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Natural Sciences and Mathematics
January 2012
Reverse drag revisited: Why footwall deformation
may be the key to inferring listric fault geometry
Phillip G. Resor
Wesleyan University, [email protected]
David D. Pollard
Stanford University
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Recommended Citation
Resor, Phillip G. and Pollard, David D., "Reverse drag revisited: Why footwall deformation may be the key to inferring listric fault
geometry" (2012). Division III Faculty Publications. Paper 389.
http://wesscholar.wesleyan.edu/div3facpubs/389
This Article is brought to you for free and open access by the Natural Sciences and Mathematics at WesScholar. It has been accepted for inclusion in
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[email protected].
Note to Reader of the Accepted Manuscript:
Pollard and Resor, in press, “Reverse drag revisited: Why footwall deformation may be
the key to inferring listric fault geometry”, Journal of Structural Geology.
The following corrections were recommended during review of the manuscript proofs
and will be incorporated in the final publication:
Line 272
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replace “80” with “65”
replace “8‐18 km” with “8 to 18 km”
replace “1‐2.5 m” with “1 to 2.5 m”
replace “180‐220 m” with “180 to 220 m”
replace “striking 136‐144 and dipping 50‐55
136 to 144 and dipping 50 to 55
replace “0.49‐0.66 m” with “0.49 to 0.66 m”
” with
striking
Figure Captions:
Figure 6 and 7
should read “b) Best fit antithetic (α = -20) and vertical (α = 0) …”
Figure 9
should read “… Di Lucio et al. (2010)”
Accepted Manuscript
Reverse drag revisited: Why footwall deformation may be the key to inferring listric
fault geometry
Phillip G. Resor, David D. Pollard
PII:
S0191-8141(11)00181-7
DOI:
10.1016/j.jsg.2011.10.012
Reference:
SG 2660
To appear in:
Journal of Structural Geology
Received Date: 28 June 2011
Revised Date:
19 October 2011
Accepted Date: 27 October 2011
Please cite this article as: Resor, P.G., Pollard, D.D., Reverse drag revisited: Why footwall deformation
may be the key to inferring listric fault geometry, Journal of Structural Geology (2011), doi: 10.1016/
j.jsg.2011.10.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to
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Graphical Abstract
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*Revised Manuscript
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Reverse drag revisited: Why footwall deformation may be the key to inferring 2 listric fault geometry. 3 4 Phillip G. Resor1* and David D. Pollard2 5 Email: [email protected], [email protected] 6 Phone (Resor): 1‐860‐685‐3139 7 Fax (Resor): 1‐860‐685‐3651 8 9 1 Department of Earth and Environmental Sciences, Wesleyan University, 265 M
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1 Church St., Middletown, CT 06459, USA 11 12 2Department of Geological & Environmental Sciences, 450 Serra Mall, Braun Hall, 13 Building 320, Stanford University, Stanford, CA 94305‐2115, USA 14 15 * corresponding author 16 17 KEYWORDS 18 reverse drag; normal faulting; listric fault; structural modeling 19 20 To be submitted to the Journal of Structural Geology 21 AC
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22 Abstract Although reverse drag, the down warping of hanging wall strata toward a 24 normal fault, is widely accepted as an indicator of listric fault geometry, previous 25 studies have shown that similar folding may form in response to slip on faults of 26 finite vertical extent with listric or planar geometry. In this study we therefore seek 27 more general criteria for inferring subsurface fault geometry from observations of 28 near‐surface deformation by directly comparing patterns of displacement, stress, 29 and strain around planar and listric faults, as predicted by elastic boundary element 30 models. In agreement with previous work, we find that models with finite planar, 31 planar‐detached, and listric‐detached faults all develop hanging wall reverse‐drag 32 folds. All of these model geometries also predict a region of tension and elevated 33 maximum Coulomb stress in the hanging wall suggesting that the distribution and 34 orientation of near‐surface joints and secondary faults may also be of limited utility 35 in predicting subsurface fault geometry. The most notable difference between the 36 three models, however, is the magnitude of footwall uplift. Footwall uplift decreases 37 slightly with introduction of a detachment and more significantly with the addition 38 of a listric fault shape. A parametric investigation of faults with constant slip 39 ranging from nearly planar to strongly listric over depths from 1 to 15 km reveals 40 that footwall fold width is sensitive to fault geometry while hanging wall fold width 41 largely reflects fault depth. Application of a graphical approach based on these 42 results as well as more complete inverse modeling illustrates how patterns of 43 combined hanging wall and footwall deformation may be used to constrain 44 subsurface fault geometry. AC
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45 46 1. Introduction A fundamental unanswered question in continental tectonics is how extension on 48 normal faults exposed at the surface is accommodated in the middle to lower crust. 49 Many geologic cross sections envision listric fault geometries with faults flattening 50 at depth (e.g. Janecke et al., 1998; Brady et al., 2000), while seismological evidence 51 indicates that most, if not all, historic moderate‐to‐large normal‐faulting 52 earthquakes have ruptured relatively high angle faults with nearly constant dip 53 throughout the seismogenic crust (e.g. Stein and Barrientos, 1985; Jackson and 54 White, 1989). We propose that this discrepancy may stem in part from 55 misconceptions, rooted in the assumptions inherent to section balancing methods, 56 regarding the origins of reverse drag. M
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47 Reverse drag folds, in which hanging wall strata are warped down toward a 58 normal fault to create a fault‐parallel half monocline, are a common structural 59 element of extensional terranes (Schlische, 1995; Janecke et al., 1998). Since 60 Laubscher (1956) and Hamblin (1965) first proposed the relationship, the 61 formation of reverse drag has been widely attributed to slip on faults with listric 62 (concave up) cross‐sectional profiles (Shelton, 1984) (Fig. 1a). Others have noted, 63 however, that reverse drag folds form in response to slip on planar faults of finite 64 down dip extent (Barnett et al., 1987) (Fig. 1b). Schlische (1995) proposed 65 reserving the term reverse drag for folds of the latter group while calling the former 66 group rollover folds. In practice, however, the two terms are used nearly 67 synonymously (Jackson, 1997; Peacock et al., 2000) and, in cases where the down AC
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68 dip fault geometry is not well known, a clear means of differentiating between these 69 two causes of reverse drag has yet to be identified. 71 72 2. Previous Modeling of Reverse Drag RI
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70 Structural modeling provides a means to predict unknown elements of a structural system (e.g. fault geometry, fold geometry, small scale structures) from 74 available observations (e.g. Hilley et al., 2010), as well as to understand tectonic 75 processes responsible for the development of the system. Both physical (e.g. Mandl, 76 1988; Mcclay, 1990; Withjack et al., 1995) and mathematical (e.g. Crans et al., 1980; 77 Gibbs, 1983; White, 1992; Grasemann et al., 2005; Maerten and Maerten, 2006) 78 models have yielded many insights into processes of structural geology, including 79 the development of reverse drag folds. Mathematical models, however, are 80 generally best suited for parametric studies due to the relative ease with which 81 geometry, boundary conditions, and material behavior may be modified. 82 Mathematical models may be further divided into two subcategories: 1) Kinematic 83 models that use area (Chamberlin, 1910) or line length (Dahlstrom, 1969) balance 84 as a proxy for conservation of mass, assuming that rock volumes are essentially 85 incompressible and exhibit plane strain in modeled 2D cross sections; and 2) 86 Geomechanical models that employ a complete mechanics including conservation of 87 mass, momentum, and energy; kinematic equations; and constitutive relations 88 (Fletcher and Pollard, 1999). Although kinematic models have been most widely 89 applied in past studies of reverse drag, we argue here that mechanical models are 90 most appropriate for a more general exploration into the causes of reverse drag. AC
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Kinematic models of reverse drag generally posit that hanging wall folding occurs 92 due to a “space problem” associated with slip on a listric fault surface and associated 93 detachment (Hamblin, 1965). The hanging wall is down warped or down faulted to 94 fill the space that would be created if a rigid block were translated along the 95 detachment (Fig. 1a). In most models the footwall block is assumed to be rigid. The 96 area “lost” due to down warping is assumed to be balanced by an equal area 97 displaced above the detachment surface. The depth to this detachment may 98 therefore be calculated by measuring the “lost” area and estimating the magnitude 99 of hanging wall extension (e.g. Gibbs, 1983; Groshong, 1994; Poblet and Bulnes, M
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91 100 2005). Kinematic models based on line length or area balance may be used in this 101 way to evaluate the plausibility of cross sections, but don’t provide a direct means to 102 predict fault‐fold relationships. Kinematic models achieve the goal of predicting structural geometry by positing a TE
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103 specific geometric “mechanism” of folding that maintains area balance. Many such 105 mechanisms have been proposed for the development of reverse drag including 106 vertical shear (Verrall, 1981; Gibbs, 1983, 1984), inclined shear (White et al., 1986; 107 White, 1992; Xiao and Suppe, 1992; Withjack and Peterson, 1993), flexural slip 108 (Davison, 1986), slip lines (Williams and Vann, 1987), and rigid block rotation 109 (Moretti et al., 1988). These geometric mechanisms are generally proposed based 110 on an inferred section‐scale deformation mechanism, for example antithetic inclined 111 shear as a proxy for distributed antithetic faulting (Rowan and Kligfield, 1989). 112 Reviews of the efficacy of kinematic models have found, however, that in well‐
113 constrained experimental and natural examples antithetic inclined shear provides AC
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114 the best fit between fault and fold geometry regardless of the geometry and 115 orientation of hanging wall structures (Dula, 1991; Hauge and Gray, 1996). 116 Kinematic models thus have a number of shortcomings in terms of understanding the processes leading to the development of reverse drag. The geometric 118 mechanism that best fits the observed fault and fold geometry may not directly 119 reflect natural deformation mechanisms (Hauge and Gray, 1996). Furthermore, 120 imposition of a consistent geometric mechanism in these kinematic models 121 precludes heterogeneous deformation that is likely to occur in both physical analog 122 experiments and natural examples (Hauge and Gray, 1996; Withjack and Schlische, 123 2006). Finally, kinematic models require listric geometry to generate reverse drag 124 and thus preclude one mechanism of reverse drag: folding due to slip on finite 125 planar faults (Barnett et al., 1987; Grasemann et al., 2005). M
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117 In contrast to kinematic models, geomechanical models predict hanging wall 127 folding as a natural consequence of heterogeneous displacement fields associated 128 with slip on listric, planar, and even anti‐listric faults (Reches and Eidelman, 1995). 129 Reverse drag, rather than forming due to slip over a listric fault surface, may form 130 more generally as a response to the heterogeneous displacement field associated 131 with slip on faults of finite extent (Fig. 1b) (Barnett et al., 1987; Grasemann et al., 132 2005). Models invoking elastic (King et al., 1988; Reches and Eidelman, 1995; 133 Willemse et al., 1996; Gupta and Scholz, 1998; Grasemann et al., 2005), elastic‐
134 plastic (Reches and Eidelman, 1995; Bott, 1997), and viscous (Grasemann et al., 135 2003) rheologies have all revealed that reverse drag folds may form in response to 136 slip on finite planar faults. Finite fault models have been used to successfully model AC
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structures from outcrop scale (e.g. Gupta and Scholz, 1998; Grasemann et al., 2005) 138 to crustal scale (e.g. Willemse et al., 1996) and over time spans from single 139 earthquakes (e.g. Stein and Barrientos, 1985) to millions of years (e.g. Stein et al., 140 1988). 141 RI
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137 Geomechanical models also have been used to investigate the causes and consequences of listric faulting. Models based on soil plasticity theory suggest that 143 listric geometry may form in response to flow of overpressured shales on shallowly 144 dipping delta slopes (Crans and Mandl, 1980; Crans et al., 1980). Elastic and elastic‐
145 plastic finite difference models have been used to explore the effects of fault 146 geometry, stratigraphic layering, and inversion on hanging wall deformation above 147 listric faults (Erickson et al., 2001). Sequential restoration based on an elastic finite‐
148 element approach has provided insights into the process of secondary faulting 149 associated with hanging wall folding over an experimental listric fault (Maerten and 150 Maerten, 2006). Geomechanical models thus have proven useful in developing our 151 understanding of deformation associated with both finite planar and listric faults 152 and may therefore prove useful in identifying characteristics to distinguish between 153 the two commonly proposed mechanisms of reverse drag. M
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142 155 models for specific earthquakes (Stein and Barrientos, 1985) or geologic structures 156 (Willsey et al., 2002) a more general comparison between listric and planar models 157 has yet to be published. In this paper we explore the patterns of displacements and 158 stress resulting from two‐dimensional elastic models of planar to listric faults and 159 identify characteristics that may be used to differentiate between these end‐
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member fault geometries. This parametric study reveals that the absence of 161 footwall up warping rather than the presence of hanging wall down warping may be 162 the most distinctive hallmark of listric faulting. Numerous cross sections that infer 163 listric faulting from hanging wall rotation alone thus warrant a closer look, and 164 potential revisions may help resolve the apparent disagreement between 165 interpretations based on earthquake seismology and geologic cross sections. 166 167 3. Elastic modeling of deformation around planar and listric faults SC
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160 To directly compare patterns of displacement, strain, and stress around planar 169 and listric faults we employ a linear‐elastic boundary element method approach. 170 Linear elastic modeling provides a relatively straightforward mechanically based 171 approach to model geologic structures. Linear elastic behavior is defined by two 172 material parameters (e.g. Young’s modulus and Poisson’s ratio) that may be 173 measured directly in the laboratory for representative rock samples (e.g. Jaeger et 174 al., 2007). In laboratory experiments, however, rocks exhibit significant departures 175 from linear elastic behavior, particularly at large strains and over long time periods. 176 Linear elastic models have thus been used to great success for modeling 177 deformation due to slip during earthquakes, but their application to geologic 178 structures has been viewed with greater skepticism. Several studies of well‐
179 constrained outcrop and subsurface examples, however, have shown a remarkable 180 correlation between modeled and observed patterns of fault related deformation 181 (Gupta and Scholz, 1998; Healy et al., 2004; Fiore et al., 2007) and elastic models AC
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182 have provided many insights into the processes of faulting and folding in the upper 183 crust (e.g. Chinnery, 1961; Pollard and Segall, 1987; King et al., 1994). 184 Mechanical models based on the boundary element methods (BEM) are particularly well suited to problems of faulting, as only the faults must be 186 discretized, rather than the entire rock volume. Models of fault‐related deformation 187 may thus be defined by a series of fault elements and the boundary conditions on 188 these elements. In this paper we employ the BEM code TWODD (Crouch and 189 Starfield, 1983), based on the solution of a two‐dimensional displacement 190 discontinuity within a linear elastic half space. The original FORTRAN code of 191 Crouch and Starfield (1983) has been rewritten in Matlab (Martel and Langley, 192 2006) and modified for this study to permit both stress and displacement 193 discontinuity (rather than absolute displacement) boundary conditions. We 194 approximate the elastic behavior of the earth in all of the models using a Young’s 195 modulus of 30 GPa and a Poisson ratio of 0.25 (Pollard and Fletcher, 2005). Two 196 suites of models were developed in order to: 1) explore differences between planar 197 and listric models; and 2) identify characteristics of fold geometry that may be 198 indicative of down dip fault geometry. Finally, we apply these models in an inverse 199 sense to three natural examples to find the best‐fit fault geometry associated with 200 deformation due to a single earthquake and deformation accumulated over millions 201 of years. 202 203 3.1 Effects of fault geometry on off‐fault displacements and stresses AC
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204 We have constructed a series of three models to explore variations in the displacement and stress fields associated with slip on isolated planar, planar 206 detached, and listric detached fault geometries (Fig. 2a). Each model incorporates a 207 fault with a 60° dip at the surface that extends to 10 km depth, typical of earthquake 208 focal depths in regions of extension (Jackson and Mckenzie, 1983), to approximate a 209 fault extending from the surface to the brittle ductile transition. The model results 210 can be normalized by depth so that the absolute choice of this parameter is not 211 critical (see section 3.2). All model surfaces are discretized into linear elements that 212 are ~ 100 m in length. A constant slip magnitude of 100 m is imposed across the 213 fault surface using displacement discontinuity boundary conditions. The resulting 214 slip/height ratio is thus ~10‐2 consistent with geologic estimates of slip to length 215 ratios (Cowie and Scholz, 1992; Dawers et al., 1993). Although slip on real faults 216 may vary with changing fault dip, we have chosen constant slip, or a dislocation‐like 217 model, as a first order approximation, absent prior knowledge of the slip 218 distribution with depth. The abrupt termination of slip at the lower fault tip leads to 219 a nonphysical condition in models without detachment elements (where slip can 220 transfer onto detachment elements). An appropriate alternative choice of boundary 221 conditions might be a constant stress drop, or crack‐like model, (e.g. Willsey et al., 222 2002) that would lead to tapering slip at the fault tip. The relative difference 223 between the predicted displacements at the earth’s surface for crack and dislocation 224 models are relatively minor (Resor, 2008). Stresses associated with the fault tip in 225 dislocation models decay with a 1/r rather than a 1/√r singularity of the crack 226 models (Segall, 2010). Since our focus is on near surface stresses, far from the fault AC
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tip, the stress differences will also be minimized. For the planar detached model a 228 series of horizontal free‐slipping elements is added below the hanging wall 229 extending 100 km from the lower fault tip. At the far end of this detachment a 230 vertical traction free boundary is added to permit free motion of the hanging wall 231 block. The listric model is constructed in a similar manner to the planar detached 232 model, but with straight elements approximating a smoothly curving fault surface 233 defined by a third‐order Hermite polynomial with a 60‐degree dip at the surface and 234 a horizontal dip at the lower tip. SC
The vertical displacement field (Fig. 2b) for each of these models shows down M
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227 warping of the hanging wall near the surface trace of the fault leading to the 237 development of a reverse‐drag fold (deformed blue surface in Fig. 2a). Hanging wall 238 subsidence, observed at the free‐surface 250 meters from the fault, increases from 239 57 m in the isolated planar fault model to 64 m in the planar detached model to 76 240 m in the listric detached model. The width of the down warped region, defined by 241 the 10‐meter (10% of slip) vertical displacement contour at the surface, increases 242 from 14.75 to 17.25 km with introduction of the detachment. Inclusion of a listric 243 rather than planar fault above the detachment leads to a further increase in the fold 244 width to 18.75 km. Contours of displacement magnitude are approximately vertical 245 in the hanging wall of each model. EP
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236 The most notable difference between the three models in terms of vertical 247 displacements is the change in footwall uplift. Footwall uplift, observed 250 meters 248 from the fault at the free surface, decreases from 28 m to 21 m with introduction of 249 the detachment and to 9 m with the addition of a listric fault shape. The width of the ACCEPTED MANUSCRIPT
uplifted region, again defined by the 10 m contour, also decreases with introduction 251 of the detachment from 11.25 km to 7.25 km. Footwall uplift in the listric model 252 does not exceed 10 m and the fold width is thus undefined or zero. Color‐filled 253 contours of footwall displacement magnitude are largely parallel to the fault surface. 254 The presence of a hanging wall reverse drag fold thus appears to be a poor indicator 255 of down dip fault geometry, but the extent and magnitude of down warping relative 256 to footwall up warping are more indicative. The presence of a well‐developed 257 footwall fold (up warp) may thus be the best indicator of planar geometry. SC
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250 Alternatively, it might be possible to estimate subsurface fault geometry from 259 patterns of secondary fracture mode, orientation, and intensity. The magnitude and 260 orientation of stresses associated with discontinuous or irregularly shaped faults 261 has proven useful in predicting the type and orientation of secondary structures 262 around these faults (Segall and Pollard, 1980; Maerten et al., 2002). Specifically, 263 areas of tensile least compressive stress (Fig. 2 c and d) and positive Coulomb stress 264 (Fig. 2 e and f) have been used to predict regions where jointing and faulting, 265 respectively, are likely to occur. The models illustrate the orientation and magnitude 266 of stress due to slip on the fault itself and do not include a remote tectonic stress, 267 such as a most compressive vertical stress and least compressive horizontal stress 268 typical of regions of extension (Anderson, 1951). The models thus illustrate the 269 change in stress associated with fault slip rather than the complete stress state. 270 Adding a homogeneous remote stress field would tend to reduce the variability in 271 principal stress and Coulomb stress orientations with the remote stress field 272 dominating (i.e. vertical most compressive stress and 80 degree dipping maximum AC
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273 Coulomb shear planes for a coefficient of friction of 0.85) where stress changes are 274 low relative to the remote stress. 275 We focus the discussion here on patterns of near‐surface stress as we are interested in identifying means to differentiate listric and planar faults based on the 277 abundance and orientations of structures observed at or near the earth’s surface 278 during faulting. All models predict a region of increased tension in the hanging wall 279 starting approximately above the lower fault tip and extending further into the 280 hanging wall. This region of tensile stress change reaches 10 MPa at the surface at 281 just under 7 km from the fault trace for the planar models and at just over 7 km for 282 the listric fault. The far limit of this region is located 18 km from the trace for the 283 isolated planar fault model and expands away from the fault with introduction of a 284 detachment (to 26 km) and listric fault geometry (to 30 km). This tensile principal 285 stress is oriented parallel to the earth’s surface. These stress patterns suggest that 286 enhanced vertical jointing should be common in all reverse drag folds irrespective of 287 underlying fault geometry. SC
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Regions of positive maximum Coulomb stress change, calculated for a coefficient EP
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276 of friction 0.85 (Byerlee, 1978), largely follow the patterns of positive least 290 compressive stress due to the normal stress dependency of the Coulomb stress. A 291 second region of positive Coulomb stress greater than 10 MPa develops near the 292 free‐surface in the near‐fault region of the planar fault models extending from 10 293 km into the footwall to 3 km into the hanging wall for the isolated planar model and 294 from 5 km into the footwall to 3 km into the hanging wall for the planar detached 295 model. This second region of elevated Coulomb stress is associated with horizontal AC
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compression (Fig. 2d, i and ii) and would likely be counteracted by tectonic tension 297 not included in these models. Synthetic and antithetic normal faults thus are likely 298 to form in the hanging wall of planar and listric normal faults. Overall, the modeled 299 stress fields suggest that the distribution, abundance, and orientation of secondary 300 structures are likely to be of limited use in differentiating between reverse drag 301 associated with planar and listric faults. 304 SC
303 3.2 Quantifying the relationship between fault and fold geometries In order to develop criteria that may be used to distinguish between listric and M
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296 planar normal faults based on patterns of reverse drag folding we have constructed 306 a second suite of models to explore the combined effects of fault shape and fault 307 depth on the width of hanging wall and footwall folds (Fig. 3). Model fault profiles 308 are constructed using Hermite polynomials so that the surface location and dip as 309 well as lower fault location and dip are the only parameters required to describe 310 fault geometry. All modeled faults are assumed to end at a horizontal detachment 311 surface and thus have a zero degree dip imposed at their lower tip. Fault profiles 312 range from nearly planar when the lower tip is located at the down dip projection 313 from the surface to highly listric when the lower tip is offset horizontally in the 314 hanging wall direction. To quantify the degree of listricity we introduce the term 315 “listric shift” defined as the distance between a planar and listric fault at an 316 equivalent depth (Fig. 3a). AC
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305 317 The use of third‐order Hermite polynomials to describe all of the fault traces 318 leads to certain geometric limitations. Specifically, the nearly planar faults are not ACCEPTED MANUSCRIPT
truly planar as their lower tips still curve smoothly (as approximated by planar 320 elements) to meet the detachment, and the maximum listric shift for faults dipping 321 60 degrees at the surface is limited to the fault depth as greater shifts lead to faults 322 that extend below the detachment and then rise up to their lower tip. The use of 323 Hermite polynomials to describe all faults, however, has the benefit of allowing the 324 construction of a wide range of fault geometries with the same number of degrees of 325 freedom, making comparison of models relatively straightforward. SC
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319 Model results are presented as contour plots of hanging wall and footwall fold width in fault depth and listric shift parameter space (Fig. 3 b and c, respectively). 328 We define a fold as initiating at the point where the surface is displaced by 10 m 329 (10% of total fault throw). This criterion significantly exceeds the vertical 330 resolution of field mapping using differential GPS (Maerten et al., 2001; Resor, 331 2008). Alternatively, fold width could be defined as the point where a certain 332 minimum dip is reached (1‐2 degrees for typical field compass measurements). 333 Using a displacement‐based fold definition, however, has the added benefit that fold 334 width is independent of slip magnitude because the model equations (Crouch and 335 Starfield, 1983), or Green’s functions, are linear in slip. TE
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327 337 indicating a strong sensitivity to fault tip depth and only a weak sensitivity to down 338 dip geometry (listric shift). As discussed in section 3.1, the presence and width of 339 reverse drag folding is thus a poor predictor of listric fault geometry. Contours of 340 footwall fold width (Fig. 3c); however, have a relatively steep negative slope 341 indicating dependence on both fault depth and down dip geometry. Therefore, by ACCEPTED MANUSCRIPT
combining observations of footwall and hanging wall folding both the depth and 343 down dip geometry of faulting can be constrained. Normalizing by one of the 344 geometric parameters (e.g. hanging wall fold width) reduces the problem to a single 345 parameterized curve. The ratio of footwall to hanging wall folding is also sensitive 346 to fault dip (Resor, 2008) so that a suite of curves must be constructed over a range 347 of surface dips (Fig. 4). Points along a single contour thus represent the ratio 348 between footwall and hanging wall fold width (intersecting contours of Fig. 3 b and 349 c). These points are located in hanging wall normalized detachment depth – listric 350 shift space. This plot can be used in conjunction with observations of footwall and 351 hanging wall fold width and surface fault dip to uniquely determine the best‐fit fault 352 depth and listric shift. In the following section we apply this graphical approach as 353 well as a more complete numerical inverse scheme to solve for a best‐fit fault depth 354 and geometry for three examples of normal fault related deformation. 355 356 4. Application to studies of earthquake deformation and geologic structures 357 To illustrate how down dip geometry of normal faults can be inferred from EP
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342 observations of surface or near surface deformation patterns around these faults we 359 present brief analyses of three examples: 1) The 1983 Borah Peak earthquake, 360 Idaho, USA; 2) the Frog fault and Lone Mountain monocline, Arizona, USA; and 3) 361 the 2009 L’Aquila earthquake, Italy. These examples were chosen to explore 362 deformation due to a single earthquake (1 and 3) as well as deformation 363 accumulated over geologic time spans (2) where data were available to constrain 364 both hanging wall and footwall deformation. The Borah Peak and frog fault AC
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examples have historical significance as they have been used in arguments for 366 planar (Stein and Barrientos, 1985) and listric (Hamblin, 1965) faulting, 367 respectively; while the L’Aquila example is representative of the suite of data 368 available for modeling recent earthquakes. For each example we briefly review the 369 previous work, estimate fault geometry using the mechanics‐based graphical 370 approach described in the previous section, then use a more complete mechanically‐
371 based inverse approach to solve for the best‐fit geometry, and finally compare these 372 results to previous investigations of these structures as well as to kinematic 373 inversions. The mechanically‐based inversions use the equations of the elastic 374 boundary element model (TWODD, Crouch and Starfield, 1983) we used for forward 375 modeling deformation in the previous sections to find a set of model parameters 376 that minimize the misfit to the surface displacement data in a least squares sense 377 (see section 3 above for an explanation of the boundary element code TWODD). The 378 problem is non‐linear in terms of surface dip, fault depth and listric shift. We 379 therefore explore model parameters using a grid search over a range of fault 380 parameters. The model that results in the lowest squared residual in comparison to 381 the observed displacement data is considered to be the best‐fit fault model. For the 382 kinematic inversions we find a best‐fit second order polynomial to the observational 383 data from the hanging wall and then construct potential model faults following the 384 approach of White (1992) (Fig. 5). The results are very sensitive to the choice of the 385 shear angle. This angle may be independently estimated when the deformation of 386 two beds is observed (White, 1992) or layer parallel strain is measured within a 387 single bed (Groshong, 1990); however, for the relative displacement of a single AC
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layer, the case we are interested in, the solution is non‐unique. We therefore 389 construct two models that are likely to bound the range of reasonable values of 390 shear angle, one with vertical shear planes and the other with shear planes dipping 391 70° in the opposite direction (antithetic) to the main fault. These analyses serve to 392 illustrate the potential and pitfalls of inferring normal fault geometry using 393 graphical and more complete two‐dimensional mechanically‐based inverse 394 modeling. In each case the results of the mechanics‐based modeling are 395 significantly different from those of the kinematic approach and are generally in 396 better agreement with independent data of down dip geometry. 397 398 4.1 1983 Borah Peak earthquake, Idaho SC
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388 The 1983 Borah Peak earthquake (MW 6.9) is the largest historical earthquake in Idaho and one of the largest normal faulting events in the Basin and Range tectonic 401 province (Pancha et al., 2006). The event had a hypocentral depth of 13.7 km with a 402 preferred nodal plane striking 138° southeast, dipping 62° southwest, with a nearly 403 pure dip‐slip rake of ‐83° (Dziewonski et al., 1984). The earthquake produced a 36 404 km long zone of surface ruptures along the Lost River fault dipping 60‐80° with a 405 maximum net throw of 2.5‐2.7 m (Crone et al., 1987). Surface deformation 406 associated with the earthquake (Fig. 6) was measured with an uncertainty of < 20 407 mm by combining 1933, 1948, and 1984 leveling surveys that obliquely crossed the 408 zone of surface ruptures (Stein and Barrientos, 1985). These observations reveal a 409 broad region of hanging wall down warping reaching a maximum value of ~1.3 m 410 and a narrower lower‐magnitude region of footwall uplift reaching a maximum of AC
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~0.2 m. Barrientos and coauthors (Stein and Barrientos, 1985; Ward and 412 Barrientos, 1986; Barrientos et al., 1987) used several different inversion methods 413 to analyze these data and consistently found that planar models provided a better fit 414 to the data than equivalently framed listric models. Their best‐fit model for the 415 event consisted of two fault segments along strike, with the main fault plane 416 extending to 14 km depth with a 49° dip and 2.1 m of slip (Barrientos et al, 1987). The mechanics‐based graphical approach outlined in section 3.2 and illustrated SC
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411 in Fig. 4 may be used to make a preliminary estimate of the down dip geometry of 419 the Lost River fault from the observed vertical ground displacements (Fig. 6). This 420 approach requires an estimate of the fault dip at the ground surface (~70° based on 421 ground rupture observations) and the footwall/hanging wall fold ratio (2.6 km / 422 13.3 km or 0.20 where folds are defined by the point where ground displacement 423 exceeds 10% of the 2 m slip magnitude). These parameters yield a depth/ hanging 424 wall fold width ratio of 0.60 and a listric shift/ hanging wall fold width ratio of 0.27. 425 The resulting depth (8.0 km) is shallower than the focal depth of 13.7 km. This 426 error may stem in part from the oblique route the leveling observations take across 427 the rupture (Fig. 2 of Stein and Barrientos 1985). The geodetic observations 428 therefore reflect effects of the southeast tip of the rupture as well as the lower fault 429 tip and would be more appropriately addressed using a three‐dimensional (3D) 430 model. This 3D effect will lead to a narrower fold and thus shallower fault. The 431 relatively small listric shift of ~3.6 km is fairly consistent with previous analyses 432 suggesting planar faulting, but is also likely to suffer from the 3D effects discussed 433 above. These results suggest that the graphical method yields reasonable first order AC
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434 estimates of fault geometry; however, 3D effects can significantly affect results. 435 More accurate results may be obtained by inverting the entire data set rather than 436 the three parameters used in the graphical approach. Two‐dimensional mechanically‐based inverse modeling of the entire geodetic RI
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437 data set for the Borah Peak earthquake (Fig. 6a) results in a best fit dip of 44° 439 (evaluated from 40‐65° in 1° increments), a lower tip depth of 13.7 km (evaluated 440 over 8‐18 km in 100 m increments), a listric shift of 1.3 km (evaluated from 0 to the 441 fault depth in 100 m increments), and a slip magnitude of 1.3 m (evaluated from 1‐
442 2.5 m in 0.1 m increments). Although these results also suffer from the 3D effects 443 described above, they are much closer to previous estimates of the fault parameters 444 and highlight a second limitation of the graphical approach, its extreme sensitivity 445 to the estimate of surface dip. Although surface ruptures of the Borah Peak 446 earthquake had 60‐80 degree dips (Crone et al., 1987), the distribution of 447 aftershocks (Richins et al., 1987) and geodetic inversions (Stein and Barrientos, 448 1985; Barrientos et al., 1987), including this work, suggest that the fault has a 449 significantly shallower dip in the subsurface. The best‐fit model has elements that 450 are ~1 km in length and may therefore be unable to resolve any near‐surface 451 steepening. M
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438 Although the kinematic approach is not typically applied to model earthquake 453 deformation, the observed pattern of hanging wall subsidence might lead one to 454 infer that the causative fault is listric. The best‐fit kinematic models (Fig. 6b) yield 455 highly listric fault profiles (necessitated by the near absence of coseismic subsidence 456 at distances greater than ~20 km from the surface rupture) with ~6 km and ~12 km ACCEPTED MANUSCRIPT
depth to detachment for antithetic inclined and vertical shear, respectively (Fig. 6b). 458 The inferred detachment depth for the antithetic shear model is well above the 459 depth of the earthquake hypocenter; however, the vertical shear model yields a 460 more reasonable result that is also consistent with the distribution of aftershocks 461 (Fig. 6b). In the case of coseismic deformation, the basic assumption of the 462 kinematic model that volume is conserved (enforced through maintenance of the 463 shear normal component of slip) is inconsistent with observations that horizontal 464 and vertical deformation associated with normal fault earthquakes are of limited 465 spatial extent (e.g. Anzidei et al., 2009; Cheloni et al., 2010). 466 467 4.2 The Frog fault and Lone Mountain monocline, Arizona SC
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457 The Frog fault and associated Lone Mountain monocline were used as an illustration in Hamblin’s 1965 paper on the origins of reverse drag and have been 470 mapped and described in detail by Resor (2008). The Frog fault is one of a series of 471 normal faults that accommodate Basin and Range extension across the western 472 Grand Canyon (Huntoon and Billingsley, 1981; Billingsley and Wellmeyer, 2003; 473 Huntoon, 2003). The onset of normal faulting is generally inferred to be Miocene or 474 younger (Fenton et al., 2001; Amoroso et al., 2004; Karlstrom et al., 2007) with 475 distributed extension continuing to the present day (Kreemer et al., 2010). 476 Geomorphic, sedimentary, and thermochronologic evidence indicate that the area 477 was eroded to near present‐day levels, with the possible exception of the canyons, 478 by this time (Elston and Young, 1991; Dumitru et al., 1994; Pederson et al., 2002; 479 Flowers et al., 2008). AC
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The Frog fault dips ~70° to the southwest at the surface, down dropping Paleozoic
481 strata in the hanging wall up to ~225 meters. Slip on the fault, as evidenced by slickenline
482 orientations, is nearly pure dip-slip. A system of folds parallels the Frog fault, including
483 an upper half-monocline in the hanging wall (the Lone Mountain monocline of
484 Billingsley and Wellmeyer (2003)) and a lower half-monocline in the footwall. GPS
485 surveying of Esplanade Fm. strata has been used to quantify the relative vertical
486 deformation across the structures (Fig. 7) with a precision of ~3.5 m (Resor, 2008). The
487 dip of hanging wall beds increases systematically toward the fault over ~1.5 km
488 generating ~200 m of structural relief near the center of the fault. The dip of footwall
489 beds decreases away from the fault over a distance of ~0.5 km with structural relief
490 typically less than 25 meters. Resor (2008) used elastic dislocation modeling to find
491 parameters for single and multiple planar faults that best fit the GPS survey data. The
492 best-fit single fault had a 52° dip, extending 1300 m down dip (to 1024 m depth) with
493 188 m of slip. This study did not investigate potential listric geometries.
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480 Once again, the mechanics-based graphical approach may be used to make a
preliminary assessment of the down dip geometry of the Frog fault from the observed
496 patterns of bed elevations (Fig. 7). The width of the hanging wall half monocline,
497 defined by the 10% slip criterion, is 1030 m. The footwall fold width is 520 m. The
498 hanging wall to footwall fold ratio is thus 0.5. For a fault dipping 70° at the surface this
499 ratio results (Fig. 4) in a depth/ hanging wall fold width ratio of 0.635 (654 m fault depth)
500 and a listric shift/ hanging wall fold width ratio of 0.05 (listric shift of 52 m). This result
501 suggests that fault-related folding associated with the Frog fault is best fit by slip on a
502 nearly planar fault. The mechanics-based graphical approach, however, again results in
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an estimate for the depth to the lower tip of the fault that is shallower than previous
504 estimates. As highlighted in the discussion of the Borah Peak earthquake above, this
505 result may reflect the sensitivity of the graphical method to the estimate of the surface
506 fault dip. Inverse modeling of the entire data set may therefore yield more reasonable
507 estimates of fault parameters.
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503 In the case of the Frog fault, mechanics-based inverse modeling results in best-fit
solutions with very low slip (in comparison to field observations) due to the absence of
510 near-fault data (which were removed for this study to reduce the effects of displacement
511 associated with a secondary fault). To overcome this problem we constrained slip to be
512 greater or equal to 180 m for acceptable parameter space. The best-fit fault with this
513 constraint (Fig. 7a) has a dip of 72 degrees (evaluated from 52° to 74° in 2° increments),
514 a depth of 1.1 km (evaluated from 0.6 to 2 km in 100 m increments), a listric shift of 0
515 km (evaluated from 0 to the fault depth in 100 m increments), and slip of 180 m 516 (evaluated from 180‐220 m in 10 m increments). For comparison, the best‐fit 517 kinematic model (Fig. 7b) yields a listric fault with a detachment depth of ~800 m 518 for antithetic inclined shear and ~1600 for vertical shear. All models thus yield 519 depth to detachment, or lower fault tip, that is unusually shallow (~1 km). Resor 520 (2008) suggested that this might reflect an effect of the basement/cover contact 521 (Fig. 7) or processes of strain localization that are not captured by either the 522 kinematic or elastic models. Only the kinematic models, however, predict listric 523 fault profiles. Field observations reveal that the fault surface dips steeply (65‐85°) 524 to at least 400 m below the Esplanade Fm. footwall cutoff (Resor, 2008). At these 525 depths the antithetic inclined shear model predicts a fault dip of <40° and the AC
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vertical shear model predicts a dip of ~66°. The vertical shear model thus predicts a 527 fault dip that falls at the lower end of the range of observed dips; however, without 528 prior knowledge of the fault dip this choice of shear angle would not be obvious.
529 530 4.3 2009 L’Aquila Earthquake, Italy 531 RI
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526 The April 6, 2009 MW 6.3 L’Aquila earthquake was the latest of a series of moderate normal faulting events associated with 2‐4 mm/year of active extension 533 across the central and southern Apennines (D'Agostino et al., 2001; Hunstad et al., 534 2003; D'Agostino et al., 2008). The event had a hypocentral depth of 9.5 km 535 (Chiarabba et al., 2009) with a preferred regional centroid moment tensor (RCMT) 536 nodal plane striking 134° and dipping 56° southwest with a rake of ‐97° (Pondrelli 537 et al., 2010). The coseismic ground deformation field was observed by synthetic 538 aperture radar interferometry (InSAR) (Atzori et al., 2009; Walters et al., 2009; 539 Lanari et al., 2010; Papanikolaou et al., 2010) as well as GPS (Anzidei et al., 2009; 540 Cheloni et al., 2010). These geodetic data have been used in various inversion 541 schemes to find the best‐fitting planar fault surface for the event. These models 542 yield results that are largely consistent with seismological results indicating a fault 543 striking 136‐144° and dipping 50‐55° southwest with a rake of ‐98 to ‐105° and 544 0.49‐0.66 m of slip (Anzidei et al., 2009; Walters et al., 2009; Cheloni et al., 2010). 545 The L’Aquila earthquake, however, produced only minor surface rupturing, 546 extending 2.5 km along the trace of the Paganica fault with a maximum throw of 0.1 547 m (Emergeo Working Group, 2010). These results suggest that coseismic slip was 548 largely confined to the subsurface and inversions that permit variable slip on the AC
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549 fault surface have thus shown significant improvement in fitting the data in 550 comparison to similarly framed uniform slip models (Walters et al., 2009; Cheloni et 551 al., 2010). The largely blind nature of the L’Aquila event precludes use of the mechanics‐
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552 based graphical approach since the models used to generate the plots assume 554 constant slip on surface breaking faults. Instead we limit our investigation into the 555 down dip geometry of the Paganica fault to a mechanics‐based inversion of a two‐
556 dimensional profile through the InSAR–derived ground displacement observations 557 of Walters et al. (2009) (Fig. 8). Furthermore, we modify the approach taken in the 558 last two examples to permit variable slip across the fault surface. We iterate over 559 ranges of fault dip at the earth’s surface as well as lower fault tip depth and listric 560 shift as before, however, for each trial model geometry we solve directly for the 561 best‐fit variable slip using a damped least squares approach (Harris and Segall, 562 1987). The best‐fit model for the Paganica fault (Fig. 8) has a 50° dip at the surface 563 (evaluated from 46° to 60° in 2° increments) and extends to a 7 km depth (evaluated 564 from 6 to 16 km in 1 km increments) with a listric shift of 3 km (evaluated from 0‐
565 the fault depth in 1 km increments). Slip at the surface is 12 cm, decreases to a 566 minimum at ~2 km depth and gradually increases to 139 cm maximum at ~6 km 567 depth where the fault dips 36°. Slip tapers to zero at 7 km depth. The moment‐
568 weighted (slip*length) dip of the fault surface is 35°, shallower than the preferred 569 nodal plane from the RCMT. 570 571 AC
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553 These results are the first attempt to model the L’Aquila earthquake with a non‐
planar fault surface and while they suggest that the event may have ruptured a ACCEPTED MANUSCRIPT
listric fault, this result requires more thorough investigation including an 573 assessments of model resolution, trade‐offs, and uncertainties (e.g. Funning et al., 574 2005). For example, we have chosen the smoothing parameter to minimize the 575 moment after the approach of Funning et al. (2005); however, a slightly smoother 576 solution leads to a deeper more‐listric fault with a lower maximum slip value. As 577 discussed for the Borah Peak earthquake, three‐dimensional effects may also 578 significantly impact the model results. These effects are even greater for small to 579 moderate magnitude earthquakes where the along‐strike length of the rupture is 580 relatively small. The distribution of relocated aftershocks (Di Luccio et al., 2010) 581 from a 10‐km‐wide swath centered ~ 4.5 km northwest of the profile we selected 582 for modeling suggests that the fault extends as a nearly planar surface to ~11 km 583 depth inconsistent with our best fit listric model (Fig. 8). Di Luccio et al., however 584 also interpret an older thrust fault crossing the Paganica fault at ~7 km depth which 585 appears to have been reactivated during the earthquake sequence (Di Luccio et al., 586 2010) suggesting that a non‐planar geometry should not be dismissed without 587 further investigation. 588 589 5. Implications for predicting subsurface fault geometry SC
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572 Since Chamberlin (1910) introduced the concept of area balance to estimate the 591 depth to which deformation extends (depth to detachment), section balancing and 592 kinematic methods that ensure balanced sections have enjoyed widespread use in 593 structural modeling. In regions of extension a variety of methods have been used to 594 estimate not only the depth to detachment, but also the geometry of faults in the ACCEPTED MANUSCRIPT
subsurface. These methods, however, have several limitations, the most significant 596 for our purposes being the fact that they presume a listric fault geometry in order to 597 generate reverse drag folds when mechanical modeling and field observations have 598 shown that such folds may also form over finite planar faults. 599 RI
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595 We have constructed a series of two‐dimensional elastic models that confirm results of previous studies showing that reverse drag folds are likely to form over 601 faults with a variety of down dip profiles (Reches and Eidelman, 1995). Our 602 modeling results suggest, in fact, that a relative paucity of footwall up warping, 603 rather than the presence of hanging wall down warping, may be the hallmark of 604 listric faulting. Footwall uplift over geologic time scales is generally ascribed to 605 isostatic adjustment associated with unloading of the crust during extension, as it is 606 assumed that uplift due to coseismic fault slip will be largely relaxed during 607 continued extension of the crust (Jackson and Mckenzie, 1983). Models of an elastic 608 upper crust over a viscoelastic lower crust and mantle, however, have shown that 609 the relaxation is not likely to be complete (King et al., 1988). The preservation of 610 structures, such as the Lone Mountain monocline with wavelengths and amplitudes 611 that are small in comparison to the crustal thickness appears to corroborate these 612 results. Footwall uplift therefore appears to reflect not only an isostatic effect, but 613 also the process of fault slip itself. The modeling results presented here suggest that 614 the presence of footwall uplift may in fact be an indicator of a planar or nearly‐
615 planar fault surface. The width of the hanging wall fold rather than providing direct 616 information about fault geometry as inferred in kinematic models, appears to 617 correlate with the depth of the lower fault tip or detachment for both planar and AC
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618 listric fault models. Combined observations of hanging wall and footwall 619 deformation may thus be used to constrain fault shape as well as depth. 620 Based on the results of elastic models we have suggested a simple graphical approach to derive a first‐order estimate of fault depth and listric shift using the 622 surface dip of a fault and the ratio of footwall to hanging wall fold widths. 623 Application of this approach to two natural examples suggests that the method’s 624 sensitivity to the choice of surface fault dip may potentially lead to inaccurate 625 results. The ratio between footwall uplift and hanging wall down warping may thus 626 provide a quick‐check method for estimating down dip fault geometry; however, 627 inverse modeling of data describing the entire fold profile is likely to provide more 628 accurate results. In the Borah Peak earthquake example inversion of the full data 629 set resulted in an estimate of fault depth that was more consistent with 630 seismological estimates. SC
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621 A second issue in each of the case studies, however, is the impact of three‐
dimensional effects. In the earthquake examples, the relative proximity of the 633 lateral rupture tips has likely affected the deformation patterns along the modeled 634 profiles, while the two dimensional models assume that the fault is effectively 635 infinite along strike. Similarly, fault slip may be heterogeneously distributed over 636 geologic time scales, particularly along segmented normal faults (e.g. Willemse et al., 637 1996). We therefore advocate use of fully three‐dimensional inverse modeling (e.g. 638 Maerten et al., 2005) in most cases. An additional advantage of applying mechanics‐
639 based inversions is that geophysical inverse theory provides tools to estimate model 640 trade‐offs and uncertainties (Menke, 1984). Surface deformation data generally AC
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641 have poor resolving power at depths of 10‐15 km depth (e.g. Funning et al., 2005) 642 and the ability to infer down‐dip geometry from a set of observations should thus be 643 evaluated before drawing conclusions from modeling. Interestingly, the patterns of deformation in the mechanical listric models are RI
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644 consistent, to a first approximation, with some common assumptions of kinematic 646 models. For example, kinematic models typically assume a rigid footwall. In the 647 mechanical model of deformation around a listric fault footwall uplift is less than 648 10% of the slip magnitude (Fig. 2b), which may effectively be treated as non‐
649 deforming for some purposes. In a second example, inclined shear models assume 650 that principal strains are consistently oriented (constant shear angle) throughout 651 the hanging wall of a listric fault. In the elastic models stress (and therefore strain) 652 orientations are also relatively consistent in the hanging wall of the listric fault (Fig. 653 2d), although they diverge significantly from this orientation at the free surface and 654 above the detachment plane. These similarities may explain the ability of kinematic 655 models to reasonably explain deformation above apparently listric faults (Hauge 656 and Gray, 1996). These kinematic methods, however, still fail to explain folding 657 associated with slip on planar faults and their general use for predicting subsurface 658 fault geometry should therefore be avoided. M
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645 The models we present here are meant to address basement‐involved extension 660 involving relatively stiff rocks above a thermally activated brittle ductile transition. 661 The case of thin‐skinned extension over viscous salt or shale layers warrants further 662 investigation using appropriate constitutive laws for these conditions. The main 663 conclusion; however, that reverse drag is insufficient evidence to infer listric ACCEPTED MANUSCRIPT
664 geometry, has been confirmed for a wide range of material behaviors (Reches and 665 Eidelman, 1995; Grasemann et al., 2003; Grasemann et al., 2005). 667 668 6. Conclusions RI
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666 Although a causal link between reverse drag folding and listric faulting is widely accepted in the geologic literature, this relationship is largely based on kinematic 670 models that pre‐suppose this relationship. A parametric study of planar and listric 671 geometries using the elastic boundary element method reveals that the absence of 672 footwall up warping rather than the presence of hanging wall down warping may be 673 the most distinctive hallmark of listric faulting. By combining observations of 674 footwall and hanging wall deformation we can thus estimate both the depth to the 675 lower fault tip and the geometry of the fault surface. Application of this approach to 676 three natural examples suggests that normal faults in regions of crustal extension 677 may have both planar and listric geometries. A more thorough investigation of well 678 constrained geologic and earthquake examples may thus help to resolve the long‐
679 standing disagreement over normal fault geometry. 681 682 M
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669 Thanks to S. Martel for providing the Matlab version of TWODD at 683 http://www.soest.hawaii.edu/martel/Martel.BEM_dir/. Also thanks to R. Walters 684 and F. Di Luccio for kindly providing data for the L’Aquila case study. Thanks to 685 Nancye Dawers, Rick Groshong, and Martha Withjack for their insightful reviews 686 and constructive comments that helped to improve this manuscript. This work was ACCEPTED MANUSCRIPT
inspired by the first author’s PhD studies at Stanford University, supported in part 688 by the ARCO Stanford Graduate Fellowship and the Stanford Rock Fracture Project. 689 690 References 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 Amoroso, L., Pearthree, P. A., Arrowsmith, J. R., 2004. Paleoseismology and neotectonics of the Shivwits section of the Hurricane Fault, northwestern Arizona. Bulletin of the Seismological Society of America 94, 1919‐1942. Anderson, E. M., 1951. The Dynamics of Faulting and Dyke Formation with Application to Britain. Oliver and Boyd Ltd., Edinburgh. Anzidei, M., Boschi, E., Cannelli, V., Devoti, R., Esposito, A., Galvani, A., Melini, D., Pietrantonio, G., Riguzzi, F., Sepe, V., Serpelloni, E., 2009. Coseismic deformation of the destructive April 6, 2009 L'Aquila earthquake (central Italy) from GPS data. Geophys. Res. Lett. 36, L17307. Atzori, S., Hunstad, I., Chini, M., Salvi, S., Tolomei, C., Bignami, C., Stramondo, S., Trasatti, E., Antonioli, A., Boschi, E., 2009. Finite fault inversion of DInSAR coseismic displacement of the 2009 L'Aquila earthquake (central Italy). Geophys. Res. Lett. 36, L15305. Barnett, J. A. M., Mortimer, J., Rippon, J. H., Walsh, J. J., Watterson, J., 1987. Displacement geometry in the volume containing a single normal fault. American Association of Petroleum Geologists Bulletin 71, 925‐937. Barrientos, S. E., Stein, R. S., Ward, S. N., 1987. Comparison of the 1959 Hebgen Lake, Montana and the 1983 Borah Peak, Idaho, earthquakes from geodetic observations. Bulletin of the Seismological Society of America 77, 784‐808. Billingsley, G. H., Wellmeyer, J. L., 2003. Geologic map of the Mount Trumbull 30' X 60' quadrangle, Mohave and Coconino Counties, northwestern Arizona. Geologic Investigations Series U. S. Geological Survey, Report: I 2766. Bott, M. H. P., 1997. Modeling the formation of a half graben using realistic upper crustal rheology. Journal of Geophysical Research 102, 24,605‐24617. Brady, R., Wernicke, B., Fryxell, J., 2000. Kinematic evolution of a large‐offset continental normal fault system ; South Virgin Mountains ; Nevada. Geological Society of America Bulletin 112, 1375‐1397. Byerlee, J., 1978. Friction of rocks. Pure and Applied Geophysics 116, 615‐126. Chamberlin, R. T., 1910. The Appalachian folds of central Pennsylvania. Journal of Geology, 228‐251. Cheloni, D., D’Agostino, N., D’Anastasio, E., Avallone, A., Mantenuto, S., Giuliani, R., Mattone, M., Calcaterra, S., Gambino, P., Dominici, D., Radicioni, F., Fastellini, G., 2010. Coseismic and initial post‐seismic slip of the 2009 Mw 6.3 L’Aquila earthquake, Italy, from GPS measurements. Geophysical Journal International 181, 1539‐1546. Chiarabba, C., Amato, A., Anselmi, M., Baccheschi, P., Bianchi, I., Cattaneo, M., Cecere, G., Chiaraluce, L., Ciaccio, M. G., De Gori, P., De Luca, G., Di Bona, M., Di Stefano, R., Faenza, L., Govoni, A., Improta, L., Lucente, F. P., Marchetti, A., Margheriti, AC
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Reches, Z. e., Eidelman, A., 1995. Drag along faults. Tectonophysics 247, 145‐156. Resor, P. G., 2008. Deformation associated with a continental normal fault system, western Grand Canyon, Arizona. Geological Society of America Bulletin 120, 414‐430. Richins, W. D., Pechmann, J. C., Smith, R. B., Langer, C. J., Goter, S. K., Zollweg, J. E., King, J. J., 1987. The 1983 Borah Peak, Idaho, earthquake and its aftershocks. Bulletin of the Seismological Society of America 77, 694‐723. Rowan, M. G., Kligfield, R., 1989. Cross section restoration and balancing as aid to seismic interpretation in extensional terranes. AAPG Bulletin 73, 955‐966. Schlische, R. W., 1995. Geometry and Origin of Fault‐Related Folds in Extensional Settings. Aapg Bulletin‐American Association of Petroleum Geologists 79, 1661‐1678. Segall, P., 2010. Earthquake and Volcano Deformation. Princeton University Press, Princeton. Segall, P., Pollard, D. D., 1980. Mechanics of discontinuous faults. Journal of Geophysical Research 85, 4,337‐4,350. Shelton, J. W., 1984. Listric normal faults; an illustrated summary. AAPG Bulletin 68, 801‐815. Stein, R. S., Barrientos, S. E., 1985. Planar high‐angle faulting in the Basin and Range; geodetic analysis of the 1983 Borah Peak, Idaho, earthquake. Journal of Geophysical Research 90, 11,355‐11,366. Stein, R. S., King, G. C. P., Rundle, J. B., 1988. The Growth of Geological Structures by Repeated Earthquakes .2. Field Examples of Continental Dip‐Slip Faults. Journal of Geophysical Research‐Solid Earth and Planets 93, 13319‐13331. Verrall, P., 1981. Structural interpretations with applications to North Sea problems, Course notes, Joint Association of Petroleum Exploration Courses (JAPEC), London. Walters, R. J., Elliott, J. R., D'Agostino, N., England, P. C., Hunstad, I., Jackson, J. A., Parsons, B., Phillips, R. J., Roberts, G., 2009. The 2009 L'Aquila earthquake (central Italy): A source mechanism and implications for seismic hazard. Geophys. Res. Lett. 36, L17312. Ward, S. N., Barrientos, S. E., 1986. An inversion for slip distribution and fault shape from geodetic observations of the 1983, Borah Peak, Idaho, earthquake. Journal of Geophysical Research 91, 4,909‐4,914. White, N., 1992. A Method for Automatically Determining Normal‐Fault Geometry at Depth. Journal of Geophysical Research‐Solid Earth 97, 1715‐1733. White, N. J., Jackson, J. A., McKenzie, D. P., 1986. The relationship between the geometry of normal faults and that of the sedimentary layers in their hanging walls. Journal of Structural Geology 8, 897‐909. Willemse, E. J. M., Pollard, D. D., Aydin, A., 1996. Three‐dimensional analyses of slip distributions on normal fault arrays with consequences for fault scaling. Journal of Structural Geology 18, 295‐309. Williams, G., Vann, I., 1987. The geometry of listric normal faults and deformation in their hangingwalls. Journal of Structural Geology 9, 789‐795. Willsey, S., Umhoefer, P., Hilley, G., 2002. Early evolution of an extensional monocline by a propagating normal fault: 3D analysis from combined field AC
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study and numerical modeling. JOURNAL OF STRUCTURAL GEOLOGY 24, 651‐669. Withjack, M. O., Islam, Q. T., Lapointe, P. R., 1995. Normal Faults and Their Hanging‐
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Figure Captions 973 Figure 1. Two proposed mechanisms for development of reverse drag folds. a) Fault 974 bend folding associated with slip on a listric fault. Folding (iii) is posited to occur in 975 response to a space problem (ii) in association with slip on a listric fault (after 976 Hamblin, 1965). b) Folding due to slip on a finite planar fault. Folding (ii) occurs in 977 response to displacement gradients along the fault and within the surrounding rock 978 volume (after Barnett et al., 1987). 979 980 Figure 2. Two‐dimensional linear elastic boundary element models of (i) planar, (ii) 981 planar detached and (iii) listric detached faulting. a) Model geometry and boundary 982 conditions. Red solid lines are faults with 100 m constant slip imposed. Red dashed 983 lines are free slipping detachment surfaces. Red dotted lines are free surfaces 984 located 100 km to right of model. Blue line is the free surface of the elastic half 985 space after model displacement (exaggerated 10x). b) Color‐filled contours of the 986 vertical displacement field (gray contours at 10 m intervals). ±10 m contours used 987 to define fold width are highlighted in dark gray. c) Color‐filled contours (tension 988 positive) of the magnitude of least compressive stress with gray +10 MPa contour. 989 d) Tick marks showing orientation of least compressive stress. e) Color‐filled 990 contours of the magnitude of maximum Coulomb stress with gray +10 MPa contour. 991 f) Tick marks showing the orientations of maximum Coulomb shear planes. 992 993 Figure 3. Parametric exploration of varying fault geometry from nearly planar to 994 highly listric. a) Model geometry and boundary conditions. See Fig. 2 for general AC
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explanation. Listric shift is defined as the horizontal distance between the planar 996 extension of a listric fault’s surface dip and the lower tip of the listric fault at the 997 detachment depth. Model results in b and c are for faults with a 60° dip at the 998 surface and 100 m of constant slip imposed on the fault surface. b) Contours of 999 hanging wall fold width for varying depth to detachment and listric shift. Fold width RI
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995 is defined by location where vertical displacement exceeds 10 m. c) Contours of 1001 footwall fold width, as defined in b, for varying depth to detachment and listric shift. 1002 1003 Figure 4. Plot of model results normalized by hanging wall fold width for faults with 1004 surface dips of 50°, 60°, and 70°. The suite of model results for each value of surface 1005 fault dip is reduced to a single parameterized curve with points along the curve 1006 corresponding to values of the ratio between footwall and hanging wall fold width. 1007 LR and FF are results for the Lost River fault (Borah Peak earthquake) and Frog 1008 fault, respectively (discussed in section 4). The jagged nature of the contours is a 1009 result of the numerical method by which the plots are generated – exploring a grid 1010 of parameter values and then contouring the results. See Fig. 3 for a general 1011 explanation of the models. 1012 1013 Figure 5. Inclined shear model after White (1992). a) Undeformed state. The model 1014 assumes that deformation occurs by shear on planes inclined at angle α from the 1015 vertical. s is the slip vector. The component of slip perpendicular to the shear 1016 direction (h’) is kept constant. ab and cd are shear‐parallel line segments prior to 1017 deformation. b) Deformed state. After slip the line segments, a’b’ and c’d’, have AC
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been translated in the shear perpendicular direction by a distance of h’ and in the 1019 shear parallel direction in order to maintain their original length. The inverse 1020 construction is achieved by estimating ab from the deformed state and then using 1021 this length to infer the fault position at b’ and then continuing in this manner 1022 incrementally away from the surface trace of the fault. 1023 1024 Figure 6. Example 1 of fault geometry fitting using mechanical and kinematic 1025 inverse modeling approaches. Black dots are observational data. Darker gray line is 1026 best‐fit fault geometry, and light gray line is modeled surface displacements. Thin 1027 black line is topographic profile vertically exaggerated 2x. a) Elastic BEM model of 1028 the subsurface geometry of the Lost River fault based on best fit to Borah Peak 1029 earthquake coseismic displacements measured by leveling (Stein and Barrientos, 1030 1985, see their Fig. 2 for section location). b) Best fit antithetic (a=‐20) and vertical 1031 (a=‐20) shear models for the Borah Peak data. Open circles are aftershocks from 1032 Richens et al. (1987) their profile B‐B’, located ~ 5 km north of the location where 1033 the leveling profile crosses the Lost River fault. Topography along leveling line from 1034 Stein and Barrientos (1985). 1035 1036 Figure 7. Example 2 of fault geometry fitting using mechanical and kinematic 1037 inverse modeling approaches. See Fig. 6 for general explanation. a) Elastic BEM 1038 model of the subsurface geometry of the Frog fault based on best fit to relative 1039 elevation of the Esplanade Fm. in the Lone Mountain monocline surveyed using 1040 differential GPS (Resor, 2008, see their Fig. 6, section c‐c’, intermediate fault block AC
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data removed). b) Best fit antithetic (a=‐20) and vertical (a=‐20) shear models for 1042 Lone Mountain monocline. Approximate depth to basement is from Billingsley and 1043 Wellmeyer (2003). Topography from National Elevation Dataset. 1044 1045 Figure 8. Example 3 of fault geometry fitting using mechanical and kinematic 1046 inverse modeling approaches. See Fig. 6 for general explanation. a) Elastic BEM 1047 model of the subsurface geometry of the Paganica fault based on best fit to InSAR‐
1048 derived line of sight displacements during the L’Aquila earthquake. Gray circles are 1049 aftershock hypocenters from Di Lucio et al. (2009) their section 3. Displacement 1050 data projected from 9 km wide swath onto section line oriented N36E. Topography 1051 from Shuttle Radar Topography Mission. b) Slip distribution of best‐fit model shown 1052 in a. AC
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Figure 1. Two proposed mechanisms for development of reverse drag folds. a) Fault bend folding
associated with slip on a listric fault. Folding (iii) is posited to occur in response to a space problem (ii)
in association with slip on a listric fault (after Hamblin, 1965). b) Folding due to slip on a �inite planar
fault. Folding (ii) occurs in response to displacement gradients along the fault and within the surrounding rock volume (after Barnett et al., 1987).
Figure 2
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Figure 2. Two-dimensional linear elastic boundary element models of (i) planar, (ii) planar detached
and (iii) listric detached faulting. a) Model geometry and boundary conditions. Red solid lines are
faults with 100 m constant slip imposed. Red dashed lines are free slipping detachment surfaces. Red
dotted lines are free surfaces located 100 km to right of model. Blue line is the free surface of the
elastic half space after model displacement (exaggerated 10x). b) Color-�illed contours of the vertical
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e) Color-�illed contours of the magnitude of maximum Coulomb stress with gray +10 MPa contour. f)
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conditions. See Figure 2 for general explanation. Listric shift is
de�ined as the horizontal distance between the planar extension
of a listric fault’s surface dip and the lower tip of the listric fault
at the detachment depth. Model results in b and c are for faults
with a 60° dip at the surface and 100 m of constant slip imposed
on the fault surface. b) Contours of hanging wall fold width for
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corresponding to values of the ratio
between footwall and hanging wall fold
width. LR and FF are results for the Lost
River fault (Borah Peak earthquake) and
Frog fault, respectively (discussed in section
4). The jagged nature of the contours is a
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Figure 5. Inclined shear model after White
(1992). a) Undeformed state. The model
assumes that deformation occurs by shear
on planes inclined at angle α from the
vertical. s is the slip vector. The component
of slip perpendicular to the shear direction
(h’) is kept constant. ab and cd are shearparallel line segments prior to deformation.
b) Deformed state. After slip the line
segments, a’b’ and c’d’, have been translated
in the shear perpendicular direction by a
distance of h’ and in the shear parallel
direction in order to maintain their original
length. The inverse construction is achieved
by estimating ab from the deformed state and
then using this length to infer the fault
position at b’ and then continuing in this
manner incrementally away from the surface
trace of the fault.
h’
b
α
b)
RI
PT
d
a
s
h’
a’
c’
bedding
deformed
SC
b
b’
AC
C
EP
TE
D
M
AN
U
d’
Figure 6
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ACCEPTED MANUSCRIPT
−5
SW
Coseismic displacement
from leveling
−10
NE
0
5
displacement
exaggerated 1000x
−20
−10
0
10
distance from fault (km)
b) inclined shear models
NE
5
α=-20º
SC
−20
−10
0
10
distance from fault (km)
M
AN
U
−30
α=0º
displacement
exaggerated 1000x
TE
D
−10
0
EP
−5
SW
AC
C
0
Vertical
Displacement (m)
depth (km)
−30
Figure 6. Example 1 of fault geometry �itting using mechanical
and kinematic inverse modeling approaches. Black dots are
observational data. Dark gray line is best-�it fault geometry, and
light gray line is modeled surface displacements. Thin black line
is topographic pro�ile vertically exaggerated 2x. a) Elastic BEM
model of the subsurface geometry of the Lost River fault based
on best �it to Borah Peak earthquake coseismic displacements
measured by leveling (Stein and Barrientos, 1985, see their
Figure 2 for section location). b) Best �it antithetic (α=-20) and
vertical (α=-20) shear models for the Borah Peak data. Open
circles are aftershocks from Richens et al. (1987) their pro�ile
B-B’, located ~ 5 km north of the location where the leveling
pro�ile crosses the Lost River fault. Topography along leveling
line from Stein and Barrientos (1985).
RI
PT
0
Vertical
Displacement (m)
depth (km)
a) elastic model
Figure 7
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ACCEPTED MANUSCRIPT
−1.0
~ depth to basement
−1.5
−2.0
−1.0
0
1.0
distance from fault (km)
b) inclined shear model
−1.5
−2.0
α=-20º
α=0º
~ depth to basement
−1.0
0
1.0
distance from fault (km)
1.5
2.0
0
0.5
1.0
1.5
TE
D
−1.0
EP
−0.5
1.0
NE
footwall deformation
not modelled
No vertical
exaggeration
AC
C
0
0.5
Vertical
Displacement (km)
depth (km)
SW
2.0
0
RI
PT
−0.5
Bedding surface from
GPS surveying
No vertical
exaggeration
M
AN
U
0
NE
Vertical
Displacement (km)
depth (km)
SW
SC
a) elastic model
Figure 7. Example 2 of fault geometry �itting using mechanical
and kinematic inverse modeling approaches. See Fig. 6 for
general explanation. a) Elastic BEM model of the subsurface
geometry of the Frog fault based on best �it to relative elevation
of the Esplanade Fm. in the Lone Mountain monocline surveyed
using differential GPS (Resor, 2008, see their Figure 6, section
c-c’, intermediate fault block data removed). b) Best �it antithetic
(α=-20) and vertical (α=-20) shear models for Lone Mountain
monocline. Approximate depth to basement is from Billingsley
and Wellmeyer (2003). Topography from National Elevation
Dataset.
Figure 8
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ACCEPTED MANUSCRIPT
SW
NE
Coseismic displacement
(exaggerated 10000x)
descending InSAR
ascending InSAR
0.5
−40
−30
−20
−10
distance from fault (km)
0
0
0.0
−7
0.5 1.0
slip (m)
RI
PT
−10
10
EP
TE
D
M
AN
U
SC
Figure 8. Example 3 of fault geometry �itting using mechanical and kinematic inverse modeling approaches. See
Fig. 6 for general explanation. a) Elastic BEM model of the subsurface geometry of the Paganica fault based on best
�it to InSAR-derived line of sight displacements during the L’Aquila earthquake. Gray circles are aftershock hypocenters from Di Lucio et al. (2009) their section 3. Displacement data projected from 9 km wide swath onto
section line oriented N36E. Topography from Shuttle Radar Topography Mission. b) Slip distribution of best-�it
model shown in a.
AC
C
depth (km)
−5
0
b)
depth (km)
0
LOS displacement (m)
a)