Fault Geometry and Kinematics Within the Terror Rift, Antarctica THESIS

Home , Fault (geology), Growth fault, Neotectonics, Rift, Terry Wilson (scientist)

Fault Geometry and Kinematics within the Terror Rift, Antarctica

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

William B. Blocher

Graduate Program in Earth Sciences

The Ohio State University

2017

Master's Examination Committee:

Dr. Terry Wilson, Advisor

Dr. Thomas Darrah

Dr. Derek Sawyer

Copyrighted by

William B. Blocher

2017

Abstract

The Terror Rift is the youngest expression of the intraplate West Antarctic

Rift System that divides the Antarctic continent. Previous studies of the Terror

Rift have ascribed a variety of interpretations to its structure, and especially to the regional anticline known as the Lee Arch, which has been explained as a transtensional flower structure, a rollover anticline, and as the result of magmatic inflation.

Fault mapping and the documentation of stratal dips in this study have revealed a Terror Rift structure characterized by north-south folds and a complex distribution of faults. Nearly all faults have normal sense dip separation. A continuous zone of west-dipping faults with relatively high-magnitude normal separation are interpreted to be the border fault system defining the eastern margin of Terror Rift. Reconstruction of listric ramp-flat geometry of this border fault system explains intrarift fold and fault patterns well. Zonation of structures indicates that the listric rift detachment faults are segmented along the rift axis.

This new model for rift structure indicates orthogonal rift extension in the ENE-

WSW direction, with low strains of <10% calculated from bed-length balancing.

Acknowledgments

To my advisor, Dr. Terry Wilson, for her inexhaustible knowledge and patience.

And to my parents, Richard and Janice, for their unconditional love and support.

Vita

2007 ...... Whetstone H.S.

2010 ...... Associate of Science, Columbus State

2013 ...... B.S. Geology, Ohio State University

2013 to present ...... Graduate Teaching Associate,

School of Earth Sciences,

The Ohio State University

Fields of Study

Major Field: Earth Sciences

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Table of Contents

Abstract ...... i

Acknowledgments ...... ii

Vita ...... iii

Introduction ...... 1

Background ...... 4

Investigation of Terror Rift Structure ...... 4

Elements of Rift Architecture ...... 7

Folds in an Extensional or Transtensional Context ...... 11

Flower Structures ...... 11

Rollover structures ...... 13

Problem Statement...... 17

The Orthogonal Model ...... 17

The Transtensional Model ...... 20

Methods ...... 21

Seismic Data...... 21

Fault Interpretation ...... 23

Fault Correlation Criteria ...... 25

Dip Analysis ...... 31

Dip Signatures ...... 34

Cross-Section Methods ...... 36

Bed-length Balancing ...... 36

Fault Geometry Construction through Dip Domains ...... 37

Results ...... 43

Stratal Dip Patterns ...... 43

Map view stratal dip patterns ...... 43

Stratal dip patterns with depth ...... 47

Fault Patterns ...... 55

Fault identification, correlation, and mapping ...... 55

Fault geometry...... 59

Fault motions: sense and separation ...... 61

Fault Profile Reconstruction ...... 65

Extension Estimates from Balanced Sections ...... 75

Interpretive Maps ...... 81

Discussion ...... 90

References ...... 98

Appendix A: Matlab Scripts ...... 102

Rollover.m ...... 102

Cartesian_intersect.m ...... 104

Projectpointtoline.m ...... 104

planefit.m ...... 105

fwaz.m ...... 105

haversine.m ...... 105

Appendix B: Plane Fitting ...... 107

Round Earth Complications ...... 108

Sensitivity Testing ...... 114

List of Figures

Figure 1. Reconstructed Cretaceous West Antarctic Rift System from Siddoway (2008).

Area of this study is in the southwestern Ross Sea...... 1 Figure 2. Rift basins of the Ross Sea. TR denotes the Terror Rift, the subject of this study. From Hall et al. (2007) ...... 2 Figure 3. An interpretation of Terror Rift structures constructed from the first iteration of seismic interpretation by Cooper et al. (1987) ...... 3 Figure 4. The reflection seismology dataset within the study area comprises data acquired during many different surveys...... 4 Figure 5. A map from (Salvini et al., 1997) delineating Terror Rift fault patterns and depicting Terror Rift architecture as being the result of an en echelon termination of regional strike-slip faults...... 5 Figure 6. Interpretive maps from Hall et al. (2007) (left) and Salvini et al. (1999) (right) highlighting the different explanations for the same observed architecture, esp. in the proximity of Drygalski Ice Tongue...... 6 Figure 7. Profile geometries of a symmetric (A) and an asymmetric (B) rift (Acocella,

2010)...... 7 Figure 8. Offset fault segments becoming linked together as strain progresses (Fossen &

Rotevatn, 2016) ...... 8 Figure 9. Different structures that occur at releasing and restraining bends in a translational tectonic system (Cunningham & Mann, 2007)...... 9 Figure 10. A frame of a simulation of the propagation of a listric normal fault and syn- tectonic deposition. A syn-tectonic growth wedge (cyan) overlies pre-tectonic strata

(black). (Allmendinger, 2002) ...... 10 vii

Figure 11. From Vanderpluijm & Marshak (2004), various species of strike-slip fault bend structures. Positive (left) and negative (right) flower structures. Note the antiformal dip-pattern of the positive flower structure – a defining characteristic...... 12 Figure 12. A normal fault with a shallower dip at depth. Coulomb collapse is a sub- seismic scale brittle deformation that, in this case, serves to fill the space that would be void space were the hanging-wall to rigidly translate. (Tearpock & Bischke, 1991) ...... 13 Figure 13. Modified from McClay (1990), a section of an analogue experiment in which a sandbox was subjected to 50% extension along a listric fault, photographed (top) and annotated (bottom). Note the antithetic stratal dip of the inboard strata, and the multiple generations of keystone faults. Also note the fanned syn-rift strata, annotated as solid white...... 14 Figure 14 Modified from McClay (1990), a doubly-listric or ramp-flat controlling fault geometry requires a more complex hanging-wall shape change. This collapse structure features the same inboard antithetic stratal dip, and several new structures, like a ramp syncline and lower rollover anticline and keystone collapse. Also note that, as a result of the lower degree of extension (25%), fewer generations of keystone faults developed. .. 15 Figure 15. From McClay (1990), a ramp-flat normal fault, annotated. The first row of annotations demarcates zones that differ in their dominant kinematics. The second and third rows are two orders of structural zone classifications...... 16 Figure 16 From Fossen (2010) - Showing a hypothetical progression from individual fault segments to hard-linked and ultimately coalesced faults...... 18 Figure 17. Structural map from Hall (2007), with proposed accommodation zones denoted by the yellow polygons...... 19 Figure 18. An en echelon assemblage of normal faults terminating a right-lateral strike slip fault. Taken from (Harding et al., 1985) ...... 20

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Figure 19. Similar separations in profile resulting from different kinematics superimposed on different structures...... 23 Figure 20. Two examples of stereographic representations of the coplanarity of pairs of lines. The blue arbitrary lines fall on the traces' mutually perpendicular plane, demonstrating their coplanarity. The red arbitrary lines do not plot on their traces' mutually perpendicular plane, demonstrating that these traces are skew, and likely don’t correlate...... 27 Figure 21. Antithetic (with opposite fault dips) and synthetic (with similar fault dips) transfer zones (Morley, 1999). Contour lines represent depth to a stratum. The dip of this surface is only perpendicular to the trace of the fault that controls it where its contours are parallel to the trace of the fault. Where the contours turn from parallel and interfere, stratal dip is not directly perpendicular to any fault trace...... 32

Figure 22. An interpreted profile exhibiting a clear growth history...... 35

Figure 23. Line NBP0401-148, interpreted between pin lines (red) ...... 36 Figure 24. A “coarse” reconstruction from Tearpock & Bischke (1991) (top) and a “fine” reconstruction via the modified algorithm created for this study (bottom). This modified algorithm, outlined below, trades a little stability for more detail...... 38 Figure 25. The mechanics behind the “refraction” of the controlling fault initial condition.

...... 41 Figure 26 A section of line NBP0401-148m, depicting anticlines and synclines as they occur NW of Drygalski Ice Tongue...... 44 Figure 27. Measured dips of horizon Rg throughout the study area, with interpreted fold axial traces. Of note are the relatively uniform dips within McMurdo Sound and the regionally developed anticlines and synclines outboard of the Drygalski Ice Tongue. .... 46

Figure 28. (above) Superimposed dip magnitudes with depth at all 109 intersections .... 47

Figure 29. (below) A polar plot of the data intersection 105, depicting progressively steepening dips (1° - 7°) to the west with depth...... 47 Figure 30. The locations of the 109 intersections along which structural dip profiles were collected. These profiles can be found on the subsequent pages...... 48

Figure 31. Structural dip polar plots, sheet 1 of 4 ...... 49

Figure 32. Structural dip polar plots, sheet 2 of 4 ...... 50

Figure 33. Structural dip polar plots, sheet 3 of 4 ...... 51

Figure 34. Structural dip polar plots, sheet 4 of 4 ...... 52

Figure 35. The symbols that appear on the polar plots...... 52

Figure 36. Polar plot locations, colored to reflect the degree of growth they show...... 53 Figure 37. A structural dip profile piercing unconformity Ri and entering an interval of parallel strata...... 54

Figure 38. An example of a fault trace (red) ...... 55 Figure 39. Identification-phase map, depicting interpreted fault traces. Apparent dip direction, separation magnitude, and unconformity surface that truncates each fault are denoted by symbol types, size and color...... 56

Figure 40. Map of faults correlated between seismic profiles...... 57 Figure 41. An isometric view (black) featuring three profiles. Traces interpreted on the red and purple profiles project nearly exactly through a compatible fault trace on the intermediate blue profile. This is a strong argument for the correlation of all three traces.

...... 58 Figure 42. A histogram of calculated fault strikes with a bin size of 7.5°. Fault strikes reported per the right hand rule, so the blue bins represent east-dipping faults, and the red west-dipping...... 59

Figure 43. Average fault dips ± one standard deviation (grey wedge), organized by truncation by different unconformity surfaces...... 60 Figure 44. (above) An annotated figure from (Rosetti et al., 2006), identifying a fault as showing a sense-reversal with depth...... 63

Figure 45. (below) a closer view of that same fault...... 63

Figure 46. An anomalously steep disruption in reflectors observed on PD905...... 64 Figure 47. (above) Locations of profiles on which well-developed rollover structures yielded reasonable, stable controlling fault projections...... 65 Figure 48. line NBP0401-148, annotated. The red line denotes the dip domain profile that served as one of the inputs into the controlling fault projection algorithm. The solid black line is the controlling fault, both its initial condition where the fault intersects the seafloor (interpreted directly) and the projection of this fault to depth. This calculated projection eventually became unstable as it projected downward and to the west, as compounding refraction errors overwhelmed the solution. The dashed segment at the bottom of the figure represents the last acceptable segment. Listric splays to the east (right) of the controlling fault are denoted by black dashed lines. The integrated slip on these faults supplies the main controlling fault with the slip necessary to have developed the observed rollover structure...... 65 Figure 49. The fault (or faults) that produce (or compound to produce) a rollover structure must meet a certain throw threshold...... 66

Figure 50. Line IT90a-60 and its projected controlling fault in TWT...... 70

Figure 51. Line NBP0401-148 and its projected controlling fault in TWT...... 71

Figure 52. Line NBP0401-150a and its projected controlling fault in TWT...... 72 Figure 53. Line and IT90a-65 + NBP0401-142m and its projected controlling fault in

TWT...... 73

Figure 54. Line US-407 and its projected controlling fault in TWT...... 74

Figure 55. Profiles selected for balancing...... 75

Figure 56. Interpreted and restored NBP0401-148. Rh is dark green, Rg is pale green... 77

Figure 57. Interpreted and restored NBP0401-150a. Rh is dark green, Rg is pale green. 78

Figure 58. Interpreted and restored US-407. Rh is dark green, Rg is pale green...... 79 Figure 59. A section of line IT90a_57. Within zones where reflectors are disrupted by volcanic intrusions, the contribution to strain by intrusion is very difficult to ascertain. In terms of cross-sectional area, some fraction of these zones belongs to actual intrusive material, and the percent strain along any section containing these zones is highly sensitive to this elusive figure. Thus, profiles with more than a negligible presence of these areas make for poor candidates for bed length balancing...... 80

Figure 60. An interpretive map of faults in the Terror Rift ...... 83 Figure 61. Previously proposed structural solutions to the apparent right lateral shift of the Lee Arch anticline and Discovery Graben synclince...... 84 Figure 62. Torsion of the central fault block in the dip direction demonstrates that this is an overlapping synthetic transfer, or “relay ramp” ...... 86 Figure 63. McClay zones above a ramp-flat normal fault. Modified from McClay (1990)

...... 87 Figure 64. An interpretive map of Terror Rift structures, classified into structural zones.

...... 88 Figure 65. (above) Proposed locations of subsurface tear faults. This map features the same color scheme as the larger structural zone map (Figure 64)...... 89

Figure 66. A schematic block model of proposed footwall structure. (not to scale) ...... 89

Figure 67. Terror Rift structural map defining the extent of four structural zones...... 92

Figure 68. Border fault geometries that control the structures in the different zones...... 94

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Figure 69. Interpretive maps from Hall et al. (2007) (left), Storti et al., (2008) (center),

Salvini et al. (1999) (right), and this study (below)...... 95 Figure 70. A geometrical construction featuring the quantities relevant to the fitting of a plane to points A, B, and C, here outlined...... 108 Figure 71. A truer representation of the first steps of the plane-fitting algorithm. Per concession 1, the bearings θ푆 and θ푆 are defined as the initial bearing from the midpoints of great circle arcs to their endpoints...... 111 Figure 72. A section of a spherical earth, and a graphical depiction of how plunging lines (in this case, from C to A) are handled in the spherical geometry plane-fitting algorithm.

The log spiral arc from C to A crosses every intermediate radial line at the same angle 푎

...... 112 Figure 73. Three different assignments of A, B, and C, to the same three points, and the subsequent construction of a strike line (Step I of the plane-fitting algorithm) ...... 114 Figure 74. One standard deviation of calculated strikes returned from three different variable assignment schemes as a function of triplet baseline. On this figure, Figure 75 (right), and Figure 76 (below), the vertical blue line represents a typical triplet baseline...... 115 Figure 75. Results from the same simulation, shown as the actual values of those strikes for each variable assignment scheme. 115

Figure 76. While some functions the plane fitting routine calls become unstable at large distances, others become sensitive at short distances. Even in a purely cartesian space, when fitting a line to two points, a small error in the position of one point has a greater effect when those points are close together. The sensitive domains of the various helper functions define the stable domain of baselines across which plane fitting can be achieved with a high degree of confidence...... 115

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Introduction

Since the Mesozoic, Antarctica has experienced a period of extension, during Gondwana supercontinental breakup and then internally within the

Antarctic plate. In the continental interior, a widespread extensional phase that took place between 105 Ma and 80 Ma resulted in the creation of the West

Antarctic Rift System (or “WARS”) (e.g. Fitzgerald 2002), shown in Figure 1.

From the Bellinghausen Sea, this zone of extensional structures runs southwest into central West Antarctica, turns due south to meet the Transantarctic

Mountains, and follows the eastern side of this range on its way out toward the

Ross Sea.

Within the Ross Sea, a series of

extensional basins formed, with some

workers invoking a history of Cretaceous

rifting followed by Paleogene rifting (e.g.

Cooper et al., 1987; Salvini et al., 1997)

Figure 1. Reconstructed Cretaceous West Antarctic Rift System from Siddoway (2008). Area of this study is in the southwestern Ross Sea.

and others suggesting a

multi-phase history of

extension (e.g. Wilson

and Luyendyk, 2009).

The most recent

episode of this rifting

resulted in what is

known as the Terror Rift

of the Victoria Land

Basin, contained

between the volcanic

Figure 2. Rift basins of the Ross Sea. TR denotes the provinces of Mount Terror Rift, the subject of this study. From Hall et al. (2007) Melbourne and Ross

Island (Figure 2). The Terror Rift was originally identified and described by

Cooper et al. (1987), who were working with seismic reflection profiles with ~100 km line spacing. Cooper et al. (1987) defined two north-south trending structures within the Terror Rift: the Lee Arch to the east and Discovery Graben to the west

(Figure 3). The Lee Arch is a structural high that runs along the eastern margin of the Terror Rift. It is a heavily faulted anticline, bound on either side by antithetic systems of normal faults. To the west, it transitions into the Discovery Graben, its synclinal counterpart, which has been dropped down by normal faults.

The Neogene

extensional

deformation that

formed the Terror Rift

is not well understood,

and there isn't yet

consensus regarding its

causes and kinematics –

one might not expect Figure 3. An interpretation of Terror Rift structures constructed from the first iteration of seismic such an intraplate rift interpretation by Cooper et al. (1987) within a plate that is almost entirely surrounded by divergent boundaries, as the Antarctic plate has been since the breakup of Gondwanaland. Furthermore, nearly the entirety of the

WARS is either under the West Antarctic Ice Sheet or under the Ross Sea, impeding detailed investigation by standard approaches. An understanding of the

Terror Rift is crucial to any larger understanding of the WARS as a whole and the history of the dynamics of the Antarctic plate. This study aims to use seismic reflection data and geometric principles known about rifts to better establish

Terror Rift geometry and kinematics.

Background

Investigation of Terror Rift Structure

In February 1984, The USGS collected multichannel-seismic reflection data in the western Ross Sea area. These data showed that the bathymetric depression to which we now refer as Discovery Graben, is in fact a rift basin bounded by normal faults. The wide spacing of these exploratory seismic profiles did not allow detailed mapping of Terror Rift structure.

The Italian Antarctic

research program acquired an

extensive grid of new seismic

lines in the western Ross Sea in

1989 and 1990, with more closely

spaced grids in selected areas

(Figure 4). Systematic structural

mapping from these lines allowed

Salvini et al. (1997) to develop a

more detailed mapping of fault

geometry for Terror Rift (Figure Figure 4. The reflection seismology dataset within the study area comprises data 5), leading to a new kinematic acquired during many different surveys. model for the rift described below

as the ‘transtensional model’.

In 2004, research vessel

RVIB Nathaniel B. Palmer set out

to collect a variety of geophysical

data, including a significant

contribution to the set of seismic

reflection data examined in this

study, in the interest of

augmenting the available seismic

data across the Terror Rift (Figure

4). Structural studies using the Figure 5. A map from (Salvini et al., 1997) delineating Terror Rift fault patterns and NBP04-01 data, in conjunction depicting Terror Rift architecture as being the result of an en echelon termination of with other seismic profiles, to regional strike-slip faults. develop an understanding of aspects of the structure of the Terror Rift and its evolution through time were completed by Hall (2006) and Magee (2011). Structural interpretations based on these studies have led to kinematic models of orthogonal rifting.

The orthogonal rift model suggest that Terror Rift opened in the classic

Andersonian style, developing N-S trending normal faults in response to ~east- west extension in the horizontal direction perpendicular to Terror Rift fault traces (Cooper et al., 1987; Hall et al., 2007). Terror Rift faults are interpreted to

5 be normal faults that are segmented and linked by cross-strike accommodation zones (Figure 6; Hall et al., 2007).

Figure 6. Interpretive maps from Hall et al. (2007) (left) and Salvini et al. (1999) (right) highlighting the different explanations for the same observed architecture, esp. in the proximity of Drygalski Ice Tongue.

The transtensional kinematic model, as proposed by Salvini et al. (1997), links the Terror Rift with regional right-lateral strike-slip faults, suggesting that

Terror Rift normal faults formed between, or at the tips of, the strike-slip faults

(Figure 6). Other 'flavors' of the transtensional model ascribe a significant right- lateral shear component to Terror Rift faults, suggesting that they have a Riedel

6 relationship to the master strike-slip faults, having reactivated first stage WARS

Cretaceous – Paleogene normal faults in the Neogene (Storti et al., 2008).

Elements of Rift Architecture

In general, a rift is a zone of deformation that facilitates the extension of the lithosphere. They are generally linear zones, considered orthogonal when their long axis is perpendicular to the direction of extension or transtensional when this angle is oblique. In either case, extension across a rift is defined by a set of faults along which hanging-wall blocks subside.

In the context of plate tectonics and the Wilson cycle, the development of a rift precedes the formation of a new mid-ocean ridge. Alternatively, a rift can

“fail,” halting extension.

In cross-section, a rift can be symmetric, with antithetic faults forming a horst and graben architecture, flanked on either side by prominent border faults dipping in toward the axis of the rift.

Alternatively, it can be more asymmetric, with imbricate faults generally dipping one direction, and synthetic border faults on one Figure 7. Profile geometries of a symmetric (A) and an asymmetric flank taking on a significant portion of the (B) rift (Acocella, 2010). strain while any antithetic faults take on little (Figure 7).

The faults that make up a rift are commonly segmented along their length.

This occurs because faults nucleate at discrete points and propagate outward.

Fault segments tend to propagate in more or less the same orientation, controlled by the same regional stresses, but are slightly offset from each other in map view

(Figure 8). Rift faults commonly curve along strike in map view, where their propagating tips influence the local stress field

(e.g. Rosendahl, 1987). The region of overlap or underlap where one fault passes its strain to another is called an accommodation zone or transfer zone (e.g. Rosendahl, 1987; Morley,

1995). As strain along overlapping or Figure 8. Offset fault underlapping faults progresses, the elasticity of segments becoming linked together as strain progresses the rock between them can become overwhelmed (Fossen & Rotevatn, 2016) and a new set of faults may form that directly link these fault segments (e.g.

McClay et al., 2004) (Figure 8).

Orthogonal rifts are primarily made up of normal faults, while transtensional rifts feature faults with an oblique sense of motion, as their regional strike is at an oblique angle to the overall direction of extension (e.g.

Withjack and Jamison, 1986). Thus, in a transtensional rift, one might also expect to find the same translational structures that occur in strike-slip systems.

(Figure 9).

Figure 9. Different structures that occur at releasing and restraining bends in a translational tectonic system (Cunningham & Mann, 2007). Whether orthogonal or transtensional, symmetric or asymmetric, rifts tend to be made up of faults that are listric, defined by a progressive shallowing of dip with depth. At depth, where they are shallowest, listric faults commonly join together to form a single surface of high strain called a detachment or decollement. This curved geometry means that the fault blocks of a rift must move in ways more complex than simply subsiding. Where fault blocks are small enough to move rigidly, a rotation is imparted on them by the curved faults they slip along. Larger fault blocks may take on an internal shape change to reconcile their motion along curved faults.

The deposition of sedimentary rock is a function of sediment supply and accommodation space (Steele, 1998; Gawthorpe and Leeder, 2000). Rifts, characterized by fault-generated subsidence, provide the accommodation space in that equation. The deposition of sediments into the accommodation space

9 produced by a rift can occur during extension and subsidence (syn-tectonic), or entirely afterwards (post-tectonic). Post-tectonic strata are deposited horizontally, onlapping onto any rotated fault blocks that form the bottom of the accommodation space. Syn-tectonic strata, deposited continuously during block rotation and subsidence will record this rotation. Any stratum experiences the deformation that occurs after it is deposited. Thus, in the case of a syn-tectonic interval, progressively older strata experience progressively more deformation.

The result, in the case of syn-tectonic deposition over a subsiding, rotating fault block, is a growth wedge – a fan of strata, each originally deposited horizontally, but rotated progressively more with depth (Figure 10).

Figure 10. A frame of a simulation of the propagation of a listric normal fault and syn-tectonic deposition. A syn-tectonic growth wedge (cyan) overlies pre-tectonic strata (black). (Allmendinger, 2002) Volcanism commonly co-occurs with rifting (e.g. Karson and Curtis, 1989).

As the lithosphere extends laterally, it also thins, and this thinning relieves enough overburden to initiate decompression melting in the mantle. In the case of a rift that progresses to a mid-ocean ridge, this decompression melting is ultimately the process by which upwelling material forms new oceanic crust, but

10 even intracontinental rifts co-occur with volcanics, as decompression melting produces a supply and faulting produces conduits (e.g. White, 1992).

Folds in an Extensional or Transtensional Context

The Terror Rift is characterized by a regional anticline-syncline pair identified and named in Cooper et al. (1987). At a glance, the Lee Arch Anticline and Discovery Graben syncline might seem out of place in a rift basin, as the common dynamic interpretation of folding is formation as the result of buckling

– folding due to layer-parallel compression. In the context of the Terror Rift, an extensional or transtensional regime characterized by east-west extension, the east-west compression necessary to produce these folds via buckling has not taken place. However, different dynamic schemes can produce folds, and this study will address two of them: the flower structure and the rollover structure.

Flower Structures

A transtensional rift will contain faults along which there is at least a component of strike-slip motion. A planar strike-slip fault, having a straight trace on the map, can accommodate right or left-lateral relative motion of the fault blocks it divides without requiring any shape change of these blocks. Conversely, fault blocks cannot simply translate past each other in cases where the map trace of the strike-slip fault deviates from a straight line.

A bend in the trace of a strike-slip fault may be a releasing or restraining bend, depending on its geometry as it relates to the sense of slip along the fault.

At these bends, local deformation is required to accommodate block translation producing different structures depending on block and displacement geometries.

A flower structure is a zone of deformation that occurs at a bend in a strike-slip fault and accommodates the shape-change required for the translation of blocks along the non-planar fault (e.g. Harding, 1985). They are considered

“positive” or “negative” depending on whether the geometry of the bend necessitates a local contractional or extensional strain (Figure 11). A positive flower structure is the result of local contraction, where a fault bend is oriented such that it restrains fault motion. It characteristically comprises a population of reverse faults and a generally antiformal pattern of stratal dips.

Rollover structures

Another way folding occurs within an extensional or transtensional regime is the rollover structure (e.g. Schlische, 1995). Like flower structures, rollover structures form to facilitate a shape change required by motion along non-planar faults.

Rollover structures are formed due to the forced folding and brittle collapse of a hanging-wall block slipping along a normal fault that is curved in profile (Figure 12). A typical occurrence of a normal fault with profile curvature is a listric normal fault, with a dip that gradually shallows with depth. With this fault geometry, were the hanging- wall to translate without itself internally deforming, void space would be produced, as steeper parts of the fault would take on some component of opening-mode, being forced to match the slip vector of the lower dip, deeper parts of the fault. At very small scales and shallow depths, Figure 12. A normal fault with a where the cohesion of the rock shallower dip at depth. Coulomb collapse is a sub-seismic scale brittle overwhelms the comparably small deformation that, in this case, serves to fill the space that would be void influence of gravity, such a void space is space were the hanging-wall to rigidly translate. (Tearpock & Bischke, 1991) 13 allowed to exist, providing spaces that mineral veins fill over time, but at larger scales, cohesion – a material property and a constant – becomes small in comparison to the increased influence of gravity. Instead, a combination of folding and brittle collapse act to fill this area that would otherwise be void. The brittle collapse can be modeled by Coulomb failure surfaces with an orientation that is usually between 60° and 70° (a material property of the substrate)

(Tearpock & Bischke, 1991).

In the case of a listric normal fault cutting originally horizontal stratigraphy, internal hanging-wall deformation results in a monoclinal dip pattern, with a single fold limb dipping into the fault and progressively shallowing with distance from the fault.

Figure 13. Modified from McClay (1990), a section of an analogue experiment in which a sandbox was subjected to 50% extension along a listric fault, photographed (top) and annotated (bottom). Note the antithetic stratal dip of the inboard strata, and the multiple generations of keystone faults. Also note the fanned syn-rift strata, annotated as solid white.

One or more sets of keystone faults may develop (e.g. Maudault and Brun,

1998). Keystone faults are sets of antithetic normal faults that develop at the crest of rollover folds in response to the neutral surface kinematic scheme (Figure 13).

Because the shallow strata of a rollover fold adopt a larger radius of curvature along a larger arc-length than the deeper strata do, they take on an additional, secondary extensional strain. This additional strain is taken up by these keystone faults, which have a slip-vector that diminishes to zero with depth (i.e. towards the locus of curvature of the rollover fold). Keystone faults initially occur at a location dependent on the geometry of the listric normal fault, but of course are

Figure 14 Modified from McClay (1990), a doubly-listric or ramp-flat controlling fault geometry requires a more complex hanging-wall shape change. This collapse structure features the same inboard antithetic stratal dip, and several new structures, like a ramp syncline and lower rollover anticline and keystone collapse. Also note that, as a result of the lower degree of extension (25%), fewer generations of keystone faults developed.

15 fixed to and move along with the hanging-wall. Thus, in cases where the hanging- wall moves sufficiently far that a set of keystone faults is no longer located where they can facilitate the neutral surface kinematics, a new set of keystone faults is likely to develop, superimposed upon the old one.

More complex normal fault geometries produce more complex rollover collapse structures. A ramp-flat or doubly-listric normal fault produces the same inboard rollover fold that a more simple listric fault does, but the subsequent inflections of the fault produce additional hanging-wall structure, including a rollover anticline and keystone collapse for the second listric depth, and an intermediate “ramp zone” featuring a ramp syncline (Figure 14, Figure 15)

Figure 15. From McClay (1990), a ramp-flat normal fault, annotated. The first row of annotations demarcates zones that differ in their dominant kinematics. The second and third rows are two orders of structural zone classifications.

Hand-in-hand with the concept of the rollover structure is the concept of the growth fault (e.g. McClay, 1990). It is a result of the syntectonic deposition of sediment into the accommodation space generated by the subsidence of a hanging-wall fault block that is rotating, again due to a listric fault geometry. Any deposited stratum experiences all of the local deformation that takes place after it is deposited, and so, in the case of continuous deformation and deposition, younger strata experience less total deformation. This results in fanned or wedge- shaped syntectonic strata sitting on parallel pretectonic strata (Figure 14).

Problem Statement

Both the orthogonal and transtensional models make predictions for fault and stratal dip patterns in the seismic reflection profiles. Many of these predictions are shared between the models, but each model makes geometrical predictions that could be very diagnostic. Furthermore, both the transtensional and orthogonal models must be compatible with all features of the Terror Rift in order to be considered – notably, the Lee Arch anticline and Discovery graben syncline and their internal structures, and the apparent right-lateral offset in these features at two different latitudes.

The Orthogonal Model

The orthogonal rift model implied by Cooper et al. (1987) and developed by Hall et al. (2007) maintains that the observed rift opened in a direction perpendicular to its regional strike, and that the structure of the Lee Arch and

Discovery Graben are controlled by antithetic normal faults with little to no

17 oblique motion. The orthogonal model shares with the transtensional model the obligation to explain the conspicuous anticline in the Lee Arch. The orthogonal model explains the Lee Arch anticline as a rollover anticline, controlled by a high slip-magnitude, listric normal fault flanking the arch and dipping in towards its axis (Hall, 2006; Magee, 2011). This anticline is heavily faulted, with most of the population of faults truncating into the same unconformity that truncates the anticline. In the orthogonal model, these are explained as keystone faults – secondary faults that form as a result of the forced folding of a sequence.

The Lee Arch and Discovery graben trend generally north-south, but around latitude -75.75°, and then again at -76.75°, there are what appear to be right-lateral offsets of around 25 km (depicted as yellow regions on Figure 17). These offsets are interpreted as a segmentation of Terror Rift faults, having nucleated at multiple points and propagated along strike and linking across a accommodation zone where they began to overlap, as shown in Figure 16.

Figure 16 From Fossen (2010) - Showing a hypothetical progression from individual fault segments to hard-linked and ultimately coalesced faults. 18

Figure 17. Structural map from Hall (2007), with proposed accommodation zones denoted by the yellow polygons.

The Transtensional Model

Though strike-slip faults themselves may be difficult to directly observe in seismic reflection profiles due to their characteristic steep dip angles and small separation in vertical profiles, they are likely to create positive and negative flower-structures – zones of normal or reverse faults that occur around releasing or restraining bends in a single strike-slip fault, or as a step between offset strike- slip fault segments. The identification of a flower structure, especially a positive flower structure (consisting of reverse faults – difficult, otherwise, to explain in a rift setting) would give credibility to the transtensional model.

A transtensional model might explain the normal-sense dip-separation on the

Terror Rift seismic profiles as normal faults at a horsetail termination (Figure 18)of the right-lateral strike-slip faults said to be entering the Victoria Land Basin from the northwest (Salvini et al., 1997).

Figure 18. An en echelon assemblage of normal faults terminating a right-lateral strike slip fault. Taken from (Harding et al., 1985)

Methods

Seismic Data

This study primarily involves the interpretation of seismic reflection profiles in a data set that is the product of several different geophysical cruises

(Figure 4). All seismic data used in this study are available in the Antarctic

Seismic Data Library System (http://sdls.ogs.trieste.it) and have been imported into Petrel© software, by Schlumberger, for interpretation.

Seismic reflection profiles are pseudo-images of the subsurface that are formed by transmitting a sound into the subsurface and listening for the echo.

The impulse is generated for the marine surveys used in this study by an airgun, a piston that rapidly forces air into the seawater, creating a bubble that projects a loud sound of known characteristics (amplitude, frequency, duration, etc.) as it forms and collapses. This impulse travels out in all directions. When it encounters an interface across which there is a difference in acoustic impedance

(opposition to the transmission of sound), such as the water-sediment interface that is the seafloor, or an interface between strata of differing mechanical properties, part of the impulse is reflected, and part is refracted and travels through the interface at a new angle determined by Snell's law – the greater the difference in impedance, the greater the portion of the signal that is reflected.

Prominent reflectors, corresponding to regional unconformities and changes in lithology, have been identified many times throughout the legacy of studies applied to this seismic dataset. Many of these reflectors bear different

21 names from one study to the next. No new reflectors are identified or interpreted in the course of this study. Instead, reflectors originally identified in Fielding et al. (2006) and mapped by Hall (2006) were used.

The three reflectors most closely examined in this study are Rg, Rh, and

Ri, corresponding to ages of ~13 ma, ~7.6 ma, and ~4.5 ma respectively – an interval thought to capture the renewed rifting phase of the Terror Rift, which followed another rifting phase and associated passive subsidence (Fielding et al.,

2006; Wilson et al., 2012). There are no regionally mapped reflectors above Ri, due to the prevalence of glacial clinoforms and erosion at the seafloor. However, in the proximity of Ross Island, where additional accommodation space was created in isostatic response to the loading by Ross Island volcanoes, additional reflectors were mapped, including Rj, corresponding to an age of ~3.2 ma and defining the bottom of an interval of sediment that thickens toward Ross Island

(Chen, 2015).

Fault Interpretation

Faults in seismic reflection profiles can sometimes be directly interpreted, as they might bring one rock type next to another, and therefore define a plane across which there is an impedance contrast. Alternatively, and more commonly, fault traces may be inferred due to the truncation or separation of other reflectors. This is possible on faults where there is a significant dip-slip component to fault motion, or otherwise in profiles with significant stratal dips into or out of the profile. When there is neither, which is to say, in the case of shallowly-dipping strata cut by strike slip faults, like those predicted by the transtensional model, these faults are nearly invisible in profile. A vertical profile of a fault cutting through strata will only show dip-separation, and the magnitude of this separation is the result of contributions of the actual slip vector, stratal dip, and the angles between the profile, strike of the fault, and strike of the strata.

These effects are as likely to diminish each other as they are to compound, as highlighted in Figure 19.

Figure 19. Similar separations in profile resulting from different kinematics superimposed on different structures.

While a classical Andersonian strike-slip fault is near vertical, the transtensional model of the Terror Rift predicts that many of the interpreted strike-slip faults are reactivated normal faults, and thus have normal fault 23 geometry (Salvini et al., 1997). At depth, however, normal faults typically become listric, their dip angle becoming shallower as the fault ultimately joins up with a shallow-dip detachment – a surface that accommodates a very large amount of extensional strain. Unfortunately, the depth at which this occurs is not visible in the current set of seismic reflection data, as it is obscured by the shallow, loud seafloor multiple.

No matter the method, the mapping of faults in 3D from 2D seismic reflection profiles is interpolation, and somewhat subjective. Conventionally, a fault trend is defined in map view when its trace is identified in two or more proximal profiles. This method is sound only as long as the spacing between the study profiles is generally less than the spatial variability of the faults being studied. The nearest neighbor to a given seismic line in the dataset used in this study is anywhere from 10 to 50 km away. Traditional profile to profile correlation of faults may fail to capture spatial variability of the faults that takes place on scales less than this spacing. To this end, this “first-order” fault trace correlation has been supplemented with other parallel analyses, such as the extensive mapping of stratal dips.

Fault Correlation Criteria

It is necessary to develop a set of criteria for the profile to profile correlation of fault traces. The introduction of objectivity into this process is critical, not just for consistency’s sake, but because of a number of cognitive biases that cause problems when a less structured approach is taken. For example, there is a tendency to default to interpretations in which fault surfaces are perpendicular to the subject profile, when there is no automatic reason this should be the case. Furthermore, some fault trace qualities that an incautious interpreter might heed may have little objective significance. For example, the similarity or difference in apparent dip between two fault traces does not dictate whether or not those traces could be two expressions of the same fault surface, but a subjective interpreter might be inclined to correlate traces of near-equal apparent dip as they may be immediately visually similar.

The criteria used as the framework for fault correlation in this study are:

1. Correlated traces should be nearly coplanar

• A pair of fault traces is a pair of lines situated in 3D space, and as

such, can either be coplanar or skew. The local coplanarity of a pair

of traces is a necessary (but not suffucient) condition for those lines

belonging to the same fault surface. This is a criterion that

diminishes in importance with distance, as fault surfaces needn’t

25 truly be perfectly planar. Throughout the course of this study, coplanarity is qualified in this way:

Any two lines share a plane with which they are both parallel, and this plane can easily be found via stereographic projection.

However, to be coplanar is to be contained in the same instance of this mutually parallel plane, and in the dimension reduction to stereographic space, length (and therefore location and the idea of

“containment”) is sacrificed for the sake of being able to completely describe 3D orientation on a 2D surface. In stereographic space there is no direct way to determine how coplanar two lines are.

However, if two lines are truly contained in the same plane, then any arbitrary line from a point on one to a point on the other is also in that plane, and therefore parallel to it, falling on the same great circle arc in stereographic space. Conversely, arbitrary lines

constructed between two skew lines can never be parallel to both,

and thus will never lie on the same great circle arc. In the context of

this study, the coplanarity of two lines was judged by constructing

the plane that contains both lines and arbitrary lines constructed

between them.

2. Correlated fault traces should exhibit the same sense of motion.

• In 3D, sense describes the relative motion of fault blocks, with

respect to the attitude of the fault surface. In profile, sense is the

direction of separation

of reflectors, with

respect to the direction

of plunge of the fault

trace. However, despite

the reduction in

dimension, this “profile”

definition is still

meaningful. Looking Figure 20. Two examples of past the ambiguities stereographic representations of the coplanarity of pairs of lines. The blue introduced in Figure 19, arbitrary lines fall on the traces' mutually perpendicular plane, demonstrating their “profile sense” or coplanarity. The red arbitrary lines do not plot on their traces' mutually “apparent sense” can at perpendicular plane, demonstrating that these traces are skew, and likely don’t least be used to correlate.

disqualify trace pairs outright, as there is no trick of perspective, no

two ways to cut a single fault with vertical seismic profiles that

would exibit a reversal of sense of displacement of the same

reflector, barring the possibility of rotational fault motion.

Geological explanations for true spatial sense reversals do exist, e.g.

in the rare case of rotational-mode faulting, the case of a fault being

incompletely reactivated with a different sense. These possibilities

were considered on a case by case basis thoughout this study, and

only in cases where other criteria were strongly in favor of

correlation.

3. Traces that exhibit more significant separation are allowed to project

across larger gaps.

• Separation is the magnitude of apparent reflector offset in profile.

On it’s own, separation observed in profile does not elucidate the

slip vector of a fault, but where stratal dips are shallow (as is the

case throughout the study area) it can be said that the magnitude of

slip along the fault is at least equal to the magnitude of separation

(in the case of pure dip slip) or more (in the case of oblique slip).

Neither separation nor slip should be required to match between

correlated traces, as slip magnitude truly varies laterally along a

fault, ultimately diminishing to zero at the tip-line or transferring to

another fault, but separation does play a role in correlation. Though

slip does vary from place to place on a fault, it does so continuously,

and at a rate limited by the rigidity of the substrate. Therefore,

where separation magnitude on a trace is high, one can expect the

fault it belongs to to be laterally extensive, and therefore manifest in

traces on more distant profiles. In cases where a fault doesn’t taper

to a free tip, but instead transfers abruptly to another structure, this

assumption does not hold. Either through direct linkage to a

transfer structure, or across an unlinked accommodation zone, the

strain of a fault can be transferred to another vector over a short

distance. Where dip data is available, however, it can help to predict

these structures.

4. Correlated traces should exhibit the same cross-cutting relationships with

regional unconformity surfaces.

• In this study area, fault traces are truncated above either by the

seafloor or by one of several prominent unconformity surfaces. As

these unconformities represent erosional surfaces, they provide an

upper bracket on the geologic age of faults they truncate. A useful

coarse filter in the correlation of faults is the comparison of

truncation relationships, which should be consistent along their

entire length. However, faults are often reactivated, and needn’t

necessarily be reactivated along their entire length, so there is a

conceivable case in which you might expect the traces of a single

fault surface to truncate into different horizons in different places.

As in the case of sense-reversal, this possibility is only considered

when other criteria are strongly adhered to.

5. Sequence alignment

• A term borrowed here from molecular genetics, sequence alignment

in the abstract is the arrangement of sequences of objects (e.g.

nucleotides or fault traces) in order to identify regions of similarity.

A fault trace has a particular plunge, exhibits a particular separation

and sense of motion, and has particular cross-cutting relationships.

But instead of examining one trace at a time and searching for good

correlation candidates, one can take advantage of the fact that a

particular fault trace is likely one in a sequence of fault traces, each

with different attributes, and where the group of faults to which

they belong crosses another nearby profile, that sequence is likely to

appear again. Trace-set X, trace-set Y, and trace-set Z may each be

individually correlated with a middling degree of confidence, but if

their traces occur in the same sequence on multiple profiles, this

ascribes more confidence to their correlation.

6. Local stratal dips

Detailed in Scott et. al (1994) is a justification for the analysis of stratal dips as a proxy for rift kinematics and fault correlation. Listric normal faults produce rotation of hanging wall strata that results in dip directions parallel to

30 the direction of maximum extension. Beyond that, variations in dip among pre- rift basin strata can signify the presence of faults that are in anomalous orientations or that change with depth.

Dip Analysis

It’s important to know the orientation of strata throughout the study area for the sake of helping to resolve otherwise ambiguous fault separations. A dip map is a key component to this study because the dips of these strata and the faults that cut them are genetically related (e.g. Scott et al., 1994).

Fault geometry can be informed by the examination of dipping strata in their proximity, and the invocation of known relationships between faults and the dip of strata they cut. Where the slip vector along a listric normal fault is constant, the resultant rotation of hanging-wall strata produces dips directly toward the fault. A fault will tend to have a constant slip vector, save in the proximity of free-tips or transfer zones, where strain accommodated by the fault changes continuously and, in these areas, stratal dips do not unambiguously reveal fault orientation, as they are sensitive to the particulars of the geometry of the transfer zone.

Figure 21. Antithetic (with opposite fault dips) and synthetic (with similar fault dips) transfer zones (Morley, 1999). Contour lines represent depth to a stratum. The dip of this surface is only perpendicular to the trace of the fault that controls it where its contours are parallel to the trace of the fault. Where the contours turn from parallel and interfere, stratal dip is not directly perpendicular to any fault trace. The local geometry of faults can be directly and unambiguously observed only where those faults are pierced by the vertical lines of intersection of two seismic reflection profiles. Unlike faults, the strata they cut are spatially ubiquitous and their geometry can always be determined at seismic reflection profile intersections where the data quality allows it. Where fault geometry controls this stratal dip, that geometry can be elucidated by dip observations.

Thus, a field of stratal dip information is a substantial aid toward fault trace correlation.

But while stratal attitudes can be determined with high accuracy in places where three proximal, noncollinear points on a reflector can be identified (e.g. 32 where two profiles cross), the assumption that the reflector is locally planar breaks down as you try to interpolate over larger and larger gaps. Also, across too small a gap, interpolation (plane fitting) becomes too sensitive to the resolution of the seismic data (or any other source of error in the location of the subject stratum).

While it may be appropriate to fit splines or surfaces holistically to other data sets, this study takes an approach to dip mapping that honors the well constrained geometry in these goldilocks zones - small enough that a plane fitting is meaningful, and large enough that the fitting isn’t overly sensitive. Typical plane-fitting operand point baselines in this study were between 1 and 3 km. Via an algorithm outlined in appendix B, a classic three-point problem was applied near profile intersections throughout the study area. These interpretation methods are sensitive to the differences among the various seismic facies that occur within the study area. Some packages of reflectors are parallel and continuous, whereas others comprise clinoforms formed with original, sedimentary dip. Intervals with a nonzero sedimentary dip must be isolated from strata with dips entirely controlled by tectonics (‘structural dip’).

Both the transtensional model and orthogonal model for the Terror Rift make predictions for the species of structures that should comprise it, and these structures exhibit some characteristic ways in which stratal dip should vary both laterally and with depth. By creating a 3D field of dip data, the various “dip signatures” of these different structures might become visible.

The vertical line defined by the intersection of two seismic reflection profiles pierces many different strata, and the way that these strata change in dip amount and dip direction with depths can be telling. In this study, these “stacks” of dip data are determined by picking three points per reflector at each suitable profile intersection and performing a series of geographic operations on these points that ultimately returns the strike, dip, and dip direction of the plane defined by each set of three points.

Dip Signatures

The structures sought in this study area have characteristic signatures in the geometry of the strata that comprise them. As an example, normal growth faults are common in the area and have a distinct signature that is very noticeable in stacks of stratal attitudes. These are faults that slip over a period of time during which there is also sediment being deposited. The result is a sequence of stratal attitudes that records a history of the rotation of the fault blocks that the fault divides.

In Figure 22, the vertical dashed line marks the intersection line of 2 seismic profiles that pierces several significant interfaces. First (from top to bottom) the line crosses through the seafloor into unconformable strata with a slight apparent dip to the right of the profile, then through the green reflector and into a sequence of strata with steadily increasing dips, then through the red horizon and into a sequence of reflectors with relatively steep (~6°) but uniform dip, and finally into the fault. This records the clear geologic history of ongoing fault motion and its timing relative to deposition of strata. The parallel strata below the red horizon were deposited entirely before the first episode of motion, the parallel strata above the green horizon were deposited before the second episode, and the fanned strata in between these horizons were deposited during the first episode, as the hanging wall was rotating and accommodation space was continuously being created.

Figure 22. An interpreted profile exhibiting a clear growth history.

Growth faults, in the abstract, feature sequences of uniformly dipping pretectonic strata under fanned syntectonic strata. This can be considered the

“dip signature” of a normal growth fault.

Cross-Section Methods

Bed-length Balancing

A seismic reflection profile that crosses multiple fault blocks in an orthogonal rift will have the following property: A segment of a reflector within pre-tectonic strata should have the same length in the deformed and retrodeformed state of the section. Furthermore, the segment should have the same length as any parallel reflector segment directly above or below it. This is called bed-length balancing, and is a powerful tool for constructing and testing structural cross section interpretations. Just as a single reflector segment

(defined by two points along its length) can be balanced, so can a segment of a sequence of reflectors representing a conformable sedimentary sequence.

Figure 23. Line NBP0401-148, interpreted between pin lines (red)

Retrodeformation is accomplished by the removal of fault displacement and folding of a target horizon. This was done in Adobe Illustrator®, which has integrated tools for measuring lengths of both straight and curved lines. Fault heave, the horizontal component of displacement along a fault, can confidently be restored, even in a two-way-time domain, but the restored extent of folded layers 36 requires a conversion to depth. Assuming a single acoustic velocity for rock 0f 2 m/s, a curved reflector can be rescaled in the vertical direction such that its vertical and horizontal scales are equal. At this point, its length is measured and it is redrawn as a horizontal line segment.

Fault Geometry Construction through Dip Domains

Another cross-section method employed in this study is the projection to depth, below the seafloor multiple, of the geometry of possible large, normal faults that may have formed hanging wall folds such as a rollover anticline – what we perhaps see in the Lee Arch anticline. If such a fault were planar, the strata would not have to change shape in order to translate along it. As deep as they can be interpreted visually, the traces of the normal faults in this region typically appear nearly straight. However, if the structure within the Lee Arch anticline is a rollover and keystone structure, then the controlling fault must be non-planar.

More than that, it must have the specific shape that would deform nearly planar, flat lying strata into exactly the shape we see in profile today. The geometry of the fault would determine the geometry of the strata uniquely, and can therefore be determined by looking at the geometry of the folded and faulted strata, in the context of geometrical and rheological rules for how they can or would deform.

Construction of dip domains - segments of packages of reflectors that have nearly equal dip and dip directions – and the projection of those dip domains to depth along Coulomb collapse angles can uniquely define the single controlling fault geometry that would produce those dip domains via an

algorithm described in Tearpock & Bischke (1991). Such dip domains are often

created by hand, and thus tend to approximate long segments of reflectors as

being within the same dip domain. For this study, reflectors have been

interpreted across the Lee Arch anticline and these point data have been passed

to a program – written for this project (Appendix A) – that treats the segment

connecting any two subsequent data points as a dip domain, and projects these

dip domains to depth with a given Coulomb angle, thus, every point on the

Figure 24. A “coarse” reconstruction from Tearpock & Bischke (1991) (top) and a “fine” reconstruction via the modified algorithm created for this study (bottom). This modified algorithm, outlined below, trades a little stability for more detail. 38 reflector maps directly to a point on the predicted fault. This allows for more resolving power of the subsurface structure and the ability to tinker with constants, such as the Coulomb collapse angle, for purposes of sensitivity testing.

The algorithm for the reconstruction of controlling fault geometry is as follows:

I. Create a domain profile

i. A domain profile is a continuous theoretical trace of a surface

representative of stratal dips. In essence, it is a reflector with the

effect of any secondary faulting removed. This trace should

intersect the subject fault and reflect the dips of pre-tectonic strata.

II. Load domain profile into an x, z space

i. The profile is exploded into a multipart data set of points, with

attributes of latitude, longitude, and depth.

ii. The data is projected to a planar profile. The trace of the seismic

profile on which the domain profile was constructed is unlikely to

be straight. A new, planar profile is defined running through the

first and last data points in the series of points comprising the

domain profile, and all intermediate data points are projected along

strike to this profile. Ideally, this profile is nearly perpendicular to

strike, but fault projection is still as possible and meaningful even at

moderate obliquity, because rollover anticlines can be

approximated closely being cylindrical folds, which is to say they

can be defined by a single cross sectional profile swept along a

single axis (strike, in this case). Such a geometry cut by vertical

profiles of different orientations will only differ in appearance on

those profiles by some factor of horizontal scaling. For this reason,

the product of fault projection on a moderately oblique profile can

ultimately be projected with confidence along strike back to a

profile that is perpendicular to strike. This is only as valid as it is

true that rollover anticlines have this cylindrical geometry, which is

an approximation that breaks down over long distances. Thus,

profiles that are nearly perpendicular to structural strike are still

ideal.

III. Construct dip domains

i. Per the Tearpock and Bischke algorithm, a dip domain is an area of

equal stratal dip bound laterally by Coulomb collapse lines (a failure

angle that is a rheological property of the substrate, typically

between 60 and 85 degrees). A given Coulomb collapse line thus

divides two dip domains, one inboard (toward the fault), and one

outboard. These Coulomb collapse lines are drawn in the synthetic

or antithetic direction (with respect to the fault) depending on

whether the strata in the outboard domain are dipping in the

synthetic or antithetic direction. The width of a dip domain is a

result of the fitting of tangents to a marker bed. The primary

difference between the Tearpock and Bischke algorithm and the one

employed in this study is that the restored marker bed (the domain

profile) is further, and automatically, discretized, such that a dip

domain is defined as being bound by a Coulomb collapse line

passing through one data point and another through its immediate

neighbor – that dip domain’s defining dip (or apparent dip) being

the slope between those data points.

IV. Project controlling fault geometry

i. Consider the data points comprising the

domain profile to be named [P0, P1, P2,…

Pn], where P0 is coincident with the fault.

Each point after P0 has an associated

Coulomb collapse line S passing through

it, so these can be named [S1,… Sn]. Figure 25. The mechanics behind the “refraction” of the Each point P can be mapped to controlling fault initial condition. exactly one other “sister point” F on

the fault, and the location of each point F can be determined

sequentially. F0 is coincident with P0. F1 is located where the

projection downward of the initial condition of the fault intersects

S1. A simple interpretation might place F2 where this projection

continues straight on to S2, but planar faults require no hanging-

wall shape change, so this couldn’t explain the transition into a new

dip domain across S1. Were the controlling fault planar, its trace

would continue downward along the black dashed line that appears

on Figure 25. Instead, it is deflected upward along S1 by vector D1.

The resulting shape-change on the stratum (red) is a deflection by

that same vector along the same Coulomb collapse line. Thus, the

fault trace “refracts” in a predictable fashion as it crosses each

collapse line.

This process is a sensitive one. As the projected fault trace refracts across collapse line after collapse line, any errors quickly compound. Because of this, the problem is easily perturbed by the initial condition of the fault, which is the location and near-surface dip angle of the proposed ‘controlling fault’, and by the shape of the domain profile. A reconstruction is considered ‘stable’ if it retains its general morphology and geological feasibility when its input parameters are perturbed. Stable results were only achieved once a small ‘skip interval’ was defined. A skip interval is an integer defined in the interest of reducing the number of refractions a reconstruction makes by having the program pass over a number of data points, only evaluating those data points whose indices are multiples of the skip interval. This sacrifices detail for the sake of added stability.

Results

Stratal Dip Patterns

Map view stratal dip patterns

Vertical structural dip profiles – stacks of calculated attitudes of different horizons along a vertical piercing line – were collected at 109 seismic line intersections. Each dip profile plots the computed dip magnitude and direction for regional reflectors mapped northward from the McMurdo Sound region by

Hall (2006) and other unnamed reflectors that were selected to sample stratal dips at ar0und 100 ms intervals. ‘Structural dips’ refers to tilting of strata due to post-depositional displacements, distinguished from ‘sedimentary dips’ that form during deposition, for example clinoforms. The stratal dip information exhibits patterns both in map view and with depth, and was collected to directly assist with fault mapping and to reveal rift structure dip signatures characteristic of regional basin-bounding border faults and intervening accommodation zones.

Where regionally continuous, large-displacement normal border faults control rift basin stratal tilting, dip directions are perpendicular to fault strikes over long distances. Thus, stratal dip can constrain fault strike direction. Because the regional reflector denoted as ‘Rg’ provides the most extensive stratal dip information spatially across the study area, that horizon exemplifies map-view stratal dip patterns along Terror Rift (Figure 27). Only the McMurdo Sound region shows a uniform dip pattern, dip directions and magnitudes have considerable variability further north along the rift. Within McMurdo Sound,

43 there is a uniform ENE dip direction and dips increase with depth (Figure 27,

Figure 30, & Figure 31). This consistent dip direction and subtle growth pattern with depth can be explained by a small number of extensive, NNW-striking and

WSW-dipping listric normal faults. The location of these major faults must lie below the volcanoes of Ross Island and the seafloor volcanic bodies extending northward from it.

In contrast, further north, in the area outboard of the Drygalski Ice

Tongue, the spatial pattern of stratal dips is more complex. Overall dip directions remain dominantly oriented ENE-WSW. Changes in dip direction define 4 regional NNW-trending anticlines and synclines (Figure 26, Figure 27).

Figure 26 A section of line NBP0401-148m, depicting anticlines and synclines as they occur NW of Drygalski Ice Tongue.

This fold pattern is more complex than the ‘Lee Arch’ and ‘Discovery

Graben’ previously defined by Cooper et al. (1987) and the ‘regional anticline’ and

‘regional syncline’ previously defined by Hall (2006). Fold axial traces parallel normal fault strikes, described in following sections, suggesting a genetic link.

Regional fold sets are not characteristic, however, of normal faults with smoothly curved listric fault profiles. Fault-fold relations are addressed in the following section on ‘Dip Domain Analysis’.

The rift stratal dips in the Drygalski sector also show deflections in dip direction on scales <10 km (Figure 27), suggesting a stratal response to local effects, rather than systematic rotations of dip directions in response to changes in regional fault strike. Fault mapping, presented below, yielded a population of faults with only a moderate range of strikes, yet the range of dip directions in the proximity of Drygalski Ice Tongue is much larger. This, along with the high spatial density and low correlation multiplicity of faults in this area, suggests a high frequency of transfer zones between offset, underlapping or overlapping faults. The stratal dip patterns support fault interpretations invoking a more widespread, diffuse distribution that that proposed in Hall et al. (2007).

Figure 27. Measured dips of horizon Rg throughout the study area, with interpreted fold axial traces. Of note are the relatively uniform dips within McMurdo Sound and the regionally developed anticlines and synclines outboard of the Drygalski Ice Tongue.

Stratal dip patterns with depth

An increase in dip with depth, a typical signature of syndepositional, normal- displacement growth faults, is observable on many dip profiles (Figure 28). In the simplest case, this signature is defined by the steady increase in dip magnitude with depth toward a single dip azimuth, reflecting the deposition of sediment throughout the continuous rotation of a subsiding hanging-wall fault block (Figure Figure 28. (above) 29). Some dip profiles pierce intervals of Superimposed dip magnitudes with depth at all 109 pretectonic (parallel) strata at greater depth, in intersections which case multiple data points in the deepest Figure 29. (below) A polar plot of the data intersection 105, part of the series have the same, relatively high depicting progressively steepening dips (1° - 7°) to the dip magnitude. west with depth. These data have been organized into polar plots. Each polar plot corresponds to one profile intersection, depicting orientations of pierced strata by their dip direction and magnitude (Figure 31- Figure 34).

Figure 30. The locations of the 109 intersections along which structural dip profiles were collected. These profiles can be found on the subsequent pages.

Figure 31. Structural dip polar plots, sheet 1 of 4

Figure 32. Structural dip polar plots, sheet 2 of 4

Figure 33. Structural dip polar plots, sheet 3 of 4

Figure 34. Structural dip polar plots, sheet 4 of 4

Figure 35. The symbols that appear on the polar plots.

Figure 36. Polar plot locations, colored to reflect the degree of growth they show.

Just as growth signatures can be identified on these plots (as in Figure 29) so can variants of this signature.

Figure 37 depicts a profile intersection that pierces down through an angular unconformity. The thin, onlapping growth wedge (Figure 37, upper) was thin enough that it wasn’t sampled, but the pattern of dips becoming suddenly steeper across the unconformity is very clear on this intersection’s polar plot, with a cluster of points corresponding to the many sampled, parallel strata.

Figure 36 is a colored map of polar plot locations, with their corresponding points colored to indicate how clear a growth signature they convey. A plot was considered Figure 37. A structural dip to show a strong growth signature if it depicted progressively profile piercing unconformity Ri steeper dips towards a single azimuth. When such a pattern and entering an interval of occupied only a small range of dip magnitudes, or when it parallel strata. was generally weakly expressed, it was classified as a “moderate” growth signature. The spatial trend of this classification shows very strong growth in

McMurdo Sound, and in the north near Cape Washington. This growth is very steady and all in one direction, suggesting regional control, while the growth near

Cape Washington was as likely to be toward the east as it was the west, suggesting local fault control.

Fault Patterns

Fault identification, correlation, and mapping

The intersection of a fault and a reflection seismology profile is denoted a ‘fault trace’. These appear in profile as a line or curve into which reflectors truncate (Figure 38); 429 fault traces were identified in this study. The result of the trace identification phase is summarized on Figure 39.

Every fault trace corresponds to a fault, Figure 38. An example of a fault but any one fault may comprise any number of trace (red) traces. The initial fault mapping phase involved the correlation of fault traces between adjacent seismic lines per the criteria outlined in the Methods section.

The result of this process is Figure 40, in which correlating fault traces were connected into faults.

Each fault can be described as having a correlation “multiplicity” – the number of linked fault traces that comprise it. This number is a consequence of both fault and survey geometry, and thus doesn’t independently reveal anything about these faults. Of the 429 identified traces, 52 did not correlate to a second trace, i.e. belonged to faults of multiplicity of 1. The extent of these faults is unknown, and they do not appear on the correlation map (Figure 40).

Figure 39. Identification-phase map, depicting interpreted fault traces. Apparent dip direction, separation magnitude, and unconformity surface that truncates each fault are denoted by symbol types, size and color.

Figure 40. Map of faults correlated between seismic profiles.

The remaining 337 fault traces were assembled into 143 faults of multiplicity 2 or greater. Faults of multiplicity 2 are likely underrepresented in this phase of fault interpretation, as they inherently make for correlations of lower confidence.

Two traces are much more likely to be expressions of the same fault if a third compatible trace can be observed between them (Figure 41). A match to a third trace was commonly the deciding factor for traces that otherwise only weakly correlate because, for example, Figure 41. An isometric view (black) featuring three profiles. Traces they have different slip magnitudes, or interpreted on the red and purple profiles project nearly exactly through are somewhat skew. Thus it’s almost a compatible fault trace on the intermediate blue profile. This is a certain that I have interpreted some strong argument for the correlation of all three traces. multiplicity 2 faults as two multiplicity 1 segments, and in general, erred on the side of under-correlating.

Fault geometry

Where faults are planar, their strike can be defined as the bearing from one trace to a correlative one at two points of equal depth or, where a fault is curved, this value is the average. This calculation can be achieved by way of a subset of the formulas invoked in the plane fitting algorithm (Appendices A and

B). Similarly, their dip can be computed, given this calculated strike and the plunge of one of its traces.

Calculated fault strikes vary from

345° to 005°, with a slight systematic shift

clockwise from south to north (Figure 40). A

rose diagram histogram (Figure 42) shows

that, relative to the overall average strike of

355°, the population of east-dipping faults

strikes just counter-clockwise and the west-

dipping just clockwise of the average. This is

most likely due to data coverage relative to Figure 42. A histogram of calculated fault strikes with a bin the geography, rather than any structural size of 7.5°. Fault strikes reported per the right hand rule, so the rotation. Generally, the Terror Rift exhibits blue bins represent east-dipping faults, and the red west-dipping. an antithetic geometry, with east-dipping faults in the west, and west-dipping faults in the east (Figure 39, Figure 40), and a roughly equal number of each is observed. However, more east-dipping faults are visible in the south because the presence of overprinting volcanics of Ross

Island obscures west-dipping faults, and more west-dipping faults are visible in

the central and northern Terror Rift due to lack of data in the

vicinity of the Drygalski Ice Tongue in the west. Thus, west-

dipping faults are obscured where the regional strike is closer

to 000°, and east-dipping faults are obscured where the

regional strike is closer to 350°.

Fault dips ranged from 60° - 70°, with little variation

as a function of location. However, a subtle systematic trend

of shallowing of fault dip can be observed as a function of

depth, as defined by grouping faults that are truncated by

successive unconformity surfaces (Figure 43). This is

consistent with the idea that many of these faults are listric

at depth, as a fault may rotate not just a fault block, but

other, older, inactive faults on its hanging-wall side.

Of the faults that reach the seafloor, the average dip

and the standard deviation are higher due to a small number

of very steep (~80°) normal faults superimposed on the rift,

especially within Cape Roberts Basin. The large standard

deviation among faults that are truncated by surface Rg is

due to the fact that these faults have been subject to multiple

Figure 43. Average fault dips ± one standard deviation (grey wedge), organized by truncation by different unconformity surfaces.

60 rotations due to multi-phase fault slip and, depending on how a fault is oriented relative to a rotation, it can either be made steeper or shallower.

Fault motions: sense and separation

Of the 429 identified traces, all but two showed clear normal sense offset.

In addition to the overwhelming majority of normal sense separation, stratal dip signatures indicate that many faults are growth faults. Interpreted traces show a range of separation magnitudes, limited at the low end by the resolution of the seismic data, around 10 meters, and upwards to around 140 meters, with a small number of traces show a separation of nearer 400 meters. Large-magnitude fault traces occur with a steadily increasing frequency from south to north, and, to a lesser extent, from west to east.

The separation along a single fault trace can vary with depth. Just as progressively older strata in a growth wedge have experienced more rotation, so have they experienced more offset. Thus, the separation magnitude along a length of a trace bounding a growth wedge is greatest at the bottom of the wedge and diminishes continuously upwards. Similarly, an inactive fault may be buried by sediment and later reactivated, propagating up through the younger sediment, so a trace may have a high separation magnitude at depth, and a suddenly smaller one above an unconformity surface. The magnitudes reported here and on the trace identification map (Figure 39) are the measured separations of Rg across each trace. Where Rg was not visible above the seafloor multiple, the deepest visible reflector was considered instead. Therefore, the separation

61 magnitude ascribed to a fault trace is actually the high end of its range of magnitudes.

The number of individual fault traces exhibiting a range of separation magnitudes increases from south to north. Whether due to continuous, syndepositional growth faulting, or to episodic fault reactivation, fault traces that show variable separation with depth are common. North of Drygalski Ice Tongue, virtually every fault that reaches near the seafloor offsets regional unconformity

Ri by a smaller amount than it does the strata below it, indicating that the majority of the youngest faults in that area formed via the reactivation, or continuous activation, of preexisting faults.

The two traces that do not clearly show normal separation (denoted with a yellow symbol on the trace identification map (Figure 39)) do not clearly exhibit reverse sense offset either, nor do they correlate to a second trace, making their interpretation yet more difficult. The first of these occurs on line IT90a-61b, between Drygalski Ice Tongue and Cape Washington. It was originally identified and described in Rosetti et al. (2006), in which it is claimed that it exhibits both normal and reverse separation at different depths. Such a thing could happen by the reactivation of a preexisting fault with the opposite sense, or, as Rosetti et al. proposed, as a result of strike slip motion. If a stratigraphic interval varies laterally in thickness, a strike slip fault could bring sections of different

62 thicknesses against each other, producing an apparent reversal in sense. Rosetti et al. suggest that this is taking place within the interval bound by horizon Ri below and the seafloor above.

Horizon Ri shows clear normal Figure 44. (above) An annotated figure from (Rosetti et al., 2006), identifying a fault as sense separation. However, showing a sense-reversal with depth. the interval above it is made up of hummocky, poorly Figure 45. (below) a closer view of that same defined reflectors, and none exhibits clear reverse or fault. normal separation. The reflector ascribed a reverse sense of separation by Rosetti et al. (2006) is the seafloor itself, and indeed, there is a scarp-like feature above this fault trace, but it is not clear that this must be a fault scarp as opposed to some other type of seafloor geomorphic feature.

The second trace that fails to show clear normal- sense separation occurs entirely between seismic traces

381 and 382 on line PD9045. Because it doesn’t cross a single seismic trace, its plunge must be almost exactly

90°. That fact, along with the unusually prominent fault scarp suggest that this is actually not a fault at all, but some form of acquisition or processing error. In either case, it doesn’t correlate to a second trace, and didn’t advance to later stages of interpretation.

Figure 46. An anomalously steep disruption in reflectors observed on PD905.

Fault Profile Reconstruction

Five different E-W trending profiles which exhibited potential rollover geometries were selected for analysis, to assess whether observed regional folds could be produced by controlling normal faults of geologically plausible geometries.

Controlling faults were reconstructed on five different profiles: IT90a-60,

NBP0401-148, NBP0401-150a, the Figure 47. (above) Locations of profiles on which well-developed combination of IT90a-65 and NBP0401- rollover structures yielded reasonable, stable controlling fault 142m, and finally US-407. These profiles projections.

Figure 48. line NBP0401-148, annotated. The red line denotes the dip domain profile that served as one of the inputs into the controlling fault projection algorithm. The solid black line is the controlling fault, both its initial condition where the fault intersects the seafloor (interpreted directly) and the projection of this fault to depth. This calculated projection eventually became unstable as it projected downward and to the west, as compounding refraction errors overwhelmed the solution. The dashed segment at the bottom of the figure represents the last acceptable segment. Listric splays to the east (right) of the controlling fault are denoted by black dashed lines. The integrated slip on these faults supplies the main controlling fault with the slip necessary to have developed the observed rollover structure.

65 were chosen because they each met several conditions: first, they each exhibit features that make them good candidates for rollover structures from the outset, most notably either one or two anticlines, but also other features defined by

McClay (1995) such as ramp synclines and crestal collapse faults (keystone faults).

Second, the profiles comprise well defined reflectors, little obscured by volcanics. Volcanic-free profiles are limited spatially to the latitudes between

Cape Washington and Ross Island. Third, the profiles are close to parallel to the dip direction, and close to straight. The algorithm takes steps to meaningfully project input data to a synthetic profile that is perfectly planar and parallel to dip, but these steps introduce less error if less projection is required.

Fourth, and finally, any proposed rollover structure must have a candidate for a controlling fault “initial condition” – the part of the controlling fault that is visible above the seafloor multiple and that is fed into the algorithm – and this Figure 49. The fault (or faults) that fault must meet or exceed a certain produce (or compound to produce) a rollover structure must meet a certain amount of offset. No volume of rock throw threshold. making up a rollover structure moves up, relative to the footwall. Crestal anticlines develop because their crests move down less than their limbs do. Thus, the change in elevation between the highest occurrence of a reflector on the

66 hanging-wall and the point where that same reflector meets the fault (red interval on Figure 49) defines the minimum amount of throw the fault must have experienced in order to produce an observed rollover structure. Where one fault cannot meet this minimum, several may compound to do so, as is illustrated on

Figure 49 and is thought to be the case for the rollover structure on line

NBP0401-148 (Figure 48).

Dip domain profiles constructed for each seismic cross section were based off of the changing attitude of horizon Rh. Rh was here chosen for the same reason that this surface was chosen for balancing – it is old enough that it experienced almost all of the extension that produced these rollover structures, while not being so deep that it is lost in the seafloor multiple. A profile’s construction began with fitting a spline directly to Rh where it is truncated by the fault, and then following Rh throughout the hanging wall, but restoring the displacement of secondary faults and assuming a parallel fold geometry of the rollover. A reconstructed fault profile was deemed ‘stable’ when a plausible variation in input parameters produced nearly the same reconstruction, and

‘unstable’ where small variation in the input parameters produced very different fault reconstructions.

The solutions yielded by this process varied from line to line, but generally fell into three versions of ramp-flat, listric geometry. In the north, the fault reconstruction for profiles NBP0401-148m and IT90a-60 share characteristics,

67 as do the reconstructed faults for US-407 and IT90a-65 + NBP0401-142m in the south, while the NBP0401-150a fault projection is a third variation.

The solutions produced for the southern lines US-407 and IT90a-65 +

NBP0401-142m were the least stable, with relatively small changes in input parameters producing unreasonable results. It is possible that this instability was at least partially due to the fact that their profile geometries were the least ideal of the five lines examined; IT90a-65 + NBP0401-142m has a 5 km gap between the two lines, and US-407 deviates the most from being a straight line.

Controlling faults reconstructed for these profiles feature a listric depth of ~9 km

(Figure 53, Figure 54), and all reasonable variations of these solutions feature some degree of “step” in the fault at ~7 km depth, necessitated by the fact that these profiles exhibit a poorly developed ramp syncline (e.g. at trace #345 on

IT90a-65 (Figure 53).

The solution for NBP0401-150a (Figure 52) was more stable, allowing for a smaller skip interval and smoother predicted fault trace. This trace has an inflection point and a listric depth at 4.1 km. The nature of this inflection point – whether or not there is a ramp-flat geometry along this profile – is hard to determine because of the limited westward extent of the seismic data due to the

Drygalski Ice Tongue. This profile exhibits a single anticline and crestal collapse, with a dip change toward a syncline at its western end, suggesting another inflection of the projected fault.

The solutions for NBP0401-148m (Figure 51) and IT90a-60 (Figure 50) were very stable. Both depict a well-defined ramp-flat listric geometry with listric depths stepping down from ~2.6 km to ~4 km across a 12 km long step – a geometry that resulted in the clear collapse structure visible in the profiles, each with an upper and lower crestal anticline and intermediate ramp syncline.

A sixth profile, IT90a-57 was initially selected for this process, but no solution was ultimately produced. Despite the profile itself having an acceptable geometry and exhibiting a clear crestal collapse anticline, an objective construction of a domain profile proved impossible due to the insufficient amount of collapse structure that wasn’t obscured by the volcanics in the proximity of Cape Washington, and to the lack of visible fault traces that made good candidates for the controlling fault. Based on the domain profile, however, the controlling fault would be similar to the other northern lines, NBP0401-148 and IT90a-60, with two different listric depths separated by a well-defined step.

All of the fault reconstructions yield ramp-flat and listric geometries, changing from south to north in listric depths and number of steps.

Figure 50. Line IT90a-60 and its projected controlling fault in TWT.

Figure 51. Line NBP0401-148 and its projected controlling fault in TWT.

Figure 52. Line NBP0401-150a and its projected controlling fault in TWT.

Figure 53. Line and IT90a-65 + NBP0401-142m and its projected controlling fault in TWT.

Figure 54. Line US-407 and its projected controlling fault in TWT.

Extension Estimates from Balanced Sections

Bed length balancing on three different sections yielded unambiguous results, whereas balanced sections in other locations proved prohibitively difficult to construct. Those profiles for which reconstructions were created were

NBP0401-148, NBP0401-150a, and US-407. The horizon flattened in each case was “Rh”. This horizon was chosen primarily because of its lateral continuity, and inspection shows that it has seen most of the extension that the seafloor-to- multiple interval has experienced.

Other prominent reflectors tend to terminate into either the seafloor, or the seafloor multiple.

Extension magnitudes along these profiles since Rh were Figure 55. Profiles selected for balancing. measured to be around 7.2% in the cases of NBP0401-148 and US-407, and 1.4% in the case of NBP0401-144m150a

(Figure 56, Figure 58, and Figure 57 respectively).

The overall extension amount calculated by this method is a function of two geometric effects that act in opposition to each other. The primary

75 contributor to extensional strain on these profiles, and everywhere in the study area, is the integrated heave of the numerous normal faults visible on every profile. However, there is also folding developed in the strata throughout the study area, and from a geometric bed-length-conservation standpoint, this folding results in contraction rather than extension.

The best explanation for the development of folding, as demonstrated in the previous section, is the shape change required of hanging-wall strata translating along a listric, ramp-flat normal fault. This bending does not actively oppose extension, i.e. was not dynamically compressive, but does result in a strain “debt”, and a significant measurable portion of the integrated fault heave on these profiles goes toward paying the strain debt that this folding creates. By bed length restoration, NBP0401-148 has experienced 7.2% extension since Rh.

At the same time, the integrated fault heave between the same pin lines equals

10.1% of the restored length, meaning that the missing 2.9 percentage points, amounting to about 29% of the integrated fault heave, went into opposing the profile-parallel contraction necessitated by the development of the folds.

Figure 56. Interpreted and restored NBP0401-148. Rh is dark green, Rg is pale green.

Figure 57. Interpreted and restored NBP0401-150a. Rh is dark green, Rg is pale green.

Figure 58. Interpreted and restored US-407. Rh is dark green, Rg is pale green.

The low strain value, 1.4%, measured along NBP0401-150a is not enough to produce the rollover anticline potentially observed there. One possible explanation for this is that the controlling fault or faults that formed this rollover are not imaged on this line, but are instead east of its extent. There are no obvious candidates for such faults to the east, however.

In the north, towards Cape Washington and the south, towards Ross

Island, the contribution to extension by volcanic intrusion increases, evidenced by zones of disrupted reflectors and/or antiformal reflector patterns (“pull-ups”) due to generally faster acoustic travel times (Figure 59) . It is not possible to quantitatively estimate the volume of intruded volcanic material within these zones, hence the strain due to intrusion can’t be calculated. At the same time, the

Figure 59. A section of line IT90a_57. Within zones where reflectors are disrupted by volcanic intrusions, the contribution to strain by intrusion is very difficult to ascertain. In terms of cross-sectional area, some fraction of these zones belongs to actual intrusive material, and the percent strain along any section containing these zones is highly sensitive to this elusive figure. Thus, profiles with more than a negligible presence of these areas make for poor candidates for bed length balancing.

80 volcanic intrusions obscure the geometry of fault traces and the continuity of reflectors that are critical to section balancing. Lacking these data, a given profile has a range of possible restored lengths, and the size of this range increases as volcanic intrusive zones make up a larger proportion of the profile. Therefore, only sections crossing the central portion of the Terror Rift where volcanics are largely absent were balanced in this study.

Interpretive Maps

A final interpretative fault map (Figure 60) was produced by the synthesis of the fault correlation, fault geometry, dip analysis, previously detailed, and known geometric characteristics of rift fault patterns. The map of Terror Rift faults (Figure 60) differs from the map produced in the correlation phase in a few important ways. First, it is the first product of this study intended to represent interpreted regional fault location and geometry, whereas the correlation phase map primarily depicts correlation relationships with only general geometric interpretation in the spaces between seismic profiles. Second, where every product heretofore has been created in as objective a way as possible, this fault map is necessarily interpretative, in an effort to resolve rift architecture.

Nonetheless, certain rules and assumptions were maintained in its construction. First, the locations and qualities (apparent dip, separation, etc.) of fault traces interpreted in the identification phase are strictly maintained, whereas the linkages into faults established in the correlation phase are modified to a limited degree in this interpretation. Second, a fault’s projection into

81 interprofile space depends on its separation magnitude and its proximity to other faults. A low-separation fault in a densely faulted area can reasonably be expected to have a shorter extent than a large-separation, more isolated fault. Finally, faults were allowed to curve, but as a curvature that is concave in the dip direction is a more common occurrence of normal faults, convex curvature was invoked sparingly.

At smaller scales, stratal dips, and even apparent dips, served to inform the linkage of faults. The variety of species of possible fault linkages and transfers corresponds to a variety of dip patterns (Figure 21), and even incomplete information can help elucidate these fault interactions.

The pattern of faults throughout the Terror Rift is, in some ways, a classic rift graben architecture, with east-dipping faults dominating the west, and west- dipping faults dominating the east. However, these faults are not strictly segregated into this antithetic geometry – faults of either orientation can be found throughout the region, but the pattern of their occurrence varies dramatically along regional strike.

Strata in McMurdo Sound are largely unfaulted, while at the same time exhibiting steady growth to the ENE, both within and outside of the flexural moat surrounding the volcanoes of Ross Island (Chen, 2015). Moving north, the first faults appear, with a diffuse distribution of east and west dipping faults with short extents. The first appearances of the Lee Arch anticline and Discovery

Graben syncline occur just north of Ross Island, but are not well defined at the

82 latitude of Nordenskjold Ice Tongue. Local faults take on a new degree of organization at this latitude, forming longer, simpler, and fewer segments. This is also accompanied by a northward decrease in the abundance of volcanic intrusions.

Figure 60. An interpretive map of faults in the Terror Rift

An apparent shift takes place between Nordenskjold Ice Tongue and Drygalski Ice Tongue, most conspicuously the apparent 25 km offset of the well-defined Lee Arch anticline. This apparent shift has been observed in previous studies and different interpretations have Figure 61. Previously proposed structural been invoked to explain it (Figure solutions to the apparent right lateral shift of the Lee Arch anticline and Discovery 61). In the same zone there is a Graben synclince. different fault distribution characterized by many, short, low-separation faults, and a more chaotic pattern of stratal dips.

Directly east of Drygalski Ice Tongue, the population of faults again becomes organized into one of longer, simpler faults, with antithetic dips in toward the axis of a now gentler, broader Lee Arch anticline. No Discovery

Graben syncline is visible here, but if the same 25km shift is applied the axis would lie just east of the extent of the seismic dataset in this area, motivating previous studies to infer its location there (Figure 3 & Figure 17).

The axis of the Lee Arch Anticline continues northward from here, largely uninterrupted, until it is finally lost in the volcanics of Cape Washington. The

Lee Arch anticline becomes narrower and tighter just north of Drygalski Ice

Tongue. In this northern area of the map, the assemblage of faults flanking the

Lee Arch anticline exhibit some of the largest normal separations observed in the study area – on the order of 150 m. The Discovery Graben syncline is again imaged to the west of the Lee Arch anticline, because the seismic dataset once again expands to the west. Although the syncline is at a distance consistent with the previously inferred apparent 25 km offset north of Nordenskjold Ice Tongue, to the west there is another anticline and syncline pair, buried under a thick interval of sediment sitting on top of regional unconformity Ri. These folds are directly in line with the projections of the Lee Arch anticline and Discovery

Graben syncline to the south. The occurrence of these four parallel, continuous folds, with the western fold pair directly along the trace of the southern fold pair, calls into question the apparent 25 km dextral offset. Instead the folds are interpreted here as continuous, forming part of a set of 4 parallel folds without offset.

Although no dextral shift in Terror Rift structures and, thus, no large-scale accommodation zone or strike-slip fault is mapped in this study, at small scale there are multiple transfer zones between the short faults that characterize this zone, and as evidenced by the diverse dip directions. In general, seismic line spacing precludes determination of the geometry of these small-scale transfer zones. In the case depicted in Figure 62, however, there is clearly a ‘relay ramp’ transferring displacement between two synthetic, overlapping normal faults.

Figure 62. Torsion of the central fault block in the dip direction demonstrates that this is an overlapping synthetic transfer, or “relay ramp”

While Terror Rift fault and fold structures change in nature from south to north, it is flanked by a population of continuous, moderate-high separation west-dipping normal faults. East of this fault zone, faults become scarce, and strata are parallel, continuous, and nearly horizontal. The faults that make up this group correlate directly across the “structural shifts” that occur north and south of Drygalski Ice Tongue, and among them are the faults that are the best candidates for the rollover controlling listric faults on the profiles for which these faults were reconstructed. This fault zone is interpreted as the ‘border fault’ zone bounding the margin of Terror Rift and along-axis changes in rift structure may be attributed to changes in the geometry of these controlling border fault surfaces.

Hanging-wall deformation and a classification of this deformation into various zones was described in the ‘Rollover structures’ section (Figure 15) These zones, shown again here in profile, are mapped along the axis of interpreted rollover structures in Terror Rift (Figure 63, Figure 64).

Figure 63. McClay zones above a ramp-flat normal fault. Modified from McClay (1990) 87

Figure 64. An interpretive map of Terror Rift structures, classified into structural zones.

The along-axis changes in the

structural zones defining rollover

morphology and changes in fault continuity

and pattern can be explained by changes in

the controlling listric detachments surfaces

at depth. Faults along the eastern border

fault zone, where these listric controlling

faults reach the surface, are mostly

continuous. The changes in along-axis

structural patterns is likely due to changes Figure 65. (above) Proposed in geometry of listric surfaces at depth, with locations of subsurface tear faults. This map features the same color the three main along-axis segments divided scheme as the larger structural zone map (Figure 64). by subsurface tear faults. Above these tear

faults, accommodation zones on the hanging-wall facilitate the transition

between rollover styles.

Figure 66. A schematic block model of proposed footwall structure. (not to scale)

The definition of the southern rollover structure diminishes to the south, until finally, within McMurdo Sound, no semblance of this structure persists and strata define a steady, moderate growth pattern, dipping to the ENE. At the same latitudes, the eastern border fault zone becomes lost in the volcanics north of

Ross Island. The persistence of growth strata within McMurdo Sound implies the persistence of some sort of major, west-dipping fault. A simple projection of the border fault system to the south underneath Ross Island (Figure 64) supplies the mechanism for the ENE rotation of strata in McMurdo Sound and provides a possible conduit for the intrusive material found in this area.

Discussion

The overwhelming dominance of normal faults in the Terror Rift is significant to its kinematics, and in particular to the formation of the Lee Arch and Discovery Graben. In either a transtensional or orthogonal extensional setting, an anticline begs explanation.

Cooper (1997) attributed the formation of the arch to inflation by magmatic intrusions, however the expanded seismic dataset images the Lee Arch anticline farther to the north, into an area where magmatic intrusions are sparse.

Salvini (1997) alternatively proposed that the Lee Arch is a positive flower structure. While this would explain the antiformal geometry of the arch, reverse faults are a defining characteristic of positive flower structures and are conspicuously absent. By the same token, negative flower structures often feature

90 a cluster of antithetic normal faults, but would fail to explain the arch’s antiformal geometry.

Hall (2006) proposed that the Lee Arch is a rollover anticline and identified similarities between cross sections of the Terror Rift and of sandbox analogue experiments conducted by McClay (1990). Structural mapping in this study shows that the Terror Rift is characterized by a parallel set of regional folds, consisting of a syncline-anticline-syncline-anticline sequence. Reconstruction of subsurface fault geometry using the dip domain approach shows that the fold sequence is well explained by deformation of the hanging wall to form rollover structures during displacement along an extensional listric fault with ramp-flat geometry. Mapped fault patterns within the rift can be explained by the keystone fault architecture formed in rollover structures.

Hall et al. (2007) also identified and proposed several accommodation zones – notably, one in the proximity of the Drygalski Ice Tongue, across which the structures of the Terror Rift appear to exhibit a dextral offset of 25 km (see

Figure 17). This offset had previously been explained as strike-slip offset (Salvini et al., 1997). Faults mapped in this study continue directly across the Drygalski accommodation zone with a high degree of confidence. A continuous zone of faults that define the eastern margin of Terror Rift are likely candidates for the controlling surfaces of the Lee Arch and Discovery Graben due to their high slip magnitudes, locations, and orientations. If the Drygalski offset were a dextral strike-slip fault, any faults genetically related to the Lee Arch and Discovery

Graben would, of course, also be offset by the strike-slip fault. Instead, the lateral continuity of these faults suggests that there is no apparent offset near the

Drygalski Ice Tongue and, instead, this accommodation zone facilitates changes in styles of hanging-wall deformation from south to north, in response to changes in subsurface listric controlling fault geometry. A second change in style of hanging-wall deformation between the Drygalski Ice Tongue and Cape

Washington is interpreted to mark another accommodation zone due to changes in the geometry of the subsurface listric controlling fault.

The structures of the central Terror Rift can be broken into four segments, from south to north, the McMurdo,

Nordenskjold, Drygalski, and Washington structural zones (Figure 67).

The Washington Figure 67. Terror Rift structural map defining the extent of four structural zones. region exhibits a rollover

92 geometry with well-defined upper and lower antiformal crestal collapses flanking a ramp syncline. This geometry corresponds to a shallow ramp-flat listric normal fault, with listric detachment depths of 2.6 and 4 km.

In the Drygalski region, the upper crestal antiform takes on a much more gentle interlimb angle and longer wavelength. No ramp syncline or lower crest are imaged, due to this larger upper crest, and the narrower swath of seismic reflection profiles in this region. The reconstructed controlling fault features a larger radius of curvature and listric depth of 4 km, with no ramp-flat “step”, although one may be present under the Drygalski ice tongue itself, where a ramp syncline might be observed.

In the Nordenskjold region, the upper rollover anticline once again becomes more narrow and tighter, and a lower collapse anticline can be observed, but the intermediate ramp syncline is poorly defined. Unlike the well- defined ramp-flat style collapse imaged in the Washington segment, the rollover structure in the Nordenskjold segment doesn’t imply a controlling fault with two well-developed listric depths. Instead, reconstructions primarily show a single, deep listric depth at around 9 km and only a small “step” at around 7 km, resulting in the poorly defined ramp syncline separating the upper crestal collapse from the lower crestal collapse.

The transition from the Nordenskjold to McMurdo segment is more gradational, and does not suggest the existence of a large-scale, subsurface tear between them. Instead, the collapse structure of the Nordenskjold segment loses

93 definition to the south, as faults decrease in slip magnitude and volcanic intrusions take on a larger portion of the extensional strain.

Finally, in the McMurdo segment, no rollover structure is observed at all.

Within McMurdo Sound, faults are largely absent, yet the strata in this segment exhibit steady growth to the ENE, still suggesting the presence of a regional, west-dipping, listric fault flanking the east side of the rift. Linking this proposed fault with the established border fault zone sends its projection through a region dense in volcanic intrusions and ultimately under Ross Island, which itself is volcanic in origin. Figure 68 summarizes the geometry of the listric controlling fault systems in a profile view, looking along fault strike, showing the differences in ramp flat geometries of these segments.

Figure 68. Border fault geometries that control the structures in the different zones.

Figure 69. Interpretive maps from Hall et al. (2007) (left), Storti et al., (2008) (center), Salvini et al. (1999) (right), and this study (below). The west-dipping

normal-displacement border

fault zone identified in this

study forms the eastern limit of

deformation associated with the

Terror Rift. Only minor and

isolated faults occur to the east

of this zone. The Terror Rift

border fault zone is not a single

fault where it reaches the

surface but, instead, is

characterized by splays of an

imbricate fan fault system at the

95 surface, which in aggregate provide sufficient displacement to produce the observed rollover anticline folds.

In comparison to previous studies, a similar regional strike was determined for overall Terror Rift structure, turning from NNW in the south towards nearer due north in the north. In this study the regional border fault system was mapped continuously along the eastern Terror Rift margin, rather than being offset as interpreted in previous studies (Figure 69).

The ubiquitous normal-sense fault displacement, the strong case for rollover origin of regional folds within the rift, and the absence of flower structures and other features typical of strike-slip and transtensional fault systems, all support an orthogonal rift model for Terror Rift, with regional extension in the ENE-

WSW direction. Total extensional strain across Terror Rift near the Drygalski Ice

Tongue sector is low, on the order of <10%. This value is comparable to the values determined in Magee (2011) by similar methods. Bed-length balancing was also completed in Henrys et al., (2008), determining a magnitude of extension of

10-15 km, corresponding to extension percentages of ~13% - ~20%.

Vertical dip profiles commonly show strong growth signatures, sometimes across the entire interval spanning from Rg (~13 ma) to the seafloor, suggesting that strata within this interval is syntectonic. This is corroborated by inspection, as a general eastward thickening of intervals above Rg is observed. The most prominent growth wedges – those that occupy larger ranges of stratal dips, corresponding to greater syn-depositional fault slip – occur between Rh (~7.6

96 ma) and Ri (~4.5 ma). So, while there was syn-depositional extension during the entire interval above Rg, the bulk of this extension occurred after Rh and before

Ri. A large number of faults cut Ri, with most having a small separation magnitude in the interval from Ri to the seafloor, and a larger slip magnitude in the interval below Ri, suggesting that they have been reactivated. At the same time, the Ri-seafloor strata have a similar spatial dip pattern as those within the

Rg-Ri interval, imparted by the rollover structure (dipping in towards ramp synclines, out away from rollover anticlines). This suggests that after the period of erosion and non-deposition corresponding to Ri itself, extension continued in the same regional orientation.

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Appendix A: Matlab Scripts

Rollover.m clear skip=; filename=; %####################

T=readtable(strcat(filename,'.csv'));

T.Time=T.Time.*7; figure (1) hold on plot(T.X(1),T.Y(1),'*') plot(T.X,T.Y); squished = [T.X(1), T.Y(1)]; toX = [0, 0]; for i=2:length(T.ID) squished=cat(1,squished,projectpointtoline([T.X(1), T.Y(1)],[T.X(end), T.Y(end)],[T.X(i),T.Y(i)])); toX_int=projectpointtoline([T.X(1), T.Y(1)],[T.X(1)+1,T.Y(1)],squished(i,:)); toX=cat(1,toX,[toX_int(1)-T.X(1),toX_int(2)-T.Y(1)]); end plot(squished(:,1),squished(:,2),'-r'); hold off ry=-1.*twospeedTTD(1384,-1.*T.Time); rx=toX(:,1); figure (2) hold on plot(rx,ry,'-g'); plot(rx(1),ry(1),'*'); hold off flip=input('Flip? (algorithm assumes controlling fault to be at first point) (y/n): ','s'); %%%flip='y'; if flip=='y' ry=flipud(ry); rx=flipud(rx); rx=(rx-rx(1)).*-1;

102 end figure (3) hold on plot(rx,ry,'-g'); plot(rx(1),ry(1),'*'); hold off alpha=input('Fault alpha at first point: '); strike=input('Structure strike: ');

CCA=input('Collapse angle: '); beta=ang_between_az(midaz([T.lat(1), T.lon(1), 0],[T.lat(end), T.lon(end), 0]),strike); CCA_alpha_rad=atan(tan(dtr(CCA)).*sin(dtr(beta))); for i = 2:length(ry)-1 dy(i)=ry(i+1)-ry(i); dx(i)=dy(i)./sin(CCA_alpha_rad); end running_alpha=-1.*dtr(alpha); sx(1)=rx(1); sy(1)=ry(1); figure (4) hold on plot(rx,ry,'-g'); plot(rx(1),ry(1),'*'); plot(sx(1),sy(1),'or'); for i = 1+skip:skip:length(ry) if dy(i-skip) > 0 out=cartesian_intersect([sx(i-skip),sy(i-skip)],running_alpha,[rx(i), ry(i)],CCA_alpha_rad); sx(i)=out(1)+dx(i-skip); sy(i)=out(2)+dy(i-skip); else out=cartesian_intersect([sx(i-skip),sy(i-skip)],running_alpha,[rx(i), ry(i)],0- CCA_alpha_rad); sx(i)=out(1)-dx(i-skip);

103

sy(i)=out(2)+dy(i-skip);

end

running_alpha=atan((sy(i)-sy(i-skip))./(sx(i)-sx(i-skip))); end plot(sx(1:skip:end),sy(1:skip:end),'-b');

Cartesian_intersect.m function [ out ] = cartesian_intersect( p, p_slope, q, q_slope ) %slope in RADIANS y1=p(2); a=tan(p_slope); x1=p(1); c=y1-a.*x1; y2=q(2); b=tan(q_slope); x2=q(1); d=y2-b.*x2; out=[((d-c)./(a-b)),((a*d-b*c)./(a-b))]; end

Projectpointtoline.m function [ out ] = projectpointtoline( v1, v2, p ) % Project x onto a-b in cartesian space

%dot product of v1-v2, v1-p e1=[v2(1)- v1(1), v2(2)- v1(2)]; e2=[p(1)- v1(1) p(2)- v1(2)]; dp=dot(e1,e2); %length of v1-v2, v1-p l1=sqrt(e1(1)^2+e1(2)^2); l2=sqrt(e2(1)^2+e2(2)^2); cos_val = dp / (l1*l2); %length of v1-p' proj_val = cos_val * l2; out=[v1(1)+proj_val*e1(1)/l1,v1(2)+proj_val*e1(2)/l1];

104 planefit.m function out = planefit(a,b,c) f=(c(3)-a(3))/(b(3)-a(3)); cprime=intpt(a,b,f); strike=midaz(c,cprime); alpha=plunge(c,a); theta=midaz(c,a); if alpha<0 alpha=-alpha; theta=az_wrap(theta+180); end beta = ang_between_az(strike,theta); delta = rtd(atan2(tan(dtr(alpha)),sin(dtr(beta)))); dip_az=az_wrap(strike+90); check=ang_between_az(dip_az,theta); if check>90 dip_az=az_wrap(dip_az+180); strike=az_wrap(strike+180); end out=[strike,delta,dip_az]; end fwaz.m function [out1] = fwaz(n,p) %fwaz(n,p) %calculates the forward azimuth (initial bearing) from n to p, %where n and p are coordinate pairs in decimal degrees lat1=dtr(n(1)); lon1=dtr(n(2)); lat2=dtr(p(1)); lon2=dtr(p(2)); out1=az_wrap(rtd(atan2(sin(lon2-lon1)*cos(lat2),cos(lat1)*sin(lat2)- sin(lat1)*cos(lat2)*cos(lon2-lon1)))); end haversine.m function [out1]=haversine(n,p) n=n*pi/180; p=p*pi/180;

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R = georad((n(1)+p(1))/2); % Earth's radius in m delta_lat = p(1) - n(1); % difference in latitude delta_lon = p(2) - n(2); % difference in longitude a = sin(delta_lat/2)^2+cos(n(1))*cos(p(1))*sin(delta_lon/2)^2; c = 2 * atan2(sqrt(a), sqrt(1-a)); out1 = R * c; end plunge.m function [out] = plunge(x,y) out = rtd(atan2((y(3)-x(3)),haversine([x(1),x(2)],[y(1), y(2)]))); end

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Appendix B: Plane Fitting

Several geographical mathematical operations have been utilized throughout the course of this study, primarily for the fitting of planes to sets of three points (A, B, & C) in 3D space, and returning the strike, dip, and dip directions of those planes. The algorithm applied to this problem is as follows:

I. Determine the strike of the plane (θS)

i. Locate C’ - a point at the same elevation as C that lies on the

line that passes through A and B.

ii. The strike of the plane is the bearing from C to C’ (or from C’

to C)

II. Determine the dip of the plane (훿)

i. The dip of a plane can be found from its strike and the

attitude of any plunging arbitrary line contained in the plane

by the following equation:

tan⁡(훼) 훿 = arctan⁡( ) sin⁡(훽)

where α is the plunge of the arbitrary line, and β is the

difference in bearings between the strike of the plane and the

trend of the arbitrary line.

i. The arbitrary line 퐶퐴 is as good as any other

1. The plunge of 퐶퐴 (α) relates to the map

distance between C and A (d) and the

107

difference in their elevations (ΔE)

ΔE by α = arctan⁡( ) 푑

2. The trend of 퐶퐴 (θa) is the bearing from C to A, Figure 70. A geometrical construction featuring the or from A to C, whichever case is the down- quantities relevant to the fitting of a plane to points A, B, and C, plunge direction. β is the angle between this here outlined. bearing and strike.

III. Determine the dip direction of the plane (θd)

i. The dip direction of a plane is perpendicular to its strike. Thus, either θ푑 = θ푆 + 90°, or θ푑 = θ푆 − 90°, one of two horizontal rays running directly away from strike. One of these rays points in the direction in which the plane climbs the fastest – the other in the direction it falls the fastest (the dip direction). The ray corresponding to the dip direction is the one that forms the smaller angle with the down-plunge bearing of the arbitrary line (θa).

Round Earth Complications

Working out of a dataset of points of latitude, longitude, and depth, this

study has the same obligation as any to address the complications of working in

spherical coordinates, to which the default prescription is to project such data

and work in a faux-cartesian space, accepting that one or more spatial properties

108 are less than perfectly preserved. However, this study has an extra obligation on top of that, which is to reconcile the fact that, strictly speaking, concepts such as strike (a property of a plane - the (single possible) bearing of a horizontal line contained within it) or dip (that plane’s inclination from the horizontal plane) are undefined outside of cartesian space, because they rely on axioms that aren’t true, but are instead very, very close to being true at human and even rift basin scales.

Strike is the bearing of a horizontal line in a plane. If to be “horizontal” is to be neither climbing nor falling in elevation, then a straight line is only horizontal in places where a normal dropped down from it intersects the Earth perpendicularly (only one place per line on a sphere or ellipsoid). We could sacrifice straightness for horizontality, allowing these lines to follow the curvature of the earth, but a curved line uniquely defines a plane from the outset

– only vertical planes could have a strike line at all. And even then, when you went to measure the bearing of this line, you would find that despite never turning left or right, the line’s bearing changes continuously.

The primary stumbling block here is not the inaccuracy imposed on the plane fitting problem and related problems by the curvature of the Earth.

Instead, it’s the fact that the data and the products are fundamentally incompatible with each other. There is no way to tell a computer to produce a straight, horizontal line that passes through two lattitude, longitude ordered pairs.

109

Since the data and products are incompatible, one or both need to be modified before a path from data to product can be constructed. One way to do this is by projecting the data onto a flat deveopable surface, while another is to modify the definitions of the products. The first method offers several advantages, not the least of which is that the algorithm just outlined in could be directly applied to any points A, B, and C for an unambiguous solution. This solution could be precise out to an arbitrary number of decimal places and, more importantly, would be the same regardless of which point was called A, which B, and which C. These are desireable properties for any solution to have, but the problem is that is the accuracy, and therefore meaningfullness of these solutions can’t be guaranteed. In fact they can be guaranteed to be wrong, if only slightly, but more importantly, the magnitude of this projection-imparted error would vary systematically as a function of the distance from the origin of the projection and (effectively) randomly as a function of the spatial configuration of the three operannd points. As this study is chiefly concerned with subtle spatial variability in stratal geometry, these errors (especially any systematic error) are undesirable, if for no other reason than that their magnitudes can’t be simply quantified.

Instead for the purposes of this study, select definitions have been modified in such a way that reconciles the operands and the algorithm without sacrificing the meaningfulness of the products.

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The fundamental issues to overcome are that straight lines have neither constant plunge, nor constant bearing outside of a cartesian space. To resolve these issues, I make two concessions:

1. “The” bearing to point Q from P is taken to be the initial bearing to Q from

the midpoint between P and Q.

Figure 71. A truer representation of the first steps of the plane-fitting algorithm. Per concession 1, the bearings θ푆 and θ푆 are defined as the initial bearing from the midpoints of great circle arcs to their endpoints.

Again, the reality is that a the great circle arc connecting P and Q sweeps out a range of bearings along its length. The size of this range depends primarily on the distance between P and Q. At distances typical of operand points in this study, this range is usually on the order of .05° in size, which is to say that a person standing at Q (with an arbitrarily precise compass) and measuring the bearing to

P would notice that it was something like 180.05° from the bearing from P to Q.

Instead of accepting either of the endpoints of the range, the initial bearing from

P to Q or that from Q to P, as the single “true” bearing , the midpoint between P

111 and Q is determined, and the initial bearing from this intermediate point to P or

Q is considered the bearing to P or Q.

2. Straight lines are taken to be arcs of logarithmic spirals with origins at the

center of the earth.

Figure 72. A section of a spherical earth, and a graphical depiction of how plunging lines (in this case, from C to A) are handled in the spherical geometry plane-fitting algorithm. The log spiral arc from C to A crosses every intermediate radial line at the same angle 푎

Logarithmic spirals are defined in polar coordinates by the equation

푟 = 푎 ∙ 푒푏∙휃 where a and b are constants particular to the shape of the spiral.

When b = 0, the curve is a circle of radius a. Otherwise, it is a curve that passes through (a,0) and spirals down to the origin, crossing every radial line at exactly the same angle, 휓 = arccot⁡(푏). For this reason, they are also known as equiangular spirals. Applying this to profiles of a spherical Earth, radial lines can be thought of as vertical lines. Those spherical surfaces (or circles in section) that cross all of these vertical lines perpendicularly can be thought of as horizontal datums. As arcs of log spirals cross all radials at ψ, they also must cross all horizontal datums at α = 90° − 휓 = arctan⁡(푏). In the context of this study, this is what it means to be a plunging line. Where these spirals cross a given horizontal

112 datum (e.g. one with a radius of 6358 km, the radius of the Earth in the area of study) they have a local radius of curvature that is directly proportional to the term 푅 ∙ √1 + tan2(훼), i.e. a radius of curvature that is a minimum of R (the radius (or “height”) of that datum) when 훼 = 0° and that goes to infinity as 훼 →

90°, which is to say, where such an arc intersects the surface of the Earth, it can’t be more curved than the surface of the Earth, and it gets straighter as α increases.

All spatial equations employed in this study either assume an eliptical earth with an equatorial radius of re = 6,378,137.0 m, and polar radius of rp =

6356752.3 m, or a spherical earth with a radius equal to the local ellipsoidal earth radius given by

2 2 2 2 √푟푒 ∙ cos(휑) + 푟푝 ∙ sin⁡(휑) 푅 = 2 2 2 2 푟푒 ∙ 푐표푠(휑) + 푟푝 ∙ sin⁡(휑) where φ is the local latitude.

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Sensitivity Testing

The plane fitting algorithm previously outlined is almost exactly that which is applied in this study, with the exception that, behind the scenes, lines and bearings are treated in the special way just detailed. The algorithm is not symmetrical – the points A, B and C, each have different roles, though the of these labels to the particular operand points is arbitrary. This aspect of the algorithm confers an advantage in that the three points can be assigned these labels in six different ways, and the cluster of solutions can say something about the their meaningfullness. Figure 73. Three different assignments of A, B, and C, to the same three points, and the subsequent construction of a strike line (Step I of the plane-fitting algorithm) Toward this end, a test was conducted wherein a particular triplet of points was passed through the algorithm six different times with all possible different label assignments. In a cartesian space, this would trivially yield the same solution six times, but after being adapted for spherical coordinates, the algorithm produces three slightly varied solutions, one for each assignment of the label “C”. The deviation of these solutions is taken to be inversely related to their meaningfulness. This process was then repeated after incrementally increasing the spacing of the operand points. The results are shown here on figures 75, 76.

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Figure 75. One standard deviation of calculated strikes returned from three different variable assignment schemes as a function of triplet baseline. On this figure, Figure 74 (right), and Figure 76 Figure 74. Results from the same (below), the vertical blue line represents a simulation, shown as the actual typical triplet baseline. values of those strikes for each variable assignment scheme.

115