Each histogram is divided into 100 equally spaced bins. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model. (b) Plot of the course taken by the lowest energy chain of the APT algorithm...... 58
Figure 2-6: Comparison of original synthetic model (solid lines) and best-fitting P/S < 1 solution (dashed lines)...... 59
Figure 2-7: Contours of two-dimensional histograms of APT results. The lowest contour in each plot is 1000 models. Contour intervals, clockwise from top left, are: 4000, 2000, 2000, and 1000 models. The full parameter space was divided into 100 bins in each dimension, but only the regions of maximum probability are shown here...... 60
Figure 2-8: Histograms of APT results fit to (a) surface contacts only and (b) dip data only. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model...... 61
Figure 2-9: Histograms of APT results fit to (a) surface contacts only and (b) dip data only, without propagating errors. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model...... 62
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Figure 2-10: Histograms of APT results. (a) σ = 2.5° for dips and 2.5 length units for contacts. (b) σ = 10° for dips and 10 length units for contacts. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model...... 63
Figure 2-11: (a) Histograms of APT model results for the Waterpocket Monocline. The total number of models is 80,000. Histograms are divided into 100 bins. (b) Plot of the course taken by the lowest energy chain of the APT algorithm...... 64
Figure 2-12: Modeled cross sections of the Waterpocket Monocline, using our best fit model (solid line) and Cardozo’s (2005) best fit model (dashed line). The background is from Figure 4 of Cardozo (2005), which is in turn derived from Bump (2003)...... 65
Figure 3-1: Major features of the New Zealand plate boundary showing sampling sites. Haumuri Bluff, Motunau Beach, and Glenafric are the sites of marine terraces within the North Canterbury Fold and Thrust Belt (NCFTB) that were dated in this study. Cape Kidnappers, Whanganui, and All Day Bay were sampled for the amino acid racemization calibration. The extent of basement rocks in the inset is from Rattenbury et al. (2006) and Forsyth et al. (2008)...... 105
Figure 3-2: Aerial views of marine terraces at (A) Glenafric; view is to south, (B) Motunau Beach; view is to west, and (C) Haumuri Bluff; view is to south. Sample sites (GA, MB, and HB) and terrace units (Qt) are labeled...... 106
Figure 3-3: Three North Canterbury field sites: (a) Glenafric, (b) Motunau Beach area, and (c) Haumuri Bluff. Mapping of terraces is based on previous maps by Yousif (1987), Barrell (1989), Warren (1995), Pettinga and Campbell (2003), Rattenbury et al. (2006), and Forsyth et al. (2008), unpublished QMAP record sheets provided by GNS Science (http://data.gns.cri.nz/metadata/srv/eng/search), and interpretation of aerial photographs from Land Information New Zealand. The background is a hillshade of the 10 m DEMs from Rattenbury et al. (2006), and Forsyth et al. (2008). .. 108
Figure 3-4: Representative stratigraphy for the marine terraces dated in this study, showing the locations of samples relative to the bedrock surface. The thickness of stratigraphic units varies laterally on each terrace, including between sample sites. Stratigraphic columns are based on Carr (1970), Ota et al. (1996), and this study. For uncertainties in ages, see Tables 3-2, 3-3, and 3-5...... 109
Figure 3-5: Sample age estimates for the three study areas in relation to marine isotope stages. Interstadials are white; stadials are shaded. Stage boundaries are from Lisiecki and Raymo (2005) and substage boundaries are from Williams et al. (2015). Error bars represent 95% confidence intervals for amino acid racemization (AAR) calibration curves, 2 standard error for infrared stimulated luminesce (IRSL), and 2- sigma range of calendar-year calibrations for radiocarbon ages (14C). Labels next to points indicate the terrace designation in Table 3-1 for this study. Data are listed in Tables 3-2, 3-3, and 3-5...... 110
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Figure 3-6: Amino acid calibration curve for racemization of aspartic acid (Asp) based on the best-fitting SPK1 model and lognormal distribution (Appendix A). Dark shading represents 95% confidence intervals for mean age, and light shading represents 95% prediction intervals. Data are listed in Table 3-4...... 111
Figure 4-1: Parallel fault propagation fold uplift rate (a) and fold shape (b) compared to trishear uplift rate (c) and fold shape (d). Uplift rates are measured at 1000 m elevation for both structures and are relative to the rate of slip on the fault. Both faults have slip = 500 m, fault dip = 20°, and P/S = 1.64. ϕ = 30° for the trishear model...... 166
Figure 4-2: Sea level curves for MIS 7 to present, with our preferred highstand sea levels and ages used for uplift rate calculations. The curves shown are the coral terrace- based sea level curves of Lambeck and Chappell (2001), the V19-30 δ18O-based sea level curve of Siddall et al. (2007), and Holocene sea level for Canterbury from Clement et al. (2016). For Clement et al. (2016), we have used their glacio-isostatic modeled curve as the upper bound and a curve drawn around their lowest data points as the lower bound. Peaks within MIS 5a and c are numbered as in Table 4-1...... 167
Figure 4-3: Map of the Haumuri Bluff area. (A) Location within North Canterbury. (B) Extent of marine terraces along the Hawkswood Range. (C) The anticline at Haumuri Bluff. The maps (B and C) are based in large part on Warren (1995), with alterations based on other published maps and this study, as described in the text...... 169
Figure 4-4: (A) The inner edge of the Amuri Bluff (Qt6) terrace in the cliff on the north side of Haumuri Bluff. (B) Inner edge elevation estimates. Red x’s are DGPS points. Blue x’s are the inner edge as determined by the two methods. The dotted and dash- dotted lines represent the reconstructed terrace riser profile prior to cliff retreat or scarp diffusion. The dashed black line is an approximation of the strath surface as 6 m below the tread. The grey lines is the modern topographic profile...... 170
Figure 4-5: Correlation diagram for Haumuri Bluff marine terraces...... 171
Figure 4-6: (a) Topographic profile and data for cross section A-A′. Geologic contacts are colored by the unit below the contact, with the same colors as in Figure 4-5. (b) Best-fitting cross section for Model 1 for the Haumuri Bluff anticline. (c) Best- fitting cross section for Model 2. P/S and ϕ change when the propagating fault tip reaches the marked points...... 173
Figure 4-7: Histogram representing probability density functions for key trishear parameters for the anticline at Haumuri Bluff. P/S upper and lower refer to the values of P/S when the fault tip is above or below its position at the time that P/S changes. Histograms of additional parameters can be found in Appendix E...... 175
Figure 4-8: Geologic Map of the Kate Anticline, with locations of seismic lines and GPS surveys. Seismic lines numbered 1-5 are lines TAG06-260-01 to -05. K1 is Kate-01. GPS surveys and profile points 4-1 and 4-2 are parts 1 and 2 of survey 4. The map is based in large part on Yousif (1987), with alterations based on other published maps and this study, as described in the text...... 177
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Figure 4-9: Kate anticline terrace profiles...... 178
Figure 4-10: Correlation diagram for Kate Anticline marine terraces along topographic profiles / GPS surveys 1 and 4...... 179
Figure 4-11: Structure contour maps of the Kate Anticline for the tops of the Waikari and Ashley Formations...... 180
Figure 4-12: Kate Anticline cross sections A-A′ and B-B′, showing data along each. Torlesse was not used in the inversion but is shown here for reference. Structure contour points are used in inversions only for Model 2. Extra structure contour points were added in cross section A at the elevations at which the corresponding horizons flatten in cross section B, to conform with the Model 2 assumption that structure is constant along strike...... 182
Figure 4-13: (a) Forelimb dip vs. backlimb dip for fixed-axis fault-propagation folding compared to the range of limb dips observed at Kate Anticline. In this model there is only one possible forelimb dip for each backlimb dip (Suppe and Medwedeff, 1990). (b) Relationship between fold limb dips and fault segment dips for parallel fault- propagation folds compared to the range of limb dips observed at Kate anticline. (c, d, and e) Predicted ratios of uplift rates between different structural positions on a parallel fault propagation fold as a function of the dips of the fold’s limbs, compared to the limb dips observed at Kate Anticline for (c) crest and backlimb, (d) forelimb and backlimb, and (e) hanging wall above the upper and lower fault segments...... 183
Figure 4-14: Fault slip rate ratios from MCMC results for the Kate Anticline for trishear and parallel fault-propagation folds (kink-band folds) for both models of Kate Anticline structure. Left column: ratio of rate of fault slip required to restore terrace Qt1 to rate of fault slip required to restore terrace Qt4, both in cross section A. Right column: ratio of fault slip required to restore terrace Qt4 in cross section A to fault slip required to restore terrace Qt4 in cross section B...... 185
Figure 4-15: Fault slip rates predicted for different fault and fold models for the Haumuri Bluff and Kate Anticlines. For Haumuri Bluff, slip rate is calculated using the average uplift rates from the formation of terraces Qt4 and Qt6 to present and for the interval between the formation of terraces Qt4 and Qt6, assuming ages of 106.9 ka for Qt4 and 71.3 ka for Qt6. For Kate Anticline, slip rate is calculated using the average uplift rates from the formation of terraces Qt4, Qt3, and Qt1 to present, assuming ages of 106.9 ka for Qt4, 124.5 ka for Qt3, and 216 ka for Qt1...... 187
Figure 4-16: Age of initiation of folding predicted for different fault and fold models for the anticline at Haumuri Bluff and Kate Anticline. Age is calculated using the different slip rates shown in Figure 4-15...... 189
Figure 4-17: Rates of shortening predicted for different fault and fold models for the anticline at Haumuri Bluff. Shortening rate is calculated using the different slip rates shown in Figure 4-15...... 191
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Figure 5-1: Major folds and thrust faults of the North Canterbury Fold and Thrust Belt. Note that the non-inverted normal fault that crosses cross section D-D′ is only one of a number of such faults in the offshore region (Barnes et al., 2016), which for simplicity are not shown here. Locations of faults and folds are from Yousif (1987), Litchfield (1995), Warren (1995), Rattenbury et al. (2006), Forsyth et al. (2008) and Barnes et al. (2016), and distribution of basement is from Rattenbury et al. (2006) and Forsyth et al. (2008)...... 236
Figure 5-2: Focal mechanisms in the NCFTB from the GeoNet catalogue (http://info.geonet.org.nz/download/attachments/8585256/GeoNet_CMT_solutions.c sv?api=v2), accessed February 9, 2016. Focal mechanisms are labeled with their centroid depths...... 237
Figure 5-3: Contours of detachment depth in km as a function of backlimb width and fault dip for a fold formed by rigid rotation of a basement block above a circular listric fault...... 238
Figure 5-4: Forward models of the Montserrat Anticline, using listric faults. (a) The detachment is at 6 km and the inclined shear angle is 70° (from horizontal). (b) The detachment is at 11 km and shear is vertical...... 239
Figure 5-5: (a) Shortening on the Glendhu Fault vs. detachment depth. (b) Slip on the uppermost fault segment (where the fault becomes straight instead of listric) of the Glendhu Fault vs. detachment depth. Note that in most models this is the part of the fault that reaches the surface, but in some of the models with the shallowest detachments, the fault is still listric at the surface, in which case this value is less meaningful...... 240
Figure 5-6: RMS errors of best-fit models for the Montserrat Anticline, for the three data types used to constrain the models: the locations of lithologic contacts (tops of the Torlesse basement and Broken River formation), bedding dips, and the mapped location of the Glendhu Fault at the surface...... 241
Figure 5-7: (a) Comparison of models for the Leithfield fault (cross section B-B′). The forward model used as a starting point for the Markov chains has a detachment at 10.5 km, but detachment depth was fit for as a model parameter. (b) Best-fit model (dashed lines) pre-growth strata from InvertTrishear compared to data (colored lines, from Barnes et al., 2016), when fitting an elliptical fault. (c) Best-fit model growth strata (dashed lines) compared to data (colored lines, from Barnes, 1996). Growth strata are labeled according to the stratigraphy of Barnes (1995; 1996)...... 243
Figure 5-8: (a and b) Histograms of shortening on the Leithfield Fault, when fitting (a) a listric fault of arbitrary shape and (b) an elliptical listric fault. In each case, the Markov chain was allowed to run for six million models, with the first one million removed as a burn-in period, and the remaining models subsampled at intervals of 50, so N = 100,000 for the histograms. (c and d) The path taken by the Markov chain for (c) the listric fault of arbitrary shape and (d) the elliptical listric fault. The starting value in both cases was 80 m. Note that we used an adaptive parallel tempering algorithm, such that sixteen chains were run in parallel, and only the
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untempered, “lowest energy” chain is shown here and is used to produce the histograms. See Miasojedow et al. (2013) for details of the algorithm...... 244
Figure 5-9: (a) Total shortening since formation of growth strata (unconformity surfaces) vs. age of growth strata for the Leithfield Fault. The shortening on the oldest layer is within a few meters (average 7 m) of the shortening on pre-growth strata. (b) Shortening rates with 2σ errors bars for the intervals between formation of the folded unconformities...... 245
Figure 5-10: Listric models for the anticline at Haumuri Bluff. Line C-C′ corresponds to Haumuri Bluff cross section A-A′ in Chapter 4, though here we have extended faults farther to the northwest. In both sections, we have drawn in a likely geometry for the Hundalee Fault, but it is not part of the model. (a) Two non-horizontal segments connected by a short listric segment. Here, the listric segment is only a few meters wide and is essentially a fault bend. The syncline axis, however, follows the shear angle (here 68°) and does not bisect the fault bend as in the fault-parallel flow models in Chapter 4. (b) A detachment at 11 km depth is required. Note that in this model, a very narrow trishear zone (0.2° apical angle) takes the place of the upper part of the fault. ϕ (the half apical angle) was allowed to change during fault propagation and decreased after initial folding to produce this result...... 247
Figure 5-11: Detachment depth vs. shear angle (measured from horizontal so that 90° is vertical) for two offshore Cretaceous-aged normal faults in North Canterbury...... 248
Figure 5-12: Forward model for cross section D-D′, based on seismic line GG07-103 as interpreted by Barnes et al. (2016). The trishear angle is 60°, and the propagation to slip ratio is 0.5 for the first 500 m of slip on the detachment, 1 for the next 500 m of slip, and 5 for the remainder, until the tip reaches the top of the Kowai Formation (yellow). The inclined shear angle is 80° from horizontal. (a) The initial, non- inverted normal fault and half-graben. The geometry of the basement-cover contact has been simplified slightly, and offsets on small, synthetic faults have been removed. The listric fault geometry was determined from the half-graben geometry and dip of the upper part of the fault as described in the text. (b) The fold after 500 m of slip on the detachment. The backlimb dip is about 4°. (c) The fold after 1000 m of slip on the detachment. The backlimb dip is about 10°. (d) The fold after 2000 m of slip on the detachment. The backlimb dip is about 20°...... 249
Figure 5-13: Pegasus Bay Fault (cross section E-E′). (a) Restored section from Barnes et al. (2016) with a listric fault fit to the half-graben geometry. (b) Modern day structure as interpreted by Barnes et al. (2016) with the listric fault from (a) extended into the cover. The fault as interpreted by Barnes et al. (2016) is shown dashed. (c) The fold that results from 750 m of heave applied to the fault in (b). The backlimb dip is now about 20°...... 251
Figure 5-14: Regional cross section. (a) Deformed state. (b) Restored state. HBF: Hurunui Bluff Fault, AF: Mt. Alexander Fault, MHF: Moores Hill Fault, OF: Omihi Fault, HF: Hamilton Fault, KF: Kate Fault, CB: Culverden Basin, W: Waipara Valley...... 253
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Figure 5-15: Cross sections (a, c) for folds formed by slip on a planar fault in an elastic half-space and resulting dips (positive to the left) within the hanging wall (b, d). The reference horizons are initially at depths of 1 km (red) and 1.5 km (blue). To simulate fault propagation, slip is applied in 1 m increments and the fault tip is propagated forward at the same rate (P/S = 1) from an initial depth of 2 km, stopping when it reaches the surface. The model fault is 10 km wide, with the cross section taken in the middle, its dip is 60°, and its base is at 11 km depth. Young’s modulus is 8x1011 Pa and Poisson’s ratio is 0.25 (suggested values from Toda et al., 2011). Fault slip is 1.5 km in (a) and (b) and 3 km in (c) and (d)...... 254
Figure E-1: Parameter histograms for Model 1 for Haumuri Bluff...... 313
Figure E-2: Parameter histograms for Model 2 for Haumuri Bluff...... 314
Figure F-1: Seismic line Kate-01 in time and depth. The Kate-1 well is shown in the depth section for comparison...... 316
Figure F-2: Seismic line TAG06-260-01 in time and depth...... 317
Figure F-3: Seismic line TAG06-260-02 in time and depth...... 318
Figure F-4: Seismic line TAG06-260-03 in time and depth...... 319
Figure F-5: Seismic line TAG06-260-04 in time and depth ...... 320
Figure F-6: Seismic line TAG06-260-05 in time and depth. Note that there are two possibilities for the basement-cover unconformity...... 321
Figure G-1: Alternative structure contour map...... 322
Figure G-2: Slip rate ratio (analogous to Figure 4-14)...... 323
Figure G-3: Y-coordinate (elevation relative to sea level) of the fault tip...... 324
Figure G-4: Total slip on the fault. (For kink-band models in which slip is not conserved across fault bends, this is the slip on the lower segment of the fault...... 325
Figure G-5: Contours of probability for dips of the upper and lower fault segments. These two parameters are highly correlated...... 326
Figure G-6: Restored elevation of the top of the Waikari formation ...... 327
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LIST OF TABLES
Table 2-1: Grid used for the grid search algorithm...... 65
Table 2-2: Means (μ) and standard deviations (σ) calculated from the grid search results for the synthetic section...... 66
Table 2-3: Means (μ) and standard deviations (σ) calculated from the APT results for the synthetic section...... 66
Table 2-4: Parameter space for the Waterpocket Monocline inversion...... 66
Table 2-5: Mean values of model parameters for different uncertainties in dips (σdip) and contact location points (σpoint)...... 67
Table 2-6: Standard deviations for model parameters for different uncertainties in dips (σdip) and contact location points (σpoint)...... 68
Table 3-1: Names and ages of marine terraces based on previous work and this study ...... 112
Table 3-2: Analytical data used to calulate luminesence ages...... 113
Table 3-3: Results of radiocarbon analyses...... 114
Table 3-4: Dataset used for amino acid racemization calibration...... 115
Table 3-5: Amino acid racemization ages...... 116
Table 4-1: Sea level highstands for marine terrace uplift rate calculations (with 2σ uncertainties)...... 192
Table 4-2: Uplift rate calculations for the terraces at Haumuri Bluff...... 192
Table 4-3: Interval uplift rates at Haumuri Bluff...... 193
Table 4-4: Uplift rates at Claverley and Dawn Creek...... 193
Table 4-5: Model parameter space for cross section A-A′...... 194
Table 4-6: Uplift rates for the terraces at the Kate Anticline...... 194
Table 4-7: Velocity model for depth conversion of Kate Anticline seismic data...... 195
Table 4-8: Model parameter space for Kate Anticline cross sections...... 196
Table 5-1: Earthquakes shown in Figure 5-1. Events in which both nodal planes have rake in the range 45° ≤ rake ≤ 135°, indicating a predominantly reverse sense of slip, are highlighted in yellow. Events in which the rake of one nodal plane is in this range are highlighted in orange...... 255
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Table 5-2: Comparison of models for the anticline at Haumuri Bluff...... 257
Table A-1: Sample Locations ...... 260
Table A-2: AAR Data ...... 261
Table A-3: Input data for AAR Model ...... 280
Table B-1: Haumuri Bluff GPS Survey ...... 281
Table B-2: Claverley GPS Survey ...... 283
Table B-3: Dawn Creek Survey ...... 285
Table B-4: Kate GPS Survey 1 ...... 287
Table B-5: Kate GPS Survey 3 ...... 288
Table B-6: Kate GPS Survey 4 ...... 288
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ACKNOWLEDGEMENTS
Funding for this project was provided by Geological Society of America, American
Association of Petroleum Geologists, and Sigma Xi graduate research grants, the Penn State
Department of Geosciences Scholten-Williams-Wright fellowship and P.D. Krynine awards, a
Shell geosicences energy research facilitation award, and an NSF East Asia and Pacific Summer
Institutes (EAPSI) fellowship, as well as a Penn State Deike grant awarded to Don Fisher. Access to the Move Software Suite was provided by Midland Valley’s Academic Software Initiative.
Without the generous support of these organizations, this research would not have been possible.
I thank my advisor, Don Fisher, for his guidance and support throughout this project.
Likewise, I thank Tom Gardner, for his advice and assistance throughout. I thank also my advising committee, Kevin Furlong, Roman DiBiase, and Derek Elsworth, whose valuable input has helped to shape this dissertation. I would also like to thank Darrell Kaufman, whose expertise in amino acid geochronology has been crucial in our efforts to date marine terraces, and Uwe
Rieser and Ningsheng Wang for processing our IRSL samples and providing advice on their interpretation. I thank Phil Barnes for hosting me for an EAPSI fellowship and providing valuable insight into faulting and folding in North Canterbury, and I thank Francesca Ghisetti and Jarg
Pettinga for their insight and advice as I have worked to understand the structural geology of this region. I thank Mark Quigley for assistance in the field and valuable discussions. I would also like to thank my undergraduate professors at Williams College, particularly Paul Karabinos and
Bud Wobus, for encouraging my interest in geology and preparing me for graduate school.
I would like to thank too the many North Canterbury landowners who allowed me access to their land and expressed interest in my research. I hope that they have remained safe through the recent Kaikoura earthquake sequence, and I hope that the insight my research provides into the active faults of the region may ultimately help those who live near them.
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I thank my fellow graduate students in the Department of Geosciences, who have been my friends and peers during my time here. In particular, I thank Rebecca VanderLeest for assistance in the field and research collaboration, for being both a colleague and a friend, and for her unwavering optimism. And I thank Christine Regalla, who has provided valuable guidance from her own experience in navigating life as a graduate student and who through her graduate research first introduced me to trishear. In addition, I would like to thank Mary-Kate Stewart, an undergraduate at Trinity University whose thesis research on the terraces at Kate Anticline contributed to my own work on that structure.
Finally, I would like to thank my family. My parents, Bernie and Rebecca, have long encouraged me to pursue my interests in science, and their support and advice have been invaluable throughout my life and through the challenges of pursuing a Ph.D. My siblings, James and Elizabeth, have likewise supported and encouraged me through my time at Penn State. My grandmother too has provided a valuable voice of support. All of them have been invaluable in getting me here, and they have stood by me alike through the successes and setbacks of graduate life.
Chapter 1
Introduction
1-1. Fault-Related Fold Kinematics
Fault-related folding is a fundamental geological process that occurs throughout the world in a variety of tectonic settings (Brandes and Tanner, 2014). In these structures, folding occurs in response to slip on a fault, so there is a direct relationship between fault motion and fold growth. Fault-propagation folds, the focus of this work, are a type of fault-related folds in which deformation occurs ahead of a propagating blind fault tip. Slip on the fault at depth is accommodated by folding of the overlying rock units. Since the fault tip propagates upward during fold growth, total separation on the fault decreases updip. Understanding fault-related folds, including fault-propagation folds, and interpreting fault and fold geometry and deformational history are of critical importance in many applications (Groshong et al., 2012;
Woodward, 2012; Brandes and Tanner, 2014) including resource exploration (e.g. Roure and
Sassi, 1995; Sciamanna et al., 2004; Toro et al., 2004; Fowler et al., 2004), earthquake hazard analyses (e.g. Allmendinger and Shaw, 2000; Champion et al., 2001; DeVecchio et al., 2012), and understanding tectonic histories and processes (e.g. Morell et al., 2008; Regalla et al., 2010;
Poblet and Lisle, 2011; Calamita et al., 2012; McQuarrie and Ehlers, 2015). Of particular interest to this study are basement-involved fault-propagation folds – a style of folding that occurs in many parts of the world (Narr and Suppe, 1994), in which thrust faults uplift basement rock and deformation is not confined to the layered, generally weaker cover rocks.
Balanced cross sections are a primary tool in studying fault-related folds. Cross sections of geologic structures have been used for over a hundred years since the work of Chamberlin
2 (1910) to make predictions of subsurface geology (Groshong et al., 2012), and the concept of a balanced cross section, in which bed length and/or area are conserved during deformation, has been in use since Dahlstrom (1969). Modern understanding of fault-related folding is based in large part on kinematic models, which have been developed since the 1980s. Kinematic models not only facilitate cross section balancing and restoration, but also make predictions about the relationship between faulting and folding, help to distinguish between types of folds that are formed by different underlying physical processes, and allow fold geometry and the movement of material points to be reconstructed over time. They include the widely-used kink-band models for fault-bend and fault propagation folding (Suppe, 1983; 1985; Suppe and Medwedeff, 1990) and the trishear model for fault-propagation folding (Erslev, 1991) (Figure 1-1). These different approaches rely on different assumptions about the underlying processes of deformation and predict different fold shapes. Kink-band models assume that deformation occurs by layer-parallel slip (Suppe, 1983); they conserve bed length and produce angular fold hinges and self-similar fold growth. Trishear assumes distributed deformation within a triangular zone ahead of the fault tip; it does not conserve bed length or thickness and results in curved fold limbs and progressive limb rotation.
Basement-involved fault-propagation folds pose unique challenges for kinematic modeling (Narr and Suppe, 1994). The kink-band fold kinematic models that are widely applied to thin-skinned deformation (Suppe et al., 1983; Suppe and Medwedeff, 1990) are difficult to apply to basement-involved folds, since basement rocks typically don’t deform by the bedding parallel slip assumed in those models. This difficulty inspired the development of the trishear model (Erslev, 1991). The simplest models for basement-involved folding involve deformation of the cover above rigid basement blocks; this is accurate for some natural examples, but in others, basement deformation is observed (Narr, 1993). Fault geometry within basement can also be
3 difficult to constrain, and in at least some cases, basement-involved faults are thought to be listric
(Erslev, 1986).
Kinematic models can be formulated in terms of material velocity (Waltham and Hardy,
1995; Hardy and Poblet 1995; Hardy, 1995; Hardy, 1997; Hardy and Ford, 1997; Zehnder and
Allmendinger, 2000). This approach can be used to forward model or restore cross sections by moving individual points according to the velocity field for the chosen kinematic model. In addition, the velocity description of deformation allows kinematic models to make predictions about rock uplift rates (Hardy and Poblet, 2005), which raises the possibility of using rock uplift rates and fold geometry to jointly constrain the kinematics of actively-growing fault-related folds.
A single balanced cross section is often a non-unique solution (Thorbjornsen and Dunne,
1997; Groshong et al., 2012; Brandes and Tanner, 2014), since more than one deformational path can lead to the same or similar final geometry. Recent work has highlighted the uncertainties in balanced cross sections and in interpretations of geological data (Bond et al., 2007; Judge and
Allmendinger, 2011; Allmendinger and Judge, 2013). When a kinematic model for fault-related folding is used to produce a balanced cross section, there may be uncertainty in the choice of model and in the values of model parameters (e.g. fault tip position, fault dip, total slip, and model dependent parameters such as trishear angle and the ratio of fault propagation to fault slip).
The problem of constructing a balanced cross section to fit available data (surface geologic data, interpreted seismic reflection data, well data, etc.) can be considered as a data inversion problem.
As such, it can be approached using methods that are applied to other inverse problems, such as
Monte Carlo simulations and Markov chain Monte Carlo methods, which involve testing various combinations of model parameters to find those that best fit the data. This is a relatively new and understudied approach to cross section balancing and to the study of fold kinematics, which offers a way to address the problem of non-uniqueness. Data inversion methods are not typically necessary for kink-band kinematics, since these models predict exact relationships between fault
4 and fold shapes, which can be solved with the help of graphs relating key model parameters such as fault dips and fold interlimb angles (Suppe, 1983; Suppe and Medwedeff, 1990). Trishear cannot be solved by these graphical methods, however, and it has therefore provided an impetus for the use of inverse techniques. Computational methods that have been used by previous researchers include grid searches (Allmendinger, 1998), optimization methods (Cardozo and
Aanonsen, 2009), simulated annealing (Cardozo et al., 2011), and Monte Carlo simulations
(Regalla et al., 2010). Optimization and simulated annealing methods are well-suited to finding a best-fit model, while grid searches and Monte Carlo simulations are better for exploring the full parameter space but can be too slow computationally for large, multi-dimensional parameter spaces. The limitations of these techniques suggest that other methods may be useful, such as
Markov chain Monte Carlo simulations – a commonly used data inversion technique that has not previously been applied to trishear. Much work remains to be done in developing and comparing data inversion techniques for fold kinematics, characterizing the uncertainty in kinematic models and model parameters, and investigating what data are needed to constrain or distinguish among different kinematic models.
1-2. North Canterbury Fold and Thrust Belt
In this dissertation, we investigate the kinematics of basement-involved fault-propagation folding and develop methods for fitting kinematic models to data, using the North Canterbury
Fold and Thrust Belt (NCFTB) on New Zealand’s South Island (Figure 1-2) as a case study.
Basement involved fault-propagation folds are the primary structures in the NCFTB, and marine terraces are uplifted on the limbs of actively-growing folds along the coast. These features make the NCFTB an ideal location to study not only the kinematics of fault-propagation folding but also the role that uplift rates, calculated from the marine terraces, can play in constraining
5 kinematic models. It is furthermore a region where the rates of deformation and the structure of faults have significant implications for the rates of slip on faults with seismic potential (as indicated by the 2016 Kaikoura earthquake sequence) as well as for the role of the NCFTB in accommodating deformation within a complex plate boundary region.
1-2.1. Tectonic Setting
New Zealand lies along the boundary between the Pacific and Australian plates (Figure
1-2). Along the east coast of the North Island and the northernmost South Island, the Pacific plate subducts beneath the Australian plate along the Hikurangi subduction zone, a process that began at about 25-30 Ma (Furlong and Kamp, 2009). Farther south, the buoyant Chatham Rise inhibits subduction, and the plate boundary through most of the South Island is formed by the Alpine
Fault. Southwest of the Alpine Fault, the Australian Plate subducts beneath the Pacific Plate along the Puysegur subduction zone.
The North Canterbury fold and thrust belt (NCFTB) is in the northeastern part of the
South Island. As such it lies in the region of transition between subduction and transpressional continental collision. The southern end of the Hikurangi subduction zone lies offshore, near the
Kaikoura peninsula, while the Southern Alps and the Alpine Fault lie to the west. The edge of the subducted slab has been interpreted by some authors to lie beneath the northern part of the fold and thrust belt (Reyners and Cowan, 1993; Anderson and Webb, 1994), although an alternative interpretation by Furlong (2007) places it farther north. North of the NCFTB, a series of four major strike-slip faults, termed the Marlborough Fault System, transfer slip from the Hikurangi subduction zone to the Alpine Fault. An additional zone of strike slip faulting, termed the Porters
Pass-to-Amberley Fault Zone, lies southwest of the NCFTB and has been interpreted as an immature analogue of the Marlborough Fault System (Cowan et al., 1996). The NCFTB itself
6 accommodates a small amount of shortening within this complex plate-boundary region. This is best illustrated by the work of Wallace et al., (2012), who used geodetic and fault slip data to construct an elastic-block model of deformation in the plate boundary transition region. In their model, they include a component of permanent, distributed strain within the blocks containing the
NCFTB. Without this permanent strain, modeled slip rates in the Porter’s Pass to Amberley Fault
Zone, and to a lesser extent on the Hope Fault, would be inconsistent with geologic slip rates
(Wallace et al., 2007; 2012), as estimated from paleoseismic studies (Cowan et al., 1996; Howard et al., 2005; Langridge et al., 2003; Langridge et al., 2005).
1-2.2. Geology of the North Canterbury Fold and Thrust Belt
The NCFTB is one of several structural domains in Canterbury, New Zealand (Figure 1-
3), as defined by Pettinga et al. (2001). It is characterized by thrust faulting and associated folding, which has produced topography characterized by hills and valleys. The northwestern boundary of the domain is marked by the Hope Fault, where the NCFTB borders the
Marlborough Fault System. To the west the NCFTB borders the West Culverden Fault zone, which is characterized by west-dipping thrust faults and associated folds; these faults have been interpreted as back-thrusts off the Alpine Fault (Pettinga et al., 2001). The two domains are separated by the Culverden Basin, a wide, flat depression. To the south, the NCFTB borders the
Canterbury Plains, where folding and faulting are hidden beneath thick alluvial gravels and are not expressed topographically. Both the NCFTB and surrounding regions are seismically active, as has been demonstrated by the 2010-2011 Canterbury earthquake sequence within the
Canterbury Plains and the ongoing 2016 Kaikoura earthquake sequence, which has involved faults in both the NCFTB and the region of the Marlborough Fault System.
7 The basement rock of the NCFTB, which outcrops at the surface in the crests of many anticlines, consists of Late Jurassic to Early Cretaceous indurated greywacke of the Pahau terrane, a part of the Torlesse composite terrane. It is overlain by a sedimentary sequence including the mid-Cretaceous to Paleogene marine sandstones and siltstones of the Eyre Group, the distinctive Eocene to Oligocene Amuri limestone, and the late Oligocene to Pliocene marine sedimentary rocks of the Motunau Group, with the youngest units dating from around the beginning of uplift of the NCFTB. (Browne and Field, 1985; Warren, 1995, Rattenbury et al.,
2006; Forsyth et al., 2008). The lowest units of the cover sequence are syn-rift sediments, which fill half-grabens (Nicol, 1993; Barnes et al., 2016) formed by normal faulting during the
Cretaceous separation of New Zealand from Gondwana (Laird and Bradshaw, 2004). A period of minor contractional deformation (with fold amplitudes of tens of meters over wavelengths of kilometers) during the Oligocene has been inferred from the unconformity on top of the Amuri limestone (Lewis, 1992) and from fracture orientations (Nicol, 1992), but there is generally no apparent angular unconformity between Eyre and Motunau Group rocks. Apart from this minor event, the region has not experienced deformation between the end of Cretaceous rifting and the beginning of the current episode of deformation. Folding of the cover sequence therefore reflects the modern deformational regime.
Structurally, the North Canterbury fold and thrust belt is characterized by asymmetric, fault-related anticlines. These folds are clearly basement-involved, since Torlesse basement rock is uplifted and exposed in the cores of many of these anticlines (Figure 1-4). These structures have been mapped by various authors, including Wilson (1963), Yousif (1987), Nicol (1991),
Cowan (1992), Warren (1995), Litchfield (1995), Finnemore (2004), Rattenbury et al. (2006), and
Forsyth et al. (2008). Offshore folds and faults, similar in form but smaller in amplitude, have been interpreted from seismic reflection data by Barnes (1993; 1996) and Barnes et al. (2016).
Where fault dips can be determined, they are generally steep (about 40°-70°) within the cover
8 sequence (Pettinga et al., 2001; Litchfield et al., 2014; Barnes et al., 2016). Some faults have been identified as inverted normal faults, reactivated as reverse faults (Nicol, 1993; Barnes, 2016).
Fault geometry within the basement is unknown, although several authors have postulated listric fault geometries and a possible mid-crustal detachment (Nicol, 1991; Cowan, 1992; Barnes,
1993; Litchfield, 1995; Litchfield et al., 2003; Campbell et al., 2012). The onset of deformation within the NCFTB has been estimated at about 0.8-1.2 Ma (Nicol et al., 1994; Barnes, 1996;
Barnes et al., 2016).
Uplifted marine terraces occur along large parts of the coast of the NCFTB. They are found on the limbs of mapped anticlines (Rattenbury et al., 2006; Forsyth et al., 2008), where they provide a record of uplift associated with fold growth. While the terraces have been described and mapped by a number of authors (Jobberns, 1926; 1928; Jobberns and King, 1933;
Wilson, 1963; Suggate, 1965; Carr, 1970; Bull, 1984; Yousif, 1987; Barrell, 1989; Ota et al.,
1984; 1996; Rattenbury et al., 2006; Forsyth et al., 2008), very few have previously been numerically dated (Ota et al., 1984; 1996 provide the only dates prior to this study), and estimated ages for undated terraces have varied considerably between studies. If their ages can be established accurately and the differences among previous estimates can be resolved, then the marine terraces can provide a valuable record of fold growth and deformation rates within the
NCFTB.
1-3. Overview of Chapters
Overall, this dissertation addresses two closely related problems: 1) How can we determine the most appropriate kinematic model for a given fault-related fold and provide quantitative estimates of model parameters that control fold growth? 2) What can kinematic models and uplifted marine terraces reveal about the kinematics and deformational processes,
9 rates of deformation, and tectonic significance of fault-propagation folding in the North
Canterbury Fold and Thrust Belt. With the first problem, we address the non-uniqueness of solutions that is a major difficulty in the use of fault-related fold kinematic models. To do this, we use Markov chain Monte Carlo simulations to fit kinematic models to data, and we investigate the use of marine terrace uplift rates in constraining fold kinematics. The North Canterbury fold and thrust belt provides a location where the methods we develop can be tested on actively growing fault-related anticlines and where uplifted marine terraces can be used to calculate uplift rates.
Furthermore, our study of structures in North Canterbury provides valuable insight into the rates of fault slip in a seismically active region, the role of the North Canterbury fold and thrust belt in accommodating plate-boundary deformation, and the kinematics of basement-involved folding.
Chapter 2
In Chapter 2, we develop and test methods for fitting trishear kinematic models to structural data. We derive an equation for the rate of change of bedding dip within the trishear zone, based on the trishear velocity equations of Zehnder and Allmendinger (2000). We also demonstrate the benefits of using Markov chain Monte Carlo methods for fitting trishear models to data with a focus on the adaptive parallel tempering method (Miasojedow et al., 2013). Finally, we show that probability distributions for trishear model parameters may be multimodal and address how to identify such distributions. This chapter has been published as Oakley and Fisher
(2015) in the Journal of Structural Geology. As part of the work for this chapter, we developed a computer program, called InvertTrishear, for fitting trishear models to data. Since publication of
Oakley and Fisher (2015), we have continued to further develop this program and have developed methods to make it more computationally efficient (Appendix C), and we have used it in Chapters
4 and 5.
10 Chapter 3
In Chapter 3, we develop a new marine terrace chronology for three locations in North
Canterbury, New Zealand. These terraces were largely undated previously, but knowing their ages is crucial to using them as a constraint on fold kinematic models and to calculating rates of fault slip and shortening in the NCFTB. For the youngest terraces, we used radiocarbon dating, while for older terraces we used infrared stimulated luminescence (IRSL) and amino acid racemization (AAR). The Bayesian fitting method of Allen et al. (2013) was used to calibrate
AAR ages, from samples of known age, in the first application of this method to a Pleistocene dataset. Our terrace ages differ significantly from the age estimates of most previous studies that did not have the benefit of numerical dating. The dated terraces are mostly from MIS 5a and 5c, with one from MIS 3 and one Holocene terrace. Small terrace remnants at high altitude, which remain undated, likely represent the MIS 5e and 7 sea level highstands. Notable results, beyond the new terrace chronology, include the lack of MIS 5e ages among the most extensive terraces and the partial reoccupation of the outer edge of one exceptionally wide MIS 5a terrace during the
MIS 3 sea level highstand. This chapter has been published in Quaternary Research as Oakley et al. (2017).
Chapter 4
In Chapter 4, we investigate the value of marine terraces in constraining fault- propagation fold kinematic models. Terrace ages from Chapter 3 and inner edge elevations determined from differential GPS (DGPS) surveys are used to calculate uplift rates for marine terraces on the forelimb of an anticline at Haumuri Bluff and on the backlimb of the Kate
Anticline. We use structural data from published geologic maps and new field work, seismic
11 reflection surveys, and well data to fit fault-propagation fold models for these anticlines using
InvertTrishear. We incorporate marine terraces uplifted by fold growth directly into the inversion, fitting for the fault slip necessary to restore them. At Haumuri Bluff, we show that the lack of folding of the terraces allows us to reduce a multimodal probability distribution for the propagation to slip ratio, similar to the multimodal distributions seen in Chapter 2, to a unimodal distribution, and we test a model in which P/S is allowed to change during fold growth. At Kate
Anticline, we compare the relative uplift rates of marine terraces on different parts of the fold and show that these tend to favor a trishear model over a kink-band model. For both locations, we use the results of our models to calculate fault slip rates, shortening rates, and the likely age at which folding began, which we compare to previously published work.
Chapter 5
In Chapter 5, we investigate the possibility of listric fault geometries and a mid-crustal detachment, as suggested by previous workers in North Canterbury, but not previously tested by kinematic modeling. We show that rigid rotation of a basement block on a circular listric fault does not provide an adequate model for the style of deformation observed in North Canterbury. A kinematic model incorporating trishear ahead of the fault tip and inclined simple shear in the hanging wall above a steeply-dipping, non-circular listric fault, however, can produce the observed fold styles. Detachment depth is not uniquely determined by the surface fold structure, but a mid-crustal detachment is consistent with observations. As part of this chapter, I investigate two approaches to fitting listric fault models to data using InvertTrishear, apply the technique to on- and offshore folds, use growth strata to determine the shortening history of an offshore fold, and use the anticline at Haumuri Bluff to compare the listric fault model to the simpler model
12 used in Chapter 4. Finally, we construct a balanced cross section across the on-land part of the
North Canterbury fold and thrust belt, using listric faults, and estimate regional shortening.
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23
Figure 1-1: Examples of fault-related folds, following three different kinematic models: (a) kink- band fault bend fold, (b) kink-band parallel fault-propagation fold, and (c) trishear fault- propagation.
24
Figure 1-2: The Pacific-Australian plate boundary through New Zealand. MFS: Marlborough
Fault System, NCFTB: North Canterbury Fold and Thrust Belt.
25
26 Figure 1-3. North Canterbury Fold and Thrust Belt and surrounding structural domains. NCFTB:
North Canterbury Fold and Thrust Belt, MFS: Marlborough Fault System, WCFZ: West
Culverden Fault Zone, PPAFZ: Porters Pass-Amberley Fault Zone. Structural domains follow the nomenclature of Pettinga et al. (2001). Earthquakes are from the GeoNet earthquake catalogue
(http://quakesearch.geonet.org.nz/), accessed February 11, 2017. Earthquakes of magnitude ≥ 6 are labeled with the year in which they occurred.
27
Figure 1-4: Major structures and geology of the North Canterbury Fold and Thrust Belt.
Modified from T. Gardner, unpublished.
28 Chapter 2
Inverse trishear modeling of bedding dip data using Markov chain Monte Carlo methods
Abstract
We present a method for fitting trishear models to surface profile data, by restoring bedding dip data and inverting for model parameters using a Markov chain Monte Carlo method.
Trishear is a widely-used kinematic model for fault-propagation folds. It lacks an analytic solution, but a variety of data inversion techniques can be used to fit trishear models to data.
Where the geometry of an entire folded bed is known, models can be tested by restoring the bed to its pre-folding orientation. When data include bedding attitudes, however, previous approaches have relied on computationally-intensive forward modeling. This paper presents an equation for the rate of change of dip in the trishear zone, which can be used to restore dips directly to their pre-folding values. The resulting error can be used to calculate a probability for each model, which allows solution by Markov chain Monte Carlo methods and inversion of datasets that combine dips and contact locations. These methods are tested using synthetic and real datasets.
Results are used to approximate multimodal probability density functions and to estimate uncertainty in model parameters. The relative value of dips and contacts in constraining parameters and the effects of uncertainty in the data are investigated.
29 2-1. Introduction
Trishear is a kinematic model for fault-propagation folds, in which a triangular zone of distributed deformation occurs ahead of the fault tip. It is capable of reproducing features such as non-uniform forelimb dips and footwall synclines (Erslev, 1991) — features that are not explained by kink band models (Suppe, 1985; Suppe and Medwedeff, 1990) but are frequently observed in the field. Deformation within the trishear zone can be described in terms of a velocity field (Hardy and Ford, 1997; Zehnder and Allmendinger, 2000) in which the velocity of a particle at any given point can be calculated from its position relative to the fault tip. By integrating this velocity through the slip of the fault as the fault tip propagates, the geometry of a trishear fold can be modeled from the initial stratigraphy and the trishear model parameters: tip position, fault dip, fault slip, trishear zone apical angle, propagation to slip ratio, and the concentration factor, which if greater than one concentrates deformation toward the center of the trishear zone. No analytic solution is known, but the integration can be performed numerically.
Balanced cross sections, although fundamental to structural geology, are often non- unique, and multiple interpretations may be possible. This is especially likely when subsurface data are absent. Interpretations based on trishear kinematics are by no means exempt from these concerns, and the six or seven (if the concentration factor is included) free parameters in the model make it likely that there will be a range of reasonable interpretations. Interpretations, such as shortening estimates based on a balanced cross section, may have substantial uncertainty
(Judge and Allmendinger, 2011) and will be more useful if this uncertainty can be quantified.
Trishear folds, which must be modeled numerically, are amenable to methods that require testing a large number of potential cross-sections. Such techniques have been used to find a best-fit trishear model for given data beginning with the work of Allmendinger (1998), who used a grid search over the parameter space. The grid search approach has subsequently been employed by
30 other authors (e.g. Allmendinger and Shaw, 2000; Allmendinger et al., 2004; Cardozo, 2005; Lin et al., 2007), principally with the aim of identifying a best-fit model.
Quantifying uncertainty in trishear parameters presents a greater challenge than finding a best-fit model alone, but it is essential to a full understanding of a structure. Although not commonly done, formal uncertainty can be estimated from grid search results, as we demonstrate below. Other methods to quantify uncertainty in trishear model parameters have been proposed by Cardozo and Aanonsen (2009) and Regalla et al. (2010). Cardozo and Aanonsen (2009) use the randomized maximum likelihood (RML) method, in which the best fit is found by optimization for many different realizations of the data and the results are plotted as histograms for each trishear parameter. Regalla et al. (2010) use a Monte Carlo simulation and plot histograms of all models for which the objective function is below a certain threshold. Cardozo et al. (2011) have also shown that simulated annealing can be used to estimate the range of possible models. In this study, we propose the use of Markov chain Monte Carlo methods. Such methods, while not previously applied to trishear, are commonly used for data inversion and are capable of both finding a best fit and estimating uncertainty (Tarantola, 2005).
Trishear kinematics is fully reversible and so can be used to test possible cross-sections by restoration (Allmendinger, 1998). A bed trace, imaged in seismic data or exposed in outcrop along the length of the fold, can be restored in this manner, and the model results can be compared to a predicted geometry using an objective function (such as distance from a point to a best fit straight line) in order to evaluate the goodness of fit. Mapped contacts along a surface profile can also be restored and matched across the fold or to a known original depth, but dip measurements pose more of a problem. The approach used in previous work to match dip measurements to trishear models (Cardozo, 2005; Regalla et al., 2010) has been to forward model beds, interpolate dips between modeled points, and attempt to match the observed dips.
31 In summary, work by various authors has demonstrated the value of the inverse method, using a variety of inversion algorithms, in fitting trishear models to data. More limited, but significant, work has shown that the technique can be applied to datasets consisting of dips or of mixed dip and point data and has proposed some methods by which error in trishear parameters can be estimated. We build on this body of knowledge by developing a method for the direct restoration of dip data, testing Markov Chain Monte Carlo techniques for trishear data inversion and error estimation, and investigating the relative value of contact positions and bedding dips in constraining a trishear model.
2-2. Methods
2-2.1 Velocity Equations
The trishear velocity field proposed by Zehnder and Allmendinger (2000), while not the only possible velocity field, is among the simplest and most widely used, often in its linear form
(s = 1 in Eq. (1)) (Hardy and Allmendinger, 2011). This formulation is supported by the finite element modeling of Cardozo et al. (2003) for an elastoplastic material, which produces a similar fold shape. This velocity field is defined with reference to a coordinate system (x,y), for which the origin is at the fault tip and moves with it as the fault propagates. A second coordinate system
(ζ,η) has its origin fixed at the initial fault tip position. The trishear zone is a triangular region defined by the angle (φ) between its boundaries and the x-axis. The tangent of φ is denoted m. In some cases, the trishear zone may be asymmetric, requiring two φ values (Zehnder and
Allmendinger, 2000), but we will focus on the symmetric case. The hanging wall velocity outside the trishear zone is labeled v0. Figure 2-1 shows the geometry of the trishear zone with the
32 coordinate axes and important variables indicated. The trishear velocity field of Zehnder and
Allmendinger (2000) is:
1 v |y| s v = 0 [sgn(y) ( ) + 1] x 2 mx
(1+s) (1) v m |y| s v = 0 [( ) − 1] y 2(1 + s) mx
−xm ≤ y ≤ xm, s ≥ 1 where sgn(y) indicates the sign of y, and s is a concentration factor. Increasing s concentrates deformation toward the center of the trishear zone. When s = 1, the field is termed "linear"
(Zehnder and Allmendinger, 2000) or "homogeneous trishear" (Erslev, 1991). This velocity field properly describes the velocities in the (ζ,η) coordinate system. In the (x,y) coordinate system there is an additional component of relative motion in the x direction, equal to –v0(P/S), due to fault propagation. P/S is the ratio of fault propagation to fault slip. In summary, there are seven parameters that determine the form of a trishear fold: the two coordinates of the fault tip (xt, yt), the fault ramp angle (fault dip), the total slip on the fault, P/S, ϕ, and s.
This velocity field can be used directly to restore a folded bed by incrementally moving points along the bed. For dip data, the forward modeling approach (Cardozo, 2005; Regalla et al.
2010), requires moving a large number of points for each dip, which is time-consuming. If dips are instead restored directly, via an equation for the rate of change of dip, dip data can be inverted in the same manner as point data. This approach is much more efficient, allows a restoration approach to be used with datasets that contain both dip and point (bed or contact) data, and also reduces the errors inherent in interpolating between points.
The velocity field of Eq. (1), or any trishear velocity field, can be used to define a strain rate tensor in two-dimensions:
33 휕v 휕v x x 휕x 휕y 퐞̇ = (2) 휕v 휕v y y [ 휕x 휕y ]
If a dip in cross-section is defined by an angle that it makes with the x-axis in the trishear coordinate system, θ, then the strain rate tensor can be transformed into a coordinate system aligned with the dip:
(3) 퐞̇ ′ = 퐚T퐞̇퐚 where ė′ is the strain rate tensor in a Cartesian coordinate system with its x′-axis parallel to the dip, and a is the rotation matrix:
cos θ − sin θ 퐚 = [ ]. (4) sin θ cos θ
The off diagonal term ė′21 of the rotated strain rate tensor will then be the rate of rotation of a line parallel to the x′-axis (Allmendinger et al., 2012) and will therefore be equal to the rate of change of dip.
∂v ∂v ∂v ̇ ′ x y 2 x 2 θ = 퐞̇ 21 = − cos θ sin θ + cos θ − sin θ ∂x ∂x ∂vy (5) ∂vy + cos θ sin θ ∂y
For the trishear velocity field of Eq. (1), this simplifies to
1+s 1−s 2 2s 2s v0 |y| 1 |y| (6) θ̇ = − [√m ( ) cos θ − sgn(y)√ ( ) sin θ] 2sx mx m mx
Note that Eq. (6) uses the convention that θ is positive counterclockwise up from the x- axis, meaning that in cross-section with x positive to the right, dips down to the left will be positive, while those down to the right will be negative. This convention can be readily switched if desired. Eq. (6) can be used along with Eq. (1) to incrementally move and rotate a point, thus allowing a dip measurement to be restored to its pre-deformational position and dip.
34
2-2.2 Uncertainty Analysis
For any inversion of bed or dip data, each model test produces a measure of misfit according to the objective function. This misfit can be translated into a probability, P(d | p), that is the probability of obtaining the observed data (d) given the parameters of the tested model (p).
If errors in the data are uncorrelated and normally distributed with uncertainty σ, this probability will be:
χ2 − (7) P(퐝|퐩) ∝ e 2 where χ2 is the sum of squares of the errors divided by σ2. Assuming no a priori information and ignoring uncertainties in the theoretical relationship between d and p, P(d | p) will be proportional to P(p | d) or the probability of the model parameters given the data (Tarantola and
Valette, 1982, their Eq. 6.9). If one does have prior information, this can be multiplied in to the final probability.
Uncertainties are typically estimated for data in the deformed state, but the inverse (cross- section restoration) method produces an error for the undeformed state. It is therefore necessary to propagate errors in position or dip through deformation in the trishear zone. If trishear deformation is solved numerically in a series of timesteps of length Δt or slip increments of length Δx = v0*Δt, then the uncertainties after each step can be derived from Eq. (1) for points moving through the trishear zone and from Eq. (6) for dips. For points, the uncertainties in x and y, here termed δx and δy, are:
35
훿푥푡+Δ푡
1 2 1−푠 2 (8) 푣 Δ푡 |푦| 푠 푣 Δ푡 |푦| 푠 = √(1 − 0 푠푔푛(푦) ( ) ) 훿푥2 + ( 0 ( ) ) 훿푦2 2푠푥 푚푥 푡 2푚푠푥 푚푥 푡
and
훿푦푡+Δ푡
1 2 1+푠 2 (9) 푣 Δ푡 |푦| 푠 푚푣 Δ푡 |푦| 푠 = √(1 + 0 푠푔푛(푦) ( ) ) 훿푦2 + ( 0 ( ) ) 훿푥2 2푠푥 푚푥 푡 2푠푥 푚푥 푡
Since |y| ≤ mx within the trishear zone, the change in δx and δy with each time step will usually be small, especially for large values of x.
The uncertainty in the dip, δθ at time step t+Δt will be a function of the previous δθ and also of uncertainties in the position (δx, δy), since 휃̇ is a function of θ, x, and y. The equation for error propagation for θ is therefore
휕휃̇ 휕휃̇ 휕휃̇ 훿휃푡+Δ푡 = √(1 + Δt ) 훿휃푡 + Δt 훿푥푡 + Δt 훿푦푡 휕휃 휕푥 휕푦 (10)
where
1 휕휃̇ 푣 |푦| 푠 1 푦 푥 = 0 푠푔푛(푦) ( ) [ ( − ) sin(2휃) + cos(2휃)], 휕휃 sx 푚푥 2 푥 푦 (11)
휕휃̇ 푣 |푦| 1/푠 푦 = 0 푠푔푛(푦) ( ) [(1 + 2푠) ( ) cos2 휃 휕푥 2푠2푥2 푚푥 푥 (12) 푥 − (2 + 2푠) sin 휃 cos 휃 + sin2 휃] 푦
36 and
1 휕휃̇ 푣 |푦| 푠 = − 0 푠푔푛(푦) ( ) [(1 + 푠) cos2 휃 휕푦 2푠2푥2 푚푥 (13) 푥 푥 2 − 2 ( ) sin 휃 cos 휃 + (1 − 푠) ( ) sin2 휃] 푦 푦
where, as in Eq. (6), θ is considered positive for dips down to the left in the cross section and negative for dips down to the right.
If the uncertainties in position of dip measurements are considered to be negligible relative to the uncertainty in dip, then Eq. (10) simplifies to
1 푣 Δt |푦| 푠 1 푦 푥 훿휃 = (1 + 0 푠푔푛(푦) ( ) [ ( − ) sin(2휃) 푡+Δ푡 sx 푚푥 2 푥 푦 (14)
+ cos(2휃)]) 훿휃푡.
This might occur if the positions of dip measurements are well constrained by GPS but more significant uncertainty exists in the dips themselves. It is also suggested by the fact that Eq. (11) has an x-1 term where Eqs. (12) and (13) have x-2, which will make the effect of uncertainty in position much smaller than that of uncertainty in dip for large values of x. For this paper, we use this simplification.
This interpretation of the error in the restored model in terms of probability means that each model tested is sampled from a density function over the parameter space, which when normalized is the probability density function for the parameters. Given that this density function exists in six dimensions when solving for fault tip position, total slip, ramp angle, P/S, and ϕ, it cannot be visualized directly, but by integrating over this density function, one can calculate the marginal probability density functions for each parameter as well as statistics such as expected
37 value, standard deviation, and covariance (Tarantola and Valette, 1982). The probability density function allows the uncertainty and range of possible values for each parameter to be characterized, whether or not they follow a Gaussian distribution. With a grid search algorithm, one can readily integrate over the grid of probabilities using quadrature. Alternatively, the treatment of the error as a probability lends itself to solution by Markov Chain Monte Carlo methods, which sample directly from the final probability distribution.
To test these new techniques, we have developed a program for trishear inverse modeling, using Eqs. (1) and (6) to restore points and dips respectively and Eqs. (8), (9), and (14) to propagate errors. The program is capable of using several inversion techniques, including grid search and multiple forms of Markov Chain Monte Carlo simulation. We apply this program first to a synthetic section and then to a natural example.
2-3. Results
2-3.1. Synthetic Section
To test the techniques presented here, we first use data derived from a synthetic cross- section. This approach allows us to test the algorithms in a controlled case before applying them to natural, and thus more uncertain, structures. The synthetic section (Figure 2-2) was created using the FaultFoldForward v. 6.1.1 program by Allmendinger
(http://www.geo.cornell.edu/geology/faculty/RWA/programs/faultfoldforward.html), so that the program used for inversion could be validated against this older, more-tested forward-modeling trishear program. Length is in generic units, which could be scaled as appropriate to be representative of folds from outcrop to map scale. Beds were initially horizontal at elevations
38 from 50 to 250 units, spaced every 10 units. Modeled points were spaced every 1 unit along the beds, and fault slip proceeded in 1 unit increments. Model parameters are shown in Figure 2-2.
Data were extracted from the model along a line at a 10° slope, as shown in Figure 2-2.
This line was chosen as one that would intersect all 21 modeled beds and pass from the hanging wall into the trishear zone. Contact locations, where the line intersected a bed, were taken from every fifth bed starting with the lowest, and dips were calculated at every bed, using the slope between the two nearest modeled points. In this manner, a simulated profile of dip and contact data was produced. The simulated profile was then perturbed to simulate measurement error, using a Gaussian distribution with standard deviation of 5° for dips and 5 units (horizontally and vertically) for contact positions. The positions of dip measurements were not altered.
Inversion was first performed using a grid search. Grid parameters are shown in Table 2-
1. The concentration factor, s, was assumed to be 1, and a model was generated to fit for all six other parameters. In total, 482,599,971 models were tested. This took 56,247 s (about 15.6 hours), running on a desktop computer with an Intel Core i7-4790K four-core, 4 GHz processor. All times quoted in this paper are for this same computer. For the grid search, eight processes were run in parallel using the OpenMP API for parallel processing, for a total processor time of about
125 hours. The undeformed stratigraphy was assumed to be flat with known bed elevations, so dip errors were simply the restored dips, and errors for the contacts were the distance between each restored contact point and a line representing the expected position of the corresponding horizon, which for initially horizontal stratigraphy is the difference in elevation between the point and the horizon. Probabilities were calculated using uncertainties in the data of 5° for dips and 5 units for contacts, which were propagated as described above. The results were integrated to calculate probability densities (Figure 2-3) and statistics (Table 2-2).
Several important results should be noted. First, despite the large number of models run, several of the marginal probability distributions come to sharp peaks, suggesting that the grid is
39 still not fine enough to accurately capture the curvature of the distribution. Second, several of the distributions are multimodal, having peaks not only at the correct solutions, but also at alternative solutions. Since means and standard deviations of the multimodal distribution are not very useful in estimating model parameters, we also list these statistics for the two groups of solutions, dividing them into those for which P/S ≤ 1 and those for which P/S ≥ 1.
Given the difficulties with the grid search approach, there is a need for a faster approach that is still capable of accurately estimating the probability density functions for the model parameters. Markov chain Monte Carlo methods are frequently used in data inversion to achieve this purpose and are suited to the probabilistic treatment of the model errors described above. One difficulty with the basic Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) in this case is that without knowing the uncertainty in the final parameter values, it can be difficult to choose a good proposal distribution from which the random walk of the algorithm will choose new model parameters. To overcome this difficulty, we first used the Robust Adaptive
Metropolis (RAM) algorithm (Vihola, 2012), which adapts the proposal distribution in order to achieve an ideal acceptance rate for proposed models in the Markov chain. Histograms of the results of the RAM algorithm are shown in Figure 2-4 for a run consisting of 1 million models, which took only 665 s (or ~11 minutes). The initial model was randomly chosen from within the parameter space, which has the same maximum and minimum values as the grid search parameter space. Dashed lines in Figure 2-4 show the correct values of model parameters that were used to make the original model. Model peaks lie near these correct values. Where the RAM algorithm falls short is that the second set of peaks, as seen in Figure 2-3, are not observed here. Thus the algorithm has not captured the full range of models that can fit the data.
To reproduce the multimodal distributions seen in Figure 2-3, we made use of the adaptive parallel tempering (APT) algorithm of Miasojedow et al. (2013). This method is another type of Markov chain Monte Carlo algorithm, in which multiple chains are run targeting
40 increasingly higher "temperature" versions of the probability distribution, meaning that the higher level chains move more easily through the parameter space. Chains are allowed to swap states, thus allowing the lower level chains to explore multiple probability maxima, and both the temperature distribution and the proposal distribution for each level are adapted to achieve an ideal acceptance rate. For this, we follow Vihola (2012) and Miasojedow et al. (2013) in using a target acceptance rate of 23.4% for both parts of the algorithm. Adaptation of the proposal distributions is performed using the RAM method. The results of the lowest level, which is untempered, are taken to represent the probability density of the model parameters.
Histograms of the results of the APT algorithm are shown in Figure 2-5a and statistics of the distributions are given in Table 2-3 for a run consisting of 8 energy levels, each with 1 million models, starting with a randomly chosen initial model. To run these models took 1224 s (~20 minutes), or longer than the RAM method but much less time than the grid search. Parallel processing was used to run models at different energy levels at the same time, but unlike the grid search, this algorithm requires frequent returns to sequential processing. Figure 2-5a and Table 2-
3 show that this method is able to reproduce the multimodal distributions seen in the grid search results (Figure 2-3 and Table 2-2), but it does so with far fewer models and with a higher resolution in areas of interest. The lowest energy chain switches regularly between the two modes
(Figure 2-5b), allowing both to be accurately represented in the histograms. A comparison of the original model used to produce the synthetic section with the best-fitting P/S < 1 model (Figure 2-
6) shows that both solutions are similar. The fit is best near the surface profile line, where it is constrained by data. If a different line is used, the same effect is seen: alternative models match the original model well along the chosen line but may differ from it away from that line. The difference between the two models in Figure 2-6 is greatest in the forelimbs of the folds, where the curvature is greatest and where the shape of the fold is most sensitive to the trishear parameters. Good data in the forelimb are thus necessary to distinguish between possible models.
41 Since trishear parameters may covary, contours of two-dimensional histograms are plotted in Figure 2-7. The division between models with P/S < 1 and P/S > 1 appears clearly in plots including P/S and in the yt vs. xt plot, where the two sets of models define two distinct regions in which the fault tip may be found. There is a positive correlation between xt and yt, showing linear regions in the cross-section space where the tip is most likely to be found. There is also a strong negative correlation between ramp angle and total slip, which is likely a product of the need to restore the contacts by a set vertical distance to their undeformed elevations. An approximately inverse correlation between P/S and ϕ has been noted by previous authors
(Allmendinger et al., 2004; Cardozo, 2005; Hardy and Allmendinger, 2011). In Figure 2-7, we observe this behavior when P/S > 1, but we see a positive correlation between the two variables when P/S < 1. Thus, more generally, P/S approaches 1 as ϕ increases.
The use of both dips and contacts along the surface profile provides more data than either alone. APT model runs of 1 million models each were produced using each type of data separately. When using just the points for the five contacts, the probability distributions are much wider, indicating a loss of precision (Figure 2-8a). Peaks remain near the correct values, but may be offset somewhat, as is particularly noticeable for total slip. The bimodal nature of the distribution remains visible in the xt and P/S histograms, but the two peaks are less distinct from each other. When using just dip data, precision is also lost and the losses in accuracy are more severe (Figure 2-8b). The histogram for ϕ, for instance, has its mode at about 45° and remains high at the boundary of the parameter space, but it is low at the correct value of 30°.
To compare the effect of propagating the uncertainty to that of assigning an uncertainty to the restored data, we ran the APT algorithm for the bed and dip data individually without propagating uncertainty. Uncertainties of 5° and 5 length units were then used to calculate a probability from the errors in the restored data. Results are shown in Figure 2-9a and Figure 2-9b.
42 For contacts, the histograms in Figure 2-9a are not very different from those in Figure 2-8a, indicating that the effects of propagating errors are minimal. As noted above, Eqs. (8) and (9) suggest that the change in uncertainty is likely to be small with each slip increment, and these results support that assumption. For dips, however, this assumption is not valid. Histograms for total slip, ramp angle, and ϕ in Figure 2-9b all have peaks at the limit of the allowed parameter space for high values of slip and low values of ramp angle and ϕ. These results are not close to the correct values that were used to make the model, and are considerably less accurate than the results obtained with error propagation (Figure 2-8b).
The data uncertainty will have an important effect on the final probability distribution. In this example, the uncertainties of 5° for dips and 5 units for contact positions were those used to perturb the data in the first place, but in a natural example they will need to be estimated based on the quality of the measurements. Figure 2-10a and Figure 2-10b show the results, from the APT algorithm, when uncertainties of half and twice these values respectively are used instead. As expected, the uncertainty in model parameters increases and decreases with the uncertainty in the data. When using half the original uncertainties, the difference between the two peaks in the histograms is greater than in the original model, with the correct value at P/S ≈ 1.5 clearly preferred over the P/S ≈ 0.5 solution. When the uncertainties are twice their original values, on the other hand, the two peaks, in addition to being broader (less precise), are more nearly equal in amplitude. Thus differences in the uncertainty of the data will affect not only the precision of the results, but also the degree to which alternative models appear likely.
2-3.2. Waterpocket Monocline
Cardozo (2005) uses the Waterpocket Monocline, a Laramide structure in southern Utah, as an example in the paper that first introduced a method for fitting trishear models to surface
43 profile data, including dip data. As we are proposing new methods for this purpose, we apply our methods to this same example for comparison. The data are taken from Bump (2003) and consist of dips and formation contacts mapped along a topographic profile as well as contact points on a footwall stratigraphic column, derived from well logs and projection from the surface. We assume that the undeformed stratigraphy is flat, with the bed contacts at the elevations shown in the footwall stratigraphic column of Bump (2003). While Cardozo's (2005) method used only the dips as measured at the contacts, we use all the dips shown in the profile, since by our method they can be restored individually. We also use a larger parameter space (Table 2-4) than that used in Cardozo's (2005) grid search. We use the APT algorithm with 100,000 models and 10 energy levels. This run took about 57 minutes; this was significantly slower than the synthetic model, due to the much larger slip distances, and thus larger number of steps for deformation within the trishear zone. The first 20,000 models were removed as a burn-in period before plotting histograms, since this was approximately the number needed for the Markov chain to find the regions of high probability after starting from a randomly chosen value within the large parameter space. Uncertainties are 5° for dips and 25 m for contacts. 5° is an estimate of the likely uncertainty in dip measurements, while 25 m is an approximation of the uncertainty in our digitization of contact positions from the surface profile in Cardozo (2005).
2-Figure 11a shows histograms of the inversion results for the Waterpocket monocline.
Like those for the synthetic section, these results show two significant modes, one with P/S > 1 and one with P/S < 1. Unlike the synthetic section inversion, in which the Markov chain switched frequently between the two solutions (Figure 2-5b), in this case the chain switched from the P/S >
1 solution to the P/S < 1 solution only once, due to the great distance between the two (Figure 2-
11b). A further test (with one million models) showed that after this switch, the chain did not switch back. Thus the relative magnitude of the two peaks is not in this case a good indicator of their relative probability; it is only the fact that both exist and provide good solutions that should
44 be noted. In fact, the best models in the P/S <1 group produced somewhat higher probabilities than the best P/S > 1 models.
The P/S < 1 solutions correspond to ramp angles very near vertical. Models in this category have an average ramp angle of 89.9°. While mathematically valid as a solution for trishear kinematics, these solutions are mechanically unrealistic. Reverse slip on an essentially vertical fault is inconsistent with the subhorizontal direction of maximum principal stress inferred for the Laramide orogeny (Erslev and Koenig, 2009). This geometry is also inconsistent with other Laramide faults, which seismic and well data show to dip less steeply (e.g. Smithson, 1984;
Stone, 1993), consistent with horizontal shortening.
Results with P/S > 1 are relatively similar to the best fit model of Cardozo (2005). The means and 1σ uncertainties for the parameters (xt, yt, slip, ramp angle, ϕ, P/S) are: (6637 ± 11.9 m, 1682 ± 19.9 m, 3157 ± 78.9 m, 43.9° ± 1.4°, 58.0° ± 1.3°, 2.36 ± 0.013), and the best-fitting model is (6638 m, 1688 m, 3173 m, 43.7°, 57.8°, 2.37). For comparison, Cardozo's (2005) grid search best fit is (6600 m, 1270 m, 3800 m, 35°, 52.5°, 2.25). These values are outside the uncertainty range estimated for our parameters but still within the same general area of the parameter space. Given the different inversion methods (APT vs. grid search), different objective functions (probability vs. chi-square statistic), differences in the parameter space (our yt and ramp angle means are outside the range tested by Cardozo (2005)) and use of additional dip data from the Bump (2003) profile, it is not surprising that there is some difference between the results.
Figure 2-12 shows a comparison of the folds produced by our best-fit model and that of Cardozo
(2005), plotted on top of the surface profile from Cardozo (2005).
In the Waterpocket example, and in many other natural cases, assigning an uncertainty to the data may be difficult. In addition to measurement errors, which may or may not be well known, the complexities of a natural structure mean that it is unlikely to perfectly follow the trishear model. To explore the effects of different uncertainty values on the results, we ran APT
45 inversions for uncertainties in dip of 2.5°, 5°, and 10° and uncertainties in contact position of 12.5 m, 25 m, and 50 m, for a total of 9 different combinations. Each inversion used 100,000 models and 8 energy levels. Ramp angle was limited to values less than 70°, in order to have only a single peak in which to measure uncertainty, and the rest of the parameter space was as in Table
2-4. The first 50,000 models were removed as burn-in time, as some runs took nearly this long to reach the region of interest, leaving the last 50,000 models from each run to be used to calculate the means and standard deviation. Means are shown in Table 2-5 and standard deviations in Table
2-6. As expected, standard deviation increases as uncertainty increases, but the means are not very sensitive to the change. In many cases, the means for the other uncertainty values are within the 1 σ uncertainty of the means for the 5° and 25 m uncertainties, and in no cases do they jump to a wholly new part of the parameter space.
2-4. Discussion
The restoration approach for dip data and the APT inversion algorithm show substantial promise in solving trishear problems. The superiority of APT over the grid search is straightforward. It can find the probability distributions of model parameters with higher resolution than the grid search and with many fewer models and thus less time required. The large number of models used in the grid search for Figure 2-3 would not be possible if the number of data or slip distance were significantly larger, since these are the two main factors controlling how many times Eqs. (1) and (6) must be evaluated. Even with this many models, the grid search produces far less resolution in the areas of interest in the probability density functions than does the APT algorithm. If one were searching for additional parameters beyond the six in this example, the grid search would become even more impractical. Such additional parameters might include s, depth to detachment, or geometry of a listric fault as in Cardozo and Brandenburg
46 (2014). APT can identify both local and global probability maxima, but in comparison to optimization methods, it has the advantage of also being able to determine uncertainty in model parameters, and it is less likely to be susceptible to the local minima that those methods can be trapped by (Cardozo and Aanonsen, 2009). Nonetheless, if only a best fit is desired and the problem is well constrained, an optimization method will likely be faster. Other methods may also be chosen depending on the situation. For example, Cardozo and Aanonsen (2009) successfully used a randomized maximum likelihood method to estimate uncertainties in four parameters for a trishear model of the Santa Fe Springs anticline based on seismic reflection data.
Markov chain Monte Carlo techniques that are simpler, and faster, than APT may also be useful in many situations. The results shown in Figure 2-4 suggest that the RAM method would be sufficient if the parameter space were sufficiently constrained that a multimodal distribution was not likely. Like these other approaches, the APT algorithm can also be used to find best-fit models and to estimate uncertainty in well-constrained problems, but it is particularly suited to exploring a large, poorly constrained parameter space and to reproducing multimodal probability distributions.
The multimodal nature of the probability density function for our synthetic model is important to note. If it were not known a priori which solution is correct, as would be the case for a natural example, it would be necessary to consider both possible solutions as valid. Such results might indicate the need for additional data in order to distinguish between the two models. In this case, the two models are divided most clearly by their P/S values. In published studies of real structures, most P/S values are greater than 1 (Pei et al., 2014), and it may be that values less than
1, while mathematically possible are uncommon in nature. More studies of natural trishear folds and a better understanding of the physical underpinnings of the propagation-to-slip ratio are needed to show whether or not this is the case. At present, however, care must be taken when assuming a particular trishear model for a fold if more than one model is possible.
47 Bedding and contact locations along a surface profile work best when used together to constrain a trishear model. Dips alone produce a less accurate result (Figure 2-8b) and either data type alone results in significantly lower precision. Different data can help to constrain different parameters, as for example dips alone provide a much more precise constraint on P/S (Figure 2-
8b) than do contacts alone (Figure 2-8a), while as noted above, dips provide a poor constraint on
ϕ.
The inaccuracy in fitting dip data alone is much greater without error propagation (Figure
2-9b) than with (Figure 2-8b). The results shown in Figure 2-9b are a more direct representation of the error in the restored dips than are those shown in Figure 2-8b. This is because in Figure 2-
9b the dip errors are converted into probabilities with the same σ for all models. Thus this approach is analogous to methods commonly used for restoring beds (Allmendinger, 1998) that seek to minimize the error in the restored beds. The maximum probability model in Figure 2-9b is also the model with the lowest RMS error in the restored dips. Specifically, RMS error for the restored dips for this model is 1.6°, while the model parameters (xt, yt, slip, ramp angle, ϕ, P/S) are (501, 187, 392, 11.1, 10.5, 0.79). In contrast, the best-fit model in Figure 2-8b has a restored dips RMS error of 9.9° with parameters of (574, 173, 231, 46.5, 45.0, 1.28). Although not perfect, these parameters are closer to the correct values shown in Figure 2-2, despite the much higher
RMS error in the restored dips. For comparison, the correct parameter values give an RMS error of 4.1° in restored dips, due to the fact that the original dips were perturbed to simulate measurement errors. For beds or contacts, the lack of major differences between Figure 2-8a and
Figure 2-9a indicates that error propagation is not critical and minimizing RMS error in the restored beds is appropriate. For dips, however, error propagation is necessary. While both show significant losses of accuracy and precision over an inversion with both dip and contact data, the results with error propagation (Figure 2-8b) are clearly more accurate than those without (Figure
2-9b). Therefore, simply minimizing RMS error in restored dips is not appropriate. Propagating
48 errors will produce a result consistent with the uncertainties in the measurements in the deformed state but with the greater efficiency that inverse modeling achieves over forward modeling.
The Waterpocket monocline example shows that our methods are applicable to a natural structure. As with the synthetic section, we found two solutions, one with P/S > 1 and one with
P/S < 1. We were able to discard the P/S < 1 solution, however, on the basis of mechanical requirements and analogy to other Laramide folds. This result highlights the fact that while trishear inverse modeling is a powerful mathematical tool, unrealistic solutions are possible. By using a method of inversion that allows us to identify multiple solutions, we are able to discover the full range of kinematically-possible trishear models, and then to evaluate them on other grounds to narrow the solution further.
Our results for the P/S > 1 case are, within reason, similar enough to Cardozo et al.
(2005), given the differences between the two approaches. Our standard deviations are relatively small and do not include Cardozo’s (2005) model within the estimated uncertainty, despite the fact that this model is visually a reasonable fit to the data (Figure 2-12). Thus a strict application of these error estimates may underestimate the range of acceptable models in some cases. The imperfect match between theory (the trishear model), with its simplifications, and the details of a natural structure may be part of the reason for this and is difficult to account for quantitatively.
Regardless of the exact uncertainty values that may be appropriate, the results of a test of different data uncertainties (Table 2-5) indicate that our mean values remain good estimates of the parameter values.
Nonetheless, our results for the Waterpocket monocline show advantages to using the
APT algorithm as opposed to the grid search and to restoring the data instead of using batch forward models. The APT algorithm allows us with a similar number of models to consider a larger parameter space than the grid search while sampling with much higher precision in the area of interest. For example, when the vertical separation between hanging wall and footwall contacts
49 is well constrained, as in this case and in the synthetic section, there is a clear covariance between total slip and ramp angle. Indeed, our best-fit total slip differs from that of Cardozo (2005) by 627 m, but the uplift of the hanging wall differs by only 11 m due to our correspondingly higher best- fit ramp angle. There is therefore a range of total slip and corresponding ramp angles that will produce approximately the correct uplift, and the grid search finds only the one(s) that happen to lie closest to the grid, while the APT algorithm can search with greater resolution for the best fit and can reveal the covariance between the two parameters.
The second benefit of our new approach comes from the dip restoration algorithm we employed, using Eq. (6). The approach used by Cardozo (2005) required the forward modeling of six beds, each of which would consist of a large number of points to be moved individually. If the dips not at formation contacts were to be used as well or if the undeformed elevations were not well known, even more beds would have had to have been modeled, with dips interpolated between them, as in the approach used by Regalla et al. (2010) for the Futaba fault in Japan. By restoring the dips directly, we have only to move and rotate a single point for each data point and we are able to restore equally well dip measurements that do and do not correspond to formation contacts.
2-5. Conclusions
Using the trishear velocity equations of Zehnder and Allmendinger (2000), we have derived an equation for rate of change of dip, Eq. (6), which we have applied to both synthetic and natural datasets. This equation can be used to restore the dips of folded beds by alternately moving the point where the dip was measured and rotating the dip. The error in the restored dip can be used with an estimated uncertainty to calculate probability for each model. Using this approach, dips and points on a bed or contact can be inverted jointly, and each model tested will
50 sample from a probability distribution over the multidimensional parameter space. The traditional grid search algorithm can be used not only to find a best-fit model, but also to estimate the marginal probability densities for each parameter, which can be calculated by integrating over the probability density function. Markov chain Monte Carlo methods, however, provide much faster solution times than the grid search and higher resolution in the areas of interest in the probability density function. In particular, we have found the adaptive parallel tempering algorithm
(Miasojedow, 2013) well suited to the trishear problem, as it is able to efficiently explore the multi-dimensional parameter space, identify multiple solutions (as represented by multimodal probability distributions), and estimate uncertainty in parameter values.
Dip and contact data along a surface profile provide a better constraint on the trishear model than either alone. Dip data, in particular, may provide a poor fit when used alone. It is also necessary to treat errors in dip data in terms of probabilities and to propagate uncertainties through the trishear deformation, as simply minimizing RMS error in restored dips is likely to be inaccurate.
Program Availability
The data inversions described in this paper were performed using a program,
InvertTrishear, written for the purpose. The program can be downloaded at www.davidosoakley.com/trishear.html, or can be obtained by contacting the corresponding author.
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54
Figure 2-1. The geometry of the trishear zone, trishear coordinate system, and model parameters.
55
Figure 2-2. The synthetic cross section and the model parameters used to create it. The bold line is the profile along which data were taken.
56
Figure 2-3: Probability density functions for synthetic model parameters generated from a grid search. The dashed lines show the parameter values used to create the original model.
57
Figure 2-4: Histograms of results from the RAM algorithm. Each histogram is divided into 100 equally spaced bins. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model.
58
Figure 2-5: (a) Histograms of results from the APT algorithm. Each histogram is divided into
100 equally spaced bins. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model. (b) Plot of the course taken by the lowest energy chain of the APT algorithm.
59
Figure 2-6: Comparison of original synthetic model (solid lines) and best-fitting P/S < 1 solution
(dashed lines).
60
Figure 2-7: Contours of two-dimensional histograms of APT results. The lowest contour in each plot is 1000 models. Contour intervals, clockwise from top left, are: 4000, 2000, 2000, and 1000 models. The full parameter space was divided into 100 bins in each dimension, but only the regions of maximum probability are shown here.
61
Figure 2-8: Histograms of APT results fit to (a) surface contacts only and (b) dip data only. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model.
62
Figure 2-9: Histograms of APT results fit to (a) surface contacts only and (b) dip data only, without propagating errors. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model.
63
Figure 2-10: Histograms of APT results. (a) σ = 2.5° for dips and 2.5 length units for contacts.
(b) σ = 10° for dips and 10 length units for contacts. The total number of models is 1 million. The dashed lines show the parameter values used to create the original model.
64
Figure 2-11: (a) Histograms of APT model results for the Waterpocket Monocline. The total number of models is 80,000. Histograms are divided into 100 bins. (b) Plot of the course taken by the lowest energy chain of the APT algorithm.
65
Figure 2-12: Modeled cross sections of the Waterpocket Monocline, using our best fit model
(solid line) and Cardozo’s (2005) best fit model (dashed line). The background is from Figure 4 of Cardozo (2005), which is in turn derived from Bump (2003).
Table 2-1: Grid used for the grid search algorithm.
Parameter Minimum Maximum Step Size xt 300 700 10 yt 50 250 10 Total Slip 0 400 10 Ramp Angle 10 50 2 ϕ 10 50 2 P/S 0 3 0.1
66 Table 2-2: Means (μ) and standard deviations (σ) calculated from the grid search results for the synthetic section.
Full Model P/S ≤ 1 P/S ≥ 1 Parameter μ σ μ σ μ σ xt 531 39.6 483 9.0 562 10.4 yt 131 25.6 103 8.1 150 12.7 Total Slip 195 20.9 196 20.3 194 21.3 Ramp Angle 32.4 3.8 32.2 3.8 32.5 3.8 ϕ 32.7 2.9 33.0 3.3 32.6 2.6 P/S 1.05 0.54 0.41 0.13 1.48 0.099
Table 2-3: Means (μ) and standard deviations (σ) calculated from the APT results for the synthetic section.
Full Model P/S < 1 P/S > 1 Parameter μ σ μ σ μ σ xt 529 38.1 484 8.9 559 8.7 yt 130 24.2 103 7.8 148 10.3 Total Slip 193 20.8 193 20.0 194 21.3 Ramp Angle 32.7 4.0 32.8 3.9 32.6 4.1 ϕ 32.3 3.2 32.6 3.1 32.1 3.2 P/S 1.05 0.54 0.400 0.13 1.48 0.10
Table 2-4: Parameter space for the Waterpocket Monocline inversion.
Parameter Minimum Maximum xt 5000 8000 yt -1000 2000 Total Slip 2000 8000 Ramp Angle 0 90 ϕ 0 90 P/S 0 5
67
Table 2-5: Mean values of model parameters for different uncertainties in dips (σdip) and contact location points (σpoint).
σdip σpoint 2.5 5 10
xt 12.5 6633 6639 6637 25 6619 6637 6638 50 6573 6630 6637
yt 12.5 1659 1697 1706 25 1610 1683 1704 50 1521 1654 1690 Slip 12.5 3177 3155 3186 25 3246 3159 3181 50 3397 3178 3150 Ramp Angle 12.5 43.5 44.0 43.5 25 42.3 43.9 43.6 50 39.9 43.6 44.1 ϕ 12.5 57.7 58.0 57.4 25 56.7 58.0 57.5 50 54.6 57.7 58.1 P/S 12.5 2.36 2.37 2.37 25 2.35 2.36 2.37 50 2.31 2.36 2.36
68
Table 2-6: Standard deviations for model parameters for different uncertainties in dips (σdip) and contact location points (σpoint).
σdip σpoint 2.5 5 10
xt 12.5 9.0 8.1 8.4 25 21.7 12.3 12.6 50 21.3 20.3 17.7
yt 12.5 14.9 12.0 11.9 25 28.8 19.3 18.7 50 26.6 30.9 28.1 Slip 12.5 49.1 74.2 84.7 25 64.9 80.2 112 50 85.0 98.7 126 Ramp Angle 12.5 0.84 1.3 1.4 25 1.0 1.4 1.9 50 1.2 1.7 2.2 ϕ 12.5 0.79 1.2 1.4 25 1.0 1.3 1.8 50 1.2 1.6 2.1 P/S 12.5 0.009 0.009 0.010 25 0.015 0.013 0.015 50 0.016 0.018 0.020
69 Chapter 3
Quaternary marine terrace chronology, North Canterbury, New Zealand using amino acid racemization and infrared stimulated luminescence
Abstract
Extensive marine terraces along the North Canterbury coast of the South Island of New
Zealand record uplift in this tectonically active area. Although the terraces have been studied previously, applications of Quaternary geochronologic techniques to the region have been limited. We use infrared stimulated luminescence (IRSL), amino acid racemization (AAR), and radiocarbon to determine ages of terraces at three locations – Glenafric, Motunau Beach, and
Haumuri Bluff. We develop an AAR calibration curve for the mollusk species Tawera spissa from sites of known age, including the sedimentary sequence of the Whanganui Basin. Bayesian model averaging of the results is used to estimate ages of marine shells from the North
Canterbury terraces. By using both IRSL and AAR, we are able to confirm ages using two independent dating methods and to identify one IRSL result that is likely in error. We develop new age estimates for the marine terraces of North Canterbury and propose correlations between sites. This terrace chronology differs significantly from most previous studies, highlighting the importance of numerical dating. The most extensive terraces are from MIS 5a and 5c, with partial reoccupation of one terrace during MIS 3, while MIS 5e terraces are notably lacking among those dated.
70 3-1. Introduction
Marine terraces, formed during sea-level highstands and subsequently uplifted above sea level, represent the combined imprint of the glacio-eustatic sea-level cycle and vertical uplift on coastal geomorphology (Lajoie, 1986; Pedoja et al., 2014). When terraces form along an uplifting coast, they can be correlated with known highstands, provided that the ages of the terraces can be determined. This study focuses on the North Canterbury region of New Zealand, where uplift related to active faulting and folding has produced extensive marine terraces along the coast.
These terraces have been the subject of a number of previous studies (Jobberns, 1926; 1928;
Jobberns and King, 1933; Carr, 1970; Bull, 1984; Yousif, 1987; Barrell, 1989; Ota et al., 1984;
1996), many of which have proposed possible correlations with sea-level highstands, and estimates of their ages have been used in studies that address rates of deformation (Nicol et al.,
1994) and the level of seismic hazard (Barrell and Townsend, 2012). Despite this body of work, the terraces remain poorly dated (Ota et al., 1984, 1996), with little geochronologic data to support proposed correlations with sea-level highstands. Previous workers have had few options for dating the terraces; most are too old for radiocarbon dating, and corals suitable for uranium series dating have not been found in New Zealand terraces (Pillans, 1990), despite the success of this technique in terrace studies in other parts of the world (e.g. Muhs, 1992). More modern techniques, such as luminescence dating and amino acid racemization, however, provide the opportunity to date the North Canterbury marine terraces and to test previous age estimates.
In this study, we report the results of IRSL and AAR analyses of sediments and shells from marine terraces. We use AAR analyses of shells from deposits of known age, predominantly from the Whanganui Basin of the southern part of the North Island, New Zealand, to calibrate the rate of AAR. This study expands on previous AAR studies in the region by analyzing the extent of AAR in multiple amino acids preserved within the intra-crystalline fraction of the shell
71 carbonate, and by using newly developed Bayesian statistical methods for developing an AAR age equation and calculating age uncertainties. We then apply this AAR age model in combination with IRSL to date marine terraces of unknown age along the coast of North
Canterbury, South Island. We demonstrate the value of AAR and IRSL as independent, but complementary methods in establishing a terrace chronology, we provide new ages for the North
Canterbury terraces that differ from previous estimates, and we develop an AAR chronology for the species Tawera spissa extending back to MIS 17 in the region.
3-2. Background
3-2.1 AAR Calibration Sample Sites
In order to calibrate the AAR age model, mollusk shells were collected from six previously studied sites, the ages of which are known independently from methods other than
AAR. Four of these sites are in the Whanganui Basin (Figure 3-1) where the ages of stratigraphic units are constrained by methods independent of AAR. The late Cenozoic marine sedimentary sequence that accumulated in near-shore Whanganui Basin was uplifted (Kamp et al., 2004) and exposed along the coast. Bowen et al. (1998) previously developed an aminostratigraphy for the
Pleistocene beds of the sequence using the amino acids D-alloisoleucine/L-isoleucine (A/I) in the molluscan species Tawera spissa and Austrovenus stutchburyi. This work demonstrated the utility of the technique in combination with tephrochronology and marine isotope stage (MIS) periodicity, and it showed that the two venerid bivalves racemize at similar rates.
Our four Whanganui samples were collected from three locations along the coast.
(1) Coastal bluffs along Castlecliff Beach (site 11 of Bowen et al., 1998) expose gently dipping early to middle Pleistocene marine strata whose ages are well constrained by multiple
72 lines of evidence including tephrochronology (Kohn et al., 1992; Alloway et al., 1993; Shane et al., 1996; Pillans et al., 2005) and cyclostratigraphy (Beu and Edwards, 1994; Carter and Naish,
1998; Naish et al., 1998). We collected T. spissa and other taxa from the Kupe Formation (MIS
17) and the Shakespeare Cliff Sand (MIS 11). Samples from several of the older formations at
Castlecliff Beach were also analyzed for AAR (Appendix A), but no T. spissa were found and data are not included in the calibration.
(2) Road- and river-cuts near the mouth of Whanganui River at Landguard Bluff expose a complicated stratigraphy of marine and terrestrial sediment (Pillans et al., 1988). We sampled T. spissa and other taxa from the Waipuna Delta Conglomerate (site 4 of Bowen et al., 1998), which is correlated with the Ngarino marine terrace (Beu and Edwards, 1984). An MIS 7c age for the conglomerate has been established by Pillans et al. (1988) on the basis of stratigraphy, paleontology, and identification of a tephra of known age.
(3) Also in the Whanganui area, sediments underlying the Hauriri marine terrace are exposed at Waverley Beach (site 2 of Bowen et al., 1998) where they have been correlated with
MIS 5a (Pillans, 1983) based on their elevation relative to terraces of known age.
Our other samples were collected from two sites not within the Whanganui Basin.
(1) Approximately 150 km to the east, an uplifted marine terrace at Cape Kidnappers (site
1 of Bowen et al., 1998) exposes Holocene marine sediment with T. spissa shells that were previously radiocarbon dated to around 2.5 ka (Hull, 1987). Although we could not re-locate T. spissa shells, we collected shells of Venerupis largillierti and analyzed them for both AAR and radiocarbon.
(2) In the Otago region on the South Island, we collected T. spissa shells at about 3 m asl exposed in a coastal bluff at All Day Bay underlying a prominent terrace. Given its elevation, the fact that this segment of coast is tectonically stable (Pillans, 1990), and the presence of other MIS
5e terraces in this area (Pillans, 1990), an MIS 5e age is indicated for this sample.
73 3-2.2. North Canterbury Marine Terraces – Tectonic Setting and Previous Work
The North Canterbury terraces lie within the North Canterbury fold-and-thrust belt, a distinct structural domain within the larger Canterbury region (Pettinga et al., 2001). The fold- and-thrust belt lies on the Pacific plate near the transition between the Hikurangi subduction zone, which forms the plate boundary to the north, and the transpressive Alpine Fault, which forms the plate boundary through most of the South Island (Figure 3-1). It is bounded to the north by the strike-slip Hope Fault, to the west by the Southern Alps, and to the south by the Canterbury
Plains (Pettinga et al., 2001).
Marine terraces (Figure 3-2) occur along much of the coastline within the fold-and-thrust belt, most prominently on the limbs of actively growing, basement-cored anticlines. Typically, there is one wide terrace, from several hundred meters to as much as 3.5 km in width (Figure 3-2a and b), with two or three smaller, more dissected terrace remnants above it (Carr, 1970; Yousif,
1987; Barrell, 1989). The marine terraces of North Canterbury were first systematically described by Jobberns (1926, 1928), and Jobberns and King (1933). These early works established the marine origin of the terraces. Carr (1970) mapped terraces in the southern part of North
Canterbury and proposed correlations with interglacial periods of Suggate (1965). Bull (1984) correlated the terraces at Kaikoura Peninsula with the U-series-dated New Guinea terrace sequence of Bloom et al. (1974) and Chappell (1983), based on the altitudinal spacing of the terraces. Yousif (1987) used remote sensing to map terraces, and assigned ages based on correlation to sea level high stands, using relative elevation, relationships to fluvial aggradation gravels inferred to date from periods of glacial advance, and proposed correlations between knickpoints and terraces. Barrell (1989) mapped and proposed ages for terraces in the vicinity of
Motunau Beach based on elevation and correlation to sea level highstands. Ota et al. (1984, 1996) were the first to apply numerical geochronological techniques (AAR and 14C) to the North
74 Canterbury terraces, working in the northern part of the area. Most recently, terraces capped by marine deposits were mapped at 1:250,000 scale (Rattenbury et al., 2006; Forsyth et al., 2008), and were assigned to MISs on the basis of elevation and degree of dissection. Table 3-1 summarizes the ages assigned to the terraces in these previous studies.
The AAR dating by Ota et al. (1996) and 14C dating by Ota et al. (1984) are the only numerical ages for the North Canterbury terraces (Table 3-1). Previous luminescence dating in the Canterbury region has focused on loess (Berger et al., 2001; Almond et al., 2007) and glacial or glaciofluvial sediments (Rother et al., 2010; Shulmeister et al., 2010; Rowan et al., 2012), although IRSL dating has proven useful in dating marine sediments in North Westland (Rose,
2011). The previous work in Canterbury has shown promise but also difficulties for the technique
(Rowan et al., 2012), including some age reversals in loess IRSL ages (Berger et al., 2001;
Almond et al., 2007). Feldspar IRSL, which we use in this study, does not suffer from the low signal intensities observed in quartz in this region (Rowan et al., 2012). It can be susceptible to anomalous fading (Huntley and Lamothe, 2001; Huntley and Lian, 2006), but this phenomenon has not been observed in previous South Island studies (Almond et al., 2001; Hormes et al., 2003;
Preusser et al., 2005; Rother et al., 2010; Shulmeister et al., 2010; Rose, 2011). Given the scarcity of numerical ages for the North Canterbury marine terraces, we dated samples from three locations along the North Canterbury coast, where marine terraces are prominent: Glenafric, the
Motunau Beach area, and Haumuri Bluff (Figure 3-3).
3-2.3. Glenafric
At the southern end of the North Canterbury fold and thrust belt near Glenafric Beach and the adjacent Glenafric farm, marine terraces are uplifted along the limb of a small anticline
(Figure 3-3a), named the Kate anticline (Wilson, 1963; Yousif, 1987). The most extensive marine
75 terrace at this site (Qt4 in Figure 3-3a) is termed the Tiromoana terrace after Carr’s (1970) name for the Quaternary sediments (marine and terrestrial) that overlie the wavecut platform in this area. It is tilted up to the northeast (Figure 3-2a), with the terrace tread rising from about 45 m to about 80 m in elevation over a distance of ~6 km. The bedrock strath is overlain by marine sand and gravel, alluvium, and loess (Figure 3-4). Previous age estimates for the terrace have ranged from MIS 3 to 5 (Table 3-1), although it has not been numerically dated. Remnants of higher terraces form small notches and hilltops, and have been grouped into two (Carr, 1970) or three
(Yousif, 1987) additional terraces on the basis of elevation.
We collected two IRSL samples from the principal terrace (Qt4), one from the marine sediment and one from the overlying loess (Figure 3-4). We collected T. spissa shells for AAR analyses from the same site as the marine sediment IRSL sample and from an additional site on the same terrace and same stratigraphic position a few tens of meters away. Sample sites are shown in Figure 3-3a.
3-2.4. Motunau Beach
An extensive marine terrace occupies the coast from south of Motunau Beach north to
Stonyhurst Creek (Figures 3-2b and 3-3b, terrace Qt6), along one limb of the Montserrat anticline. At up to 3.5 km wide, this terrace is the widest in North Canterbury. The marine and terrestrial sediments overlying the wave-cut bench were termed the Motunau Formation by Carr
(1970), and the terrace has been termed the Motunau Coastal Plain or Motunau Terrace (Jobberns and King, 1933; Carr, 1970; Barrell, 1989). Previous age estimates have varied, with most work assigning its formation to some time in MIS 5 (Table 3-1), but again there has been no numerical dating. As at Glenafric, higher terraces are preserved as notches in the hillside, which were
76 grouped into two levels by Yousif (1987) and three by Barrell (1989), but suitable materials were not found for dating.
We collected two IRSL samples from marine sand of the main terrace where the overlying sediments (Figure 3-4) are exposed in road cuts. One sample (MB2) was taken near
Motunau Beach village and the other (MB1) at Boundary Creek (Figure 3-3b). T. spissa shells for
AAR analyses were collected from four sites near the southwest end of the terrace (Figure 3-3b), from between 0.3 and 2.8 km from the former sea cliff.
3-2.5. Haumuri Bluff
Haumuri Bluff is the northernmost of our three North Canterbury study sites. Two prominent marine terraces (Qt4 and Qt6 in Figure 3-3c), capped by marine sediments and loess, are found at this location, and their correlatives have been mapped along the coast to the south
(Ota et al., 1984) on the flanks of the Hawkswood anticline. Ota et al. (1996) dated the upper terrace (Tarapuhi terrace / Qt4) to MIS 5c on the basis of an AAR age of 135 ± 35 ka and the presence of a cold-water molluscan fauna considered to be incompatible with MIS 5e. Correlating to the sea level curve of Chappell and Shackleton (1986), they gave this terrace an age of 100 ± 3 ka and assigned the lower of the two prominent terraces (Amuri Bluff terrace / Qt6) to 59 ± 3 ka, based on the same sea-level curve, given the presence of another terrace (Kemps Hill terrace) immediately to the southwest (Table 3-1, Figure 3-3c), which is between the Tarapuhi and Amuri
Bluff terraces in elevation.
We collected samples for IRSL dating from the marine sediments on both Tarapuhi (Qt4) and Amuri Bluff (Qt6) terraces, and we collected shells for AAR analyses from the Tarapuhi terrace at the location sampled by Ota et al. (1996) (Figure 3-3c and 3-4). In addition, we found a small terrace at about 4.2 m elevation cut into the Torlesse basement rock northwest of Haumuri
77 Bluff (Figure 3-3c), which we call the Torlesse terrace. Leukoma crassicosta shells from this terrace were sampled for 14C and AAR analyses.
3-3. Methods
3-3.1 IRSL
IRSL samples were collected from sand and silt lenses or layers within the marine sand and gravel units of the terraces, or in the case of sample GA1, from loess. A metal tube was inserted into the sediment to collect each sample. The ends of the tube were then covered with aluminum foil and the tube was wrapped in black plastic and taped shut to ensure darkness and moisture retention. For coastal and aeolian sediments, the luminescence signal is generally reset by exposure to sunlight during deposition (Kaufman et al., 1996; Singarayer et al., 2005), although incomplete or non-resetting was considered for samples that produced anomalously old ages.
IRSL dating of feldspar grains was conducted by the Luminescence Dating Laboratory at
Victoria University, Wellington, using the Single Aliquot Regenerative method (SAR), as described by Murray and Wintle (2000). The blue luminescence of fine-grained (4-11 μm) feldspar was measured during infrared stimulation. Preheating was done at 270° C for 30 s. Blue luminescence centered about 410 nm was detected using an EMI 9235QA photomultiplier with
Schott BG39 and Kopp5-58 filters. Preheating and the use of 410 nm blue luminescence overcomes the potential age underestimation caused by the thermally unstable 290 nm emissions peak (Krbetschek et al., 1996; Clarke and Rendell, 1997), which has been an issue for some samples from Westland, New Zealand (Rose, 2011), for which UV luminescence was measured instead. Fading tests were performed using the method of Huntley and Lamothe (2001), but no
78 anomalous fading was detected. Equivalent dose (De) values were determined for 8 or 12 aliquots and the arithmetic mean of these was used for the age calculation.
Dose rates were determined from radionuclide content, water content, and a-value and from the expected dose rate from cosmic rays. Radionuclide contents were measured in the lab using gamma spectrometry. Gamma rays were counted for a minimum of 24 h, using a high- resolution, broad-energy gamma spectrometer. Spectra were analyzed with GENIE2000 software.
U, Th, and K concentrations were determined by comparison to standard samples. The dose rate from radionuclides was calculated from the activity concentrations of 40K, 208Tl, 212Pb, 228Ac,
214Bi, 214Pb, and 226Ra, using the conversion factors of Guérin et al. (2011). Water content for each sample was calculated as weight of water divided by dry weight of the sample, with an assumed
25% uncertainty. The a-value, a measure of the contribution of alpha radiation to the luminescence signal (Aitken and Bowman, 1975), was estimated at 0.05 ± 0.01 for all samples.
The dose rate from cosmic rays was calculated from burial depth, geomagnetic latitude, and altitude, using the formulae and correction factors of Prescott and Hutton (1994).
3-3.2. AAR
Shells of marine bivalve mollusks of the family Veneridae were chosen for the analysis, with the species T. spissa used wherever possible. This species is common in New Zealand
(Powell, 1979) and has proven suitable for use in previous AAR studies (Ota et al., 1996; Bowen et al., 1998). Additional analyses were conducted on some other marine mollusk species, which provide a supporting dataset for both the calibration and the previously undated samples
(Appendix A). AAR analyses were conducted at the Northern Arizona University Amino Acid
Geochronology Laboratory using reverse-phase high performance liquid chromatography (RP-
HPLC) (Kaufman and Manley, 1998). Ratios of right- (D) and left-handed (L) isomers were
79 measured for eight amino acids: aspartic acid (Asp), glutamic acid (Glu), serine (Ser), alanine
(Ala), valine (Val), phenylalanine (Phe), isoleucine (Ile), and leucine (Leu). In contrast, Ota et al.
(1996) and Bowen et al. (1998) used ion-exchange HPLC, which focuses entirely on D- alloisoleucine /L-isoleucine (A/I). Our results are not quantitatively comparable to the previously published results for this reason, and because we pre-treated the shells to isolate the intra- crystalline fraction of amino acids, following the techniques of Penkman et al. (2008).
AAR calibration curves were calculated using the Bayesian fitting method and R scripts of Allen et al. (2013). We tested four different functions for relating sample age to D/L ratio: apparent parabolic kinetics (APK) (Mitterer and Kriausakul, 1989), simple power law kinetics
(SPK) (Goodfriend et al., 1995), constrained power-law kinetics (CPK) (Manley et al., 2000), and time-dependent reaction kinetics (TDK) (Allen et al., 2013). For each function, we tested two possibilities for the D/L value at time 0 (R0): that it is fixed at 0 and that it is allowed to vary as a fitted parameter (denoted by 0 or 1, respectively, after the function name). For the uncertainty in the age predicted from the D/L ratio, we tested both lognormal and gamma distributions, as suggested by Allen et al. (2013). Of the eight amino acids measured, Ser was not used because its
D/L ratio did not increase monotonically with age, and the TDK and CPK models were not tested for Ala and Leu because we measured D/L values greater than 1 for these amino acids. In total,
96 combinations of amino acid, age-D/L function, and probability distribution were tested.
Following the methods described in Allen et al. (2013), we used the Bayesian Information
Criterion (BIC) (Schwarz, 1978) to identify a best-fit model and for Bayesian model averaging
(Hoeting et al., 1999), which included all models within six BIC units of the best-fit model (Allen et al., 2013).
80 3-3.3. Radiocarbon
Holocene shells from two locations were analyzed for 14C in order to date the young terraces and to help calibrate the rate of AAR. These included analyses using gas proportional counting by Beta Analytic, and accelerator mass spectrometry (AMS) using both graphite targets for full-precision ages and carbonate targets for lower-precision ages (Bush et al., 2013) at the
University of California Irvine, Keck Carbon Cycle AMS facility. Radiocarbon ages were calibrated using Calib 7.0.4 and the Marine13 (Reimer et al., 2013) database. We used a ΔR value of 25 ± 35 14C yr (calib.qub.ac.uk/marine/, Higham and Hogg, 1995) for the North Canterbury samples and -25.8 ± 15 14C yr (Higham and Hogg, 1995) for the Cape Kidnappers samples.
3-4. Results
3-4.1. IRSL
The IRSL ages (Table 3-2) indicate the marine terraces formed during various stages of the last interglacial cycle, with most corresponding to substages of MIS 5 (Figure 3-5) We use two standard error uncertainties in order to compare these results to the 95% confidence intervals for the AAR ages. For sample GA2, from the beach facies at Glenafric, the luminescence growth curve approached saturation, resulting in a minimum age estimate far older than any of the other samples. This indicates that the luminescence signal of the sediment had not been zeroed during transport and is unlikely to reflect the true age of the terrace. Sample MB1 showed disequilibrium in the uranium decay chain, which affects the estimated dose rate and thus the age of the sample.
Three different ages (Table 3-2) have been calculated for this sample using three different estimates of uranium content, based on (1) 234Th, (2) 226Ra, 214Pb, and 214Bi, and (3) 210Pb.
81 3-4.2. Radiocarbon
The 14C results (Table 3-3) support late Holocene ages for samples from the low- elevation terraces at Cape Kidnappers and Haumuri Bluff. At Cape Kidnappers, one shell from sample CK4 was dated at 529–455 and 515–325 cal yr BP, and is significantly younger than the previously determined age of 2300 yr BP for the terrace (Hull, 1987). We assume that the shells in our collection were deposited on top of the terrace following its abandonment, perhaps by storm waves. These shells were collected from sand about 1 -2 m above the bedrock contact, so they may not be comparable to Hull’s (1987) samples, which were collected from within or immediately above the basal gravel. At Haumuri Bluff, we rejected the 14C age for sample HB5-
D as it is inconsistent with samples HB5-B and HB4, which were collected from the same site and the same marine gravel unit, and it consisted of a small shell fragment, which was noted by the laboratory as being very small or nonhomogeneous.
3-4.3. AAR Calibration and Ages
Six collections of shells were used for the AAR calibration dataset (Table 3-4). The age for the Cape Kidnappers sample is based on our 14C results (Table 3-3, sample CK4). Although younger than Hull’s (1987) age for the terrace, this age is paired with our AAR analyses. The ages for the other five samples are based on their independently determined MIS or substage as described above, using the midpoint of the age range (Lisiecki and Raymo, 2005; Williams et al.,
2015) as the best age and half the age range as the uncertainty, with assumed Gaussian distribution.
The AAR calibration focuses on Asp D/L (Table 3-4) because the best-fitting models rely on this amino acid, which is among the most abundant in the shells, and analytically the most
82 reproducible. The best-fitting model (lowest BIC) is simple power-law kinetics (SPK1) with R0 =
0.190 for Asp, with a lognormal distribution, and with coefficient a = 1.53x106 years and exponent b = 4.83 (Appendix A). The range of possible SPK models can be used to assign confidence intervals to the predicted ages (Figure 3-6). Seven other models are within six BIC units from the best-fit model, the cutoff used for Bayesian averaging (Appendix A). Six of these models also use Asp, while one uses Leu. Four use SPK, two TDK, and one CPK. Four use lognormal distributions, and three use gamma distributions.
None of the models within six BIC units of the best fit is an APK model. This is in agreement with the results of Allen et al. (2013), who generally found APK to provide a poor fit when using the Bayesian fitting procedure. It should be noted, however, that APK can provide a good fit to the three samples of MIS 5e age or younger. We prefer not to use a model that is such a poor fit to our complete dataset, however, when the model that best fits the entire dataset is still a good fit to the three youngest data points (Figure 3-6). Moreover, we aim to generate an AAR age model that can be applied over as wide a range of ages as possible, and SPK provides this, while APK does not.
Differences in temperature history among the samples could account for some of the unexplained variance in the AAR calibration. Specifically, the sample from All Day Bay is from significantly farther south than the other samples and thus presumably experienced lower postdepositional temperatures, as indicated by the modern temperature (Table 3-4). Lower temperatures reduce racemization rates, and thus we would expect the D/L value for this sample to be lower than the trend that is largely determined by results from farther north. The best-fit model, in fact, essentially intersects the D/L value for the MIS 5e sample within errors (Figure 3-
6), so we conclude that the effect of the temperature difference is not significant, and that the calibration model applies to both islands of New Zealand. The sample from Cape Kidnappers likely experienced similar temperatures to Whanganui (Bowen, 1998; Table 3-4), but the shells
83 are of the species Venerupis largillierti, which along with T. spissa is in the family Veneridae.
Differences in racemization rates among Venerid bivalves are estimated at about ± 10% (Lajoie et al., 1980). To test the sensitivity of the calibration model to possible taxonomic effects, we tried adjusting the D/L value of this sample by ± 10%. The differences in predicted ages from these adjusted D/L values are minor compared to the uncertainties in the age prediction.
To calculate the age of each of the 10 previously undated samples, we used the average
D/L measurements for that sample (n = 5) and the Bayesian model averaging of age predictions from all models within 6 BIC units of the best-fit model. For each sample, a weighted mean age was calculated, along with a 95% confidence interval (Table 3-5). These uncertainties, which take into account the eight models with BIC values below the cutoff, are larger than the confidence intervals for SPK1 alone (Figure 3-6).
3-5. Discussion
3-5.1 Glenafric
The 99 ± 9 ka IRSL age of sample GA1 for the loess on the Qt4 terrace suggests an MIS
5c age for the terrace (Figure 3-5). Since the sample was collected from loess, at about 7 m above the top of the marine sediments, its age is a minimum for the terrace. About 5 m of alluvium separates the two layers. Although its rate of deposition is unknown, it was likely deposited rapidly following sea level retreat, as the low slope of the abandoned terrace would likely have caused aggradation of streams flowing from the adjacent steep hills. Loess accumulation rates are also uncertain, but rates of up to 1 mm/yr have been reported elsewhere in Canterbury (Almond et al., 2007). Given such rates of deposition, and the uncertainties in the IRSL age and in the timing of sea level highstands, this result is consistent with an MIS 5c age for the terrace. An MIS 5e age
84 cannot be entirely ruled out, but there are no obvious stratigraphic unconformities within the cover sediments (Figure 3-4), suggesting continuous deposition. The three AAR samples from the
+42 marine sand and gravel overlying the bedrock strath produce age estimates ranging from 111 -42
+43 to 79 -29 ka (95% confidence intervals) (Table 3-5). The youngest (GA5) is most consistent with an MIS 5a age, although the uncertainty overlaps with MIS 5c or 5e as well. This sample exhibits unusually high inter-shell variability (Table 3-5), and moreover, MIS 5a is younger than the IRSL age of overlying loess. The other two age estimates lie between MIS 5c and 5e, with uncertainties large enough to encompass both substages. One possibility is that the samples date to MIS 5c but could include shells reworked from substage 5e. The Asp D/L values for this terrace (average of
0.568 for the three samples) are very close to those of the Tarapuhi terrace at Haumuri Bluff
(average of 0.564), suggesting that they are the same age, and the age of the latter terrace is more confidently established at MIS 5c (see below).
Given the evidence from both IRSL and AAR data, an MIS 5c age is most likely for the main terrace (Tiromoana terrace) at Glenafric. The 60 ka (MIS 3) age estimate of Yousif (1987), which was based on elevation relative to other terraces, is therefore likely too young, and no MIS
3 terrace appears to be preserved at this location. If the youngest terrace was formed during MIS
5c, then higher terrace remnants identified by Carr (1970) and Yousif (1987) may represent MIS
5e and MIS 7 (Table 3-1).
3-5.2. Motunau Beach
Both the IRSL data (Table 3-2) and AAR data (Table 3-5) suggest that at least two different ages of marine sediment are represented by the Motunau Beach samples (Figure 3-5), despite the fact that they were all collected from the Motunau coastal plain, which appears nearly continuous on its surface. The 54 ± 6 ka IRSL age for sample MB2 suggests an MIS 3 age for
85 this marine unit (Qt7). AAR samples MB3 and MB4, which were collected near MB2 support
+41 +38 this interpretation, with estimated ages of 53 -21 and 54 -24 ka. AAR samples MB5 and MB6,
+42 +43 on the other hand, have estimated ages of 68 -26 and 83 -30 ka, which suggest MIS 5a, although
MIS 3 or 5c cannot be ruled out given the range of error for those samples. IRSL sample MB1 gives a range of possible ages, due to disequilibrium in the uranium decay chain (Table 3-2).
Such disequilibrium can occur as a result of addition or removal of a mobile daughter product within the decay chain, and can lead to overestimation of the sample age (Olley et al., 1996). This age range is consistent with an MIS 5e or 5c age for the sample. It is inconsistent with IRSL sample MB2 and AAR samples MB3 and MB4 (Figure 3-5). The age for MB1 lies within the
95% confidence intervals of MB6 and MB5, but a younger age for the AAR samples is more likely. Given the AAR data for the other Motunau Beach samples, the youngest of the three possible ages calculated for MB1 (105 ± 12 ka) is the most likely, but the discrepancy between the two dating methods is larger here than at our other sites.
The contrast in ages between the older samples (MB1, MB5, and MB6) and the younger samples (MB2, MB3, and MB4) suggests partial reoccupation of the seaward edge of an older
(MIS 5) terrace during the MIS 3 highstand. In this scenario, the lowest, seawardmost part of the older terrace was transgressed during the MIS 3 highstand, but this transgression did not reach the previous sea cliff. Such partial reoccupation can occur when a sea-level highstand reaches the level of the lower part of a previously formed terrace but does not reach all the way to the previous inner edge (Kelsey and Blockheim, 1994). At Motanau Beach, reoccupation occurs only near the seaward edge of the widest terrace within North Canterbury. Alternately, a terrace may have been incised into the lower part of the MIS 5 terrace during MIS 3, but close enough in elevation to the older terrace tread that, if any sea cliff formed, it was buried by subsequent colluvial processes and no riser was preserved. These interpretations are in agreement with Carr
86 (1970), who first suggested that the Motunau coastal plain may preserve the record of two marine transgressions.
The older part of the terrace, before reoccupation or incision, would have formed during either MIS 5a (preferred age from AAR samples MB5 and MB6) or MIS 5c-e (IRSL age MB1).
The IRSL age is suspect due to the disequilibrium of the U-decay chain, and even the youngest of our three age estimates for this sample might be too old. Further, given the reoccupation or incision of the terrace during MIS 3, an MIS 5c or MIS 5e age for the main terrace would be improbable and MIS 5a would be more likely. Based on this consideration, the AAR results, and the problematic IRSL results from MB1, we prefer an MIS 5a age for the terrace.
Partial reoccupation of the seaward edge of an MIS 5a terrace during the MIS 3 sea-level highstand is surprising in an uplifting tectonic setting. In this case, however, it is possible given the slope of the MIS 5a terrace and the sea level at this time. The farthest inland MIS 3 sample,
MB4, was collected at an elevation of 40 m, immediately on top of the wave-cut platform.
Sample MB5, from the MIS 5a terrace, was at 70 m elevation, above a wave-cut platform at 67 m. (MB2 and 3 are offset by a small fault and MB6 is slumped, so those samples are not useful for this analysis.) The two samples are 1.2 km apart as measured perpendicular to the terrace tread, for a slope of 1.3°, which is low enough to be the original slope of a marine terrace
(Trenhaile, 2002), although some tectonic tilting cannot be ruled out. Extrapolating this slope inland to the terrace riser (600 m from MB5) would put the buried shoreline angle of the terrace at about 80 m. The actual elevation may be as high as 100 m due a steeper slope near the inner edge (Carr, 1970), but this has been attributed to tectonic warping (Carr, 1970; Barrell, 1989), although control on the bedrock elevation near the inner edge is poor away from the Motunau
River. Thus the MB4 sample site was likely at ~40 m below the sea level at the peak of MIS 5a or about -60 m relative to modern sea level, possibly more if the steep slope near the inner edge is not tectonic in nature. Taking a typical age of about 80 ka for MIS 5a and assuming a constant
87 rate of uplift, the sample site would then have been at about -35 m at 60 ka and -23 m at 50 ka.
Estimates of MIS 3 sea levels vary widely, with extremes ranging from at or above present-day levels to -85 m (Rodriguez et al., 2000). Our results are consistent with other evidence for relatively high sea levels of -20 to -35 m in New Zealand (Pillans et al., 1983; Berryman, 1993;
Rose, 2011), and southern Australia (Cann et al., 1993; Murray-Wallace et al., 1993) and for similar sea levels in a number of locations worldwide (e.g. Cabioch and Ayliffe, 2001; Mallinson et al., 2008; Wright et al., 2009; Doğan et al., 2012; Wang et al., 2013). They are, however, higher than MIS 3 sea levels inferred from the Huon Peninsula terrace record (Chappell et al.,
1996; Yokoyama et al., 2001; Chappell, 2002), for which the highest MIS 3 sea level is -46 to -35 m at 51.8±0.8 ka (Chappell, 2002), as well as most oxygen isotope records (Siddall et al., 2008).
If MIS 3 sea level at Motunau Beach did not exceed these lower levels, then the simplest explanation for our results would be that uplift rates have been higher since MIS 3 than between
MIS 5a and MIS 3.
The location of the inner edge of the Motunau Beach terrace appears to be lithologically controlled (Jobberns, 1928), coinciding with the more resistant lower part of the Mt. Brown
Formation and the Waikari Formation. As such, multiple sea-level highstands likely reached a similar extent inland, each eroding away previously formed terraces. The MIS 5a terrace may have overprinted the 5c terrace in this manner, whereas the lower sea level of MIS 3 was not able to reach as far inland. Above the main terrace, aligned notches and hilltops suggest older terrace remnants (Carr, 1970; Yousif, 1987; Barrell, 1989). We tentatively correlate these with similar remnants at Glenafric (Table 3-1), which we have assigned to MIS 5e and 7.
88 3-5.3 Haumuri Bluff
Both IRSL and AAR data support an MIS 5c age for the Tarapuhi terrace (Qt4, Figure 3-
5), which is in agreement with Ota et al. (1996). The difference between the two ages is only 1 ka, well within the error of both methods (Figure 3-5; Table 3-5). For the lower terrace (Amuri
Bluff terrace / Qt6), however, Ota et al.’s (1996) estimated age of 59 ± 3 ka is younger than our
IRSL age of 74 ± 9 ka. While not coinciding with the peak of any major sea-level highstand
(Siddall et al., 2007), this age suggests deposition during the end of MIS 5a (Figure 3-5).
The Kemps Hill terrace (Qt5), which lies between the Tarapuhi (Qt4) and Amuri Bluff
(Qt6) terraces in elevation (Ota et al., 1986) must have formed sometime between the two. Most likely, it formed during the early part of MIS 5a, since the Tarapuhi and Amuri Bluff terraces appear to be from the later parts of MIS 5c and 5a, respectively (Figure 3-5), but a late 5c age cannot be ruled out.
For the low (4.2 m asl) Torlesse terrace northwest of Haumuri Bluff, both 14C and AAR give Holocene ages (Figure 3-5). Two of the three 14C ages, HB4 (3.37–3.12 ka) and HB5-B
(3.42–3.16 ka) agree, while HB5-D (6.99–5.41 ka) is outside the two sigma range of the other two (Table 3-3). Although the older age is closer to the AAR estimate, we rejected HB5-D because the 14C laboratory reported analytical issues as described above. The younger 14C ages
+16 +17 nonetheless overlap the 95% confidence intervals for the 6 -4 and 8 -5 ka age estimates for the two AAR samples (Table 3-5). The proportionately large uncertainty in the AAR age of Holocene shells likely reflects the difficulty in applying a calibration model based primarily on Pleistocene data, given the change in temperature and thus racemization rate at the close of the Pleistocene, and in this case, possible differences in racemization rate between T. spissa and L. crassicosta.
While we prefer the late Holocene 14C ages over the middle Holocene AAR ages for the shells of the Torlesse terrace, we note that the marine unit that forms the Conway Flat surface, located 6
89 km to the southwest at 12 m asl also dates to the middle Holocene (7.0-8.6 ka; Ota et al., 1984).
On the other hand, the AAR calibration curve seems to have overestimated the age of the Cape
Kidnappers sample, which is likewise Holocene and not T. spissa, so the same effects may be responsible here.
3-5.4 Terrace Correlations Across the Study Areas
We have numbered the terraces in our three study areas as Qt1 to Qt8 from oldest to youngest (Table 3-1) using our geochronology to correlate terraces between localities. We correlate the Tiromoana terrace at Glenafric with the Tarapuhi terrace at Haumuri Bluff (both
Qt4). Further, the Motunau terrace, which was previously thought to be the same age as or older than the Tiromoana terrace (Table 3-1), is now considered to be younger (Qt6), and is correlated with the Amuri Bluff terrace at the Haumuri Bluff locality.
The lack of an extensive MIS 5e terrace at all locations is surprising, considering the highstand was higher and lasted longer than subsequent highstands (Lambeck and Chappell,
2001; Siddall et al., 2007; Dutton and Lambeck, 2012) and that 5e marine terraces are common globally (Pedoja et al., 2014) and in other parts of New Zealand (Pillans, 1990). Nor do we find terraces of the same age at all three sites, with our best age estimates indicating MIS 5c terraces at
Glenafric and Haumuri Bluff, MIS 5a terraces at Motunau Beach and Haumuri Bluff, and MIS 3 terrace reoccupation only at Motunau Beach. These observations can be explained by the destruction of older terraces by wave erosion during subsequent interstadial periods, and by their dissection by fluvial processes during stadial periods. A stretch of cliffed coastline near Motunau
Beach, for example, has retreated by 1.6 m/yr since 1950, (Barrell, 1989; Foster, 2009), although it was stable before that time, so the long-term average rate is likely less. As noted above, small terrace remnants above the main terraces at Glenafric and Motunau Beach likely date from MIS
90 5e, based on their position relative to dated terraces (Figure 3-5). That they are so poorly preserved compared to the MIS 5c or 5a terraces suggests that under the right conditions
(presumably of rapid erosion and uplift), an MIS 5e terrace can be largely destroyed during these subsequent sea level highstands. Differences in lithology, exposure to wave attack, and uplift rates among the three sites may explain the differences in terrace preservation at each location.
This local variability, the lack of extensive MIS 5e terraces, and the unusual MIS 3 partial reoccupation of the seaward edge of one terrace all present a picture of a complex regime of terrace formation and preservation. Because of this complexity, terrace ages in North Canterbury, and anywhere else that similar conditions may prevail, can be difficult to discern without the aid of numerical dating.
While the inner edge of each terrace likely formed during the peak of the corresponding sea-level highstand, there is some evidence that the sediments and shells that we dated were deposited as sea level regressed. The most probable ages for IRSL samples HB2 and HB1 at
Haumuri Bluff are from the young ends of MIS 5c and 5a, respectively (Figure 3-5), although this is not certain within 95% confidence. In addition, at Motunau Beach, the preferred ages for samples MB6, MB5, MB4, and MB3 are progressively younger with distance away from the terrace inner edge (Table 3-5 and Figure 3-3b), suggesting that they were deposited as the sea level retreated during MIS 5a (MB6 and MB5) and again during MIS 3 (MB4 and MB3), although the uncertainties in the AAR results are large enough that the opposite cannot be completely ruled out. These two lines of evidence suggest deposition during regression, but further investigation is needed to confirm or refute this hypothesis.
91 3-5.5. Comparison of Dating Techniques
AAR and IRSL are independent and complimentary techniques for dating Quaternary marine terraces. The range of factors that are incorporated into AAR age uncertainties, including inter-shell variability and calibration statistics, tend to yield larger errors compared to those that are included in IRSL age uncertainties, which focus on analytical precision rather than geological uncertainties such as incomplete zeroing of the luminescence signal. In this study, the conservative 95% confidence intervals on the AAR calibration estimated by Bayesian averaging span more than one MIS substage. In practice, our results show good agreement between the most likely stage or substage estimates from AAR and those of IRSL, except in the older part of the
Motunau Beach terrace where the IRSL sample had analytical issues. Despite the differences in modern temperature among the sample sites (Table 3-4), the fit of the model to the calibration data did not show a temperature effect (Figure 3-6), and the primarily North Island based model proved applicable to the North Canterbury data set. AAR is also useful for correlating terraces independently of the errors associated with numerical-age calibration, such as the correlation of the Qt4 terrace from Glenafric with that from Haumuri Bluff. When IRSL and AAR results converge, we have considerably more confidence in the terrace age than if only one method were used. In this study, the methods agreed for terrace Qt4 at Glenafric and Haumuri Bluff, and Qt7 at
Motunau Beach. When the two methods do not agree, such as at terrace Qt6 in the Motunau
Beach area, and when there is reason to doubt one of the results, as with the radioisotope disequilibrium in sample MB1, then we have additional evidence by which to exclude that result.
This study includes the first application of the Bayesian method of Allen et al. (2013) to develop an AAR calibration curve for a Pleistocene dataset. It is also the first AAR study in New
Zealand to analyze the intracrystalline fraction of amino acids. For Asp D/L, which correlates most strongly with sample age in the calibration dataset, the inter-shell variability based on the
92 intracrystalline amino acids was consistently lower than for shells analyzed using conventional techniques and A/I in previous studies. Specifically, for the four stratigraphic units that were analyzed both in this study and by Bowen et al. (1998), the average coefficient of variation (x̅ /) was 4% for intracrystalline (bleached) Asp D/L versus 9% for unbleached A/I.
3-6. Conclusions
Using OSL and AAR analyses, we have developed a new chronology for the marine terraces of North Canterbury, and we have developed an AAR age model for aspartic acid racemization in T. spissa that is applicable over a range of latitudes within New Zealand. This work provides the geochronological context for subsequent neotectonic studies of the region, for which marine terraces serve as important markers of uplift rate. More specifically:
1. The method of Allen et al. (2013) proved successful in developing an AAR calibration curve for a mostly Pleistocene dataset, using Bayesian model fitting and Bayesian averaging of different possible models.
2. IRSL and AAR work well together for dating Quaternary marine terraces. Estimated ages from the two methods agree sufficiently well in most cases for a reliable MIS assignment, and disagreement between them in one case helped to confirm the inaccuracy of a suspect IRSL result.
3. Numerical dating of terraces is necessary to establish reliable ages. Our results suggest that using elevation or degree of fluvial dissection to date and correlate terraces is insufficient in this region, since terraces of different ages are preserved at different sites. Our terrace ages and correlations differ significantly from most previous work (Table 3-1), which was based on such methods. Where previous numerical dating did exist for Qt4 at Haumuri Bluff (Ota et al., 1996), however, our results agree with that previous work.
93 4. Our new geochronologic results indicate that the Tiromoana terrace at Glenafric and the Tarapuhi terrace at Haumuri Bluff were both formed during MIS 5c. The Amuri Bluff terrace at Haumuri Bluff is assigned to MIS 5a. The Motunau coastal plain consists principally of an MIS
5a terrace, which is correlated with the Amuri Bluff terrace, but it has been partially reoccupied during MIS 3. This variety of ages for the widest, most continuous terrace at each site shows that terrace preservation is locally variable.
5. Unexpectedly, our results do not support an MIS 5e age for any of the terraces sampled, although this age is common for terraces worldwide. We attribute this to rapid erosion rates.
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104
105 Figure 3-1: Major features of the New Zealand plate boundary showing sampling sites. Haumuri
Bluff, Motunau Beach, and Glenafric are the sites of marine terraces within the North Canterbury
Fold and Thrust Belt (NCFTB) that were dated in this study. Cape Kidnappers, Whanganui, and
All Day Bay were sampled for the amino acid racemization calibration. The extent of basement rocks in the inset is from Rattenbury et al. (2006) and Forsyth et al. (2008).
106
Figure 3-2: Aerial views of marine terraces at (A) Glenafric; view is to south, (B) Motunau
Beach; view is to west, and (C) Haumuri Bluff; view is to south. Sample sites (GA, MB, and HB) and terrace units (Qt) are labeled.
107
108 Figure 3-3: Three North Canterbury field sites: (a) Glenafric, (b) Motunau Beach area, and (c)
Haumuri Bluff. Mapping of terraces is based on previous maps by Yousif (1987), Barrell (1989),
Warren (1995), Pettinga and Campbell (2003), Rattenbury et al. (2006), and Forsyth et al. (2008), unpublished QMAP record sheets provided by GNS Science
(http://data.gns.cri.nz/metadata/srv/eng/search), and interpretation of aerial photographs from
Land Information New Zealand. The background is a hillshade of the 10 m DEMs from
Rattenbury et al. (2006), and Forsyth et al. (2008).
109
Figure 3-4: Representative stratigraphy for the marine terraces dated in this study, showing the locations of samples relative to the bedrock surface. The thickness of stratigraphic units varies laterally on each terrace, including between sample sites. Stratigraphic columns are based on Carr
(1970), Ota et al. (1996), and this study. For uncertainties in ages, see Tables 3-2, 3-3, and 3-5.
110
Figure 3-5: Sample age estimates for the three study areas in relation to marine isotope stages.
Interstadials are white; stadials are shaded. Stage boundaries are from Lisiecki and Raymo (2005) and substage boundaries are from Williams et al. (2015). Error bars represent 95% confidence intervals for amino acid racemization (AAR) calibration curves, 2 standard error for infrared stimulated luminesce (IRSL), and 2-sigma range of calendar-year calibrations for radiocarbon ages (14C). Labels next to points indicate the terrace designation in Table 3-1 for this study. Data are listed in Tables 3-2, 3-3, and 3-5.
111
Figure 3-6: Amino acid calibration curve for racemization of aspartic acid (Asp) based on the best-fitting SPK1 model and lognormal distribution (Appendix A). Dark shading represents 95% confidence intervals for mean age, and light shading represents 95% prediction intervals. Data are listed in Table 3-4.
Table 3-1: Names and ages of marine terraces based on previous work and this study
Glenafric Motunau Beach Haumuri Bluff
Carr (1970) Yousif (1987) QMAPb Carr (1970) Yousif (1987) Barrell (1989) QMAP Ota et al. (1984; 1996) Warren QMAP This Study (1995) Name Age Name Age Name Agea Name Age Name Age Name Age Name 1984 Agea 1996 Name Name Age Name Age (ka) (MIS) (ka) (ka) (MIS) Age (MIS) (MIS) Name Agea (ka) Leonard Waiwheran M4 125 mQb 6-13 Vulcan 210 mQb 6-13 Qt1 7c Formation Terrace 3 Leonard Waiwheran M3 105 mQb 6-13 Stonyhurst Waiwheran M4 125 Vulcan 118- mQb 6-13 Qt2 7a Formation Formation Terrace 2 124 Bob's Flat Terangian M2 80 mQb 6-13 Stonyhurst Waiwheran M3 105 Vulcan 106 Q7b 7 Qt3 5e Formation Formation Terrace 1 mQb 6-13 Tiromoana Oturian M1 60 Q5b 5 Tarapuhi Waiwheran 100 ± 3 TrigT Q9b 9 Qt4 5c Formation Terrace Formation Kemps Terangian 81 ± 3 Kemps Q7b 7 Qt5 5a-c Hill (upper) Hill Terrace 72 ± 3 Formation (lower) Motunau Terangian- M2 80 Motunau 83 Q5b 5 Amuri Oturian 59 ± 3 Wenlock Q5b 5 Qt6 5a Formation Waimean Terrace Bluff Formation Terrace Estuarine Oturian- Qt7 3 Beds Otiran Amberley Aranuian M0 Holo Q1b 1 Conway Holocene Rafa Q1b 1 Qt8 1 Formation cene Flat (c. 8 ka) Formation Terrace
a Terrace ages from Carr (1970) and Ota et al. (1984) use the glacial-interglacial stage names of Suggate (1965). From youngest to oldest, the interglacials are Aranuian (present), Oturian, Terangian, and Waiwheran. b QMAP geologic maps covering this area are Rattenbury et al. (2006) and Forsyth et al. (2008). Gray boxes indicate that a terrace is not present in a given area or was not described by a given study.
Table 3-2: Analytical data used to calulate luminesence ages.
a Sample Laboratory Depth Water U Th K Cosmic Total Dose De (Gy) Luminescence Number Code Below Content (ppm) (ppm) (ppm) Ray Dose Rate Age (ka) (WLL-) Surface (%) Rate (Gy/ka) (2 SE)b (m) (Gy/ka) GA1 1113 3.5 12.5 2.55 8.01 1.63 0.1310 3.19 ± 0.15 314.86 ± 4.60 99 ± 9 ±0.15 ±0.11 ±0.04 ±0.0066 GA2 1083 6 0.202 2.09 8.71 1.99 0.0970 3.15 ± 0.18 >728.85 ± 15.53 >231 ± 28c ±0.17 ±0.14 ±0.05 ±0.0048 MB1d 1112 4 15.6 1.40 4.27 1.39 0.1230 2.18 ± 0.11 270.90 ± 3.94 124 ± 13 ±0.14 ±0.06 ±0.03 ±0.0062 2.01 2.37 ± 0.12 270.90 ± 3.94 114 ± 12 ±0.10 2.68 2.57 ± 0.13 270.90 ± 3.94 105 ± 12 ±0.15 MB2 1082 2.5 0.136 1.40 4.55 1.55 0.1490 2.40 ± 0.10 130.0 ± 4.4 54 ± 6 ±0.17 ±0.07 ±0.03 ±0.0075 HB1 1114 4.4 22.5 2.21 9.44 2.39 0.1171 3.53 ± 0.21 260.82 ± 3.04 74 ± 9 ±0.15 ±0.13 ±0.05 ±0.0059 HB2 1109 5.3 15.2 1.98 8.09 2.09 0.1051 3.33 ± 0.16 316.50 ± 6.12 95 ± 10 ±0.15 ±0.09 ±0.04 ±0.0053 a U content was calculated in three ways: from 234Th, from 226Ra, 214Pb, and 214Bi, and from 210Pb. For samples for which the three values are within error of each other, only the median value is given here. For MB1, all three are given (in the order listed here). b Uncertainties of 1 standard error were reported by the luminescence laboratory. However, we use 2 standard error uncertainties in the text for comparison with the 95% confidence intervals on the AAR ages. c This is a minimum age. The growth curve for sample GA2 trends toward saturation
Table 3-3: Results of radiocarbon analyses.
Laboratory Lab Code Sample Radiocarbon Age Calendar Ageg (yr Codea (UAL)d Number (yr BPe) BPe) 350130b 10174 HB4 3410 ± 30 3371–3121
CaCO3 target 10936-B HB5-B 3450 ± 35 3420–3162 f CaCO3 target 10936-D HB5-D 5790 ± 350 6993–5409
CaCO3 target 13468-A CK4-A 785 ± 45 515–325 163843c 13468-A CK4-A 845 ± 15 529–455
a CaCO3 targets analyzed at University of California, Irvine using the procedure of Bush et al. (2013) are considered range-finder ages and are not assigned a laboratory code. b Beta Analytic code. Analyzed by beta counting. All other samples analyzed by accelerator mass spectrometry. c UCIAMS code d Northern Arizona University Amino Acid Geochronology Laboratory identifier. e Years before present, using 1950 CE as the present datum. f This sample was very small and/or inhomogeneous. g Two-sigma age ranges determined using Calib 7.0.4 and the Marine13 database.
Table 3-4: Dataset used for amino acid racemization calibration.
Lab Location Lat Long MATa nb Asp D/L MISc Median Uncertaintyd Code (°) (°) (°C) Age (ka) (UAL) mean std (ka) 13468 Cape Kidnappers -39.6 177.1 14.6 6 0.220 0.016 1 0.56 0.04 10933 Waverley -39.8 174.6 14.0 5 0.538 0.014 5a 79 8 9448 All Day Bay -45.2 170.9 11.1 5 0.606 0.015 5e 123.5 6.5 9433 Landguard -40.0 175.0 14.0 5 0.644 0.041 7 217 26 9417 Shakespeare Cliff -39.9 174.8 14.0 5 0.767 0.016 11 399 25 9414 Kupe Formation -39.9 174.8 14.0 5 0.848 0.007 17 694 18
Note: All analyses on Tawera spissa except mollusks from Cape Kindappers, which are Venerupis largillierti, which is in the same family as T. spissa. See Appendix A for details of individual shells and all amino acids measured. a Mean annual temperature. Data are from NIWA for the period 1981-2010 from stations closest to the sample site, including Napier for Cape Kidnappers, Whanganui for Landguard, Shakespeare Cliff and Kupe Formation, and Dunedin for All Day Bay. For comparison, the MAT at Christchurch is 12.2 °C. (http://www.niwa.co.nz/sites/www.dev2.niwa.co.nz/files/sites/default/files/mean_monthly_air_temperature.xlsx) b Number of individual shells analyzed and used to calculate sample mean. c Marine oxygen isotope stages (MIS) based on ages assigned by previous workers, as described in the text. d 95% confidence, based on 14C results for Cape Kidnappers and MIS for the rest. See text for details.
Table 3-5: Amino acid racemization ages.
Lab Code Sample n a Asp D/L 2.5th Estimated 97.5th (UAL) percentile Age (ka) percentile mean std (ka) (ka) 9455 GA3 5 0.576 0.016 70 109 197 10938 GA4 5 0.585 0.015 69 111 153 10937 GA5 5 0.542 0.043 50 79 122 10929 MB3 5 0.495 0.019 32 53 94 10928 MB4 5 0.500 0.012 30 54 92 10930, 10932 MB5 5 0.524 0.027 42 68 109 10931 MB6 5 0.548 0.017 53 83 126 10175, 10178 HB3 5 0.564 0.025 60 94 136 10174 HB4 5 0.319 0.024 3 8 25 10936 HB5 5 0.289 0.018 2 6 22 a Number of individual shells analyzed and used to calculate sample mean.
117 Chapter 4
Uplift Rates of Marine Terraces as a Constraint on Fault-Propagation Fold Kinematics: Examples from the Hawkswood and Kate Anticlines, North Canterbury, New Zealand
Abstract
Marine terraces on growing fault-propagation folds provide valuable insight into the relationship between fold kinematics and uplift rates, providing a means to distinguish among otherwise non-unique kinematic model solutions. Here, we investigate this relationship at two locations in North Canterbury, New Zealand: the Kate Anticline and Haumuri Bluff, at the northern end of the Hawkswood Anticline. At both locations, we calculate uplift rates of previously dated marine terraces, using GPS surveys to estimate terrace inner edge elevations.
We then use Markov chain Monte Carlo methods to fit fault-propagation fold kinematic models to structural geologic data, and we incorporate marine terrace uplift into the model as an additional constraint. At Haumuri Bluff, we find that marine terraces, as originally flat surfaces to be restored, can help to eliminate certain models that could fit the geologic data only. At Kate
Anticline, we compare uplift rates at different structural positions and find that the spatial pattern of uplift rates is more consistent with trishear than with a parallel-fault propagation fold kink- band model. Finally, we use our model results to compute new estimates for fault slip rates, ages of the folds, and shortening rates at the two locations.
118 4-1. Introduction
A variety of kinematic models exist for fault-related folds. For fault-propagation folds, the two most commonly used types are kink band models (Suppe and Medwedeff, 1990) and trishear models (Erslev, 1991). Kinematic models can be used to predict subsurface fold and fault geometry and provide estimates of shortening and fault slip. They have found applications in structural geology and tectonics, petroleum geology, and earthquake hazard evaluation (Brandes and Tanner, 2014). With multiple possible kinematic models, and with multiple parameters for each model, determining the best model for any given fault-propagation fold can be difficult, and solutions are typically non-unique. Different interpretations can make significantly different predictions of total fault slip or rate of slip, with important consequences for seismic hazards of active faults. Examples include the different models proposed for the Santa Monica Mountains-
Channel Islands anticline (Davis and Namson, 1994; Seeber and Sorlien, 2000) and the contrasting kink-band (Mueller et al., 1999) and trishear (Champion et al., 2001) models of the
Reelfoot fault in Tennessee. Moreover, the trishear model, even if it can be chosen with confidence, can have significant uncertainty in model parameters (Cardozo and Aanonsen, 2009;
Cardozo et al., 2011; Oakley and Fisher, 2015).
In this study, we investigate the kinematics of two fault-propagation folds in North
Canterbury, New Zealand, where uplifted marine terraces allow us to determine uplift rates.
These terraces further help us to constrain trishear model parameters and choose between trishear and kink-band models. We consider multiple possible models for each structure in order to test the benefits and limitations of using the marine terraces as a constraint on fold kinematics. We show that terraces can serve as a constraint on models in two ways: by serving as originally flat surfaces and markers of deformation and by recording the relative rates of uplift at different structural positions which can be tested against kinematic models. We incorporate marine terraces
119 into a Markov Chain Monte Carlo simulation as a data type alongside folded lithologic contacts and measurements of bedding dip, inverting for the model parameters that can best fit data of all three types.
4-2. Background
4-2.1 Fault-Propagation Fold Kinematics
The first quantitative kinematic models for fault-propagation folding were the kink-band models of Suppe (1985) and Suppe and Medwedeff (1990). Like the kink-band model of fault- bend folding (Suppe, 1983), these models are characterized by angular fold hinges. The resulting folds grow self-similarly by kink-band migration, and an exact relationship exists between fold limb dips and fault geometry. Fault dip, total slip, and fault tip position are the only free parameters, while other variables such as propagation-to-slip ratio are functions of these (Hardy,
1997; Allmendinger et al., 2012). Two kink-band fault-propagation fold models are described by
Suppe and Medwedeff (1990): parallel fault-propagation folding, in which bed thickness and bed length are conserved, and fixed-axis fault-propagation folding, in which material cannot move through the front anticlinal surface and area is conserved but bed thickness and length are not.
An alternative kinematic model, trishear, was proposed by Erslev (1991). In this model deformation occurs in a triangular zone of distributed shear ahead of the propagating fault tip.
The model produces curved forelimbs, footwall synclines, changes in bed thickness, and progressive limb rotation, but it requires numerical solution and has a minimum of six free parameters. Fold limb dip is not a unique function of fault geometry and varies in space and over the course of fold growth. Due to the number of free parameters, there is typically a range of trishear models that can fit a given dataset (Cardozo and Aanonsen, 2009; Cardozo et al., 2011),
120 and the probability distributions for model parameters are sometimes multimodal (Oakley and
Fisher, 2015). Using trishear to make useful interpretations of real-world structures therefore depends on being able to constrain the model as much as possible.
Although neither kinematic model directly incorporates physical properties of the deforming rocks, each reflects certain assumptions about the processes of deformation. Kink- band models assume that deformation occurs primarily by layer-parallel slip (Suppe, 1983), while trishear simulates distributed deformation within a triangular zone without regard to bedding orientation. Natural examples have been documented that illustrate both styles of folding.
Mechanical stratigraphy has been found to be a key factor in determining which type of folding is observed in both natural examples (Cardozo et al., 2005) and modeling studies (Johnson and
Johnson, 2002; Hardy and Finch, 2007; Hughes and Shaw, 2015). This is not to say that the choice of which model to use is always straightforward. Fold and growth strata geometries characteristic of both types of folding can occur in the same fold and thrust belt (Zapata and
Allmendinger, 1996; Erslev and Mayborn, 1997), and the same structure may be interpreted in different ways by different authors. Examples of folds that have been interpreted using both kink- band and trishear models include the Sante Fe Springs anticline in Los Angeles (Shaw and
Shearer, 1999; Allmendinger and Shaw, 2000) and the Sant Llorenç de Morunys fold in Spain
(Ford et al., 1997; Suppe et al., 1997).
4-2.2 Velocity Description of Folding
Fold kinematic models, including both kink-band and trishear fault propagation folding, can be described in terms of the velocity of material at any point in the fold (Hardy, 1995; 1997;
Waltham and Hardy, 1995; Hardy and Poblet, 1995; 2005; Hardy and Ford, 1997; Zehnder and
Allmendinger, 2000), which can be used to model folds or restore cross sections. With the
121 velocity description of folding, kinematic models make predictions not only about the final shape of a fold, but also about the rates of fold growth. A model that can accurately reproduce both fold shape and rates of deformation likely provides a better description of physical reality than one that reproduces fold shape alone.
Hardy and Poblet (2005) recognized the value in using the velocity description of fold kinematics to predict uplift rates and derived equations for uplift rates above fault-propagation folds for both kink-band (parallel and fixed-axis) and trishear models. Kink-band models predict zones of constant uplift rates, bounded by the fold axial surfaces that separate the backlimb, crest, and forelimb (Figure 4-1). Trishear predicts a constant uplift rate in the hanging wall outside the trishear zone and an exponential decrease within the trishear zone (Figure 4-1).
Using fold kinematics to predict uplift rates raises the possibility that uplift rate can be used to distinguish among different kinematic models (e.g. trishear, parallel, and fixed-axis models) and also to constrain the range of possible values for the parameters of a given model.
This second consideration is particularly significant for trishear, due to the large number of model parameters. Previous work of this sort has been limited and has mostly focused on using deformed fluvial terraces as markers of deformation (Gold et al., 2006; Scharer et al., 2006;
Amos et al., 2007; Wilson et al., 2009). We expand upon previous work by: (1) investigating the potential of marine terrace uplift rates for constraining fold kinematics, (2) demonstrating the ability of terraces to help constrain the large number of parameters of the trishear model, and (3) integrating geologic and geomorphic data into a single model that can be inverted for fault geometry, fault slip, and kinematic model parameters.
122 4-2.3 North Canterbury Fold and Thrust Belt
The North Canterbury region of New Zealand’s South Island provides an excellent opportunity to test the utility of marine terrace uplift rates in understanding fault-propagation fold kinematics. The region is the site of active basement-involved, fault-related folding, but little work has been done on the kinematics of folding in the area. Uplifted marine terraces occur extensively on the limbs of anticlines along the coast.
The North Canterbury Fold and Thrust Belt is a distinct structural domain within the
Canterbury region (Pettinga et al., 2001). Numerous active thrust faults have been mapped in the area (Rattenbury et al., 2006; Forsyth et al., 2008; Barrell and Townsend, 2012; Litchfield et al.,
2014; Barrell, 2015). Folds are typically asymmetric, with backlimb dips between 10° and 30°, most often around 20°, and steep to overturned forelimbs (Wilson, 1963; Yousif, 1987;
Rattenbury et al., 2006; Forsyth et al., 2008). In many cases these folds occur in clear association with mapped faults, which cut the forelimbs of the folds, and even where faults are not seen, blind thrusts may be present. This fold geometry and the association with faulting indicate that fault- propagation folding is most likely the dominant style of deformation.
Marine terraces occur in many places along the North Canterbury coast (Jobberns, 1928;
Carr, 1970; Yousif, 1987; Ota et al., 1984; 1996; Rattenbury et al., 2006; Forsyth et al., 2008;
Oakley et al., 2017). A spatial relationship between folds and the locations of terraces is readily apparent from geologic maps (Yousif, 1987; Warren, 1995; Rattenbury et al., 2006; Forsyth et al.,
2008). In this study, we focus on folds at two locations discussed in Oakley et al. (2017): the Kate
Anticline and Haumuri Bluff. The marine terraces at Kate Anticline are the southernmost of the
North Canterbury terraces, adjacent to the Canterbury Plains, and are uplifted on the backlimb of an asymmetric anticline. At Haumuri Bluff, farther north, terraces are formed on the forelimb of an anticline.
123 4-3. Methods
4-3.1 Inner Edge Identification
The inner edge of a marine terrace, where the wave-cut platform meets the paleo sea cliff, represents the maximum transgression of the sea during the sea-level highstand that formed the terrace. Its elevation is therefore crucial to calculating uplift rate. Inner edge elevations were identified from Differential GPS (DGPS) surveys, measurements of sediment thickness, and analysis of DEMs. DGPS surveys (Appendix B) were used to produce profiles of the terrace surface and to measure the elevation of points on the bedrock strath, where it was exposed. 10 m/pixel DEMs from Rattenbury et al. (2006) and Forsyth et al. (2008) and LiDAR DEMs provided by Environment Canterbury were also used to produce profiles of terrace surfaces.
Measurements of sediment thickness from this study and previous studies (Carr, 1970; Ota et al.,
1984) were used to help estimate bedrock strath elevation from surface elevation.
The bedrock shoreline angle at the terrace inner edge along the paleo sea cliff is almost invariably covered by colluvial sediments. It was necessary, therefore, to estimate the elevation of the inner edge based on the surface morphology and on the bedrock elevation away from the inner edge. Measurements and estimates of bedrock elevation were used to approximate the wave cut platform, strath surface, as a straight line in cross section. The intersection of this line with a line approximating the original sea cliff was then taken as the inner edge. Uncertainty in this inner edge elevation was estimated from uncertainty in DGPS points, scatter of DGPS points about the estimated wave cut platform, and differences between DGPS and LiDAR or DEM elevations.
To approximate the original location of the sea cliff, assuming subsequent cliff retreat due to erosion, a line was fit to the steepest part of the modern terrace riser and then moved
124 forward such that the area (in cross section) of eroded material equaled the area of the colluvial wedge at the base of the riser. An alternative method, assuming that the sea cliff has instead eroded by transport limited scarp diffusion was also used for comparison. In this case, a diffusion profile was fit to the observed profile of the modern terrace riser and used to calculate the position and slope of the original sea cliff (Hanks and Andrews, 1989, their Eq. 3). This method gave similar inner edge elevations to the cliff retreat method but was less broadly applicable, as not all terrace riser profiles had the smooth diffusion profile shape. For that reason, we relied primarily on the first method.
4-3.2 Uplift Rate Calculations
Uplift rates were calculated from inner edge elevation, paleo-sea level, facies depth, and terrace age. Each terrace is assumed to have formed during a highstand of sea level (Lajoie, 1986;
Pedoja et al., 2014), for which we use the correlation of terraces to sea level highstands of Oakley et al. (2017). Paleo-sea levels were determined by taking the sea level and its uncertainty for the appropriate highstand from a sea level curve, as described below. The inner edge is assumed to have formed at the maximum sea level of the highstand and its age is taken as the age of the highstand as given by the sea level curve. The shoreline angle at the inner edge of a marine terrace corresponds to mean higher high water or mean high water spring tide (Matsu’ura et al.,
2014), so this is taken as the facies depth (or elevation). We adopt a value of 1±0.2 m for MHWS, based on modern tidal data for Lyttelton and Kaikoura (http://www.linz.govt.nz/sea/tides/tide- predictions/standard-port-tidal-levels and http://www.linz.govt.nz/sea/tides/tide-predictions).
Uplift rate is calculated by the equation
퐸 − 퐹 − 푆 푈 = 푇
125 where U is uplift rate, E is modern elevation, F is facies elevation, S is sea level at the time of terrace formation, and T is the age of the terrace. E, F, and S are measured positive up with zero at modern sea level. All uncertainties that we report for these quantities are 2σ uncertainties and assume normally-distributed errors.
4-3.3 Paleo-Sea Level
Paleo-sea level is critical to accurate calculations of uplift rate and can be a significant source of uncertainty. No sea level curve specific to New Zealand is available, except for the
Holocene (Clement et al., 2016), and so we rely on estimates from other sources. Since New
Zealand was far from major continental ice sheets, glacio-isostatic effects are assumed to be small, and the difference between local and eustatic sea level is not likely to be a major source of error in our uplift rate calculations. During the last interglacial (MIS 5e), for example, New
Zealand sea level is likely to have departed from eustasy by not more than about 2 m due to glacio-isostatic effects (Creveling et al., 2015). Hydro-isostatic effects are similarly small – less than 1 m of subsidence for the Holocene highstand in Canterbury (Clement et al., 2016).
The geochronologic results of Oakley et al. (2017) allow us to assign terraces to interstadial MIS stages and substages. There may be fluctuations of sea level within each substage (Figure 4-2) which we cannot accurately distinguish among on the basis of our dating techniques, and which may or may not be of sufficient length and magnitude to produce individual marine terraces. These substage fluctuations are most significant for the Haumuri Bluff area, where three to four terraces were formed within the MIS 5a-5c timespan.
For calculating terrace uplift rates, we have compiled a list of paleo-sea level highstands and ages for MIS 5 and 7 (Table 4-1). For MIS 5c and 5a, we use the sea level curve of Lambeck and Chappell (2001), which is based on the detailed, coral terrace-based sea level record of the
126 Huon Peninsula, New Guinea. This curve shows two distinct peaks in MIS 5c and four in MIS 5a.
We calculate the sea-level for each peak as the midpoint between the maximum and minimum values of the ice-volume-equivalent (eustatic if no other factors contribute) sea-level curve. To help account for possible glacio- and hydro-isostatic effects, we take the difference of this midpoint and the upper and lower limits for both ice-volume-equivalent and Huon Peninsula relative sea-levels (Lambeck and Chappell, 2001, their Figs. 3A and 1B respectively), using the maximum positive and negative differences and averaging the two to arrive at the uncertainties shown in Table 4-1. For MIS 5e, we use a value of 6±4 m, which encompasses the range of eustatic and relative sea levels reported in several studies (Hearty et al., 2007; Dutton and
Lambeck, 2012; Kopp et al., 2013). For MIS 7a, c, and e, we take the midpoints of the sea level and age ranges of Siddall et al. (2007), which are themselves based on a compilation of multiple studies, and we take half the range as the uncertainty.
In the Holocene, South Island sea levels are thought to have first attained present mean sea-level between 7-6.4 ka (Clement et al., 2016). Most significant for this study, however, is the 3.1-3.4 ka age of the Torlesse terrace near Haumuri Bluff. Sea level data from Canterbury are absent for this time period, but those from Otago and the southwest North Island show sea levels mostly below present (minimum of -1.4 ± 0.8 m, weighted average -0.8 ± 0.3), while a glacial isostatic adjustment model predicts sea level about 1.5 m above present (Clement et al., 2016).
Given these conflicting estimates, we adopt a sea level of 0 ± 1.5 m, which encompasses both possibilities.
4-3.4 Structural Data Sources
Previously published geologic maps (Wilson, 1963; Yousif, 1987; Warren, 1995;
Litchfield, 1995; Pettinga and Campbell, 2003; Forsyth et al., 2008; Rattenbury et al., 2006)
127 provide significant data on geologic structure. Additionally, unpublished QMAP record sheets and a database of strike and dip measurements (including many from the above-mentioned sources) were provided by GNS Science. Finally, field mapping of key areas was conducted to supplement this body of existing data (Appendix D). Geologic maps of the two study areas are based most closely on Yousif (1987) for the Kate Anticline and Warren (1995) for Haumuri
Bluff, but are modified based on other sources, original mapping, and analysis of DEMs
(including the LiDAR dataset mentioned above) and aerial photos (from Land Information New
Zealand). For the Kate Anticline, an additional source of data came from reflection seismic lines
(Exploration Solutions Ltd., 2006; Geosphere (Excel), 2008) and the Kate-1 well (Styles et al.,
2008). The seismic reflection data include six lines: the five TAG06-260-01 to 05 lines
(Exploration Solutions Ltd, 2006) and the Kate-01 line acquired in 2005 and subsequently reprocessed (Geosphere (Excel), 2008) following drilling of the Kate-1 well. The Kate-1 well was a hydrocarbon exploration well, which targeted the anticlinal crest within the Broken River
Formation but drilled the forelimb of the fold due to an inaccurate pre-drilling interpretation of the structure (Styles et al., 2008).
4-3.5 Trishear Markov chain Monte Carlo Simulations
Markov chain Monte Carlo simulations were used to fit trishear fold models to data using the InvertTrishear program (Oakley and Fisher, 2015; http://davidosoakley.com/trishear.html). In addition to the structural data listed above, points along the bedrock straths of marine terraces were included in the models. Error in the restored position of terrace points was calculated for each model as the distance from the restored points to a line passing through the expected paleo- sea level at the position of the inner edge. Since the observed present day slopes of the wave-cut platforms were all <1° and seaward, the restored slope of the terraces was assumed to have been
128 the same as the present-day slope. The adaptive parallel tempering method (Miasojedow et al.,
2013) was used to invert for the best model and the uncertainty in model parameters, and marginal probability density functions were approximated by histograms of results. For most of the analyses in this paper, six million models were tested, with the first one million discarded as a burn-in period and the remainder subsampled every 50 models, for a total of 100 thousand models in each histogram. For model 2 at Haumuri Bluff, due to slow mixing of the Markov Chain, 11 million models were tested, with the first one million discarded, and the rest were subsampled every 100 models so that the final number would be the same as in the other analyses. The program was updated from its original version to incorporate a semi-analytic solution to the equations of trishear deformation (Appendix C), which allows for much faster calculations as well as consideration of more complicated fault geometries.
For all the models described in this paper, we consider a fault composed of two segments with different dips, like those shown in Figure 4-1, but without requiring that the lower segment is horizontal as shown there. This is the minimum complexity of fault geometry needed to produce an anticline, since fault-propagation folding above a single straight fault segment would produce a monocline. It is possible that the fault may be more complex, having a listric form or composed of many segments. We have no direct evidence of such complexity, however, and our purpose of comparing different kinematic models and constraining trishear parameters is best served if fault geometry is consistent among the models to be compared, so we consider here the simplest reasonable fault geometry. For kink-band folding the orientation of the backlimb synclinal axis can be calculated from the dips of the fault segments (Suppe and Medwedeff,
1990). Trishear, which is a model for deformation of the forelimb, does not make any explicit predictions about the backlimb syncline and the effect of fault bends. For this case, we use fault- parallel flow (Ziesch et al., 2014), in which the synclinal axis bisects the two fault segments and fault slip is conserved across fault bends. Because fault-parallel flow, like trishear and unlike
129 flexural-slip fault bend folding, does not consider the orientation of bedding or preserve line length, it is a reasonable choice for folds in which trishear deformation occurs.
Uncertainties were assigned to each data type and used to calculate a probability for each model, assuming normally distributed errors. For mapped lithologic contacts, we used uncertainties (1 σ) of 60 m horizontally and 20 m vertically. These are based on guidelines given by Allmendinger and Judge (2013), but are doubled to account for the fact that most of the contacts used were only approximate. For contacts imaged by seismic reflection, we follow
Cardozo and Aanonsen (2009) in using the spherical variogram model (Davis, 2002) to represent uncertainty in seismic sections and its correlation along the length of a bed. For this, we used an uncertainty of 50 m, based on a comparison of the converted depths that would result from using the velocity model of Barnes et al. (2016) versus our velocity model (described below), and we used a correlation length of 20 km, which approximately follows the guideline of Cardozo and
Aanonsen (2009) that it should be 10 times the length of the bed being restored. Uncertainties in dip measurements (projected onto a cross section) were taken to be 5°. This follows the practice in other studies that have addressed uncertainty in balanced cross sections (e.g. Regalla et al.,
2010, Allmendinger and Judge, 2013), and is consistent within 1-2° with the observed standard deviations of dips in the backlimbs (assumed homoclinal) of folds in the cross sections considered in this study. Uncertainties in terrace points were estimated independently for each terrace, based on GPS point vertical uncertainties and discrepancies among GPS points or between GPS and
DEM or LiDAR data. Uncertainties in paleo-sea levels are as given in Table 4-1.
Uncertainties were propagated through cross section restoration as described in Oakley and Fisher (2015). We correct an error in Oakley and Fisher (2015, their Eq. 7), in which the factor of 1/σ in the probability density function of the normal distribution was neglected on the assumption that it should cancel out when calculating the ratio of two probabilities for the
Markov chain. The propagation of uncertainty through the restoration means that σ for the
130 restored state data will not always be the same for the current and proposed models at each step of the Markov chain, even though σ for the deformed state data is constant. The effect of this change is minor, however.
4-4. Haumuri Bluff Results
4-4.1 Setting
Haumuri Bluff lies at the northeastern end of the Hawkswood Range (Figure 4-3). The range is composed mostly of Torlesse basement greywacke. The overlying Cretaceous through
Pliocene or early Pleistocene sedimentary sequence is preserved only at the northern and southern ends of the range, including at Haumuri Bluff where they define the anticlinal structure of the range. Elsewhere, the Plio-Pleistocene Greta Formation directly onlaps basement along the flanks of the range. Marine terraces at Haumuri Bluff occur on the more steeply-dipping forelimb of the anticline. They have been correlated with other terraces along the seaward side of the range (Ota et al., 1984). A fault responsible for terrace uplift has not been identified but is likely offshore and/or blind. While a possible onland fault zone south of the Conway River was described by
McConnico (2012), it is inland of some of the terraces and therefore not likely to be responsible for terrace uplift, and a northward continuation has not been observed at Haumuri Bluff.
4-4.2 Terrace Inner Edge Elevations
Three marine terraces at Haumuri Bluff were dated by Oakley et al. (2017): the MIS 5c
Tarapuhi terrace (Qt4), the MIS 5a Amuri Bluff terrace (Qt6), and the Holocene Torlesse terrace
(Qt8). The only terrace for which we could directly measure bedrock elevation within a few
131 meters of the inner edge was the Torlesse terrace (4.2 m elevation). For the rest, we had to calculate the likely elevation from other measurmements. The Tarapuhi terrace is preserved as a hilltop, so no terrace riser is present. An isolated outcrop of bedrock standing above the strath surface was identified, however, and is interpreted as a former sea stack. The elevation of the bedrock strath at this point was taken as equivalent to an inner edge. Since the inner edge was covered by colluvium, the elevation of the strath was measured in a nearby cliff.
A clear terrace riser separates the Tarapuhi terrace (Qt4) from the Amuri Bluff terrace
(Qt6). The bedrock shoreline angle can be seen only in an inaccessible cliff face (Figure 4-4a), where it was estimated from photographs at about 38-44 m above sea level. The inner edge elevation was also estimated from a DGPS and LiDAR profile over the terrace surface (Figure 4-
4b), with the thickness of sediment overlying the terrace estimated at 6 m from measurements in
Ota et al. (1984) and Oakley et al. (2017). The DGPS profile can be fit well by a diffusion model
(RMS error of 1.4 m) and by the area-based cliff retreat method, both of which give an inner edge elevation of 40.7 m (Figure 4-4b). Uncertainty in the inner edge elevation is estimated at ±5 m and comes principally from uncertainty in the sediment thickness away from the point where it was measured (where the bedrock strath is at ~34 m), and in the slope of the bedrock strath, which we assume to be parallel to the terrace tread (Figure 4-4b).
While not dated in Oakley et al. (2017), the Kemps Hill terrace (Qt5) lies between the
Tarapuhi and Amuri Bluff terraces in elevation and is in places divided into an upper and lower level (Ota et al., 1984). Near Haumuri Bluff, the Kemps Hill terrace is cut into the hillside immediately southwest of the Tarapuhi Terrace (Figure 4-3), with a tread elevation of 96 m (from
LiDAR) at the base of the terrace riser, but the bedrock strath is not exposed. Kemps Hill terrace is also found south of Okarahia Stream, with a tread at ~106 m and a strath at ~98 m (Ota et al.,
1984). Ota et al. (1996) estimate elevations of 90 m and 110 m for the lower and upper levels.
These estimates appear to be based on extrapolation from more prominent terraces to the south,
132 assuming that the Kemps Hill terrace decreases in elevation to the north along with the rest of the terraces, but if we take the terrace immediately south of Okarahia Stream to be Kemps Hill upper, then its elevation is closer to 98 m. We use 90 ± 5 m and 98 ± 10 m for the two terraces.
4-4.3 Uplift Rates
The Tarapuhi terrace most likely correlates to the first of the two MIS 5c highstands of
Lambeck and Chappell (2001), which we term MIS 5c-1 (at 106.9 ka), as it is the significantly higher of the two. The two levels of the Kemps Hill terrace, being close in elevation, are a good fit to the double peak at the beginning of MIS 5a, as shown in Lambeck and Chappell (2001) and reproduced in Figures 4-2 and 4-5. Finally, the Amuri Bluff terrace would have formed during either MIS 5a-3 or 5a-4. It may have been occupied by both, as they attained similar sea levels, and the difference in the resulting uplift rate between the two choices is less than 0.1 mm/yr. Here we use MIS 5a-4, as it is closer in age to the HB1 IRSL sample (Oakley et al., 2017), but either choice is possible. Other possible choices for the terrace correlations are, however, unlikely, as they result in negative or extremely high uplift rates.
Uplift rate at Haumuri Bluff has decreased over time (Table 4-2; Table 4-3; Figure 4-4).
Our estimates for the average uplift rate of each terrace to the present (Table 4-2) are, except for the Kemps Hill Lower terrace, within error of those of Ota et al. (1996). This is probably a coincidence, because our ages and paleo-sea levels for all except the Tarapuhi terrace are significantly different. The average uplift rate of the Tarapuhi terrace is 1.8 mm/yr (Table 4-2), but the maximum rate, during the time interval between formation of the Tarapuhi (Qt4) and
Kemps Hill Upper (Qt5) terraces, reached 3.7±1.4 mm/yr (Table 4-3), which is the fastest known in the Canterbury or Marlborough regions (Ota et al., 1984; 1996; this study). The decrease in uplift rate to 1.0±0.5 mm/yr since formation of the Torlesse terrace (Table 4-2 and Table 4-3)
133 presumably reflects a decrease in fault activity and fold growth rate. The alternative possibility that the terraces are separated by a fault is not supported by structural or geomorphic evidence, as there is no mapped fault or observed fault scarp in this location.
South of Haumuri Bluff, uplifted marine terraces occur for 20 km along the coast, with elevations increasing southward at a gradient that is highest for the oldest terraces (Ota et al.,
1984). Summit heights of the Hawkswood Range likewise increase to the south. This pattern can be explained most simply by a gradient in fault slip that is highest in the central part of the fault and lower towards the ends, which is commonly observed in thrust belts (Elliott, 1976; Cowie and Scholz, 1992; Dawers and Anders, 1995; Davis et al., 2005). To quantify the uplift rates elsewhere in the range, we conducted two additional GPS surveys, from which we estimate inner edge elevations and uplift rates (Table 4-4), although control on bedrock strath elevations was poor in these areas. These were conducted at Claverley and Dawn Creek, which are respectively
2.5 km and 14 km southwest of Haumuri Bluff. We relied on the terrace correlations of Ota et al.
(1984) and Warren (1995) to correlate these terraces with dated ones at Haumuri Bluff. Uplift rates at Claverley are within error of those at Haumuri Bluff, while uplift rates at Dawn Creek are significantly faster, with the youngest terrace (Qt6) having an uplift rate of 1.9±0.2 mm/yr, compared to only 1±0.2 mm/yr at Haumuri Bluff. As at Haumuri Bluff, uplift rate has decreased with time at both locations, suggesting that uplift has slowed along the entire Hawkswood Range.
4-4.4 Structure
Haumuri Bluff lies on the forelimb of an asymmetric anticline (Figure 4-3). Forelimb dips decrease with distance to the southeast, which is suggestive of trishear folding and would be difficult to explain with a kink-band model, which predicts a constant forelimb dip. The structure is likely similar to that of the larger Hawkswood Range, although the lack of cover strata on most
134 of the range makes this difficult to confirm. There are some along strike changes, however: forelimb dips are steeper along the coast on the northeast side of Haumuri Bluff (~35°-10°, decreasing to the southeast over 1.2 km) than along Okarahia Stream on the southwest side (81° to 19°, also decreasing to the southeast), and the strike of bedding changes by about 20°-30° from southwest to northeast. Backlimb bedding along the coast mostly strikes between 220°-250° and dips between 20°-30°, reaching a syncline just south of the town of Oaro, which is well exposed in the coastal cliffs. The other limb of the syncline rises to the northwest until it is truncated by the Hundalee Fault, which brings up Torlesse basement in its hanging wall.
Plio-Pleistocene Greta Formation rocks fill the syncline and form an angular unconformity with the older cover units (Figure 4-3), although bedding in the Greta formation still dips to the NW. Mollusk shells collected by Beu (1979) indicate that these rocks were deposited within a submarine canyon at 600-800 m depth. Therefore, the apparent unconformity between Greta and older units in the syncline may simply be due to its deposition as canyon fill, and we cannot say whether or not it was deposited after the onset of folding. If it was deposited after folding began, then folding would have to predate the Nukumaruan (early Pleistocene) age of the fauna (Beu, 1979). The site described by Beu (1979) lies near the axis of the syncline
(Warren, 1995) at ~240 m elevation. Since the shells were deposited at 600-800 m depth, at a time when sea level would have been no more than about 100 m below to 50 m above present
(Rohling et al., 2014), the syncline must have undergone somewhere between ~800-1100 m of uplift, and cover strata should all restore to depths of >550 m below sea level.
4-4.5 Trishear Models
We constructed cross section A-A′ across the anticline and adjacent syncline (Figure 4-3) and projected data from the coast between Oaro and the northeast side of Haumuri Bluff onto the
135 cross section (Figure 4-6a). A point at the location of the shells collected by Beu (1979) was also projected onto the cross section and required to restore to -700 ± 100 m (1-σ error). Two points from each of the Amuri Bluff (Qt6) and Tarapuhi (Qt4) terraces were projected onto the cross section, with one point from each being the inner edge. For both terraces, these points lie on lines with less than 1° seaward slope, suggesting that they have not been tectonically tilted. In order to investigate the value of terraces in assessing trishear models, we compared models that satisfy only the geologic information related to bedding attitude and position of contacts with models that include the terraces as an additional constraint.
While an anticline can be produced by a simple step fault with a horizontal detachment, that would not explain the uplift of the backlimb syncline that is indicated by the Oaro shell sample, and it would require a very shallow (<2 km) detachment, despite most regional seismicity being deeper (Cowan, 1992; Reyners and Cowan, 1993). Instead, we consider a fault consisting of two segments, neither of which is required to be horizontal. We test two model variants with
InvertTrishear, exploring a wide parameter space (Table 4-5). In model 1 (Figure 4-6b, Figure 4-
7), we assume that P/S and φ are constant during fold growth. In model 2 (Figure 4-6c, Figure 4-
7), we allow the values of P/S and φ to change during fold growth, and we introduce additional constraints to limit deformation in the otherwise unconstrained offshore footwall region.
Model 1 (Figure 4-6b; Figure 4-7) fits the available data and can uplift the two terraces without deforming them when they are included as constraints. One problem with this model’s results, however, is that there is significant lowering of the footwall in front of the fold, which extends out to indefinite distances. In the best-fit model (Figure 4-6b), for example, the restored state elevation of the basement-cover contact is -1187 m, but in the footwall of the deformed state cross section, it reaches a depth of about -1600 m. This is a consequence of the trishear model, which occurs whenever ϕ (the half-apical angle of the trishear zone) is greater than the fault dip.
This causes the lower boundary of the trishear zone to extend down into the earth and the effects
136 of trishear deformation to extend indefinitely far from the fault tip. While the formation of a footwall syncline is reasonable, lowering of the footwall at great distances from the fault is most likely unrealistic, as deformation of this sort is not seen in numerical modeling of trishear-like folds (Johnson and Johnson, 2002; Cardozo et al., 2003; Hardy and Finch, 2006; 2007; Hughes and Shaw, 2015). In this case, the lack of any data from the offshore footwall results in this part of the model being underconstrained.
In order to prevent this lowering and achieve a result more consistent with the modeling work cited above, we introduced two additional constraints: (1) a requirement that ϕ must be less than or equal to the fault dip at all times, which prevents the trishear zone from extending downwards, and (2) creating an artificial point in the footwall far from the fold – at a position of
(20 km, -1 km) – where no deformation is expected, which is given an expected restored depth -1 km, so that it should not move substantially. The first constraint is the most critical one, as it prevents footwall lowering, but the second constraint helps to limit far-field deformation of any sort, including uplift far from the fault, in the otherwise unconstrained footwall. Introducing these two constraints to model 1 fails to provide any model that can fit all the data well, however. In model 2 (Figure 4-6c; Figure 4-7), we allow P/S and ϕ to each change once during the course of fault propagation, which produces models consistent with the footwall constraints. A change in
P/S during fold growth was previously included in forward modeling of some folds in offshore
North Canterbury by Barnes et al. (2016), so it is reasonable that it could occur on land as well.
In both models, when we do not include the terraces as a constraint in the model, we see two or more peaks in the histograms of model parameters (Figure 4-7). Like the examples in
Oakley and Fisher (2015), these are most clearly separated by P/S (for model 2, by P/S after the change), which is <1 for one group of models and >1 for the other groups. Including the terraces and requiring them to restore to the known paleo-sea levels as described above, however, causes
137 the P/S < 1 peaks to disappear, leaving only one set of models that is consistent with both the structural data and the terraces. The P/S < 1 models represent faults with a deep, blind tip, above which the marine terraces would be within the trishear zone and would be substantially deformed and tilted. The lack of deformation of the terraces, therefore, rules out these models. In constrast, the P/S > 1 models typically have tips that propagate to the surface (offshore), which results in folding of the forelimb in the trishear zone during the early stages of fold growth, followed by translation along the fault without further folding by the time the terraces form. In model 2, with
P/S allowed to change, low P/S values are favored in the early stages of faulting, but the lack of terrace deformation requires that P/S must increase once the tip reaches -2230±460 m. An increase in P/S results in initial folding in a wide trishear zone above a deep fault tip, followed by tip propagation to the surface so that the terraces are not folded.
Both models predict the presence of an offshore fault, which reaches the seafloor off
Haumuri Bluff. If this is correct and if the same fault is responsible for uplift of the entire
Hawkswood Range, then it likely extends along the ~40 km length of the range. Though not known to have ruptured in historical times, such a fault could pose a seismic hazard to the region.
This fault may be a splay off the Hundalee Fault, which lies on the landward side of the range, and which did rupture in the recent Kaikoura earthquake sequence (Clark et al., 2016). Our cross sections (Figure 4-6) suggest that the two would intersect at depth, although this is dependent on the geometry of the Hundalee Fault, which is not known except that it is thought to be steep at the surface (Warren, 1995) and is estimated to dip at between 40° and 70° (Litchfield et al., 2014).
Model 2 is the more likely of the two models considered here, since it avoids far-field footwall deformation that is thought to be unrealistic, but data offshore of Haumuri Bluff would be needed to distinguish between them for certain. Despite their differences, however, key results are similar for both models. Both predict a similar offshore fault, and both show that marine
138 terraces, as syn-folding markers of deformation, can be used to rule out certain classes of trishear models.
4-5 Kate Anticline Results
4-5.1 Setting
The Kate Anticline (Figure 4-8) lies at the boundary between the North Canterbury Fold and Thrust Belt and the Canterbury Plains and uplifts the southernmost of the North Canterbury marine terraces. Marine terraces here have been mapped and described by multiple authors
(Jobberns, 1928; Carr, 1970; Yousif, 1989; Forsyth et al., 2008; Stewart, 2016; Oakley et al.,
2017), and the lowest and widest terrace has been dated to MIS 5c (Oakley et al., 2017). It is south of the Cass Anticline, with the Teviotdale Syncline between the two. Unlike in the Cass
Anticline, Torlesse basement is not exposed in the crest of the Kate Anticline and there is no surface fault along its forelimb.
4-5.2 Terrace Inner Edge Elevations
Terrace inner edge elevations were estimated along four topographic profiles (Figure 4-8;
Figure 4-9), with profile 4 split into two parts, in order to cover multiple terraces. Profile 2 was based purely on DEM and LiDAR data, while the others followed DGPS surveys. Inner edge estimates agree within 2-3 m with earlier work by Carr (1970) (Figure 4-9). Inner edge elevations were determined using the area balance method, or from hilltop elevations for higher terraces.
139 4-5.3 Uplift Rates
The Qt4, 3, 2, and 1 terraces are correlated to the MIS 5c, 5e, 7a, and 7c sea level highstands respectively (Oakley et al, 2017). There is a spatial pattern to uplift rates (Table 4-6;
Figure 4-10), which increase along the coast from southwest to northeast (topographic profiles 1 to 4). Correlation diagrams (Figure 4-10) suggest an overall decrease in uplift rate over time, but a constant rate is not ruled out at the 95% confidence level, unlike at Haumuri Bluff.
4-5.4 Structure
Seismic reflection lines across the Kate Anticline (Exploration Solutions Ltd, 2006;
Geosphere (Excel), 2008) were interpreted (Appendix F) to help delineate the fold geometry in the subsurface. Three prominent reflectors in the Kate-01 seismic line were correlated to the tops of the Ashley mudstone, Waikari Formation, and Torlesse basement based on the synthetic seismogram of Styles et al. (2008). These same three horizons were then identified in the other seismic lines where possible. Checkshot data and horizon depths from Styles et al. (2008) were fit to a velocity model (Table 4-7), which was used to depth convert the six seismic sections. The relatively high seismic velocities between the Waikari Formation and Ashley mudstone are due to the presence of carbonate units (Amuri limestone and Omihi Formation) in this interval. The
Waikari formation as identified by Styles et al. (2008) is thinner and deeper than would be expected from mapped surface geology (Yousif, 1987). This difference may be no more than a difference in how the contact between the Waikari and Tokama formations – two lithologically similar units – is defined: Sandstone content relative to siltstone is highest at the top of the
Waikari formation in the mudlog of Styles et al. (2008) and decreases abruptly in the Tokama
Formation, while in Yousif’s (1987) description, the upper part of the Waikari formation in this
140 area has little sand. Regardless of the precise identification of this horizon, it forms a prominent reflector, which we refer to here as the Waikari Formation.
Only seismic line TAG06-260-02 images both limbs of the fold, but it is of limited use for this study because it lies in a structurally complex region at the plunging west end of the anticline. The backlimb of the fold is best imaged in the Kate-01 line, where it dips at 18-20° and rises sharply from nearly flat bedding to the south. Lines TAG06-260-03 and 04 show approximately the same structure, but less well imaged. In Tag06-260-05, a section of the fold crest or upper backlimb dips at no more than 10°. Surface dip measurements on the backlimb are mostly between 10 and 20°, without any clear systematic change along strike. Nearly flat bedding was observed near the Teviotdale Stream mouth, which corresponds to the flat beds seen in seismic data. The forelimb is best imaged in line TAG06-260-05, where it is curved with a maximum dip of 39°. Dip-meter measurements of ~25° within the Tokama Formation (Styles et al., 2008) are consistent with this seismic data if they are from a less steep part of the curved forelimb. Surface dip measurements in the forelimb of the fold reach only about 20°, but some of the youngest parts of the Greenwood Formation may be syn-folding growth strata, as we observed a slight upward decrease in dip within beds near the Teviotdale Stream. The backlimb, with its sharp bend is suggestive of kink-band folding, while the curved forelimb with spatially variable dips suggests that trishear may be a better model.
The stratigraphic thickness of the Eyre Group (the units between the top of the Torlesse and top of the Ashley mudstone) is somewhat variable in our interpretations. It is thinner in the forelimb (~260 m) than in the backlimb (mostly 300-350 m, but up to 420 m), and in the backlimb of TAG06-260-02, it thickens northward from ~300 to 420 m. While some of these differences may be due to uncertainties in depth conversion or interpretation, they would be consistent with the presence of an inherited half-graben in the hanging wall, with the bounding normal fault either reactivated as a reverse fault or localizing the formation of a new thrust.
141 Similar structures have been documented offshore in North Canterbury (Barnes et al., 2016).
Since the top of the Torlesse would not restore to a straight line in this interpretation, and also due to uncertainty in identification of the Torlesse in TAG06-260-05 (see Appendix F), we do not include the basement-cover contact in our models, but instead rely on the Ashley and Waikari horizons to define the shape of the fold.
One surprising aspect of the structure, which complicates our kinematic modeling, is that the hanging wall syncline imaged in line Kate-01 is about 200 m lower than the footwall syncline
(Teviotdale Syncline) imaged in TAG06-260-05, despite the fact that the hanging wall should be uplifted relative to the footwall. This presents two possibilities: either the elevations of both synclines decrease along strike to the southwest by >200 m between TAG06-260-05 and Kate-01 or the footwall syncline truly is elevated relative to the hanging wall syncline. We consider both structural models in our kinematic analysis below, where they are termed Model 1 and Model 2 respectively. Model 1 would be consistent with the fact that strata dip to the southwest in the region immediately southwest of Kate Anticline (Wilson, 1963; Yousif, 1987; Figure 4-11), as the fold could be superimposed on this regional dip, but it is difficult to reconcile with the fact that in seismic line TAG06-260-02 (further southwest than Kate-01) the footwall syncline is only about 60 m lower than in TAG06-260-05. Model 2 could be explained if the Kate Anticline refolds the lower part of the backlimb of the Cass Anticline, resulting in a structurally elevated syncline between the two anticlines.
We have constructed a structure contour map on the tops of the Ashley Mudstone and
Waikari Formation (Figure 4-11), which reveals an asymmetric anticline and syncline pair, with an s-shaped curve to the fold axes, approximately paralleling the coast. This interpretation assumes that the structure is as constant as possible along strike, with the Kate-01 and TAG06-
260-05 seismic lines providing the primary sources of data on the backlimb and forelimb geometries respectively. As such it is consistent with Model 2 for our kinematic modeling and is
142 used to help define the fold structure when testing that model. In areas away from seismic lines, offshore, or where seismic data quality is poor, the structure is uncertain, and other interpretations are possible. (See Appendix G for one alternative.)
4-5.5 Relationship Between Structure and Uplift Rates
All the terraces between the Teviotdale River and the bend in the coast, where uplift rates are low, lie in the region of nearly flat bedding (Figure 4-11). Northeast of the bend in the coast, terraces lie mostly on the dipping backlimb of the fold. Some of the higher terraces cross into the crest and forelimb of the fold near Kate Creek and Dovedale Stream. The fact that uplift rates are lowest where bedding is near flat and are higher on the backlimb of the fold suggests that uplift rate is a function of structural position. Specifically, in the two-segment fold models we are using, the lower uplift rates and flat beds would lie above the lower segment, while the higher uplift rates and steeper dips would lie above the upper, more steeply-dipping segment. The fact that there is not a major change in uplift rates from the backlimb to the crest or forelimb places constraints on the possible fold model.
We focus our analyses on two structural cross sections (Figure 4-12), A- A′ and B- B′, which lie along seismic lines Kate-01 and TAG06-260-05, respectively. Terrace inner edge elevations along the structural cross sections are the same within error as along topographic profiles 1 and 4. The pattern of uplift rates allows us to test possible kinematic models by comparing the observed ratios of uplift rate above different parts of the fold (forelimb, crest, backlimb, or flat) to those predicted by the kinematic models. Specifically, we consider two tests, relying on different and independent assumptions. First, along section A-A′, Qt4 has an uplift rate of 0.8 ± 0.1, and Qt1 has an uplift rate of 0.9 ± 0.1, for a ratio of UQt1 / UQt4 = 1.2 ± 0.2. Qt4 lies above the backlimb of the fold, and Qt1 lies near the boundary between the forelimb and crest of
143 the fold. Thus, for a fold model to be correct, it should predict the observed ratio of uplift rates between these two structural positions. The ratio of 1.2±0.2 is very similar to the ratio of 1.3±0.3 for the same two terraces along cross section B where they do not cross the crest of the anticline, suggesting that the rate of fault slip has either been constant or decreased slightly over time.
Secondly, while Qt4 is in the fold backlimb along cross section A-A′ and therefore above the upper fault segment, along B-B′ it is in the flat region and above the lower fault segment. Taking into account the covariance in paleo-sea level and facies depth, the ratio of the two uplift rates
(UA / UB) is 1.5 ± 0.1. (It is similar for the other terraces: 1.6 ± 0.2 for Qt3, and 1.4± 0.1 for Qt1.)
Therefore, a fold model should predict this ratio of uplift rates. Since terraces of the same age are compared, this test does not require a constant uplift rate over time. It does, however, require that we either assume fault slip rates and fault geometry are constant along strike or account for how along-strike changes affect uplift rates. We apply each of these tests to our two different models of Kate Anticline structure.
4-5.6 Kinematic Models
We test kinematic models for cross sections A and B, for trishear and kink-band kinematics, and for Models 1 and 2 of fold structure. For Model 1, assuming a pre-folding dip to the southwest, we restore the beds in each cross section to flat but do not require them to restore to the same depths in the two cross sections. For data (Figure 4-12), we use the horizons interpreted in the seismic data for the forelimb in cross section A and the backlimb and flat beds in section B, along with dip data for the parts of the fold not imaged in the seismic data for each cross section. We do not use the structure contours for this model to avoid including interpretations not constrained by data or model assumptions, and we do not use any data from the Cass Anticline, since we are considering it as separate from the Kate Anticline. For Model 2,
144 we assume that the restored depth of each horizon is constant between the two cross sections, and that the Kate Anticline has refolded the lower part of the backlimb of the Cass Anticline. To keep depths constant (within a few 10s of meters) between the two cross sections, we first fit models to cross section A-A′ and then use the mean and standard deviations of the resulting depths of the two horizons to assign a Gaussian prior probability to these parameters when fitting for cross section B-B′. To model the refolding of the Cass Anticline backlimb, we fit data from the backlimb of the Cass Anticline and the forelimb of the Kate Anticline together to dipping lines
(with the same dip for both horizons). Meanwhile, the crest and backlimb of the Kate Anticline along with the flat beds to the south are restored to a flat line. Because we require data over all parts of the fold, and in order to ensure that the forelimb and backlimb synclines are at approximately the same elevation in both cross sections, we use the structure contours of Figure
4-11 to reconstruct the approximate shape of the fold in areas not imaged in the seismic sections for each cross section (Figure 4-12). The expected restored depths of the units are poorly constrained, but we limit the Ashley and Waikari horizons to -1500 m for both models. The top of the Ashley mudstone is ~300 m above the top of the basement (but varies as described above).
Regionally, gravity evidence suggests a basement-cover contact at 1-1.5 km depth in the Waipara
Valley (Loris, 2000) while offshore seismic data puts it at 1-2 km (Barnes et al., 2016). The complete ranges of model parameters for Markov chain Monte Carlo simulations are shown in
Table 4-8.
Testing kink-band kinematic models for Model 2 of fold structure is difficult because the equations of Suppe and Medwedeff (1990) and Hardy (1997) are based on the assumption of conservation of bed length for beds that all have the same initial dip, which is not the case here, due to the pre-existing structure of the Cass Anticline. We use the velocity field that will preserve the length of originally flat beds, but we note that bed length will not be conserved in the part of the fold that is initially dipping as part of the Cass Anticline backlimb.
145 Since kink-band models of fault-propagation folding predict specific relationships between fold limb dips, fault dips, and relative uplift rates at different structural postions (Suppe and Medwedeff, 1990; Hardy and Poblet, 2005), we can compare these models to observations using plots of these relationships. Since this approach assumes that pre-folding bedding is horizontal, it is only applicable to structural Model 1. The ranges of dips observed for the Kate
Anticline (~10°-20° for the backlimb and 20°-40° for the forelimb) rule out fixed-axis fault propagation folding (Figure 4-13a), but parallel fault-propagation folding is possible for fault dips of 40.7°-69.7° for the lower segment and 53.4°-71.4° for the upper segment (Figure 4-13b). The uplift rate ratio of 1.2 ± 0.2 for the Qt1 and Qt4 terraces along cross section A-A′ is incompatible with the predicted ratio of crest to backlimb uplift rates (Figure 4-13c), but it is consistent with the predicted ratio of forelimb to backlimb uplift rates (Figure 4-13d). Qt4 is approximately above the crest-forelimb boundary at the level of the Waikari Formation (Figure 4-12), so it is probably above the fold crest at the surface since this domain widens upward (Figure 4-1b). For the uplift rates over the upper and lower parts of the two segment fault in cross sections A and B respectively, assuming that fault dip and slip rate are the same between the two cross sections, the predicted uplift rate ratio is within error of the observed ratio of 1.5±0.1 for high values of both limb dips (Figure 4-13e). Thus the first test of uplift rate ratios is difficult to reconcile with parallel fault-propagation fold kinematics, while the second test does not rule it out.
For trishear, we cannot simply plot uplift rate ratios as a function of limb dip, because additional parameters such as P/S and ϕ are not captured in such plots and because uplift rate is not constant throughout the forelimb (Figure 4-1). Instead, we use Markov chain Monte Carlo simulations to fit models to data from both cross sections, in which we calculate the amount of fault slip necessary to restore each terrace, and we plot histograms of the ratios of fault slip rates.
For comparison, and as an additional check on the results described above, we also fit parallel fault-propagation fold models to the data in the same manner. This approach is different from
146 Figure 4-13c-e, in that we are comparing the slip rates needed to produce the observed uplift rates, rather than the uplift rates predicted by a constant slip rate, but the two are closely related.
In contrast to the assumption that fault geometry is constant along strike in Figure 4-13e, any of the model parameters (fault dip, total slip, fault tip position, etc.) may vary between the two cross sections since we are fitting each cross section separately.
For Model 1, parallel fault-propagation folding models produce slip rate ratios for Qt1 to
Qt4 of 0.85±0.14 (mean and two standard deviation error) (Figure 4-14a), while for trishear the ratio is 1.3±0.5 (Figure 4-14b), albeit with a skewed distribution that peaks at 1.2. Neither result favors a constant fault slip rate, but parallel fault-propagation folding suggests that fault slip has increased over time, while trishear suggests that it has decreased. As noted above, along GPS transect 1 (and cross section B-B′), which does not cross the fold crest or forelimb, the same Qt1 to Qt4 uplift rate ratio is 1.3 ± 0.3, which supports a decrease in fault slip rate over time, not an increase, and thus favors the trishear model. For the second test, comparing the fault slip needed to restore Qt4 in cross section A to that needed to restore Qt4 in cross section B, both parallel fault-propagation folding (Figure 4-14b) and trishear (Figure 4-14d) are consistent with a constant slip rate between the two cross sections, or with a range of other values, so this test fails to distinguish between the two kinematic models.
For Model 2, the results are similar to Model 1. For the first test, the slip rate ratios are
0.73±0.15 for parallel fault-propagation folding (Figure 4-14e) and 1.5±0.3 for trishear (Figure 4-
14g). As in model 1, trishear is consistent with the observed ratios of uplift rates, given a slight decrease in slip rate over time, while parallel fault-propagation folding would require that fault slip rate has increased over time, which the available evidence does not support. For the second test, the ratios are 0.50 ± 0.27 for parallel fault-propagation folding (Figure 4-14 f) and 0.64±0.52 for trishear (Figure 4-15h), though this histogram is multimodal. Both suggest that the fault slip rate in cross section A is greater than in cross section B, although the trishear result is within error
147 of a constant slip rate between the two locations. This somewhat favors trishear, but it also suggests that if Model 2 is correct, then the assumption of constant slip between the two sections may not be valid.
Overall, in both Models 1 and 2 for the structure of the Kate Anticline, the first test favors trishear over parallel fault-propagation folding, while the second is more ambiguous.
These results show that comparing marine terrace uplift rates at different structural positions can help to distinguish between trishear and kink-band kinematic models, although not all such comparisons will produce a clear distinction. Of the two structural models considered to explain the difference between forelimb and backlimb syncline elevations, Model 2 is probably more likely to be correct, due to the similar forelimb syncline elevations in TAG06-260-05 and -02, but uncertainty remains about the complete three-dimensional structure of the fold due to data gaps and poor seismic imaging of some parts of the structure.
4-6. Discussion
4-6.1 Marine Terraces as a Constraint on Fold Kinematics
Our work has shown that marine terraces can serve as a constraint on fold kinematic models in two ways. First, because terraces are surfaces that are known to have formed horizontally or at a very low seaward slope, models that fail to restore terraces to this geometry at the appropriate paleo sea-level can be ruled out. At Haumuri Bluff, this principle was used to rule out certain models that could fit the geologic structure, in the process reducing multimodal probability distributions to unimodal ones. That trishear can result in multimodal probability distributions was previously recognized in Oakley and Fisher (2015), and the constraints imposed by marine terraces allow us to eliminate some classes of solutions. In this case, the uplifted
148 terraces were not tilted or folded, despite being on the forelimb of a fold, so a model was needed that restored them without deforming them. If deformed terraces had been observed, then models would be needed capable of restoring that deformation. In this case, we treat erosional platforms much like growth strata, which can also help to constrain fold kinematics (Storti and Poblet,
1997).
Secondly, we have investigated how different uplift rates at different structural positions can help to constrain fold kinematics. Kinematic models predict specific ratios of uplift rates between different positions on a structure, which can be compared to what is observed. By showing that certain models fail to predict the observed ratios of uplift rates, we have been able to rule out some models that could otherwise have been applied to the Kate Anticline. Our first test
– comparing the uplift rates on the backlimb and crest of the fold along a single cross section – proved to be the most useful. This test favored trishear over kink-band folding and exploited a key difference between the two possibilities: trishear allows the uplift rate to be the same above both crest and backlimb (Figure 4-1), and even above regions of the forelimb that have left the trishear zone, while parallel fault-propagation folding produces distinct uplift rate domains with a rapidly uplifting fold crest (Figure 4-1). The lack of a major difference in uplift rate between crest and backlimb therefore favors trishear.
Our second test of uplift rate ratios at Kate anticline – comparing uplift rates over the upper and lower segments of the fault ramp in two different cross sections – proved more ambiguous and unable to distinguish between trishear and kink-band models. This may be due to the fact that models were fit independently to the two cross sections, with the resulting estimates for parameters such as ramp angle and total slip varying independently between them. Even though we kept the restored bed depths nearly constant between the two cross sections, other parameters such as total slip, and fault dips were able to vary. Further, this test was predicated on the assumption that the rate of slip is the same along both cross sections, which is not certain.
149 Overall, the failure of this second test to reliably distinguish between the two kinematic models shows the limits of our approach. A three-dimensional model, in which model parameters are not independent between the two parts of the fold, might be better suited to this test. The observation that led us to try the second test remains valid – more rapid uplift occurs over the fold backlimb, while slower uplift occurs where bedding flattens to the south of the backlimb syncline, strongly suggesting structural control – but using the details of that structural control to deduce fold kinematics is difficult in this case.
4-6.2 Implications for Trishear and Fold Kinematics
The trishear model is particularly well suited to basement-involved folding (Erslev, 1991;
Hardy and Allmendinger, 2011). North Canterbury is no exception, as trishear has proved suitable to model basement-involved fault-propagation folds in this region. Trishear was used previously in North Canterbury by Barnes et al. (2016) to model the smaller, mostly monoclinal folds of
Pegasus Bay, offshore North Canterbury. Here, we have shown that the same kinematic model can explain onshore folds as well, with the addition of a fault bend to produce the more pronounced backlimbs seen in the onshore folds.
Our preferred model for the anticline at Haumuri Bluff requires changes in P/S and φ during fold growth. Changes in P/S were also used by Barnes et al. (2016) in some of their forward models of offshore folds. An increase in P/S during trishear fault-propagation folding represents the process of fault breakthrough, which is a common feature of fault-propagation folds that is not unique to trishear (Suppe and Medwedeff, 1990; Mercier et al., 1997;
Allmendinger, 1998). It is also similar to break-thrust folding (Fischer et al., 1992) in that folding is followed by rapid fault propagation through the fold limb, but in this case both phases are explained by the same kinematic model. Allmendinger (1998) suggests that changing P/S through
150 time may be common, and some studies that employ forward models of trishear folds have included such changes, including the offshore North Canterbury work of Barnes et al. (2016).
Such parameter changes have previously been considered too complex to invert for
(Allmendinger et al., 2004), however. Our work suggests that it may be worthwhile to consider this possibility and that computing speed and data inversion algorithms are now advanced enough to allow such complex trishear models to be tested. The significance of the change in ϕ is less clear than that in P/S. It tends to occur very near the fault bend (Figure 4-6c). At least in part, this is a response to the requirement that ϕ must be less than or equal to the ramp angle; if the fault tip begins propagating in the lower segment and then bends upwards, then ϕ must be low when the tip is in the lower segment but is able to increase when the tip is above the bend.
4-6.3 Implications for North Canterbury Neotectonics
Our results provide new models of two actively-growing anticlines in North Canterbury.
The trishear models make predictions about the geometry of faults that have not been observed directly, which could potentially be tested in the future. We predict a fault near the coast at
Haumuri Bluff, which has propagated up to the seafloor, while for Kate Anticline, we predict a blind fault underlying the anticline, which in most of our models has its tip at 1-2 km below sea level (see Appendix G). Some questions remain about the structure of North Canterbury faults.
As noted above, fault geometries could be more complex than the two segment models we have used here. Our proposed faults, which become more gently dipping with depth, may be a simplification of more complex listric fault geometries. To what depth the faults extend is also not known. Parts of the structure of the Kate Anticline remain uncertain, in regions not covered or poorly imaged by the available seismic reflection data. Nonetheless, our work provides information on the possible geometry of the faults, where none was previously available.
151 In MCMC simulations that incorporate marine terraces, one of the parameters is the amount of fault slip necessary to restore each terrace to the elevation at which it formed. Given that the age of the terrace is known, this allows us to calculate the rate of slip on the fault. For
Haumuri Bluff, we calculate slip rates based on the uplift rates of terraces Qt4 and Qt6 and the uplift rate for the interval between the formation of those two terraces (Figure 4-15), while for
Kate Anticline we calculate slip rates based on the uplift rates of terraces Qt1 and Qt4 for A-A′ and Qt1, Qt3, and Qt4 for B-B′ (Figure 4-15). Uplift rates at Haumuri Bluff and to a lesser extent
Kate Anticline have not been constant over time, and this is reflected in the slip rates. The lowest slip rates (<1 mm/yr) are predicted by the kink-band models for Kate Anticline, while trishear models predict higher slip rates. Since our analysis favors trishear over kink-band models for
Kate Anticline, it likewise favors the faster rates of fault slip implied by trishear. These rates differ among the different terraces, the two cross sections, and the two structural models, but overall a range of fault slip rates of about 1-2 mm/yr is most consistent with all combinations
(Figure 4-15). A similar range of slip rates is suggested by the youngest terrace at Haumuri Bluff
(Figure 4-15), with faster slip rates, up to ~4 mm/yr in the past. Both the Kate and Hawkswood
Anticlines have been previously recognized as locations of seismic hazard by Barrell and
Townsend (2012). They estimated “vertical slip rates”, more analogous to our uplift rates, of 0.5 mm/yr for Kate Anticline and 0.5 or 1.5 mm/yr for the Hawkswood Anticline. These are generally lower than our uplift and fault slip rates, suggesting that the seismic hazard may be greater than previously realized, although the higher of the two Hawkswood estimates is similar to our Qt6 slip rate for that structure.
Dividing the total slip in each model by the slip rate, we can calculate the age at which folding began. Previous estimates of the age of the North Canterbury fold and thrust belt include
0.8 ± 0.4 Ma (Nicol et al., 1994), 0.8 Ma (Barnes, 1996), 1-2 Ma (Litchfield et al., 2003), and 1.2
± 0.4 Ma (Barnes et al., 2016). All these estimates are based on work done in the southern part of
152 the North Canterbury fold and thrust belt (both on- and offshore). We have estimated ages for each of our models for both anticlines, based on the different slip rates above (Figure 4-16). Since we know that slip rates have not been constant over time, these estimates must be treated with caution. Again there are differences based on model choices and the different terrace uplift rates, as well as between kink-band and trishear models for the Kate Anticline (Figure 4-16), with kink- band models predicting a younger initiation of folding. Nonetheless, for both anticlines, age estimates are broadly in agreement with previous work, with ages greater than 2 Ma virtually ruled out, and ages closer to 1 Ma or less more likely. That this is true at both locations, in multiple cross sections, using different estimates of uplift rate indicates that it is a robust result, even considering the many uncertainties involved. What caused folding to begin in this region at c. 1 Ma is not entirely clear, but a similar age has been estimated by some workers for the beginning of slip on the Hope Fault (Wood, 1994; Langridge et al., 2013), and Bull (1991) has suggested that uplift of the Seaward Kaikoura range began only within the past 1 Ma. So, one possibility is that folding in North Canterbury may have become necessary to accommodate a component of plate boundary strain not taken up by the Hope Fault as the Hope Fault became the locus of maximum strike slip deformation in the region during the Pleistocene (Little and
Roberts, 1997).
Rates of shortening across the folds can also be estimated from model results, which we have done by multiplying the fault slip rate by the cosine of the dip of the lower fault segment in each model. These rates are more speculative than slip rate and age estimates, since they depend on assumptions about fault structure at depth and hanging-wall kinematics outside the trishear zone. In a trishear model, fault ramp angle and slip on the upper part of the fault are constrained by the shape of the forelimb and by the total structural relief on the fold, while the lower ramp angle is mostly a function of the backlimb dip. We have used fault parallel flow to describe deformation in the hanging wall of a trishear fold, which conserves total slip between the two
153 segments of the fault, but alternative choices such as conservation of bed length or constant heave would result in a different shortening rate without changing the rate of slip on or dip of the upper part of the fault. In addition, if the fault flattens into a detachment at depth, shortening estimates could be different. (See Chapter 5 for further discussion of these issues.) Since we have focused on constraining trishear parameters and on comparing trishear to kink-band models, these aspects of folding have not been our primary concern. As with slip rates and ages, shortening rates vary among the different models tested. For trishear models, rates of about 1-2 mm/yr are consistent with both cross sections for Kate Anticline (and higher rates only for cross section B), while rates of 1-4 mm/yr, decreasing over time, are seen at Haumuri Bluff. For comparison, the elastic block model of Wallace et al. (2012) estimates a shortening rate of 3-4 mm/yr at azimuth
343.75°±24.39° across the Pegasus Block, which includes most of the North Canterbury fold and thrust belt. The lines of section used in this study trend at 329° (Haumuri Bluff), 315° (Kate
Anticline cross section A), and 0° (Kate Anticline cross section B). If long-term shortening rates have been the same as these geodetic rates, then the two anticlines studied appear to account for a significant portion (a quarter or more) of total shortening. For Haumuri Bluff this is not unreasonable, as the zone of deformation is relatively narrow and only one other mapped fault
(the Hundalee Fault) lies between there and the Hope Fault. For Kate Anticline, it is possible if this anticline is the current locus of deformation. The region of deformation is wider than in the north, however, with at least five other anticlines and associated faults subparallel to the Kate
Anticline (Rattenbury et al., 2006; Forsyth et al., 2008) as well as structures offshore (Barnes et al., 2016), so the Kate Anticline cannot have accommodated such a significant fraction of total shortening at all times during growth of the fold and thrust belt.
154 4-7. Conclusions
1) We have determined new uplift rates for marine terraces at Haumuri Bluff and Kate
Anticline. Uplift rates at Haumuri Bluff at MIS 5c were the highest that have been seen anywhere in North Canterbury (3.7±1.4 mm/yr), but the rate had decreased substantially by MIS 5a
(1.0±0.5 mm/yr). Uplift rates at Kate Anticline show a more modest decrease with time, within error of a constant rate, but they vary more significantly in space, with the uplift rate of the lowest and most extensive terrace ranging from 0.5±0.1 mm/yr to 0.8±0.1 mm/yr from southwest to northeast.
2) Marine terraces provide constraints on fold kinematics in two ways: by serving as originally flat surfaces that must be restored to a known elevation and by allowing a comparison of uplift rates at different structural positions on a fold.
3) Using marine terraces, we were able to constrain trishear models at Haumuri Bluff and to reduce probability distributions that were otherwise bimodal in P/S to unimodal distributions.
At Kate Anticline, comparison of backlimb and crest uplift rates along a single cross section favored trishear over kink-band kinematics, but comparison of uplift rates between two different cross sections proved less useful, likely due to differences resulting from fitting model parameters independently for the two cross sections.
4) We estimate fault slip rates, ages of initiation of faulting and folding, and shortening rates derived from Markov chain Monte Carlo simulation results. Fault slip rates (~1-2 mm/yr at
Kate Anticline and 1-4 mm/yr at Haumuri Bluff) are at the high end of or higher than previously recognized rates, age estimates of ~1 Ma are generally consistent with previous work, and rates of shortening (~1-2 mm/yr for Kate Anticline and 1-4 mm/yr for Haumuri Bluff), while somewhat uncertain, are on the high end of what could be expected given regional geodetic shortening.
155 4-8. References
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Allmendinger, R.W. and Shaw, J.H., 2000. Estimation of fault propagation distance from fold
shape: Implications for earthquake hazard assessment. Geology 28, 1099-1102.
Allmendinger, R.W., Zapata, T., Manceda, R., and Dzelalija, F., 2004. Trishear kinematic
modeling of structures, with examples from the Neuquén Basin, Argentina, in K.R.
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Figure 4-1: Parallel fault propagation fold uplift rate (a) and fold shape (b) compared to trishear uplift rate (c) and fold shape (d). Uplift rates are measured at 1000 m elevation for both structures and are relative to the rate of slip on the fault. Both faults have slip = 500 m, fault dip = 20°, and
P/S = 1.64. ϕ = 30° for the trishear model.
167
Figure 4-2: Sea level curves for MIS 7 to present, with our preferred highstand sea levels and ages used for uplift rate calculations. The curves shown are the coral terrace-based sea level curves of Lambeck and Chappell (2001), the V19-30 δ18O-based sea level curve of Siddall et al.
(2007), and Holocene sea level for Canterbury from Clement et al. (2016). For Clement et al.
(2016), we have used their glacio-isostatic modeled curve as the upper bound and a curve drawn around their lowest data points as the lower bound. Peaks within MIS 5a and c are numbered as in
Table 4-1.
168
169 Figure 4-3: Map of the Haumuri Bluff area. (A) Location within North Canterbury. (B) Extent of marine terraces along the Hawkswood Range. (C) The anticline at Haumuri Bluff. The maps (B and C) are based in large part on Warren (1995), with alterations based on other published maps and this study, as described in the text.
170
Figure 4-4: (A) The inner edge of the Amuri Bluff (Qt6) terrace in the cliff on the north side of
Haumuri Bluff. (B) Inner edge elevation estimates. Red x’s are DGPS points. Blue x’s are the inner edge as determined by the two methods. The dotted and dash-dotted lines represent the reconstructed terrace riser profile prior to cliff retreat or scarp diffusion. The dashed black line is an approximation of the strath surface as 6 m below the tread. The grey lines is the modern topographic profile.
171
Figure 4-5: Correlation diagram for Haumuri Bluff marine terraces.
172
173 Figure 4-6: (a) Topographic profile and data for cross section A-A′. Geologic contacts are colored by the unit below the contact, with the same colors as in Figure 4-5. (b) Best-fitting cross section for Model 1 for the Haumuri Bluff anticline. (c) Best-fitting cross section for Model 2.
P/S and ϕ change when the propagating fault tip reaches the marked points.
174
175 Figure 4-7: Histogram representing probability density functions for key trishear parameters for the anticline at Haumuri Bluff. P/S upper and lower refer to the values of P/S when the fault tip is above or below its position at the time that P/S changes. Histograms of additional parameters can be found in Appendix E.
176
177 Figure 4-8: Geologic Map of the Kate Anticline, with locations of seismic lines and GPS surveys. Seismic lines numbered 1-5 are lines TAG06-260-01 to -05. K1 is Kate-01. GPS surveys and profile points 4-1 and 4-2 are parts 1 and 2 of survey 4. The map is based in large part on
Yousif (1987), with alterations based on other published maps and this study, as described in the text.
178
Figure 4-9: Kate anticline terrace profiles.
179
Figure 4-10: Correlation diagram for Kate Anticline marine terraces along topographic profiles /
GPS surveys 1 and 4.
180
Figure 4-11: Structure contour maps of the Kate Anticline for the tops of the Waikari and Ashley
Formations.
181
182 Figure 4-12: Kate Anticline cross sections A-A′ and B-B′, showing data along each. Torlesse was not used in the inversion but is shown here for reference. Structure contour points are used in inversions only for Model 2. Extra structure contour points were added in cross section A at the elevations at which the corresponding horizons flatten in cross section B, to conform with the
Model 2 assumption that structure is constant along strike.
183
Figure 4-13: (a) Forelimb dip vs. backlimb dip for fixed-axis fault-propagation folding compared to the range of limb dips observed at Kate Anticline. In this model there is only one possible forelimb dip for each backlimb dip (Suppe and Medwedeff, 1990). (b) Relationship between fold limb dips and fault segment dips for parallel fault-propagation folds compared to the range of limb dips observed at Kate anticline. (c, d, and e) Predicted ratios of uplift rates between different structural positions on a parallel fault propagation fold as a function of the dips of the fold’s limbs, compared to the limb dips observed at Kate Anticline for (c) crest and backlimb, (d) forelimb and backlimb, and (e) hanging wall above the upper and lower fault segments.
184
185 Figure 4-14: Fault slip rate ratios from MCMC results for the Kate Anticline for trishear and parallel fault-propagation folds (kink-band folds) for both models of Kate Anticline structure.
Left column: ratio of rate of fault slip required to restore terrace Qt1 to rate of fault slip required to restore terrace Qt4, both in cross section A. Right column: ratio of fault slip required to restore terrace Qt4 in cross section A to fault slip required to restore terrace Qt4 in cross section B.
186
187 Figure 4-15: Fault slip rates predicted for different fault and fold models for the Haumuri Bluff and Kate Anticlines. For Haumuri Bluff, slip rate is calculated using the average uplift rates from the formation of terraces Qt4 and Qt6 to present and for the interval between the formation of terraces Qt4 and Qt6, assuming ages of 106.9 ka for Qt4 and 71.3 ka for Qt6. For Kate Anticline, slip rate is calculated using the average uplift rates from the formation of terraces Qt4, Qt3, and
Qt1 to present, assuming ages of 106.9 ka for Qt4, 124.5 ka for Qt3, and 216 ka for Qt1.
188
189 Figure 4-16: Age of initiation of folding predicted for different fault and fold models for the anticline at Haumuri Bluff and Kate Anticline. Age is calculated using the different slip rates shown in Figure 4-15.
190
191 Figure 4-17: Rates of shortening predicted for different fault and fold models for the anticline at
Haumuri Bluff. Shortening rate is calculated using the different slip rates shown in Figure 4-15.
192 Table 4-1: Sea level highstands for marine terrace uplift rate calculations (with 2σ uncertainties).
Sea Level Age Highstand (m) (ka) 5c-1 -23±11 106.9±3.0 5c-2 -45±14 99.9±4.0 5a-1 -37±9 90.6±2.0 5a-2 -31±9 84.0±1.6 5a-3 -52±8 77.6±2.0 5a-4 -50±10 71.3±2.0 5e 6±4 124.5±5.5 7a -10±5 197.0±4.0 7c -10±5 216.0±4.0 7e -10±5 232.5±2.5
Table 4-2: Uplift rate calculations for the terraces at Haumuri Bluff.
MIS T (ka) E (m) S (m) F (m) U (m/ka) Tarapuhi (Qt4) 5c-1 106.9±3.0 173.1±2 -23±11 1±0.2 1.8±0.1 Kemps Hill upper (Qt5) 5a-1 90.6±2.0 98±10 -37±9 1±0.2 1.5±0.2 Kemps Hill lower (Qt5) 5a-2 84.0±1.6 90±5 -31±9 1±0.2 1.4±0.1 Amuri Bluff (Qt6) 5a-4 71.3±2.0 40±5 -50±10 1±0.2 1.2±0.2 Torlesse (Qt8) 1 3.3±0.15 4.2±0.2 0±1.5 1±0.2 1.0±0.5
T = terrace age; E = terrace inner edge elevation; S = paleo-sea level; F = facies depth; U = uplift rate. Uncertainties are 2-σ errors for all quantities.
193 Table 4-3: Interval uplift rates at Haumuri Bluff.
Older Terrace Younger Terrace Uplift Rate Name Age (ka) Name Age (ka) (m/ka) Tarapuhi 106.9±3.0 Kemps Upper 90.6±2.0 3.7±1.4 Kemps Upper 90.6±2.0 Kemps Lower 84.0±1.6 2.1±2.7 Kemps Lower 84.0±1.6 Amuri Bluff 71.3±2.0 2.4±1.3 Amuri Bluff 71.3±2.0 Torlesse 3.3±0.13 1.3±0.2 Torlesse 3.3±0.15 Present 0 1.0±0.5
Table 4-4: Uplift rates at Claverley and Dawn Creek.
MIS T (ka) E (m) S (m) F (m) U (m/ka) Claverley Tarapuhi (Qt4) 5c-1 106.9±3.0 171±10 -23±11 1±0.2 1.8±0.1 Kemps Hill upper (Qt5) 5a-1 90.6±2.0 111±10 -37±9 1±0.2 1.6±0.2 Kemps Hill lower (Qt5) 5a-2 84.0±1.6 93±10 -31±9 1±0.2 1.5±0.2 Conway Flat1 (Qt1) 1 8±0.4 10±2 0±1.5 1±0.2 1.1±0.3 Dawn Creek Tarapuhi (Qt4) 5c-1 106.9±3.0 233±10 -23±11 1±0.2 2.4±0.2 Kemps Hill upper (Qt5) 5a-1 90.6±2.0 180±15 -37±9 1±0.2 2.4±0.2 Kemps Hill lower (Qt5) 5a-2 84.0±1.6 123±10 -31±9 1±0.2 1.8±0.2 Amuri Bluff (Qt6) 5a-4 71.3±2.0 84±5 -50±10 1±0.2 1.9±0.2
194 Table 4-5: Model parameter space for cross section A-A′.
Parameter Minimum Maximum xtip (m) 0 20000 ytip (m) -5000 20000 total slip (m) 0 5000 fault ramp angle 0 90 bend or detachment elevation -10000 0 phi 0 90 elevation of phi change -10000 0 P/S 0 20 elevation of P/S change -10000 0 s 1 1 slip to restore each terrace (m) 0 3000 restored bed elevations1 (m) -2000 300
1Except for the artificial far-field point in model 2, which was required to be at -1000 m.
Table 4-6: Uplift rates for the terraces at the Kate Anticline.
Profile / Terrace MIS T (ka) E (m) S (m) F (m) U (m/ka) GPS Survey 1 Tiromoana (Qt4) 5c-1 106.9±3 33±1 -23±11 1±0.2 0.5±0.1 1 Bob's Flat (Qt3) 5e 124.5±5.5 85±11 6±4 1±0.2 0.6±0.1 1 Leonard (Qt2) 7-a 197±4 123±10 -10±5 1±0.2 0.7±0.1 1 Leonard (Qt1) 7-c 216±4 138±10 -10±5 1±0.2 0.7±0.1 2 Tiromoana (Qt4) 5c-1 106.9±3 40±6 -23±11 1±0.2 0.6±0.1 3 Tiromoana (Qt4) 5c-1 106.9±3 54±8 -23±11 1±0.2 0.7±0.1 4 Tiromoana (Qt4) 5c-1 106.9±3 63±4 -23±11 1±0.2 0.8±0.1 4 Bob's Flat (Qt3) 5e 124.5±5.5 128±8 6±4 1±0.2 1.0±0.1 4 Leonard (Qt1) 7-c 216±4 194±13 -10±5 1±0.2 0.9±0.1
195 Table 4-7: Velocity model for depth conversion of Kate Anticline seismic data.
1 2 Unit at Top of Age of Units V0 k Layer Greenwood Miocene -Pliocene 1766 1.7712 Formation
Waikari Oligocene - 3204 0.9417 Formation Miocene
Ashley Cretaceous- 2222 1.6593 Mudstone Eocene
Torlesse3 Permian- 5500 0 Cretaceous
1Velocity at top of the layer (m/s). 2Rate of change of velocity with depth below the top of the layer. 3We did not fit for velocity in the Torlesse, as there was insufficient data to do so. This velocity is from Barnes et al. (2016).
196 Table 4-8: Model parameter space for Kate Anticline cross sections.
Parameter Minimum Maximum xtip (m), cross section A 0 3000 xtip (m), cross section B 1500 4500 ytip (m) -5000 -500 total slip (m) 0 3000 fault ramp angle 0 90 bend elevation -10000 0 phi1 0 90 P/S 0 20 s 1 1 slip to restore each terrace (m) 0 3000 restored bed elevations (m) – Wakari -1500 -500 and Ashley restored bed elevation (m) – Torlesse -2000 -500 restored Cass Anticline backlimb dip -20 -12 (°)2 restored Cass Anticline backlimb y- -2500 -500 intercept – Waikari and Ashley1 restored Cass Anticline backlimb y- -3000 -500 intercept – Torlesse1
1As in Model 3 for the anticline at Haumuri Bluff, we require phi to be ≤ the fault ramp angle. In this case, however, that requirement seems to be less significant. 2Model 2 only.
197
Chapter 5
Kinematic Modeling of Listric Faulting and Basement-Involved Folding in the North Canterbury Fold and Thrust Belt, South Island, New Zealand
Abstract
Deformation within the North Canterbury fold and thrust belt, South Island, New Zealand is dominated by basement-involved, fault-related folding, but fault structure below the basement-cover unconformity is largely unknown. Previous workers have suggested that faults may be listric at depth and may root in a mid-crustal detachment. Here we investigate whether listric faulting can explain the structure of fault-related folds in North Canterbury, and we test methods for fitting listric fault models to data using Markov chain Monte Carlo simulations. We find that rigid basement block rotation on a circular listric fault – a model that has been applied to basement-involved folding elsewhere in the world
– does not adequately explain the style of deformation occurring in North Canterbury. Instead, we develop a model of faulting on steeply-dipping, elliptical listric faults, using inclined shear kinematics together with trishear. This model is able to produce the different styles of folding seen in the onshore and offshore parts of the fold and thrust belt and is consistent with evidence from earthquake sequences in the region that suggest that reverse faults dip steeply within basement. It does not allow us to constrain detachment depth from surface structure, but it is consistent with previously suggested depths of 10-12 km. We use this model to construct a balanced cross section across the onland portion of the North
Canterbury fold and thrust belt. The estimated shortening is consistent with the regional geodetic shortening rate and estimates of the age of the onset of folding, albeit at the low end of those estimates.
198
5-1. Introduction
The North Canterbury fold and thrust belt (NCFTB, Figure 5-1) is a distinct structural domain within the northeastern South Island of New Zealand (Pettinga et al., 2001). Deformation within this region is characterized by fault-related, basement-cored asymmetric anticlines, trending NE-SW. Folds are mostly NW-vergent in the southwestern part of the fold belt and SE-vergent in the northeast. Faults appear to be mostly steep near the surface (Pettinga et al., 2001; Litchfield et al., 2014; Barnes et al.,
2016) and include inherited normal faults, which have been reactivated as thrusts (Barnes et al., 2016).
Fault structure at depth and below the basement-cover unconformity, however, is largely unknown.
Several previous workers have suggested that North Canterbury faults may be listric at depth and may flatten into a regional, mid-crustal detachment (Nicol, 1991; Cowan, 1992; Barnes, 1993; Litchfield,
1995; Litchfield et al., 2003; Campbell et al., 2012), but kinematic models incorporating these features have not been tested. North Canterbury fault-related folds are an example of basement-involved folds, a style of deformation that occurs in many parts of the world. As in North Canterbury, interpreting fault structure and kinematics of basement-involved folding is often difficult (Narr and Suppe, 1994), and listric fault models are among those that have been proposed to explain these structures (e.g. Erslev, 1986;
Seeber and Sorlien, 2000). Here, we investigate whether kinematic models incorporating listric faults and mid-crustal detachments can reproduce the structure of North Canterbury basement-involved folds. We address the differences between the styles of folding in the onshore and offshore parts of the North
Canterbury fold and thrust belt, and we consider the relationship between the style of folding in the North
Canterbury fold and thrust belt and the 2010-2011 Canterbury and 2016 Kaikoura earthquake sequences.
Finally, we develop a structural model for the style of faulting and folding in North Canterbury, which has implications for understanding regional tectonics, the kinematics of basement-involved folding, and the use of Markov chain Monte Carlo (MCMC) techniques in kinematic modeling of fault-related folds.
199
5-2. Background
5-2.1 Folding and Faulting in North Canterbury
Deformation in the North Canterbury fold and thrust belt (Figure 5-1) is characterized by basement-involved folding (Pettinga et al., 2001; Rattenbury et al., 2006; Forsyth et al., 2008). Basement in this region consists of Mesozoic greywacke of the Torlesse terrane, which is overlain by a Cretaceous to recent sedimentary sequence (Rattenbury et al., 2006; Forsyth et al., 2008). The asymmetric fold shapes and the association of folding with thrust faulting suggest fault-propagation folding as the primary mode of deformation, but the geometry of the thrust faults and the depth to which they extend are poorly known.
Folds vary in wavelength from about 3 to 10 km, with many having a wavelength of about 5 km.
There is a distinct difference between the shapes of onshore and offshore folds. Onshore folds have steep to overturned forelimbs and backlimbs that typically dip at about 20° and seldom less than 10° or greater than 30° (Wilson, 1963; Yousif, 1987; Rattenbury et al., 2006; Forsyth et al., 2008). Offshore folds, by contrast, are nearly monoclinal (Barnes et al., 2016). Backlimb dips are typically no more than a few degrees, and forelimbs, while steeper, are not typically overturned. Shallow Pleistocene growth strata show rotation of both limbs and indicate that uplift is occurring up to 25 times more slowly offshore than onshore (Barnes, 1996). The differences between onshore and offshore folds could indicate either a fundamentally different process of folding between the two regions or, more simply, a single process in which the offshore folds are less developed than those onshore, due to a slower rate of deformation and/or a more recent onset of folding.
Faults are typically steep within the cover sequence. Offshore faults imaged in seismic reflection surveys often have dips in excess of 60° and are mostly blind, with tips within the Miocene-Pleistocene cover sequence (Barnes et al., 2016). Onshore faults in many places reach the surface, although they are seldom exposed in outcrop so their dips are difficult to measure exactly. Blind faults likely also exist
200 onshore, as suggested for the Kate Anticline in Chapter 4. Estimates of onshore fault dip angles at the surface are mostly in the range of 40-70° (Pettinga et al., 2001; Litchfield et al., 2014). Many of these steep faults are inverted normal faults, which first formed during Late Cretaceous to Eocene rifting and extension (Laird and Bradshaw, 2004) and were reactivated as thrusts. Reactivated normal faults have been documented both onshore (Nicol, 1993; Ghisetti and Sibson, 2012) and offshore (Barnes et al.,
2016). Even the offshore Motunau Fault, which is thought to be newly-formed (Barnes et al., 2016), dips at a high angle similar to the reactivated normal faults. Newly-formed thrust faults at the lower dips
(~30°) expected from Andersonian theory may also exist, but the only low-angle thrust fault documented in the NCFTB is a relatively small fault exposed in the Waipara River (Bradshaw and Newman, 1979), which is not associated with any anticline. Some faults of this type have been imaged by Ghisetti and
Sibson (2012) in the Ashley River region, to the southwest of the NCFTB, but only within the cover sequence. Overall, the evidence suggests that most, and perhaps all, of the anticlines within the NCFTB are formed as fault-propagation folds by steep faults, which typically result from the inversion of pre- existing normal faults.
The geometry of faults in the North Canterbury fold and thrust belt is not well known below the basement-cover contact. Previous authors have speculated that they may be listric at depth and may shallow into a mid-crustal detachment (Nicol, 1991; Cowan, 1992; Barnes, 1993; Litchfield, 1995;
Litchfield et al., 2003; Campbell et al., 2012). This fault geometry would be similar to models for basement-involved folding in other parts of the world (Lynn et al., 1983; Erslev, 1986; Seeber and
Sorlien, 2000), which also invoke listric faults and mid-crustal detachments.
Evidence for a detachment in the North Canterbury region (here including the Canterbury Plains as well as the fold and thrust belt) comes mainly from the distribution of seismicity within the crust. The crustal structure of the region is thought to consist of a quartzofeldspathic upper crust, composed of greywacke and schist, overlying a mafic lower crust, which may represent a portion of the Hikurangi
Plateau subducted at c. 100 Ma (Reyners and Cowan, 1993; Eberhart-Phillips et al., 2010; Reyners et al.,
2011; 2013). The depth of a possible North Canterbury detachment has been placed at 10-12 km by most
201 authors on the basis of work by Cowan (1992) and Reyners and Cowan (1993). These authors observed a gap in microseismicity between 12 and 17 km depth and interpreted this as a ductile, aseismic layer. In these models, then, the detachment is not necessarily a detachment fault, in the manner of thin-skinned tectonics, but a shear zone at the brittle-ductile transition. A similar maximum depth of 10-12 km for microseismicity has been observed along the Alpine Fault (Leitner et al., 2001) and among aftershocks to the 1994 Mw 6.7 Arthur’s Pass earthquake (Bannister et al., 2006). Patterns of seismicity during the
Canterbury earthquake sequence are also similar (though with some deeper events) and are discussed in more detail below. Farther south, deep structure in the Mackenzie Basin imaged by seismic reflection
(Long et al., 2003) shows evidence for listric fault geometries within basement and a possible detachment at 10-12 km depth. There, as in North Canterbury, faults are developed within Torlesse greywacke, but the degree to which the two regions can be compared is uncertain.
5-2.2. Canterbury and Kaikoura Earthquake Sequences
Earthquakes within the North Canterbury fold and thrust belt and adjacent regions can provide insight into fault structure at depth. These come in large part from two major earthquake sequences: the
2010-2011 Canterbury earthquake sequence in the Canterbury Plains, and the 2016 Kaikoura earthquake sequence, which ruptured faults in the NCFTB and in the Marlborough fault zone to the north. Events within the NCFTB structural domain iteslf (from the GeoNet catalogue) include both reverse and strike slip mechanisms (Figure 5-2). The majority of these events are from the recent Kaikoura sequence and are concentrated in the northern part of the fold belt, with less data in the central region. Events of 18-27 km depth would fall in the lower seismogenic layer of Reyners and Cowan (1993) and are unlikely to reflect surface structures. The majority of the events, however, are within the upper ~12 km of the crust. Among these shallower events, reverse faulting mechanisms have often steep nodal planes, ranging from 24°-85° in dip (Table 5-1), with an average of 58° for the steeper nodal plane and 37° for the less steep one among reverse mechanisms. The frequency of strike-slip mechanisms, in addition to reverse and oblique ones, is
202 somewhat surprising since the regional map patterns do not suggest large strike-slip offsets within the
NCFTB, but it is consistent with the strike-slip faulting of the Canterbury earthquake sequence. Strike- slip faults may, at least in part, form the lateral boundaries of thrust-bounded blocks (Campbell et al.,
2012). Further analysis of the Kaikoura earthquake sequence, including relocation to better constrain event depth, will be needed to fully understand its implications for structure of the region.
The Canterbury earthquake sequence, which began with the 2010 Darfield earthquake, provides a large dataset of seismicity. Although within the Canterbury plains structural domain (Pettinga et al.,
2001), faults revealed by these events may provide an analogy to those in the fold and thrust belt to the north, as the styles of faulting appear to be similar (Campbell et al., 2012). Although the majority of the events in this sequence were strike-slip, some had reverse mechanisms (Sibson et al., 2011; 2012;
Bannister and Gledhill, 2012; Herman et al., 2014). Two events and aftershock sequences within the
Canterbury earthquake sequence are of particular interest. The first is the rupture of the Eastern Hororata
Fault in the Darfield earthquake. This fault experienced a dominant reverse sense of slip on a fault plane dipping northwest at 45°-50° (Beavan et al., 2010; 2012; Holden et al., 2011; Elliott et al., 2012; Syracuse et al., 2013). Geodetic and seismic waveform inversions indicate that slip was restricted to above 10 km depth (Beavan et al., 2010; 2012; Holden et al., 2011; Elliott et al., 2012). Aftershocks mostly occurred at
5-11 km depth, although a few at 14-17 km depth could also be associated with this fault (Syracuse et al.,
2013). The second set of events of interest are the Pegasus Bay aftershocks, which began on December
23, 2011 with three events of Mw 5.4-5.9 (Ristau et al., 2013). The fault plane orientation (strike, dip) for the Mw 5.9 event was calculated as (060°, 69°) by Beavan et al. (2012) and (057°, 51°) by Ristau et al.
(2013). Predominantly reverse aftershocks relocated by Ristau et al. (2013) extend to 15 km depth but do not define a clear fault plane.
Evidence from the Canterbury earthquake sequence suggests that reverse slip occurred on relatively steep planes, within a similar range to the 40°-70° dips estimated for faults in North Canterbury.
Evidence for a detachment depth is more ambiguous. Most events are above 10 km depth and consistent with a detachment at ~10-12 km. The presence of a few deeper events complicates this picture, although
203 whether they occur on the same faults as shallower events is less clear, and the detachment depth may not be the same in the Christchurch area as in the fold and thrust belt farther north. Reyners et al. (2013) suggest that a brittle-ductile transition occurs at ~11 km depth in much of the region, but not in the vicinity of Banks Peninsula, due to the presence of a strong basaltic plug. Similar results are reported by
Ellis et al. (2016): a midcrustal ductile layer is absent near Banks Peninsula, but present elsewhere.
If reverse-slip events from the Canterbury and Kaikoura earthquake sequences are taken as a guide to the geometry of reverse faults throughout North Canterbury, then we can expect faults in the fold and thrust belt that are steep at the surface to remain steep within the upper crust and any detachment to be below 10 km depth. Fault dips may decrease some with depth, as the fault dips discussed above are mostly less than the >60° dips seen near the surface in many North Canterbury faults (Barnes et al.,
2016), which would be consistent with a listric geometry.
5-2.3. North Canterbury Fold Kinematics
Few studies have used the principles of fault-related fold kinematics to balance and restore cross sections of North Canterbury fault-related folds. Barnes et al. (2016) restored cross sections of the offshore part of the fold and thrust belt, using trishear kinematics on steep reverse faults. Because of the nearly monoclinal nature of the offshore folds, they were able to restore backlimb deformation with small amounts (<2°) of angular shear. For onshore folds, with typical backlimb dips of about 20°, this approach is insufficient. Forelimb deformation can still be modeled with trishear (VanderLeest, 2015; Chapter 4 of this dissertation), but kinematic models to produce the required backlimb dip require some form of fault bend. Vanderleest (2015) showed that the Montserrat Anticline can be modelled with a ramp from a horizontal detachment, but this simple model predicts a detachment depth of less than 2 km and a fault dip of less than 30° – results that are inconsistent with the evidence presented above for deep detachments and steep fault dips. An alternative, presented in Chapter 4, is a fault composed of two non-horizontal straight segments. This allows for steep dips near the surface, which must shallow at depth. Model 2 for
204 the anticline at Haumuri Bluff, for example, estimated fault dips of 54.4° ± 13.5° for the upper segment and 23.4° ± 7.0° for the lower segment, with the bend at a depth of 2630 ± 240 m (2 standard deviation uncertainties).
In this chapter, we investigate listric faulting as a further alternative model. While the two- segment model of Chapter 4 is able to reproduce the observed fold geometry with a fairly simple model and without a shallow detachment, additional considerations suggest that it may not be sufficient to explain all aspects of fault-related folding in North Canterbury. It requires a bend within the upper crust, with the lower fault segment having a dip typically less than the ~40°-70° inferred from the seismic focal mechanisms and fault slip models discussed above. It also does not produce progressive rotation of the fold backlimb in response to reverse faulting, as is seen in offshore Pleistocene growth strata (Barnes,
1996). Listric fault models provide the opportunity to reproduce those features, at the cost of increased model complexity. They also allow us to investigate the question of detachment depth and to draw comparisons to basement-involved folding in other parts of the world, where listric faulting is known or suspected.
5-3. Methods
To test the possibility of listric faulting in the North Canterbury fold and thrust belt, we consider the fault-related fold kinematic models of trishear (Erslev, 1991; Zehnder and Allmendinger, 2000), inclined simple shear (Gibbs, 1983; White et al., 1986; Waltham and Hardy, 1995), and rigid block rotation (Erslev, 1986; Seeber and Sorlien, 2000; Amos et al., 2007). Midland Valley’s Move software was used to construct forward models of fault related folds. For cross sections incorporating both trishear and simple shear, we modelled forelimb and backlimb deformation separately in Move and joined the resulting deformed beds at their points of intersection.
The InvertTrishear program of Oakley and Fisher (2015) was used to fit fold models to data by
Markov chain Monte Carlo simulations, using the adaptive parallel tempering method of Miasojedow et
205 al. (2013). The program was modified to allow inclined simple shear to be used in conjunction with trishear and to model listric fault geometries, with shear angle as a model parameter to be fit. A semi- analytic method for calculating displacements in the trishear zone (Appendix C), which is faster than the incremental slip methods of Zehnder and Allmendinger (2000) and Oakley and Fisher (2015), was valuable in running the large number of models required to explore the large parameter spaces of listric fault models.
Listric faults in both Move and InvertTrishear were approximated by a series of straight segments. Two approaches were used to fit for the geometry of the faults in InvertTrishear. In the first approach, we fit individually for the dip of each segment and the elevation of each bend, with the only requirement being that the fault must be concave-up. This approach allows the most freedom for the shape of the fault, but it introduces a large number of free parameters, which can make it difficult to efficiently search the parameter space, even with a Markov chain Monte Carlo simulation. It also requires that a starting model be specified, since the requirement for fault segments and bend elevations to decrease monotonically means that few randomly generated models will be acceptable. In the second approach, we required the listric fault to have an elliptical shape, with the two axes of the ellipse being horizontal and vertical. The fault was allowed to become straight at some maximum and minimum elevations, with the elliptical portion connecting the two straight segments. In this case we had only to fit for five fault-geometry parameters (in addition to the fault tip position): dips of the upper and lower straight segments, lengths of the two semi-axes of the ellipse, and elevation of the top of the lower straight segment. If a horizontal detachment was expected, the dip of the lower straight segment was required to be zero and its depth was the detachment depth. The ellipse was divided into a user-specified number of straight segments, with a constant angle between adjacent segments.
206
5-4. Results
5-4.1 Listric Fault Kinematic Models
Basement-involved listric reverse faults can be modeled by simple rigid rotation of a basement block (Erslev, 1986; Seeber and Sorlien, 2000). This model requires that the listric faults have a circular shape, and it involves no internal deformation of the basement block. This listric-faulting model was applied by Amos et al. (2007) in the Mackenzie Basin of south Canterbury, where Torlesse greywacke forms the basement rock, although they interpreted listric faulting as largely confined to the cover sequence. Such shallow faulting would be inconsistent with the observations in North Canterbury and may not be characteristic of the Mackenzie Basin either (Ghisetti et al., 2007), but the listric model could still be applicable.
The model of rigid rotation on a circular fault predicts specific relationships between the width of the fold backlimb, depth to detachment, and dip of the fault, as measured at the same stratigraphic level
(Seeber and Sorlien, 2000, their equations 1-3). Figure 5-3 shows that with this kinematic model the 10 km or greater detachment depths expected from geophysical considerations require backlimb widths of 10 km or more and are incompatible with the smaller North Canterbury folds. A fold with a 5 km wide backlimb and a fault that dips at 60° at the basement-cover contact (typical values for North Canterbury) would have a detachment at only 2.9 km below the contact. Therefore, while the rigid rotation listric fault model could produce fold geometries of the sort seen in North Canterbury, it cannot do so with faults that remain steep to depth or with the expected detachment depths in excess of 10 km.
An alternative model for listric faulting is deformation of the hanging wall by vertical or inclined simple shear. This kinematic model is commonly used for deformation in the hanging walls of listric normal faults (Gibbs, 1983; White et al., 1986; Waltham and Hardy, 1995), but it can also be applied to reverse faults. Its application to inverted listric faults is supported by analogue modeling (Yamada and
McClay, 2003), and it can be used in combination with trishear kinematics (Cardozo and Brandenburg,
207
2014). The model requires internal deformation of the basement, but it does not require circular fault geometries and therefore can be applied to faults that are longer in the vertical dimension than the horizontal. We focus on this simple shear (vertical or inclined) kinematic model for the remainder of the chapter.
5-4.2. Montserrat Anticline
For an onshore example of a possible listric fault, we consider the Montserrat anticline (Figure 5-
1, cross section A-A′), using structural data from VanderLeest (2015, her cross section A-A′). The anticline is formed by uplift on the Glendhu fault. The fold is at least 7.4 km wide (the distance from the
Glendhu fault to the coast), making it one of the larger anticlines in the region. Its full width is uncertain, as the location of the backlimb syncline is offshore and unknown.
We constructed two possible forward models of the anticline using listric fault geometries. These were created for VanderLeest et al. (submitted manuscript, “Growth and Seismic Hazard of the
Montserrat Anticline in the North Canterbury Fold and Thrust Belt, South Island, New Zealand”) and are reproduced in Figure 5-4. One model has a detachment depth of 6 km and an inclined shear angle of 70°
(from horizontal), while the other has a detachment depth of 11 km and vertical shear. Despite their differences, the two models both provide a reasonable fit to the data. The greatest error, which is seen in both models, is in fitting the basement-cover contact to its mapped location in the forelimb. This discrepancy may indicate that the fault formed by reactivation of a pre-existing normal fault, in which case the initially flat geometry of the basement-cover contact, which was used for these models, would be incorrect. The fact that both models provide a similarly good fit to the data (locations of lithologic contacts, dips of bedding, and the location of the Glendhu Fault at the surface), however, suggests that the depth to detachment for an inclined-shear listric fault is not adequately constrained by the available data.
The two cross sections make very different predictions for shortening (here defined as slip along the detachment): 1800 m with the detachment at 6 km, but only 700 m with the detachment at 11 km depth.
208
Fault slip at the surface differs less, however: 2360 m with the detachment at 6 km and 1980 m with it at
11 km. The model with a detachment depth at 11 km is more consistent with the geophysical evidence discussed above for steep fault planes that extend to depths greater than 10 km, but both could conceivably produce the observed fold structure.
Listric fault models were fit to the data for Montserrat Anticline using InvertTrishear. Fitting individually for the position and orientation of each fault segment proved problematic. In addition to the difficulty of exploring the large parameter space, this model tends to produce unlikely results, including faults that reach low dips at shallow depths inconsistent with what we expect from regional patterns of seismicity (e.g. 10° by 4 km depth) and sharp bends in the offshore (and thus unconstrained by data) part of the backlimb. Instead, we assumed an elliptical shape for listric faults, as described above. The elliptical fault model requires that the listric fault has a smooth curvature and therefore produces straight or gently curved anticline backlimbs. Further, because of the reduced number of parameters, we can test models from a random starting position, which allows us to test for reproducibility of results and thereby to ensure that the program has found a global probability maximum.
When no constraints are placed on detachment depth or fault dip, the best-fitting models are those with shallow (~2 km depth) detachments and maximum fault dips of ~20°, which are similar to the simple step model of VanderLeest (2015). In order to find a model that is consistent with the evidence for a deeper detachment and for steeper fault dips, including evidence of a steep dip to the Glendhu Fault (a dip of 48° was measured near the southwest end of the fault by Yousif, 1987), we required the dip on the fault to be >40°, and we specified the detachment depth, trying several different depths. We also found that the restored state depth of the basement-cover contact was difficult to fit, as the Markov chain tended to converge on the deepest allowed value (tested for maximum depths of 1.5, 2, and 3 km). Difficulty in fitting for the restored depths of folded units was encountered also in Chapter 4: a depth estimate from fossils in the Pliocene Greta Formation from Beu (1979) was needed to constrain the restored depths of units at Haumuri Bluff, while uncertainty about the restored depths of horizons at Kate Anticline hindered our ability to compare two cross sections. For this reason, we suggest that the restored depths of folded
209 units cannot be reliably fit simply by using fold shape to constrain fault dip and total slip without prior knowledge of the expected depth. In this case, we limit the depth of the basement-cover contact to 1-1.5 km. This choice is based on the depths to basement in surrounding regions, as seen in gravity (Loris,
2000) and seismic reflection (Barnes et al., 2016) surveys, and on the ~1.3 km thickness of sediment from the top of the basement to the top of the Mt. Brown formation as measured in the backlimb of the
Montserrat anticline. Mt. Brown formation occurs at the surface in the footwall of the Glendhu Fault along the chosen line of section.
Given these constraints, we fit elliptic fault models to the data. Detachment depth was fixed for each inversion, with depths of 5 to 15 km tested in increments of 1 km. Shortening (slip on the detachment) decreases as detachment depth increases (Figure 5-5). Slip on higher parts of the fault can be calculated from slip on the detachment, given the dips of each segment and the shear angle (Hardy, 1995;
Waltham and Hardy, 1995). For the uppermost fault segment, there is little dependence on detachment depth, however, except for the (unlikely) models with the shallowest detachments (Figure 5-5). This confirms our inference from the forward models that slip on the upper part of the fault is better constrained. RMS errors for the best-fitting models vary some among the different detachment depths
(Figure 5-6), mostly reflecting trade-offs between the three data types used to constrain the model, but they do not show a consistent improvement in model fit towards shallower or deeper detachments, reflecting our earlier observation from forward modeling that a large range of detachment depths can fit the available data.
5-4.3. Leithfield Fault
For an offshore example, we consider the Leithfield fault (Figure 5-1, cross section B-B′). This structure was interpreted by Barnes et al. (2016) in multiple seismic sections, and we use their interpretation from the GG06-04 seismic section offshore of Motunau Beach (their Figure 10a). This section lies very close to profile 13 of Barnes (1996), which images shallow growth strata above the fault,
210 dating from marine isotope stages (MIS) 19 to present. By projecting profile 13 onto the GG06-04 line of section, we obtain a cross section that includes both growth and pre-growth strata. The fault dips at about
70° within the cover strata, but it is not well imaged within the basement.
We first made a rough forward model of the fault in Move (Figure 5-7a). This fault flattens into a detachment at 10.5 km depth and is composed of 29 straight segments that approximate a listric geometry.
In this model, we use vertical shear with 80 m of heave, which translates into 254 m of slip on the upper part of the fault, which is similar to the 240 m slip of Barnes et al. (2016).
We fit two models to this data using InvertTrishear. As before, in the first model, we fit for the dip of each fault segment and depth of each fault bend individually, while in the second model we required the listric fault to have an elliptical shape, and we fit only for the semi-axis lengths, detachment depth, and maximum dip. In both models, we required the fault to shallow into a horizontal detachment, but the detachment depth was not fixed. In addition to fitting for the fault geometry and trishear parameters, we fit for the slip required to restore each of the eight horizons interpreted by Barnes 1996 in the growth strata, and for the restored-state dip and y-intercept of each of the eight growth and four pregrowth layers. Since a low regional dip can be seen in the seismic sections beyond the limits of this anticline, we did not require the beds to restore flat, although we limited the restored-state dip to a maximum of 1° for growth strata and 6° for pre-growth. We fit for inclined shear angle within the range
60° to 90°, allowed P/S to change once during fold growth, and allow a maximum detachment depth of 11 km. The total number of free parameters was 96 when fitting all segments individually and 44 when fitting an elliptical fault geometry. With such large numbers of parameters, it was necessary to start from a specified initial model, for which the forward model from Move was used (and an elliptical approximation of it for the elliptical fault case). In both cases, 29 segments were used to approximate the listric fault, as in the forward model.
When fitting for the position and orientation of each segment independently, we see that the data are better fit with a slight increase from our initial estimate of slip on the detachment (Figure 5-8a). The complexity of the model also means that a large number of models need to be tested (~2 million in Figure
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5-8c) before the Markov chain finds a region of high probability. The ability to test models quickly, using the methods detailed in Appendix C, is beneficial in doing this efficiently. When the listric fault is required to have an elliptical shape, the number of model parameters is greatly reduced, but it is still necessary to specify a starting model (an elliptical approximation of the previous forward model) in order to achieve a result in a reasonable amount of time. In this case, the estimated slip is slightly less (Figure 5-
8b) and the Markov chain has less trouble mixing in its early stages (Figure 5-8d). In both cases, it is unlikely that the Markov chain is able to explore the full parameter space, given the large number of parameters. For example, only a relatively narrow range of shortening values has been explored in each case (Figure 5-8), and the final detachment depths are not far from the starting value (Figure 5-7a).
Despite this fact and the need for (and dependence of the results on) a starting model, the APT algorithm does allow us to refine our initial forward model, allowing us to find a model that provides a reasonably good fit to the data (Figure 5-7) and that provides an improved fit to the data, compared to the initial model.
Since we have fit for the fault slip to restore the growth strata, we can use the results of our
Markov chain Monte Carlo simulation to plot rates of slip through time. The growth beds in the cross section have been identified by Barnes (1995; 1996) as unconformities correlated to sea level low stands during stadial periods. Unconformities A, D, and E through J of Barnes (1995) are present in this cross section. We have assigned these to the ages for the ends of the interstadial periods that they represent, using the ages of Lisiecki and Raymo (2005). The results (Figure 5-9), based on the elliptical fault model, show some variation in time, but most intervals are within error of each other. A period of particularly fast shortening at about 230 ka, followed by particularly show shortening, comes from the deformation of unconformity D, which is thought to be within MIS 7 (Barnes et al., 1995). This horizon is identified only in the backlimb of the fold, while most are seen in both limbs; this may have resulted in its total uplift and tilting being underconstrained. Alternatively, if this horizon is instead unconformity C and therefore marks the base of MIS 5 (Barnes et al., 1995), then the slip rate is 0.09±0.04 mm/yr, which is within error of the preceding and following intervals. Given these two possibilities, we cannot with confidence assert
212 that this unconformity truly indicates a period of rapid uplift. On the other hand, as seen in Chapter 4, deformation rates at some locations have changed substantially over time, and this could be another such example.
5-4.4. Haumuri Bluff
There are reasons to think that the anticlines discussed in Chapter 4 (the Kate Anticline and the northern end of the Hawkswood Anticline at Haumuri Bluff) may not be representative of the style of folding discussed in this chapter. The Kate Anticline has one of the shortest fold wavelengths seen in
North Canterbury and is therefore difficult to fit to a fault extending to 10 km or greater depth (see further discussion below). We have speculated that the anticline at Haumuri Bluff (Figure 5-1, cross section C-
C′) may be a splay off the nearby Hundalee Fault (Chapter 4), although this is not certain. To investigate how our interpretations of the anticline at Haumuri Bluff (Chapter 4) would change under the assumption of listric faulting and inclined simple shear, we test two possible models using an elliptical listric fault. In the first case, we consider a listric fault segment between two non-horizontal straight segments. Since this does not require the lower part of the fault to be horizontal, it is most applicable if the fault is indeed a splay off the Hundalee Fault. In the second case, we consider a model in which a horizontal detachment at
11 km depth is required.
When a detachment is not required, the best-fitting models have only a very short listric segment between two straight segments (Figure 5-10a). The vertical extent of the listric section for these models averages only 190 m. They are therefore similar to the models presented in Chapter 4 that consist of only two fault segments, with the principal difference being the use of inclined shear rather than fault-parallel flow kinematics in the hanging wall. When a deep detachment is required, the result is a listric fault that would likely be roughly parallel to the Hundalee Fault (Figure 5-10b) and could intersect it at depth. Key parameters of both models, as well as model 3 of Chapter 4, are compared in Table 5-2. Magnitudes and rates of shortening are greatest for the fault parallel flow model of Chapter 4 and least for the model with
213 an 11 km detachment, although the error margins mostly overlap. Slip on the uppermost part of the fault, which is most important for marine terrace uplift, varies less between the models. Estimates for the age at which faulting began from the listric models are lower than, but within error of, those from Chapter 4, and are very similar to previous studies on other folds in North Canterbury (Nicol et al., 1994; Barnes, 1996;
Barnes et al., 2016).
5-4.5. Depth to Detachment for Normal Faults
The geometry of normal faults can be used to estimate the depth to detachment, given an inclined shear angle (White et al., 1986). Inversion of normal faults in North Canterbury may or may not extend to the reactivation of inherited detachments, but to consider this possibility, we estimate detachment depths for two normal faults. The first is an unnamed, non-reactivated normal fault (cross section D-D′ in Figure
5-1) in seismic line GG07-103 of Barnes et al. (2016; their Fig. 6a, the half-graben-forming normal fault in the middle of the section), while the second is the normal fault that was reactivated as the Pegasus Bay fault (cross section E-E′ in Figure 5-1), in the restored cross section of Barnes et al. (2016; their Fig. 10d, seismic line GG06-03). A key difference between the two examples is the width of the adjacent half- graben, which is about 6 km for the Pegasus Bay fault and 10 km for the non-reactivated fault. For both faults, we used Move’s “Construct Fault Geometry” tool to calculate detachment depth as a function of inclined shear angle (Figure 5-11), based on the geometry of the upper part of the fault and a simplified interpretation of the basement-cover contact within the adjacent half-graben. Detachment depths of 10-12 km can be produced by shear on steep shear planes in both cases. At detachment depth of 11 km, for example, the shear angle is 82° for the non-inverted fault (Figure 5-12a) and 90° (vertical) for the Pegasus
Bay Fault (Figure 5-13a). These are similar to the shear angles found when fitting models to the reverse faults discussed above – e.g. 85°-90° for the Montserrat Anticline with a detachment at 11 km depth,
82°±3° for the elliptic-fault model for the Leithfield Fault, and 70°-90° for the Hawkswood Anticline when a detachment is required.
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5-4.6. Forward Models
To test whether onshore and offshore fold styles could be formed by different amounts of displacement on geometrically and kinematically similar listric faults, we construct forward models to test how folds above offshore faults would evolve with greater fault slip. We first use the non-inverted normal fault described above, to model what fold would be produced if it were inverted, using the same listric fault geometry as for the normal fault. Figure 5-12 shows the evolution of the fold that would be formed from the initial half-graben (Figure 5-12a), to a nearly monoclinal fold (Figure 5-12b), an intermediate form (Figure 5-12c), and an anticline with a 20° dip to the backlimb (Figure 5-12d). The propagation to slip ratio (P/S) of the fault is increased from 0.5 to 5 with increasing fault slip, to reflect the changes in
P/S that have been inferred for other faults in the NCFTB (Barnes et al., 2016; Chapter 4). Figure 5-12b and c are analogous to folds seen offshore, while Figure 5-12d is analogous to the anticlines seen onshore.
The non-inverted fault is probably not a perfect model for the folds seen in the North Canterbury fold and thrust belt. Besides the fact that it has been reactivated (and is southeast of the region of reactivated faults), it is wider (~10 km) than many of the folds in the region, and the throw on the normal fault (1100 m) is greater than on any of the documented inverted normal faults. Consequently, the 2 km of shortening on the detachment and 3 km of slip within the cover, which are needed to invert such a large structure, are likely greater than for actual inverted faults.
For a second example, which may be a better template for onshore folds formed by normal fault inversion, we constructed a forward model for the Pegasus Bay Fault, using the listric fault geometry determined above from the half-graben shape (Figure 5-13). In this case, due to the smaller size of the structure only about 750 m of shortening are required to produce a 20° backlimb dip. In both cases, the most important result is that inclined simple shear on a listric fault can produce folds of the styles seen offshore and onshore with the same fault geometry and kinematic model. There is, therefore, most likely no fundamental difference between the two regions, only different amounts of shortening and fault slip.
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5-4.7. Regional Cross Section
As we have shown that the listric fault model can be used to model individual folds, it can also be used to create regional cross sections. To demonstrate this, we have constructed a cross section across the onland portion of the North Canterbury Fold and Thrust Belt (Figure 5-14), from the Culverden Basin in the northwest to Kate Anticline in the southeast (section F-F′ in Figure 5-1). The major faults crossed by the section are from northwest to southeast: the Hurunui Bluff fault, the Mt. Alexander fault, the Moores
Hill fault, the Omihi fault, the Hamilton fault, and the blind fault inferred to underlie the Kate anticline.
The cross section was constructed in Move and is based on mapping by Wilson (1963), Yousif (1987),
Litchfield (1995), Pettinga and Campbell (2003), Finnemore (2004), Rattenbury et al., (2006), Forsyth et al. (2008), and this study, as well as a database of strike and dip measurements provided by GNS Science,
Waipara valley gravity and seismic data from Loris (2000) and Finnemore (2004), and Kate anticline seismic reflection and well data from Exploration Solutions Ltd, (2006), Geosphere (Excel) (2008), and
Styles et al. (2008), as interpreted in Chapter 4.
We assume a detachment at 11 km depth (based on a range of 10-12 km from previous literature) and use trishear combined with simple shear kinematics. For consistency, we keep the shear angle vertical for all faults (i.e. constant heave). Faults have been restored sequentially from northwest to southeast, on the assumption that the northwestern structures, being closest to the Southern Alps and having been more completely stripped of cover strata, are the oldest, while the Kate Anticline, known from marine terraces to be actively uplifting is the youngest. This is probably a simplification (discussed more below), as more than one fault could be active at a time. Except for the fault under the Kate Anticline, faults are interpreted to be steep (60°-75°) at the surface and to remain steeper than 40° to depths of at least 9 km, before shallowing into the detachment at 11 km. By keeping the faults steep to depth, we both make them consistent with the geophysical data discussed above and minimize interactions among the anticlines, which would otherwise tend to produce multiple dip panels in fold backlimbs as they are partly refolded by multiple faults (although refolding may occur in some cases – see Chapter 4). Using vertical shear also
216 minimizes interactions between faults. The only fold which we were not able to model successfully with a steep to depth listric fault is the Kate Anticline. Instead, we were only able to model this fold using a fault bend within the upper crust. This result is somewhat surprising, given the evidence discussed in Chapter 4 suggesting that the Eyre Group thickens toward the fault, which would be indicative of an inherited normal fault. In Figure 5-14, we have interpreted this region as a small half graben bounded by a normal fault that localized thrust faulting but was not reactivated. Thrusts localized by inherited normal faults, shortcut ramps, and other styles of deformation besides simple reactivation of a normal fault are known to occur in inverted normal fault systems (Bonini et al., 2012), so it would not be surprising to find them in
North Canterbury. Although we used vertical simple shear instead of fault parallel flow, in order to be consistent with the rest of the cross section, this interpretation is qualitatively similar to the fault bend interpretation of Chapter 4, and it is not clear which of these kinematic models is more nearly correct for deformation in the hanging wall of this fault. Both interpretations use trishear kinematics for forelimb deformation.
Two other faults have been explicitly interpreted as reactivated normal faults: the Hurunui Bluff fault and the Hamilton fault, as the structural and stratigraphic data tended to favor that interpretation. For the Hamilton fault, this is supported by the relatively thick Eyre Group sediments in its hanging wall, while for the Hurunui Bluff fault it helps in fitting the basement-cover unconformity structure contours of
Litchfield (1995). This does not rule out the possibility that other structures are also reactivated normal faults. The Kate anticline and Omihi fault do not bring up basement, while the anticlines in the northwestern half of the cross section (for which we have principally relied on the structure contours of
Litchfield, 1995) have mostly been stripped of their cover sediments. In both cases, this makes identification of inverted half-grabens difficult. Further, many of the normal faults interpreted by Barnes et al. (2016) have only small displacement, sometimes less than 100 m, which could be difficult to recognize when inverted.
The restored strata do not have a perfectly flat, layer-cake geometry (Figure 5-14b). This is to be expected. The Eyre group, of course, thickens toward inherited normal faults. The Amuri limestone dies
217 out to the west, as seen in regional geologic maps (Litchfield, 1995; Rattenbury et al., 2006; Forsyth et al., 2008). The basement-cover unconformity also rises to the northwest, forming a feature that has been recognized by previous workers and termed the Hurunui High (Litchfield, 1995). This leads to thinning of the lower part of the cover sequence in the northwest.
Total shortening across this cross section is 2348 m. If extended offshore, the section would also cross the Waikuku fault and the Pegasus Bay fault. Shortening on offshore structures, however, is much less than onshore (Barnes et al., 2016 and Figure 5-8, this study) and is unlikely to add more than a few hundred meters to the total amount. The fault that accommodates the most shortening is the Hurunui Bluff fault, at 560 m. Fault slip at the surface is significantly greater, more than 1 km for most of the faults, although heave is constant due to the choice of vertical shear.
5-5. Discussion
5-5.1. Structure of the North Canterbury Fold and Thrust Belt
Fault propagation folding on listric faults, with trishear kinematics ahead of the fault tips and inclined simple shear in their backlimbs, provides a viable model for the principal style of folding in
North Canterbury. The model is able to produce structures similar to the near-monoclinal folds that are seen offshore (Barnes, 1996; Barnes et al., 2016) and the anticlines that are found onshore. It provides a direct connection to the geometry of reactivated Cretaceous normal faults, with the same listric fault geometries providing for both half-graben formation and later folding. It allows for fault planes that remain steep to depth, like those suggested by focal mechanisms, fault slip models, and aftershock distributions of reverse faults in the Canterbury earthquake sequence, and it allows for a detachment at
10-12 km depth, as suggested by Cowan (1992) and Reyners and Cowan (1993).
The physical principle underlying inclined simple shear kinematics is that slip occurs on small, subparallel planes within the hanging wall (White et al., 1986). Steeply-dipping faults accommodating
218 small displacements are seen in the hanging walls of some offshore normal faults (Barnes et al., 2016) in both synthetic and antithetic orientations. Additional small faults may exist at subseismic scales and could accommodate reverse displacements in the current tectonic regime. An alternative hypothesis is that shear occurs along bedding planes within the Torlesse greywacke itself. Although bedding orientations within the Torlesse are variable, due to significant mesoscale deformation, there are some overall trends. In the southern part of the fold and thrust belt, bedding in the Torlesse strikes predominantly WSW-ENE and dips near vertically (Litchfield, 1995) and is therefore reasonably well oriented to help accommodate steeply-inclined shear in NE-SW striking folds. Evidence of bedding parallel shear, of unknown age, is noted by Litchfield (1995). Farther north, including within the Hawkswood Range, however, bedding within Torlesse strikes NW-SE (Hall et al., 2004) and so likely does not serve as inclined shear planes.
Most likely, a variety of small faults, some of which may be bedding parallel, serve to accommodate deformation within the heavily-deformed Torlesse rocks.
The same model may not be applicable to every fault and fold in the North Canterbury fold and thrust belt. Specifically, newly-formed, moderately dipping faults could reasonably exist in combination with steep inverted normal faults, as is seen for example in the northwestern South Island (Ghisetti and
Sibson, 2006). Faults that bend to shallow dips within the cover or upper basement have been seismically imaged in the Canterbury Plains (Ghisetti and Sibson, 2012; Jongens et al., 2012). The onshore NCFTB has experienced greater deformation than the Canterbury Plains and the offshore region, which might be expected to have caused more new faults to form. The Kate Anticline, for instance, is difficult to fit with a fault extending to the expected 10-12 km depth, and it may be formed above a smaller, less steep fault.
The anticline at Haumuri Bluff may be interpreted as either a steep fault with a detachment at ~11 km depth, or a splay off the Hundalee Fault, with the latter case providing a slightly better fit to the data. In either case, however, faults that do not directly fit the model of steep listric faults with a deep detachment are either related to or of subsidiary importance to those that do. Faults underlying the largest anticlines with the most structural relief, such as the Glendhu Fault and associated Montserrat Anticline or the
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Hamilton Fault and associated Cass Anticline have been interpreted with the deep, steep listric fault model.
Detachment depth cannot be uniquely determined from the surface structure of folds in North
Canterbury. This is, at least in part, a consequence of the inclined shear kinematic model, in which predicted detachment depth depends critically on the shear angle (White et al., 1986), which is also unknown. Our preference for a detachment depth at 10-12 km is based on regional geophysical evidence from previous studies. Further geophysical studies within North Canterbury would be needed to confirm a detachment for certain, however, and to confirm that fault geometries are similar to the Canterbury plains faults that we have used as a model.
The choice of detachment depth has considerable implications for the amount of shortening across structures in the region (Figure 5-4; Figure 5-5a), and so estimates of shortening can only be trusted to the degree that the detachment depth is known. Slip on the upper parts of the faults, however, is much better constrained (Figure 5-5b; Table 5-2). By extension, slip rates, such as those derived from marine terrace studies (VanderLeest, 2015; Chapter 4; Table 5-2), will also be best constrained in this region and more uncertain at greater depths. Slip is greatest on the steeper, uppermost fault segments in the simple shear kinematic model, so if this model is correct, then those parts of the fault where slip is best constrained likely also contribute the most to seismic slip.
Our preferred model is similar to the model of Campbell et al. (2012) for structures of the northern and eastern Canterbury Plains. In that model, listric thrust faults rooted in a detachment at 12 km depth uplift basement blocks at regularly spaced intervals. Each block is bounded by a major frontal thrust and deformed by additional internal faults. In our regional cross section (Figure 5-14), we have interpreted the five major faults as bounding a series of separate, narrow blocks, where only the Kate
Fault might be an internal fault to a block bounded by the Hamilton Fault. The spacing is not entirely regular, however, with the Waipara Valley in particular dividing the folds into two groups. It is possible that some of the faults merge at depth within basement above the detachment and are therefore not independent. The Hamilton and Omihi faults are perhaps the best candidates due to their close spacing,
220 although the Hamilton Fault, not the frontal Omihi Fault, has the greater slip and the most evidence of normal fault reactivation.
5-5.2. Nature of the Proposed Detachment
Two possibilities exist for the nature of a detachment. One is that it is a discrete low-angle fault plane, perhaps inherited from earlier tectonic events, such as from Cretaceous rifting. Such a detachment could conceivably exist at any depth. The other possibility is that the detachment is a ductile or semi- ductile shear zone. This is what is suggested by Cowan (1992) and subsequent authors who have adopted a detachment at 10-12 km depth on the basis of the distribution of seismicity. This is a reasonable depth for the brittle-ductile transition in a quartzo-feldspathic crust (Scholz, 2002; Reyners et al., 2013), so the lack of seismicity is suggested to represent the depth of the brittle-ductile transition (Cowan, 1992;
Reyners and Cowan, 1993; Bannister et al., 2006; Campbell et al., 2012). In this case, faults root into a ductile shear zone rather than a single discrete detachment plane, and the representation of the detachment as a single fault, as in Figure 5-14, should be regarded as a necessary simplification for the purpose of kinematic modeling. 10-12 km may also be too shallow a depth if aftershock distributions extending down to ~15 km (Ristau et al., 2013; Syracuse et al., 2013) are taken as evidence of the detachment or brittle-ductile transition depth. The brittle and ductile regimes will be separated by a semi-brittle zone, however (Scholz, 2002), and the number of aftershocks drops off substantially below about 12 km
(Syracuse et al., 2013), which might represent sparse seismicity within such a semi-brittle zone. Deeper events near Banks Peninsula may also result from the presence of a stronger basaltic layer there (Reyners et al., 2013; Ellis et al., 2016).
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5-5.3. Alternative Models
The kinematic model that we have presented – steeply dipping, elliptical listric faults rooting in a detachment at 10-12 km depth – aims to account for the available data, but as we have seen, kinematic modeling does not on its own require this detachment depth. Kinematically, a detachment is not required at all to produce an anticline. Fault propagation folds incorporating either an angular bend (as in Chapter
4) or a listric geometry in which dip does not decrease all the way to horizontal can also produce an anticline. Such faults present difficulties in constructing a regional cross section, however, as the sum of the hanging wall uplifts becomes too great to match observations.
Focal mechanisms and aftershock distributions that require fault planes to remain steep at depths approaching 10 km do not in themselves require a listric fault, even a steep one; they could be explained as well, if not better, by a planar fault. Faults in our kinematic models are already planar or nearly planar within the upper few kilometers of the crust with greater curvature near the detachment, so they are intermediate between planar and circular listric fault geometries, but they do still require a listric shape at depth. Trishear or other fault-propagation fold kinematic models applied to a planar fault will produce a monocline, and are indeed used to model basement-involved monoclines (e.g. Bump, 2003), but this would not explain the formation of anticlines in the NCFTB. To consider whether this is only a limitation of the kinematic model, we briefly consider the folding that would result from faulting on a steep (60° dip) planar fault extending from 11 km depth to the near surface, using an elastic halfspace model
(Okada, 1992). This does produce an asymmetric fold shape (Figure 5-15), but backlimb dips reach a maximum of only about 5° after 1.5 km of fault slip and 11° with 3 km of slip. Although not an exhaustive explanation of possible mechanical models, this does suggest that the difficulty in using a planar fault to produce the >20° backlimb dips seen in anticlines of the NCFTB extends beyond the limitations of kinematic modeling.
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5-5.4. Evolution of the North Canterbury Fold and Thrust Belt
Our calculations of the age of onset of folding at Haumuri Bluff (Table 5-2) show that the use of a listric kinematic model does not substantially change the conclusions arrived at with a fault-parallel flow model in Chapter 4. Our results remain consistent with previous studies (Nicol et al., 1994; Barnes,
1996; Barnes et al., 2016) in supporting an age of about 0.8-1.2 Ma for the onset of folding in North
Canterbury.
The modern, geodetically-determined rate of shortening across the North Canterbury fold and thrust belt is about 3-4 mm/yr at azimuth 343.75°±24.39° (Wallace et al., 2012, taking the value for their
Pegasus Block). If these rates were sustained over the course of 0.8-1.2 Ma, then anywhere from 2.4 to
4.8 km of shortening might be expected. Our estimate of 2350 m of shortening from the cross section in
Figure 5-14, which trends at 341°, is consistent with the low end of these estimates. The cross section does not include shortening on the offshore Waikuku and Pegasus Bay faults, which probably adds another 200-400 m if they are similar to our model of the Leithfield fault. In addition, this is only one model, and therefore subject to some uncertainty. We cannot quantify this uncertainty for an entire cross section as we can for a single fault. In our cross section of the anticline at Haumuri Bluff, however, a total shortening of 1040 m had an uncertainty of 240 m, or 23% of the total. An uncertainty of the same percent of the total for the shortening in Figure 5-14 would be 540 m. Shortening uncertainties from balanced cross sections that are similarly high percentages of the total have been reported by Judge and
Allmendinger (2011) and Allmendinger and Judge (2013). If uncertainty in the detachment depth is considered, the uncertainty would be even greater. We also note that the shortening on the individual faults within Figure 5-14 tends to be somewhat less than that estimated from analyses of the Montserrat and Haumuri Bluff anticlines individually (Figure 5-5a; Table 5-2), which may suggest that the shortening estimate from Figure 5-14 is on the low end of the possible range. They are also lower than estimates of shortening from shortening rates on individual faults in Chapter 4, but this is most likely due to the use of a different fault geometry and of a kinematic model (inclined shear) in this chapter that does
223 not conserve slip along the fault. In summary, estimates of total shortening are consistent within error with the estimates of shortening rates from the model of Wallace et al. (2012) extrapolated to the estimated age of the onset of folding in North Canterbury.
In Figure 5-14, we have modeled displacement sequentially on the six faults from northwest to southeast. Sequential modeling of folds is a necessary approximation in constructing a cross section, but the reality is likely more complex. Estimates of the time at which folding began along the coast (Nicol et al., 1994; Chapter 4; this chapter) are similar to estimates for offshore folds farther southeast (Barnes,
1996; Barnes et al., 2016), and inland faults are still thought to be seismically active (Barrell and
Townsend, 2012). Uplift along all faults may have begun at a similar time, with deformation rates decreasing from northwest to southeast. Such an interpretation would be similar to models of basin inversion by Buiter and Pfiffner (2003), in which uplift decreases from one edge of the model to the other but normal fault reactivation does not occur in a simple linear sequence.
5-5.5. Basement-Involved Fold Kinematics
Our preferred kinematic model for North Canterbury folds may have implications for basement- involved folding in other locations. We have shown that the rigid rotation, circular-listric fault model of
Erslev (1986) and Seeber and Sorlien (2000) is not a good fit for the basement-involved structures of the
North Canterbury fold and thrust belt. Instead, we favor a model that incorporates trishear ahead of fault tips, inclined shear kinematics in listric fault hanging walls, and elliptical listric faults that typically have longer vertical than horizontal axes. The success of this model in North Canterbury suggests that it may be applicable to other regions. For example, if the Torlesse-basement-involved folds of the Mackenzie
Basin follow similar fold kinematics, this might help to explain the discrepancy between the shallow detachments inferred there from the circular-fault model (Amos et al., 2007) and the steeper faults imaged by Ghisetti et al. (2007). How broadly applicable the model is remains to be tested, however. The greywacke of the Torlesse likely does not deform in the same manner as the stronger, less anisotropic
224 crystalline rocks that form the basement of many basement-involved folds, such as those of the Laramide orogeny that inspired the model of Erslev (1986). While more work will be needed to determine what factors determine which model of basement-involved folding is best applied to a given setting, inclined simple shear and steep, non-circular listric faults should be considered as a possibility in regions beyond
North Canterbury.
5-5.6. Fitting Listric Fault Models
We have extended the capability of the InvertTrishear program to allow listric fault models to be fit to data, and we have tested two methods of doing so, approximating the listric fault in both cases by a series of straight segments. Fitting for the orientation of each segment and depths of each bend allows the maximum freedom in fault shape. The large number of model parameters, however, means that Markov chain Monte Carlo simulations require a specified starting model and are unlikely to be able to find a global probability maximum. They can be used, however, to refine a starting model and find at least a local probability maximum. Requiring the listric fault to have an elliptical shape greatly reduces the number of parameters, making it easier to search the entire parameter space and making the model less focused on minor complexities of the fault geometry. It also ensures that the curvature of the fault is smooth, which in turn ensures the same of the anticlinal backlimb produced. There is no reason to suppose that a listric fault must be elliptical in shape, and a variety of other mathematical functions could be used, but this shape provides a simple model for a curved fault, and it includes the often-used circular listric fault as one end member (eccentricity = 0) and approaches a planar fault as the other end member
(as eccentricity approaches 1).
The difficulty we had in constraining the detachment depth for a listric fault should be taken into consideration when applying this model. With sufficient information including dip of the fault at the surface, width of the fold, slope and curvature of the backlimb, and inclined shear angle, the detachment depth most likely can be determined from the surface structure of an anticline in much the same way it
225 can be determined for a normal fault from the structure of a half-graben. In practice, the problem is likely to be under-constrained in many cases. In particular, the inclined shear angle is difficult to determine from field observations, and as it does for normal faults (White et al., 1986), it will co-vary with detachment depth. Knowing either inclined shear angle or detachment depth a priori is probably necessary to fit an accurate model to data, and this is in turn necessary for an accurate determination of shortening (Figure 5-
5a, Figure 5-10).
We have also demonstrated the value of using growth strata in InvertTrishear, using the Leithfield
Fault as an example. Growth strata provide an additional constraint on the model and give us valuable information about the rates of fold growth (Figure 5-9), but including them also increases the number of model parameters, making the inversion more difficult. At a minimum, one must fit separately for the slip necessary to restore each growth bed. In the case of the Leithfield Fault, we also had to fit for the restored-state elevation and restored state dip (within narrow constraints), to help account for an apparent slight regional southeast dip of the strata. Overall, we have demonstrated the use of listric faults and growth strata in inversion for a fault-kinematic model and have also identified some of the limits of this approach.
5-6. Conclusions
1) Trishear fault propagation folding combined with inclined shear in the hanging walls of steep listric faults provides a model for folding and faulting in North Canterbury, New Zealand that is consistent with available geological and geophysical data and is capable of producing the folding styles observed both on- and offshore without the need for different styles of faulting in the two domains. Most of these faults are thought to be inverted normal faults, but a smaller number of newly-formed faults may also exist.
226
2) Listric faulting by rigid rotation of the hanging wall on a circular listric fault does not provide an adequate model of the style of folding in North Canterbury, given the geological and geophysical constraints.
3) Surface structure alone does not uniquely determine a detachment depth for the North
Canterbury fold and thrust belt, and shortening is highly dependent on the choice of detachment depth.
However, the surface structure is consistent with a detachment depth at about 10-12 km as proposed by previous workers (Cowan, 1992; Campbell et al., 2012). Fault slip near the surface is not strongly dependent on detachment depth.
4) Listric faults can be fit to data using Markov chain Monte Carlo methods. Assuming an elliptical shape to the faults minimizes the number of new model parameters, while still being able to model faults that are steep through most of the 10-12 km above the detachment.
5) Total shortening across the North Canterbury fold and thrust belt is likely to be in the range of
2-3 km, which is consistent with shortening rates in the region and the likely age at which faulting began.
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235
236
Figure 5-1: Major folds and thrust faults of the North Canterbury Fold and Thrust Belt. Note that the non-inverted normal fault that crosses cross section D-D′ is only one of a number of such faults in the offshore region (Barnes et al., 2016), which for simplicity are not shown here. Locations of faults and folds are from Yousif (1987), Litchfield (1995), Warren (1995), Rattenbury et al. (2006), Forsyth et al.
(2008) and Barnes et al. (2016), and distribution of basement is from Rattenbury et al. (2006) and Forsyth et al. (2008).
237
Figure 5-2: Focal mechanisms in the NCFTB from the GeoNet catalogue
(http://info.geonet.org.nz/download/attachments/8585256/GeoNet_CMT_solutions.csv?api=v2), accessed
February 9, 2016. Focal mechanisms are labeled with their centroid depths.
238
Figure 5-3: Contours of detachment depth in km as a function of backlimb width and fault dip for a fold formed by rigid rotation of a basement block above a circular listric fault.
239
Figure 5-4: Forward models of the Montserrat Anticline, using listric faults. (a) The detachment is at 6 km and the inclined shear angle is 70° (from horizontal). (b) The detachment is at 11 km and shear is vertical.
240
Figure 5-5: (a) Shortening on the Glendhu Fault vs. detachment depth. (b) Slip on the uppermost fault segment (where the fault becomes straight instead of listric) of the Glendhu Fault vs. detachment depth.
Note that in most models this is the part of the fault that reaches the surface, but in some of the models with the shallowest detachments, the fault is still listric at the surface, in which case this value is less meaningful.
241
Figure 5-6: RMS errors of best-fit models for the Montserrat Anticline, for the three data types used to constrain the models: the locations of lithologic contacts (tops of the Torlesse basement and Broken River formation), bedding dips, and the mapped location of the Glendhu Fault at the surface.
242
243
Figure 5-7: (a) Comparison of models for the Leithfield fault (cross section B-B′). The forward model used as a starting point for the Markov chains has a detachment at 10.5 km, but detachment depth was fit for as a model parameter. (b) Best-fit model (dashed lines) pre-growth strata from InvertTrishear compared to data (colored lines, from Barnes et al., 2016), when fitting an elliptical fault. (c) Best-fit model growth strata (dashed lines) compared to data (colored lines, from Barnes, 1996). Growth strata are labeled according to the stratigraphy of Barnes (1995; 1996).
244
Figure 5-8: (a and b) Histograms of shortening on the Leithfield Fault, when fitting (a) a listric fault of arbitrary shape and (b) an elliptical listric fault. In each case, the Markov chain was allowed to run for six million models, with the first one million removed as a burn-in period, and the remaining models subsampled at intervals of 50, so N = 100,000 for the histograms. (c and d) The path taken by the Markov chain for (c) the listric fault of arbitrary shape and (d) the elliptical listric fault. The starting value in both cases was 80 m. Note that we used an adaptive parallel tempering algorithm, such that sixteen chains were run in parallel, and only the untempered, “lowest energy” chain is shown here and is used to produce the histograms. See Miasojedow et al. (2013) for details of the algorithm.
245
Figure 5-9: (a) Total shortening since formation of growth strata (unconformity surfaces) vs. age of growth strata for the Leithfield Fault. The shortening on the oldest layer is within a few meters (average 7 m) of the shortening on pre-growth strata. (b) Shortening rates with 2σ errors bars for the intervals between formation of the folded unconformities.
246
247
Figure 5-10: Listric models for the anticline at Haumuri Bluff. Line C-C′ corresponds to Haumuri Bluff cross section A-A′ in Chapter 4, though here we have extended faults farther to the northwest. In both sections, we have drawn in a likely geometry for the Hundalee Fault, but it is not part of the model. (a)
Two non-horizontal segments connected by a short listric segment. Here, the listric segment is only a few meters wide and is essentially a fault bend. The syncline axis, however, follows the shear angle (here 68°) and does not bisect the fault bend as in the fault-parallel flow models in Chapter 4. (b) A detachment at 11 km depth is required. Note that in this model, a very narrow trishear zone (0.2° apical angle) takes the place of the upper part of the fault. ϕ (the half apical angle) was allowed to change during fault propagation and decreased after initial folding to produce this result.
248
Figure 5-11: Detachment depth vs. shear angle (measured from horizontal so that 90° is vertical) for two offshore Cretaceous-aged normal faults in North Canterbury.
249
Figure 5-12: Forward model for cross section D-D′, based on seismic line GG07-103 as interpreted by
Barnes et al. (2016). The trishear angle is 60°, and the propagation to slip ratio is 0.5 for the first 500 m of slip on the detachment, 1 for the next 500 m of slip, and 5 for the remainder, until the tip reaches the top of the Kowai Formation (yellow). The inclined shear angle is 80° from horizontal. (a) The initial, non- inverted normal fault and half-graben. The geometry of the basement-cover contact has been simplified slightly, and offsets on small, synthetic faults have been removed. The listric fault geometry was determined from the half-graben geometry and dip of the upper part of the fault as described in the text.
(b) The fold after 500 m of slip on the detachment. The backlimb dip is about 4°. (c) The fold after 1000 m of slip on the detachment. The backlimb dip is about 10°. (d) The fold after 2000 m of slip on the detachment. The backlimb dip is about 20°.
250
251
Figure 5-13: Pegasus Bay Fault (cross section E-E′). (a) Restored section from Barnes et al. (2016) with a listric fault fit to the half-graben geometry. (b) Modern day structure as interpreted by Barnes et al. (2016) with the listric fault from (a) extended into the cover. The fault as interpreted by Barnes et al. (2016) is shown dashed. (c) The fold that results from 750 m of heave applied to the fault in (b). The backlimb dip is now about 20°.
Figure 5-14: Regional cross section. (a) Deformed state. (b) Restored state. HBF: Hurunui Bluff
Fault, AF: Mt. Alexander Fault, MHF: Moores Hill Fault, OF: Omihi Fault, HF: Hamilton Fault,
KF: Kate Fault, CB: Culverden Basin, W: Waipara Valley.
254
Figure 5-15: Cross sections (a, c) for folds formed by slip on a planar fault in an elastic half- space and resulting dips (positive to the left) within the hanging wall (b, d). The reference horizons are initially at depths of 1 km (red) and 1.5 km (blue). To simulate fault propagation, slip is applied in 1 m increments and the fault tip is propagated forward at the same rate (P/S = 1) from an initial depth of 2 km, stopping when it reaches the surface. The model fault is 10 km wide, with the cross section taken in the middle, its dip is 60°, and its base is at 11 km depth.
Young’s modulus is 8x1011 Pa and Poisson’s ratio is 0.25 (suggested values from Toda et al.,
2011). Fault slip is 1.5 km in (a) and (b) and 3 km in (c) and (d).
255 Table 5-1: Earthquakes shown in Figure 5-1. Events in which both nodal planes have rake in the range 45° ≤ rake ≤ 135°, indicating a predominantly reverse sense of slip, are highlighted in yellow.
Events in which the rake of one nodal plane is in this range are highlighted in orange.
First Nodal Plane Second Nodal Plane Centroid Depth Event ID Lat. Long. Strike Dip Rake Strike Dip Rake Mw (km) 2122842 -43.3 172.9 49 85 12 318 78 175 4.5 20 2128373 -43.3 172.9 283 60 80 123 31 107 3.9 18 2290108 -43.3 172.8 227 78 -115 114 28 -26 4.4 27 2737094 -42.3 173.6 59 46 -79 224 45 -101 4.2 27 2883904 -43.0 172.7 65 72 159 162 70 19 3.6 10 3051685 -43.1 173.4 241 60 82 78 31 104 3.7 24 3186228 -43.2 172.9 57 66 90 237 24 90 3.8 27 3372442 -42.5 173.7 276 87 164 7 74 3 3.9 8 3372454 -42.5 173.8 94 82 -164 2 74 -8 3.8 4 3460266 -42.5 173.8 254 74 -136 150 48 -21 3.8 2 2013p433056 -42.6 173.2 50 75 159 146 70 16 3.8 8 2014p686520 -43.2 172.8 41 53 94 215 37 85 4.1 9 2016p858000 -42.3 173.0 219 38 128 354 61 64 7.8 16 2016p858340 -42.5 173.3 240 85 159 332 69 5 4.9 12 2016p858704 -42.4 173.7 12 66 61 246 38 137 4.9 4 2016p858803 -42.7 173.0 216 46 103 17 45 77 4.4 6 2016p858815 -42.6 173.2 43 85 99 161 11 29 4.9 3 2016p859051 -42.3 173.8 247 86 -145 155 55 -5 4.6 11 2016p859493 -42.5 173.3 37 47 109 190 46 71 3.9 7 2016p859524 -42.6 173.3 159 88 -9 249 81 -178 6.5 10 2016p859628 -42.4 173.8 56 75 114 177 28 34 5.1 10 2016p860053 -42.4 173.5 70 75 147 170 58 18 4.8 11 2016p860119 -42.3 173.8 34 87 -152 302 62 -4 4.4 9 2016p860567 -42.4 173.8 278 82 -147 182 57 -10 5.1 4 2016p860592 -42.6 173.3 253 88 -124 160 34 -3 4.6 11 2016p860816 -42.4 173.6 101 76 153 198 64 16 4.5 14 2016p861086 -42.7 173.1 191 71 41 85 52 155 4.3 9 2016p861102 -42.4 173.5 168 74 -17 263 74 -163 4.2 14 2016p861383 -42.8 173.3 46 67 129 162 44 34 4.0 5 2016p861719 -42.5 173.2 52 65 97 216 25 75 4.1 4 2016p862211 -42.8 172.9 40 48 103 202 44 76 4.3 14 2016p863064 -42.5 173.3 20 63 57 255 41 137 4.3 6 2016p863486 -42.6 173.4 244 69 147 348 59 25 3.9 15
256
2016p863743 -42.4 173.9 232 66 134 344 49 32 4.3 18 2016p864224 -42.5 173.6 23 71 126 137 40 30 4.5 5 2016p864229 -42.6 173.6 253 86 -161 161 71 -4 4.2 6 2016p864749 -42.3 173.9 28 55 115 169 42 59 3.8 9 2016p865572 -42.6 173.3 20 61 55 256 44 137 4.0 13 2016p866561 -42.5 173.5 317 82 20 224 70 171 4.6 13 2016p869914 -42.3 173.8 30 59 75 238 34 114 4.1 10 2016p871889 -42.7 173.4 34 72 87 222 18 98 5.3 2 2016p872063 -42.5 173.1 16 56 85 206 34 98 4.4 4 2016p873325 -42.3 173.9 88 74 149 187 60 19 4.1 10 2016p876450 -42.6 173.5 208 83 24 115 66 172 4.7 11 2016p881669 -42.9 173.0 256 85 -158 164 69 -6 5.5 4 2016p881704 -42.9 173.0 249 83 -167 157 77 -7 4.3 11 2016p881727 -42.9 173.0 3 67 68 230 31 132 4.4 4 2016p881756 -42.9 173.0 20 54 88 203 36 92 4.4 6 2016p881916 -42.9 173.0 349 53 68 203 43 116 4.3 4 2016p882181 -42.9 172.9 242 52 94 55 38 85 3.8 5 2016p882187 -43.0 173.0 354 64 67 218 34 129 4.6 4 2016p890609 -42.5 173.3 53 60 118 186 40 51 3.9 7 2016p914194 -42.6 173.2 173 89 -18 264 72 -179 4.1 10 2016p946148 -42.4 173.9 5 59 82 201 32 103 4.2 14 2016p976987 -42.5 173.9 233 55 106 27 38 69 4.5 11 2016p977108 -42.4 173.6 69 71 147 171 59 23 4.2 10 2017p016396 -42.6 173.3 220 51 125 352 51 55 4.4 11 2017p058654 -42.4 173.5 54 80 158 149 68 11 3.9 18
257 Table 5-2: Comparison of models for the anticline at Haumuri Bluff.
Chapter 4, Listric, No Listric, Detachment at Model 2 Detachment 11 km Depth
Maximum Fault Dip (°) 54±14 52±9 64±10 Minimum Fault Dip (°) 23±7 34±8 0 Total slipa (m) 2290±1240b 2030±580c 1810±1200c uppermost fault segment Total slipa (m) 2290±1240b 1660±340 1040±240 lowermost fault segment Total shorteningd (m) 2110±1220b 1370±370 1040±240 Restored-state basement depth 1230±350 1350±220 1330±180 (m) Restored-state Oaro fossils depth 660±350 700±190 700±180 (m) Qt4 shortening rated,e (m/ka) 2.1±0.5 1.6±0.4 1.2±0.2 Qt6 shortening rated,e (m/ka) 1.4±0.4 1.1±0.3 0.9±0.4 Qt4 slip ratee (m/ka) 2.3±0.4 2.3±0.8c 2.1±1.6c upper segment Qt6 slip ratee (m/ka) 1.6±0.3 1.6±0.8c 1.5±1.6c upper segment Qt4 slip ratee (m/ka) 2.3±0.4 1.9±0.3 1.2±0.2 lower segment Qt6 slip ratee (m/ka) 1.6±0.3 1.3±0.3 0.9±0.4 lower segment Fold Agef (ka) 1000±450b 870±160 850±170 Qt4 slip rate Fold Agef (ka) 1460±670b 1260±280 1210±370 Qt6 slip rate Best-fit model RMS error (m) 12 15 19 for location of contacts Best-fit model RMS error (°) for 4.6 4.9 6.7 bedding dips Best-fit model RMS error (m) 3.3 2.2 2.4 for points on marine terraces
All values are means with 2σ errors. a Actual total separation along the fault will vary with position due to tip propagation, so these are maximum values, for a point below the initial tip position. b The total slip histogram for model 3 of Chapter 4 is bimodal, and therefore so are the total shortening and fold age histograms. See Chapter 4 for more details.
258 c A small number of listric fault models have very high ratios of fault slip on the uppermost fault segment to slip on the lowermost segment, resulting from steep (approaching vertical) uppermost fault segments. This results in particularly high standard deviations, especially for the 11 km detachment models. Removing these results and focusing only on the main peaks of each histogram gives values of total slip, Qt4 slip rates, and Qt6 slip rates respectively of 2020±420 m,
2.3±0.3 m/ka, and 1.6±0.3 m/ka for the models without a detachment and 1760±330 m, 2.1±0.2 m/ka, and 1.4±0.2 m/ka for the models with a detachment at 11 km depth. d Shortening is calculated as the slip on the detachment for the model with a detachment and as the slip on the lowermost fault segment times the cosine of the dip of that segment for the other two models. e All rates are calculated for the Qt4 and Qt6 terraces (see Chapter 4) and are the average rates from the formation of that terrace to the present day. f Fold age is calculated using the fault slip rates for the two terraces, assuming that the specified rate represents the average rate since folding began.
Appendix A
Supplementary Tables for Chapter 3
Three Excel spreadsheets containing information on sample locations, AAR data, and inputs and outputs for the program used to fit an AAR-age model to the calibration datasets were included as online supplements to Oakley et al. (2017), “Quaternary marine terrace chronology,
North Canterbury, New Zealand using amino acid racemization and infrared stimulated luminescence.” Abbreviated copies of those tables are included here, but as they are not all suitable to fit on a page, the reader is referred to the online supplements to the paper for complete versions.
260 Table A-1: Sample Locations
NZTM WGS84 Name Easting (m) Northing (m) Latitude Longitud Elevation (m) Type e HB1 1641788 5287724 -42.5618 173.5091 40 IRSL HB2 1641534 5288634.604 -42.5536 173.5059 165.8 IRSL HB3 1641534 5288634.604 -42.5536 173.5059 166 AAR HB4 1641182.6 5289312.313 -42.5475 173.5016 4.2 AAR, C14 HB5 1641182.6 5289312.313 -42.5475 173.5016 4.2 AAR, C14 GA1 1590281.1 5227847.079 -43.102 172.8806 67 IRSL GA2 1590324 5227891.53 -43.1016 172.8811 60 IRSL GA3 1590324 5227891.53 -43.1016 172.8811 60 AAR GA4 1590189.1 5227939.339 -43.1012 172.8794 58.7 AAR GA5 1589936.9 5228169.949 -43.0991 172.8763 61.5 AAR MB1 1607727.5 5238589.697 -43.0053 173.0948 62.2 IRSL MB2 1605879 5233917.429 -43.0474 173.0722 34.1 IRSL MB3 1605791.5 5233816.45 -43.0483 173.0711 31.3 AAR MB4 1604841.7 5234259.24 -43.0443 173.0594 40.0 AAR MB5 1602861.2 5234075.422 -43.046 173.0351 70 AAR MB6 1601180.4 5232594.978 -43.0593 173.0145 33 (slumped AAR block) CK4 5604054 1950329.468 -39.6416 177.0822 4 AAR, C14 Kupe 5582000 1754000 -39.9 174.8 AAR Shakespear 5582000 1754000 -39.9 174.8 AAR e Cliff Landguard 5575000 1774000 -39.96 175.03 30 ± AAR All Day 4992000 1434000 -45.21 170.88 3 AAR Bay Waverley 5590231.9 1738253.759 -39.8267 174.6155 1 AAR
Elevations with 1 decimal place given were measured by DGPS or tape measure. Elevations with no decimal places are approximate, based on non-differential DGPS and DEMs, and may have uncertainties of several meters.
Table A-2: AAR Data
excluded DL DL DL DL DL DL DL DL UAL Sample Site Unit Taxon ex Asp Glu Ser Ala Val Phe Ile Leu Castlecliff Talochlamys 9413A NZ-4C Beach Omapu gemmulata 0.663 0.591 0.204 0.940 0.707 0.797 0.774 0.653 Castlecliff Talochlamys 9413B NZ-4C Beach Omapu gemmulata 0.689 0.627 0.160 0.959 0.708 0.863 0.796 0.700 Castlecliff Talochlamys 9413C NZ-4C Beach Omapu gemmulata 0.688 0.510 0.185 0.892 0.562 0.790 0.677 0.647 Castlecliff Talochlamys 9413D NZ-4C Beach Omapu gemmulata 0.674 0.545 0.211 0.925 0.631 0.807 0.680 0.655 Castlecliff Talochlamys 9413E NZ-4C Beach Omapu gemmulata 0.680 0.499 0.188 0.908 0.594 0.770 0.613 0.625 Castlecliff 9414A NZ-6A Beach Kupe Tawera spissa 0.845 0.621 0.439 1.069 0.562 0.844 0.892 1.295 Castlecliff 9414B NZ-6A Beach Kupe Tawera spissa 0.851 0.630 0.624 1.076 0.598 0.866 0.969 1.450 Castlecliff 9414C NZ-6A Beach Kupe Tawera spissa 0.838 0.635 0.454 1.068 0.590 0.852 0.989 1.242 Castlecliff 9414D NZ-6A Beach Kupe Tawera spissa 0.858 0.667 0.579 1.068 0.620 0.881 0.943 1.253 Castlecliff 9414E NZ-6A Beach Kupe Tawera spissa 0.850 0.644 0.835 1.084 0.589 0.871 0.940 1.406 Castlecliff Nucula 9415A NZ-2C Beach Okehu nitidula 0.866 0.846 0.413 1.054 0.862 0.987 1.063 1.053 Castlecliff Nucula 9415B NZ-2C Beach Okehu nitidula 0.852 0.820 0.452 1.067 0.850 0.866 1.002 0.971
262
Castlecliff Nucula 9415C NZ-2C Beach Okehu nitidula 0.851 0.818 0.386 1.051 0.832 0.978 1.067 0.978 Castlecliff Nucula 9415D NZ-2C Beach Okehu nitidula 0.875 0.842 0.657 1.066 0.909 1.001 1.093 0.981 Castlecliff Nucula 9415E NZ-2C Beach Okehu nitidula 0.852 0.806 0.485 1.097 0.839 0.962 1.091 0.945 Kaimatira Castlecliff Pumice Talochlamys 9416A NZ-3B Beach Sand gemmulata 0.688 0.645 0.176 0.976 0.769 0.878 0.869 0.763 Kaimatira Castlecliff Pumice Talochlamys 9416B NZ-3B Beach Sand gemmulata 0.675 0.655 0.158 0.995 0.801 0.831 0.985 0.819 Kaimatira Castlecliff Pumice Talochlamys 9416C NZ-3B Beach Sand gemmulata 0.693 0.649 0.186 0.993 0.774 0.849 0.931 0.826 Kaimatira Castlecliff Pumice Talochlamys 9416D NZ-3B Beach Sand gemmulata 0.658 0.581 0.174 0.976 0.745 0.913 0.598 0.738 Kaimatira Castlecliff Pumice Talochlamys 9416E NZ-3B Beach Sand gemmulata 0.702 0.673 0.168 0.973 0.775 0.920 0.968 0.755 Castlecliff Shakespeare 9417A NZ-7B Beach Cliff Tawera spissa 0.740 0.522 0.574 0.995 0.480 0.750 0.763 1.029 Castlecliff Shakespeare 9417B NZ-7B Beach Cliff Tawera spissa 0.770 0.533 0.583 0.991 0.488 0.749 0.737 1.152 Castlecliff Shakespeare 9417C NZ-7B Beach Cliff Tawera spissa 0.770 0.526 0.567 0.982 0.499 0.751 0.737 1.182 Castlecliff Shakespeare 9417D NZ-7B Beach Cliff Tawera spissa 0.775 0.545 0.534 1.001 0.567 0.778 0.713 1.140 Castlecliff Shakespeare 9417E NZ-7B Beach Cliff Tawera spissa 0.782 0.555 0.574 1.011 0.520 0.795 0.760 1.207
263
Kaimatira Castlecliff Pumice 9418A NZ-3A Beach Sand Paphies delta x 0.452 0.265 0.552 0.534 0.195 0.362 0.257 0.444 Kaimatira Castlecliff Pumice 9418B NZ-3A Beach Sand Paphies delta 0.746 0.671 0.553 1.076 0.746 0.822 0.924 1.104 Kaimatira Castlecliff Pumice 9418C NZ-3A Beach Sand Paphies delta 0.740 0.663 0.501 1.059 0.704 0.828 0.903 1.053 Kaimatira Castlecliff Pumice 9418D NZ-3A Beach Sand Paphies delta 0.718 0.604 0.207 1.038 0.719 0.821 0.897 0.969 Kaimatira Castlecliff Pumice 9418E NZ-3A Beach Sand Paphies delta 0.756 0.568 0.141 1.055 0.711 0.748 0.851 0.945 Castlecliff Scalpomactra 9419A NZ-4A Beach Omapu scalpellum 0.746 0.568 0.369 1.089 0.462 0.761 0.708 0.739 Castlecliff Scalpomactra 9419B NZ-4A Beach Omapu scalpellum 0.750 0.566 0.492 1.066 0.464 0.707 0.626 0.720 Castlecliff Scalpomactra 9419C NZ-4A Beach Omapu scalpellum 0.748 0.556 0.475 1.076 0.476 0.752 0.668 0.769 Castlecliff Scalpomactra 9419D NZ-4A Beach Omapu scalpellum 0.735 0.539 0.233 1.049 0.419 0.676 0.658 0.753 Castlecliff Scalpomactra 9419E NZ-4A Beach Omapu scalpellum 0.782 0.664 0.169 1.095 0.710 0.932 1.000 0.945 Castlecliff 9420A NZ-2A Beach Okehu Mactra carteri 0.772 0.703 0.524 1.097 0.763 0.877 0.881 1.155 Castlecliff 9420B NZ-2A Beach Okehu Mactra carteri 0.747 0.660 0.591 1.076 0.705 0.851 0.788 1.024 Castlecliff 9420C NZ-2A Beach Okehu Mactra carteri 0.739 0.683 0.398 1.089 0.744 0.878 0.876 1.128
264
Castlecliff 9420D NZ-2A Beach Okehu Mactra carteri 0.779 0.719 0.178 1.099 0.806 0.920 0.998 1.204 Castlecliff 9420E NZ-2A Beach Okehu Mactra carteri 0.759 0.696 0.362 1.094 0.754 0.877 0.889 1.061 Castlecliff Talochlamys 9421A NZ-2B Beach Okehu gemmulata 0.736 0.747 0.155 1.036 0.888 0.952 1.017 0.927 Castlecliff Talochlamys 9421B NZ-2B Beach Okehu gemmulata 0.723 0.734 0.159 1.069 0.879 0.897 1.004 0.929 Castlecliff Talochlamys 9421C NZ-2B Beach Okehu gemmulata 0.732 0.719 0.150 1.054 0.886 0.935 0.968 0.939 Castlecliff Talochlamys 9421D NZ-2B Beach Okehu gemmulata 0.729 0.731 0.131 1.044 0.912 0.979 1.039 0.949 Castlecliff Talochlamys 9421E NZ-2B Beach Okehu gemmulata 0.687 0.796 0.141 1.071 0.943 0.924 1.115 0.955 Castlecliff Shakespeare Pleuromeris 9422A NZ-7C Beach Cliff zelandica 0.809 0.576 0.410 0.900 0.679 0.801 0.863 0.939 Castlecliff Shakespeare Pleuromeris 9422B NZ-7C Beach Cliff zelandica 0.797 0.600 0.534 0.926 0.703 0.819 0.926 0.921 Castlecliff Shakespeare Pleuromeris 9422C NZ-7C Beach Cliff zelandica 0.793 0.579 0.292 0.910 0.683 0.748 0.905 0.947 Castlecliff Shakespeare Pleuromeris 9422D NZ-7C Beach Cliff zelandica 0.803 0.583 0.320 0.921 0.676 0.789 0.838 0.923 Castlecliff Shakespeare Pleuromeris 9422E NZ-7C Beach Cliff zelandica 0.805 0.592 0.535 0.904 0.672 0.783 0.871 0.978 Castlecliff Zenatia 9423A NZ-4B Beach Omapu acinaces 0.804 0.696 0.141 1.045 0.752 0.883 0.986 0.984 Castlecliff Zenatia 9423B NZ-4B Beach Omapu acinaces 0.783 0.630 0.122 1.055 0.726 0.816 0.784 0.956 Castlecliff Zenatia 9423C NZ-4B Beach Omapu acinaces 0.785 0.692 0.459 1.070 0.719 0.860 0.909 0.971
265
Castlecliff Zenatia 9423D NZ-4B Beach Omapu acinaces 0.792 0.671 0.271 1.102 0.731 0.865 0.770 0.982 Castlecliff Zenatia 9423E NZ-4B Beach Omapu acinaces 0.795 0.693 0.516 1.045 0.735 0.872 1.039 0.965 Castlecliff Divalucina 9424A NZ-1C Beach Butlers huttoniana 0.904 0.817 0.524 1.119 0.678 0.937 0.954 0.938 Castlecliff Divalucina 9424B NZ-1C Beach Butlers huttoniana 0.904 0.788 0.349 1.124 0.653 0.863 0.984 0.935 Castlecliff Pratulum 9425A NZ-1B Beach Butlers pulchellum 0.726 0.675 0.206 1.069 0.610 0.877 0.868 0.941 Castlecliff Pratulum 9425B NZ-1B Beach Butlers pulchellum 0.752 0.681 0.452 1.084 0.621 0.894 0.890 0.949 Castlecliff Pleuromeris 9426A NZ-6B Beach Kupe zelandica 0.845 0.662 0.682 1.005 0.784 0.895 1.013 1.136 Castlecliff Pleuromeris 9426B NZ-6B Beach Kupe zelandica 0.847 0.657 0.939 1.015 0.801 0.918 0.994 1.253 Castlecliff Pleuromeris 9426C NZ-6B Beach Kupe zelandica 0.853 0.679 0.897 1.019 0.793 0.915 0.953 1.310 Castlecliff Pleuromeris 9426D NZ-6B Beach Kupe zelandica 0.846 0.649 0.436 1.023 0.787 0.873 0.927 1.286 Castlecliff Pleuromeris 9426E NZ-6B Beach Kupe zelandica 0.856 0.667 0.282 0.997 0.803 0.899 1.030 1.184 Castlecliff Cyclomaetra 9427A NZ-6D Beach Kupe tristis 0.733 0.683 0.264 1.127 0.752 0.883 1.005 1.208 Castlecliff Cyclomaetra 9427B NZ-6D Beach Kupe tristis 0.726 0.664 0.429 1.079 0.704 0.842 0.859 1.047 Castlecliff Cyclomaetra 9427C NZ-6D Beach Kupe tristis 0.747 0.660 0.373 1.097 0.718 0.873 0.926 1.108 Castlecliff Cyclomaetra 9427D NZ-6D Beach Kupe tristis 0.730 0.690 0.363 1.076 0.718 0.813 0.855 0.940
266
Castlecliff Cyclomaetra 9427E NZ-6D Beach Kupe tristis 0.746 0.689 0.259 1.087 0.740 0.840 0.875 1.102 Castlecliff 9428A NZ-6E Beach Kupe Calloria spp. 0.879 0.834 0.267 0.989 0.838 0.877 1.053 0.966 Castlecliff 9428B NZ-6E Beach Kupe Calloria spp. 0.879 0.831 0.168 1.015 0.823 0.789 1.060 0.878 Castlecliff 9428C NZ-6E Beach Kupe Calloria spp. 0.890 0.858 0.116 1.019 0.799 0.811 1.082 1.052 Castlecliff 9428D NZ-6E Beach Kupe Calloria spp. 0.865 0.825 0.134 1.012 0.804 0.727 0.965 0.816 Castlecliff 9428E NZ-6E Beach Kupe Calloria spp. 0.873 0.840 0.182 0.741 0.864 0.848 1.032 1.021 Castlecliff Shakespeare Sigapetella 9429A NZ-7D Beach Cliff tennis x 0.558 0.489 0.034 0.896 0.474 0.535 0.535 0.658 Castlecliff Shakespeare Sigapetella 9429B NZ-7D Beach Cliff tennis 0.669 0.608 0.158 0.955 0.607 0.657 0.606 0.753 Castlecliff Shakespeare Sigapetella 9429C NZ-7D Beach Cliff tennis 0.647 0.624 0.092 1.007 0.569 0.635 0.678 0.769 Castlecliff Shakespeare Sigapetella 9429D NZ-7D Beach Cliff tennis 0.660 0.617 0.098 0.982 0.560 0.748 0.693 0.783 Castlecliff Shakespeare Sigapetella 9429E NZ-7D Beach Cliff tennis x 0.511 0.393 0.044 0.651 0.431 0.439 0.445 0.480 Castlecliff Sigapetella 9430A NZ-6C Beach Kupe tennis 0.687 0.675 0.082 1.016 0.640 0.732 0.847 0.863 Castlecliff Sigapetella 9430B NZ-6C Beach Kupe tennis 0.667 0.623 0.061 1.001 0.641 0.770 0.810 0.825 Castlecliff Sigapetella 9430C NZ-6C Beach Kupe tennis 0.706 0.702 0.211 1.056 0.683 0.895 0.932 0.908 Castlecliff Sigapetella 9430D NZ-6C Beach Kupe tennis x 0.392 0.469 0.028 0.563 0.369 0.642 0.477 0.675
267
Castlecliff Sigapetella 9430E NZ-6C Beach Kupe tennis 0.717 0.653 0.059 1.009 0.642 0.766 0.840 0.847 Ostrea 9431A NZ-8B Landguard chilensis 0.754 0.622 0.188 1.105 0.538 0.659 0.918 1.171 Ostrea 9431B NZ-8B Landguard chilensis 0.653 0.431 0.322 1.091 0.308 0.600 0.520 0.973 Ostrea 9431C NZ-8B Landguard chilensis 0.719 0.553 0.220 1.065 0.557 0.714 0.841 1.073 Ostrea 9431D NZ-8B Landguard chilensis 0.737 0.537 0.220 1.129 0.507 0.722 0.870 1.134 Ostrea 9431E NZ-8B Landguard chilensis 0.658 0.519 0.264 1.104 0.477 0.703 0.789 0.996 Zethalia 9432A NZ-8F Landguard zelandica 0.643 0.403 0.503 0.903 0.241 0.790 0.403 0.621 Zethalia 9432B NZ-8F Landguard zelandica 0.604 0.376 0.222 0.877 0.237 0.661 0.336 0.523 Zethalia 9432C NZ-8F Landguard zelandica 0.695 0.420 0.558 0.916 0.245 0.823 0.425 0.591 Zethalia 9432D NZ-8F Landguard zelandica 0.667 0.395 0.356 0.812 0.263 0.665 0.365 0.541 Zethalia 9432E NZ-8F Landguard zelandica 0.666 0.419 0.424 0.905 0.257 0.774 0.400 0.582 9433A NZ-8C Landguard Tawera spissa 0.693 0.363 0.757 0.840 0.257 0.567 0.416 0.814 9433B NZ-8C Landguard Tawera spissa 0.619 0.334 0.696 0.812 0.240 0.508 0.342 0.783 9433C NZ-8C Landguard Tawera spissa 0.616 0.332 0.609 0.793 0.246 0.531 0.352 0.821 9433D NZ-8C Landguard Tawera spissa 0.685 0.359 0.724 0.865 0.246 0.611 0.417 0.881 9433E NZ-8C Landguard Tawera spissa 0.608 0.329 0.680 0.823 0.245 0.546 0.382 0.771 Castlecliff Shakespeare Paphies 9434A NZ-7A Beach Cliff donacia 0.737 0.579 0.640 0.992 0.617 0.767 0.751 0.905 Castlecliff Shakespeare Paphies 9434B NZ-7A Beach Cliff donacia 0.697 0.536 0.644 1.030 0.591 0.724 0.696 0.935
268
Castlecliff Shakespeare Paphies 9434C NZ-7A Beach Cliff donacia 0.693 0.553 0.579 1.036 0.674 0.751 0.829 0.956 Castlecliff Shakespeare Paphies 9434D NZ-7A Beach Cliff donacia 0.715 0.562 0.383 1.024 0.605 0.745 0.708 0.812 Castlecliff Shakespeare Paphies 9434E NZ-7A Beach Cliff donacia 0.715 0.566 0.422 1.005 0.560 0.740 0.699 0.841 Castlecliff Shakespeare Ostrea 9435A(orE) NZ-7E Beach Cliff chilensis 0.741 0.487 0.227 1.083 0.486 0.636 0.743 0.901 Castlecliff Shakespeare Ostrea 9435A(orE) NZ-7E Beach Cliff chilensis 0.757 0.577 0.175 1.096 0.545 0.661 0.813 1.030 Castlecliff Shakespeare Ostrea 9435B NZ-7E Beach Cliff chilensis 0.736 0.507 0.169 1.114 0.499 0.606 0.670 0.895 Castlecliff Shakespeare Ostrea 9435C NZ-7E Beach Cliff chilensis 0.754 0.501 0.276 1.127 0.493 0.677 0.751 0.890 Castlecliff Shakespeare Ostrea 9435D NZ-7E Beach Cliff chilensis 0.722 0.514 0.201 1.076 0.500 0.655 0.691 0.842 Castlecliff Dosimia 9436A NZ-1A Beach Butlers greyi 0.800 0.754 0.506 1.093 0.858 0.971 1.112 1.252 Castlecliff Dosimia 9436B NZ-1A Beach Butlers greyi 0.825 0.815 0.238 1.091 0.894 1.006 1.102 1.189 Castlecliff Dosimia 9436C NZ-1A Beach Butlers greyi 0.782 0.751 0.359 1.092 0.887 0.947 1.189 1.263 Castlecliff Kaikokopu Cyclomaetra 9437A NZ-5A Beach (?) tristis 0.708 0.675 0.288 1.102 0.751 0.855 0.965 1.141 Castlecliff Kaikokopu Cyclomaetra 9437B NZ-5A Beach (?) tristis 0.774 0.718 0.365 1.114 0.767 0.874 0.974 1.149 Castlecliff Kaikokopu Cyclomaetra 9437C NZ-5A Beach (?) tristis 0.709 0.658 0.665 1.100 0.691 0.872 0.884 1.115 Castlecliff Kaikokopu Cyclomaetra 9437D NZ-5A Beach (?) tristis 0.737 0.705 0.394 1.108 0.766 0.868 0.951 1.110
269
Castlecliff Kaikokopu Cyclomaetra 9437E NZ-5A Beach (?) tristis 0.707 0.639 0.372 1.102 0.687 0.819 0.846 1.006 Castlecliff Kaikokopu Barnea 9438A NZ-5B Beach (?) similis 0.673 0.651 0.301 0.989 0.789 0.837 1.104 1.039 Castlecliff Kaikokopu Barnea 9438B NZ-5B Beach (?) similis 0.734 0.748 0.356 1.037 0.830 0.929 1.101 1.230 Castlecliff Kaikokopu Barnea 9438C NZ-5B Beach (?) similis 0.695 0.660 0.264 1.021 0.769 0.888 1.005 1.238 Castlecliff Kaikokopu Barnea 9438D NZ-5B Beach (?) similis 0.687 0.684 0.140 1.045 0.818 0.871 1.040 1.251 Castlecliff Kaikokopu Barnea 9438E NZ-5B Beach (?) similis 0.699 0.729 0.242 1.084 0.837 0.906 1.084 1.306 9439A NZ-8A Landguard Gari stangeri 0.652 0.409 0.602 0.693 0.312 0.602 0.459 0.657 9439B NZ-8A Landguard Gari stangeri 0.677 0.373 0.787 0.682 0.326 0.642 0.538 0.685 9439C NZ-8A Landguard Gari stangeri 0.593 0.341 0.614 0.813 0.249 0.608 0.347 0.579 9439D NZ-8A Landguard Gari stangeri 0.602 0.355 0.373 0.654 0.267 0.587 0.431 0.614 Pleuromeris 9440A NZ-8D Landguard zelandica 0.633 0.449 0.305 0.798 0.414 0.584 0.622 0.749 Pleuromeris 9440B NZ-8D Landguard zelandica 0.795 0.659 0.077 0.963 0.750 0.735 0.886 0.976 Pleuromeris 9440C NZ-8D Landguard zelandica 0.763 0.551 0.084 0.926 0.639 0.631 0.788 0.867 Pleuromeris 9440D NZ-8D Landguard zelandica x 0.576 0.363 0.033 0.797 0.532 0.369 0.542 0.509 Pleuromeris 9440E NZ-8D Landguard zelandica 0.754 0.561 0.199 0.868 0.648 0.610 0.758 0.849 Glycymeris 9441A NZ-8E Landguard modesta 0.734 0.522 0.360 0.853 0.425 0.645 0.557 0.710 Glycymeris 9441B NZ-8E Landguard modesta 0.696 0.538 0.343 0.854 0.436 0.530 0.551 0.698
270
Glycymeris 9441C NZ-8E Landguard modesta 0.835 0.560 0.316 0.872 0.447 0.698 0.547 0.742 Glycymeris 9441D NZ-8E Landguard modesta 0.625 0.393 0.498 0.647 0.270 0.577 0.359 0.503 Glycymeris 9441E NZ-8E Landguard modesta 0.936 0.391 0.570 0.670 0.259 0.595 0.366 0.509 9442A NZ-9C Ross Amalda spp. x 0.516 0.698 0.077 0.992 0.705 0.733 0.760 0.957 9442B NZ-9C Ross Amalda spp. 0.660 0.858 0.169 1.079 0.822 0.804 1.008 1.018 9442C NZ-9C Ross Amalda spp. 0.686 0.820 0.140 1.005 0.750 0.809 0.927 1.002 9442D NZ-9C Ross Amalda spp. 0.613 0.791 0.069 1.024 0.768 0.787 0.952 1.034 unidentified 9443A NZ-9B Ross clams 0.832 0.867 0.339 1.115 0.995 0.964 1.256 1.087 unidentified 9443B NZ-9B Ross clams 0.872 0.925 0.588 1.153 0.876 1.101 1.284 1.104 unidentified 9443C NZ-9B Ross clams 0.858 0.922 0.229 1.153 0.825 1.009 1.249 1.168 unidentified 9443D NZ-9B Ross clams 0.878 0.937 0.254 1.164 0.916 0.991 1.256 1.080 unidentified 9443E NZ-9B Ross clams 0.844 0.970 0.856 1.159 0.851 1.015 1.267 1.045 9444A NZ-9A Ross Patro undatus 0.714 0.988 0.164 1.141 1.037 1.099 1.309 1.211 9444B NZ-9A Ross Patro undatus 0.702 0.798 0.139 1.107 0.837 0.914 1.171 1.019 9444C NZ-9A Ross Patro undatus 0.708 0.863 0.164 1.111 0.963 0.933 1.165 1.116 9444D NZ-9A Ross Patro undatus 0.770 0.884 0.150 1.152 0.999 0.595 1.177 0.995 9444E NZ-9A Ross Patro undatus 0.685 0.819 0.109 1.095 0.823 0.890 1.172 1.101 9448A ADB-2B All Day Bay MIS 5e Tawera spissa 0.608 0.324 0.709 0.706 0.232 0.457 0.349 0.558 9448B ADB-2B All Day Bay MIS 5e Tawera spissa 0.624 0.309 0.679 0.711 0.216 0.477 0.368 0.573 9448C ADB-2B All Day Bay MIS 5e Tawera spissa 0.587 0.298 0.700 0.658 0.209 0.421 0.358 0.588 9448D ADB-2B All Day Bay MIS 5e Tawera spissa 0.615 0.310 0.708 0.667 0.227 0.452 0.318 0.653 9448E ADB-2B All Day Bay MIS 5e Tawera spissa 0.597 0.302 0.706 0.694 0.208 0.415 0.309 0.615
271
Paphies 9450A ADB-2A All Day Bay MIS 5e subtriangulata 0.465 0.250 0.552 0.516 0.197 0.336 0.243 0.439 Paphies 9450B ADB-2A All Day Bay MIS 5e subtriangulata 0.476 0.261 0.591 0.529 0.195 0.359 0.259 0.476 Paphies 9450C ADB-2A All Day Bay MIS 5e subtriangulata 0.741 0.667 0.526 1.073 0.762 0.819 0.935 1.088 Paphies 9450D ADB-2A All Day Bay MIS 5e subtriangulata 0.467 0.265 0.552 0.509 0.197 0.362 0.248 0.458 Paphies 9450E ADB-2A All Day Bay MIS 5e subtriangulata 0.454 0.264 0.536 0.531 0.202 0.350 0.285 0.439 Tiromoana Terrace Pholadidea 9452A MAC-3A Glenafric (Qt4) suteri 0.578 0.333 0.609 0.649 0.282 0.543 0.362 0.597 Tiromoana Terrace Pholadidea 9452C MAC-3A Glenafric (Qt4) suteri 0.573 0.347 0.378 0.652 0.291 0.578 0.402 0.681 Tiromoana Terrace Pholadidea 9452D(orB) MAC-3A Glenafric (Qt4) suteri 0.519 0.287 0.563 0.498 0.217 0.494 0.299 0.548 Tiromoana Terrace Pholadidea 9452D(orB) MAC-3A Glenafric (Qt4) suteri 0.584 0.349 0.554 0.610 0.286 0.581 0.390 0.730 Tiromoana Terrace Pholadidea 9452E MAC-3A Glenafric (Qt4) suteri 0.539 0.297 0.452 0.520 0.219 0.491 0.312 0.596 Tiromoana Terrace Irus (Notirus) 9453A MAC-3B Glenafric (Qt4) reflexus 0.607 0.347 0.668 0.710 0.236 0.549 0.313 0.490 Tiromoana Terrace Irus (Notirus) 9453B MAC-3B Glenafric (Qt4) reflexus 0.596 0.350 0.674 0.710 0.247 0.526 0.330 0.534
272
Tiromoana Terrace Barnea 9454A MAC-3E Glenafric (Qt4) similis 0.649 0.328 0.543 0.605 0.228 0.575 0.292 0.519 Tiromoana GA3 (MAC- Terrace 9455A 3A) Glenafric (Qt4) Tawera spissa 0.550 0.266 0.602 0.678 0.218 0.437 0.306 0.613 Tiromoana GA3 (MAC- Terrace 9455B 3A) Glenafric (Qt4) Tawera spissa 0.577 0.283 0.673 0.670 0.285 0.578 0.436 0.907 Tiromoana GA3 (MAC- Terrace 9455C 3A) Glenafric (Qt4) Tawera spissa 0.587 0.295 0.669 0.685 0.209 0.470 0.306 0.719 Tiromoana GA3 (MAC- Terrace 9455D 3A) Glenafric (Qt4) Tawera spissa 0.592 0.283 0.622 0.647 0.184 0.451 0.260 0.678 Tiromoana GA3 (MAC- Terrace 9455E 3A) Glenafric (Qt4) Tawera spissa 0.574 0.269 0.622 0.644 0.193 0.431 0.277 0.683 Tarapuhi HB3 (NZ13- Haumuri Terrace 10175 08) Bluff (Qt4) Tawera spissa 0.576 0.315 0.680 0.680 0.268 0.411 0.321 0.506 Torlesse HB4 (NZ13- Haumuri Terrace Leukoma 10174A 05) Bluff (Qt8) crassicosta 0.298 0.126 0.346 0.219 0.072 0.178 0.072 0.176 Torlesse HB4 (NZ13- Haumuri Terrace Leukoma 10174B 05) Bluff (Qt8) crassicosta 0.348 0.132 0.546 0.220 0.073 0.186 0.067 0.171 Torlesse HB4 (NZ13- Haumuri Terrace Leukoma 10174C 05) Bluff (Qt8) crassicosta 0.299 0.133 0.520 0.218 0.068 0.176 0.072 0.215
273
Torlesse HB4 (NZ13- Haumuri Terrace Leukoma 10174D 05) Bluff (Qt8) crassicosta 0.309 0.134 0.479 0.219 0.072 0.175 0.075 0.205 Torlesse HB4 (NZ13- Haumuri Terrace Leukoma 10174E 05) Bluff (Qt8) crassicosta 0.343 0.126 0.522 0.195 0.068 0.147 0.066 0.153 Tarapuhi HB3 (NZ13- Haumuri Terrace unknown 10176A 08) Bluff (Qt4) bivalve 0.570 0.300 0.672 0.668 0.211 0.391 0.225 0.440 Tarapuhi HB3 (NZ13- Haumuri Terrace unknown 10176B 08) Bluff (Qt4) bivalve 0.574 0.312 0.584 0.700 0.218 0.449 0.269 0.555 Tarapuhi HB3 (NZ13- Haumuri Terrace unknown 10176C 08) Bluff (Qt4) bivalve 0.531 0.309 0.624 0.702 0.223 0.387 0.246 0.488 Tarapuhi HB3 (NZ13- Haumuri Terrace unknown 10176D 08) Bluff (Qt4) bivalve 0.533 0.287 0.616 0.633 0.240 0.408 0.253 0.433 Tarapuhi HB3 (NZ13- Haumuri Terrace 10177B 09) Bluff (Qt4) Paphies sp. 0.549 0.214 0.619 0.467 0.233 0.428 0.213 0.330 Tarapuhi HB3 (NZ13- Haumuri Terrace 10177C 09) Bluff (Qt4) Paphies sp. 0.576 0.227 0.589 0.417 0.229 0.443 0.213 0.358 Tarapuhi HB3 (NZ13- Haumuri Terrace 10177D 09) Bluff (Qt4) Paphies sp. 0.583 0.269 0.662 0.601 0.293 0.481 0.319 0.401 Tarapuhi HB3 (NZ13- Haumuri Terrace 10177E 09) Bluff (Qt4) Paphies sp. x 0.469 0.169 0.575 0.426 0.189 0.323 0.156 0.249
274
Tarapuhi HB3 (NZ13- Haumuri Terrace 10177F 09) Bluff (Qt4) Paphies sp. 0.646 0.298 0.678 0.664 0.367 0.721 0.406 0.538 Tarapuhi HB3 (NZ13- Haumuri Terrace 10177G 09) Bluff (Qt4) Paphies sp. 0.612 0.279 0.682 0.677 0.378 0.582 0.447 0.483 Tarapuhi HB3 (NZ13- Haumuri Terrace 10178A 09) Bluff (Qt4) Tawera spissa 0.590 0.336 0.646 0.824 0.308 0.478 0.369 0.599 Tarapuhi HB3 (NZ13- Haumuri Terrace 10178B 09) Bluff (Qt4) Tawera spissa 0.568 0.312 0.635 0.715 0.276 0.398 0.310 0.458 Tarapuhi HB3 (NZ13- Haumuri Terrace 10178C 09) Bluff (Qt4) Tawera spissa 0.523 0.258 0.623 0.670 0.239 0.361 0.272 0.520 Tarapuhi HB3 (NZ13- Haumuri Terrace 10178D 09) Bluff (Qt4) Tawera spissa 0.563 0.291 0.673 0.631 0.234 0.353 0.267 0.452 Motunau MB4 Terrace, (Motunau Motunau Estuary 10928A Beach 10) Beach Sector (Qt7) T. spissa 0.518 0.265 0.698 0.650 0.220 0.563 0.260 0.369 Motunau MB4 Terrace, (Motunau Motunau Estuary 10928B Beach 10) Beach Sector (Qt7) T. spissa 0.500 0.230 0.655 0.557 0.193 0.427 0.214 0.337 Motunau MB4 Terrace, (Motunau Motunau Estuary 10928C Beach 10) Beach Sector (Qt7) T. spissa 0.504 0.247 0.673 0.559 0.198 0.345 0.246 0.354
275
Motunau MB4 Terrace, (Motunau Motunau Estuary 10928D Beach 10) Beach Sector (Qt7) T. spissa 0.485 0.237 0.662 0.594 0.222 0.476 0.240 0.429 Motunau MB4 Terrace, (Motunau Motunau Estuary 10928E Beach 10) Beach Sector (Qt7) T. spissa 0.494 0.231 0.667 0.573 0.206 0.366 0.256 0.347 Motunau MB3 Terrace, (Motunau Motunau Estuary 10929A Beach 7) Beach Sector (Qt7) T. spissa 0.507 0.232 0.658 0.616 0.217 0.490 0.264 0.475 Motunau MB3 Terrace, (Motunau Motunau Estuary 10929B Beach 7) Beach Sector (Qt7) T. spissa 0.490 0.226 0.654 0.651 0.224 0.615 0.296 0.390 Motunau MB3 Terrace, (Motunau Motunau Estuary 10929C Beach 7) Beach Sector (Qt7) T. spissa 0.478 0.237 0.541 0.644 0.219 0.348 0.289 0.457 Motunau MB3 Terrace, (Motunau Motunau Estuary 10929D Beach 7) Beach Sector (Qt7) T. spissa 0.523 0.249 0.665 0.686 0.226 0.375 0.275 0.469 Motunau MB3 Terrace, (Motunau Motunau Estuary 10929E Beach 7) Beach Sector (Qt7) T. spissa 0.479 0.227 0.648 0.628 0.217 0.354 0.281 0.381 MB5 Motunau (Motunau Motunau Terrace 10930A Beach 4) Beach (Qt6) T. spissa 0.509 0.254 0.561 0.609 0.229 0.452 0.251 0.515
276
MB6 Motunau (Motunau Motunau Terrace 10931A Beach 2) Beach (Qt6) T. spissa 0.569 0.286 0.599 0.662 0.251 0.469 0.297 0.581 MB6 Motunau (Motunau Motunau Terrace 10931B Beach 2) Beach (Qt6) T. spissa 0.545 0.268 0.594 0.662 0.238 0.506 0.276 0.553 MB6 Motunau (Motunau Motunau Terrace 10931C Beach 2) Beach (Qt6) T. spissa 0.524 0.254 0.575 0.604 0.220 0.409 0.267 0.440 MB6 Motunau (Motunau Motunau Terrace 10931D Beach 2) Beach (Qt6) T. spissa 0.547 0.269 0.627 0.654 0.264 0.463 0.318 0.486 MB6 Motunau (Motunau Motunau Terrace 10931E Beach 2) Beach (Qt6) T. spissa 0.556 0.277 0.590 0.660 0.250 0.432 0.279 0.524 MB5 Motunau (Motunau Motunau Terrace 10932A Beach 5) Beach (Qt6) T. spissa 0.494 0.242 0.545 0.592 0.230 0.384 0.260 0.470 MB5 Motunau (Motunau Motunau Terrace 10932B Beach 5) Beach (Qt6) T. spissa 0.564 0.279 0.633 0.577 0.233 0.570 0.278 0.446 MB5 Motunau (Motunau Motunau Terrace 10932C Beach 5) Beach (Qt6) T. spissa x 0.441 0.191 0.361 0.532 0.156 0.318 0.185 0.206 MB5 Motunau (Motunau Motunau Terrace 10932D Beach 5) Beach (Qt6) T. spissa 0.517 0.249 0.601 0.614 0.220 0.407 0.226 0.477 MB5 Motunau (Motunau Motunau Terrace 10932E Beach 5) Beach (Qt6) T. spissa 0.537 0.292 0.580 0.732 0.232 0.562 0.259 0.484 10933A Waverley Waverley MIS 5a T. spissa 0.520 0.287 0.690 0.675 0.265 0.348 0.320 0.389
277
10933B Waverley Waverley MIS 5a T. spissa 0.539 0.300 0.621 0.747 0.230 0.460 0.228 0.384 10933C Waverley Waverley MIS 5a T. spissa 0.548 0.316 0.630 0.681 0.257 0.433 0.287 0.391 10933D Waverley Waverley MIS 5a T. spissa 0.553 0.322 0.647 0.765 0.246 0.499 0.246 0.404 10933E Waverley Waverley MIS 5a T. spissa 0.528 0.305 0.707 0.726 0.247 0.468 0.272 0.477 MB5 Motunau (Motunau Motunau Terrace 10934A Beach 4) Beach (Qt6) unknown 0.562 0.294 0.564 0.671 0.212 0.458 0.245 0.484 MB5 Motunau (Motunau Motunau Terrace 10934B Beach 4) Beach (Qt6) unknown 0.520 0.312 0.531 0.635 0.219 0.481 0.291 0.544 MB5 Motunau (Motunau Motunau Terrace 10934C Beach 4) Beach (Qt6) unknown 0.524 0.291 0.553 0.699 0.210 0.705 0.230 0.430 MB5 Motunau (Motunau Motunau Terrace 10934D Beach 4) Beach (Qt6) unknown 0.500 0.282 0.515 0.652 0.213 0.487 0.266 0.463 MB5 Motunau (Motunau Motunau Terrace 10934E Beach 4) Beach (Qt6) unknown 0.515 0.298 0.528 0.651 0.205 0.569 0.282 0.484 10935A Waverley Waverley MIS 5a unknown 0.541 0.370 0.678 0.622 0.275 0.452 0.306 0.344 10935B Waverley Waverley MIS 5a unknown 0.509 0.368 0.560 0.541 0.243 0.422 0.254 0.384 10935C Waverley Waverley MIS 5a unknown 0.511 0.354 0.693 0.620 0.253 0.454 0.257 0.332 10935D Waverley Waverley MIS 5a unknown 0.543 0.386 0.670 0.618 0.261 0.504 0.249 0.376 10935E Waverley Waverley MIS 5a unknown 0.531 0.382 0.611 0.584 0.286 0.526 0.299 0.417 Torlesse Haumuri Terrace Leukoma 10936A HB5 (NZ14-1) Bluff (Qt8) crassicosta 0.299 0.139 0.471 0.248 0.105 0.244 0.070 0.179 Torlesse Haumuri Terrace Leukoma 10936B HB5 (NZ14-1) Bluff (Qt8) crassicosta 0.268 0.124 0.477 0.217 0.064 0.190 0.084 0.256
278
Torlesse Haumuri Terrace Leukoma 10936C HB5 (NZ14-1) Bluff (Qt8) crassicosta 0.300 0.135 0.415 0.218 0.089 0.211 0.073 0.205 Torlesse Haumuri Terrace Leukoma 10936D HB5 (NZ14-1) Bluff (Qt8) crassicosta 0.306 0.145 0.516 0.226 0.090 0.223 0.083 0.225 Torlesse Haumuri Terrace Leukoma 10936E HB5 (NZ14-1) Bluff (Qt8) crassicosta 0.271 0.126 0.393 0.247 0.080 0.176 0.067 0.151 Tiromoana GA5 (GPS1 Terrace 10937A point 7) Glenafric (Qt4) T. spissa? 0.608 0.297 0.655 0.677 0.259 0.497 0.295 0.617 Tiromoana GA5 (GPS1 Terrace 10937B point 7) Glenafric (Qt4) T. spissa? 0.495 0.249 0.579 0.648 0.232 0.551 0.301 0.531 Tiromoana GA5 (GPS1 Terrace 10937C point 7) Glenafric (Qt4) T. spissa? 0.530 0.246 0.599 0.626 0.242 0.431 0.272 0.472 Tiromoana GA5 (GPS1 Terrace 10937D point 7) Glenafric (Qt4) T. spissa? 0.521 0.260 0.603 0.642 0.225 0.436 0.296 0.466 Tiromoana GA5 (GPS1 Terrace 10937E point 7) Glenafric (Qt4) T. spissa? 0.557 0.275 0.654 0.697 0.273 0.399 0.284 0.488 Tiromoana KA4 (GPS1 Terrace 10938A point 3) Glenafric (Qt4) unknown 0.576 0.261 0.652 0.616 0.235 0.472 0.303 0.508 Tiromoana KA4 (GPS1 Terrace 10938B point 3) Glenafric (Qt4) unknown 0.602 0.288 0.666 0.675 0.262 0.510 0.294 0.473
279
Tiromoana KA4 (GPS1 Terrace 10938C point 3) Glenafric (Qt4) unknown 0.586 0.268 0.639 0.560 0.235 0.479 0.292 0.566 Tiromoana KA4 (GPS1 Terrace 10938D point 3) Glenafric (Qt4) unknown 0.566 0.257 0.647 0.590 0.224 0.485 0.275 0.416 Tiromoana KA4 (GPS1 Terrace 10938E point 3) Glenafric (Qt4) unknown 0.597 0.282 0.643 0.666 0.228 0.472 0.308 0.561 Cape Venerupis 13468A CK4 Kindnappers MIS 1 largillerti 0.219 0.103 0.364 0.144 0.034 0.094 0.039 0.083 Cape Venerupis 13468B CK4 Kindnappers MIS 1 largillerti 0.214 0.104 0.359 0.143 0.034 0.098 0.036 0.076 Cape Venerupis 13468C CK4 Kindnappers MIS 1 largillerti 0.240 0.108 0.413 0.159 0.036 0.103 0.040 0.078 Cape Venerupis 13468D CK4 Kindnappers MIS 1 largillerti 0.225 0.107 0.377 0.156 0.036 0.101 0.043 0.083 Cape Venerupis 13468E CK4 Kindnappers MIS 1 largillerti 0.193 0.096 0.305 0.118 0.028 0.087 0.036 0.067 Cape Venerupis 13468F CK4 Kindnappers MIS 1 largillerti 0.229 0.108 0.387 0.157 0.034 0.098 0.042 0.091
280
Table A-3: Input data for AAR Model
This is the data table used as input for the program. A copy of the Cape Kidnappers sample is included in the data of unknown age so that the lowest D/L value will be the same for both groups, which is necessary to overcome a problem that otherwise comes up when calculating R0 from ln.d.
Asp_D Glu_D Ser_D Ala_D Val_D Phe_D Ile_D Leu_D ageMedia specimen_no taxon L L L L L L L L n ageOld ageYng Kidnappers spissa 0.22 0.104 0.367 0.146 0.034 0.097 0.039 0.08 556.5 593 520 Waverley spissa 0.538 0.306 0.659 0.719 0.249 0.442 0.271 0.409 79000 87000 71000 ADB spissa 0.606 0.309 0.7 0.687 0.218 0.444 0.34 0.597 123500 130000 117000 Landguard spissa 0.644 0.343 0.693 0.827 0.247 0.552 0.382 0.814 217000 243000 191000 Shakespeare spissa 0.767 0.536 0.566 0.996 0.511 0.765 0.742 1.142 399000 424000 374000 Kupe spissa 0.848 0.639 0.586 1.073 0.592 0.863 0.947 1.329 694000 712000 676000 CK_copy spissa 0.22 0.104 0.367 0.146 0.034 0.097 0.039 0.08 NA NA NA NZ13-05 spissa 0.319 0.13 0.483 0.214 0.071 0.172 0.07 0.184 NA NA NA NZ14-1 spissa 0.289 0.134 0.454 0.231 0.086 0.209 0.075 0.203 NA NA NA NZ13-08-09 spissa 0.564 0.302 0.651 0.704 0.265 0.4 0.308 0.507 NA NA NA MB2 spissa 0.548 0.271 0.597 0.648 0.244 0.456 0.287 0.517 NA NA NA MB4-5 spissa 0.524 0.263 0.584 0.625 0.229 0.475 0.255 0.478 NA NA NA MB7 spissa 0.495 0.234 0.633 0.645 0.22 0.436 0.281 0.435 NA NA NA MB10 spissa 0.5 0.242 0.671 0.587 0.208 0.435 0.243 0.367 NA NA NA GPS1-3 spissa 0.585 0.271 0.649 0.621 0.237 0.484 0.294 0.505 NA NA NA GPS1-7 spissa 0.542 0.265 0.618 0.658 0.246 0.463 0.29 0.515 NA NA NA MAC-3C spissa 0.576 0.279 0.638 0.665 0.218 0.474 0.317 0.72 NA NA NA
281
Appendix B
GPS Data From Terrace Profiles
Table B-1: Haumuri Bluff GPS Survey
ID Longitude Latitude Height (m) Comment 1 173.508669516 -42.564559404 41.902 2 173.508401498 -42.564218460 42.167 3 173.508215004 -42.564009068 42.939 4 173.508010801 -42.563756344 43.704 5 173.507842197 -42.563546860 44.472 6 173.507670978 -42.563309755 44.658 7 173.507528956 -42.563123295 44.616 8 173.507356546 -42.562922239 44.911 9 173.507158870 -42.562660565 45.279 10 173.506927696 -42.562337910 45.052 11 173.506555223 -42.561787340 45.648 12 173.506354265 -42.561553841 45.763 13 173.506203671 -42.561385706 45.745 edge of a large gulley so break 14 173.505598181 -42.560676699 47.256 other side of gulley 15 173.505456458 -42.560501943 48.713 16 173.505348805 -42.560348974 51.033 fence 17 173.505274137 -42.560358427 51.417 other side of fence 18 173.505174698 -42.560248225 53.443 on likely alluvium at slope base 19 173.505085897 -42.560135693 55.925 20 173.504991530 -42.560030406 58.663 21 173.504953607 -42.559921072 61.619 22 173.504868706 -42.559790159 66.271 23 173.504778313 -42.559675649 72.034 24 173.504703662 -42.559572991 77.898 25 173.504644691 -42.559493431 82.798 26 173.504542133 -42.559398063 89.832 27 173.504488960 -42.559275114 97.110 28 173.504414401 -42.559213451 102.282 29 173.504383853 -42.559139760 108.182 30 173.504375843 -42.559081872 111.628 31 173.504320165 -42.559017956 115.508
282
32 173.504237108 -42.558936257 120.750 base of a small slump 33 173.503981881 -42.558812914 128.761 34 173.503857768 -42.558749036 135.572 35 173.503779691 -42.558646967 140.494 36 173.503644657 -42.558515120 145.757 37 173.503508465 -42.558418013 149.346 38 173.503439398 -42.558346783 151.348 39 173.503398350 -42.558248586 153.992 40 173.503317168 -42.558138491 158.755 41 173.503214886 -42.558007789 162.426 42 173.503097476 -42.557880989 165.004 43 173.502971705 -42.557678943 168.618 44 173.502917759 -42.557616429 169.451 45 173.502700553 -42.557480436 169.580 end 46 173.498775181 -42.550676592 176.943 47 173.498661473 -42.550663952 177.392 48 173.498533404 -42.550658079 178.021 49 173.498367638 -42.550641311 178.823 50 173.494717476 -42.552250990 181.790 51 173.494341038 -42.552528985 181.343 52 173.493868821 -42.552792895 180.017 53 173.493452915 -42.553058329 179.509 54 173.493196523 -42.553230066 181.543 55 173.492961976 -42.553358689 186.661 56 173.492766524 -42.553474037 191.445 57 173.492650970 -42.553524362 195.824 58 173.492488740 -42.553578597 203.777 59 173.492424114 -42.553607516 205.171 60 173.492366245 -42.553636444 204.724 61 173.492429049 -42.553644838 204.855 62 173.492497532 -42.553663514 202.810 63 173.492613303 -42.553677135 198.160 64 173.492742386 -42.553669662 194.226 65 173.492870543 -42.553675736 192.120 66 173.492995862 -42.553673401 190.709 67 173.493276194 -42.553635031 189.673 68 173.494480563 -42.553655338 186.064 69 173.495410750 -42.553206873 187.852 70 173.496169739 -42.552958939 187.662 71 173.496846467 -42.552795893 187.637 72 173.497180369 -42.556209415 162.780 73 173.497121988 -42.555919095 164.487 74 173.497035749 -42.555688916 165.643
283
75 173.496773337 -42.555410526 163.856 76 173.496697124 -42.555289484 163.645 77 173.497017042 -42.554929610 163.938 78 173.496892709 -42.554733904 164.538 79 173.496681478 -42.554138951 171.037 80 173.496615363 -42.553788005 173.470 81 173.496572627 -42.553655715 175.222 82 173.496483816 -42.553430445 181.375 83 173.496405053 -42.553230275 182.465 84 173.496200655 -42.552926006 187.652 85 173.499729363 -42.551374814 178.879 86 173.499985798 -42.551424609 179.676 87 173.500492594 -42.551555602 178.488 88 173.501158789 -42.551655634 177.046 89 173.501632487 -42.551781418 175.979 90 173.501694071 -42.552126420 175.856 91 173.501746145 -42.552244099 176.448 92 173.501926127 -42.552348160 178.207 93 173.502131653 -42.552458382 180.452 94 173.502404634 -42.552533989 180.407 95 173.505921059 -42.553570180 165.763 96 173.505918090 -42.553672396 171.739
Table B-2: Claverley GPS Survey
ID Longitude Latitude Height (m) Comment 1 173.482358648 -42.579934136 53.917 2 173.481961692 -42.579767901 55.260 3 173.481817634 -42.580234797 55.904 4 173.481337018 -42.579997322 57.655 5 173.480970327 -42.579739754 58.687 6 173.480964243 -42.579708004 58.515 7 173.480314493 -42.579375861 60.485 8 173.479731758 -42.578971510 62.326 9 173.479654903 -42.578984185 62.765 10 173.478876297 -42.578398941 65.245 11 173.478595192 -42.578161577 66.070 12 173.478312741 -42.578089212 66.998 gulley slope edge 13 173.477339701 -42.578570452 72.520 14 173.477058070 -42.578536836 73.713
284
15 173.476827984 -42.578504980 75.816 16 173.476507527 -42.578491283 78.306 17 173.476442059 -42.578487935 79.102 18 173.476237240 -42.578426744 83.706 19 173.476086636 -42.578363731 87.582 20 173.475967173 -42.578304866 91.506 21 173.475944210 -42.578293743 91.909 22 173.475836754 -42.578275806 93.558 23 173.475490438 -42.578233509 92.522 24 173.475085463 -42.578194765 96.585 25 173.474749010 -42.578093577 98.447 26 173.474358959 -42.577924727 99.797 27 173.473929591 -42.577624285 103.681 28 173.473404722 -42.577259430 106.048 29 173.473069955 -42.576981973 113.332 30 173.473100654 -42.576876674 114.459 31 173.473542198 -42.576700392 120.749 32 173.473844063 -42.576637569 121.600 33 173.474310704 -42.576499037 122.249 34 173.476776162 -42.577602793 72.678 35 173.476724455 -42.577571632 73.425 36 173.476640052 -42.577543519 74.590 37 173.476534013 -42.577512068 77.273 38 173.476436567 -42.577476866 80.298 39 173.476337337 -42.577396346 84.602 40 173.476136526 -42.577316497 94.695 41 173.476049097 -42.577224401 99.785 42 173.476002513 -42.577155804 101.340 43 173.475949234 -42.577068116 102.463 44 173.475852066 -42.576968998 105.531 45 173.475832151 -42.576933013 107.183 46 173.475706452 -42.576855168 109.075 47 173.475480459 -42.576746719 113.061 48 173.475403644 -42.576684280 113.933 49 173.475116755 -42.576503145 116.447 50 173.474790652 -42.576336259 120.193 51 173.474542599 -42.576169161 124.553 52 173.474390801 -42.576090682 127.107 53 173.474299410 -42.576041799 129.634 54 173.474244706 -42.576019541 131.161 55 173.474094656 -42.575950550 135.777 56 173.474009014 -42.575910536 138.791 57 173.473927786 -42.575855857 142.600
285
58 173.473884716 -42.575839142 145.762 59 173.473837655 -42.575796000 148.616 60 173.473757867 -42.575780773 152.824 61 173.473672679 -42.575754342 157.492 62 173.473619088 -42.575725060 161.419 broken up bedrock here 63 173.473546830 -42.575688359 164.574 64 173.473442176 -42.575653334 169.286 65 173.473329640 -42.575594897 173.725 66 173.473225015 -42.575531664 176.311 67 173.473059930 -42.575438427 178.669 68 173.472845477 -42.575320821 180.255 69 173.472543155 -42.575210629 182.227 70 173.472308979 -42.575101237 183.284 71 173.472119478 -42.575012446 184.227 72 173.471995728 -42.574906950 183.857 73 173.471874383 -42.574832792 182.468 74 173.482741831 -42.580453206 22.706 75 173.482791889 -42.580491646 18.616 76 173.482844219 -42.580533676 16.123 77 173.482948219 -42.580634208 12.934 78 173.483052135 -42.580691921 10.937 79 173.483218613 -42.580891516 9.559 80 173.483330511 -42.581028932 9.251
Table B-3: Dawn Creek Survey
GPS Vertical Horizontal Point_ID Longitude Latitude Height (m) Precision Precision Comment 1 173.42113173 -42.67929749 54.950 0.4 0.2 2 173.42075977 -42.67927495 56.641 0.3 0.1 3 173.42063206 -42.67926539 59.087 0.2 0.1 4 173.42031911 -42.67923449 66.746 0.3 0.1 5 173.42004556 -42.67929007 69.504 0.2 0.1 6 173.41978126 -42.67932621 68.643 2.9 1.6 7 173.41930440 -42.67928586 77.786 0.3 0.1 8 173.41915681 -42.67927136 78.302 0.4 0.4 edge of a little gulley 9 173.41793201 -42.67873709 83.445 0.7 0.5 other side of gulley 10 173.41737359 -42.67885205 83.319 1.3 0.4 11 173.41676558 -42.67895016 85.331 0.9 0.3 12 173.41623873 -42.67891164 85.981 0.8 0.1
286
13 173.41569712 -42.67890246 89.025 0.7 0.2 14 173.41523642 -42.67894464 89.395 0.3 0.1 15 173.41500818 -42.67896175 90.870 0.2 0.1 base of riser 16 173.41485464 -42.67896787 92.350 0.2 0.1 17 173.41467706 -42.67898449 94.297 0.4 0.2 18 173.41449021 -42.67896716 97.447 0.1 0.1 19 173.41428020 -42.67893984 102.084 0.2 0.2 20 173.41401208 -42.67892863 107.150 0.5 0.2 21 173.41383137 -42.67890751 109.007 0.1 0.1 22 173.41355940 -42.67892837 110.240 0.6 0.2 23 173.41315066 -42.67901141 112.684 0.6 0.3 24 173.41267892 -42.67899102 115.155 0.2 0.1 25 173.41204343 -42.67892331 117.883 0.1 0.1 26 173.41138105 -42.67895929 120.310 0.1 0.1 27 173.40990303 -42.67907192 121.728 0.1 0.1 28 173.40775311 -42.68006257 128.441 0.2 0.1 29 173.40659188 -42.68020801 127.834 0.4 0.2 30 173.40534474 -42.68047545 144.015 0.1 0.1 31 173.40460910 -42.68070386 148.125 0.1 0.1 32 173.40279806 -42.68088617 155.367 0.1 0.1 33 173.40188335 -42.68056499 162.718 0.1 0.1 34 173.40164054 -42.68046786 164.450 0.1 0.1 35 173.40122607 -42.68025045 165.439 0.1 0.1 36 173.40085318 -42.67987325 165.218 0.1 0.1 37 173.40149401 -42.67885784 163.817 0.1 0.1 38 173.40212124 -42.67833765 160.605 0.1 0.1 39 173.40224361 -42.67521503 166.945 0.1 0.1 40 173.40192780 -42.67528741 167.725 0.2 0.1 41 173.40163113 -42.67536768 174.603 0.2 0.2 42 173.40146382 -42.67542426 179.158 0.2 0.1 43 173.40126550 -42.67552822 184.160 0.2 0.1 44 173.40105760 -42.67568355 188.190 0.2 0.2 45 173.40086406 -42.67585157 189.040 0.2 0.2 46 173.40078579 -42.67596702 188.520 0.2 0.2 47 173.39974470 -42.67608975 194.567 0.1 0.1 48 173.39904438 -42.67683973 198.452 0.1 0.1 49 173.39857778 -42.67730614 197.356 0.1 0.1 50 173.39835054 -42.67730721 199.571 0.1 0.1 51 173.39814972 -42.67730952 202.504 0.1 0.1 52 173.39792193 -42.67729303 208.954 0.1 0.1 53 173.39779382 -42.67722796 214.494 0.1 0.1 54 173.39755700 -42.67721321 221.757 0.1 0.1 55 173.39733531 -42.67719784 228.108 0.1 0.1
287
56 173.39720240 -42.67715690 231.227 0.1 0.1 57 173.39714733 -42.67715837 231.905 0.1 0.1 58 173.39707578 -42.67711330 232.965 0.1 0.1 59 173.39696724 -42.67710383 233.536 0.1 0.1 end
Kate Anticline GPS Surveys were conducted by Mary Kate Stewart, an undergraduate student at
Trinity University, and Tom Gardner. They were included in Stewart’s senior thesis and are reproduced here.
Table B-4: Kate GPS Survey 1
Point Elevation Vertical Horizontal ID Longitude Latitude (m) Error (m) Error (m) Comment 1 172.7997 -43.138 58.958 0.1 0.1 tread possibly some sort of 2 172.7991 -43.1378 50.739 0.1 0.1 unconformity 3 172.8107 -43.1321 31.803 0.2 0.2 unconformity marine sed thickness top 4 172.8111 -43.1321 40.996 0.7 0.4 tread 5 172.8107 -43.1322 32.068 0.1 0.1 retakee 6 172.8155 -43.1322 2.67 0.8 0.6 holocene inner edge 7 172.8154 -43.1319 22.737 0.6 1.2 riser 8 172.8152 -43.1319 32.643 0.5 0.6 unconformity 9 172.815 -43.1313 45.44 0.9 0.4 tread thickness 10 172.8151 -43.1309 49.968 0.6 0.5 inner edge 11 172.8146 -43.13 71.124 12 6.3 yousif m2? 12 172.8151 -43.1296 84.815 11.3 6.3 unconformity 13 172.8154 -43.1294 92.51 10 5.9 top tread thickness 14 172.8154 -43.1283 84.184 10.6 5.9 unconformity 15 172.8157 -43.1273 89.39 13.9 6.5 unconformity 16 172.8159 -43.1267 104.806 13.2 6 tread thickness of m4? 17 172.816 -43.1256 123.175 9.8 6 M4? inner edge 18 172.8161 -43.1243 145.346 9.6 5.9 highest tread 19 172.8119 -43.1309 42.029 8.9 6 unconformity m1? river terrace 20 172.8047 -43.1283 48.868 10.6 8.3 unconformity 21 172.7929 -43.1256 102.884 8.8 6 unconformity marine
288
Table B-5: Kate GPS Survey 3
Elevation Vertical Horizontal Point ID Latitude Longitude (m) Error (m) Error(m) Comment 1 172.8591 -43.118 40.271 0.4 0.2 bedrock unconformity 2 172.8587 -43.1183 62.909 0.5 0.2 thickness of marine cover mis5 3 172.8584 -43.1192 44.019 0.5 0.3 4 172.8579 -43.1197 41.449 0.6 0.2 5 172.8581 -43.1195 57.148 0.3 0.1 ht of marine sed tread 6 172.8576 -43.1165 53.782 1.2 0.6 unconformity 7 172.8596 -43.1133 89.291 0.2 0.1 8 172.8645 -43.1071 68.862 0.7 0.9
Table B-6: Kate GPS Survey 4
Point Elevation Vertical Horizontal ID Longitude Latitude (m) Error (m) Error (m) 1 172.8807 -43.1018 56.579 3.2 1.8 2 172.8803 -43.1016 71.587 3.5 2 3 172.8794 -43.1012 58.696 4.7 2.7 4 172.8796 -43.1008 73.956 3.9 1.5 5 172.8788 -43.0999 74.741 2.9 1.3 6 172.878 -43.0996 53.917 8.8 3 7 172.8763 -43.0991 61.46 4.5 2.7 8 172.8767 -43.0982 76.99 3.1 2.2 9 172.8791 -43.0957 84.727 4.1 2.9 10 172.8795 -43.0935 92.17 5 3.1 11 172.8793 -43.093 102.752 4.8 3.7 12 172.879 -43.0925 119.42 6.9 4.5 13 172.878 -43.0918 135.626 5 2.8 14 172.8774 -43.0911 151.133 1.8 1.1 15 172.8771 -43.0905 171.518 2.5 1.6 16 172.8763 -43.0901 193.919 2.5 1.6
289
Appendix C
A New, Mostly Analytic Trishear Solution
Points
The typical way to model trishear deformation is to move points along a bed using a velocity field. Since the velocity is a function of position, one has to move each point a small increment, recalculate the velocity, move another small increment, and so on. This is typically alternated with propagating the fault tip.
The trishear velocity field of Zehnder and Allmendinger (2000) is:
1 s v0 |y| v = [sgn(y) ( ) + 1] x 2 mx
(1+s) (1) s v0m |y| v = [( ) − 1] y 2(1 + s) mx
−xm ≤ y ≤ xm, s ≥ 1 x and y are in a coordinate system with its origin at the fault tip and its x-axis aligned with the fault. V0 is the rate of slip on the fault, and P/S is the propagation to slip ratio. So the fault tip is propagating at a rate of V0(P/S). Since the coordinate system is moving with the fault tip, the full rate of change of x and y are:
1 s dx v0 |y| 푃 = [sgn(y) ( ) + 1] − 푣 dt 2 mx 0 푆 (2) (1+s) s dy v0m |y| = [( ) − 1] dt 2(1 + s) mx
The term s is a concentration factor, which concentrates deformation in the center of the trishear field. Almost all papers that have used this velocity field have used s = 1, and this is supported
290 by physical modeling (Cardozo et al., 2003). So far, for what follows, we assume that s = 1. It may be possible to derive something similar even if it doesn’t, but we haven’t tried that yet.
Given s = 1, equation 2 becomes:
dx v0 y 푃 = [ + 1] − 푣 dt 2 mx 0 푆 (3)
dydt=v0m4ymx2−1
We can rewrite these in polar coordinates (r,θ) as
dx v0 tan 휃 푃 = [ + 1] − 푣 dt 2 m 0 푆 (4) 2 dy v0m tan 휃 = [( ) − 1] dt 4 m
To calculate the rates of change of the polar coordinates with time, we use the formulas:
dr dx 푑푦 = cos 휃 + sin 휃 dt dt 푑푡 (5) dθ 1 푑푦 푑푥 = ( ) ( cos 휃 − 푠푖푛휃) dt r 푑푡 푑푡
Where dx/dt and dy/dt are as given in equation 4. Combining the two equations in 5, we get:
푑푥 푑푦 dr cos 휃 + sin 휃 = 푟 (푑푡 푑푡 ) (6) dθ 푑푦 푑푥 푐표푠휃 − sin 휃 푑푡 푑푡
To avoid having to deal with so many trigonometric functions, we can do a change of variables to u = tan(θ). This gives us:
푑푥 푑푦 푑푟 푑푟 dθ 푟 + 푢 = = (푑푡 푑푡 ) (7) 푑푢 푑휃 du 2 푑푦 푑푥 √1 + 푢 − 푢 푑푡 푑푡 We can also rephrase equation 4 in terms of u:
291
dx v0 u 푃 = [ + 1] − 푣 dt 2 m 0 푆 (8) 2 dy v0m 푢 = [( ) − 1] dt 4 m
Substituting equation 8 into equation 7 and simplifying eventually gives us:
3 2 푃 푑푟 푢 + (2 − 푚 )푢 + 푚 (2 − 4 ) = −푟 ( 푆 ) (9) 푃 푃 푑푢 푢4 + 푚 (2 − 4 ) 푢3 + (푚2 + 1)푢2 + 푚 (2 − 4 ) 푢 − 푚2 푆 푆
Separating variables and integrating both sides (integration was done in Mathematica, so the details aren’t shown) gives us:
1 푃 (10) ln(푟) = ln(1 + 푢2) − ln (푚2 + 2푚푢 − 4푚 푢 + 푢2) + ln (푐 ) 2 푆 1
Where c1 is a constant of integration. Equation 10 simplifies to:
푐 √1 + 푢2 푟 = 1 (11) 푃 푚2 + 2푚 (1 − 2 ) 푢 + 푢2 푆 where
2 푃 2 푟0 (푚 + 2푚 (1 − 2 ) 푢0 + 푢0) (12) 푐 = 푆 1 2 √1 + 푢0
And u0 and r0 are the original position of the point.
We can now go back to our dθ/dt equation from (5) and convert it to du/dt.
du √u2 + 1 푑푦 푑푥 (13) = ( ) ( − 푢) dt 푟 푑푡 푑푡
We can then substitute equation 11 for r and equation 8 for dx/dt and dy/dt.
2 du 푣0 푃 (14) = − (푢2 + 2푚 (1 − 2 ) 푢 + 푚2) dt 4푚푐1 푆
Since we seldom actually know time (t) but instead know slip (S = v0t) we can change this to:
292
2 du 1 푃 (15) = − (푢2 + 2푚 (1 − 2 ) 푢 + 푚2) dS 4푚푐1 푆
We can solve this for S as a function of u by integration:
1 S = −4푚푐 ∫ 푑푢 (16) 1 푃 2 (푢2 + 2푚 (1 − 2 ) 푢 + 푚2) 푆
This is an integral of a form:
1 ∫ 푑푢 (17) (푎푢2 + 푏푢 + 푐)2
Where
푎 = 1
푃 (18) 푏 = 2푚 (1 − 2 ) 푆
푐 = 푚2
Again, this integral can be solved. In this case, the solution (Gradshteyn and Ryzhik, 1980) is:
2푎푢 + 푏 2푎 1 푆 = −4푚푐 [ + ∫ 푑푢] + 푐 (19) 1 (4푎푐 − 푏2)(푎푢2 + 푏푢 + 푐) 4푎푐 − 푏2 푎푢2 + 푏푢 + 푐 2 and
1 ∫ 푑푢 푎푢2 + 푏푢 + 푐
2 −1 2푎푢 + 푏 tan 2 √4푎푐 − 푏2 √4푎푐 − 푏2 푓표푟 4푎푐 − 푏 > 0 (20)
1 2푎푢 + 푏 − √푏2 − 4푎푐 = ln | | √푏2 − 4푎푐 2푎푢 + 푏 + √푏2 − 4푎푐 푓표푟 4푎푐 − 푏2 < 0 2 − 2 { 푎푥 + 푏 푓표푟 4푎푐 − 푏 = 0 and c2 can be calculated by solving equations 19 and 20 when S = 0 and u = u0.
Thus equations 19 and 20 allow us to calculate the slip necessary to move a point initially at (u0, r0) to some final u. If we know the final u we want (such as u = m, to calculate the slip necessary for a point to leave the trishear zone), then we can use these equations to calculate the necessary
293 slip. Alternately, if we want to know where a point ends up after a given amount of slip, we can solve equations 19 and 20 for u, using a root-finding method such as Newton’s method. In that case, this is still not an analytic solution for u, but it is typically a lot faster than slipping an increment at a time. In either case, once we have solved for u, we can use equation 11 to solve for r from u.
As an aside, 4ac-b2 is > 0 if P/S < 1 and < 0 if P/S > 1. This provides a mathematical explanation for the fact that trishear results often have a P/S > 1 and and a P/S < 1 solution.
Dips
Building off this result, one can derive an equation for final dip, as a function of initial dip, u, r, u0, and r0. It is rather complicated but does give an analytic solution, once u has been solved for.
The tangent of the initial dip (δ0) is simply the slope of the line in the (x,y) coordinate system:
dy (21) tan(훿 ) = 0 dx The derivative dy/dx can be converted into (r,u) coordinates using the chain rule.
2 dr0 푟0푢0 u0 + 푟0 − 2 du0 푢0 + 1 (22) tan(훿0) = 푑푟0 푟0푢0 − 2 푑푢0 푢0 + 1
This can be rearranged to solve for dr0/du0.
dr0 푟0 푟0푢0 (23) = + 2 du0 tan(훿0) − 푢0 푢0 + 1
To calculate the final dip, we will need to solve for dr/du, from which we can calculate dy/dx and then dip. We calculate dr/du by taking the derivative of Equation 11.
294
dr dc √1 + 푢2 푐 푢 = 1 + 1 푃 푃 du du 푚2 + 2푚 (1 − 2 ) 푢 + 푢2 1 + 푢2 (푚2 + 2푚 (1 − 2 ) 푢 + 푢2) 푆 √ 푆 (24) 푃 푐 1 + 푢2 2푢 + 2푚 (1 − 2 ) 1√ ( 푆 ) − 푃 2 (푚2 + 2푚 (1 − 2 ) 푢 + 푢2) 푆
It is important to note that Equation 24 is not the same as Equation 9, even though both are for dr/du. Equation 9 gives the rate of change of r with u for a given point as slip changes. In that case, r0 and u0 are constant, so dc1/du is 0. In Equation 24, we are interested in the rate of change of r with u along a folded line, with slip constant but r0 and u0 changing as r and u change. In this case, dc1/du is not zero. We can calculate dc1/du using the chain rule.
푑푐 푑푐 푑푢 푑푆 1 = 1 0 (25) 푑푢 푑푢0 푑푆 푑푢
dS/du is the reciprocal of Equation 15. dc1/du0 can be calculated by taking the derivative of
Equation 12.
2 푃 2 푃 푑푐 푑푟 (푚 + 2푚 (1 − 2 ) 푢0 + 푢0) 푟0 (2푚 (1 − 2 ) + 2푢0) 1 = 0 푆 + 푆 푑푢 푑푢 2 2 0 0 √1 + 푢0 √1 + 푢0 (26) 푃 푢 푟 (푚2 + 2푚 (1 − 2 ) 푢 + 푢2) 0 0 푆 0 0 − 3 2 2 (1 + 푢0)
where dr0/du0 is as given in Equation 23. Finally, du0/dS is the reciprocal of dS/du0, which can be calculated from Equation 19, in which c1 and c2 are functions of u0.
−1 −1 푑푢0 푑푆 푆 푑푐1 푑푆 (27) = ( ) = − ( − | ) 푑푆 푑푢0 푐1 푑푢0 푑푢 푢=푢0
where dc1/du0 and dS/du are given by Equation 26 and the reciprocal of Equation 15, respectively.
295
Equations 27, 26, and 15 can be substituted in to Equation 25, which can then be substituted into Equation
24 to solve for dr/du. The final dip (δ) can be calculated from dr/du by the inverse of the procedure used to derive Equation 23 from Equation 21. The final equation is:
푑푟 푟푢2 푢 + 푟 − 푦 푑푢 2 훿 = tan−1 ( ) = tan−1 ( 푢 + 1) (28) 푑푟 푟푢 푥 − 푑푢 푢2 + 1
Error Propagation
From the equations above, we can derive equations to propagate uncertainty in positions and dips.
For the position of a point, with an initial covariance matrix Σ0 and final covariance matrix Σ, we use the standard equation:
푇 (29) 횺 = 푱횺ퟎ푱
where J is the Jacobian matrix for the functions relating r and u to r0 and u0. Σ and Σ0 are for r and u (or r0 and u0) can be derived from or transformed into the corresponding equations for x and y as needed. To calculate J, we need to calculate the derivatives of r and u with respect to r0 and u0:
푑푢 푑푢
푑푢 푑푟 푱 = 0 0 (30) 푑푟 푑푟
[푑푢0 푑푟0]
The various derivatives can be calculated by application of the chain rule. We calculate du/du0 as
푑푢 푑푢 푑푆 (31) = 푑푢0 푑푆 푑푢0
where du/dS is given by Equation 15 and dS/du0 is the reciprocal of Equation 27. We can similarly calculate du/dr0 using the chain rule:
296
푑푢 푑푢 푑푆 푑푐1 = (32) 푑푟0 푑푆 푑푐1 푑푟0
We know du/dS from Equation 15. dS/dc1 can be calculated from Equation 19 and is just
푑푆 푆 (33) = 푑푐1 푐1
Likewise, dc1/dr0 can be calculated from Equation 12 and is
푑푐 푐 (34) 1 = 1 푑푟0 푟0
Substituting Equations 15, 33, and 34 into Equation 32 gives
푑푢 푆 (35) = 푑푟0 푟0
With du/du0 and du/dr0, we can calculated dr/du0 and dr/dr0:
푑푟 푑푟 푑푢 (36) = 푑푢0 푑푢 푑푢0 and
푑푟 푑푟 푑푢 (37) = 푑푟0 푑푢 푑푟0
We have already solved for dr/du in Equation 24, and du/du0 and du/dr0 are given by Equations 31 and 35, respectively. Therefore, we can solve for the remaining two derivatives. Substituting Equations 31, 35,
36, and 37 into Equation 30 provides the Jacobian matrix, which can be used to propagate uncertainty in the position of a point.
297
To propagate the uncertainty in a dip measurement, we need to calculate the derivative dδ/dδ0. If there is uncertainty in the position of the dip measurement, the covariance between position and dip will also have to be considered, but here we follow Oakley and Fisher (2015) in treating uncertainty in dip independently of position. Taking the derivative of Equation 28, we find
푑훿 1 −푟 푑 푑푟 (38) = ( ) ( ) ( ) 푑훿 1 + tan2 훿 푑푟 푟푢 2 푑훿 푑푢 0 ( − ) 0 푑푢 푢2 + 1
This requires that we solve for the derivative of dr/du with respect to δ0. dr/du is given by Equation 24, so this is
푑 푑푟 √1 + 푢2 푑 푑푐 (39) ( ) = ( 1) 푃 푑훿0 푑푢 푚2 + 2푚 (1 − 2 ) 푢 + 푢2 푑훿0 푑푢 푆
This in turn requires that we calculate the derivative of dc1/du with respect to δ0. We can expand this using the chain rule to
푑 푑푐 푑 푑푐 푑푢 푑푆 푑 푑푐 푑푢 푑푆 푑푐 푑 푑푢 푑푆 (40) ( 1) = ( 1 0 ) = ( 1 ) 0 + 1 ( 0) 푑훿0 푑푢 푑훿0 푑푢0 푑푆 푑푢 푑훿0 푑푢0 푑푆 푑푢 푑푢0 푑훿0 푑푆 푑푢
The two derivatives in this can be calculated from Equations 26 and 27:
푑 푑푢 푆 푑푐 푑푆 −2 푆 푑 푑푐 (41) ( 0) = − ( 1 − | ) ( ( 1 )) 푑훿0 푑푆 푐1 푑푢0 푑푢 푢=푢0 푐1 푑훿0 푑푢0 and
2 푃 2 푑 푑푐 푚 + 2푚 (1 − 2 ) 푢 + 푢 푑 푑푟 (42) ( 1 ) = 푆 ( 0 ) 푑훿 푑푢 2 푑훿 푑푢 0 0 √1 + 푢0 0 0 where, taking the derivative of Equation 23,
298
2 푑 푑푟0 푟0 sec 훿0 (43) ( ) = − 2 푑훿0 푑푢0 (tan 훿0 − 푢0)
By solving (in reverse order) Equations 38 through 43, one can calculate dδ/dδ0, which can be used to propagate uncertainty in bedding dip through the process of restoration.
References
Cardozo, N., Bhalla, K., Zehnder, A.T., Allmendinger, R.W., 2003. Mechanical models of fault
propagation folds and comparison to the trishear kinematic model. Journal of Structural Geology
25, 1-18.
Gradshteyn, I.S. and Ryzhik, I.M., 1980. Table of Integrals, Series, and Products: Corrected and Enlarged
Edition. New York: Academic Press, 1160 p.
Oakley and Fisher, 2015. Inverse trishear modeling of bedding dip data using Markov chain Monte Carlo
methods. Journal of Structural Geology 80, 157-172.
Zehnder, A.T., Allmendinger, R.W., 2000. Velocity field for the trishear model. Journal of Structural
Geology 22, 1009-1014.
299
Appendix D
Structural Data
This table lists strike and dip measurements taken during the course of field work in North Canterbury, New Zealand. Latitude and longitude use the WGS84 datum. Altitude and Position Uncertainty are as given by the GPS used to measure position. 99 indicates that this value was not recorded. Strike and dip follow the right hand rule. Quality is a subjective rating of the quality of the measurement, where 1 is poor and 3 is good. 99 indicates that a quality rating was not recorded.
Position Latitude Longitude Altitude Uncertainty Strike Dip Quality -43.08203 172.7911 124 99 196 76 99 -43.08226 172.7915 125 99 345 20 99 -43.08196 172.79092 120 4 223 38 99 -42.92569 173.2666 12 6 176 21 1 -42.92667 173.25372 15 99 240 22 1 -42.97826 173.17331 -5 10 268 46.5 1.5 -42.97832 173.17379 -5 8 240 36 3 -42.9785 173.17567 -2 8 242 41 3 -42.97764 173.1812 -1 8 35 4 1 -42.97831 173.17327 9 11 255 25.5 2.5 -42.97796 173.17164 11 11 307 0.5 99 -42.67872 173.43214 8 99 253 18 2 -42.51984 173.50473 19 8 185 25 2.5 -42.52429 173.50407 18 6 181 19 99 -42.52578 173.50228 13 7 224 25 99 -42.53317 173.50087 22 99 250 38 99 -42.55013 173.50463 3 99 86 35 99 -42.55013 173.50463 3 99 49 35 99 -42.55013 173.50463 3 99 46 32.5 2.5 -42.55051 173.50496 6 7 55 26 2.5 -42.5749 173.49269 22 99 10 52 2 -42.57276 173.49391 26 7 11 63 2 -42.5695 173.49582 30 4 25 66 3 -42.5682 173.4972 33 7 28 66 3 -42.56778 173.49783 34 7 17 51 2.5
300
-42.56431 173.4995 16 6 39 72 99 -42.56447 173.50083 15 9 38 47 99 -42.56447 173.50083 15 9 205 29 99 -43.12584 172.77915 50 12 75 21 2 -43.12584 172.77915 50 12 56 14.5 1.5 -43.12584 172.77915 50 12 12 20 99 -43.08846 172.8009 185 4 341 31 1.5 -43.0888 172.80133 167 6 172 63 1 -43.03115 172.90046 181 99 95 4 1 -43.09726 172.89153 3 17 260 22.5 1.5 -43.09725 172.89341 3 6 358 30.5 1 -43.09724 172.89376 3 8 81 24 2.5 -43.0988 172.90162 8 6 1 16 1 -43.09854 172.90263 6 4 160 14 2.5 -43.10222 172.88157 7 9 279 2.5 2.5 -43.09566 172.88956 6 4 278 14 99 -42.75262 173.31039 25 8 211 10 1 -42.75262 173.31039 25 99 190 44.5 1 -42.72489 173.3195 113 7 277 24 1 -42.72896 173.32051 72 5 33 49 2.5 -42.72896 173.32051 72 5 9 32 2 -42.72896 173.32051 72 5 46 58 2 -42.73236 173.3246 58 4 58 20 1 -42.73403 173.32897 105 2 99 85.5 1.5 -42.73486 173.33149 141 5 247 43 99 -42.72924 173.32096 47 6 27 28 2 -42.72924 173.32096 47 6 16 21 2 -42.72945 173.32077 52 5 333 17 1.5 -42.73126 173.32005 36 9 165 24 2.5 -42.73118 173.32019 40 9 142 25 1.5 -42.72871 173.32077 69 5 209 25 1 -42.7277 173.32099 94 4 80 23 2.5 -42.7277 173.32099 94 4 89 25 2 -42.7272 173.32105 110 4 141 16 1.5 -42.72528 173.32033 118 6 300 47 0.5 -42.80467 173.30536 122 8 166 44 1.5 -42.84091 173.3217 8 7 205 34 99 -42.8451 173.31561 36 6 135 27 1.5 -42.8451 173.31561 36 6 84 36 1.5 -42.84458 173.31531 32 7 68 21 1.5 -42.86711 173.30814 7 7 218 16 99 -42.86775 173.30831 7 7 214 21 2.5 -42.86938 173.3097 18 12 137 3 1.5
301
-42.86943 173.30966 9 13 102 10 2 -42.86946 173.30959 2 16 245 26 1 -42.86905 173.30982 0 20 181 21 2 -42.8691 173.30979 6 18 185 25 1.5 -42.86882 173.31026 9 32 206 26 99 -42.8693 173.3094 16 7 249 30 2 -42.86945 173.30994 4 5 285 18 3 -42.86939 173.30995 4 4 243 25 2.5 -42.86932 173.30988 3 3 225 31 3 -42.86932 173.30984 5 5 229 58 2 -42.86931 173.30981 4 4 231 33.5 2.5 -42.8692 173.30966 4 5 240 29 99 -42.86923 173.30961 5 4 224 26 99 -42.8689 173.30948 4 4 227 23.5 99 -42.86885 173.30944 3 5 235 23 99 -42.86889 173.30926 4 5 227 24 99 -42.86824 173.30905 3 5 232 27 3 -42.86822 173.30822 17 7 235 25.5 99 -42.85283 173.31108 21 7 78 13 2.5 -42.85283 173.31106 22 5 60 14 2 -42.85282 173.31115 21 7 50 16 99 -42.85277 173.31118 22 7 80 13 99 -42.85273 173.31088 28 8 67 11 1.5 -42.85258 173.3114 23 6 92 20 2.5 -42.85264 173.31151 24 6 74 16 99 -42.85262 173.31161 24 6 87 27 99 -42.85281 173.31175 24 6 60 13 3 -42.85277 173.31177 22 7 91 12 2 -42.85077 173.31072 18 5 56 37 1.5 -42.85033 173.30939 22 5 169 21 1 -42.85057 173.30852 26 7 56 31 1 -42.85076 173.30729 30 6 78 34 1.5 -42.85096 173.30801 41 5 25 15 99 -42.85082 173.30694 26 5 84 32 1.5 -42.85133 173.30321 38 7 76 22 0.5 -42.85147 173.30188 39 9 83 16 1.5 -43.09777 172.88975 4 14 247 12 99 -43.03218 173.02713 79 7 48 33 99 -43.03214 173.02715 80 6 50 20 99 -42.55808 173.50022 44 11 44 34 99 -42.55722 173.50051 29 13 180 54 99 -42.56433 173.49956 2 9 34 77 99 -42.56771 173.49761 11 12 8 57 99
302
-42.56948 173.49572 4 10 23 67 99 -43.00046 172.71455 78 37 70 31 1.5 -42.94911 173.24499 -28 11 151 22 1 -42.94934 173.24481 21 23 64 17 2 -42.94939 173.24473 15 18 70 18 99 -42.94947 173.24473 5 16 63 18 99 -42.93323 173.26016 15 17 274 12 99 -42.91002 173.28342 -5 16 136 12 99 -42.91044 173.28269 0 12 33 26 99 -42.56427 173.50536 4 8 50 19 1.5 -42.56421 173.50381 2 9 50 21 99 -42.56427 173.50358 4 11 30 22 99 -42.62325 173.33786 56 7 199 47 1 -42.62363 173.33777 76 9 223 52 99 -42.62487 173.33742 84 16 218 51 99 -42.64697 173.45032 -3 9 81 7 99 -42.65887 173.44265 7 7 354 14 99 -42.52007 173.50477 14 10 153 28 99 -42.52583 173.50228 15 7 225 31 99 -42.87973 173.31554 -1 13 70 21 99 -42.87504 173.31063 -6 6 203 21 99 -42.87243 173.30998 1 5 235 10 99 -42.872 173.31002 3 15 207 15 99 -42.87129 173.31053 3 25 221 20 99 -42.55363 173.50746 155 9 27 25 99 -42.55388 173.50783 158 11 54 11 1 -42.55355 173.50748 132 12 19 39 2.5 -42.51775 173.48217 49 30 50 73 99 -42.51777 173.48219 49 30 236 86 99 -42.62118 173.32354 97 8 122 21 99 -42.75261 173.31093 33 26 237 41 99 -42.86241 173.3056 53 8 42 22 1 -42.8624 173.30557 42 6 14 18 1.5 -42.8625 173.30534 37 8 20 22 1 -42.87514 173.30483 96 19 226 9 1 -43.049 173.07231 8 9 215 24 1.5 -43.04594 173.07011 47 10 53 11 1.5 -42.98241 173.08332 158 7 250 24 1 -42.88111 173.30004 145 13 167 23 2 -42.8737 173.30957 11 7 320 29 1 -42.87345 173.30978 10 5 243 26 2 -42.87157 173.31023 7 8 235 19 99 -42.8759 173.3111 -2 10 110 31 1
303
-43.04789 173.06159 7 11 90 12 2 -43.00889 173.09222 14 43 299 8 1 -43.03605 173.0313 124 4 63 22 3 -43.03622 173.03177 127 4 75 25 2.5 -43.03432 173.02797 95 5 74 17 1.5 -43.04771 173.07962 79 27 80 13 3 -43.04794 173.07968 84 28 71 16 3 -43.05079 173.08048 15 6 55 16 2.5 -43.05046 173.08082 13 7 75 10 2 -43.04895 173.08327 3 22 6 11 1.5 -43.04851 173.08428 4 8 152 20 1.5 -43.0483 173.08403 0 7 164 7 2 -43.04771 173.08509 9 8 66 21 2.5 -43.0476 173.08516 3 9 85 9 2 -43.04729 173.08499 -2 10 46 22 2 -43.04668 173.0847 -8 7 14 13 2 -42.58015 173.48962 9 5 252 35 99 -42.56 173.50913 2 7 62 21 99 -42.56098 173.50933 -2 6 54 10 99 -42.56413 173.50048 44 5 53 53 1 -42.94833 173.2425 67 15 201 10 1 -42.94831 173.24067 104 22 92 25 1 -42.94809 173.24069 102 9 88 24 1.5 -42.94791 173.24079 95 9 58 10 1.5 -42.94705 173.23903 153 9 84 27 1.5 -42.94655 173.23664 122 15 269 13 0.5 -42.94663 173.23639 111 7 274 12 2.5 -42.94688 173.23658 154 5 110 10 1 -42.94705 173.23637 170 5 210 12 1 -42.94699 173.23597 169 6 256 5 1 -42.94669 173.23533 180 5 30 14 1 -42.94655 173.23521 185 9 21 14 2 -42.94656 173.23482 169 22 38 5 1.5 -43.07665 172.9648 107 10 91 28 2 -43.08078 172.9553 132 8 56 40 2.5 -43.09377 172.86986 196 12 271 25 1.5 -42.55015 173.50469 8 4 54 24 99 -42.55221 173.5089 -6 11 271 32 99 -42.51643 173.5076 2 5 188 14 2 -42.51622 173.50771 6 5 30 25 99 -42.51614 173.50773 4 7 50 36 2 -42.51591 173.50772 5 4 208 46 2 -42.5158 173.50775 4 6 210 34 2
304
-42.67447 173.28551 39 9 164 23 0.5 -42.6773 173.28811 59 11 169 14 1 -42.67729 173.28789 61 18 249 32 1 -42.66399 173.2632 64 8 105 7 2 -42.66398 173.26332 66 7 142 4 2 -42.65276 173.24284 83 8 60 8 2.5 -42.65271 173.24217 116 5 115 15 1 -42.64741 173.23693 114 7 10 2 3 -42.64497 173.22778 164 7 354 34 2.5 -42.64733 173.21688 188 6 121 6 1.5 -42.62206 173.16459 141 8 153 16 1.5 -42.61975 173.17603 152 8 292 23 3 -42.61905 173.1737 148 10 254 14 1 -42.61864 173.17377 148 18 310 27 2 -42.62009 173.17797 147 11 280 19 3 -43.0528 173.01617 120 11 20 52 99 -43.0528 173.01617 120 11 23 51 99 -42.55234 173.50289 165 7 5 37 3 -42.55609 173.50337 166 8 29 43 0.5 -42.60001 173.21162 247 18 64 51 0.5 -42.58978 173.23397 204 6 225 45 2 -42.59021 173.23444 203 6 231 32 99 -42.58986 173.23492 197 5 240 26 99 -42.59024 173.23501 190 11 210 39 3 -42.59012 173.23669 200 12 220 33 2 -42.59089 173.23678 202 11 215 44 1.5 -42.59192 173.23535 196 26 231 38 3 -42.59278 173.23612 202 4 217 36 1 -42.59285 173.23756 209 6 233 29 2.5 -42.59017 173.23097 198 10 237 31 2 -42.58904 173.23072 205 5 228 28 1.5 -42.58731 173.22794 235 13 290 30 1.5 -42.58886 173.22777 202 5 243 35 1 -42.58737 173.22798 219 7 298 53 2 -42.58784 173.21911 216 9 185 31 1 -42.5873 173.21972 213 9 168 26 2 -42.58696 173.22014 197 10 219 19 1.5 -42.58786 173.2159 224 50 70 40 99 -42.5832 173.20207 237 50 45 40 99 -42.56129 173.50218 92 4 209 33 0.5 -42.55492 173.50406 167 9 209 32 1.5 -42.56945 173.49579 22 7 22 62 2 -42.56937 173.49581 20 7 15 75 1.5
305
-42.56934 173.49589 23 7 24 66 99 -42.56779 173.49776 26 6 4 59 3 -42.56394 173.49948 -8 6 17 81 2.5 -42.56393 173.49950 19 11 46 45 99 -42.56403 173.49996 24 6 35 60 2 -42.56433 173.50034 20 4 37 56 3 -42.55886 173.50255 158 4 155 22 0.5 -42.55916 173.50246 159 4 181 53 0.5 -42.85396 173.31022 13 14 58 16 1 -42.85418 173.30975 5 23 44 11 99 -42.85428 173.30951 5 25 40 16 2 -42.85443 173.30954 5 31 9 16 99 -42.85499 173.30939 9 7 70 17 99 -42.85478 173.30830 7 50 297 21 0.5 -42.85483 173.30728 12 9 77 12 2 -42.85487 173.30695 13 7 76 15 3 -42.85478 173.30701 17 8 72 12 3 -42.85546 173.30642 21 7 77 15 2 -42.85584 173.30583 0 6 62 10 2.5 -42.85617 173.30625 17 12 78 11 2 -42.85623 173.30622 26 9 84 38 99 -42.85651 173.30550 9 15 98 17 99 -42.85498 173.30467 37 6 90 22 1 -42.85538 173.30353 22 5 85 12 3 -42.85538 173.30354 16 9 78 15 3 -42.85545 173.30362 0 5 65 11 99 -42.85575 173.30319 13 8 70 18 2 -42.85586 173.30211 29 5 87 16 1.5 -42.85467 173.30569 30 31 17 17 99 -42.65082 173.33295 129 9 235 26 1.5 -42.54475 173.46447 70 5 118 43 1 -42.54371 173.46695 125 8 262 47 2 -42.54363 173.46906 72 9 267 35 1 -42.84459 173.31398 134 11 71 16 1 -42.86711 173.30812 8 13 212 21 3 -42.86707 173.30805 9 13 153 32 1.5 -42.86723 173.30806 10 13 211 20 3 -42.67838 173.42200 41 10 31 31 1.5 -42.67805 173.42223 33 10 13 27 1.5 -42.67813 173.42280 31 26 22 21 1 -42.67846 173.42289 49 8 58 23 1 -42.67869 173.42290 21 5 37 30 2 -42.67879 173.42288 18 6 47 21 2
306
-42.67866 173.42305 19 31 70 15 0.5 -42.67891 173.42313 19 25 95 35 2 -42.67895 173.42316 19 25 34 18 1.5 -42.67887 173.42317 19 26 40 22 3 -42.67879 173.42327 50 8 28 22 2 -42.67852 173.42322 58 13 24 20 2 -42.67862 173.42368 58 9 5 33 1 -42.67853 173.42386 35 7 15 20 1 -42.67868 173.42391 27 10 23 25 2.5 -42.67869 173.42452 29 6 48 24 99 -42.67856 173.42219 47 10 36 30 1.5 -42.67797 173.42150 48 7 250 26 1 -42.67883 173.42152 48 13 358 27 0.5 -42.67863 173.42116 31 9 41 27 99 -42.67851 173.42100 56 11 209 27 1 -42.67830 173.42100 39 30 232 26 99 -43.04274 173.02352 140 16 35 27 0.5 -43.04307 173.01861 311 4 208 36 1 -43.04325 173.01829 306 4 220 30 1 -43.04193 173.01810 295 6 107 19 2 -43.04167 173.01766 312 5 217 12 1.5 -43.04171 173.01772 310 5 81 22 1.5 -43.04208 173.01821 292 10 41 12 1.5 -43.04229 173.01837 285 7 152 9 2 -42.98784 173.03942 200 5 175 18 2 -42.98789 173.03936 200 5 230 26 1.5 -42.99781 173.06182 226 11 21 25 2.5 -42.99781 173.06182 226 11 25 25 99 -42.99787 173.06178 189 14 357 41 2.5 -42.99787 173.06178 189 14 19 33 99 -42.99759 173.06405 226 17 24 31 2.5 -42.99748 173.06396 228 10 324 16 2.5 -42.99742 173.06394 233 11 34 17 1 -42.99510 173.05765 229 0 3 27 1 -42.99405 173.05657 209 0 89 21 1 -42.99403 173.05658 214 0 78 27 1 -42.98679 173.07118 126 7 128 9 1 -43.06187 172.97025 227 0 42 16 3 -43.06404 172.97783 270 0 166 16 99 -43.06404 172.97783 270 0 168 17 99 -43.06349 172.97891 280 4 34 40 2.5 -43.06318 172.97933 271 11 89 24 99 -43.06258 172.97945 277 5 100 18 99
307
-43.06251 172.97969 276 5 98 24 99 -43.06379 172.98225 285 12 44 18 3 -43.06168 172.97918 277 11 280 23 3 -43.06187 172.98089 297 14 245 5 2 -43.06184 172.98074 295 7 2 7 99 -43.06142 172.98064 303 7 351 18 99 -43.06145 172.98056 302 7 66 7 99 -43.06126 172.98055 303 7 71 7 99 -43.06046 172.98053 318 4 120 18 99 -43.06019 172.98058 319 8 125 8 99 -43.05944 172.98039 308 4 124 26 1 -43.05817 172.98208 324 4 284 23 2.5 -43.05821 172.98174 321 4 284 41 2.5 -43.05815 172.98209 326 4 304 44 2 -43.05583 172.98369 314 4 144 19 1 -43.05547 172.98412 328 5 21 14 2 -43.05339 172.98414 320 5 254 21 1.5 -43.05354 172.98397 317 11 355 68 1 -43.05492 172.97923 256 5 170 15 1 -43.04991 172.97344 182 4 40 24 0.5 -43.05068 172.97262 184 4 61 14 2 -43.04902 172.97141 177 4 20 30 1 -43.05189 172.96457 185 7 108 24 3 -43.05192 172.96480 193 6 44 11 3 -43.05198 172.96472 186 6 103 5 2.5 -43.05034 172.96913 171 8 310 6 1 -43.05035 172.96914 171 7 127 4 2 -43.04874 172.97227 167 6 349 23 2 -43.04867 172.97222 162 4 340 39 2 -43.04801 172.95517 231 5 235 35 3 -43.04808 172.95517 234 4 40 82 1 -43.04733 172.95467 245 7 31 9 1 -43.04668 172.95540 251 5 25 89 2 -43.04620 172.95636 275 5 208 82 99 -43.04555 172.95689 288 5 24 71 1.5 -43.04522 172.95721 293 9 34 82 99 -43.04362 172.95647 229 7 223 55 1 -43.04200 172.95842 232 5 41 46 3 -43.04206 172.95848 232 8 28 60 2 -43.04206 172.95848 232 8 32 55 2 -43.04519 172.95442 212 7 177 89 0.5 -43.04361 172.95682 231 5 20 62 1 -43.04125 172.96076 267 4 207 85 99
308
-43.04006 172.96123 252 8 31 67 1 -43.03665 172.96820 203 8 53 52 2 -43.03665 172.96823 202 0 30 51 99 -43.03594 172.96927 231 7 90 74 2 -43.03587 172.96921 233 6 79 65 3 -43.03592 172.96912 234 7 82 69 3 -43.03608 172.96917 237 6 87 70 99 -43.03861 172.96330 224 6 20 51 2.5 -43.04483 172.96500 194 4 0 11 1 -43.04483 172.96500 194 4 349 25 0.5 -43.03423 172.95261 180 12 285 20 99 -43.03417 172.95278 146 7 340 32 2 -43.03417 172.95278 146 7 352 25 3 -43.03396 172.95279 134 4 325 34 1 -43.03452 172.95049 135 4 200 27 1 -43.03440 172.94916 142 5 300 24 2 -43.03362 172.95290 114 11 213 40 1.5 -43.03007 172.95322 253 5 240 5 1 -43.03004 172.95320 252 4 110 12 99 -43.03008 172.95324 254 5 320 3 99 -43.03260 172.95834 195 4 243 52 1.5 -43.03322 172.95768 154 4 194 66 1.5 -43.03461 172.95767 120 9 216 87 99 -43.03430 172.95909 118 5 48 35 99 -43.03212 172.96178 134 7 62 83 1.5 -43.03097 172.96355 103 6 28 24 3 -43.03095 172.96358 103 7 24 25 99 -43.03070 172.96464 98 7 28 23 2.5 -43.00969 173.02788 310 4 304 76 2 -43.00951 173.02751 309 4 25 60 99 -43.00836 173.02659 265 4 357 59 99 -43.00281 173.02406 139 12 283 41 1 -43.00299 173.02302 142 12 200 27 1 -43.00349 173.02174 139 8 208 68 2.5 -43.00347 173.02194 134 8 182 52 1.5 -43.00602 173.01811 143 5 220 68 1 -43.00721 173.01739 137 7 228 52 3 -43.00654 173.01723 123 12 214 66 1 -43.00370 173.01845 114 8 264 37 99 -43.00370 173.01845 114 8 282 38 99 -43.00075 173.01667 112 5 40 30 3 -43.00036 173.01742 106 7 42 26 99 -42.99858 173.02009 109 5 30 22 99
309
-43.00058 173.01190 96 12 187 30 3 -43.10627 172.87583 1 8 85 12 2 -43.10595 172.87618 3 4 81 18 2 -43.09186 172.85021 240 11 95 14 1.5 -43.09231 172.85174 187 12 270 28 2 -43.09393 172.85372 198 6 10 48 1 -43.09663 172.85870 142 5 280 31 1 -43.10670 172.85329 296 6 222 14 1 -43.10682 172.85327 306 5 239 20 2.5 -43.10689 172.85678 278 5 20 15 1 -43.10664 172.85677 265 6 216 13 2.5 -43.04269 173.02521 142 5 26 22 99 -43.05703 173.01347 95 11 317 12 1 -43.00094 173.09267 34 9 26 18 2 -42.99835 173.08583 47 57 90 15 2.5 -42.99901 173.08638 68 12 52 12 2 -42.99901 173.08642 67 14 36 18 2.5 -42.99863 173.08686 78 13 25 26 2 -42.99863 173.08686 78 13 55 20 2.5 -42.99885 173.08767 67 18 74 32 2.5 -43.09294 172.86945 221 7 100 33 1 -43.10311 172.87541 28 6 352 30 1 -43.10387 172.87464 14 18 46 19 0.5 -43.10337 172.87344 39 6 125 35 1 -43.10196 172.86924 84 4 173 12 1 -43.07991 172.80148 313 5 62 59 2 -43.08168 172.80255 337 5 65 21 1 -43.08358 172.80759 368 6 66 29 2 -43.08119 172.81539 449 5 220 71 1 -43.08108 172.81547 454 4 16 36 0.5 -42.67812 173.42120 55 8 98 31 3 -42.67809 173.42116 52 8 72 15 3 -42.67760 173.41966 40 12 285 25 1 -42.67784 173.41909 35 8 349 29 3 -42.67657 173.41705 110 5 6 17 3 -42.67660 173.41714 83 8 8 2 99 -42.67628 173.41667 66 10 10 20 1 -42.67541 173.41669 59 6 86 10 99 -42.67541 173.41669 59 6 145 90 99 -42.67536 173.41632 104 6 60 14 99 -42.67492 173.41583 79 0 64 33 99 -42.67468 173.41572 76 6 70 22 1 -42.67463 173.41536 63 25 25 21 99
310
-42.67419 173.41511 63 6 46 28 2 -42.67418 173.41509 61 6 77 16 2.5 -42.67385 173.41502 70 7 57 27 2.5 -42.67382 173.41597 74 9 25 3 99 -42.60806 173.45984 45 4 85 56 3 -42.60806 173.45995 46 4 75 51 3 -42.60786 173.46011 41 4 82 56 3 -42.60776 173.45999 41 5 101 40 3 -42.60874 173.45539 11 5 59 41 3 -42.60863 173.45522 21 5 59 41 3 -42.60774 173.45375 23 4 131 44 3 -42.67155 173.02811 176 7 74 36 3 -42.66990 173.03141 163 4 142 31 99 -42.66984 173.03141 166 4 129 25 99 -42.66185 173.04553 153 7 123 65 2 -42.66186 173.04541 156 5 45 17 1 -42.66173 173.04554 171 5 311 24 2 -42.66182 173.04649 178 7 22 14 2 -42.58816 173.10055 222 6 182 31 1 -42.58809 173.10147 217 6 230 37 99 -42.58808 173.10151 213 6 226 38 99 -42.58652 173.10170 223 6 146 24 99 -42.58811 173.10497 219 5 196 33 99 -42.58534 173.10448 233 5 195 41 99 -42.59820 173.10434 229 5 45 34 99 -42.50489 173.16273 438 6 220 8 2 -42.49153 173.18876 477 5 225 42 99 -42.48758 173.19059 483 6 240 44 99 -42.48751 173.19060 488 5 233 42 99 -43.00163 173.09225 51 5 43 32 3 -43.02498 172.70878 140 9 53 23 1.5 -43.02498 172.70878 140 9 75 11 1.5 -43.02500 172.70876 138 7 130 33 1 -43.01673 172.70332 156 4 120 13 1 -43.01672 172.70336 155 4 168 2 1.5 -43.01739 172.70380 159 5 50 13 2.5 -43.00042 172.71352 175 4 20 18 1 -42.99984 172.71410 202 7 50 29 2.5 -42.99981 172.71416 199 7 40 22 2.5 -42.99538 172.71964 217 4 20 31 2 -42.99510 172.71995 218 5 47 18 1.5 -42.99498 172.71993 219 5 4 20 2 -42.99538 172.71976 212 6 51 16 3
311
-42.98984 172.72020 220 4 343 14 99 -42.98991 172.72041 232 5 343 21 2 -42.98550 172.71924 230 5 85 26 2 -42.90443 172.72131 247 5 193 26 3 -43.04931 173.07278 15 10 340 17 2 -43.04827 173.06193 12 10 327 18 2 -43.04785 173.06155 10 10 30 19 2 -43.04751 173.06099 7 10 56 5 2 -43.04730 173.06048 12 11 226 24 3 -43.04182 172.87314 143 9 56 25 3 -43.05316 172.86452 256 4 266 49 3 -43.05592 172.86134 294 8 82 32 3 -43.05563 172.86539 251 5 65 26 2 -43.05571 172.86558 252 7 0 0 2 -43.05561 172.86401 237 7 85 14 3 -43.05504 172.85855 282 5 73 66 3 -43.05482 172.85824 281 5 62 73 3 -43.05587 172.85711 261 5 142 14 2.5 -43.06066 172.85357 267 5 82 16 1.5 -43.06054 172.84925 249 5 92 17 1.5 -43.06291 172.84174 247 5 32 7 2 -43.06381 172.83955 244 5 162 7 2 173.50473 -42.52013 8.962 99 288 22 2 173.50475 -42.52041 7.762 99 190 7 1 173.50475 -42.52055 8.153 99 152 14 1 173.04111 -42.98466 254.178 0.2 295 27 1.5 173.04169 -42.98471 261.125 0.1 325 26 2 173.04234 -42.98419 271.281 0.2 285 45 2 173.04375 -42.98301 287.919 0.1 355 21 2 -42.51996 173.5046 4.388146 5 156 13 99 -42.55015 173.50464 0.226782 3 15 26 3 -42.55028 173.50464 0.487435 3 1 23 2.5 -43.13242 172.81052 11.435457 3 51 4 99 -43.13146 172.80858 8.53813 5 129 20 2 -43.13142 172.8086 13.807621 4 141 22 2 -43.13194 172.80894 22.333988 6 91 20 0.5 -43.1313 172.80856 29.634722 3 130 16 2 -43.13195 172.80808 67.326942 99 155 14 1.5 -43.13202 172.80804 65.163536 4 85 14 99 -43.1321 172.80836 75.328896 4 105 11 1.5 -43.13117 172.80919 1.797707 6 103 11 1 -43.13095 172.81176 33.265755 3 115 18 1.5 -43.13037 172.81151 30.155737 6 134 14 2
312
-43.1315 172.81033 14.306055 6 93 13 1 -43.13239 172.8105 -4.06523 4 44 11 1.5 -43.13294 172.82072 -5.465841 4 25 2 1.5 -43.13321 172.8209 -0.746009 6 80 9 1 -43.13315 172.82072 2.411268 5 81 2 1.5 -43.13291 172.82052 4.836837 11 80 6 1
313
Appendix E
Haumuri Bluff Kinematic Modeling – Supplementary Figures
Figure E-1: Parameter histograms for Model 1 for Haumuri Bluff.
314
Figure E-2: Parameter histograms for Model 2 for Haumuri Bluff.
315
Appendix F
Kate Anticline Seismic Sections
The following are time and depth sections for the six seismic lines used to determine the structure of the Kate Anticline. Solid lines mark confident interpretations, which were used for structure contouring. Dashed lines are used where the interpretation is uncertain. Blue is the top of the Waikari formation, yellow is the top of the Ashley mudstone, and red is the top of the Torlesse basement.
Topography is shown over the depth sections. Seismic data are available from New Zealand Petroleum and Minerals, http://www.nzpam.govt.nz (PR3323 and PR3875). Data were displayed and interpreted using Move.
316
Figure F-1: Seismic line Kate-01 in time and depth. The Kate-1 well is shown in the depth section for comparison.
317
Figure F-2: Seismic line TAG06-260-01 in time and depth.
Figure F-3: Seismic line TAG06-260-02 in time and depth.
319
Figure F-4: Seismic line TAG06-260-03 in time and depth.
320
Figure F-5: Seismic line TAG06-260-04 in time and depth.
321
Figure F-6: Seismic line TAG06-260-05 in time and depth. Note that there are two possibilities for the basement-cover unconformity.
322 Appendix G
Kate Anticline Kinematic Modeling – Supplementary Figures
G-1 An Alternate Structure Contour Map for the Kate Anticline
This is an alternative to the structure contour map shown in Figure 4-15. Instead of assuming that the dip of the fold backlimb is approximately constant along strike, this alternative assumes that the backlimb dips significantly less steeply (~10°) in the vicinity of cross section A-
A′. This is inspired by the apparently low angle of bedding dip at the SE end of seismic line
TAG06-260-05, which in Figure 4-11 was assumed to represent only a wider fold crest. On the other hand, this interpretation is inconsistent with bedding dips measured along the coast, which
are mostly greater than 10°.
Figure G-1: Alternative structure contour map.
323 Of our two models for the restored-state structure of the Kate Anticline, Model 1 did not rely on the structure contour map, so only Model 2 would produce different results for these structure contours. For the first test (ratio of slip rates for the two terraces along A-A′), trishear predicts a ratio of slip rates of 1.3±0.2, which is almost identical to the 1.3±0.3 that we expect
(see Chapter 4). Parallel fault-propagation folding, however, predicts a ratio of 1±0.2, which is also within error of the expected value. The second test (ratio of slip rates between cross sections
A-A′ and B-B′) is also ambiguous, with both models consistent with a constant or nearly constant rate of slip. If this interpretation of the structure of the Kate Anticline is correct, it is more difficult to distinguish between the two kinematic models on the basis of uplift rate ratios. We note, however, that with this set of structure contours it is difficult to fit a parallel fault- propagation fold model to cross section A-A′ at all. The best-fitting result from the Markov chain
Monte Carlo simulation for this case predicts a forelimb dip of 63°, which is significantly steeper than observed. For that reason, we find this interpretation unlikely.
Figure G-2: Slip rate ratio (analogous to Figure 4-14).
324
G-2 Histograms of Trishear and Kink-Band Model Parameters for the Kate Anticline
Figure G-3: Y-coordinate (elevation relative to sea level) of the fault tip.
325
Figure G-4: Total slip on the fault. (For kink-band models in which slip is not conserved across fault bends, this is the slip on the lower segment of the fault.
326
Figure G-5: Contours of probability for dips of the upper and lower fault segments. These two parameters are highly correlated.
327
Figure G-6: Restored elevation of the top of the Waikari formation
VITA
David O. S. Oakley
Education Ph.D., Geosciences, Pennsylvania State University. 2017 B.A., Geoscience and Astrophysics, Williams College 2011
Teaching Experience Teaching Assistant at The Pennsylvania State University, Geosciences Department GEOSC 465 Structural Geology Spring 2013, 2016 GEOSC 010 Geology of the National Parks Fall 2014, 2016 GEOSC 001 Physical Geology Fall 2015 GEOSC 310 Earth History Spring 2014, 2015 GEOSC 472 Field Camp 2012, 2013, 2014 GEOSC 452 Hydrogeology Fall 2012
Awards and Honors Scholten-Williams-Wright Scholarship in Field Geology 2014 Chelius Graduate Fellowship in Earth Sciences 2011 Mineralogical Society of America Undergraduate Prize 2011 Sigma Xi 2011 Phi Beta Kappa 2010
Publications Oakley D.O.S. and Fisher, D.M., 2015. Inverse trishear modeling of bedding dip data using Markov chain Monte Carlo methods. Journal of Structural Geology, v. 80, p. 157-172. Oakley D.O.S., Kaufman, D.S., Gardner, T.W., Fisher, D.M., and VanderLeest, R.A., 2017. Quaternary marine terrace chronology, North Canterbury, New Zealand using amino acid racemization and infrared stimulated luminescence. Quaternary Research 87, 151-167.
Conference Talks Oakley, D.O.S. and Fisher, D.M., 2016. A New Method for Analysis of Trishear Fault- Propagation Folds, with Application to Structures in North Canterbury, New Zealand. Geological Society of America Abstracts with Programs, Vol. 48, No. 7. Oakley, D.O.S., Gardner, T.W., and Fisher, D.M., 2015. Uplift Rates of Marine Terraces as Constraints on Fault-Propagation Fold Kinematics: Examples from Two Anticlines in North Canterbury New Zealand. Geological Society of America Abstracts with Programs, Vol. 47, No. 7, p. 719. Oakley, D., Fisher, D., and Gardner, T., 2014. Inverse Trishear Modeling of Bedding Dip Data, Geological Society of America Abstracts with Programs. Vol. 46, No. 6, p. 227. Oakley, D., Gardner, T., Fisher, D., and Kaufman, D., 2013. Uplift Rates and Structure of the North Canterbury Fold and Thrust Belt, South Island, New Zealand. Geological Society of America Abstracts with Programs, Vol. 45, No. 7, p. 710.