THE TOPOGRAPHY, , AND TECTONICS OF THE TERRESTRIAL PLANETS

by JASON ANDREAS RITZER

Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy

Dissertation Adviser: Dr. Steven A. Hauck, II

Department of Geological Sciences CASE WESTERN RESERVE UNIVERSITY

August, 2010 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

______Jason Andreas Ritzer

Ph.D. candidate for the ______degree *.

Steven A. Hauck,II (signed)______(chair of the committee)

James A. Van Orman ______

______J. Iwan D. Alexander

Ralph P. Harvey ______

______

______

June 11, 2010 (date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. Contents

1 Introduction 1

2 Lithospheric Structure and Tectonics at , 5 2.1 Introduction ...... 5 2.2 Isidis Planitia ...... 7 2.3 Approach ...... 11 2.4 Results ...... 15 2.4.1 Globally-constrained flexural models ...... 16 2.4.2 Tectonic constraints ...... 20 2.4.3 Role of external loads on tectonics ...... 30 2.5 Discussion ...... 34 2.5.1 Globally-constrained flexure models ...... 34 2.5.2 Tectonic Constraints ...... 35 2.5.3 Role of External Loads on Tectonics ...... 36 2.5.4 Thermal Gradient and Heat Flux ...... 37 2.6 Conclusions ...... 39

3 Spherical Harmonic Radial Basis Functions: A Framework for Representing Heterogeneous Geophysical Data 41 3.1 Introduction ...... 41 3.2 Representation of Lunar Gravity ...... 43

1 3.2.1 Introduction ...... 43 3.2.2 Methods ...... 47 3.2.3 Results ...... 52 3.2.4 Discussion ...... 55 3.3 Localization ...... 59 3.3.1 Introduction ...... 59 3.3.2 Methods ...... 61 3.3.3 Results ...... 62 3.3.4 Discussion ...... 63 3.4 MESSENGER ...... 65 3.4.1 Introduction ...... 65 3.4.2 Methods ...... 68 3.4.3 Results ...... 72 3.4.4 Discussion ...... 80 3.5 Summary ...... 81

4 Lobate Scarps at Mercury: Constraining Thrust Fault Geometry using MESSENGER Laser Altimetry from flybys 1 and 2 83 4.1 Introduction ...... 83 4.2 Laser Altimetry ...... 84 4.3 Model ...... 86 4.4 Results ...... 89 4.5 Discussion and Conclusions ...... 93

Bibliography 96

2 List of Tables

2.1 Summary of Thermal Gradient and Heat Flux Estimates as determined in McGovern et al. (2004)...... 16 2.2 Range of model input parameters used in this study. Nominal values are best estimates from Wieczorek and Zuber (2004) as well as those values which matched all of our constraining criteria, and were used during analysis when not varied. All combinations of these values at resolution were modeled. Ages are: Early EN, N Noachian, H, and A...... 16 2.3 Parameters common to all models...... 17 2.4 Summary of Figure 5 in (Freed et al., 2001) describing the regions of the faulting styles determined by Simpson’s shape parameter. . . . . 23 2.5 Average distances from the center of Isidis Planitia to the innermost faults mapped at Nili and Amenthes Fossae. The ID’s correspond to those seen in Figure 2.1...... 27

3 List of Figures

2.1 Colored shaded-relief map of Isidis Planitia centered near 13.1◦ N and 87.4◦ E. Blue color is low, yellow is high. Horizontal axis is longitude and vertical axis is latitude. Outlying extensional faulting zones Nili and Amenthes Fossae are mapped. Inset is from the THEMIS Day IR Global Mosaic. Faults are illustrated with colored lines along with their labels (see Table 2.5)...... 8 2.2 Free-air of Isidis Planitia. Horizontal axis is longitude and vertical axis is latitude. The large positive anomaly in the center of the basin indicates a non-compensated load. Geologic units of Tanaka et al. (2005) were adapted for this map as an overlay. For a detailed description of geologic units, see Tanaka et al. (2005)...... 10

2.3 Diagram of loading model. Parameters are Ob, original basin shape; δc, crustal thickness anomaly; H, observed topography; F , thickness of fill material; w, vertical displacement. All values are defined positive upwards except w...... 12

4 2.4 Limits on fill density as a function of lithospheric and crustal thickness. Local crustal thickness is defined as the depth of crust after impact and before infilling. The local crustal thickness is positive above these lines and negative below these lines. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the value of the global crustal thickness c in kilometers. . . 18 2.5 Limits on fill density as a function of lithospheric and crustal thickness. Local crustal thickness is defined here as the crust present before fill plus the fill. Local crustal thickness is positive above these lines and negative below these lines. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the value of the global crustal thickness c in kilometers...... 19 2.6 Maximum crustal density as a function of lithospheric thickness and compensation factor. Local crustal thickness is negative above these lines and positive below. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the

value used for compensation factor Cf ...... 21 2.7 Map of predicted faulting style based upon Simpson’s shape parame- ter at Isidis for the nominal model. Differential stresses indicated by contour lines. The horizontal and vertical axes are east longitude and latitude respectively. The contours show the stress differences from 20 to 100 MPa. Numbers in bold denote distance to the faults of measured along the transect from the center of the figure to the top left. Lighter shading represents compressive, while darker shading indicates tensile faulting styles...... 24

5 2.8 Contours of shape parameter (shown in degrees), on a plot of distance

◦ to center of basin vs. lithospheric thickness. Aψ = 60 is a lower

◦ bound for lithospheric thickness and Aψ = 75 is and upper bound. The horizontal line at 810 km is the mean distance to the NB fault in Nili Fossae. The horizontal line at 740 km is the distance to fault NBa in Nili. The locations at which the contours cross the horizontal lines can be considered lower and upper bounds on lithospheric thickness. 26 2.9 RMS misfits between predicted and observed styles of faulting. Each panel corresponds to a particular mapped fault (Table 2.5). The solid lines are the misfit of shape parameter 60◦ and the dashed lines are for shape parameter 75◦. The black lines are the misfits from the model using the global dataset, while grey lines are for the local models. . . 28 2.10 Lower and upper bounds on the thickness of the elastic lithosphere. These bounds were determined by finding the lowest RMS misfits be- tween predicted and observed styles of faulting at the NBa fault using the global model at various crustal thicknesses. See Figures 2.8 and 2.9 for an explanation of the fitting method...... 29 2.11 Differences between the maps produced by the inverse model, and the maps used as input for the forward model. The white circle is the extent of the localizing window. The crosses indicate the locations of points used to represent individual faults around Isidis: magenta is fault NA, yellow is NB, orange is NBa, and white is AA (Table 2.5). 32

6 2.12 The left two maps a and b show the predicted faulting style determined through shape parameter. a is from the global model, while b is from the local model. c shows the difference between the two models for the nominal case. The black cross in a shows the directions of the principal stresses in an extensional regime where the largest stress is perpendicular to the observed graben...... 33

3.1 Free-air gravity anomaly at the lunar surface from SELENE model SGM90d (Namiki et al., 2009). Left side view shows the near side of the , while the right shows the far side. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively. The image is shown in a azimuthal equal-area projection...... 45 3.2 Synthetic gravity data based on lunar free-air gravity anomaly. The map on the left is the gravity of the near side of the Moon resolved to degree 36, while that on the right is the far side, resolved to degree 4. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively...... 46 3.3 Contour map of the scaled Abel-Poisson reproducing kernel as a func- tion of dilation factor. The dilation factor causes the broadest base at 0.1 and the function becomes increasingly concentrated as this fac- tor nears 1. The amplitude of the function decreases as the angular distance from the knot point increases...... 50 3.4 Spherical Harmonic gravity model expanded to degree and order 4. This model was inverted directly without any regularization. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgal; where red and blue denote positive and negative anomalies, respectively...... 53

7 3.5 Map of the difference between the full gravity data as seen in Figure 3.1, and the low resolution spherical harmonic gravity model seen in Figure 3.4. Detail is missing in both hemispheres of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively...... 53 3.6 Spherical Harmonic gravity model expanded to degree and order 34. This model was constrained by Kaula’s power law. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively...... 54 3.7 Map of the difference between the full gravity data as seen in Figure 3.1, and the high resolution spherical harmonic gravity model seen in Figure 3.6. The Lunar nearside is reproduced quite accurately, while detail is missing from the far side of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively...... 55 3.8 Spherical radial basis function gravity model. This model was inverted directly without any regularization. Black dots pinpoint the locations of the model’s knot points. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively...... 56

8 3.9 Map of the difference between the full gravity data as seen in Figure 3.1, and the spherical radial basis function model seen in Figure 3.8. The Lunar nearside is reproduced quite accurately, while detail is missing from the far side of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively...... 57 3.10 Comparison of the RMS errors between the radial basis function as well as the spherical harmonic models and the subset of the gravity data used to constrain their solutions. The abscissa depends on the degree resolution of the farside, while keeping the nearside at full reso- lution. The dotted line is the RMS error of the low resolution spherical harmonic model. The solid black line is the RMS error of the high res- olution spherical harmonic model. The RMS error of the spherical radial basis function was at machine precision, and can therefore not be seen here...... 57 3.11 Comparison of the RMS errors between the radial basis function as well as the spherical harmonic models and the full gravity data. The abscissa depends on the degree resolution of the farside, while keeping the nearside at full resolution. The dotted line is the RMS error of the low resolution spherical harmonic model. The solid black line is the RMS error of the spherical radial basis function. The light gray line is the RMS error of the high resolution spherical harmonic model. The spherical radial basis function and the regularized spherical harmonic model have nearly identical interpolation error...... 58

9 3.12 The top plot shows the reconstruction of a localization window which has a radius of 30◦. The spherical radial basis function reproduces the function accurately enough that the black dots are indistinguishable from the solid black line representing the window function itself. The empty circles show the positions of the knot points of the radial basis. The bottom plot shows the reconstruction by the spherical harmonic function. There is a significant amount of signal leakage and ringing at the discontinuity...... 61 3.13 A comparison of the RMS errors of the misfit (dotted line) and inter- polation errors of the two models as a function of window size. The interpolation error of the spherical harmonic model is the light gray line, while that of the radial basis function is the solid black line seen near the abscissa. The misfit of the radial basis function model was too small to see on this graph...... 63 3.14 The predicted altitude of the MESSENGER spacecraft above the sur- face of Mercury for the first Mercurian year. The spacecraft will be consistently closer to the surface in the northern hemisphere of Mer- cury than in the ...... 66 3.15 Contour map of the solid spherical harmonic Abel-Poisson reproducing kernel as a function of radial position. The function is much more con- centrated at radii closer to the value of α. The amplitude of the func- tion decreases as the angular distance from the knot point increases. 70

10 3.16 Example maps of the difference between the anomaly of the synthetic lunar data, and both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included no synthetic error. The black dots in the top two maps show the locations of the knot points used by the radial basis. Left side views show the near side of the Moon, while the right two show the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. . . 74 3.17 These two plots show an example of the error between the values of gravity anomaly predicted by both the radial basis (top) and spherical harmonic (bottom) models at the surface through downward contin- uation. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. The radial basis model was inverted using TSVD regularization, while the spherical harmonic model was constrained using Kaula’s power law. 75 3.18 These two plots show an example of the error between the values of gravity anomaly interpolated between data points by both the radial basis (top) and spherical harmonic (bottom) models and the gravity at spacecraft altitude. The data provided to the two models included synthetic error which had a standard deviation of 5% of the range. The radial basis model was inverted using TSVD regularization, while the spherical harmonic model was constrained using Kaula’s power law. 76

11 3.19 Example maps of the recovery of the synthetic lunar data, using both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively...... 77 3.20 Example maps of the difference between the surface gravity anomaly of the synthetic lunar data, and both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. . . 78 3.21 Plot matrix comparing the continuation and interpolation errors (top row) and the coefficients of determination (bottom row) of both the radial basis and spherical harmonic models using several different types of regularization for various amounts of synthetic error. The top row of panels compares the RMS of the continuation error of the radial basis (blue) and the spherical harmonic () models, as well as the RMS of the interpolation error of the radial basis (red) and the spherical harmonic (cyan) models. The statistics representative of the goodness of fit to data by models using: Kaula’s rule (first column), Tikhonov regularization (second row), truncated singular value regu- larization (third row), and direct inversion (fourth row) comprise the matrix. Each panel is shown as a function of the standard deviation of added synthetic error which is a percentage of the range of the data. 79

12 4.1 MDIS image of scarp D overlain by the MLA track in red and the corresponding topographic profile in green. The scarp is centered at 137.1◦ W and 3.2◦ S...... 85 4.2 MDIS image of scarps A, B, and C overlain by the MLA track in red and the corresponding topographic profile in green. Scarps A, B, and C are centered at locations (56.7◦ E, 4.7◦ S), (59.3◦ E, 4.8◦ S), and (64.6◦ E, 4.8◦ S) respectively...... 86 4.3 Elastic finite element model setup. The example model in this figure has a 100 km thick crust with a surface breaking thrust fault, overlying a mantle 400 km thick. All models were 2000 km wide to minimize edge effects. Dislocation was achieved by applying a fixed acceleration load in compressive x directions along the fault plane. The walls were fixed in lateral displacement and the bottom edge was supported by a Winkler foundation of springs...... 88 4.4 A series of model profiles (solid lines) varying the amount of fault slip compared to the MLA profile (black dots). All profiles kept the same fault geometry. The track of the profile cuts across scarp C...... 90 4.5 A series of model profiles (solid lines) varying the length of the fault compared to the MLA profile (black dots). All profiles had the same dip angle of 20◦ and the amount of slip was chosen to best match the observations. The track of the profile cuts across scarp C...... 91 4.6 A series of model profiles (solid lines) varying the angle of the fault compared to the MLA profile (black dots). All profiles had the same fault length of 85 km and the amount of slip was chosen to best match the observations. The track of the profile cuts across scarp C. . . . . 91

13 4.7 Best fits between the models and data were chosen based on root mean squared errors. Here the fault angles best fitting the data was around 25◦ to 30◦ for scarp D (solid gray line) and around 20◦ to 23◦ for scarps B (dotted line) and C (solid black line)...... 92 4.8 A series of model profiles (solid lines) varying the thickness of the elastic plate in the model compared to the MLA profile (black dots). All profiles had the same geometry and amount of slip. The track of the profile cuts across scarp C...... 93

14 Acknowledgments

I would first like to thank my adviser Steve for his patience and understanding. We have learned about the mentor/mentee relationship together. I would also like to thank my committee for making time in their schedules for my dissertation. I am especially happy to be working with the MESSENGER mission team, helping any way I can to advance our knowledge of the innermost planet. I am grateful to Bruce Banerdt and Mark Wieczorek for sharing code utilized in developing this study. GMT (Generic Mapping Tools) by Wessel and (1998) was used to generate several of the figures. I also appreciate the availability of the Case Western Reserve University Information Technology Service’s High Performance Computing cluster.

15 The Topography, Gravity, and Tectonics of the Terrestrial Planets

Abstract by J. ANDREAS RITZER

The topography and gravity of some of the solid earth bodies in our solar system including the Moon, Mercury, and Mars are used to investigate the density structure, lithospheric structure and resultant tectonic deformation. The difficulties in repre- senting the gravitational field based on satellite data are explored including possible solutions including a change of representational basis function. The problem of repre- senting the gravitational field based on non-uniform data distributions is addressed. Early results and expected returns from the MESSENGER mission to Mercury in- cluding compressional tectonics are studied. Chapter 1

Introduction

Models of the physical processes that have shaped the planets of our solar system can only be trusted if they are based on physical evidence. The surface of a planetary body can be directly sensed using a variety of techniques. The interior of a planet, however, can only be constrained by analyzing and inverting geophysical data acquired at or above the surface. These measurements and subsequent inversion models are used to place physical constraints upon the models of the physical state of the interior as well as the processes that influence the dynamic evolution of a planet. Investigation of the subsurface structure of planets can only be carried out by sensing the physical properties at or above the surface. One successful method of probing the physical properties of a planet’s subsurface, is by measuring the minute vibrations or seismic waves at the surface. The analysis of these waves can constrain the speed and path with which they pass through a body. Unfortunately the techno- logical limitation of placing the sensitive seismic sensors on the surface of terrestrial planets other than the Earth is unfeasible at this time. Other methods of investigation are available, which are the primary focus of this study. The measurement of the topography and gravity made at the surface of a planet, or in orbit, can be used to constraint the physical structure of the interior.

1 The topography, or surface features of an object, are useful as a lens for seeing what the history or a planet may be. The processes that shape the surface of planets are expressed in its topography, including cratering, tectonicsm, volcanism, hydrology, and even aeolian effects. By investigating the topography, we can look into the past of the planet’s surface. The gravitational field of a planet is a particularly useful way of investigating the interior of a planet. The gravity field is directly determined by the subsurface density structure, and as such is a powerful tool for probing the interior. ’s laws dictate that the gravitational acceleration outside the planet is determined by the internal density distribution. The limitation of this method stems from the fact that the resolving power of measurements of gravity decrease with increasing distance from their source. Therefore the investigations of the subsurface of planets are mostly restricted to the outermost layers of planetary bodies. Separate analysis of topography and gravity are useful in themselves for making in- ferences about planetary interiors, however these studies often lack robust constraints. To get the most information out of the measurements available, it is preferable to com- bine the results of these two datasets in order to constrain the subsurface of planets. By using both topography and gravity as limits on models, the possibility of a well- constrained model emerges. The relationship between topography and gravity is an excellent way to determine the structure of the near subsurface of a planet, as well as the loads which may be acting upon the mechanically rigid upper layer called the mechanical lithosphere. The response to loads at the surface provides further infor- mation about the planet as the thickness of this lithosphere is directly related to the thermal state of the upper layers of the rocks. So, by measuring and analyzing the topography and gravity of a planet, while taking into account the mechanical rigid- ity of the lithosphere, an inference of the mechanical and thermal properties of the planet’s outer shell can be achieved.

2 Deformation of the lithosphere can take one of three forms or some combination thereof: elastic, brittle or viscous. The elastic response of the planet is important for estimates of the thickness of the mechanical lithosphere, which in turn provides estimates for the thermal structure. Sometimes, however, the relationship between topography and gravity is not simple and other ways of constraining deformation models are needed. The brittle deformation of the planet can provide such a con- straint. Tectonism, or faulting of the surface of a planet is due to the brittle failure of rocks. This localized deformation of the surface is expressed as either compressional, extensional or strike-slip faults. The location and geometry of these faults is a useful tool for investigating the thermal structure and history of the near subsurface of a planetary body. The faults are often directly influenced by the temperature of the rock in which they occur since the method of deformation is related to the tempera- ture. As such, only the uppermost layer of the lithosphere deforms in a brittle fashion. Knowing the depth of this brittle layer by modeling the faults is an excellent way to infer the thermal structure of a planet. The location of faulting at the surface of a rocky planet is often related to the loads which act upon the mechanical lithosphere. The loading of the lithosphere as well as the properties of the lithosphere itself can best be constrained by a combination of knowledge of the topography, gravity and tectonism of the surface. This study is an anthology of three separate projects investigating separate aspects of the topography, gravity and related tectonics of some of the terrestrial planets. The first deals with the analysis of the topography and gravity of Mars at a particular location of significant import to fundamental questions regarding the thermal history of the planet. Topography, gravity, and the faulting associated with a giant impact crater on Mars called Isidis Planitia is investigated. The second takes a look at the method for representation of gravity and seeks to find an alternative more suited for the measurement of gravity from satellites in highly elliptical orbits. The third focuses

3 on the pervasive compressional tectonism seen on the surface or Mercury, and uses recent results from the MESSENGER mission to constrain the geometry of several faults. All these studies attempt to increase our understanding of the history and evolution of the interior of the terrestrial planets.

4 Chapter 2

Lithospheric Structure and Tectonics at Isidis Planitia, Mars

2.1 Introduction

Characterizing the lithosphere of Mars is an important step towards understanding the state of its surface and interior. The lithosphere, which is the outer, mechanically rigid shell of the planet, responds directly to loads imposed upon it. Observations of the response of planetary surfaces are inherently useful because they can provide con- straints on the structure of the lithosphere and the applied loads. The structure of the lithosphere is especially important at Isidis Planitia as it lies directly on the boundary between the southern highlands, with its higher topography and thicker crust, and the adjacent lower and thinner northern plains. Leveraging recently acquired topography and gravity data for Mars (e.g., Smith et al., 1999a,b; Lemoine et al., 2001; Smith et al., 2001a; Tyler et al., 2001; Yuan et al., 2001; Konopliv et al., 2006) provides the to develop a better understanding of the lithosphere in this region as well as additional constraints on the history of lithospheric heat flow (e.g., Solomon and Head, 1990; Zuber et al., 2000; McGovern et al., 2002). Gravity and topography data

5 are important indicators of lateral variations in the properties of the crust and litho- sphere of planetary bodies. Techniques for analyzing the transfer function between gravity and topography, known as admittance, are commonly employed using models to estimate local lithospheric and crustal thicknesses (e.g., Forsyth, 1985; McGovern et al., 2002; Belleguic et al., 2005) when these datasets are spectrally correlated in spatially localized regions. Admittance modeling techniques have been successful at providing estimates of the thickness of the lithosphere beneath major volcanoes, the rim of the Hellas basin, and other surface features on Mars (McGovern et al., 2002; Belleguic et al., 2005). While these models have been successful in several regions on Mars, gravity and topography data in the northern plains, including Isidis Planitia, tend to be uncorrelated, and hence previously used admittance modeling techniques are incapable of adequately modeling lithospheric flexure of this important part of the surface (McGovern et al., 2002). However, approaches for investigating the rigidity of the lithosphere at Isidis using forward flexural modeling techniques (Comer et al., 1985) as well as flexural modeling constrained by gravity and topography data (Banerdt, 1986; Searls et al., 2006) are more capable in this situation. Isidis is the only major basin on Mars ringed by distinct circumferential graben like those of some lunar mascons (Comer et al., 1979; McGovern et al., 2002) that are associated with flexure of the basin. Therefore, in addition to gravity and topography data, the lo- cations of extensional faulting surrounding Isidis can be used as a further constraint on flexural models (Comer et al., 1985). We analyze gravity and topography data, as well as the expression of circumferen- tial faulting at Isidis in order to understand the lithospheric and crustal structure of the basin. First, we apply an extension of Banerdt’s (1986) well-known flexural model in order to characterize the range of parameters capable of satisfying the gravity and topography data. Second, we compare the stresses predicted by our flexural model to the observed tectonics around Isidis to constrain estimates of the thickness of the

6 elastic lithosphere. We also assess the role of loads outside of the basin on lithospheric flexure and faulting in the area. Finally, we estimate the thermal gradient below Isidis and the surface heat flux and compare these values with estimates for other features on Mars (McGovern et al., 2002).

2.2 Isidis Planitia

Isidis Planitia, centered near 13.1◦ N and 87.4◦ E (Figure 2.1), is a circular, flat- bottomed topographic depression approximately 1352 km in diameter (Frey, 2006) on Mars that lies near the boundary between the southern highlands and northern lowlands, also known as the dichotomy boundary. Its size and morphology are con- sistent with a multi-ring impact basin (Wichman and Schultz, 1989). Recent work on the distribution of impact craters and quasi-circular depressions (QCDs), interpreted to be the remnants of buried impact features, suggest that Isidis was likely the last of Mars major impact basins to be emplaced, near the end of the Early Noachian period approximately 3.92 GY before present (Nimmo, 2002). Geologic mapping by Tanaka et al. (2005) identified a thin surface cover over the interior of Isidis comprised of an Early Amazonian-aged ridged member that is similar morphologically to the formation and is thought to be one of the thinnest fill unit of the basins in the Northern plains (Buczkowski, 2007). The stratigraphy at the contact between the Syrtis Major region and Isidis Planitia is also interpreted to have been formed by an Early Hesperian ridged plains unit emplaced when the Syrtis Major volcanoes were formed, followed by the Vastitas Borealis formation, which is equivalent to the Tanaka et al. (2005) AIi unit, followed by another Syrtis Major flow forming a knobby terrain unit (Ivanov and Head, 2003). Bordering Isidis Planitia are several significant geologic provinces. To the im- mediate west of Isidis lies the Syrtis Major formation. The surface of Syrtis Major

7 Figure 2.1: Colored shaded-relief map of Isidis Planitia centered near 13.1◦ N and 87.4◦ E. Blue color is low, yellow is high. Horizontal axis is longitude and vertical axis is latitude. Outlying extensional faulting zones Nili and Amenthes Fossae are mapped. Inset is from the THEMIS Day IR Global Mosaic. Faults are illustrated with colored lines along with their labels (see Table 2.5).

Planum consists primarily of Hesperian-aged volcanic units with clear flow fronts and margins (Tanaka et al., 2005). The volcanic deposits primarily extend radially from two depressions, Nili Patera and Meroe Patera on the western side of the volcanic province. Bordering Isidis to the northeast is perhaps the largest known basin on Mars, . Utopia is more than two and a half times larger than Isidis at

8 approximately 3200 km in diameter (Tanaka et al., 2005; Searls et al., 2006) and its surface is comprised of Hesperian and younger Amazonian units, of which the latter dominate. The remainder of Isidis’ periphery is predominantly composed of older eroded Noachian-aged highlands terrain. The gravity signature of Isidis provides insight into possible loading scenarios for the basin. Recently acquired gravity data from the (MGS) and Mars Odyssey spacecrafts indicate that Isidis Planitia has a large, positive, free-air gravity anomaly, > 140 mGals at spacecraft altitude, centered near 12.1◦ N and 85.8◦ E (Figure 2.2) (Lemoine et al., 2001; Tyler et al., 2001; Yuan et al., 2001; Potts et al., 2004; Neumann et al., 2004; Konopliv et al., 2006). Such a large free-air anomaly suggests the strength of the lithosphere is supporting significant mass loads (Forsyth, 1985). Isidis’ gravity signature is similar to those of the large lunar mascons (ar- eas of the lunar crust characterized by an excess of mass) often interpreted to be mass anomalies of shallow origin, such as infilling of previously isostatically com- pensated basins (e.g., Booker et al., 1970; Phillips et al., 1972; Solomon and Head, 1980; Arkani-Hamed, 1998; Wieczorek and Phillips, 1999). Alternatively, mascons have been interpreted as large mantle plugs, the result of uplift of the crust-mantle boundary above the isostatic compensation level of the basin topograpy (termed over- compensation) at the time of crater formation (e.g., Neumann et al., 1996; Wieczorek and Phillips, 1999) or as some combination of overcompensation and infilling. Flexure of the lithosphere can lead to the development of faults where stresses exceed the strength of the crust. As observed in lunar mascons (Solomon and Head, 1980; Freed et al., 2001), infilled impact basins tend to produce compressional tec- tonics inside the topographic ring, a generally unobserved (from orbit) annulus of conjugate strike slip faulting around the interior, and farther away from the basins center, a ring of radial extensional faulting near the flexural bulge. The style of fault- ing indicates stress orientations and the locations of such zones demonstrate where

9 Figure 2.2: Free-air gravity anomaly of Isidis Planitia. Horizontal axis is longitude and vertical axis is latitude. The large positive anomaly in the center of the basin indicates a non-compensated load. Geologic units of Tanaka et al. (2005) were adapted for this map as an overlay. For a detailed description of geologic units, see Tanaka et al. (2005).

the stress exceeds the strength of the lithosphere. The location and style of such faulting near lunar mascons has been useful for constraining the thickness of the lu- nar lithosphere (Comer et al., 1979; Solomon and Head, 1980); on Mars, Isidis is the only major basin with clearly associated tectonic features. Lying to the northwest of the basin, Nili Fossae are the most prominent tectonic features near Isidis. The Nili Fossae region contains a series of large graben that are 810 ± 30 km to 1060 ± 30 km from the center that strike in a roughly circumferential orientation. Another set of circumferential graben called Amenthes Fossae lie 740 ± 40 km to 1120 ± 40 km to

10 the southeast of Isidis. Both of these extensional tectonic features may have formed during the middle Noachian (Tanaka et al., 2005), which is distinctly younger than the early Noachian age estimate for basin formation (Tanaka et al., 2005). Within the basin are features that resemble compressional wrinkle ridges, consistent with mascon loading scenarios (Wichman and Schultz, 1989), though recent work suggests that these features may also be lava flow fronts (Ivanov and Head, 2003).

2.3 Approach

The structure of the near subsurface of Isidis is analyzed using a model for the de- formation of a thin, elastic, spherical shell (e.g., Turcotte et al., 1981; Willemann and Turcotte, 1982; Banerdt, 1986; Phillips et al., 2001) constrained by the present- day and topography. Variations in relief at the surface and internal density interfaces, such as at the crust-mantle boundary, as well as excess mantle buoyancy, lead to net loads on the lithosphere that result in deformation. Following the gen- eral approach of Banerdt (1986), both vertical and horizontal load components are included in the elastic shell deformation model that accounts for both elastic bending and membrane support of lithospheric loads. We calculate the deformation of the lithosphere and the subsequent stress state resulting from the vertical load q(θ, φ) and horizontal load potential Ω(θ, φ) as a func- tion of colatitude, θ , and longitude, phi, in the spherical harmonic domain. We model the deformation using a typical assumption that a thin elastic shell (e.g., Turcotte et al., 1981) can approximate the lithosphere. Furthermore, we utilize the observed topography H(θ, φ) and geoid height G(θ, φ) as explicit constraints (Banerdt, 1986). The data (i.e., q, Ω, H, G, etc.) are represented by on a surface

P∞ Pl m f(θ, φ) = l=0 m=−l almYl (θ, φ) where l is the harmonic degree, m is the harmonic

m order, alm are the harmonic coefficients, and Yl (θ, φ) are the spherical harmonics. We

11 represent each geophysical parameter by these coefficients and perform the equations relating them on a degree-by-degree basis (e.g., Banerdt, 1986).

Figure 2.3: Diagram of loading model. Parameters are Ob, original basin shape; δc, crustal thickness anomaly; H, observed topography; F , thickness of fill material; w, vertical displacement. All values are defined positive upwards except w.

We use global representations of the Mars Global Surveyor (MGS) Mars Orbiter Laser Altimeter (MOLA) topography (Smith et al., 1999b, 2001a) and MGS and Mars Odyssey derived gravitational potential field (Smith et al., 1999a; Tyler et al., 2001; Lemoine et al., 2001; Yuan et al., 2001; Konopliv et al., 2006) data, using spherical harmonic gravity model MGS95J which are expanded to degree and order 95, but truncated at degree and order 70 as the noise in data are equal to the signal for higher degree terms (Konopliv et al., 2006). We extend the Banerdt (1986) model for the deformation of a thin elastic shell, which relates these data to the inferred displacement and stresses. The model setup, illustrated in Figure 2.3, assumes that when the Isidis basin formed it had an initial degree of compensation relating relief on the crust-mantle interface δc, known as the moho, to surface expression of the basin by δc = Cf ρc/∆ρOb, where Cf is a constant degree of compensation (0 = no compensation, 1 = complete isostatic compensation), ρc is an assumed average global crustal density, Ob is the pre-loading basin shape, and ∆ρ = ρm −ρc, where ρm is the average global mantle density. We extend Banerdt’s (1986) original model by

12 also considering the possibility of an additional density interface due to subsequent

filling of the basin with a material of density ρf , to a thickness of F (see also, Searls et al., 2006 for a similar formulation). The non-isostatic loads within the basin deflect the lithosphere by an amount w that is consistent with the observed topography H and geoid G. The thickness of fill, F , is:

F = H − w − Ob (2.1)

as present-day topography, H, is the sum of the pre-loading basin shape, Ob, the thickness of the fill material F , and the deflection of the lithosphere w. The gravity anomaly observed within Isidis is likely due to some combination of a non-isostatic compensation state of the basin prior to infill and the excess mass of the fill in the basin (Neumann et al., 2004). The additional density interface introduces more terms to the standard formula- tion of Banerdt (1986). The relation for net vertical load is

q = ga[ρF F + ρc(w − G) + ρcOb] + gc[∆ρ(w − Gc) − Cf ρcOb] (2.2)

where ga is the acceleration of gravity at the surface, gc is the acceleration of gravity

on the Moho, and Gc is the geoid height at the Moho. Previous studies using a Banerdt-type model (e.g., Banerdt, 1986; Searls et al., 2006) do not explicitly include

the variation of gravity with depth; ga and gc approximate this effect in manner similar to Banerdt and Golombek (2000). The geoid height at the surface is

( ) 3 R − cl+2 G = ρ F + ρ (O + w) + [∆ρw − C ρ O ] (2.3) ρ¯(2l + 1) F c b f c b R

whereρ ¯ is the mean bulk density of Mars and R is the radius to the mid-plane of the

13 elastic shell. Similarly, the geoid height at the base of the crust is

( ) 3 R − cl R − cl G = ρ F + ρ (O + w) + (∆ρw − C ρ O ) (2.4) c ρ¯(2l + 1) f R c b R f c b

With the new density interface, the horizontal load potential is then

ν F ν O Ω = ρ g L − C ρ g (L − c) b 1 − ν F a R 1 − ν f c m R ν O w + ρ g c b + [ρ g c + ρ g (L − c)] (2.5) 1 − ν c a R c m m c R

where L is the thickness of the elastic lithosphere and c is the mean crustal thickness of Mars. Banerdt (1986) derives the vertical displacement produced by both the vertical load and the horizontal load potential:

w = αq + γΩ (2.6)

where 4 R fl + 1 − ν α = − 3 2 2 (2.7) D fl + 4fl + ψ(1 − ν )(fl + 2)

3 2 , D = EL /12(1−ν ) is the flexural rigidity, E is the Youngs modulus, fl = −l(l+1), ψ = 12(R/L)2, and f (1 + ν − f /ψ) γ = α l l (2.8) fl + 1 − ν

. We solve equations 2.1 - 2.6 simultaneously, which allows us to infer the structure of the basin for assumed global parameter values as well as determine the corresponding elastic deformation of the lithosphere. The addition of the intracrustal density interface between the basin fill material and average crust is an important step toward quantifying the relationship between the amount of moho relief and basin fill material that are consistent with the ob- served gravity and topography data. However, the additional density interface also

14 introduces more parameters to the problem; in particular, we must make a priori assumptions about the density of the fill material in any given model. Furthermore, we must also calculate the thickness of the fill material and the depth of the original basin. Though the intracrustal interface widens the potential parameter space, the additional generality of the extended formulation is consistent with the geological context of Isidis Planitia.

2.4 Results

Our primary goal is to understand the range of lithospheric thicknesses and basin crustal properties that are consistent with observations. Towards this goal, we explore a broad range of plausible model parameters (e.g., L, c, ρF , ρc, etc.) and potential tradeoffs among parameters (Table 2.2 lists variable parameters varied and Table 2.3 constant parameters applicable to all models). The basic approach is to leverage a comprehensive exploration of the models plausible parameter space with two addi- tional geologic constraints. First, we make the reasonable assumption that there is no exposed mantle within the Isidis basin. Second, calculation of the deformation- induced stress field near the basin provides the opportunity to assess whether indi- vidual models are consistent with the existence, style, and orientation of faulting, particularly at Nili and Amenthes Fossae. The latter is particularly useful for con- straining estimates of elastic lithospheric thickness. Combined, this approach affords the opportunity to investigate the range of, and tradeoffs among, lithospheric and crustal properties that are physically consistent with the gravity and topography data as well as geological constraints.

15 Table 2.1: Summary of Thermal Gradient and Heat Flux Estimates as determined in McGovern et al. (2004).

Feature Surface L, km Thermal Heat Flux Load Density 2 3 Age Gradient mW/m ρl, kg/m K/km A > 70 < 8 < 24 3150 Ascraeus Mons A 2 – 80 5 – 55 13 – 140 3250 A < 100 > 5 > 13 3250 A > 20 < 10 < 28 3300 Alba Patera A-H 38 – 65 5.5 – 16 16 – 40 2950 Elysium Rise A-H 15 – 45 6 – 13 15 – 33 3250 Hebes A-H 60 – 120 5 – 9 17 – 25 2100 – 2300 A-H 80 – 200 3 – 7.5 11 – 23 2200 Capri Chasma A-H > 100 < 7 < 23 2500 Solis Planum H 24 – 37 8 – 14 20 – 35 2900 Hellas south rim H-N 20 – 31 10 – 16 25 – 40 2900 Hellas west rim H-N < 20 > 12 > 30 2650 Hellas basin N < 13 > 14 > 35 2750 Noachis Terra N < 12 > 20 > 50 2800 Isidis Planitia H 100 – 180 3.4 – 6.5 14 – 26 > 3100

Table 2.2: Range of model input parameters used in this study. Nominal values are best estimates from Wieczorek and Zuber (2004) as well as those values which matched all of our constraining criteria, and were used during analysis when not varied. All combinations of these values at resolution were modeled. Ages are: Early Noachian EN, N Noachian, Hesperian H, and Amazonian A.

Input Parameter Range Nominal Resolution c 30 – 90 km 50 km 10 km L 50 – 220 km 120 km 10 km 2 2 2 ρc 2600 – 3400 kg/m 2900 kg/m 100 kg/m 2 2 2 ρF 2600 – 3400 kg/m 3100 kg/m 100 kg/m Cf 0.1 – 1.6 1.0 0.1

2.4.1 Globally-constrained flexural models

We calculate a large suite of global flexural models explicitly constrained by present- day gravity and topography in order to understand lithospheric and crustal properties near Isidis Planitia. By considering a broad range of values for the mean crustal

16 Table 2.3: Parameters common to all models.

Constant Value 2 ρm 3500 kg/m g 3.71 m/s2 R 3389.5 km E 1.0 x 1011 Pa ν 0.25

thickness c, elastic lithospheric thickness L, mean density of fill ρF , compensation

factor Cf , and mean crustal density ρc (see Table 2.2) we are able to calculate how

fill thickness F , crustal thickness local to Isidis cl = c − Ob − δc − F at a time post- impact and prior to infilling, original basin shape Ob, and displacement w vary with these parameters in order to produce the observed gravity and topography. Though these calculations are inherently non-unique, by applying the reasonable constraint that cl > 0 we can considerably limit the range of valid models. The origin of the majority of material filling Isidis, indeed the northern plains, is unclear, though the presence of Syrtis Major to the west suggests that significant volcanic deposits are possible (e.g., Tornabene et al.). Large values for the density of the basin fill material (ρF ) would be consistent with such an idea. Figure 2.4 illustrates how fill density varies with lithospheric thickness (L) and mean global crustal thickness (c); each line is a contour of zero local crustal thickness for a given value of c. The models in the parameter space above the surface defined by these contours have a positive local crustal thickness, while those below have negative crustal thickness; therefore, those models above these lines are permissible. Assuming

−3 that c = 50 km (Wieczorek and Zuber, 2004) implies a ρF > 3000 kg m , however, based on an analysis of geoid to topography ratios and a comprehensive survey of estimates in the literature these authors found that the most robust estimate of uncertainty in the mean global crustal thickness is ± 12 km. While a crustal thickness less than 50 km would imply a well-constrained, large value for the density of the fill

17 material an increase of even 10 km in crustal thickness leads to an unconstrained

−3 value for ρF for an average crustal density of 2900 kg m .

Figure 2.4: Limits on fill density as a function of lithospheric and crustal thickness. Local crustal thickness is defined as the depth of crust after impact and before infilling. The local crustal thickness is positive above these lines and negative below these lines. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the value of the global crustal thickness c in kilometers.

A more conservative approach recognizes that while exhumation of the mantle by the Isidis-forming impact may be possible, basin infilling could obscure the mantle. Therefore, in Figure 2.5 we include the fill material as a component of the local crustal thickness, and apply the constraint that it cannot be negative. These results show

18 that a robust estimate of the minimum fill density is not feasible without knowledge of the depth of fill.

Figure 2.5: Limits on fill density as a function of lithospheric and crustal thickness. Local crustal thickness is defined here as the crust present before fill plus the fill. Local crustal thickness is positive above these lines and negative below these lines. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the value of the global crustal thickness c in kilometers.

Another factor that influences our understanding of the crust and lithosphere at Isidis is the initial state of compensation of the basin (Figure 2.6). The state of com-

pensation of the Isidis basin was varied between an under-compensation of Cf = 0.2 and an over-compensation of Cf = 1.6. The maximum value of Cf =1.6 is consistent

19 with the end-member assumption that the gravity anomaly can be explained entirely by super-isostatic relief along the Moho (Neumann et al., 2004). The clearest impli- cation of the role of Cf is that the maximum possible mean global crustal density ρc increases as Cf increases; the same is true, but to a lesser extent for increasing litho- spheric thickness. The variation with Cf is also non-linear as the relative change in ρc becomes smaller as the compensation factor increases. The maximum mean crustal density only varies by about 12% through a large range of compensation states for the nominal crustal thickness of 50 km.

2.4.2 Tectonic constraints

The distinct extensional tectonic features circumferential to the Isidis basin (Figure 2.1) provide an additional opportunity, one not afforded by other large basins on Mars, to constrain the lithospheric structure of the region at the time material was filling the basin’s interior. The strength of the lithosphere is finite and the state of stress induced by loads on the lithosphere controls the locations and style of faulting. Therefore, at Isidis the wide range of lithospheric loading models that satisfy the gravity and topography data can be further constrained by requiring successful models to match the observed locations, style, and orientation of faulting surrounding the basin. The lithospheric stress field at the surface (i.e. vertical stress is zero) is determined from the amount of deformation (w) of the elastic lithosphere, the vertical load (q), and the horizontal load potential (Ω) (see Banerdt (1986) Appendix A for details). From this stress field we estimate both where the magnitude of stress exceeds the strength of the lithosphere as well as the style and orientation of any predicted fault- ing. Movement along a fault is favored after reaching a minimum level of stress in the lithosphere. Barnett and Nimmo (2002) inferred that extensional faults on Mars greater than 150 km across, such as may exist within the system, withstood over 100 MPa differential stress (i.e., the difference between the maximum

20 Figure 2.6: Maximum crustal density as a function of lithospheric thickness and compensation factor. Local crustal thickness is negative above these lines and positive below. Nominal values were used for all parameters not shown, as given by Table 2.2. Numbers on contours indicate the value used for compensation factor Cf . and minimum principal stresses), while smaller faults are more similar to terrestrial faults and only require about 20 MPa. Nili Fossaes largest fault is ≈37 km across (Figure 2.1) consistent with a rock strength somewhere between the 20 MPa and 100 MPa limits for a reasonable minimum to initiate faulting near Isidis, though the exact differential stress required is poorly constrained. In addition to the requirement that the stresses exceed the strength of the litho- sphere in order for our models to predict faulting, it is also necessary to determine

21 the predicted style and orientation of faulting for comparison with Nili and Amen- thes Fossae. The most common method for predicting the style of faulting using stress directions is Anderson’s faulting criteria (Anderson, 1951). These criteria pre- dict one of three styles of faults: normal (extension), strike-slip (shear), or thrust (compression). Anderson’s criteria result in discrete predictions of only the major style of faulting, however, mixed modes of faulting are common in nature (e.g., trans- tensional), and therefore these criteria do not adequately address the full range of observed styles of faulting. A continuous representation of the faulting criteria was developed by Simpson (1997) in order to help alleviate this problem, and has been used to better understand and resolve the strike-slip faulting paradox of mascon load- ing on the Earths moon (Freed et al., 2001). The geometrically defined Simpson’s shape parameter

σ1 − σφ Aψ = arctan √ (2.9) 3(σ2 − P )

is a continuous metric that implicitly includes the possibility of mixed-mode faulting styles. The largest and smallest compressive principal stresses on the ra- dial/vertical plane axisymmetric to the center of the basin are σ1 and σ2 respectively,

σφ is the circumferential principal stress, and P = (σ1 + σ2 + σφ)/3 is hydrostatic pressure. The types of faulting styles predicted by shape parameter (Table 2.4) are pure as well as transitional modes of faulting. Table 2.4 lists the criteria for mapping

values of Aψ to the style and orientation of faulting; we have adopted the modified cri- teria proposed by Freed et al. (2001) which are weighted away from strike-slip faulting compared to the original criteria of Simpson (1997). The modified criteria are based upon the practical implementation of Simpson’s framework, which recognizes that in mixed modes of faulting the non strike-slip component (normal or thrust) tends to be the observable expression of the faulting from orbit unless the mixed-mode is dominantly strike-slip in style (Freed et al., 2001). Using the presence of faults surrounding Isidis as a constraint on lithospheric

22 Table 2.4: Summary of Figure 5 in (Freed et al., 2001) describing the regions of the faulting styles determined by Simpson’s shape parameter.

◦ Aψ ( ) Fault-style from shape parameter 0 – 30 Pure normal faulting 30 – 75 Strike-slip & normal 75 – 105 Pure strike-slip faulting 105 – 150 Strike-slip & thrust 150 – 180 Pure thrust faulting

structure requires that our models must meet both criteria (sufficient stress magni- tudes and the style) and orientation of faulting consistent with the modeled stress field must match the observed style of faulting at Nili and Amenthes Fossae. First, the amount of differential stress at the location of an observed fault must be greater than 20 MPa. Second, the Simpson’s shape parameter must predict dominantly nor-

◦ ◦ mal faulting circumferential to the basin (0 ≤ Aψ ≤ 75 ). An individual model is considered reasonable if both these criteria are met at the fault locations (e.g., Nili Fossae) and just as importantly a model must not predict inconsistent styles and orientations of faults (i.e. no circumferential graben inwards of the innermost faults). We compare the calculated shape parameter to the locations of observed faulting (Figure 2.7) to estimate limits on the thickness of the elastic lithosphere. Normal

◦ faulting is most definitively predicted when Aψ < 60 and this then delineates the

◦ lower bound. Modeled values of Aψ < 60 inward of the innermost faulting are not permissible, as that would imply normal faulting in the models closer to the interior of the basin than is observed. We use the innermost faults for comparison because they constrain where extensional faulting is observed. Stresses here are high enough that if compressional or strike-slip faulting were possible, it should be expressed in

◦ the landscape. We find the upper bound on lithospheric thickness using Aψ = 75 which is the point where the dominant faulting style turns from normal faulting to

23 strike-slip, therefore, the shape parameter cannot be above 75◦ outside the distance to the innermost fault otherwise the majority of Nili Fossae faulting would be predicted to be strike-slip. The example in Figure 2.8 shows that in addition to a preference for normal faulting stress differences are sufficient for faulting to occur at Nili and Amenthes Fossae, while other areas around the periphery have lower stresses.

Figure 2.7: Map of predicted faulting style based upon Simpson’s shape parameter at Isidis for the nominal model. Differential stresses indicated by contour lines. The horizontal and vertical axes are east longitude and latitude respectively. The contours show the stress differences from 20 to 100 MPa. Numbers in bold denote distance to the faults of Nili Fossae measured along the transect from the center of the figure to the top left. Lighter shading represents compressive, while darker shading indicates tensile faulting styles.

In order to quantitatively assess our lithospheric flexure models we focus on com- paring the stress magnitude and value of Aψ along individual radial transects (e.g.,

24 the black diagonal line in Figure 2.7) across Isidis that intersect the innermost normal faults. The transect in Figure 2.7 is important because it intersects what we interpret to be the innermost visible normal fault associated with Nili Fossae, fault NBa (Fig- ure 2.1). As such, we have used the location of NBa as the limiting case of normal faulting for the majority of our analysis of flexural induced tectonics at Isidis. Profiles

of the value Aψ along this transect were assembled for the entire range of lithospheric thicknesses (Figure 2.8) in order to determine whether constraints can be put on the value of lithospheric thickness. The requirement that viable models match the style

◦ of faulting of the innermost faults implies that the intersection between the Aψ = 60 contour and the distance to the innermost faults defines a lower bound for lithospheric

◦ thickness, while the intersection with the Aψ = 75 contour defines an upper bound. The fault NB is 810 km from the center of the basin (Figure 2.1 and Table 2.5), which gives a lower bound of ≈90 km and an upper bound of ≈180 km. However, these estimates are somewhat suspect because our constraint scheme fails in this scenario

◦ as the Aψ = 60 intersection predicts compressive stress nearer to the basin than is

◦ seen, while the Aψ = 75 intersection does not predict compressive stress outside of the fault. This result illustrates a potential pitfall of this technique, which is that the fault(s) used for the analysis must represent the innermost normal fault or the logic breaks down. Fortunately, both the aforementioned small fault NBa and the NB fault can be compared to each other directly as they lie on the same transect. The model predicts some unobserved extensional faults inwards of the observed fault(s) if the intersections of the distances with the contours of shape parameter are above the cusp of the contour. It is interesting to note that a model gives the location of the hidden fault NBa with almost precisely the same lithospheric thickness as the intersection with NB. We therefore assume that the bounds on lithospheric thickness are robust only if the locations of the faults are indeed the innermost bound of the faulting zone.

25 Figure 2.8: Contours of shape parameter (shown in degrees), on a plot of distance to ◦ center of basin vs. lithospheric thickness. Aψ = 60 is a lower bound for lithospheric ◦ thickness and Aψ = 75 is and upper bound. The horizontal line at 810 km is the mean distance to the NB fault in Nili Fossae. The horizontal line at 740 km is the distance to fault NBa in Nili. The locations at which the contours cross the horizontal lines can be considered lower and upper bounds on lithospheric thickness.

The shape parameter results from the radial transect in Figures 2.7 – 2.9 demon- strate the utility of comparing the predicted faulting style with the observed location of the innermost normal faults at Isidis. By extending this idea to examine how shape parameter varies along the strike of a fault, we can develop a more robust determi- nation of whether particular models are consistent with the observed faulting. We

26 Table 2.5: Average distances from the center of Isidis Planitia to the innermost faults mapped at Nili and Amenthes Fossae. The ID’s correspond to those seen in Figure 2.1.

Fault ID Name Average Distance (km) NA 774 NB 809 NBa 742 AA 725

conducted a survey of the innermost faults of Nili and Amenthes Fossae at three dif- ferent locations (Table 2.5). The RMS misfit between the shape parameter calculated

◦ ◦ by our models at those locations and the Aψ = 60 and Aψ = 75 contours provides a quantitative means for obtaining the best fit to these bounds (Figure 2.9). Nili Fossae is the more prominent of the two graben systems near Isidis, and has the most clearly defined faults. The faults mapped as NA, NB, and NBa all belong to the Nili Fossae system; however, as discussed above, the large NB fault does not lead to self-consistent interpretations, likely due to the presence of smaller, more inward faults, particularly NBa. The RMS misfit between the predicted and observed faulting location (Figure 2.9) at NA and NBa are similar despite their azimuthal and radial separation. The consistency between the results for NA and NBa highlights the spatial variability of the stress field as well as the potential strength of the approach of constraining the bounds on lithospheric thickness using observed faulting. For the nominal model parameters of Table 2.2, the lower bound on lithospheric thickness at Isidis is approximately 100 km, while the upper bound is ≈180 km. Variations in the mean global crustal thickness have little effect on the lower bound of the lithospheric thickness (Figure 2.10), however they lead to differences in the upper bound by > 20%. Amenthes Fossae to the southeast of Isidis is a more subtle set of features with smaller and fewer visible extensional faults ( and Guest, 1987; Tanaka et al.,

27 Figure 2.9: RMS misfits between predicted and observed styles of faulting. Each panel corresponds to a particular mapped fault (Table 2.5). The solid lines are the misfit of shape parameter 60◦ and the dashed lines are for shape parameter 75◦. The black lines are the misfits from the model using the global dataset, while grey lines are for the local models.

2005). Our models also predict normal faulting in this area (Figure 2.7); however, the interpretation of these results is not as straightforward as at Nili Fossae. The RMS misfit at fault AA (Figure 2.9) shows a much higher estimate for the bounds on lithospheric thickness. The RMS values at Amenthes are also higher than at

28 Figure 2.10: Lower and upper bounds on the thickness of the elastic lithosphere. These bounds were determined by finding the lowest RMS misfits between predicted and observed styles of faulting at the NBa fault using the global model at various crustal thicknesses. See Figures 2.8 and 2.9 for an explanation of the fitting method.

Nili Fossae. Though somewhat speculative, this result suggests that there may be obscured faults further inward from fault AA, which would lead to an over-estimation of lithospheric thickness. However, a detailed survey of the area revealed no clear, smaller hidden faults, as is the case of fault NBa on the Nili Fossae side of the basin, which suggests that there may have been significant surface modification at Amenthes near the basin. It is also possible that the faults at Amenthes did not initiate at the

29 same time as Nili Fossae, which would explain the difference in lithospheric thickness estimates.

2.4.3 Role of external loads on tectonics

A common approach for studying lithospheric flexure is to employ simplified load ge- ometries that make for straightforward model calculations (e.g., Comer et al., 1979; Janes and Melosh, 1990; Comer et al., 1985). These studies also suffered from a relative lack of data that would justify a more precise modeling approach. Often these models use a single axisymmetric load shape that inevitably yields axisymmet- ric stress distributions and predictions of faulting rather than concentrated zones, such as observed along the periphery of Isidis. Though the lack of circumferentially continuous faulting may be the result of burial (e.g., near Syrtis) and erosion, the non-circumferential strike of the Nili Fossae graben belies that idea. Therefore, it is important to ask how much the inclusion (or exclusion) of loads, external to Isidis (or any other feature) affects our ability to infer lithospheric structure based upon the location of faulting. Our basic approach to addressing the role of loads external to Isidis on regional tectonism is to compare the flexural response of the lithosphere near Isidis using the deformation model constrained by the global topography and gravity against an equivalent forward solution that models the response only due to loads within the impact basin. The comparison of these two models allows us to investigate the relative role of exterior loads in controlling the style and location of faulting. In order to facilitate a direct comparison between the two types of models we isolated the relevant load components within the basin from the global models and used them as inputs for the forward models. Specifically, we extract the loads on the lithosphere at Isidis by localizing the size and shape of the basin, its fill material, and compensating relief along the moho from the globally constrained model. We use

30 the local, forward model, with equivalent loading and flexure equations and identical flexural parameters, to determine the amount of deformation and lithospheric stress within and peripheral to the load that result from the loading internal to the basin. The loads in the basins interior were isolated using a spherical-cap localizing win- dow in the spherical harmonic domain defined spectrally

√ J00 = 4π (2.10)

where J00 is the zeroth order term and

r 4π Pl−1(cos ξ) − Pl−1(cos ξ) Jl0 = (2.11) 2l + 1 P0(cos ξ) − P1(cos ξ)

where Jl0 are the zonal terms, Pl are the associated Legendre polynomials of degree l, and ξ is the angular radius of the desired window (Simons et al., 1997). The window is normalized to have a value of one within the window and zero outside. Because the localization window is defined purely by zonal harmonics, it is then rotated into a position centered on Isidis and expanded into the spatial domain. We multiply the window with the global data, which localizes the data to the basin that are then trans- formed back into spherical harmonics up to degree and order 120 to minimize spectral leakage and ringing outside of the window. All of the operations on the data described above were accomplished using the freely available software archive SHTOOLS avail- able at http://www.ipgp.jussieu.fr/ wieczor/SHTOOLS/SHTOOLS.html. To generate the best possible comparison between the global and local loading models it is desirable to have loading conditions between the different model types be as similar as possible. At the center of the window (i.e. the basin) where the load magnitudes are greatest and contribute the most deformation, the localized loads are identical to loads from the global model and match well within the remainder of the localizing window (Figure 2.11, a – b). Indeed, the window used here results in local-

31 ized loads with maximum differences of less than 8% relative to the loads in the global model. There are discrepancies at the southern edge of the basin; however, these are due to the localizing window and especially the low magnitude of the signals here, which acts to magnify fractional differences. More importantly, the forward models based on these localized loads generally produce comparable topographic relief (Fig- ure 2.11, c), gravity (Figure 2.11, d), and deflections with the global model, especially within the localized region of the basin, providing confidence in our approach.

Figure 2.11: Differences between the maps produced by the inverse model, and the maps used as input for the forward model. The white circle is the extent of the localizing window. The crosses indicate the locations of points used to represent individual faults around Isidis: magenta is fault NA, yellow is NB, orange is NBa, and white is AA (Table 2.5).

The most readily apparent differences between the two models are the azimuthal

32 variations of the stresses (Figure 2.12, c). The global model that includes external loads predicts circumferential extensional stresses concentrated near the locations of graben observed on the surface (Figure 2.12, a). The localized model, however, does not produce any significant azimuthal variations in stress (Figure 2.12, b). Therefore, variations in loading inside the basin do not substantially contribute to the azimuthal variations seen in the global model, or to the observed tectonics. The radial variations of stresses (Figure 2.12, c) from the center of the load are used to compare our models to observed conditions on the surface (Figure 2.7) and the RMS misfit between these two distributions suggests a difference between the global and localized models (Figure 2.9), where the localized models consistently predict larger values for the lithospheric thickness relative to the global model at each fault investigated. Moreover, the shape of the RMS curve for the localized model at fault AA is inconsistent with the global model. Therefore, the simplified local models do not fully characterize the faulting observed and consistently over-predict the estimates of lithospheric thickness at the time of Isidis Planitia’s loading.

Figure 2.12: The left two maps a and b show the predicted faulting style determined through shape parameter. a is from the global model, while b is from the local model. c shows the difference between the two models for the nominal case. The black cross in a shows the directions of the principal stresses in an extensional regime where the largest stress is perpendicular to the observed graben.

33 2.5 Discussion

2.5.1 Globally-constrained flexure models

The inversion of topography and gravity was constrained by the assumption that there should be a positive crustal thickness beneath the basin immediately after impact. Negative crustal thickness as defined in our model, would imply that excavation of the entire crust at Isidis Planitia upon impact is insufficient to account for the gravity signal observed. Models with negative crustal thickness necessitate excavation of a portion of the mantle or a reduction in local mantle density in concert with removal of the crust. We cannot rule out excavation of the mantle at the time of impact; however, the likelihood of a localized region of low-density mantle coincident with the placement of the basin is very low. While assuming that the local crustal thickness beneath Isidis at impact should be greater than zero is a logically sound constraint, it is not strictly testable with available observations. A more robust observation is that no mantle material has been detected at the surface inside Isidis, which suggests that the more conservative constraint is to limit acceptable models to those with a positive local crustal thickness at present. The density of the fill material in the basin is more consistent with a value greater than 2900 kg m−3, rather than lower, as more models that are reasonable exist with fill densities greater than this value rather than less. A mean global crustal thickness of 50 km (Wieczorek and Zuber, 2004) coupled with the upper bound we determined for the elastic lithospheric thickness (L = 180 km) results in a fill density greater than 3050 kg m−3 (Figures 2.4 and 2.5). This result implies that materials found within Isidis have a significant igneous component. The basin’s proximity to the Syrtis Major volcanic province just west of the basin supports this interpretation, as it is a prime source region for an olivine-rich volcanic unit at least 800 m thick (Tornabene et al.) on the western edge of the basin. Kiefer (2004) estimated a surface material density in

34 the Nili Fossae region similar to the meteorite, which has a grain density of 3590 kg m−3, and could contribute to a higher fill density in the basin. Furthermore, Kiefer (2004) found that Nili and Meroe Patera have a Nakhla-like density of 3290 kg m−3, suggesting that the material flowing from the Syrtis region may be high-density lava flows. Both the volcanic sources and deposits in close proximity to the basin lend credence to the idea of a relatively high bulk density fill. This does not rule out the possibility of lower density material being present at the basin, but suggests that any such material will be more scarce than the volcanics. However, here we estimate the bulk fill density leaving open the possibility of higher density volcanic deposits interlayered with lower density sedimentary deposits that accrued episodically over time (e.g., Ivanov and Head, 2003; Hiesinger and Head, 2004). The inversion of gravity observations in our formulation depends on the first- order mass sheet approximation calculation of the geoid heights and does not take into account any finite amplitude effects due to relief on density interfaces (Wieczorek and Phillips, 1998). However, owing to the small dynamic range of the topography in the study area of this paper, the discrepancy introduced by this approximation of the gravity was less than 2% over the entire area and less than 1% at the center of the basin where the load is greatest. The effect of this assumption on our results is therefore minimal.

2.5.2 Tectonic Constraints

The existence of tectonic features clearly associated with Isidis Planitia provides a potentially important additional constraint on the state of the lithosphere compared to other basins on Mars. Indeed, the tectonic features are critical to understanding the flexure and lithospheric thickness around Isidis, as other methods (e.g., previous admittance studies) are not capable given that the basins gravity and topography are not correlated. Application of the observed tectonics as a constraint on flexural

35 models is not entirely straightforward, however, as there is a broad region of faulting (e.g., Nili Fossae), the distribution of faulting varies azimuthally, and the timing of faulting across the region is under-constrained. Basin filling ostensibly occurred over a finite time interval during which the lithospheric thickness may have changed, a common challenge of elastic flexure models. Thickening of the lithosphere due to cooling tends to freeze-in flexure, though progressive cooling during loading leads to lithospheric flexure estimates slightly thicker than if there were no cooling (e.g., Al- bert et al., 2000). Therefore, absent thermal rejuvenation of the lithosphere beneath Isidis our results represent reasonable effective lithospheric thicknesses from the time of major infilling. However, more sophisticated finite element modeling with an elas- toviscoplastic rheology will be important for unraveling the tradeoffs in load timing, cooling, and relaxation of the basin and the implications for faulting in the region.

2.5.3 Role of External Loads on Tectonics

An important aspect of modeling lithospheric flexure is adequately capturing the relevant loads; this issue is amplified when considering the orientations and style of tectonic features as a constraint. Indeed, Banerdt (1986) flexural formulation in- cludes the horizontal load potential (Ω) that, while a modest contributor to overall lithospheric deflections, is important for stress magnitudes and orientations. At Isidis Planitia, loads external to the basin play an important role in shaping flexurally in- duced tectonism in the region as demonstrated by the distinct differences between our globally constrained models and the localized forward models. Important con- tributions to these loads arise from the nearby Syrtis Major and Utopia Planitia; the global response of the lithosphere to the Plateau (Phillips et al., 2001) may also play a small role. Azimuthal variations in the predicted stress-state can be explained by the presence of external loads located heterogeneously around the periphery of the basin. In the globally constrained models, the largest stresses tend

36 to be concentrated between external loads such as Syrtis Major and Utopia Planitia (e.g., Nili Fossae). The variation in the radial direction is due to the loads external to the basin that reduce the zone over which deformation caused by Isidis occurs, hence concentrating their magnitude. Compared with the previous work of Comer et al. (1985), who estimated a lower bound on the lithospheric thickness at Isidis Planitia of 120 km and determined a best fit of between 200 and 300 km using a simplified, forward, axisymmetric model, our results suggest tighter limits and relatively thinner values for the lithospheric thickness. However, consistent with the limited data available at the time, their model only considered surface mass loads rather than including crust-mantle relief or intra-crustal density interfaces. Furthermore, by implementing a constraint that models match both the style of faulting and a threshold stress for faulting our results indicate that simplified, local forward models tend to overestimate the lithospheric thickness compared to those that include loads external to the basin in the analysis. We see this overestimation while comparing our lower bound of 100 km to the 120 km lower bound of Comer et al. (1985) along with our upper bound of 180 km compared to the 200 to 300 km best-fit lithospheric thickness of Comer et al. (1985). Given differences in the data available and modeling approach, the modest differences in results are quite reasonable.

2.5.4 Thermal Gradient and Heat Flux

An important rationale for estimating the thickness of the elastic lithosphere for a variety of geological terrains (e.g., McGovern et al., 2002) is that such determina- tions are an important step in investigating the thermal history of Mars. Using the well-established methodology of McNutt (1984), we have calculated ranges of po- tential thermal gradients at Isidis at the time of loading based upon our results for upper and lower limits on elastic thickness (L). Thermal gradients are determined

37 by translating estimates of the thickness of the elastic lithosphere found using our

model to the thickness of the mechanical lithosphere Tm. The mechanical lithosphere is defined to be the depth at which the lithosphere has no mechanical strength. We may approximate this depth as an isotherm between 550◦ C and 600◦ C depending on the assumed strain rate (10−16 – 10−17 s−1) (McNutt, 1984). The best estimates of the thickness of the crust at Isidis are significantly smaller than those of its elastic lithosphere, which allows us to assume that the ductile strength of the lithosphere is limited by the creep strength of olivine (Goetze, 1978), the likely dominant mineral in the lithospheric mantle. The relationship between the thicknesses of the elastic and mechanical lithospheres depends on the curvature of the lithosphere (McNutt, 1984; Solomon and Head, 1990; McGovern et al., 2002); greater curvature produces a larger difference between the elastic and mechanical thicknesses of the lithosphere. By calculating the second derivative of the deflection of the shell in our models we find the curvature of the lithosphere surrounding Isidis in our models is rather small,

−9 −9 −1 between 2x10 and 9x10 m , which allows us to assume that L ≈ Tm. We calcu- late the thermal gradients by assuming the surface temperature to be 220 K (Kieffer et al., 1977). For an elastic lithospheric thickness of 100 km (our lower bound), the thermal gradient is 6 – 6.5 K/km, while for a thickness of 180 km (our upper bound), it is 3.4 – 3.6 K/km (Table 2.1). Surface heat fluxes are estimated from these thermal

∂T gradients using q = k ∂z , where k is the mean thermal conductivity, assumed to be 4.0 W m−1 K−1 which is consistent for mantle material (McGovern et al., 2002) since the amount of crust beneath Isidis is minimal. For an elastic lithospheric thickness of 100 km, the heat flux is 24 – 26 mW m−2; for a thickness of 180 km, it is 13.6 – 14.4 mW m−2 (Table 2.1). The heat flux and thermal gradient estimated at Isidis are consistent with the general decline of mantle heat flux through time (McGovern et al., 2002), however the values seen at Isidis are lower than the results of Hauck and Phillips (2002) which may suggest that Isidis is an area of anomalously low heat flow.

38 The heat flux at Isidis compares with the present day values (Hauck and Phillips, 2002) for an elastic thickness of 100 km, however the heat flux calculated for our upper range of elastic thicknesses is lower than any heat flux in the nominal model of Hauck and Phillips (2002). We estimate the thermal properties of Isidis for the time of flexure corresponding to the initiation of faulting in Nili and Amenthes Fossae, dated to be of Middle Noachian age (Tanaka et al., 2005). The surface units in Isidis are significantly younger than Noachian with ages ranging from the Late Hesperian to the Early Amazonian, clearly the fill seen at the surface of Isidis cannot be respon- sible for the majority of the flexure. This result is not surprising, as the surface units appear to be quite thin (Buczkowski, 2007). When compared to the results of the McGovern et al. (2002), the thermal estimates for Isidis Planitia agree with a surface age in the Late Hesperian or Early Amazonian. The discrepancy between the ages of the tectonics and the age of the surface units suggests that surface ages can only be used as a bound on the youngest age of any results. Thus, we must be cautious when ascribing any physical properties to the timing of a feature by using the surface age as a lower bound.

2.6 Conclusions

The analysis of the tectonics at Isidis in concert with gravity and topography data has proven useful in constraining the elastic lithosphere beneath Isidis to be between 100 – 180 km thick, which is smaller than the range proposed by Comer et al. (1985). The range of lithospheric thickness determined in this study have provided a relatively narrow estimate of the thermal gradient beneath Isidis (3.4 – 6.5 K km−1), and the heat flux (13.6 – 26 mW m−2) at its surface. The ranges of thermal properties given here are consistent with those estimated by McGovern et al. (2002) however the age of the surface units at Isidis cannot be the only time at which material was

39 emplaced as the fill densities are generally too high to be sedimentary. In this study, we have inferred the majority of the fill inside Isidis to be of densities higher than crustal density, which is corroborated by the fact that there is a large effusive volcanic province directly adjacent to the basin. This suggests that the surface layer AIi, which is interpreted to be sedimentary, should be quite thin (e.g., Buczkowski, 2007). The surface age of Isidis should thus, only be used as a youngest bound for the ages of the flexure. Localized loads modeled in a forward sense compared to the inverse global models have demonstrated that loads outside the basin are important to interpretations of lithospheric flexure. Models that do not include the external loads tend to overesti- mate the lithospheric thickness beneath a feature. The elastic model used in this study was useful for constraining properties of the basin at the time of loading, which may have occurred during the Middle Noachian, however, this limited model cannot simulate any relative timing of loading events. Further investigation of this basin using a more sophisticated model with a more realistic and time-dependent rheology is warranted.

40 Chapter 3

Spherical Harmonic Radial Basis Functions: A Framework for Representing Heterogeneous Geophysical Data

3.1 Introduction

Spherical harmonics have been used ubiquitously in the geophysical community to represent an array of geophysical data (e.g., Lorell et al., 1973; Campbell and An- derson, 1989; Zuber et al., 2000; Konopliv et al., 2001; Reigber et al., 2005; Tapley et al., 2005; Konopliv et al., 2006; Namiki et al., 2009). These harmonics allow a spectral analysis of planetary data that is advantageous in that it can be filtered, and certain terms of the expansion are easily related to physical properties of planets. Inversion of the data, however, ideally requires that data be distributed equally over the surface (Weyl, 1914), and in the case of satellite measurements, from a regular surface of nearly circular orbital measurements (Freeden and Michel, 2004). In prac-

41 tice though, it is not always possible to aquire data from the full area of the surface nor in an equidistributed fashion as is the case for single satellite Doppler gravity measurements at the Moon (Konopliv et al., 1998, 2001). Satellite data may also come from various altitudes in elliptical orbits (e.g., Soffen and Snyder, 1976; Colin, 1980; Snyder, 1977; Saunders et al., 1992; McAdams et al., 2007). Both of these situations are not conducive to obtaining accurate gravity models. Investigations of specific locales on the surface of a planet sometimes requires the localization of data by isolating the study region with a windowing function. Applying windowing functions effectively to data (Simons et al., 1997; Wieczorek and Simons, 2005) to perform regional analysis of geophysical data is a highly complex and laborious process. Harmonic radial basis functions are in some respects more efficient and effective at localizing data on a sphere. We propose that the problems associated with heterogeneous data distributions can be solved by changing the modelling basis from spherical harmonics, to a system of harmonic radial basis functions. These relatively recently developed functions (Freeden, 1981; Freeden and Michel, 2004) have better space localizing characteristics and do not require an equally sampled distribution of points. The goal here is to test these basis functions and compare them against spherical harmonic representations of heterogeneous data. This paper is divided into three main parts. The first deals with the difficulties in dealing with scattered sets of data like those returned from single satellite Lunar missions. The lack of Doppler shift gravity measurements from the far side of the Moon is an incomplete spherical dataset that spherical harmonics cannot accurately represent without added constraints. The second applies the methods of harmonic radial basis functions to the problem of localizing geophysical data to specific regions of interest. Spherical harmonics are not able to represent subsets of the sphere without numerical artifacts without spectral leakage corrections (Goossens, 2010). The third

42 addresses the problem of highly elliptical orbits and determining the gravity field from them. Resolving the gravity field at different resolutions over the sphere in one representation is only possible with a spatially localized basis. In all cases spherical harmonic functions are compared against the harmonic radial bases.

3.2 Representation of Lunar Gravity

3.2.1 Introduction

Line-of-sight

Due to the tidally locked orbit of the Moon, a single spacecraft system cannot aquire line-of-sight Doppler measurements of radio tracking signals from its far side. Any knowledge of the far-side gravity field of the Moon before the SELENE mission (Namiki et al., 2009) has come from a method of using Kaula’s rule-of-thumb to predict the surface accelerations of the far-side and apply a fixed measurement vari- ance to the solution (Konopliv and Sjogren, 1994). The lack of direct observations limits the possible resolution over the far side greatly. There is a gap in the data cov- erage of about 33% of the surface on the far side of the Moon, satellite measurements are possible about 20◦ past the limb (Konopliv et al., 2001). While recent results from the SELENE mission to the Moon have eliminated this gap in coverage using a 4-way Doppler tracking system between an orbiter and a relay satellite (Namiki et al., 2009), there are many other missions, including the recent Lunar Reconissance Orbiter (Chin et al., 2007), which may suffer from a lack of global data coverage due to occultations of their spacecraft or hyperbolic trajectories (e.g. Lorell et al., 1973; Tarakanov et al., 1984; Campbell and Anderson, 1989; Yeomans et al., 1997; Kono- pliv et al., 2001). Inhomogeneous data is undesirable when using discretized globally supported functions since their solution is dependent on an equidistribution of data

43 points (Weyl, 1914; Montgomery, 2001).

Limitations of Spherical Harmonics

Spherical harmonics are in their essence, an infinite sum of weighted sinusoids sim- ilar to the Fourier expansion. These types of expansions require the evaluation of orthogonal integrals and solving for gravity based on measurements from the space outside a boundary, makes this a boundary value problem. To solve such a system using approximate integration requires a sufficiently dense equidistribution of points in space (Freeden and Michel, 2004). Without this type of distribution, some type of external constraint (i.e., Kaula’s rule (Kaula, 1966)) must be applied for stability. Constraints of this type impose a restriction on the smoothness (regularize) of the solution, but this introduces a bias on the signal which may affect our interpretations.

Reproducing Kernel

One way to avoid the pitfalls of approximate integration is to use an alternative method called a reproducing kernel. A Hilbert vector space of functions defined by a reproducing kernel is suited to interpolating inhomogeneous data as it is solved by reducing the discrepancies between the kernel and the data to a minimum norm that can be chosen to induce a smoothness on the solution. Minimum norm interpolation reduces large oscillations and instabilities in data poor regions, while reproducing the data (Freeden and Michel, 2004). Reproducing kernels can be constructed using the same methods as spherical har- monic series expansions, and can accomplish many of the same tasks, without requir- ing homogeneous data. One such method is called a radial basis function (Buhmann, 2001). In its basic form, it is simply a function in Euclidean space which does not depend on the distance from some common origin, but rather the radial distances from a series of arbitrary points, known as ”knot” points. Radial basis functions are

44 similar to spline functions, but are not required to vanish at infinity. Radial basis functions can be used to solve interpolation problems (Freeden, 1981) with scattered or incomplete data or can act as a supplement to an already existing global solution in order to reduce the residuals of the solution.

Synthetic Data

To test the radial basis function as a solution to the problem of incomplete or inho- mogeneous data, we construct a synthetic dataset of the gravity of the Lunar surface. Recent results from the SELENE mission (Namiki et al., 2009) to the Moon (Figure 3.1) provide gravity data from both sides of the Moon, which we use to create a synthetic, inhomogeneous distribution of data.

Figure 3.1: Free-air gravity anomaly at the lunar surface from SELENE model SGM90d (Namiki et al., 2009). Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively. The image is shown in a Lambert azimuthal equal-area projection.

To simulate the effect of a highly resolved Lunar nearside and a low resolution farside, we expand the SGM90d spherical harmonic model (Namiki et al., 2009) to

45 points distributed around the sphere that can be computed exactly for a chosen resolution with equally spaced sampling (Driscoll and Healy, 1994). The distribution of data that can be reproduced exactly by a spherical harmonic model depends on the number of terms which are involved in the expansion (Driscoll and Healy, 1994). The problem of dissimilar resolution between the nearside and farside of the moon is approximated by choosing a dense distribution of points for the better resolved nearside, and a less dense distribution for the farside. See Figure 3.2 for an example dataset. By changing the difference in resolution and thus the homogeneity of the distribution, we can study its effect on a model solution.

Figure 3.2: Synthetic gravity data based on lunar free-air gravity anomaly. The map on the left is the gravity of the near side of the Moon resolved to degree 36, while that on the right is the far side, resolved to degree 4. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively.

46 3.2.2 Methods

Spherical Harmonics

Surface spherical harmonics are a summation series of sinusoids with specific weights that give them spherical symmetry. The simplest form of spherical harmonics are called zonal expansions and depend only on colatitude and are built of weighted Legendre polynomials ∞ X f(θ) = flPl(θ) (3.1) l=0

where l is the degree, fl are the Legendre coefficients and the Legendre polynomials

Pl are calculated using Rodrigues’ formula (Blakely, 1996)

1 dl P = (θ2 − 1)l (3.2) l l!2l dθl

This expansion can be generalized to more dimensions in spherical coordinates and should also depend on longitude in order to cover the entire sphere. Square- integrable functions on a sphere f(θ, φ) ∈ L2 can be represented by a system of linear equations of spherical harmonics

∞ l X X f(θ, φ) = flmYlm(θ, φ) (3.3) l=0 m=−l

whereflm are the Legendre coefficients of degree l and order m, and Ylm are spherical harmonics defined by

  m cos mφPl (cos θ) if m ≥ 0 Ylm(θ, φ) = (3.4)  m sin |m|φPl (cos θ) if m < 0

m where Pl are the fully normalized associated Legendre functions (Wieczorek and

47 Simons, 2005) given by

s (n + 1 )(n − m)! (−1)m 2 P m(θ) (3.5) (n + m)! n

Given a set of data, a system of equations can be assembled using Equation 3.3, to solve for the Legendre coefficients which describe that data function.

Kaula’s Rule

Through studies of the covariance of surface gravity anomalies, Kaula (1966) found that the spherical harmonic power spectra of planets tend to decrease by a power law

β × 10−5 (3.6) l2

where β was a constant determined empirically. The Kaula rule is used to constrain the power spectra of spherical harmonic expansions of planetary gravitational po- tential, which dampens the higher order terms and limits the effect of noise in the solution. This constrains the smoothness of the solution and stabilizes a system which would otherwise be unsolvable. The Kaula rule applied to these systems is a type of regularization, akin to the truncated singular value decomposition (TSVD) or Tikhonov regularizations (Aster et al., 2005).

Spherical Addition Formula

Harmonic radial basis functions that are dependent only on the radial distance be-

tween two points ξ and η on the unit sphere Ω1 are constructed using the addition theorem of spherical harmonics

2l+1 X 2l + 1 Y (ξ)Y (η) = P (ξ · η) (3.7) lm lm 4π l l=1

48 (Freeden and Michel, 2004). If the two points do indeed lie on a unit sphere, their dot product will be simply the angular distance between them, thus creating a function that only depends on the angle between two points.

Spline Construction

Radial basis functions are constructed by choosing a norm that will produce a re- producing kernel for use as the radial basis. The reproducing kernel is calculated by ∞ X 2l + 1 K(ξ, η) = A−2P l(ξ · η) (3.8) 4π l l=0 where Al is an infinitely summable sequence which makes the series convergent and therefore makes the closed-form expression for the reproducing kernel possible. One

−1/2 such expression results from choosing Al = h , 0 < h < 1, where h is a dilation factor. Summing to infinity results in the Abel-Poisson kernel

1 1 − h2 K(ξ, η) = (3.9) 4π (L(ξ · η))3/2 where L(ξ · η) = 1 + h2 − 2h(ξ · η) (3.10)

(Freeden and Michel, 2004). The dilation factor is a mechanism for changing the angular support of the function (Figure 3.3). When the dilation factor is 1, the Abel- Poisson kernel becomes a Dirac kernel, which is infinite at its knot point and zero everywhere else.

Interpolation Problem

The Abel-Poisson function provides a way to solve the interpolation problem using a reproducing kernel. Given N data points at locations ξi with values f(ξi) on a sphere

49 Figure 3.3: Contour map of the scaled Abel-Poisson reproducing kernel as a function of dilation factor. The dilation factor causes the broadest base at 0.1 and the func- tion becomes increasingly concentrated as this factor nears 1. The amplitude of the function decreases as the angular distance from the knot point increases.

Ω we find a representation of the underlying function VN by solving

kVN k = inf kf(ξi)k (3.11)

The unique form of f(ξi) is given explicitly by

N X f(ξ) = aiK(ξi, ξ) (3.12) i=1 where coefficients a1, . . . , aN satisfy

N X f(ξi) = aiK(ξi, ξj), j = 1,...,N (3.13) i=1

50 (Freeden and Michel, 2004). A similar system of equations is also used to solve for the Legendre coefficients of spherical harmonics where the reproducing kernel is replaced with the spherical harmonic basis of Equation 3.4.

Variable Dilation Factor

Previous work using the Abel-Poisson kernel (see Amirbekyan et al., 2008) has used a constant dilation factor. However, when dealing with highly non-uniform data distributions, a single dilation factor might not be able to represent all the data accurately. The dilation factor controls how localized the Abel-Poisson function is in space. The effective support for the kernel is defined by a spherical cap of angular radius

1 − h2  σ = arcsin (3.14) K 1 + h2

(Freeden and Michel, 1999). If the data, and thus the knot points, are spread so thinly that this support does not overlap between kernels the functions lose the signal in between. If the data are overly dense, the solution becomes nearly singular and largely inaccurate as an interpolant. When using a constant dilation factor, it must be adjusted in order to achieve the best support for a distribution of data. Some distributions are too widely scattered for a single dilation factor to suffice. The Abel-Poisson reproducing kernel provides an orthogonal basis in space and as such, should not require a common dilation factor for each knot point. By individually adjusting the dilation factor to minimize the difference between the radius of the supporting spherical cap, and the average distance between neighboring knots, an optimal level of support can be achieved. The average distance between knots is calculated by triangulating the knot distri- bution using Delauney triangulation (CGA) and equating the area of the conjugate

51 Voronoi distribution (Aurenhammer, 1991) to the area of a circle, determining an average radius. The corresponding dilation factor for the radius σ is given by

√ 1 − sin σ h = √ (3.15) 1 + sin σ

Each knot point and corresponding kernel is then localized in space optimally for its local surroundings. This method provides a robust way of solving for highly scattered data with good interpolation (Wang and Dahlen, 1995).

3.2.3 Results

The effectiveness of using radial basis functions to represent irregular data on the surface of a planet is investigated by solving the interpolation problem constrained by a series of datasets which have a varying degree of inhomogeneity. The accuracy of the results is compared to solutions of spherical harmonic systems based on the same data. The distribution of data on the nearside of the Lunar surface was kept constant at a resolution ideally uniform for a spherical harmonic series truncated at degree and order 36, while the resolution of the farside was varied (e.g., Figure 3.2). Direct solutions of spherical harmonic solution matricies are only possible when truncated at the level appropriate for the lowest resolution on the surface. For an example see Figure 3.4, which is a spherical harmonic representation of the input data of Figure 3.2. The solution is missing much of the detail in both hemispheres of the surface (Figure 3.5), even on the nearside where additional detail is available. Solutions to spherical harmonic solution matrices taken to higher degrees require a smoothness constraint to be imposed upon the solution in order ensure stability. We impose a Kaula constraint on the solution and expand the spherical harmonic expansion to a degree and order slightly lower than that of the nearside resolution. As seen in Figures 3.6 and 3.7, the discrepancy between this solution and that of

52 Figure 3.4: Spherical Harmonic gravity model expanded to degree and order 4. This model was inverted directly without any regularization. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgal; where red and blue denote positive and negative anomalies, respectively.

Figure 3.5: Map of the difference between the full gravity data as seen in Figure 3.1, and the low resolution spherical harmonic gravity model seen in Figure 3.4. Detail is missing in both hemispheres of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgal, where red and blue denote positive and negative anomalies, respectively.

53 Figure 3.1 is quite low on the nearside, and detail missing from the farside can be expected due to lack of information.

Figure 3.6: Spherical Harmonic gravity model expanded to degree and order 34. This model was constrained by Kaula’s power law. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively.

The radial basis function, due to its spatially localized nature, can solve for both low and high resolution data at the same time and requires no regularization. Figures 3.8 and 3.9 show that the accuracy of radial bases for this type of data distribution is very similar to a constrained spherical harmonic function. Comparison of the accuracy of the solutions to the synthetic data shows us that the level of inhomogeneity in the dataset controls the fit. Data that are closer to a uniform distribution are better representations of the gravity field than those that are very scattered (Figures 3.10 and 3.11). When compared to the subset of data that were used as input, the formal misfit of the spherical harmonic solutions were quite inaccurate when compared to those with a Kaula regularization and the radial basis function solution (Figure 3.10). The regularization of the data by Kaula’s rule

54 Figure 3.7: Map of the difference between the full gravity data as seen in Figure 3.1, and the high resolution spherical harmonic gravity model seen in Figure 3.6. The Lunar nearside is reproduced quite accurately, while detail is missing from the far side of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. precludes an exact solution, while the radial basis function solution is exact since we can solve directly with a Cholesky factorization. The accuracy of the interpolation by both model types (Figure 3.11) is largely similar, in fact the interpolation accuracy of the radial basis function and regularized spherical harmonic solutions are nearly identical. With nearly identical interpolation accuracy, it is the level of misfit that separates the two solutions. The radial basis function solutions are unbiased and can represent the data exactly, while the smoothness constraint keeps the spherical harmonic solutions from being accurate.

3.2.4 Discussion

The nature of the spherical radial basis function makes it ideally suited for interpo- lations of scattered data. Spherical harmonics, on the other hand, must abide by the

55 Figure 3.8: Spherical radial basis function gravity model. This model was inverted directly without any regularization. Black dots pinpoint the locations of the model’s knot points. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. principle of equidistributed sampling (Driscoll and Healy, 1994) because of its global support with constant power over the entire sphere. The more localized support of the radial basis (Figure 3.3) allows for more localized representation of data. The dilation function of the Abel-Poisson kernel controls the balance between the amount of spatial and spectral localization that is desired. When the dilation factor is 1, the kernel has perfect spatial localization, which is not desirable for interpolation. By adjusting the dilation factor the desired localization of each kernel, dependent on the distribution of the data, is possible. Spherical harmonics has no similar method for controlling the level of space localization. It does have perfect spectral localization, which is useful for decomposing solutions into separate spectral components; however it may be better to have a non-biased and accurate representation of the data, rather than this ability. The advantages of using harmonic radial basis functions for representation of

56 Figure 3.9: Map of the difference between the full gravity data as seen in Figure 3.1, and the spherical radial basis function model seen in Figure 3.8. The Lunar nearside is reproduced quite accurately, while detail is missing from the far side of the Moon. Left side view shows the near side of the Moon, while the right shows the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively.

Figure 3.10: Comparison of the RMS errors between the radial basis function as well as the spherical harmonic models and the subset of the gravity data used to constrain their solutions. The abscissa depends on the degree resolution of the farside, while keeping the nearside at full resolution. The dotted line is the RMS error of the low resolution spherical harmonic model. The solid black line is the RMS error of the high resolution spherical harmonic model. The RMS error of the spherical radial basis function was at machine precision, and can therefore not be seen here.

57 Figure 3.11: Comparison of the RMS errors between the radial basis function as well as the spherical harmonic models and the full gravity data. The abscissa depends on the degree resolution of the farside, while keeping the nearside at full resolution. The dotted line is the RMS error of the low resolution spherical harmonic model. The solid black line is the RMS error of the spherical radial basis function. The light gray line is the RMS error of the high resolution spherical harmonic model. The spherical radial basis function and the regularized spherical harmonic model have nearly identical interpolation error. highly scattered datasets have been demonstrated here. These special functions re- quire no additional constraints for solution, and as such do not impose any external bias upon the solution. This feature is especially advantageous for bodies like the Moon, as they have significant power in the higher spectral degrees owing to large mascons which exhibit highly anomalous gravity anomalies inside impact basins. Ap- plying Kaula’s rule limits the power of the shorter wavelengths and can therefore mask some of the smaller features which may have anomalously large signals. When the data acquired are dense and uniform however, there is no advantage to using the radial basis as spherical harmonic expansions can represent the data accu- rately, simply, and have the additional advantage of being separable in the spectral domain. However, as will be explained in the next component of this work, there are advantages to using spherical radial basis functions when performing localized studies of any data type, even when the spherical harmonic models are accurate. An interesting future implementation would use the spherical radial basis as a

58 supplement to spherical harmonics, providing a way of co-solving solutions to these types of interpolation problems in a hybrid framework (Freeden and Michel, 2004). Spherical harmonics would be expanded to the maximum degree and order which requires no external constraints and the spherical radial basis functions would be used to reduce the remaining residuals and provide a better solution for both global and regional scales. In this way, the best global solution using spherical harmonics could be determined, while also modeling regions with more detail with appropriate resolution.

3.3 Localization

3.3.1 Introduction

Previous Work

Global spherical harmonic representations of the gravity field can provide many bulk properties of a planetary body. The spectral decomposition of a gravity field is advantageous for deriving several key parameters for describing a planet such as:

total mass M, flattening J2, and C22 which provides the polar moment of inertia C/MR2 and the period of libration (e.g., Kaula, 1966; Konopliv et al., 1998; Tapley et al., 2005; Zuber et al., 2007; Peale et al., 2009). Spherical harmonics can also provide information about time varying components of the gravity field elucidating the seasonal variations of the hydrosphere (Smith et al., 2001b). These properties were the driving force behind adopting spherical harmonics as the customary mathematical representation of global gravity fields. Analyses of regional and local phenomena affecting the gravity of a planet at specific locations on a planetary surface require a spatial localization of the surface field to a specific region of interest (see McGovern et al., 2002; Lawrence and Phillips,

59 2003; Han and Simons, 2008; Ritzer and Hauck, 2009). The spherical harmonic basis is not ideally suited for this purpose and highly complex and time-consuming methods must be employed to provide a serviceable localization (Wieczorek and Simons, 2005).

Localizing Window

We investigate the use of harmonic radial bases as a more easily computed alternative to spherical harmonics when representing local signals. Most regional gravity studies have employed a windowing technique to localize the gravity field to a specific area. The type of window can vary from a simple spherical cap (e.g., Simons et al., 1997; Ritzer and Hauck, 2009), to a multi-tapered window (Wieczorek and Simons, 2005). Localization, however, is not a natural role for spherical harmonics as they possess no spatially localizing characteristics (Freeden and Michel, 1999). Spherical harmonics suffers from leakage of signal outside of most localization windows and often displays ringing outside of the study area. We examine the use of a harmonic radial basis as a more efficient method of localizing the gravity field by demonstrating the improved characteristics of the window itself.

Test Function

The simplest localization window in a spatial sense is a simple spherical cap, where the function has a value of one inside a specific radius from a central point and zero outside. We choose to set up the window with the North Pole as its center for simplicity. A uniformly random distribution of points over the sphere with constant point density, and with those points inside the window radius as ones and those outside as zeros, produces a spherical cap window (Figure 3.12, solid black line). The ability for an interpolation function to reproduce this window tests its suitability for localization of data.

60 Figure 3.12: The top plot shows the reconstruction of a localization window which has a radius of 30◦. The spherical radial basis function reproduces the function accurately enough that the black dots are indistinguishable from the solid black line representing the window function itself. The empty circles show the positions of the knot points of the radial basis. The bottom plot shows the reconstruction by the spherical harmonic function. There is a significant amount of signal leakage and ringing at the discontinuity.

3.3.2 Methods

Using the test function described above, we solve the interpolation problem of Equa- tion 3.13 using both spherical harmonics as defined in Equation 3.4 and the Abel- Poisson reproducing kernel of Equation 3.9. The spherical cap test function is mostly the low wavelength features of the cap and the outside, with a discontinuous jump in between. Given this fact, it is clear that a spatially localizing function such as the radial basis function need not use all the data available in order to reproduce the entire test function. Those areas with low or no gradients such as the large regions of ones and zeros, do not require a large number of knot points to be reproduced. As such, we may use a subset of the data for a solution.

61 Given a set of N points {xi, . . . , xN : f(xi), . . . , f(xN )}, solution of Equation 3.13 with kernel centers (known as knots points) at all data will reproduce all the function values within machine precision. It can be shown however, that a subset of those N points {xj, . . . , xP } ⊆ {xi, . . . , xN } may be used to approximate the original function to some maximum error tolerance, . We can achieve a solution to the problem with the required error based on a subset of the data by following algorithm: As the number

Algorithm 1 Find an optimally distributed subset of data to use as knot points.

Choose symmetrically distributed subset of data xN while δmax ≥  do Solve the system of equations of Equation 3.13. Expand the solution to the positions of the larger dataset. Calculate residuals between the model and the observations at those points. Place an additional knot at the position with the maximum residual. end while

of points in the subset of data used for solution increases, the errors will converge to zero when P reaches N. However, depending on the desired tolerance, many fewer knot points than data need be used in order to achieve an acceptable solution.

3.3.3 Results

Spherical harmonics can, in theory, represent a highly localized function to any finite accuracy by including more terms in the polynomial series. Similarly, radial basis functions can represent any function to arbitrary accuracy by including more data in the solution matrix. Both bases are able to achieve the same accuracy in theory; however the goal here is to compare the computational efficiency of the two represen- tations for localizing windows. To do this we solve the system of radial basis functions by adding knot points as discussed in the algorithm above, and then use spherical harmonics to solve the interpolation problem with the same number of coefficients. The tolerance chosen for this test case was 1% error, and as can be seen in Figure 3.12, a localizing window reproduced to this tolerance is very closely matched by the

62 radial basis functions. The interpolation by spherical harmonics shows a significant amount of ringing in the solution (Figure 3.12) which would be expressed in any analysis where this window was convolved with a gravity signal. The amount of ringing and leakage in a spherical harmonic windowing function depends on the size of window desired. The accuracy of the interpolation, found by expanding the solution at the positions in the full dataset, showed that for spherical harmonics the more concentrated the window, the more ringing affected the solution (Figure 3.13). Radial basis function expansions are able to represent the same win- dowing functions to very good accuracy with RMS error values of less than 0.01%. Comparable spherical harmonic expansions can have up to 22% RMS error for their interpolated values.

Figure 3.13: A comparison of the RMS errors of the misfit (dotted line) and interpo- lation errors of the two models as a function of window size. The interpolation error of the spherical harmonic model is the light gray line, while that of the radial basis function is the solid black line seen near the abscissa. The misfit of the radial basis function model was too small to see on this graph.

3.3.4 Discussion

If the information contained in a gravity signal can be resolved using a spherical har- monic model of degree and order lobs and the window being applied to that signal for

63 localization is truncated at degree and order lwin, then the maximum degree and order to which the resulting convolved model can be trusted without aliasing is the localized

Nyquist frequency lnyq = lobs − lwin (Simons et al., 1997). Increased localization (i.e.

smaller windows) requires higher lwin, so there is always a trade-off between spatial and spectral resolution when using spherical harmonics as a windowing function. The maximum spectral resolution of a localized field can only be increased by decreasing the spatial resolution.

The Abel-Poisson reproducing kernel is not band-limited and therefore has lwin of infinity so we would expect the Nyquist frequency to also be infinite and suffer from significant spectral leakage. However, the Abel-Poisson kernel does not have equal

−1/2 weight given to all the spectral coefficients in the expansion, recall that Al = h in Equation 3.8. So the Nyquist sampling theorem depends on neighboring points rather than an equidistribution of points and can be calculated locally. By using the optimal dilation constant at each point, we can limit the smoothness of the solution spatially and control the balance between the spatial and spectral resolutions. Given the same number of solution coefficients, the Abel-Poisson kernel proves to be a far more accurate interpolation solution, and therefore a much better function for spatial localization of a signal. In fact, to achieve the same accuracy with a spherical harmonic function would require so many more coefficients, that more computational resources than were available to us for this study would be required.

64 3.4 MESSENGER

3.4.1 Introduction

MESSENGER Mission Orbit

The Mercury Surface, Space Environment, Geochemistry, and Ranging (MESSEN- GER) spacecraft enters orbit around Mercury on March 18, 2011, and has a nominal orbital mission of one year. In that time MESSENGER hopes to answer questions about Mercury’s high bulk density, crustal and core structure, and orbital parameters using several optimized experiments on-board. The structure and state of Mercury’s core is one of the guiding questions of the MESSENGER mission and requires an investigation into the gravity field through Doppler measurements of the radio fre- quency (RF) communications subsystem. One of the stated priority goals for the mission is a global spherical harmonic model to degree and order 16 (Solomon, et al. 2007). Measuring the Doppler shifted frequency of the RF signal measured at Earth by the NASA Deep Space Network provides the line-of-sight velocity of MESSENGER (Srinivasan et al., 2007). Slight variations in the velocity of the spacecraft, minus any spacecraft maneuvers or solar , are caused by small perturbations of the acceleration of gravity on Mercury. These measurements are inverted for a rep- resentation of the gravity field, typically performed in spherical harmonics (Equation 3.4). The orbit predicted (Figure 14) for MESSENGER’s orbital phase will have an inclination of over 80◦ with a periapsis near 60◦ N latitude at a minimum altitude of 200 km, though it will vary throughout the mission. The eccentricity of MES- SENGER’s orbit will be ≈ 0.7 which means that apoapsis will be above 1000 km (McAdams et al., 2007). Due to the eccentricity of the orbit the altitude of the orbit will therefore depend highly on latitude and be symmetric about the poles (Figure

65 3.14).

Figure 3.14: The predicted altitude of the MESSENGER spacecraft above the surface of Mercury for the first Mercurian year. The spacecraft will be consistently closer to the surface in the northern hemisphere of Mercury than in the south.

The ability to resolve a feature on the surface of a planet by measuring a har- monic potential field is approximately proportional to the altitude at which those measurements are taken (Dickey et al., 1997). During the MESSENGER mission the highly elliptical orbit will only allow low-altitude measurements of the surface over the northern hemisphere. The wavelength on a sphere of radius R equivalent to a

66 spherical harmonic of degree l is

2πR 2πR λ = ≈ (3.16) pl(l + 1) l + 1/2

(Wieczorek and Simons, 2005) and a feature is resolvable when the half-wavelength of a spherical harmonic expansion is smaller than the size of the feature. Therefore, if the smallest resolvable wavelength is equal to the altitude of a measurement, using spherical harmonics alone will result in a model with a maximum degree and order of 76 in the northern hemisphere while the southern will only be resolved to about degree and order 14 on average tapering to degree and order 0 near 80◦ S.

Synthetic Data

Lacking any significant information about the gravity structure of Mercury other than a very low resolution model (degree 4) developed after the last three flybys (Zuber, Smith, et al. In Press), we use a spherical harmonic representation of the gravity of the Moon Namiki et al. (2009) as a synthetic dataset to simulate the measurements which might be made during the orbital phase of MESSENGER. The data were created by expanding the spherical harmonic series to degree and order 70 at points along the predicted orbital track of MESSENGER adjusted for the smaller radius of the Moon. The data set begins after orbital insertion and samples values for the ideal acceleration of gravity at altitude for 59 days or one revolution of Mercury. Data are assumed to be taken every 10 seconds which approximates the minimum integration time of the Doppler measurements expected from the mission (Srinivasan et al., 2007). The -Earth-Mercury geometry is neglected in this analysis, so spacecraft occultations, solar conjunction periods, and specific periods communication throughout the orbit are not accounted for, leading to a best-case scenario for gravity determination from MESSENGER. Also used as input data are a set of points with values which are one

67 half phase offset from and should be between each data point in order to test the accuracy of interpolation between data points at spacecraft altitude.

3.4.2 Methods

Solid Spherical Harmonics

The methods used in the previous two sections of this paper have used surface har- monics, meaning that they only represent functions on a spherical surface Ω. Repre- sentation of a gravitational potential field requires a dependence on the radius from the center of the sphere and adds a dimension to the problem. The nature of the harmonic functions used here allows them to be solved as separable variables. The formulae used beforehand in Equations 3.4 and 3.8 can be used again with an added continuation term which depends on radius.

We begin with a sphere Ωα of radius α, approximating the general shape of most

int ext planets. Let Ωα and Ωα be the space inside and outside the sphere, respectively. Any vector x or y in these spaces can be expressed in terms of the spherical coordinates of colatitude θ, longitude φ, and radius r. These surface spherical harmonic functions can be represented below or above the

int surface Ωα by adding a continuation term to Equation 3.4. Functions in Ωα can be represented by adding the upward continuation operator (r/α)l when r ≤ α

∞ l X X  r l f int(θ, φ, r) = f Y (θ, φ) (3.17) α lm α lm l=0 m=−l

ext l+1 and Ωα by adding the downward continuation operator (α/r) when r ≥ α

∞ l X X αl+1 f ext(θ, φ, r) = f Y (θ, φ) (3.18) α lm r lm l=0 m=−l

(Blakely, 1996). The two types of harmonics are related by the Kelvin transform Kα

68 where an outer harmonic relative to the sphere Ωα is obtained from a corresponding inner harmonic by

α f ext(θ, φ, r) = f ext(x) = Kα[f int](x) = f int(¯x) (3.19) α α α |x| α where x → x¯ by α2 x¯ = x (3.20) |x|2

int In this way each point in the representation of a function in Ωα has a mapping onto

ext Ωα space (Freeden and Michel, 2004).

Solid Harmonic Radial Basis

The reproducing kernel function of Equation 3.8 can be derived similarly using the spherical addition theorem of Equation 3.7, with the addition of the harmonic con- tinuation terms identical to those in spherical harmonics. Given that measurements

ext taken from altitude will always be inside Ωalpha we restrict ourselves to outer spher- ical harmonics and the corresponding harmonic radial basis (Freeden and Michel, 2004). Using the upward continuation operator the reproducing kernel is constructed as before with

2l+1 l+1 X αl αl 2l + 1  α2   ξ η  Y (ξ) Y (η) = P · (3.21) r lm r lm 4πα2 |ξ||η| l |ξ| |η| l=1 and the harmonic radial basis function kernel is then

∞ l+1 X 2l + 1  α2   ξ η  K(ξ, η) = P · (3.22) 4πα2 |ξ||η| l |ξ| |η| l=0

−1 2 l+1 −2 here the continuation term α (α /|ξ||η|) takes the place of Al in Equation 3.8. It is evident that the continuation operator acts in a similar manner to the dilation factor (Figure 3.15), though differences in altitude rather than the spatial distribution

69 of knot points controls the spatial concentration of the reproducing kernel function. The reproducing kernel of Equation 3.22 is used to solve the interpolation problem

ext defined by Equations 3.11 – 3.13 (Freeden and Michel, 2004) over Ωα .

Figure 3.15: Contour map of the solid spherical harmonic Abel-Poisson reproducing kernel as a function of radial position. The function is much more concentrated at radii closer to the value of α. The amplitude of the function decreases as the angular distance from the knot point increases.

Ill-posed Satellite Problem

Downward continuation is an unstable process and as a result, the problem of using space borne gravity data measurements to infer the gravity structure at the surface of a planet is ill-posed, in the sense that it does not satisfy the three criteria for well-posed problems. A solution of the downward continuation problem does not nec- essarily exist without some alternative solution method, nor is the inverse necessarily continuous. The reason for this instability in the solution of downward continua- tion is that the inverse of the upward continuation operator diverges as l → ∞.

70 This makes the downward continuation problem exponentially ill-posed (Freeden and Michel, 2004). The process of downward continuation tends to amplify the high frequency infor- mation as well as noise in a solution. This leads to the undesirable result of measure- ment error and noise being augmented in relation to the lower frequency component of the signal. When solving the systems of equations used for interpolation, such as Equations 3.3 and 3.13, the solution matrices tend to have very high condition numbers, indicating ill-conditioned matrices. This means that the maximum ratio of the relative error in the solution coefficients is much higher than the relative error in the data. Poorly conditioned matrices are very sensitive to their input as they tend to enhance error (Aster et al., 2005). One way to deal with ill-conditioned problems is to impose smoothness on a solu- tion vector. First, we decompose the m × n solution matrix G into its singular values where G = USV T , U is an m × m orthogonal matrix with columns spanning the data space, S is a m × n diagonal matrix of singular values, and V is a n × n orthogonal matrix with columns spanning the model space. Second, filter factors can be applied to those singular values S by multiplying it by some sequence of numbers (Aster et al., 2005). This process is called regularization and there are several techniques which can be employed to stabilize the inversion of ill-conditioned problems. The truncated singular value (TSVD) regularization multiplies all singular values of the solution matrix by one up to some truncation value and zero for all the others. In other words, it is analogous to solving the equations y = Gx using the Moore- Penrose pseudo inverse solution

−1 T x = VnSn Un y (3.23)

and setting all Sn to zeros above some number (Aster et al., 2005). Since the singular

71 values are ordered most significant first, the TSVD is essentially a low-pass filter. Another method is called the Tikhonov regularization which modifies the problem from least-squares to

2 2 2 min kGx − yk2 + γ kxk2 (3.24) where γ is a regularization parameter, and solved by SVD with filter factors increasing quadratically (Aster et al., 2005). The Tikhonov regularization can also be easily extended to perform the same function as Kaula’s rule (Floberghagen et al., 1999) by solving a similar problem as Equation 3.24 with

2 2 2 min kGx − yk2 + γ kLxk2 (3.25) and L = l2 where l are the degrees of spherical harmonics. This formulation applies filter factors to the singular values like in Equation 3.6.

3.4.3 Results

Doppler measurements from varying altitude will result in a gravity dataset with widely varying resolving power over the sphere. The accuracy of an interpolation function in this instance should be determined by comparing the misfit error mea- suring how well the model reconstructs the data, the error introduced by downward continuation to the surface, and the error of interpolation between points at space- craft altitude. All the errors were calculated as a difference between the full resolution data expanded to degree and order 70 and the model values.

Direct Solution

An ideal solution to the problem of downward continuation is one which requires no additional regularization in order to stabilize a solution. Models with no regulariza- tion applied to the spherical harmonics, however, are not possible. Comparing the

72 errors introduced by downward continuation of a least squares solution of harmonic radial basis functions (Figure 3.16, Upper Row) to those of a spherical harmonic model (Figure 3.16, Lower Row) regularized using the SVD Kaula rule (see Equa- tion 3.22). Both models provide similar results as the pattern and magnitudes of their errors are nearly the same. A nearly identical accuracy of points between the data points at altitude is also evident, where the errors introduced by interpolation at altitude were negligible. The formal misfit between the model and data at the positions of the input was also very low. Both models showed a tendency of missing information in the southern hemisphere, as is to be expected from such an elliptical orbit. Notice that the distribution of knot points over the surface (Figure 3.16) is uneven and concentrated in the northern hemisphere. The method of adding knot points one at a time from algorithm 1 to determine their optimal locations was used. Therefore the number of coefficients needed to resolve the information in the data using radial basis functions is much lower than that of spherical harmonics. As a result, the radial basis function solutions were computed 10 times faster than the spherical harmonics solutions.

Effect of Error

The instabilities caused by downward continuation increase exponentially when deal- ing with error. To test the effect of introducing error we added a Gaussian error similar to that of Uno et al. (2009) with a standard deviation set to a percentage of the range of the input data. By varying the error introduced to the model, we can evaluate the ability of the harmonic radial basis functions to handle possible measurement errors. Accurate representations of the data without using regularization were not possible for either radial basis functions or spherical harmonics with any significant amount

73 Figure 3.16: Example maps of the difference between the surface gravity anomaly of the synthetic lunar data, and both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included no synthetic error. The black dots in the top two maps show the locations of the knot points used by the radial basis. Left side views show the near side of the Moon, while the right two show the far side. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. of error. As an example, using an error distribution with a standard deviation of 5% of the range, Figures 3.17 – 3.19 compare the different type of model error. The application of a TSVD regularization to the radial basis function model (Figure 3.17, top) results in a nearly constant amount of continuation error over all latitudes. The

74 spherical harmonic model (using a Kaula rule regularization), on the other hand performs better in the northern hemisphere (Figure 3.17, bottom), suggesting that the amount of downward continuation affects the instability.

Figure 3.17: These two plots show an example of the error between the values of gravity anomaly predicted by both the radial basis (top) and spherical harmonic (bottom) models at the surface through downward continuation. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. The radial basis model was inverted using TSVD regularization, while the spherical harmonic model was constrained using Kaula’s power law.

For example, a five percent error had relatively little effect on the error interpolat- ing at spacecraft altitude (Figure 3.18), downward continuation is the most significant effect. The radial basis functions did however have a slightly more stable interpolation at altitude, combined with the even distribution of error from downward continua-

75 Figure 3.18: These two plots show an example of the error between the values of gravity anomaly interpolated between data points by both the radial basis (top) and spherical harmonic (bottom) models and the gravity at spacecraft altitude. The data provided to the two models included synthetic error which had a standard deviation of 5% of the range. The radial basis model was inverted using TSVD regularization, while the spherical harmonic model was constrained using Kaula’s power law. tion, which indicates that the continuation itself is not to blame when using radial bases, but rather the error alone. When expanded to the entire surface of the planet, the example 5% models have a fair amount of continuation error over the surface. However, it is the spherical

76 Figure 3.19: Example maps of the recovery of the synthetic lunar data, using both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. harmonic model which shows a systematic loss of information in the southern hemi- sphere, while the radial basis functions have a more even distribution of error (Figures 3.19 and 3.20). The effect of increasing error can be seen in Figure 3.21 for several different reg- ularization types. The root mean square error of a model for any type of error is a good indicator of its accuracy. While both models had nearly identical interpolation

77 Figure 3.20: Example maps of the difference between the surface gravity anomaly of the synthetic lunar data, and both the radial basis function (top two) and spherical harmonic (bottom two) models. The data provided to the two models in this case included synthetic error which had a standard deviation of 5% of the range. Colors represent gravity in mgals, where red and blue denote positive and negative anomalies, respectively. error using several regularization methods (Figure 3.21), the continuation error of the radial basis function models was slightly better than spherical harmonics in all cases of Figure 3.21. The harmonic radial basis was significantly more accurate without any regularization as the spherical harmonic solutions were highly unstable.

78 Figure 3.21: Plot matrix comparing the continuation and interpolation errors (top row) and the coefficients of determination (bottom row) of both the radial basis and spherical harmonic models using several different types of regularization for various amounts of synthetic error. The top row of panels compares the RMS of the contin- uation error of the radial basis (blue) and the spherical harmonic (green) models, as well as the RMS of the interpolation error of the radial basis (red) and the spherical harmonic (cyan) models. The statistics representative of the goodness of fit to data by models using: Kaula’s rule (first column), Tikhonov regularization (second row), truncated singular value regularization (third row), and direct inversion (fourth row) comprise the matrix. Each panel is shown as a function of the standard deviation of added synthetic error which is a percentage of the range of the data.

The accuracies of models using regularizations were very similar, so a better mea- sure of the goodness of fit of a model is the coefficient of determination (Draper and Smith). The inverse of this statistical parameter is the fraction of error introduced by the model. The bottom row of Figure 3.21 shows that while none of the models fit the data well when no regularization is used, the radial basis function fits the field at altitude and on the surface better when some type of regularization is applied.

79 3.4.4 Discussion

The harmonic radial basis of Abel-Poisson reproducing kernels is able to downward continue satellite data from a wide range of altitudes simultaneously, while spherical harmonics does not without significant regularization. The expected accuracy of the Doppler measurements in the orbital phase is about 0.1 mm s−1, which, for a 10 s sampling interval equates to about 1 mgal of error in the gravity measurement, or about 0.25% error of Figure 3.21. Given that the continuation error drops sharply below 0.5% error in Figure 3.21, this small amount of error in the data allows for the direct solution of the system without regularization of the radial basis functions. However, any significant amount of error included in the data significantly affects the accuracy of the results. Given that the Abel-Poisson kernel can reproduce surface gravity anomalies when no error is included they are fundamentally well-equipped to deal with multi-resolution datasets such as will be expected from MESSENGER. In terms of quality of representation, spherical harmonics and harmonic radial basis approaches have similar performance in the presence of significant noise. However, a more optimized approach for dealing with noise in the geodetic problem (e.g., Kaula’s rule for spherical harmonics) using harmonic radial basis functions could improve their relative performance. Despite the similarities in data representation, the harmonic radial basis was much more efficient in solving for the gravity field. Radial basis functions required fewer coefficients to resolve the gravity field because those regions with lower resolution had fewer knot points. Spherical harmonics samples the data equally over the entire sur- face, while the harmonic radial basis only requires densities of knot points dependent on the maximum resolving power of the gravity field in any point over the sphere. If locally supported harmonic splines are used, the basis functions become zero outside a certain radius dependent on the height of the knot point. This way a sparse solution matrix and a spline space can be constructed where the entire matrix need not be

80 recalculated for additions of knots. Spherical harmonic models of the gravity field of Mercury are possible above degree and order 16 as required by the mission, however Kaula’s rule must be applied and the signal of any uncompensated mascons may be attenuated. On the other hand, radial basis functions could achieve the same results using fewer coefficients, and less filtering of the higher frequency information.

3.5 Summary

Harmonic radial basis functions proved to be much better suited for the representation of gravitational potential data scattered over the surface of the sphere. The appli- cation of these functions to highly irregular data is possible without any additional external constraints such as Kaula’s rule, while spherical harmonics were unable to be solved without. Solving for the gravity without constraints is useful for accurately representing any anomalously high regions of gravity such as mascons. Harmonic radial bases and spherical harmonics were nearly equivalent in accuracy when the data were homogeneous. The localization of data using the radial basis proved to be much more efficient and simpler than using a windowing function constructed with spherical harmonics. Localizations with much higher detail and smaller windows are therefore possible. Errors made while measuring gravity data during a highly ellipti- cal orbit requires regularization of the solution whether using radial basis functions or not. Despite this, the radial basis demonstrates promise since it does not seem to be the process of downward continuation that causes the bulk of the instability. Therefore, the regularization is more effectively a de-noising of the solution rather than a stability constraint. The goal of producing a degree and order 16 spherical harmonic model during the orbital phase of MESSENGER’s mission is certainly pos- sible using a Kaula type constraint, however interpretation of such a model may be

81 difficult due to the varying amount of resolution over the surface and there is no way to vary the resolving power of the model spatially. To fully assess the capabilities of the spherical radial basis functions at solving the multi-resolution problem, tests using more realistic data.

82 Chapter 4

Lobate Scarps at Mercury: Constraining Thrust Fault Geometry using MESSENGER Laser Altimetry from flybys 1 and 2

4.1 Introduction

Tectonic features on the surface of terrestrial bodies provide important constraints on the structure and history of their lithospheres. The most pervasive tectonic features found on Mercury during the 10 and MESSENGER missions are lobate scarps thought to be surface expressions of thrust faults (Watters et al., 2009). The scarps are seen across the entire surface imaged by the Mercury Dual Imaging System (MDIS) suggesting that a global phenomenon caused them. Lobate scarps are believed to be the accommodation of contractional strain due to the cooling of the planet during its

83 evolution (Solomon, 1977). Given that there are lobate scarps over Mercury’s entire surface, they are the most important way of constraining the thermal structure of the lithosphere. In this paper, we constrain the possible geometries of three thrust faults measured at the surface with the Mercury Laser Altimeter (MLA) (Cavanaugh et al., 2007) as lobate scarps during flybys one and two of the MESSENGER mission. We will use a forward modeling approach with a mechanical finite element analysis to determine those geometries. Given the depth of faulting we then constrain the thermal gradient and heat flux at the time of faulting.

4.2 Laser Altimetry

The MESSENGER mission to Mercury (Solomon et al., 2007) has performed 3 flybys of the planet to date. The Mercury Laser Altimeter (MLA) (Cavanaugh et al., 2007; Sun et al., 2009; Zuber et al., 2010) has ranged to the surface during the first two flybys providing absolute measurements of the topographic relief of the surface. The altimeter took measurements along a topographic profile from longitude 150◦ to 80◦ W and within 5◦ of the equator during the first flyby (M1) and detected a lobate scarp at 137.1◦ W and 3.2◦ S we call scarp D (Figure 4.1). This scarp is 1350 meters high and has a topographic depression 45 km away to the West. The second flyby (M2) acquired strong MLA signals between longitudes 20◦ and 85◦ E and between 2.5◦ and 3.5◦ S. The MLA track during this flyby cut across 3 lobate scarps within 200 km of each other (Figure 4.2). We call these scarps A, B, and C from West to East. Scarp A was not used for our analysis due to the presence of a crater along track near the lip of the scarp. Scarp B is 1430 meters in relief at 59.3◦ E and 4.8◦ S. A trough was also detected within 120 km of scarp B to the East. Further to the East scarp C is at 64.6◦ E and 4.8◦ S, with 1395 meters of relief and

84 the same topographic depression as scarp B. Ranging data was not acquired during flyby three due to the spacecraft being in safe-hold mode during the closest approach to the planet.

Figure 4.1: MDIS image of scarp D overlain by the MLA track in red and the corre- sponding topographic profile in green. The scarp is centered at 137.1◦ W and 3.2◦ S.

The angle at which these topographic profiles transect the scarps is never perpen- dicular, so any comparison to surface profiles from mechanical models must be done after the MLA tracks have been projected onto a line perpendicular to the scarp. The MLA track is 40.1◦ from normal to scarp D, while tracks across scarps B and C are 65.9◦ and 56.3◦ from normal respectively. More topographic data from the orbital phase of the mission will make this correction unnecessary as profiles will be extracted from a topographic model along several perpendicular tracks. The scarps studied here are newly observed (Zuber et al., 2010) and are only a small fraction of all scarps compared to the 82 lobate scarps detected previously (Watters et al., 2004).

85 Figure 4.2: MDIS image of scarps A, B, and C overlain by the MLA track in red and the corresponding topographic profile in green. Scarps A, B, and C are centered at locations (56.7◦ E, 4.7◦ S), (59.3◦ E, 4.8◦ S), and (64.6◦ E, 4.8◦ S) respectively.

4.3 Model

The structure of the thrust faults thought to have created the lobate scarps seen on Mercury during the MESSENGER flybys was forward modeled using mechanical finite elements. The geometry of the faults were constrained by minimizing the misfit between the forward models and the topography data. We used the commercially available MSC.Marc package that has been used in several other geodynamical studies (e.g., Brown and Phillips, 1999; Ghent et al., 2005; Dombard and McKinnon, 2006). Thrust faults in this analysis are modeled in Cartesian coordinates using an elastic plane strain formulation (Figure 4.3). We approximate the lithosphere using an elastic

86 plate with an inviscid substrate. The extent of the lateral domain of the models was chosen (2000 km width) in order to limit edge effects where the length of the model was increased until no significant change in the resulting solution was seen. The left and right edges of the model were constrained by a fixed displacement in the lateral direction while allowing free slip along the boundary. The bottom boundary was supported by a Winkler foundation which resists the displacement of the boundary as a spring with a constant equal to the gravity times the density of the inviscid substrate. Coupled with the application of a gravitational body force with the acceleration of Mercury (3.7 m s−2) and a large displacement formulation, the foundation models the isostatic response of the bottom density interface. The fault itself was modeled using gap elements, similar to slippery nodes (Melosh and Williams, 1989), as a frictionless rectangular plane. Our models assume a planar geometry, as opposed to listric, consistent with results from previous work (Watters and Schultz, 2002; Watters et al., 2002) and the nearly linear strike profile (Figs. 4.1 and 4.2) of the lobate scarps included in the analysis. We assume a surface breaking geometry, where the displacement along the fault is free, dependent only on the geometry of the fault and the fixed acceleration applied in the lateral direction along the fault plane (Figure 4.3). The application of a fixed acceleration in the horizontal direction approximates the effect of the forces resultant from horizontal shortening of the crust at Mercury’s surface. The density of the crustal material modeled was assumed to be 2900 kg m−3 and a mantle of 3500 kg m−3 however these parameters are very insensitive to variation and any range of densities could have been used. The depth of the crust was found to be insensitive during the analysis and so was kept at a constant depth of 100 km for the suite of models used for analysis. Both the crust and mantle had a Young’s modulus of 100 GPa and Poisson’s ratio of 0.25. These parameters are all consistent with values for the terrestrial lithosphere (Turcotte and Schubert, 2002).

87 Figure 4.3: Elastic finite element model setup. The example model in this figure has a 100 km thick crust with a surface breaking thrust fault, overlying a mantle 400 km thick. All models were 2000 km wide to minimize edge effects. Dislocation was achieved by applying a fixed acceleration load in compressive x directions along the fault plane. The walls were fixed in lateral displacement and the bottom edge was supported by a Winkler foundation of springs.

The model parameters are constrained by the root mean square misfit between the predicted and observed topography of the lobate scarps D, B, and C. The geometry of the fault is prescribed by fixing the dip of the fault using the gap elements and the length of the fault. The amount of displacement in the lateral direction in meters is equal to the acceleration applied at the fault in meters per squared second. While the fault is free to move in space dependent upon the flexure, the angle of the fault is fixed, therefore the dislocation of the fault imposes a torque on the flexure of the model and must be corrected for by de-trending the model profile to match the altitudes of the altimetry at the edges of the observation window.

88 4.4 Results

Good fits between the topography predicted by the models and the MLA profiles depends mostly upon the geometry of the fault rather than material properties. The fits between model and data are very insensitive to variations of the densities of the crust or mantle, as well as the crustal thickness. Variation of the mechanical properties of the lithosphere Young’s modulus and Poisson ratio also have very little effect on the results. The most sensitive parameter affecting the goodness of fit to observed topography was the amount of slip along the fault interface (Figure 4.4). For every model to fit the observed relief across the scarp, the slip must be calculated depending on the angle of the fault. Fortunately, the magnitude of acceleration applied to the faults was equal to the total amount of displacement in the horizontal direction. MSC.Marc implements the fixed acceleration as a nodal force which will produce a prescribed amount of displacement. In other words, Marc applied the force to chosen nodal values to up to the point where the displacement is equal to the acceleration. This allows these fixed acceleration forces to be independent of the mass being moved. For example an applied acceleration of 1000 m s−2 produces 2000 meters of horizontal displacement. This relation lets us calculate the acceleration needed to produce a desired amount of relief across the modeled scarp given the expression

yx a = ◦ (4.1) 2y◦

where a is the applied acceleration, y is the observed amount of vertical relief across

a lobate scarp, x◦ is the horizontal distance between the surface break of the modeled

fault and its subsurface tip, and y◦ is the depth to the fault tip. For any given scarp, the amount of acceleration need to match the scarp relief is calculated using Eq. 4.1. Comparison of predicted fault topography to the lobate scarps D, B, and C demon-

89 Figure 4.4: A series of model profiles (solid lines) varying the amount of fault slip compared to the MLA profile (black dots). All profiles kept the same fault geometry. The track of the profile cuts across scarp C. strates that there is a flexural trough associated with thrust faulting which is directly above the fault tip (Figure 4.5). The best fits for any scarp can only be achieved when the horizontal scale of the fault is equal to the distance to the trailing flexural trough observed in the MLA topography. By measuring the distance to the trough seen on the surface, the horizontal location of the fault tip is predicted. This greatly reduces the number of models necessary to find a fit to the data. While both fault displacement and fault length greatly affect the fits to the ranging data, the effect of varying the angle of the fault is fairly small. As can be seen in Figure 4.6, the differences between models dependent only on the fault angle are smaller than the variance of the data. Visually the differences are small, however the fits based on the RMSE between the predicted and observed topography (Figure 4.7) show that there is enough of a variation to show that the preferred angle for scarp D is between 24 and 29 degrees, while the preferred angle for scarps B and C is around 22 degrees. Given the fault length and these angles, the depth to the fault tip at: scarp D is 20 km, Scarp B is 45 km, and Scarp C is 30 km.

90 Figure 4.5: A series of model profiles (solid lines) varying the length of the fault compared to the MLA profile (black dots). All profiles had the same dip angle of 20◦ and the amount of slip was chosen to best match the observations. The track of the profile cuts across scarp C.

Figure 4.6: A series of model profiles (solid lines) varying the angle of the fault compared to the MLA profile (black dots). All profiles had the same fault length of 85 km and the amount of slip was chosen to best match the observations. The track of the profile cuts across scarp C.

Some previous studies (Schultz and Watters, 2001; Watters et al., 2002; Grott et al., 2007) have modeled thrust faults using an elastic half space. The rigidity of

91 Figure 4.7: Best fits between the models and data were chosen based on root mean squared errors. Here the fault angles best fitting the data was around 25◦ to 30◦ for scarp D (solid gray line) and around 20◦ to 23◦ for scarps B (dotted line) and C (solid black line). a half-space far exceeds that of an elastic plate, so any flexure associated with the deformation of the fault will be smaller than expected for a lithosphere. We investigate the validity of the half-space assumption by varying the thickness of our elastic plate model (Figure 4.8). Thinner elastic thicknesses produce more flexed solutions as expected. The models with an elastic thickness of 200 km predict topography which better fits scarp C. The geometry of the fault is the most important characteristic in determining the best fit to data.

92 Figure 4.8: A series of model profiles (solid lines) varying the thickness of the elastic plate in the model compared to the MLA profile (black dots). All profiles had the same geometry and amount of slip. The track of the profile cuts across scarp C.

4.5 Discussion and Conclusions

The results of this study show that the surface expression of deeply rooted thrust faults is mostly dependent on the kinematic offset across a fault and its length. The angle of the fault changes the shape of the back arc of a modeled fault and can be used to find the best fit to the observed topography. The estimates for the best fit depth of faulting provide a means to investigate the thickness of the brittle lithosphere given that the maximum depth to which a fault can propagate is limited by the ductile nature of the lower crust. Given this, the minimum depth of the seismogenic lithosphere is bounded by the depth of faulting. The depth of faulting can then be used to constrain the temperature structure of the lithosphere at the time of faulting. Temperature increases with depth and with it the strength of the lithosphere goes from being limited by the strength of the brittle portion of the lithosphere to the ductile portion. The maximum depth of seismicity is also controlled by temperature and for the Earth is dependent on the depth to the 250◦ to 450◦ C isotherms for

93 continental crust, and the 600◦ to 800◦ C isotherm for oceanic crust (Watters et al., 2002). The mineralogy of the crust on Mercury is unknown, however it is assumed that absent a subducting crust very little continental type rock exists on Mercury. A conservative estimate of the possible temperatures at the base of the seismogenic lithosphere bounds the temperature at between 600◦ and 800◦ C. Given the thickness of the seismogenic lithosphere l and the temperature gradient ∆T/l between the surface and base, the heat flux through the plate is the gradient multiplied by the thermal conductivity (Turcotte and Schubert, 2002). The thermal conductivity varies with pressure and on the Earth can range betw een 2 W m−1 K−1 and 4 W m−1 K−1 (Hofmeister, 1999). Assuming that the thrust faults on Mercury penetrate the entire seismogenic lithosphere (Watters et al., 2002), the temperature gradient and heat flux are calculated for the scarps detected during the two MES- SENGER flybys. Assuming the average temperature of the surface of Mercury is 440◦ C, scarp D is best fit at a depth of 20 km and a corresponding range of thermal gradients is 8 – 18 K km−1 with heat fluxes of 16 – 72 mW m−2. Scarp B is best fit at 45 km deep with a thermal gradient of 4 – 8 K km−1 and heat flux of 8 – 32 mW m−2. Scarp C is best fit when the fault tip is 30 km deep, giving a thermal gradient of 5 – 12 K km−1 and heat flux of 10 – 48 mW m−2. The range of estimates is large owing to the uncertainties in the possible crustal compositions of Mercury and the scatter in the limited MLA dataset. When MES- SENGER begins the orbital phase of its mission, the crustal composition will become better constrained, and the coverage of MLA will increase. This way the tracks chosen for analysis can be perpendicular to the scarps, and the short wavelength topography can be removed by averaging several profiles along the strike of faults. The profiles used for this study were limited to those along the track of flybys. Scarp D lies very close to a crater and the profile may be affected by the presence of ejecta. Scarps B and C lie so close to one another that a coupled solution with two combined fault

94 geometries may be necessary. The assumption that the faults penetrate the entire seismogenic lithosphere may not be robust. Also, the use of an elastic half-space as the domain is not the best assumption. Previous studies may have overestimated the thickness of the elastic lithosphere. A more sophisticated study of the thrust faults on Mercury is warranted when MESSENGER enters orbit as a temperature structure can then be imposed upon the model and we need not assume anything about the depth to isotherms. The amount of ranging data present after the two flybys however, does not merit such a complex analysis.

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