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Structural geometry of Raplee Ridge, UT Revealed Using Airborne Laser Swath
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Mapping (ALSM) Data
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Hilley, G. E. (*), Mynatt, I., and Pollard, D. D.
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Department of Geological and Environmental Sciences, Stanford University, Stanford,
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CA 94305-2115
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(*) Corresponding Author: hilley@stanford.edu
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In preparation for Journal of Structural Geology
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Abstract:
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Kinematic models are often used relate the geometry of strata deformed in fault-
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related folds to the location and geometry of faults in the subsurface. Recently, two-
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dimensional mechanical analyses that assume a viscous rheology have demonstrated that
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it is possible to use such an approach to infer loading conditions and fault geometries
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based on observations of surface deformation. In this contribution, we present a method
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that uses a three-dimensional boundary element model to infer the geometry of an
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underlying fault and the loading conditions that are most consistent with layers deflected
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by fault-related folding. The model assumes that deformation accrues in a homogeneous,
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isotropic, linear-elastic half-space as a frictionless elliptical fault slips in response to
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remote loading.
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We apply this model to the Raplee Monocline in southwest Utah, where Airborne
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Laser Swath Mapping (ALSM) topographic data detail the geometry of layers deflected
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within this fold. The spatial extent of five stratal surfaces were mapped using the ALSM
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data, elevations were extracted from the high-resolution topography, and points along
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these layers were used to infer the underlying fault geometry and remote strain
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conditions. First, we compared elevations extracted from the ALSM data to publicly
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available Digital Elevation Models (DEMs), such as the National Elevation Dataset 10-m
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DEM (NED-10) and 30-m DEM (NED-30). We found that while the spatial resolution of
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the NED dataset was far coarser than the ALSM data, the elevation values extracted at
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points spaced ~50 m apart from each mapped surface yielded similar elevations. Second,
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we used a single layer in the Raplee Monocline to infer the geometry of an underlying
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fault and the remote loading that is most consistent with the deformation recorded by this
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stratum. Using a Bayesian sampling method, we also assessed the uncertainties within,
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and covariation between the fault geometric and loading parameters of the model. Next,
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we considered elevations extracted along all five mapped layers when inferring fault
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geometry, and found broad agreement with results obtained from considering a single
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layer in isolation. Interestingly, modeled elevations matched those observed to within a
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root-mean-squared error of 16-18 m, and showed little bias with position along the fold.
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Thus, while our idealized isotropic elastic model is clearly oversimplified when
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considering the complicated layered rheology of this fold, it provides an excellent fit to
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the observations. This may suggest that comparable surface deflections can be produced
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with a variety of rheological models; constraining factors such as the fault geometry
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when performing these types of inversions may be required to ascertain the appropriate
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representation of fold rheology.
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Introduction:
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Slip along blind reverse faults deform the surrounding rock mass, often resulting
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in large deflections of the strata exposed at the surface and present in the subsurface
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{e.g.`, \Allmendinger, 1998 #347; Narr, 1994 #346}. Forward mechanical models have
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been used to understand how fault geometry and loading changes the deformation
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surrounding these faults {e.g.`, \Allmendinger, 1998 #347}. In general, those studies
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relating surface deformation to slip along faults in the subsurface either assume a fault’s
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geometry and solve for the slip distribution acting along it {Johnson, 2002 #348}, assume
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that slip along a fault is uniform and solve for the location and geometry of the fault {,
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2007 #558}, or assume that long-term deformation recorded within folded strata conserve
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their cross-sectional area and use the surface exposure of such strata to infer the location
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and geometry of subsurface faults {Wolkowinsky, 2004 #560}. The latter approach has
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been widely used to extrapolate deformed structural markers at the surface to fault
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geometry at depth; however, such methods produce non-unique inferences of the
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geometry of underlying structures {Mynatt, 2007 #558; Ziony, 1966 #559} and may
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result in physically unrealistic particle displacements as the strata are required to deform
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discontinuously {e.g.`, \Stone, 1993 #569; Stone, 2004 #570}.
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approaches are unable to infer both the geometry of underlying faults as well as the
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remote loading conditions that cause them to slip. For this reason, Johnson and Johnson
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(2002) derived a complete mechanical description of the folding process in two
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dimensions that uniquely relates surface deformation to fault geometry when the strata
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deform according to a viscous rheology. This approach is advantageous in that the
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observed and inferred structural geometries are uniquely related through the requirement
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of continuity in the surrounding medium and realistic physical rheologies may be
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specified.
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dimensional elastic model to infer subsurface geometry and remote loading conditions
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from surface deflections recorded in the geologic record.
In addition, these
More recently, Mynatt et al. {Thomas, 1993 #350} have used a three
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The well-exposed Raplee Ridge Monocline in southeastern Utah is a ~N-S
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oriented fold that is ~14-km-long by ~2-km-wide (Fig. 1). Here, the San Juan River has
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incised through the fold in the last several Ma {Mynatt, 2007 #558}, exposing a thick
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sedimentary package {Davis, 1999 #562`, and references therein}. Folding within the
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ridge likely occurred during the Laramide phase of deformation in the area during latest
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Mesozoic and early Cenozoic time. Many folds within the Colorado Plateau were formed
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due to reactivation of high-angle structures that likely date back as far as the Precambrian
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{Davis, 1999 #562`, and references therein}. While no fault is exposed at the Raplee
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Ridge Monocline and no subsurface information exists here, the presence of steeply-
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dipping beds along its west side along with the fact that this fold is analogous to many
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other Laramide folds in the Colorado Plateau for which subsurface structures have been
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imaged {Davis, 1999 #562`, and references therein} suggest that this fold has likewise
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been formed above a east-dipping high-angle reverse fault.
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The National Center for Airborne Laser Mapping (NCALM) collected high-
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resolution topographic data that images the deformed strata in this structure (Fig. 2).
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These data provide a high precision, dense array of points that define the fold’s geometry,
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which we use to infer information about the subsurface geometry of faults that may be
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responsible for the formation of the Raplee Ridge Monocline. Our previous work at
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Raplee Ridge Monocline combined the high-resolution ALSM data with an elastic
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boundary element model {Mynatt, 2007 #558} to infer the geometry of a fault that may
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have produced the fold {e.g.`, \Gratier, 1991 #571}.
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In this contribution, we compare the ALSM data to other more commonly
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available topographic data to determine the necessary spatial resolution that is required to
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infer the subsurface geometry of faults based on the deflections of exposed folded strata.
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In addition, we document a method to infer fault geometry from exposed surface strata,
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and expand this method to use a series of layers within a deformed fold to better constrain
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the geometry of the underlying structure. We use a Markov-Chain Monte Carlo method
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to provide an estimate of the variation within, and covariation between, model estimates
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of fault geometry parameters that produce deformation similar to that recorded within the
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fold. Given the assumptions embedded in our mechanical model, our approach places
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broad constraints on the geometry of the underlying fault that may be improved using our
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approach should additional prior information (such as a range in fault geometric
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parameters) be available. Even using the assumption of a homogeneous, isotropic linear-
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elastic rheology for the deforming strata, we find good agreement between observed and
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predicted deflections for our best-fit model. We speculate that our inversion method,
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used with physical models that consider more complicated rheologies, may produce
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deflections that produce a similar level of agreement than those observed. If correct, this
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indicates that when viewed in isolation from other information, such as direct measures
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of physical rock properties and prior knowledge of factors such as fault geometry, surface
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deflections may provide ambiguous information about the rock rheology at the time of
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deformation.
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Study Area:
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Raplee Ridge Monocline is located in southeast Utah outside of the town of
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Mexican Hat (Fig. 1). The details of the field site are reported in Mynatt et al. (in
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review); herein we summarize the most important aspects of the field site. The fold is
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composed of the Pennsylvanian to Permian Rico Formation, which consists of alternating
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limestones, siltstones, and sandstones that record a general trend of marine regression up-
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section. The occurrence of marine incursions during this generally regressional sequence
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has resulted in the deposition of five limey sandstones and sandy limestone layers
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encased within shales and siltstones that currently provide well-exposed stratigraphic
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markers whose surfaces reveal the geometry of the fold (Figs. 1, 2d, and 8). These layers
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are folded into an ~500-m high doubly plunging monocline whose fold axis trends
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approximately N-S. Sediments within the monocline’s forelimb dip steeply (up to 40°) to
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the west, while strata are gently dipping (< 5°) to the east within its backlimb (Figs. 1 and
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2). Exposure of different statigraphic levels varies depending on the depth of incision
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into the fold. The surface of the highest stratigraphic level used as part of this study, the
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McKim limestone, is widely exposed throughout the fold due to the fact that erosion has
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stripped virtually all of the overlying sediments from this surface over much of the fold
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(Fig. 2). Within basins incised into the fold, four additional mapable layers define the
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fold’s geometry including, from top to bottom, the Goodrich, Shafer, Mendenhall, and
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Unnamed limestones (Mynatt et al., in review). Within these basins, individual bedding
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surfaces can be identified and mapped; however, the exposure of these four lower units is
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often restricted to limited areas of the fold hinge that have been excavated by erosion.
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Overlying the McKim limestone, exposure of the Halgaito Tounge shale along the
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peripheral edges and away from the fold constrains the extent of the fold by defining
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those areas that are undeformed.
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We have no direct constraints on the geometry of the underlying fault that formed
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Raplee Ridge. Throughout the Colorado Plateau, similar monoclines are often associated
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with Laramide contraction that occurred during late Mesozoic through early Cenozoic
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time {Johnson, 2002 #348}. Where exposed, at least some of these structures appear to
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result from movement along high-angle basement faults that may have first experienced
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motion as early as the Proterozoic {e.g.`, \Willemse, 1996 #344`, and references therein}.
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In some cases, the reverse faulting that deformed Colorado Plateau strata into monoclinal
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geometries were subsequently reactivated during Basin and Range extension in the
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Miocene {e.g.`, \Huntoon, 1975 #563}. However, where this can be shown to be the
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case, normal-sense reactivation of the basement faults caused them to propagate through
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the strata above. Thus, Laramide contractional deformation is often accommodated
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within the Paleozoic and Mesozoic section by warping of the strata, whereas later
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extensional deformation typically resulted in discrete offset of these units {Ziony, 1966
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#559}. Within the Raplee Ridge Monocline, detailed field observations show that there
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are no faults that link with the underlying basement structures; thus, we infer that the
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deflected strata within the Raplee Ridge Monocline result from Laramide contraction,
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rather than Miocene extension. As a result, the western monoclinal forelimb implies that
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a reverse fault that formed the structure likely dips steeply towards the east.
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Airborne Laser Swath Mapping (ALSM) data:
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On February 24, 2005, the National Center for Airborne Laser Mapping
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(NCALM) collected topographic data from the Raplee Ridge area using an Optech 25
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kHz pulsed laser range finding system and associated Inertial Navigation Unit (INU) that
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is corrected for drift using kinematic GPS observations taken onboard the aircraft. The
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acquisition was performed to build a fold-scale geometric model to compare outcrop-
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scale measurements of fracture characteristics to fold geometry (Mynatt et al., in prep.).
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The combination of acquisition frequency and low elevation flight plan provided several
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laser range positions per square meter from which a 1 m Digital Elevation Model (DEM)
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was produced by kriging interpolation. The vegetation at the site proved sufficiently
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sparse to obviate the need for spatial filtering to remove its contribution from the DEM
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(Fig. 2). From this 1-m DEM, a shaded relief map of the fold was produced, which was
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used in the field to map the extent of the uppermost bedding plane of six particular
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exposed stratagraphic units throughout the fold (Fig. 2d). Once these surfaces were
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identified and mapped on the georeferenced images, (x,y,z) points that defined the
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geometry of each of the surfaces were extracted from the high-resolution DEM. In some
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cases, the topographic surfaces that appear to define the tops of the mapped strata have
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been modified by up to 1 m by surface processes; in addition, erosion along the edges of
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the surfaces has distributed a small amount of colluvium (typically less than 1 m thick)
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onto the tops of the exposed strata.
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estimates to be ~ 2-5 m {Bayes, 1763 #215}.
Thus, we estimate the precision of elevation
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The extraction of points from the mapped layers produced > 100,000 points.
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Consideration of this large number of points would render intractable the fold-scale
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inversion described below. For this reason, we decimated the high-resolution dataset to
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provide (x,y,z) points for each of the surfaces that were spaced no less than 50 m apart.
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We compared the ALSM measurements of these extracted points to more commonly
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available datasets to assess the precision of these other data for use in our study (Figures
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3 and 4). Specifically, we compared equivalent z values at the selected (x,y) points to
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those extracted from the National Elevation Dataset (NED) 30-m resolution DEM and
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10-m resolution DEM. The poor spatial resolution of both NED datasets relative to the
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ALSM data would prohibit accurate identification and mapping of the extent of the
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different stratigraphic levels within the fold (Fig. 3), although low-elevation air photos
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that were precisely georeferenced might be employed for such a purpose. However,
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considering the coarse resolution of both NED datasets, their extracted elevation values
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compared favorably with those obtained from the decimated ALSM dataset (Fig. 4a, c).
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To provide a quantitative metric of the differences between these three datasets, we
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calculated the residual elevation by subtracting the elevation value at each (x,y) point
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used in this study for both NED datasets from the elevation at corresponding (x,y)
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locations in the ALSM dataset (Fig. 4b, d). Elevations from both NED datasets showed
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little, if any bias in elevation relative to the ALSM data (mean residual value = 1.6 m and
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–1.1 m for the NED 30-m and 10-m datasets, respectively, relative to the ALSM dataset).
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Variation between these two datasets was modest (standard deviation in residuals from
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NED 30-m and 10-m datasets was 4.1 and 4.2 m, respectively); however, as described
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below, the average misfit for our best-fit inversions was several times this variation.
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Thus, given the uncertainties in the fold-wide inferences produced by modeling carried
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out in this study, elevations extracted using the NED 30-m and NED 10-m datasets
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should supply adequate precision when using these data as inputs to the fault geometry
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inversions described below, provided accurate locations of the stratigraphic layers can be
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defined.
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Methods:
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Inversion Methods:
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The two major goals of this study were to provide a detailed and accurate
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depiction of the continuous structural geometry of Raplee Ridge Monocline that could be
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compared to outcrop scale measurements of fracture characteristics, and to infer the
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geometry of and loading conditions acting along the reverse fault underlying the
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monocline. The finite extent of the Raplee Ridge Monocline necessitated the use of a
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three-dimensional model that could capture the along-strike variations in fault and fold
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geometry (e.g., Fig. 1 and 2). Some three-dimensional models that conserve the area and
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length of strata exist {e.g.`, \Hilley, 2008 #223; Hilley, 2008 #561}; however, as outlined
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above, such methods do not take advantage of the constraints provided by a full
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mechanical analysis. Likewise, methods that use a complete mechanical analysis to
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relate stratal deflections to the geometry of underlying strata assume two-dimensional
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plane-strain conditions {e.g.`, \Pollitz, 2003 #567}, and thus cannot adequately capture
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the three-dimensional nature of the Raplee Ridge Monocline.
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In this study, we used the three-dimensional Boundary Element Model (BEM)
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Poly3D to relate the deflection of stratal layers to underlying fault geometry and loading
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conditions (Thomas, 1993). Poly3D idealizes the rheology of Earth’s brittle upper crust
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as homogeneous, isotropic, and linear-elastic. Arbitrarily shaped planar elements are
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embedded into this elastic material to represent features such as faults and fractures along
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which displacement within the material is discontinuous. Slip or opening along these
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elements is driven either by specifying the displacement discontinuity directly, allowing
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the elements to slip under a prescribed remote loading, specifying a stress drop along the
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elements, or any combination of these boundary conditions for the normal and two shear
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components. For example, one may specify mixed-mode boundary conditions in which
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shear tractions acting across an element induce slip parallel to the element, while
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prescribing a zero displacement condition perpendicular to the element (no
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opening/closing). A set of planar elements may be assembled to produce an arbitrarily
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complex surface in three-dimensions along which any combination of displacement
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discontinuity or stress drop boundary conditions may be specified. Finally, Poly3D can
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calculate fault slip distributions along the fault(s), stresses/strains within the surrounding
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medium, and displacements within a half or full-space problem. The former conditions
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compute stresses, strains, and displacements given the presence of a traction-free surface
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that represents the interface between the solid earth and atmosphere, while the latter may
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be used to approximate faulting processes deep within the crust.
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In this study, we assume that stratal deflections are produced by slip along a
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single, elliptically shaped, frictionless planar fault underlying the monocline. Its elliptical
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geometry approximates the three-dimensional extent of many faults documented in the
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field {Metropolis, 1953 #182}, while ignoring geometric complexity that would be
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difficult to constrain using measurements of stratal deflections from the ALSM data.
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Likewise, we chose a planar fault geometry to mimic the geometry of older reactivated
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basement faults {, 2007 #558}.
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underlying this monocline is likely more complex than this simple geometry; however, as
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we show below, even this simple idealization of the underlying fault geometry produces
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deflections that closely mimic those observed at Raplee Ridge Monocline.
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additional data become available that better constrain the at-depth geometry of the Raplee
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Ridge fault, it is straightforward to assimilate this information into our modeling
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approach. Given this idealization, the geometry and location of the fault underlying the
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fold is defined by 9 parameters (Fig. 5): the down-dip length of the fault (H), the along-
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strike width of the fault (W), the depth of the fault below the surface (D), the map-view
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location of the fault relative to the coordinate system of the model (xo, yo), the dip of the
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fault (), the strike of the fault (s), and the remote conditions that drive motion along the
We acknowledge that the geometry of the fault
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Should
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fault. In this study, slip along the fault results from a prescribed remote strain tensor (xx,
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yy, xy), which we assume does not vary with depth (zz=xz=xy=0). We use this remote
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strain to compute the stresses acting along the fault elements, and compute shear
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displacements that result from this load.
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perpendicular to the fault surface, as large amounts of opening or fault-zone contraction
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are physically unrealistic at depths of > 1.5 km that likely typify the shallowest levels of
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slip along this fault during late Mesozoic time (see below).
However, we prohibit opening or closing
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Once the fault geometry and loading conditions are specified, Poly3D calculates
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stresses, strains, and displacements at specified points in the surrounding rock mass.
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These displacements can be used to calculate the deflection of initially flat surfaces, such
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as the strata currently exposed in the Raplee Ridge Monocline. We reconstructed the
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relative elevation and stratigraphic thickness of each of the six units modeled in this study
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using the measured stratigraphy {, 2007 #558} as well as ALSM-based thickness
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measurements from flat-lying portions of the stratigraphy exposed by downcutting of the
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San Juan River (Mynatt et al., in review). The likely depth of each of these units was
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then inferred by noting that during late Mesozoic time, approximately 1.2 km of
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sediments locally overlaid the section under study (IAN-INSERT REF). Thus, we were
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able to use this information to estimate the depth of the originally flat-lying strata at the
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time they were deformed.
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As these units underwent deformation, points originally located on the flat-lying
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surfaces were displaced both vertically and horizontally. Thus, the current location of the
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observed points along deflected layers do not record their initial positions faithfully,
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especially in the current case in which surface displacements are large.
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calculates the displacement of a given point that is defined prior to deformation; however,
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the displaced points measured using the ALSM dataset record each point’s final, rather
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than initial position. We calculate the appropriate displacement at each (x, y) observation
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point on the deformed-state surfaces using a two-step process. First, we calculate a
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regular grid of displacements for each stratigraphic level that results from a specified set
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of fault geometric and remote loading parameters. These regularly spaced points are
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deformed into an irregularly shaped mesh that represents the final configuration of each
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deformed surface. This process results in a set of irregularly spaced points on the
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Poly3D
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deformed surface for which we know displacements that are required to restore the points
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to a regular grid. Using the deformed configuration, we use a linear interpolation to map
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these displacement values to the (x, y) locations of each of the ALSM measurements, and
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use these interpolated displacements to restore each point to its initial location on the
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undeformed surface. In a second step, we use identical input parameters with Poly3D to
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deform these initial-state points into the final state to determine the geometry of the
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deflected surface. This results in (x, y) coordinates of the new deformed state points that
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match the locations of the observations, and allows us to directly compare the elevation
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values observed to those that are predicted by the BEM.
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The above approach is appropriate for comparing observations of the elevation of
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each deflected surface to those predicted by Poly3D. However, in some instances, such
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as our observations along the Halgaito Tounge surface, the rotation of bedding rather than
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its absolute elevation helps to constrain the fault geometry. In this case, we employ a
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similar approach to that used for elevation values; however, instead of calculating
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displacements at each of the points, we instead calculate bedding-plane rotation. As
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before, we first restore the observation locations on each of the surfaces for which
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bedding rotations are measured to their original locations. However, we use the initial-
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state locations to calculate rotation of these points as they are deformed and compare
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these rotations to observations. This was used to enforce areas of no rotation (e.g., flat-
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lying sediments as seen in front of the fold).
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Using these methods, we calculate the elevation values that would be predicted by
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a given set of fault geometry and loading parameters, and compare them to those
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observed elevation values for each layer using the following misfit function:
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2 
 obs pred 2 
m  obs
rj  rjpred  
z

z
i
i

 W r  
WRSS  W z 

  zobs 
  robs
 
 
i
j
i1 
j1



n
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(1)
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
where zobs are the ALSM-derived elevations of the deflected strata at the (x, y) locations,
zobs are the standard deviations of the elevation measurements, zpred are the predicted
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elevations from Poly3D, Wz is the weight given to the misfit between measured and
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observed elevation values, robs are the observed rotations (only flat-lying portions of the
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Halgaito Tounge member are used in areas away from the fold to constrain its extent),
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robs are the standard deviations of the rotation measurements, rpred are the predicted
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rotations from Poly3D, Wr is the weight given to the misfit between measured and
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observed rotations, and WRSS is the Weighted Residual Sum of Squares misfit function.
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As WRSS values decrease, the displacements and rotations calculated by Poly3D better
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match those observed. Thus, we can find the best-fitting fault geometric and loading
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parameters (hereafter referred to cumulatively as “model parameters”) by exploring
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various values of these model parameters, calculating the misfit defined by Eq. 1, and
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identifying the model parameters associated with the minimum value of the WRSS.
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In addition to the best-fitting model parameters, we quantify the uncertainty in
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these model estimates using a Bayesian sampling method. In this approach, the model
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parameters are viewed as a joint probability density function (pdf), the number of whose
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dimensions corresponds to the number of model parameters. Such a joint pdf may be
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used to determine both the best-fitting set of model parameters and uncertainties within
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and covariation between model parameters. These measures can be used to quantify the
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uniqueness of the best-fitting solution, and provide information about those combinations
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of model parameters that provide similar fits to the observed data. We calculate this joint
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pdf of the model parameters using Bayes’ Rule {, 2007 #558}:
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P(m | x) 
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P(x | m)P(m)
(2)
n
 P(x | m )P(m )
j
j
j1
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where x is a vector consisting of the observations, m is a vector consisting of the model

parameters, P(m|x) is the joint pdf of the model parameters given the observations,
P(x|m) is the probability of the data given the model parameters that can be derived from
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a misfit function similar to Eq. 1, P(m) is a pdf that represents the probability of
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occurrence of the model parameters in the absence of any observations, and the
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denominator normalizes the density of P(m|x) to unity {, 2007 #558}. P(m|x) is often
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referred to as the posterior density while P(m) is referred to as the prior density.
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Bayes’ Rule is used to estimate P(m|x) in our analysis as follows. First, a set of
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model parameters (m) is selected and from these choices, predicted elevations and
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rotations are computed at all points for which we have measured values of these
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quantities. The probability of observing the data given the model parameters, P(x|m) is
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computed using a modified version of Equation 1 that takes into account the number of
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Degrees Of Freedom (DOF; defined as the number of observations minus the number of
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model parameters) in the model and the uncertainty associated with each elevation or
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rotation observation {e.g.`, \Jaeger, 1969 #568`, and references therein}:
P(x | m)  exp  r2 
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(3a)
where
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 r2 
WRSS
DOF
(3b)
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
Bayes’ Rule also requires the definition of the prior density P(m). This prior
density represents the probability that a set of model parameters occurs in the absence of
376
any data to which the model is compared. This prior density may represent some
377
quantitative a priori information about various aspects of the fault geometry or loading
378
conditions, or simply may be used to incorporate expert opinion into the statistical
379
analysis. For example, at a particular site, subsurface seismic data may resolve a range of
380
permissible fault geometric parameters, such as fault strike and dip.
381
information constrains the geometry of the fault in the absence of any mechanical
382
modeling or measurements of bedding deflection. We cast these ranges in the fault
383
geometric parameters in terms of the probability of each value’s occurrence as a pdf. By
384
repeating this process for all of the model parameter values, we construct the prior pdf
385
P(m) that represents our prior knowledge of reasonable ranges in fault geometry. In the
386
case of the Raplee Ridge Monocline, we do not have additional prior information that
387
constrains the geometry of the underlying fault, and so we simply specify a uniform
388
probability of P(m) for all values of the model parameters, m. Thus, Eq. 2 reduces to
This prior
389
P(m | x) 
390
P(x | m)
(4)
n
 P(x | m )
j
j1

14
391
By evaluating P(x|m) for all permissible values of m, we can use Eq. 4 to compute the
392
probability of occurrence of the model parameters, P(m|x).
393
The simple mechanical model used to compute P(m|x) requires nine parameters to
394
be specified, and hence, m consists of a nine dimensional parameter space. If each
395
dimension of the parameter space were discretized into only ten values, the number of
396
evaluations of Poly3D that would be required to compute P(m|x) would be 109, or one
397
billion different combinations of model parameters. Such a large number of evaluations
398
is computationally infeasible given current computing technology. Indeed, it is this
399
limitation that has prevented the direct application of Bayes’ Rule to all but the simplest
400
problems with few model parameters. To circumvent this difficulty, Markov-Chain
401
Monte Carlo (MCMC) methods have been developed that sample the underlying
402
distribution P(m|x) in a computationally efficient way to provide a numerical
403
approximation of this distribution.
404
performed by the MCMC sampler, which is designed to sample P(m|x) according to the
405
pdf’s underlying probabilities.
406
identified using only a small fraction of the evaluations that would be required to
407
exhaustively explore the entire parameter space.
408
In these methods, sparse sampling of P(m|x) is
In this way, the portions with high P(m|x) can be
The sampler employed in this study is the Metropolis-Hastings MCMC method
409
{Mynatt, 2007 #558}.
In this algorithm, The Metropolis-Hastings sampler uses a
410
selection-rejection criterion to guide sampling through the parameter such that P(m|x) is
411
approximated. We first use an initial, randomly selected choice for m—the exact choice
412
for this starting point becomes less important as the simulation proceeds and the sampler
413
instead selects points based on the underlying distribution P(m|x). After this choice has
414
been made, a set of random numbers is drawn from the interval [-1, 1], one for each
415
dimension of m. These numbers are then scaled by a specified, arbitrary constant and
416
added to the previous choices for the initial model parameters, moving the samples a
417
random distance through the parameter space, m. This is the selection process. Next, this
418
new set of samples may be either accepted or rejected. If accepted, the new samples are
419
treated as a new starting point for the sampler and the selection process is repeated. If the
420
samples are rejected, the previous values of the model parameters are used to select a new
421
set of samples. Samples are accepted or rejected based on the probability of their
15
422
occurrence relative to the probability of occurrence of the previous set of samples. First,
423
the probability of occurrence for both the previous set of samples and the current
424
selection of samples is computed using Bayes’ Rule (Eq. 4). We define the probability of
425
the first sample as P1, whereas the probability of the second sample is denoted as P2. In
426
the case that P2 > P1, the new set of samples is always accepted. However, if P2 < P1, the
427
ratio of these two probabilities is computed and compared to a random number drawn
428
along the interval [0, 1]. If the ratio exceeds this random number the sample set is
429
accepted, otherwise it is rejected. Thus, the Metropolis-Hasting sampler explores the
430
parameter space m, is guided towards higher values of P(m|x) in Equation 4, and the
431
frequency of the model parameters chosen by the sampler approximates the posterior
432
density P(m|x). Initially, the values selected by the sampler will depend on the arbitrary
433
initial choices for the model parameters at the beginning of sampling; however, as
434
sampling proceeds, the memory of these initial choices tends to fade and the frequency of
435
m selected by the sampler will reflect P(m|x). Thus, we allow a “burn-in” period of
436
sampling to allow the memory of the arbitrary sampling starting location in the parameter
437
space to be erased, and we only consider the frequency of samples identified by sampler
438
after this burn-in when P(m|x) is computed. In our study, we allow a burn-in period of
439
50,000 samples to allow sample choices to become independent of the initial choices for
440
the model parameters, and collect 100,000 samples to approximate the posterior pdf,
441
P(m|x).
442
443
Modeling Raplee Ridge Monocline:
444
We used Poly3D to calculate bedding-plane deflections due to slip along a blind
445
reverse fault for two scenarios—one in which only the most well-exposed bedding
446
surface (McKim surface) was used to compare observed and modeled fold geometry, and
447
a second in which all mapped bedding surfaces were used to define the fold’s geometry
448
(Fig. 5). The first of these two scenarios is similar to the inversion presented in Mynatt et
449
al. {Figure 7; \Mynatt, 2007 #558}, excepting the fact that in this study, the misfit is
450
normalized by the number of degrees of freedom in the model. This difference does not
451
impact the values of the model parameters that best match the observed elevations;
16
452
however, the uncertainties associated with each model parameter tend to be larger (and
453
likely more realistic) than those presented in Mynatt et al. {Mynatt, 2007 #558}.
454
Elevation estimates along the mapped surfaces are available every square meter
455
using the high-resolution ALSM data. For expediency, we consider only a subset of
456
points separated by a minimum distance of 50 m when inverting for the fold’s geometry.
457
This reduces the number of computations necessary by a factor of ~2000.
458
Metropolis-Hastings inversions described above require several weeks to ~1.5 months of
459
compute time using this low-resolution dataset, and so inversions that use the full-
460
resolution data would be infeasible given today’s computing technology.
The
461
The mapped surfaces are not exposed within the forelimb syncline of the fold.
462
Thus, the observations of the mapped bedding surfaces permit a fold geometry in which
463
layers continue to dip steeply westward at great distances west of the extent of exposure
464
of these surfaces (Fig. 2b). However, surfaces exposed along unmapped higher bedding
465
surfaces within the stratigraphic section show sub-horizontal orientations, indicating that
466
the mapped units share a similar orientation in the subsurface. To account for these
467
observations, we used two different approaches.
468
considered only the observed geometry of the McKim surface, we assumed that the
469
stratigraphic thickness between the exposed units and the McKim bedding plane surface
470
within the fold’s eastern portion was similar to that observed in locations far from the
471
fold-related deformation. This thickness was used to infer the depth at which the McKim
472
surface should be located in the subsurface. These inferred (x,y,z) subsurface locations
473
were used when inverting for the fold’s geometry. In the case where we considered all
474
mapped layers in the inversion, we instead used a rotation constraint within the Halgaito
475
Tongue member, which is well exposed throughout the area surrounding the fold. In this
476
way, rather than assign subsurface locations for which the (x,y,z) locations of each of the
477
bedding planes is located, we instead required areas at the stratigraphic level of the
478
Halgaito Tongue member to experience no rotation (right-hand term in Eq. 1). This has a
479
similar effect of ensuring that the geometry of the fold is finite in areas to its west.
480
481
Results:
17
In the case of the models that
482
We first present the results of our fold inversion when only comparing elevation
483
values measured along the McKim surface to elevations of the deflected surface that were
484
predicted by Poly3D. The best-fit model (Fig. 6) shows that the observed fold geometry
485
is most consistent with a fault whose width is almost three times its down-dip length
486
(best-fitting model parameter values shown as italic numbers in each model parameter
487
panel of Fig. 7). In addition, the steep forelimb dips indicate that the tip-line of the fault
488
is likely only several hundred meters below the current surface in the fold’s center. The
489
strike of the underlying fault is approximately 3° east of north, consistent with the
490
roughly N-S trend of the observed topography. In addition, the inferred fault underneath
491
the fold is inferred to dip steeply (~47°) to the east. The Poisson’s ratio of the sediments
492
overlying the fold inferred by the model (0.28) is close to that typically assumed for
493
crustal rocks . Finally, the regional contraction in the E-W direction (xx) is expect to be
494
~3 times that in the N-S direction (yy), and is ~20 times larger than the regional shear
495
strain (xy). The absolute value of E-W strain (xx) for the best-fitting model was –0.43.
496
The best-fitting model produced surface elevations consistent with those
497
measured (Figure 6a). The quality of the fit is seen on a graph of observed versus
498
predicted values (Figure 6b) at points where elevations were extracted from the ALSM
499
data (locations of points shown in Figure 6a). The inferred subsurface points along the
500
western portion of the fold were assigned a constant value, which appear as a vertical line
501
of points at low elevations in Figure 6b. In contrast, calculated deflections vary smoothly
502
across these points, creating a mismatch between the inferred subsurface elevations and
503
those predicted by Poly3D. The residuals (defined as the modeled elevations minus the
504
observed elevations) show approximately zero mean.
505
systematically underestimated points at the crest of the fold, which skewed residuals
506
negatively (Fig. 6c). The best-fitting model yielded a root-mean-squared error of 16.1 m.
507
This value is significantly greater than the uncertainty in the elevation observations,
508
indicating that that mismatch between elevations predicted by the linear-elastic half-space
509
model and those observed is far larger than the uncertainties in elevation that we
510
measured using the ALSM. Indeed, given the high accuracy of the ALSM elevations (<
511
2-4 m), it is difficult to conceive of an idealized model (of any rheology) whose average
512
misfit might be on the order of, or less than this value. Thus, the high accuracy of our
18
In addition, the model
513
data relative to that expected from our idealized model forces us to conceptualize obs in
514
Eq. 1 as the inaccuracy that is produced by the strict model assumptions of Poly3D. In
515
this context, we adjust obs in Eq. 1 to require that r2 = 1 for the best-fitting set of model
516
parameters. As we see below, this choice for obs increases the variance of the model
517
parameters to a range that takes into account the uncertainty of using a linear elastic
518
model to calculate deflections in a material that might be better characterized by a more
519
complicated, anisotropic, and/or spatially variable rheology.
520
We used the Metropolis-Hastings MCMC method to calculate the posterior
521
probability, P(m|x), of each of the model parameters for which the inversion was
522
performed (Fig. 7). The joint posterior pdf is a nine-dimensional probability distribution,
523
with each of its axes representing the nine model parameters required for the calculation
524
of bedding-plane deflections. Such a distribution is difficult to visualize, and so we
525
present the marginal posterior pdfs by collapsing all dimensions of the joint pdf excepting
526
that of the model parameter of interest onto the dimension of this model parameter (Fig.
527
7). The resulting pdf shows the variability that characterizes each model parameter, but
528
does not capture the covariation of the different model parameters with one another. For
529
each of the model parameters, we report the value for the best-fitting scenario as italic
530
numbers, the mean of the simulated posterior pdfs as bold numbers, and the 95% range in
531
the simulated model parameter pdfs in parentheses in each of the panels. The simulations
532
reveal substantial variation in the model parameters such that a variety of their values
533
may produce similarly good fits to the observed elevations (Fig. 7). For many of the
534
model parameters, the mean values are similar to the best-fitting values, although
535
agreement is substantially less for those distributions that are highly skewed or uncertain
536
(such as , xx/xy and xx/yy). Nonetheless, the model places broad bounds on fault
537
geometries that may plausibly create the observed deflections. In some cases relatively
538
large values of Poisson’s ratio (0.23-0.49) are allowed, which implies that the single-
539
layer models favor a rheology that undergoes minimum volume change during
540
deformation. Considering the range of values produced by the simulation, the fault tip at
541
the center of the fold is expected to be only several km below the surface at the time of
542
deformation, and is currently only several hundred meters below the level of exposure.
19
543
Next, we performed our inversions using all five of the mapped layers that define
544
the geometry of the Raplee Ridge Monocline.
The best-fitting multi-layer model
545
elevations were similar to those observed (Fig. 8-10). The extensive exposure of the
546
McKim, Goodrich, and Shafer surfaces causes the model to most accurately characterize
547
its geometry (Fig. 9). Surfaces lower in the stratigraphy are not fit as well due to the fact
548
that the number of points extracted along these surfaces are fewer, and consequently their
549
weight is less in the inversion. This effect is most pronounced for the Mendenhall and
550
Unnamed surfaces, both of which show larger misfit than their counterparts higher in the
551
stratigraphic section (Figs. 9 and 10).
552
elevations for the multilayer models (Fig. 10), a slight but systematic underestimation of
553
elevations within the McKim and Goodrich surfaces are observed for elevations of ~1600
554
m. These points flank the upper-most crest of the fold along the McKim surface, and
555
their misfit arises because the fold is more cylindrical at its crest than might be expected
556
for the simple elliptical fault geometry assumed in this study. Nonetheless, the best-
557
fitting model geometry produces generally unbiased estimates of the elevations across all
558
surfaces (Fig. 10): model residuals for the McKim, Goodrich, Shafer, Mendenhall, and
559
Unnamed surface have means of -1.0, 1.2, -5.3, 1.0, and 4.6 m, respectively, with a mean
560
residual of 7.6 x 10-15 m when considering all points from all layers. The variation within
561
residuals was similar across the McKim, Goodrich, and Shafer surfaces, with standard
562
deviations equaling 16.9, 15.9 and 16.8 m, respectively. However, the fewer points
563
extracted along the less-well-exposed Mendenhall and Unnamed surfaces resulted in
564
higher standard deviations of 25.0 and 20.7 m, respectively.
When comparing observed versus predicted
565
The best-fitting model parameters deduced when using all of the mapped layers
566
(Fig. 8) shared affinity to those derived using only the McKim surface (Fig. 9 and 10).
567
The dimensions of the fold were similar, although the underlying fault was slightly wider
568
in the along-strike direction when using all mapped layers (Fig. 11, italic numbers in each
569
model parameter panel). In addition, the best-fitting model that uses all of the layers
570
favors a deeper fault tip-line at the center of the fold in comparison to the single layer
571
models. The root-mean-squared error for the best-fitting model was 18.1 m, slightly
572
larger than the single layer model.
20
573
The joint posterior distribution of model parameters estimated using the
574
Metropolis-Hasting sampler permitted sets of fault geometries and loading conditions that
575
were generally consistent with those derived from the single-layer models (Fig. 11).
576
However, inversions that utilized observations from multiple stratigraphic layers
577
generally tended to permit a wider range of model parameters than did single-layer
578
inversions (Figs. 7 and 11). Mean values of the calculated posterior pdfs were similar to
579
those calculated using the best-fitting model. As with the best-fitting modeled fault
580
depth, the range of fault depths allowable using the multilayer inversion tended to be
581
deeper than those predicted using only data from the McKim surface. Unlike single-layer
582
model results, values of Poisson’s Ratio permitted by the multilayer models spanned the
583
range from 0.03-0.48, indicating that this parameter may not be well resolved by such an
584
inversion. Nonetheless, both sets of models predict that the underlying fault dips steeply
585
to the east (Fig. 11).
586
587
Discussion:
588
This study builds on our previous work, in which we used the elevations extracted
589
from the McKim limestone in combination with Poly3D to interpolate the geometry of
590
the this surface and infer the fault geometry and regional strain . The single-layer models
591
created as part of the current study are similar to those reported in our previous work with
592
two important differences. First, our previous work fixed Poisson’s ratio to a value of
593
0.25 when determining the best-fitting and posterior probability densities of the model
594
parameters, while in the current study, we treat this ratio as a free parameter in the
595
inversion. By allowing Poisson’s Ratio to change in our models, the downdip height of
596
the fault decreased, while the depth to its upper edge increased. In addition, the larger
597
best-fitting value for Poisson’s Ratio is associated with a more steeply dipping fault .
598
Second, our previous work did not consider the number of degrees of freedom when
599
estimating P(x|m).
600
distributions to have less variance than those determined in this study. Given the large
601
uncertainties associated with the model itself (discussed below), the higher variance
602
distributions reported in this study likely provide a more realistic estimate of the
603
uncertainties in model parameters than those reported previously .
This had the effect of causing the model parameter posterior
21
604
In addition to expanding our inversion methodology to accommodate
605
observations of bedding rotations and deflections of multiple layers, we explored the
606
impact that spatial resolution and precision of different sources of elevation data might
607
have on our inversions. The standard deviation of the difference between the NED-10 and
608
NED-30 elevations when compared to the ALSM data was found to be ~25% of the RMS
609
error produced by our best-fitting modeled elevations. Thus, given the uncertainties in
610
applying such an idealized model to a complex fold, the NED-10 or NED-30 datasets
611
would have likely been adequate for this analysis. We did find the ALSM data to be
612
invaluable when identifying the extent of the bedding surfaces in the elevation data and
613
the field. In fact, many outcrops that define the fold’s geometry are not visible when
614
using the NED-10 or NED-30 datasets (Fig. 3). Thus, while the precise ALSM mapping
615
may not have increased the precision of elevation measurements used as input for our
616
inversions, it aided in the identification of those areas from which such elevation
617
measurements were extracted.
618
Our methods provide a new, mechanically based way of interpolating the
619
geometry of three-dimensional fault-related folds and imaging the geometry of the
620
underlying faults. As such, our results must be viewed in the context of the simplifying
621
assumptions of the mechanical models upon which our approach is based. In particular,
622
Poly3D idealizes the crust as a homogeneous linear-elastic half-space into which
623
elements that may accommodate slip and opening are embedded. In reality, the rocks
624
currently exposed within the Raplee monocline likely deformed anisotropically,
625
elastically or visco-plastically. In addition, our study infers the geometry of a fault that
626
has not been imaged in the subsurface. In spite of these simplifying assumptions, our
627
model predicts fold shapes consistent with elevations of specific bedding surfaces
628
observed using the ALSM data. To validate such an approach, similar analyses should be
629
performed in areas where subsurface imaging allows comparison of these imaged
630
geometries with those modeled based on surface data.
631
The success of Poly3D in accurately depicting the geometry of Raplee Ridge
632
Monocline indicates either that the approximation of linear elasticity serves as an
633
adequate characterization of the rheology of the folded materials or that different
634
rheological properties may produce similar observed fold geometries by allowing the
22
635
underlying geometry of the slipping fault to change as well. While our Metropolis-
636
Hastings sampler provides estimates of the uncertainties within model parameters that
637
arise from our characterization of the uncertainties in the data, the uncertainties
638
associated with the inappropriateness of the simplified rheology in our model is not
639
captured by our approach. In the future, numerical experiments using finite element
640
models that can consider more complicated rheologies may be used to assess the trade-
641
offs between the loading conditions (represented by the far-field strain conditions and the
642
underlying fault geometry), rheology, and observed surface deflections.
643
from a suite of different rheologies analyzed using the methods presented in this work
644
may be compared with one another to assess the degree of uncertainty that is associated
645
with the choice of the rheology of the folded material. This choice may impact the
646
parameters such as fault geometry that are inferred by our approach, and so by having
647
independent estimates of these values, it might be possible to discern the most
648
appropriate rheological model for these types of folds.
The results
649
650
Conclusions:
651
We present a method that uses the elevations of surface exposures of bedding
652
surfaces of fault-related folds to infer the underlying fault geometry, loading conditions,
653
and Poisson’s Ratio of the folded material, assuming that deformation is due to the
654
release of stress along a frictionless elliptical fault embedded in a homogeneous linear-
655
elastic half-space. This method may be applied to situations in which a single bedding
656
surface is exposed, or multiple stratigraphic surfaces have been deformed and exhumed.
657
In addition to bedding-plane elevations, the orientation of units at various locations in a
658
particular part of the stratigraphic section may be used to further constrain the geometry
659
of the fold and underlying fault. Using a Bayesian Markov-Chain Monte Carlo method,
660
we determined the fault geometry, loading conditions, and elastic parameters that best
661
explain the observed surface deflections, as well as the variation within and covariation
662
between these model parameters. This allowed us to understand the degree to which the
663
data constrain model parameters and the various combinations of model parameters that
664
may produce the observed surface deflections.
23
665
We apply these methods to the Raplee Ridge Monocline, located in southeastern
666
Utah, where high-resolution ALSM data define the geometry of exposed bedding
667
surfaces. While we found that the resolution and precision of the ALSM data are
668
unnecessary for inferring the fault geometry and loading conditions using our approach,
669
these data greatly aid the mapping of the spatial distribution of surface outcrops. Both
670
single-layer and multi-layer inversions agree remarkably well with observations, and
671
image a fault that is broadly constrained to be ~4.5-14 km in downdip height, 13-30 km
672
in along-strike width, and whose tipline was 2.0-9.5 km below the surface at the time of
673
deformation.
674
consistency of our simplified (and likely unrealistic) rheology of the folded material with
675
the observed geometry might suggest that surface deflections may reveal little about the
676
rheology of the fold when viewed in isolation from factors such as fault geometry and
677
loading conditions.
The Poisson’s Ratio was not well resolved by the inversion.
678
24
The
679
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680
681
682
683
684
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Figure Captions:
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Figure 1: Shaded relief map (color-coded for elevation), showing the location of Raplee
784
Monocline in southwestern Utah. Upper left inset shows Four Corners area; location of
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Figure 2 noted in location map.
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Figure 2: The topography of Raplee Ridge, as imaged by the (a) 1-m ALSM-derived
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data, (b) NED-10 data, and (c), NED-30 data.
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elevation and map-view scale. (d) Shaded relief map showing the mapped extent of the
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McKim surface used in single-layer inversions. The location of (x,y,z) points extracted
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and used to compare accuracy of different datasets are shown as red dots. Note that the
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mapped surface constrains the shape of the fold at a particular stratigraphic depth.
All elevation maps have the same
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Figure 3: Color-coded, shaded-relief DEM of the center of Raplee Ridge, showing the
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detailed differences in resolution between the datasets. (a) is the ALSM-derived DEM,
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(b) is the NED-10 DEM, and (c) is the NED-30 DEM. Color and map-view scales for all
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panels are identical.
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Figure 4: Comparison of elevations derived from ALSM data with those derived from
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(a) the NED-30 DEM, and (c) the NED-10 DEM. The residual, or difference between the
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NED-30 (b) or NED-10 (d) and ALSM data, are shown as histograms, with the mean and
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standard deviations of these distributions noted in each panel.
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Figure 5: Schematic model setup, showing the definition of each of the parameters used
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to define the fault geometry and loading conditions in the Poly3D model.
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parameters shown in figure are defined in text.
Model
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Figure 6: (a) Observed (filled circles) and predicted (shaded background) elevations of
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the McKim surface that are produced as an initially flat layer is deflected by movement
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along the best-fitting modeled fault geometry that slips in response to the best-fitting
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remote loading conditions. (b) Plot of observed versus predicted elevations and (c)
29
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histogram of residual elevations (observed elevations minus best-fitting modeled
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elevations) that are produced by the best-fitting modeled fault geometry and loading
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conditions.
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Figure 7: Posterior probability density functions of each of the model parameters for
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single layer (McKim surface only) model inversions. In each panel, the solid line denotes
818
the probability density of each model parameter given the elevations observed along the
819
McKim surface, while the dashed lines show the prior probability (assumed uniform in
820
this study) assumed for the model parameters. Numbers in italics represent the value of
821
each parameter for the best-fitting model, while solid bold numbers represent the mean
822
value produced by all simulations. The range shown in parentheses represents the 95%
823
bounds on the model parameter values determined from the posterior probability density
824
function.
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Figure 8: Shaded relief map showing the spatial distribution of the five bedding-plane
827
surfaces exhumed within the fold. From stratigraphically highest to lowest, they are the
828
McKim, Goodrich, Shafer, Mendenhall, and Unnamed surfaces.
829
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Figure 9: Observed (filled circles) and predicted (shaded background) elevations of the
831
McKim, Goodrich, Shafer, Mendenhall, and Unnamed surfaces produced by movement
832
along the best-fitting modeled fault geometry that slips in response to the best-fitting
833
remote loading conditions. Color and map-view scale are identical between plots. The
834
inferred depth below the surface of each of these layers at the time of deformation is
835
noted in each of the panels.
836
837
Figure 10: Observed and modeled elevations (left panels) and histogram of residual
838
elevations (defined in Fig. 6) for best-fitting fault geometry and loading conditions, when
839
considering multiple layers in the model inversion. Observed vs. predicted elevations
840
and residual histograms shown for the (a) McKim, (b) Goodrich, (c) Shafer, (d)
841
Mendenhall, and (e) Unnamed surfaces.
842
30
843
Figure 11: Posterior probability density functions of each of the model parameters for
844
multiple layer inversions. In each panel, the solid line denotes the probability density of
845
each model parameter given the elevations observed along the McKim surface, while the
846
dashed lines show the prior probability (assumed uniform in this study) assumed for the
847
model parameters. Numbers in italics represent the value of each parameter for the best-
848
fitting model, while solid bold numbers represent the mean value produced by all
849
simulations. The range shown in parentheses represents the 95% bounds on the model
850
parameter values determined from the posterior probability density function.
851
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31
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Figures:
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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7
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Figure 8
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Figure 9
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Figure 10
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Figure 11
42