The ability of models designed to use near-surface structural information to predict the deep geometry of a faulted block is tested for a thick-skinned thrust by matching the surface geometry to the crustal structure beneath the Wind River Range, Wyoming, USA. The Wind River Range is an ∼100-km-wide, thick-skinned rotated basement block bounded on one side by a high-angle reverse fault. The availability of a deep seismic-reflection profile and a detailed crustal impedance profile based on teleseismic receiver-function analysis makes this location ideal for testing techniques used to predict the deep fault geometry from shallow data. The techniques applied are the kinematic models for a circular-arc fault, oblique simple-shear fault, shear fault-bend fold, and model-independent excess area balancing. All the kinematic models imply that the deformation cannot be exclusively rigid-body rotation but rather require distributed deformation throughout some or all of the basement. Both the circular-arc model and the oblique-shear models give nearly the same best fit to the master fault geometry. The predicted lower detachment matches a potential crustal detachment zone at 31 km subsea. The thrust ramp is located close to where this zone dies out to the southwest. The circular-arc model implies that the penetrative deformation could be focused at the trailing edge of the basement block rather than being distributed uniformly throughout and thus helps to explain the line of second-order anticlines along the trailing edge of the Wind River block.
Key points: (1) The circular-arc fault model and the oblique-shear model predict a lower detachment for the Wind River rotated block to be ∼31 km subsea, consistent with the crustal structure as defined by teleseismic receiver-function analysis. The thrust ramp starts where this zone dies out. (2) The kinematic models require distributed internal deformation within the basement block, probably concentrated at the trailing edge. (3) The uplift at the trailing edge of the rotated block is explained by the circular-arc kinematic model as a requirement to maintain area balance of a mostly rigid block above a horizontal detachment; the oblique-shear model can explain the uplift as caused by displacement on a dipping detachment.
Thick-skinned basement uplifts form a major structural style around the world, commonly in the forelands of compressional orogenic belts (Pfiffner, 2017), but also within orogenic belts (Lacombe and Bellahsen, 2016) and, usually with less structural relief, as relatively isolated blocks on the cratons (Marshak et al., 2000). The backlimb of the structure is typically gently dipping and bounded on one side by a high-angle reverse fault (Fig. 1). The fault may die out upward into a steeply dipping forelimb. This style is here termed a rotated-block geometry, without specifying the size of the block or rotation mechanism, and with the understanding that the block is not necessarily rigid.
The connection between the near-surface structural geometry and the deeper crustal structure remains largely unknown for thick-skinned structures due to the lack of high-resolution deep data. Available data typically include the outcrop geometry, wells that penetrate some of the sedimentary section and possibly the top of the basement, and seismic-reflection profiles that may image down to the top of the basement. Data that directly show the connection to the deep structure (i.e., deep seismic-reflection profiles) are rare, and the seismic data remain subject to multiple interpretations. The master faults have been variously postulated to be planar, listric with detachments in the crust, anti-listric (steepening downward), or ramp-flat in shape, and to die out downward, to have formed above one or more regional detachment horizons in the crust, or to offset the Moho (Prucha et al., 1965; Smithson et al., 1979; Erslev, 1986, 2005; Stone, 1999; Yeck et al., 2014).
The Wind River Mountain Range (Figs. 2 and 3) is an excellent example of the rotated-block structural style. It is a first-order rotated block 100 km wide with a large reverse fault bounding its southwestern side. Two lines of second-order rotated-block anticlines ∼10 km wide occur along the northeastern (trailing) edge of the major structure. The COCORP (Consortium for Continental Reflection Profiling) program generated a deep seismic profile across the southeastern end of the range. This profile has been interpreted by multiple investigators and the major faults that have been inferred are compiled in Figure 3. A crustal structure interpretation based on Bouguer gravity anomalies is also shown in Figure 3. New in this study is the crustal impedance profile described later.
In the tradition of multiple working hypotheses (Chamberlin, 1897), all structural methods consistent with the rotated-block geometry (Fig. 4) are applied here to predict the shape of the Wind River master fault and the location of the lower detachment, and to provide insight into the internal deformation of the rotated block. The purpose is to find which method or methods provide the best connection between the near-surface structural geometry and the deep geometry of the master fault and position of the lower detachment. The model predictions are compared to the crustal structure defined by teleseismic receiver-function analysis (Fig. 5) to see which results are the most consistent with the crustal velocity structure. We also apply the philosophy of Box (1979) who famously wrote, “All models are wrong but some are useful.” Because the methods are based on different aspects of the observed geometry, they provide insight to different aspects of the internal deformation. Methods that work well for this profile may be applicable to other rotated basement-block structures where less deep information is available. Another question that will be addressed via the models is the possible connection between the first-order Wind River structure and the second-order structures of the same style that form the line of faulted anticlines northeast of Lander (Fig. 2) of which Sheep Mountain anticline (Fig. 3) is an example.
The predictive techniques applied are excess-area balancing and kinematic modeling (Fig. 4). Because erosion has removed much of the stratigraphic column, methods that predict the fault geometry from a single key bed are appropriate. The top of the basement is the key marker that will be used because it is the horizon most complete on the cross section. The most general structural model is the excess-area technique (Fig. 4A). It is based on the assumption that the area of the cross section remains constant during deformation and makes no prediction as to the geometry of the master fault, or if there even is one. A kinematic model is a model for the geometric evolution of a structure without regard to the forces or stresses that cause it. Kinematic models are widely used and generally successful in predicting the unresolved parts of structures, especially at the scale of oil fields (e.g., Shaw et al., 2005), but present the problem of choosing the most appropriate model (Groshong et al., 2012; Butler et al., 2018). A suitable kinematic model must be consistent with the rotated-block geometry in Figure 1. The kinematic models to be tested are the circular-arc fault model (Fig. 4B), oblique simple-shear model (Fig. 4C), and the simple-shear and pure-shear fault-bend fold models (Figs. 4D and 4E).
The models in Figure 4 share several important features. All the models maintain constant area, but each makes a different prediction as to how much area must be internally deformed for the structure to develop (gray in Fig. 4). All have the same regional elevation. In structural interpretation, the regional (Figs. 1, 3, and 4) refers to “the elevation of a particular stratigraphic unit or datum surface where it is not involved in the thrust related structures” (McClay, 1992). It represents the presumed geometry of a marker horizon before deformation. All the techniques are based on the assumption that the regional is known and planar before deformation. Every marker horizon has its own regional. For all the methods as applied here, except for the simple shear fault-bend-fold model (Fig. 4D), the trailing edge of the structure is assumed to be vertical. For the simple shear fault-bend fold model, the trailing edge is straight but tilted between the top-basement regional and the lower detachment and is vertical above the top basement.
The basement within the region is high-grade crystalline rock of Precambrian age (Bayley et al., 1973) and at the scale of the whole uplift should be fairly homogeneous and brittle (i.e., Stearns, 1971) at shallow depths. Where the basement is thrust over the sediments of the Green River Basin, the fault zone consists of ∼1–1.5 km of highly deformed and overturned Paleozoic sedimentary rock (Berg, 1962; Gries, 1983; Stone, 1993). The dip of the top basement and the sediments of the Green River Basin where they approach the Wind River thrust have been subject to multiple interpretations on the COCORP profile because the high velocity of the basement in the toe of the thrust causes a velocity pull-up of the deeper reflectors. In the processing by Smithson et al. (1980), the beds dip sharply downward as they approach the thrust, whereas in the processing of Sharry et al. 1986), the sediments dip gently upward as they approach the thrust (Fig. 3). The different seismic interpretations have led to different estimates of the dip separation on the Wind River thrust. Erslev (1986) measured a 20 km offset and, on the profile of Figure 3, the offset is between 15.1 km and 16.1 km from the regional.
Stratigraphic relationships in the cover sequence indicate that the structure began as a gentle arch in the Early Cretaceous (100 Ma), and uplift became more rapid (perhaps because the master fault reached the surface) at ca. 90 Ma and ended ca. 49–50 Ma in the Eocene (Steidtmann and Middleton, 1991). The term Laramide is widely applied to these basement-involved structures to indicate both the structural style and the age of deformation (Yonkee and Weil, 2015). The Continental normal fault zone (CF, Figs. 2 and 3) is generally considered to be post-Pliocene, related to collapse of the southern end of the mountain range (Steidtmann, et al., 1983; Steidtmann and Middleton, 1991).
The amount of eroded basement is fairly well constrained by the low limb dips of the sedimentary rocks on both sides of the exposed basement (Fig. 3). The top basement projected as a smooth curve reaches a height of 1.3 km above the erosion surface, and projected as straight lines reaches a height of 2.2 km. Apatite fission track data indicate up to ∼3 km of uplift and cooling along the range crest from 65 to 50 Ma (Peyton et al., 2012; Stevens et al., 2016; Yonkee and Weil, 2017), placing an upper limit on the erosion. This profile is well down plunge from the range crest and the width of the basement exposure is narrower, so the amount eroded should be less, consistent with the 1.3–2.2 km extrapolation of the top basement.
Choosing the correct regional (Fig. 1) is an important geological decision in the application of every predictive technique. Before deformation, the regional of the top basement would have been at the same elevation in the hangingwall and the footwall of the Wind River thrust. This elevation should be approximately preserved in the footwall where the elevation of the top of the basement is presently 6 km below sea level (Fig. 3). Because of the vertical load imposed by the Wind River thrust sheet, the footwall may have subsided isostatically. Hall and Chase (1989) place the amount at 0.6 km close to the center of the range. For the purpose of determining the sensitivity of the models, the regional will be considered to occur between 5.4 km and 6 km subsea (Fig. 3). The regional is assumed to be horizontal because the footwall is mainly horizontal on the seismic reflection profile interpretations and because the backlimb of the Wind River block returns to horizontal to the northeast. With these choices, the backlimb returns to the regional dip angle of 0° but does not return to the regional elevation at the NE trailing edge of the section.
Another geological interpretation having a significant impact on the modeling is the width of the rotated block. Normally the boundaries of a structure are placed from where it departs from the regional elevation to where it returns to the regional elevation. On Figure 3, the block does not return to the regional elevation to the northeast, requiring a boundary to be selected. The Emigrant Trail thrust is picked as the trailing edge of the Wind River block because it represents the leading edge of the next major basement block to the northeast. This fault forms the southwestern boundary of the Granite Mountains (Fig. 2), part of the Sweetwater Arch. The fault separation on Emigrant Trail thrust is comparatively small on the profile in Figure 3, but it increases to the southeast to become comparable to that of the Wind River thrust (Hall and Chase, 1989).
We utilize data from the EarthScope Transportable Array and other local seismic networks to calculate P-to-S receiver functions in order to characterize crustal thickness and identify mid-crustal structures across the region (Fig. 5). Receiver functions utilize P-to-S converted energy to determine the depth, gradient, and amplitude of velocity contrasts within the Earth, which are determined based on the amplitude of the Ps converted phase and the delay time between it and the primary P phase. Receiver functions were calculated from teleseismic events, between 25° and 95°, for all publicly available regional stations. In the receiver functions calculations, deconvolution was accomplished using an iterative time-domain deconvolution technique (Ligorría and Ammon, 1999) with a Gaussian width of 2.5, which is equal to an ∼1.2 second low-pass filter. Receiver functions with low-variance (<80% on the radial components and 60% on the tangential component) and negative initial arrivals were discarded. The cross section of receiver functions was created using the common conversion point (CCP) methodology (e.g., Dueker and Sheehan, 1997; Gilbert et al., 2003) with a bin size of 15 km and a bin sharing of 1.25 in the CCP stacks. The receiver functions were migrated to depth using the ISAP91 model (Kennett and Engdahl, 1991). The predicted timing of converted phases from the bottoms of the sedimentary basins and their multiples (arrivals that experience a reflection off of the free surface) are plotted on Figure 5. The timing of these arrivals was calculated using a digital elevation model of the depth to the Precambrian basement (Marshak et al., 2017) and applying the sediment velocities of Yeck et al. (2014) for the Bighorn basin (Vp of 3.6 km/s and a Vp/Vs of 2.1). These predicted times were then migrated to depth using the ISAP91velocity model (Kennett and Engdahl, 1991) as in our CCP stacks.
Our results are similar to the high-resolution receiver function profiles shown by Yeck et al. (2014) for the Bighorn Mountains, located to the east of the Wind River Range. In their results, Yeck et al. (2014) observe strong basin reverberations. They do a detailed analysis of the effects of basins on their results and demonstrate that in both the Powder River and Big Horn basins, Ps conversations and PpPs and PpSs+PsPs multiples are observed from the basins. Our cross section location was chosen to minimize basin effects; however, we see similar arrivals in our results. To aid our interpretation, we modeled predicted arrival times for phases converted off of the bottoms of the basins and their multiples as described above. This modeling does not account for the effects of potential inter-basin converted Ps phases but does demonstrate that most basin effects are concentrated in the shallow crust (<20 km depth) in the vicinity of the cross section. This effort demonstrates that the deeper negative arrivals (>20 km depth) likely represent primary discontinuities in velocity and not multiples from shallower discontinuities.
We interpret the Moho as a northeastward-dipping positive impedance contrast located at 52 km depth in the NE edge of the cross section and 38 km depth in the SW edge of the section. The Moho depth of 52 km beneath the Wind River basin is a bit deeper than previous estimates of Moho depth (Fig. 3) but agrees with the receiver function work of Gans (2011). The rise of the Moho to the SW is consistent with previous regional interpretations although our data shows a slightly steeper dip. We do not account for the basins in our migration, which could impact our estimates of Moho depths. The steps in the Moho shown by our data have not been observed previously and may be visible in our data because it has higher resolution than previous studies. Alternately, these steps may be artifacts of the lower station density to the SW, changes in velocity structure across the cross section, or dipping-layer effects. The stepwise rise of the Moho to the southwest requires more data before being confirmed.
Detachments are most likely to be associated with negative amplitude gradients, indicating that the velocity is decreasing downward. The implication is that the rigidity of the crust is decreasing downward, making the crust more deformable and therefore more likely to host a detachment. There are three such zones beneath the Wind River block: one between 10 and 15 km subsea, another between 29 and 35 km subsea, and one beneath the Moho. Previous work by Snelson et al. (1998) identified the fault as a decrease in velocity at shallow depths due to the thrusting of basement over sedimentary units. Smithson et al. (1978) identify a decrease in density with depth associated with faulting, which is consistent with our observation of a decrease in velocity with depth and with our negative arrival representing a primary conversion and not a basin multiple.
We first describe excess-area balancing and oblique simple shear because they are the two most general techniques. Area balance applies to any geometry regardless of whether or not a master fault is present; oblique simple shear requires a bounding master fault and can accommodate any fault geometry. The circular-arc model is based on a master fault that is the arc of a circle and the shear fault-bend fold models are based on a master fault with a ramp-flat geometry.
Measurement of lengths, angles, and areas on the cross section are required for modeling. Repeated measurements indicate that the length measurements are accurate to about ±1.0 km primarily due to finite line widths on the original cross sections; area measurements are accurate to about ±10 km2, and angles to better than ±0.5°. The measurement uncertainties are negligible compared to the geological uncertainties already mentioned, such as the location of the regional, the amount of basement eroded, or the exact dip of the master fault where used in a model. The effects of the geological uncertainties will be presented when the models are applied.
Oblique Simple Shear
The oblique simple shear kinematic model (Fig. 6) is based on the concept that as the hangingwall is displaced along a fault, it deforms internally by slip along infinitely thin, constant-length lines parallel to the shear direction. Originally proposed for normal faults with a shear direction perpendicular to the regional (Verrall, 1982; Gibbs, 1983), it was generalized to shear oblique to the regional by White et al. (1986) and to reverse faults by Yamada and McClay (2003). Application of the model requires a key bed with hangingwall and footwall cutoff locations on the master fault and a known regional. This model has been successful for a variety of extensional structures (Verrall, 1982; Gibbs, 1983; White et al., 1986; Xiao and Suppe, 1992; Withjack et al., 1995; Uzkeda et al., 2014) for which the dip of the shear angle is generally found to be ∼60° antithetic to the master fault (Xiao and Suppe, 1992; Hauge and Gray, 1996; Shaw et al., 1997). Yamada and McClay (2003) applied this method to a sandbox analog model having a listric reverse fault and found that an antithetic shear angle of 58° provided the best fit. Tanner and Krawczyk (2017) obtained a good result with this model using an angle of 60°for the rotated-block of the Hartz Mountains in Germany. The most important unknowns are the shear angle (Withjack and Peterson, 1993) and, for the COCORP profile, whether top basement marker projects to regional or projects to horizontal above regional on the backlimb.
The construction of the master fault is explained here in the context of the Wind River thrust (Fig. 6). Shear lines 0 and 1 are drawn through the hangingwall and footwall cutoffs of the master fault. The boundary displacement of the block (D) is the distance between these lines measured parallel to the regional. A set of shear lines is then constructed with spacing D. Line lengths between the marker horizon and the fault are measured, starting with t1, which is restored to its original position with the top at the regional on shear line 2. The base of t1 marks the location of the fault. Then t2 is measured, shifted to shear line 3 and the fault location marked at its base. This process is continued progressively across the profile to construct the complete fault. The shear lines can be as closely spaced as desired as long as the thicknesses are always shifted to the shear line a distance D away.
This model is based on the assumption that the master fault ramp is a segment of a circular arc (Figs. 4B, 7A) and that the hangingwall block rotates upward as displacement increases. Quantitative models for reverse displacement on a circular-arc fault have been published by Brown (1984), Erslev (1986), Seeber and Sorlien (2000), and Groshong (2006). This type of model has been successful in predicting the master fault geometry in different areas of thick-skinned reverse faulting, notably the Rocky Mountain foreland (Brown, 1984; Erslev, 1986, 1993) and the Transverse Ranges of California (Seeber and Sorlien, 2000). One version of the model for a reverse fault is the first applied to the Wind River thrust (Erslev, 1986). The model is further developed here to explore its consequences and related uncertainties. The model can be applied in two different forms, here termed the radius-displacement and the two-radius methods.
The major uncertainty in this version of the model is the location of the trailing edge of the uplift (the location of R*).
Trailing-Edge Balance Problem
As shown in Figure 7A, rigid block rotation of the original hangingwall cannot occur without a certain amount of internal deformation in the trailing edge of the hangingwall and causes a potential area balancing problem as well. As a rigid block rotates up the ramp from the lower detachment, the beds following behind it must deform as they curve to climb the ramp. The arc displacement of an arbitrary horizon (dA, Fig. 7A) increases with depth, thus to maintain constant bed length and bed thickness, the trailing pin line must rotate the same amount as the block. This top-toward-the-hinterland rotation of the trailing pin line has sometimes been interpreted in thin-skinned structures to be transmitted into the trailing block as layer-parallel shear. It seems unlikely that this mechanism will occur in crystalline basement. Therefore trailing-edge rotation would potentially cause a void to form between the trailing edge of the rotated block and the vertical boundary of the block behind it.
For this boundary displacement, deformation is not required in the translated block except near the ramp, and the hangingwall maintains constant area. A consequence of this balancing mechanism is that the boundary displacement (DB) is less than the arc displacement on the fault ramp (DA). So interpreted, the model is area balanced but not length balanced.
Shear Fault-Bend Fold Models
The shear fault-bend fold models (Figs. 4D and 4E) are based on the assumption that the master fault has a ramp-flat geometry with the lower flat occurring at the base of the deformable décollement zone (Suppe et al., 2004). The folds form in either of two styles, termed the simple-shear style or the pure-shear style. In both styles, the décollement interval is overlain by a layered cover section that deforms by flexural slip and maintains constant bed length, bed thickness, and boundary displacement. Application of the model requires the location of the regional, the location and dip of the tip of the master fault, the backlimb dip, the location of the hinge point where the backlimb flattens into the regional, and the location of the top of the décollement zone. The orientation of the trailing edge axial surface within the basement is different for each model. For the pure-shear model the dip of the trailing axial surface in the basement is given by equation 14 in Suppe et al. (2004). The simple-shear fault-bend fold model differs by having its trailing-edge axial surface continue along the same line as in the cover sequence. The displacements are determined from α (pure shear model) and αe (simple shear model) in equation 10 of Suppe et al. (2004). The models are fixed according to the geometry of the top of the décollement zone and the regional. Once these elements are selected, the models are unique.
Excess Area Balance
The excess area method places outer bounds on the position of the lower detachment (Fig. 8). Several factors cause uncertainties in the calculated result. The amount of basement material eroded from the anticline and the exact location of the regional affect the area measurement. Uncertainty in the dip separation on the fault directly affects the depth calculation. The difference in area between the two basement-surface extrapolations is small. Considerably more important is exact position of the regional. Although the difference in elevation is only 0.6 km, because it occurs at the wide base of the excess area, it has a significant impact. There is a large uncertainty in the dip separation depending on whether the displacement is equal to the dip separation at the top of the basement or has a model-derived value. The dip separation on the master fault at the top of the basement is measured along the fault from the hangingwall cutoff of the top basement to the regional. Because there are two alternative regionals, the fault separation has an associated uncertainty.
The depth to detachment has been calculated from Equation 1 for the maximum and minimum excess area values and for the maximum and minimum values of displacement. Including the uncertainties in both the area of eroded basement and in the position of the regional, the minimum excess area is 445 km2 and the maximum is 529 km2. The dip separation of the top basement is 15.1–16.1 km (Fig. 8) and an estimate of the maximum displacement of 21.5 km is taken from the circular-arc model discussed later. The range of possible lower detachments is then from 26.1 to 41.0 km below sea level. These results rule out the lower detachment being in the shallower negative amplitude gradient zone at 10–15 km subsea along with the possibility that the detachment is below the Moho (Fig. 8).
Oblique Simple Shear Model
The oblique simple shear model (Fig. 9) illustrates the results of alternative choices for the hangingwall regional. The top basement beneath the NE dipping backlimb of the Wind River block does not return to its footwall regional. Instead, there are second order structures beginning with the Sheep Mountain anticline, across which the backlimb steps up to the NE. The Emigrant trail thrust is the leading edge of the Granite Mountains block, the next major block along strike to the southeast (Fig. 2). If the step up of the basement across Sheep Mountain is not related to the Wind River rotation, then the deeper projection of the top basement (black, Fig. 9) is appropriate. In this interpretation, the Sheep Mountain and the Big Sand Draw structures must be connected to the Granite Mountains block. The result is a predicted lower detachment that flattens to horizontal to the NE (black master faults in Fig. 9). Alternatively, the Sheep Mountain structure and its backlimb are part of the Wind River rotated block and must be explained by a single model with a regional that is shallower to the NE (red, Fig. 9). This results in a lower detachment that dips gently to the NE, with no evidence as to where the detachment flattens to horizontal (red master faults in Fig. 9). Another geological explanation for the elevated basement below Sheep Mountain and the Big Sand Draw structures is provided by the circular-arc model discussed in the next section.
The largest source of uncertainty intrinsic to the oblique simple shear model is the shear angle. Because of its success in the studies cited in the Methods section, a shear angle close to 60° should be the most likely. A shear angle of 90° gives the deepest possible detachment and lower angles give shallower detachments. The boundary displacement of 14.5 km for a 90°shear angle is less than the fault separation at the top basement (15.1–16.1 km) and seems highly unlikely. The shear angle of 75.6° gives a boundary displacement of 16.1 km and thus provides a reasonable upper bound for the shear angle. Note that for a given fault-tip geometry the boundary displacement is a function of the shear angle alone. For the model that produces a horizontal lower detachment with a shear angle of 60°, the detachment is 28 km subsea (Fig. 9) and the boundary displacement is 19.1 km (Fig. 6).The kinematic assumption behind the oblique shear model implies penetrative deformation throughout the entire hangingwall (Fig. 4C). The boundary displacement of 19.1 km is more than the fault separation at the top of the basement and also implies internal deformation of the block.
Both the radius-displacement and the two-radius versions of the circular-arc model are applied to the Wind River block. The models are applied just to the concave upward part of the fault (Fig. 10). The point on the fault chosen to define radius R is in the middle of the interpreted fault traces offsetting the basement-sediment contact where the fault dip is 38.5°. The backlimb of the block returns to regional dip but not to the regional elevation. According to the circular-arc model, the backlimb must merge into the regional elevation along an arc having the same center of curvature as the master fault.
In the radius-displacement method (models 1 and 2, Fig. 10A) the length of the radius is given by Equation 2a from the displacement on the master fault and the dip of the backlimb. As discussed previously, a reasonable range of measured offsets is from 15.1 km to 20 km and Δ = 11°, giving a range for the radius of 78.7–104.2 km. A major constraint is provided by the predicted trajectory of the top basement as the block rotates. For model 1, the master fault displacement is 15.1 km and the top-basement trajectory (arc 1, Fig. 10A) fails to intersect the top basement on the backlimb, showing this model is not viable. For model 2, with a fault offset of 20 km, the top-basement (arc 2, Fig. 10A) merges smoothly into the top basement of the backlimb, making this model viable. The predicted fault flattens into a lower detachment 27 km below sea level.
In the two-radius method (model 3, Fig. 10A), the center of curvature is the intersection point of a line perpendicular to the fault near its tip (R) and the perpendicular to the regional where the beds return to regional dip (R*). Because the top basement does not return to the regional elevation, the major constraint on the exact location of R* is the predicted trajectory of the top-basement as the block rotates. The top basement curves smoothly from the lower (red) regional into the top basement as projected down dip (arc 3, Fig. 10A). The intersection of R and R* defines the center of curvature and gives a radius of 111.4 km. The predicted fault flattens into a lower detachment 30 km below sea level. The biggest source of uncertainty in this model is the dip of the master fault, which should be contained within a range of ± 2°. This generates a fairly small range of uncertainty with the lower detachment being between 28 and 31 km subsea. The predictions from models 2 and 3 are fairly similar in location of the trailing edge of the block and the predicted lower detachment location. The result shows the predictions are not very sensitive to the exact location of the center of curvature as long as the marker horizon returns to regional dip. The next phase of the interpretation is to interpret the area balance of the cross section.
The area balance of the circular-arc model offers an intrinsic explanation for the basement that occurs above the projected basement top (E1, Fig. 10B). Model 3 with C = 111.4 km is used as being the best estimate. If the hangingwall had the rigid-block geometry of the model in Figure 7A, it would extend forward to include the missing area M and rearward to include the cryptic area Cp (Fig. 10B). The missing area M is much greater than can be accounted for by erosion. The cryptic area is present but indistinguishable from the rest of the basement. The combined missing plus cryptic area equals 133 km2. The extra areas (E) not accounted for by the model of Figure 10A total 108 km2, leaving a difference of only 25 km2 unaccounted for. The two general modes of trailing-edge deformation that can achieve area balance are average displacement (Fig. 7B) and trailing-edge uplift (Fig. 7C). The mode favored here is trailing-edge uplift because it explains most of the uplift of the top basement above regional at the trailing edge of the structure (E1, Fig. 10B). The area E2 represents material that belongs to the tip of the basement block (area M) but has rotated downward during deformation. The Continental normal fault contributes to the subsidence of the block tip. Area E3 (if present, interpretations differ) could represent small amount of folding prior to fault breakthrough.
Several aspects of these results imply that the deformation is not simply rigid-block rotation. The arc displacement on the master fault is 21.5 km, which is significantly more than the dip separation of 15.1 km seen at the top of the basement. The area missing at the tip of the block and the cryptic area at the trailing edge are probably distributed throughout the block and in the uplift at the trailing edge. The trailing edge uplift (E1) is the location of the second-order anticlines forming a line of oil fields across the map (Fig. 1). The other explanations for this uplift are that the area belongs to the Granite Mountains, requiring a shallow basement thrust below Sheep Mountain that extends to the Emigrant Trail thrust, or that the master detachment does not flatten below the Wind River Basin but rather continues dipping to the northeast.
Shear Fault-Bend Fold Models
In applying the shear fault-bend fold models (Fig. 11), the most critical geological parameters are the location and dip of the trailing-edge axial surface and, for the pure-shear version, the intersection point of this axial surface and the regional. In both versions the thrust ramp is assumed to be planar. Because the top basement does not return to regional elevation on the Wind River profile, it is projected to the footwall regional to define the hinge point where it is intersected by the trailing axial surface. For the pure shear model the trailing axial surface changes its dip to Ψ at the hinge, whereas for the simple-shear version the trailing axial surface remains straight. In both models the axial surface in the décollement zone continues downward to intersect and define the base of the master fault ramp.
The lower detachment in the pure-shear model is 26 km below sea level and the boundary displacement is 18.1 km. For the simple shear model the lower detachment is 42 km below sea level. The boundary displacement for the simple shear model increases from zero at the lower detachment to its maximum value of 14.5 km at the top basement (Fig. 11A). Both predicted detachments occur where the receiver-function velocity structure indicates downward increasing velocity, which seems less likely than being within the zone of downward decreasing velocity. Therefore these models are less likely than those applied previously.
Excess-area balancing does not depend on a kinematic model and for the Wind River block indicates that the lower detachment is most likely to be in the range of 30.6–41.0 km subsea (Fig. 12A). The kinematic model predictions based on a circular-arc fault and the oblique-shear method fall within this range and put the lower detachment at 30–31.5 km subsea. The excess area balance is sensitive to the amount of basement removed by erosion and the exact position of the regional. The oblique shear model is sensitive primarily to the fault separation and is not very sensitive to the shear angle in the range of 60° to 75.6°. The circular arc model with R = 118 km yields almost the same predicted fault as the 75.6° oblique shear model. The two-radius version of the circular arc model as used here is relatively insensitive to reasonable variations in the master fault dip and the exact location of the trailing edge. The crustal density boundary at 30 km subsea determined from Bouguer gravity anomalies (A in Fig. 12B) requires a lower detachment deeper than 30 km because it is offset by the master fault. The geometry of the offset has been forward modeled using a circular-arc fault with radius 118.0 km to give the marker position labeled A′. The predicted dip of the marker matches that of the density boundary but the amount of uplift is less. Internal strain in the hangingwall might make the agreement closer. The predicted fault trajectories nicely follow the trend of the steeper faults as interpreted from seismic reflections (Fig. 12C). The model lower detachments flatten where the deeper seismic reflection faults flatten. The seismic-reflection, gravity data, and model predictions are in good agreement for the oblique-shear model with a 75.6° shear angle and the circular-arc model with a radius of 118.0 km. The lower detachment falls near the top of the downward-decreasing velocity zone derived from the receiver functions (Fig. 12A).
The boundary displacement of the most likely circular-arc model (R = 118.0 km) is 22.9 km, substantially greater than the 15.1–16.1 km offset at the top of the basement, whereas the most likely oblique shear model with a shear angle of 75.6° gives a boundary displacement of 16.1 km, equal to the larger possible observed fault offset. The reason for the large difference in displacement is that the circular-arc model displaces a considerably larger area to generate the hangingwall. The circular-arc model includes the area M at the tip of the block and Cp at the trailing edge of the block (Fig. 10B), neither of which is present in the oblique shear model. The extra area in the circular-arc model provides an intrinsic explanation for the uplift at the trailing edge of the Wind River block if the lower detachment is horizontal, whereas the oblique shear model can explain the trailing edge uplift with a dipping lower detachment. The alternative to both is a shallow uplift on a thrust ramp tied into the Granite Mountains.
Based on a comparison of the model predictions to the receiver-function velocity profile, the most likely lower detachment is near the top of the zone of downward-decreasing velocities between 29 and 35 km subsea (Fig. 12A). This zone is prominent at the NE end of the profile where the models predict the lower detachment should be and ends in the middle of the cross section, approximately where the thrust ramp begins. The location of the thrust ramp is apparently controlled by the end of a zone of weaker crust. Thrust ramps controlled by the termination of detachment zones are well known in thin-skinned thrust tectonics (e.g., Davis and Engelder, 1985; Schedl and Wiltschko, 1987; Nilfouroushan, et al., 2012). The agreement between the receiver-function detachment depth and ramp location with the structural models implies that the receiver function profile reflects in situ crustal properties and not multiples.
The upper part of the cratonic crust is generally considered to be relatively rigid and brittle (Stearns, 1971; Lacombe and Bellahsen, 2016), whereas all the kinematic models imply internal deformation from the lower detachment to the top of the basement. A circular-arc fault model can allow partial rigid rotation of the hangingwall block, but at a minimum, the entire basement must deform where it curves up the ramp from the lower detachment. Sharry et al. (1986) interpreted multiple faults within the hangingwall of the master fault on the COCORP profile. They proposed that all the faults contributed to the uplift of the block, thus providing a visible mechanism for the internal deformation of the entire basement. Laramide deformation from the thin-section to kilometer scale has been documented in the field in the northwestern Wind River Range basement by Mitra (1993) and Yonkee and Mitra (1993). These investigators found cataclastic shear zones and veins at the grain-scale and brittle high-angle reverse-fault zones with offsets ranging from 60 m to 6 km at the map scale. The spacing between the larger faults is 2–4 km. The faults die out upward into folds and represent an east-west horizontal shortening of up to 30% (Mitra, 1993).
The trailing edges of the major uplifts in the region represent a unique structural environment characterized by linear trends of second-order fault-cored anticlines. The anticlines from Sheep Mountain to Rolff Lake fall along a linear trend parallel to the Wind River master thrust (Fig. 2). These second-order anticlines (e.g., Sheep Mountain, Fig. 3) typically have a rotated-block style similar to that of the first-order structures and in the Wind River Basin many form oil fields (Fig. 2). Numerous thick-skinned oil fields around the world have this same style (Mitra and Leslie, 2003; Afifi, 2005; Nicholson and Groshong, 2006; Witte et al., 2012; Lacombe and Bellahsen, 2016). More detailed published examples from the Wind River region show the second-order anticlines to be typically 10 ± km across and bounded by 50°–70° dipping master faults (Chester and Chester, 1990; Stone, 1993; Willis and Groshong, 1993; Mitra and Mount, 1998; Craddock and Relle, 2003; Johnson and Sutherland, 2009; Yonkee and Weil, 2017). The depth to detachment predicted using the kinematic models applied here scales approximately with the width of the block, being about a third of the block width for the Wind River Mountains. This suggests that the second-order anticlines will have lower detachments in the range of ∼3 km below the top of the basement. It is unclear whether or how such a shallow detachment might link to the lower detachment of the Wind River block (Gray et al., 2019). It is important to note that the backlimbs of the second-order anticlines flanking the Wind River block do not return to the projected top-basement elevation before encountering the next structure down dip or flattening significantly (Fig. 3; Yonkee and Weil, 2017). Because the hangingwalls do not return to horizontal or to the regional elevation, there is no direct evidence for the depth of their lower detachment.
The trailing-edge-uplift version of the circular-arc model for a first-order rotated block (Fig. 7D) predicts uplift where the backlimb flattens into the regional, consistent with the Wind River rotated block as interpreted here (Fig. 10B). The required shortening dies out downward to zero at the lower detachment of the first-order block. Thus the expected lower detachment for the second-order backlimb anticlines might be the same as for the first-order uplift but the master faults of the second-order anticlines might be rather diffuse and die out downward. Because the trailing-edge deformed area in Figure 7C is not bounded by parallel pin lines, none of the kinematic models as applied here will give the correct detachment. High-resolution deep seismic profiles across the second-order anticlines might reveal significant new information about the geometry and mechanics of the complete system.
The results of this study raise numerous questions for future research. Neither the potential crustal detachment zone nor the Moho as imaged by receiver functions is continuous across the profile. Are the extent and distribution of the major uplifts controlled by the locations of separated zones of crustal weakness? Does the Moho step down from beneath the Green River basin toward the Wind River Range or does it dip smoothly? Does the lower detachment of the Wind River block return to horizontal beneath the trailing edge of the block or does it continue dipping to the northeast? The circular-arc model implies that the second order rotated blocks along the trailing edge of the Wind River block are not necessarily simply smaller versions of the first-order uplifts. What is the nature of the master faults and detachment depths for these second-order anticlines? More data will be required to answer these questions.
Multiple methods for predicting the master fault geometry have been tested on a dip profile across the first-order Wind River rotated basement block. The geometry of the Wind River master fault is equally well predicted by oblique shear with the shear angle that gives the dip separation at the top of the basement and by the two-radius version of the circular-arc model. The predicted lower detachment flattens at ∼31 km subsea. This is consistent with the faults interpreted from the COCORP seismic reflection data and with the Bouguer gravity data.
The predicted lower detachment coincides with the top of a downward-decreasing velocity zone seen in the receiver-function analysis and the thrust ramp occurs where this zone dies out to the SW. Ramp initiation where a detachment zone dies out has been documented in thin-skinned thrust belts. The agreement between the observed Wind River thrust geometry, model predictions, and the configuration of the top of the downward-decreasing velocity zone indicates that the crustal structure imaged by receiver functions is real and does not represent multiples.
The top of the basement on the trailing (NE) edge of the Wind River block does not return to its footwall regional, implying trailing-edge uplift. The uplift might be caused by displacement on a thrust connected to the major Granite Mountains block to the northeast and so not directly related to the Wind River displacement, or it might be explained as part of the circular-arc model of the Wind River block.
Internal deformation of the rotated basement block is required by both the oblique shear and circular arc models. If the displacement of the basement block is nearly rigid-block rotation as implied by the circular-arc model, the required internal deformation could be concentrated at the trailing edge of the rotated block and be responsible for the additional structural uplift and second-order structures found there.
The oblique shear model can explain the trailing-edge uplift as the result of displacement on a lower detachment that continues to dip northeast beyond the Wind River block boundary and extending beneath the Granite Mountains block.
Exploring the connection between crustal structure and rotated-block geometry was inspired by working with Paul Nicholson on the oil fields of the Arabian Platform. John Craddock and Dennis Harry provided helpful advice. Kurt Stüwe and the anonymous reviewers made excellent suggestions for improvements to an earlier version of the manuscript.