Growth of fault-cored anticlines by combined mechanisms of fault slip and buckling

Transcription

1 Growth of fault-cored anticlines by combined mechanisms of fault slip and buckling Wen-Jeng Huang* Kaj M. Johnson Department of Geological Sciences Indiana University Wen-Jeng Huang Department of Geological Sciences, Indiana University, USA 1001 E. 10 th St., Bloomington, IN Work telephone: huang22@indiana.edu Kaj M. Johnson Department of Geological Sciences, Indiana University, USA 1001 E. 10 th St., Bloomington, IN Work telephone: kajjohns@indiana.edu * Corresponding author, Phone: huang22@indiana.edu Index term: 8005 Folds and folding 8020 Mechanics, theory and modeling 8108 Continental tectonics: compressional Keywords: Folding, earthquake hazard, anticlines, fault-related fold, blind fault, fault slip rate, active fold. 1

2 ABSTRACT A primary goal of studies of blind faults underlying actively growing anticlines is assessment of earthquake hazard associated with slip on the faults. Previous studies suggest that folds grow primarily during and immediately after earthquakes by slip on the fault. In this paper we show that anticlines growing over slipping reverse faults can be significantly amplified by buckling of mechanical layering under horizontal shortening. Studies that assume folds grow solely by slip on a fault in an elastic half space may significantly overestimate fault slip. We construct boundary element models to demonstrate that the folds in a medium containing a fault and elastic layers with free slip and subjected to horizontal shortening can grow to more than twice the amplitude of folds produced in homogeneous media without mechanical layering under the same amount of shortening. The wavelengths of buckle folds can be as short as half the wavelength of folds produced by fault slip in an elastic half space. The geometry of Pitchfork Anticline on the western flank of the Big Horn Basin in Wyoming displays geometric features such as a localized anticlinal dome shape with tight hinges and amplitude that increases away from the fault tip that are characteristic features of buckle folds produced in our numerical simulations. The coseismic uplift pattern produced during a 1985 earthquake under the Kettleman Hills Anticline in central California and subsurface fold geometry inferred from seismic reflection images are consistent with folding produced by the combined mechanisms of fault slip and buckling INTRODUCTION Geodetic observations have been interpreted as evidence that anticlines over blind faults grow as a consequence of slip on faults during earthquakes. King and Stein [1983], Stein and King [1984], and Stein and Ekström [1992] proposed that a 2

3 string of Quaternary folds in central California at Coalinga and Kettleman Hills are produced by sudden, incremental growth during repeated earthquakes on underlying blind reverse faults. This idea is leveraged by observations at Coalinga Anticline that show a similarity in pattern between uplifted river terraces, current topography, and coseismic vertical displacements determined from leveling measurements before and after the 1983 Coalinga earthquake. Stein and Ekström [1992] inferred a fault slip rate for the blind fault underlying Coalinga assuming that the uplift rate of the fold directly reflects the slip rate on the fault This view that anticlines grow primarily by slip on the underlying fault is reflected in other studies of fault related folding. For example, Myers et al. (2003) use the solution for a dislocation in an elastic half-space to infer fault geometry and amount of slip on blind faults underlying active folds in the eastern Los Angeles basin. Mynatt et al. (2007) used fold geometry and a boundary element model of faulting in an elastic half-space to infer the fault geometry and loading conditions responsible for the development of Raplee Ridge, an anticlinal fold in southeastern Utah that likely formed by slip on an underlying fault during Laramide contraction (in late Cretaceous and early Tertiary time) A growing number of studies attempt to link active growth of anticlines with slip on an underlying fault. The approach in these studies is to determine the geometry and uplift rate of active fault-related folds and then infer the fault slip rate from an assumed kinematic relationship between fault slip and fold shape [e.g., Suppe, 1983]. For example, Grant et al. [1999] used dates from uplifted marine terraces in the San Joaquin Hills, California to infer slip rate on underlying blind reverse faults, assuming that the uplift rates are directly related to the total slip rate through the sine of the dip 3

4 angles of the faults. Ishiyama et al. [2004] mapped and dated uplifted alluvial terraces on the active Kuwana anticline in central Japan to obtain estimates of uplift rate. They assumed a kinematic relationship between the uplifted terrace surfaces and an underlying fault to infer the fault slip rate. Shaw and Shearer [1999] and Shaw et al. [2002] used the method of Shaw and Suppe [1996], based on kinematic constructions of fold-growth structures, to infer fault slip rates and to assess size and frequency of future earthquakes on the Puente Hills blind thrust fault, which ruptured in the 1987 Whittier Narrows earthquake under metropolitan Los Angeles [e.g., Lin and Stein, 1989; Dolan et al., 2003; Leon et al., 2007]. Allmendinger and Shaw [2000] used the trishear kinematic model to infer the total amount of fault slip on the thrust underlying the Puente Hills The main message of this paper is that the geometry of anticlines produced solely by slip on underlying reverse faults is significantly different from the geometry of anticlines produced by the combined mechanisms of fault slip and buckling of mechanical layers under horizontal shortening. Buckling of layers has the effect of amplifying and narrowing the fold produced by slip on the underlying fault. Figure 1 illustrates the buckling mechanism. The folds in Figure 1 are produced using the boundary element model described later in this paper. Folds in Figure 1a-c are produced in elastic layers embedded in an elastic half-space which are slide without resistance along contacts. Folds in Figure 1d-f are produced in passive markers in a homogeneous elastic half-space (no mechanical layers). The fault-related fold in Figure 1a was produced in a stack of layers overlying an inclined fault and subjected to 21.5% uniform far-field horizontal shortening. The maximum fault slip is 0.3 W, where W is initial fault width. The fold in Figure 1b was produced by imposing the fault slip in Figure 1a on a fault underlying the stack of layers without far-field 4

5 shortening. The fold in Figure 1c was produced by subjecting a small initial perturbation in the layering to the same amount of shortening as in Figure 1a, but without a fault. Figures 1d-1f illustrate the same loading conditions but for a medium without mechanical layering. A comparison of Figures 1b and 1e illuminates the influence of flexural slip on fold form. The fold in the mechanical layering is higher amplitude and more localized than the fold in passive markers. A comparison of Figures 1c and 1f illustrates the influence of buckling on fold form. The fold in Figure 1c is generated by buckling of layers with a small initial perturbation under horizontal shortening. The passive markers in Figure 1f are given the same initial perturbation and subjected to the same amount of shortening, but there is no visible fold. Finally, a comparison of Figures 1a and 1d demonstrates the effect of combined mechanisms of buckling and fault slip on fold form. The effect of buckling is to amplify and narrow the fold over the tip of the blind fault To date, mechanical analyses of fault-related folding have largely ignored the buckling process because either the mechanical layers required for buckle folding are absent in such analyses or passive folding of layers due to slip on a fault is considered without horizontal shortening. Perhaps the influence of buckling in fault-related folds has been largely ignored because previous studies of the mechanics of buckle folding have focused on the formation of repetitive fold forms in layered media [e.g., Johnson and Fletcher, 1994], which are rarely observed in sedimentary rocks. Yet, field observations [e.g., Erslev and Mayborn, 1997] shows that layer parallel slip appears ubiquitously in fault-cored anticlines, and therefore the role buckling accommodated by slip at layer contacts should not be neglected in analyses of fault-related folding. 5

6 We develop a boundary element model of the growth of an anticline over a fault embedded in a medium with elastic layers that slip at the contacts. We demonstrate that fault-cored folds in a mechanically layered medium can be significantly amplified and localized by buckling under horizontal compression. To demonstrate this point, we examine the subsurface geometry of the Pitchfork Anticline, a Laramide-aged (in late Cretaceous and early Tertiary time) anticline on the western flank of the Big Horn Basin of Wyoming, which exhibits features that are consistent with buckle folding. We also examine the subsurface geometry and surface deformation measurements from the active Kettleman Hills and Coalinga Anticlines in central California KINEMATICS AND MECHANICS OF FAULT-CORED ANTICLINES In this paper, we focus our attention on anticlines that form above buried reverse faults. Three classes of fault-related folds that are most relevant to this particular geometry are fault-bend folds, fault-tip/fault-propagation folds and forcedfolds. Fault-bend folds form when rock moves through a flat-ramp-flat fault geometry and generates repetition of the section and a ramp anticline [Rich, 1934 and Suppe, 1983]. Fault propagation and fault-tip folds form by shortening and shear generated at the terminations of propagating or non-propagating reverse faults. Forced folds form in a medium overlying displaced rigid basement blocks. A number of kinematic models, similar to the Suppe [1983] fault-bend-fold model with straight limbs and sharp hinges, have been constructed to capture the basic geometry of fault-tip and fault-propagation folds [e.g., Suppe and Medwedeff, 1994; Chester and Chester, 1990]. The trishear kinematic model [e.g. Erslev, 1991; Cardozo, 2008] is particularly popular in the recent literature because it produces rounded fold forms that look more like natural folds and has been used to model forced-folds and fault-tip folds. 6

7 Mechanical theories for ramp folding in homogenous linear elastic or viscous media have been developed by Elliot [1976], Wiltschko [1979], Berger and Johnson [1980, 1982], Johnson and Berger [1989], Kilsdonk and Fletcher [1989], Johnson and Fletcher [1994] and Savage and Cooke [2003]. Some mechanical models have been developed for forced-folds [Sanford, 1959; Reches and Johnson, 1978; Patton and Fletcher, 1995; Johnson and Johnson, 2000]. Discrete element modeling is becoming increasingly popular in the mechanical analysis of ramp folds [Strayer and Suppe, 2002] and fault-tip and forced folds [Finch et al., 2003; Cardozo et al., 2005]. Fault-related folding in an elasto-plastic medium has been examined by Cardozo et al. [2003]. Viscous-plastic finite-element models are used to analyze fault-bend folds by Erickson and Jamison [1995]. Each of these theoretical models investigates the passive folding of markers in materials of various rheology in response to slip on a fault. Boundary element models were developed by Cooke and Pollard [1997] and Shackleton and Cooke [2007] to analyze the contribution of frictional slip along bedding planes to fault-related folding of layered rocks. The former mainly investigated the deformation of frictional bedding planes near dipping faults under layer-parallel contraction an extension. The latter focused on evaluating the validity of the plane strain assumption in non-cylindrical folds While little attention has been afforded to the mechanics of buckling of faultrelated folds, the theory of folding of initial perturbations in isolated layers or multilayers without faulting is quite mature [e.g., Biot, 1964a and 1964b; Chappel, 1969; Fletcher, 1977; Johnson, 1977; Johnson and Fletcher, 1994; Mancktelow, 1999]. Of particular relevance to this paper are theoretical studies of the physical conditions of multilayer folding that lead to significant amplification of initially small perturbations. In linear, homogenous materials, the rate at which an initial 7

8 perturbation is amplified is a function of the number of layers in the multilayer, N, the thickness of individual layers, h, and the wavelength, L, of the initial perturbation. The rate at which initial sinusoidal perturbations are amplified by buckling under horizontal compression was quantified by Biot [1961] and Fletcher [1977] as the amplification factor. The amplification factor is a scalar quantity that determines the rate at which the amplitude of an initially small perturbation grows with increased shortening of the medium [e.g., Johnson and Fletcher, 1994] Figure 2 produced from the folding theory developed by Johnson and Pfaff [1989] shows the amplification factor as a function of the wavelength of the perturbation normalized by the thickness of a single layer. The layers have viscosity equal to the surrounding media and free slip at layer contacts. The amplification factor is shown for multilayers with two, four, or ten layers. Figure 2 illustrates that the amplification factor (i.e., the rate at which the amplitude grows) increases with the number of layers in the multilayer. Also, for a given layer thickness and number of layers, there is a so-called dominant wavelength at which the amplification factor is largest and the fold grows the fastest (the peak of the curves). Thus, Figure 2 demonstrates the rather intuitive result that very broad or very narrow initial perturbations, relative to the thickness of the layers, grow in amplitude more slowly than perturbations with a dominant wavelength, and perturbations in a multilayer with many thin layers grow in amplitude more quickly than in a multilayer of the same total thickness but composed of a few thick layers The concept of amplification factor is relevant in a general way to fault-cored folding in a multilayer under layer-parallel shortening. In this case, we expect the length of the perturbation produced by slip on the fault would be controlled by the 8

9 geometry of the fault. We would expect the rate of growth of the fault-cored fold to be a function of the shortening rate, the rate of slip on the fault, and the layer thickness and number of layers. A fault-cored fold in a medium with no mechanical layering is expected to grow more slowly than a fold overlying a fault in a medium with many mechanical layers BOUNDARY ELEMENT MODEL OF FAULT-RELATED FOLDING We develop a boundary element model to examine the amplification of faultrelated folds by buckling. The boundary element method (BEM) is different from the finite element method (FEM) in that the medium is discretized only at boundaries in the BEM whereas the entire medium is discretized in the FEM Basic Rationale In layered sedimentary rocks, mechanical interfaces between sedimentary layers may form because of differences in physical properties at the interfaces such as grain size and cementation. Soft layers interbedded with stiff layers may localize shear, allowing the stiff layers to slide past each other. These conditions are important in folding because the bedding-plane slip can allow the strata to mechanically buckle with flexural slip. We model these conditions with multiple elastic layers with frictionless contacts (Figure 3) We follow the displacement discontinuity method of Crouch and Starfield [1983] and discretize the fault surface and layer contacts using the solution for planestrain edge dislocations in an elastic half-space. The basic geometry and boundary conditions of two classes of models are illustrated in Figure 3a and Figure 3b. One model is a straight fault embedded in mechanical layers or the elastic medium under the layers. The layers and medium are subjected to horizontal shortening (Figure 3a). 9

10 The other model is a flat-ramp-flat shaped fault embedded in mechanical layers with a semi-infinite dislocation applied at the depth of the detachment at the edge of layer interfaces to approximate a uniform displacement condition of the hanging wall block above the detachment (Figure 3b). We model initially horizontal layers of finite length embedded in an otherwise homogeneous elastic half-space. In general, the layers and the fault are assumed to slip according to a Coulomb friction law, τ s C + µσ n, where τ s is shear stress, C is cohesion, µ is the coefficient of friction, and σ n is normal stress (compression is positive). However, in this paper we only examine frictionless, cohesionless contacts (C = µ = 0) because this special case simplifies the illustration of the effect of buckling on fold form. In both models we discretize the faults and layer interfaces into elements with equal length. We specify zero shear traction on all elements and zero discontinuity in normal displacement across elements. The zero shear traction condition allows the two faces of an element to slide past each other without resistance and the zero normal displacement discontinuity condition prevents faces from pulling apart or overlapping. The evolution of folding is computed by small increments of elastic deformation. During each increment of deformation, far-field strain or slip on the detachment is imposed, slip on all interfaces is computed, and new positions of layer interface and fault elements are computed The conditions for the two basic models are as follows: (1) All elements of fault and layer interfaces are specified to have zero shear traction, σ s, and zero normal displacement discontinuity. (2) The elastic medium is semi-infinite (i.e., half-space) such that the normal traction, σ n, and shear traction, σ s, are zero at the ground surface. (3) The y-coordinate of the origin of the coordinate system is fixed at the free surface and the free surface remains flat (material points that move above the free surface 10

11 disappear). (4) The elastic properties of the medium and the layers are the same. Poisson s ratio, ν, is assumed to be (5) In the model of a straight fault embedded in mechanical layers or the elastic medium under the layers, the length of the fault and the length of layer interfaces are finite. The loading condition for each increment of deformation is ff ε xx = (6) In the model of a flat-ramp-flat shaped fault embedded in mechanical layers, the length of layer interfaces is finite. A semi-infinite dislocation is applied at the depth of the detachment at the edge of the layer interfaces. The imposed semi-infinite dislocation approximates a uniform displacement of the hanging wall block above the detachment. No far-field strain is imposed in this model It is important to recognize that we have adopted the linear (infinitesimal strain) elastic solution for an edge dislocation, yet we do not restrict our analysis to small strains. We assume that each increment of deformation can be modeled with the small strain theory, ignoring nonlinear effects due to the initial stress condition at the beginning of each increment. This is equivalent to assuming that the elastic stresses in the medium surrounding the faults and layer interfaces are somehow relaxed before the beginning of the next deformation increment, although we do not explicitly implement a mechanism for relaxing these stresses. Inelastic processes for relaxing stresses include: micro-cracking [e.g. Meglis et al., 1995], grain boundary sliding [e.g. Langdon, 1970], twinning [e.g, Yamashita and Ojima, 1968], pressure solution [e.g. McClay, 1977], recrystallization, and so on [Sibson, 1986]. Because we do not account for these processes in our model, results from this analysis must be viewed with mindfulness of the assumptions. Furthermore, we assume an incremental farfield strain of ff ε xx = in all the applications in this paper which is about an order of magnitude larger strain than permissible by linear elasticity theory. However, in the Appendix A2 we show that smaller increments of deformation do not produce an 11

12 appreciable difference in the final fold form indicating that our choice of incremental far-field strain is not a severe limitation. In Appendix A3 we present the basic formulation of our method and compare it with a popular two-dimensional displacement discontinuity method Crouch and Starfield [1983] We are ignoring the influence of frictional layer contacts on fold form. Pfaff [1986] and Pfaff and Johnson [1989] demonstrated that nonlinear resistance to contact slip, such as strength or powerlaw flow, has a significant influence on fold form. They showed that rounded concentric folds form in multilayers with free slip at contacts, and sharp-hinged, straight-limbed conjugate kink bands are produced with nonlinear viscous strength at layer contacts. Preliminary experiments with our elastic BEM folding models, with contact strength, do indeed produce kink-like fault-related folds Simulations We now show fold forms produced with different fault geometries. For each simulation, we compare folds produced in mechanical layering with folds produced in passive markers with no mechanical layering. We refer to the folds in mechanical layering as fault-cored buckle folds, or ramp-cored buckle folds, and we refer to folds in passive markers as fault-cored passive folds, or ramp-cored passive folds Fault in basement underlying sedimentary layering The setting for this case is a fault in a massive rock unit (basement) underlying layers of sedimentary rocks. An anticline forms above the upper tip of the fault in response to slip on the fault. The fault does not propagate. The conditions are similar to the conditions for forced-folding in that the fault is embedded in a relatively stiff 12

13 substrate without layering, overlain by a more compliant layered medium. We model this scenario with a fault embedded in an elastic half-space underlying a stack of elastic layers. The fault initially dips 25 degrees. Figure 4 shows results for two extreme conditions for the layer interfaces: bonded (interfaces are passive markers) or freely sliding. For the bonded case, the fault-cored passive folding is shown in the left column of Figure 4 and for the freely sliding case, the fault-cored buckle folding is shown in the right column of Figure 4. We show three stages of folding with maximum fault slip of 0W, 0.14W and 0.30W, where W is the initial fault width (down-dip distance) The distinct differences between the bonded and freely sliding layers are as follows: (1) The amplitudes of folded interfaces of freely sliding layers grow faster than the amplitudes of the folded passive markers. (2) After maximum fault slip of 0.30W, the fold in the mechanical layer model is nearly symmetric with tight hinges and curved limbs while the fold in the passive markers is broader. (3) The fold wavelength, measured as the distance between synclinal hinges on the flanks of the anticline, is shorter in the folded mechanical layers than in the folded passive markers. (4) The ratio of average amplitude of folded interfaces to maximum slip on the fault in the fault-cored buckle fold is 1.1 while the ratio for the fault-cored passive fold is Therefore, given the same amount of slip on the fault, the fold amplitude in the fault-cored buckle fold is about 5 times the amplitude of the fault-cored passive fold Fault embedded in layers Figure 5 shows a model in which an originally straight fault is embedded in the layering. The fault initially dips 25 degrees. We show three stages with maximum fault slip of 0W, 0.18W and 0.30W. The fault-cored buckle fold form is different from the previous case with the fault below the layers. The crest of the anticline of the 13

14 fault-cored buckle fold in Figure 5 forms over the midpoint of the underlying fault, rather than above the fault tip as in the fault-cored buckle fold in the previous model with the fault below the layers and in the fault-cored passive folds in Figures 4 and 5. The distinct differences between the passive and buckle folds in Figure 5 are as follows: (1) The amplitudes of the fault-cored buckle fold grow faster than the amplitudes of the fault-cored passive fold. (2) After the maximum fault slip of 0.30W, the fold in the fault-cored buckle fold is more highly localized with steeper limb dips than the fault-cored passive fold. (3) The buckle fold is nearly concentric while the passive is somewhat asymmetric with a short forelimb dipping to the left. (4) The fold wavelength, measured as the distance between synclinal hinges on the flanks of the anticline, is shorter in the fault-cored buckle fold than in the fault-cored passive fold. (5) The ratio of the average amplitude of folded interfaces to the maximum slip on the fault in the fault-cored buckle fold is 0.6 while the ratio for the fault-cored passive fold is 0.3. Thus, with the same amount of fault slip, the amplitude of the buckle fold is about 2 times the amplitude of the passive fold Ramp anticline The setting for this case is fault-bend folding over a flat-ramp-flat fault embedded in the layering. The lower flat coincides with the top of a massive rock unit underlying the layering. A constant displacement is applied to the right side of the model along the detachment. In the far-field, to the left, the displacements are zero, so horizontal compression is generated in the layering We show three stages in which the total displacement on the semi-infinite dislocation is 0, 0.38W and 0.71W, where W is the width of the ramp. The ramp initially dips 25. An anticline forms above the ramp in both the bonded and freely sliding cases. However, the geometry of the anticlines is distinctly different, namely: 14

15 (1) Like the previous two models, the amplitudes of ramp-cored buckle fold grow faster than the amplitudes of the ramp-cored passive fold. (2) After total hanging wall displacement of 0.71W, the folded interfaces in the ramp-cored buckle fold are boxlike with two localized shear bands with opposite-facing limbs above the upper and lower ends of the ramp. The localized folding of the forelimb and backlimb with nearly uniform limb dips closely resembles the geometry produced in passive layers by the ramp fold model for anisotropic materials [e.g., Erickson el at., 2001]. The folding somewhat resembles the angular fault-bend fold kinematic model [e.g., Suppe, 1983], however, the relatively flat crest of the anticline is tilted in this model in contrast with the horizontal crest assumed in the kinematic fault-bend fold model. In comparison, the ramp-cored passive fold is broad and gentle. (3) The fold wavelength, measured as the distance between synclinal hinges on the flanks of the anticline, is similar in the two models. (4) The ratio of the average amplitude of folded interfaces to the maximum slip on the fault in the ramp-cored buckle fold is 0.27 while the ratio in the ramp-cored passive fold is Thus, with the same amount of fault slip, the fold amplitude in the ramp-cored buckle fold is nearly twice the amplitude of the fold in the ramp-cored passive fold Influence of buckling on fold form We have demonstrated that fault-cored fold forms can be significantly influenced by fault geometry, properties of layer contacts, and loading conditions. Hereafter in this section, we will use the setting of a fault in basement underlying sedimentary layering to examine the effects of two factors on fold forms: the free ground surface and the thickness of layers. The series of models presented below illustrate two characteristic geometric features of fault-cored buckle folds produced near the ground surface that are distinctly different from features in fault-cored 15

16 passive folds. One feature is that the amplitude of fault-cored buckle folds increases away from the fault tip to the maximum value at the ground surface. The other feature is that the wavelength of fault-cored buckle folds is about equal to the length of the fault whereas the wavelength of fault-cored passive folds is larger than the length of the fault Effect of ground surface We plot the amplitudes of folded interfaces of the folds of Figure 4 with a maximum fault slip of 0.3 W in Figure 7 in addition to the amplitudes of folded interfaces of a buckle fold and a fault-cored passive fold in a full-space medium with the same setting. The amplitude at a layer horizon is measured as the vertical difference between the crest of the anticline and the lower synclinal hinge on one of flanks of the anticline. Figure 7 shows the amplitudes of the fault-cored buckle folds are larger than the fault-cored passive folds. Furthermore, the amplitude of the two fault-cored passive folds decrease upwards whereas the amplitude of the fault-cored buckle fold near the ground surface increases upward away from the fault. This is clearly an effect of the free surface. Figure 7 shows that the amplitude of a fault-cored buckle fold far below the free surface does not steadily increase upwards from the fault Effect of layer thickness A series of fault-cored folds with different layer thickness is shown in Figure 8. The thicknesses of mechanical layers vary but the total thickness of the entire stack of layers is the same in each fold. All folds in Figure 8 are formed by shortening until fault slip reaches a maximum of 0.18 W. The ratio of their amplitudes to the amplitudes of fault-cored passive fold with the same fault slip is plotted as a function of the number of layer interfaces in Figure 9a. Fold amplitude increases as the number 16

17 of layers increases. The amplitude of the top layer interface of the fault-cored buckle fold with 25 layer interfaces is nearly 6 times the amplitude of the fault-cored passive fold under the same amount of shortening. The wavelength of the fault-cored buckle folds is plotted as a function of the number of layer interfaces in Figure 9b. The wavelength decreases as the number of layer interfaces increases. The wavelength of the fault-cored buckle fold with 10 layers or more is less than half the wavelength of the fault-cored passive fold (i.e. n = 0) FAULT-CORED ANTICLINES We now examine data from fault-cored anticlines in the western Unites States. We examine a Laramide uplift in Wyoming with abundant subsurface data and an actively growing anticline in central California. 4.1 Pitchfork Anticline, Wyoming Pitchfork Anticline is one of many anticlines in the Bighorn Basin in Wyoming that formed during the Laramide orogeny, a time during which the Rocky Mountain foreland was partitioned into isolated structural basins by incipient Precambrian uplifts. The Bighorn Basin contains a number of small anticlines, several km long, distributed around the rim of the basin shown in Figure 10. The anticlines trend west to northwest and are generally asymmetric with underlying faults dipping to the east on the west side of the basin and to the west on the east side of the basin. Stone [1983a, 1983b] terms these anticlines thrust-fold structures, apparently implying a causative relation between a reverse fault and a fold. He concludes that they initiated by reverse slip in granitic, Precambrian basement and propagated into overlying Paleozoic carbonates and Mesozoic clastics. He notes that the amount of slip on the fault decreases but the amount of folding increases up section such that the 17

18 shortening accomplished by faulting alone in the basement is accomplished by folding alone in the Mesozoic clastics We will demonstrate that many of the fold features of the Pitchfork Anticline can be explained best by a model of amplification by buckling of a fold ahead of a buried fault tip in a multilayer Form of the Anticline Figure 11 is the most nearly accurate and deepest reconstruction of a profile through Pitchfork Anticline. The geometry has been carefully controlled by surface mapping, borehole logs, and a seismic profile by Durdella [2001]. The solid lines show contacts that are considered to be known. The dashed lines in the top of the diagram extrapolate the positions of formation contacts and the ground surface at the time of folding. The dashed lines on either side of the anticline are also extrapolated positions of formation boundaries. The dashed lines in the faulted core are also extrapolated formation contacts Two of the geometric features of the anticline that Durdella [2001, p. 37] emphasized are that the limbs are rounded rather than planar and that the stratigraphic throw in the anticline increases by about 50% from the top of the Precambrian basement to the top of Cretaceous units near the current ground surface. The stratigraphic throw of a unit is measured as the vertical difference between the lowest position of the unit in the forelimb of Pitchfork anticline and the highest position of the same unit on the crest of the anticline Previous studies of the structure of the Pitchfork Anticline focused on how the fold should be classified and on speculation about formation scenarios of the structure. Petersen [1983] incorporated borehole data in order to classify the Pitchfork Anticline 18

19 and termed it a foreland detachment structure. Stone [1993] used a combination of surface, well and seismic data to construct a cross section of the fold and classified the Pitchfork Anticline as a thrust-fold. Chester and Chester [1990] devised a kinematic fault-propagation fold model of the anticline Mechanical analysis Durdella [2001] showed that mechanical models of a reverse fault in a medium with passive layering can not reproduce the primary geometric features of Pitchfork Anticline. One model considered by Durdella [2001] was based on a single, listric fault in an elastic half space with imposed slip beneath passive layering, unaccompanied by shortening. It was unsuccessful in producing even a rudimentary fold of the shape of Pitchfork Anticline. Figure 12b is a fold produced under similar conditions but using the mechanical model illustrated in Figure 4 in which passive markers above a buried fault are folded in a homogeneous elastic medium subjected to 21.5% shortening. The induced slip on the fault is similar to that inferred for the fault under Pitchfork Anticline. However, the amplitude of the model fold is much lower and the fold is much broader than the Pitchfork Anticline. Furthermore, the amplitude of the model fold increases with depth while the amplitude decreases with depth in the Pitchfork Anticline (Figure 12b) We examine the possibility that the localized fold form of Pitchfork Anticline is a result of buckle-folding under horizontal compression. We simulate Pitchfork Anticline with our fault-cored buckle fold boundary element model with a basement fault underlying a stack of mechanical layers. Figure 12c shows that the resulting fold form reproduces many of the features observed in the Pitchfork Anticline. The fold is similarly localized, the anticlinal hinge is relatively tight, and the backlimb is rounded as observed in the Pitchfork Anticline. 19

20 We have not attempted to model every detail of the geometry of Pitchfork Anticline, but the general geometric features of the fold are clearly consistent with the fold geometry of the model buckle fold and inconsistent with folds produced in the models with passive markers Coalinga and Kettleman Hills Anticlines, California Pitchfork Anticline is not actively growing so we can not directly relate slip on the fault to fold growth. We now examine the actively growing anticlines at Coalinga and Kettleman Hills in central California for which we have data relating slip on the fault to the growth of the folds. Surface displacements were recorded from moderate earthquakes in 1983 and 1985 on the faults underlying the Coalinga and Kettleman Hills anticlines Setting A 110 km-long chain of Quaternary fault-cored en echelon anticlines is located in central California approximately 30 km east of the San Andreas fault and on the western edge of the San Joaquin Valley (Figure 13). The anticlinal axes of the folds trend nearly parallel to the San Andreas fault. During , three moderate earthquakes (5.4 M w 6.5) occurred along the chain of anticlines on reverse faults underlying the folds [e.g., Stein and Ekström, 1992]. Figure 13 shows profiles across the chain of folds constructed from well and seismic reflection data [Stein and Ekström, 1992; Wentworth and Zoback, 1989; Meltzer, 1989] As discussed in the introduction, a series of papers on the earthquakes [King and Stein, 1983; Stein and King, 1984; Stein and Ekström, 1992] suggested that the Kettlemen Hills Anticlines grow primarily by slip on the underlying fault during repeated large earthquakes like the sequence. This suggestion was based 20

21 largely on the similarity between the uplift pattern determined with leveling data from the 1983 Coalinga earthquake and the geometry of the Coalinga Anticline In contrast, the peak of the vertical displacement pattern for the 1985 Kettleman Hills earthquake is offset about 3 km NE of the fold axis of the Kettleman Hills North Dome. Figure 14 shows the contours of vertical displacements from an elastic dislocation model with uniform slip that best reproduces the measurements of vertical displacement during the 1985 earthquake [Stein and Ekström, 1992]. Stein and Ekström [1992] suggested that the North Dome probably grew as a result of repeated earthquakes on a dipping fault under the fold, similar to the fault the slipped in the 1985 earthquake, but that fault has perhaps begun migrating to the northeast, generating uplift to the northeast of the anticline during the 1985 earthquake We will demonstrate that an alternative explanation is that the 1985 earthquake may very well be typical of earthquakes on the major fault underlying the anticline, but the coseismic deformation pattern does not match the fold geometry because the fold did not grow solely as a consequence of slip on the underlying fault Mechanical analysis Results of our analyses of fault-cored buckle folds with an embedded shallowly dipping fault (Figure 5) show that the crest of the anticline is well behind the fault tip, whereas models of the same type but with passive layering show the fold crest above the fault tip. Figure 14b shows the fold in a mechanically layered medium along with vertical displacement pattern at the ground surface due to slip on the buried reverse fault. The relationship between the location of the peak coseismic uplift and the axial trace of the anticline is similar to that observed from the 1985 Kettleman Hills earthquake. The peak coseismic uplift is shifted to the front limb of 21

22 the anticline, not centered on the anticline. We therefore suggest that this result indicates that the Kettleman Hills North Dome likely formed as a buckle fold overlying a reverse fault, similar to those produced in our model. The shape of the vertical displacement pattern due to earthquakes on the underlying reverse fault does not directly reflect the shape of the anticline because the anticline grows by the combined mechanisms fault slip and buckling of layers The subsurface shape of the Kettleman Hills South Dome (Figure 13c) is further evidence that buckling contributes significantly to the growth of the anticline. Figure 15 compares models of folds produced by shortening of a medium with a 45 dipping reverse fault with either passive markers or mechanical layers. The geometry of the fold produced by the model with mechanical layers is similar to the Kettleman Hills South Dome. The seismic profile and the model both show a slightly asymmetric fold localized above the dipping reverse fault. The fold produced in the homogeneous medium with passive markers clearly does not resemble the actual fold and the modeled amplitude is too small This analysis of the Kettleman Hills Anticlines is revealing, but is far from complete. To determine the extent to which the fold has recently grown between large earthquakes, either by slip on the underlying fault or buckling, one would want to examine interseismic data showing the pattern of deformation. One would also want to examine geomorphic evidence for Holocene deformation of the fold in order to determine the longer-term history of deformation. Acquisition and analysis of such data in future work would likely yield valuable insight about the processes of fold growth at Kettleman Hills. 22

23 CONCLUSIONS To demonstrate the influence of buckling on fault-cored fold growth, we have constructed boundary element models in which a medium containing a fault and elastic layers is subjected to layer-parallel shortening. Free slip is assumed on the fault and layer contacts. We compare simulations of folding in the mechanically layered elastic medium with folding of passive markers in a non-layered medium. Given the same amount of far-field shortening for both conditions, the mechanically layered medium produces more highly localized folds with higher amplitude and shorter wavelength The horizontal shortening that causes a fault in the core of an anticline to slip can cause significant amplification of the fold by buckling of the strata. Under conditions considered in this paper, the contribution to fold growth by slip on the underlying fault alone is only about 20-50% of the total growth. Therefore, studies that seek to estimate fault slip from fold geometry by assuming the fold is built by slip on the fault alone could significantly overestimate the amount of fault slip We show that the Pitchfork Anticline in Wyoming did not form as a faultcored passive fold, but may have formed by the combined mechanisms of fault slip and buckling. The BEM model with buckling of elastic layers can reproduce many of the essential geometric features of Pitchfork Anticline that cannot be reproduced by models without mechanical layering. The tight, localized anticlinal dome and increasing amplitude away from the fault tip are geometric features of Pitchfork Anticline that are characteristic of buckle folds The fault under Pitchfork Anticline has not been directly imaged, so we can only infer the relationship between faulting and folding. At Kettleman Hills Anticline 23

24 in central California, published seismic profiles show the subsurface fault and fold geometry and slip on the fault can be inferred from coseismic uplift data from the 1985 Kettleman Hills earthquake. We show that the general features of the fold form are consistent with a model that incorporates buckling of layers over a dipping, blind reverse fault. Furthermore, the general pattern of coseismic uplift centered over the forelimb of the anticline is predicted by the model ACKNOWLEDGEMENTS We thank Arvid M. Johnson for providing results of his analytical viscous folding theory to verify our BEM method and many useful suggestions that helped improve the article APPENDIX A1: Comparison of Analytical and BEM Solutions for Amplification factor To verify that our BEM program is reliable for solving folding problems, we compare amplification factors computed with the analytical solution (Johnson and Pfaff [1989]) and with the boundary-element solution. It was not intuitively obvious that a folding model (BEM) consisting of layers modeled with small boundary elements at their interfaces would be equivalent (or nearly so) to analytical folding theory [Johnson and Fletcher, 1994]. The amplification factor is related exponentially to the growth rate of amplitude of a fold. The larger the amplification factor, the faster the amplitude of a fold grows. The amplification factor for a given set of conditions is a function of the ratio of wavelength to thickness of a single layer within a multilayer. Thus, Figure A1 shows plots of amplification factors calculated with the two methods for two, four, and ten identical layers. The peak in each curve has coordinates of the maximum amplification factor and the dominant wavelength. The dominant wavelength is the wavelength that will grow the fastest. 24

25 By comparing amplification factors calculated with the BEM model and the analytical model, we see that the results are very similar (Figure A1). The similarity shows that the two models, derived in quite different ways, are almost certainly addressing the same mechanical problem. Thus we can depend on the BEM model to solve some folding problems such as those addressed in this paper that would be very difficult with the analytical folding theory A2: Examination of sizes of incremental deformations and length of layer interfaces on final fold form We are calculating large deformations using linear (infinitesimal strain) elasticity theory. As stated in the text, to make use of the linear elastic solutions, we assume large deformation can be accumulated by small increments of shortening. In each increment, we use the position of points in the medium derived from the previous increment to calculate a new position and we ignore nonlinear effects of initial stress on the increment of deformation We do not want the size of the small increments of shortening to affect the fold form, so we test the sensitivity of the final fold form to the size of the small increments of deformation. We plot fold forms in Figure A2 produced by runs with different numbers of steps with the same amount of total shortening of 21.4%. Figure A2 shows the final fold form does depend on the number of increments used, but increasing the number of steps beyond 12 does not change the final fold form appreciably. For the simulations in this paper, we use a uniform incremental far-field strain of ε xx = We also do not want the length of the layer contacts, represented by boundary elements, to affect the fold form, so we test the sensitivity of the final fold form to the 25

26 length of contacts. In the examples, the ratio of the length of contact (R) to length of the fault (W) is 3, 5 and 10 (Figure A3). Total shortening is 21.4% and the small increment of deformation is ε xx = in the three examples. Figure A3 shows no appreciable difference in final fold form in the three examples On the other hand, it is clear that the length of the layers must be greater than the wavelength of the fold (defined as the horizontal length between the synclinal troughs on either side of the anticline). The example for R/W = 3 in Figure A3, is marginal in this respect. For the conditions in the three examples, R/W values of 5 or 10 should not disturb the natural wavelength of the fold Finally, we do not want the lengths of individual boundary elements to influence the simulation of folding. An accurate numerical solution requires short boundary elements to approximate a continuous distribution of boundary conditions on the interfaces. To verify that our boundary elements are small enough to produce an accurate solution, we ran tests in which we systematically reduced the element length until a continued reduction of element length did not influence the fold form. We used element lengths of 0.05W A3: Basic formulation of Boundary element model and comparison with twodimensional displacement discontinuity method (TWODD) The numerical technique of the boundary element method has been clearly described by Crouch and Starfield [1983]. Our boundary element algorithm is largely similar to their two-dimensional displacement discontinuity method (TWODD) which was succinctly summarized by Martel and Muller [2000]. We formulate the elastic boundary element models using the solution for an edge dislocation in an isotropic, homogeneous, elastic half-space assuming infinitely long faults and bedding contacts 26

27 in the strike direction (2D plane-strain conditions). As shown in Figure 3, the bedding planes and faults are discretized into small elements with uniform slip We give a brief outline of our formulation of the boundary element model. Assume we have an N 1 vector of incremental values of the dip component of slip, s, on N elements. From the solution for a 2D edge dislocation [Crouch and Starfield, 1983], we can relate the vector of shear stresses, σ s, at the center of each element to slip on all the elements through the N N matrix, G σ, 648 σ = G s. (1) s σ We assume a coordinate system with x in the horizontal direction and y in the vertical direction. We apply increments of far-field uniform strain, 651 ff ε xx = constant, 652 ff ff ε yy = ε ν /( 1 ν ) xx, 653 ff ε xy = 0, (2) 654 with corresponding uniform far-field stress, ff ff ff σ = σ = 0, yy xy ff ff ( ε ) σ + ff xx = 2µ ε xx + λ xx ε yy, (3) where λ and µ are Lame s elastic constants, λ 2µν /( 1 2ν ) =, and ν is Poisson s ratio. We normalize all stresses by µ and assume ν = From the far-field stress we 659 compute the shear component of stress resolved onto each element, ff σ s. Assuming 27

28 cohesionless, frictionless contacts, we satisfy the condition that the shear stress is zero on each element after each increment of deformation, 662 ff σ + G s = 0. (4) s σ 663 The distribution of incremental slip on the elements that gives this condition is 664 s = G 1 σ ( ff σ ) s. (5) N N matrices x G d and y G d relating the x and y components of displacements of the endpoints of the elements to the slip on each element is constructed using the solution for the edge dislocation. Note that we only specify one boundary condition and solve for only one slip component on each element because we assume that the normal component of displacement discontinuity across elements is zero. Then the 670 incremental displacements, u x and u y, of the element endpoints during the small 671 increment of deformation are calculated as 672 u ff x = x Gds+ u x, 673 u = y G s+, (6) ff y d u y 674 with the contribution to the displacements from the far-field strain being 675 ff ff u x = x ε xx, 676 ff ff u y = y ε yy. (7) Positions of new element endpoints are calculated from the previous endpoints and the incremental displacements, and then a new increment of far-field strain is applied and the calculations in equations (1)-(7) are repeated. For each increment of far-field 28

29 strain, we fix the y-coordinate of the origin of the coordinate system at the ground surface (free surface). Note that because we assume zero resistance to sliding on the faults and layer interfaces, we do not need to consider confining pressure due to the lithostatic load. Also for simplicity, we ignore the build-up of topography and assume that erosion and deposition maintain a flat and horizontal ground surface. Any points on the interfaces that move above the ground surface are discarded We highlight some important differences between TWODD and our boundary element method. We replace the solutions of stress and displacements for an edge dislocation in an elastic whole-space used by TWODD with the solutions for an edge dislocation in an elastic half space. This implementation enables us to more efficiently analyze problems with a ground surface. Furthermore, we assume discontinuities, i.e. faults and layer contacts, remain closed in our models. While normal and shear tractions and normal and shear components of displacement are allowed to be specified as boundary conditions in either TWODD or our method, in order to keep discontinuities closed, we specify the normal component of the displacement discontinuity across faults to be zero. This displacement discontinuity boundary condition cannot be implemented in TWODD

30 REFERENCES Allmendinger, R.W., and J. H. Shaw (2000), Estimation of fault propagation distance from fold shape: implications for earthquake hazard assessment, Geology, 28(12), Berger, P. and A. M. Johnson (1980), First-order analysis of deformation of a thrust sheet moving over a ramp, Tectonophysics, 70, T9-T24. Berger, P. and A. M. Johnson (1982), Folding of passive layers and form of minor structures near terminations of blind thrusts with application to the central Appalachian blind thrust:, J. Struct. Geol., 4, Biot, M. A. (1961), Theory of folding of stratified viscoelastic media and its implications in tectonics and orogenesis, Geol. Soc. Am. Bull., 72, Biot, M. A. (1964a), Exact theory of buckling of a thick slab, Applied Scientific Research, Section A 12, Biot, M. A. (1964b), Theory of internal buckling of a confined multilayered structure, Geol. Soc. Am. Bull., 76, Cardozo, N. (2008), Trishear in 3D. Algorithms, implementation, and limitations, J. Struct. Geol., Cardozo, N., K. Bhalla, A.T. Zehnder, and R.W. Allmendinger (2003), Mechanical models of fault propagation folds and comparison to the Trishear kinematic model. J. Struct. Geol., 25, Cardozo, N., R.W. Allmendinger, and J. K. Morgan (2005), Influence of mechanical stratigraphy and initial stress state on the formation of two fault propagation folds, J. Struct. Geol., 27, Chappel, W. M. (1969), Fold shape and rheology: the folding of an isolated viscousplastic layer, Tectonophysics, 7, Chester, J. S. and F. M. Chester (1990), Fault-propagation above thrusts with constant dip, J. Struct. Geol., 12, Cooke, M. L., and D. D. Pollard (1997), Bedding-plane slip in initial stages of faultrelated folding, J. Struct. Geol., 19, Crouch, S. L. and Starfield, A. M. (1983), Boundary element methods in solid mechanics, 322 p. Dolan, J. F., S. A. Christofferson, and J. H. Shaw (2003), Recognition of paleoearthquakes on the Puente Hills blind thrust fault, California, Science, 300, Durdella, M. J. (2001), Mechanical Modeling of Fault-related Folds: West Flank of the Bighorn Basin, Wyoming: M.S. Thesis, Department of Earth and Atmospheric Sciences Library, 128 p. Ekström, G., R. S. Stein, J. P. Eaton, and D. Eberhart-Phillips (1992), Seismicity and Geometry of a 110-km-long blind thrust fault 1. The 1985 Kettleman Hills, California, earthquake, J. Geophys. Res., 97(B4), Elliott, D. (1976), The motion of thrust sheets, J. Geophys. Res., 84,

31 Erickson, S. G. and W. R. Jamison (1995), Viscous-plastic finite-element models of fault-bend folds. J. Struct. Geol., 17, Erickson, S. G., L. M. Strayer and J. Suppe (2001), Initiation and reactivation of faults during movement over a thrust-fault ramp: numerical mechanical models, J. Struct. Geol., 23, Erslev, E. A. (1991), Trishear fault-propagation folding, Geology, 19, Erslev, E. A. and Mayborn K. R. (1997), Multiple geometries and modes of faultpropagation folding in the Canadian thrust belt, J. Struct. Geol., 19, Finch, E., S. Hardy, and R. Gawthorpe (2003), Discrete element modeling of contractional fault propagation folding above rigid basement blocks, J. Struct. Geol., 25, Fletcher, R. C. (1977), Folding of a single viscous layer: Exact infinitesimalamplitude solution, Tectonophysics, 39, Grant, L. B., K. J. Mueller, E. M. Garth, H. Cheng, R. L. Edwards, R. Munro, and G. L. Kennedy (1999), Late Quaternary uplift and earthquake potential of the San Joaquin Hills, southern Los Angeles basin, California, Geology, 27(11), Ishiyama, T., K. Mueller, M. Togo, A. Okada, and K. Takemura (2004), Geomorphology, kinematic history, and earthquake behavior of the active Kuwana wedge thrust anticline, central Japan, J. Geophys. Res., 109, B12408, doi: /2003jb Johnson, A. M. and V. J. Pfaff. (1989), Parallel, similar and constrained folds. Engineering Geology, 27, Johnson, A. M., (1977), Styles of folding, Elsever Publishing Company, New York, 406 p. Johnson, A. M., and P. Berger (1989), Kinematics of fault-bend folding. Engineering Geology, 27, Johnson, A. M., and R. Fletcher (1994), Folding of Viscous Layers, 445 pp., Columbia Univ. Press. Johnson, K. M., and A. M. Johnson (2002), Mechanical analysis of the geometry of forced-folds, J. Struct. Geol., 24, Kilsdonk, B., and R. C. Fletcher (1989), An analytical model of hanging-wall and footwall deformation at ramps on normal and thrust faults, Tectonophysics, 163, King, G. C. P, and R. Stein (1983), Surface folding, river terrace deformation rate and earthquake repeat time in a reverse faulting environment: the Coalinga, California, earthquake of May 1983, California Division of Mines and Geology, Special Publication 66, King, G. C. P, R. Stein, and J. B. Rundle (1988), The growth of geologic structures by repeated earthquakes, 1. Conceptual framework, J. Geophys. Res., 93(B11), Langdon, T. G. (1970), Grain boundary sliding as a deformation mechanism during creep, Phil. Mag. 22,

32 Leon, L.A., S. A. Christofferson, J. F. Dolan, J. H. Shaw, and T. L. Pratt (2007), Earthquake-by-earthquake fold growth above the Puente Hills blind thrust fault, Los Angeles, California: Implications for fold kinematics and seismic hazard, J. Geophys. Res., 112 (B3), doi: /2006jb Lin, J. and R. S. Stein (1989), Coseismic folding, earthquake recurrence, and the 1987 source mechanism at Whittier Narrows, Los Angeles Basin, California, J. Geophys. Res., 94, Mancktelow, N. S. (1999), Finite-element modeling of single-layer folding in elastoviscous material: the effect of initial perturbation geometry, J. Struct. Geol., 21, Martel, S. J., and J. Muller (2000), A two-dimensional boundary element method for calculating elastic gravitational stresses in slopes, Pure Appl. Geophys., 157, McClay, K. R. (1977), Pressure solution and Coble creep in rocks and minerals: a review, J. Geol. Soc. London 134, Meglis, I. L., R. E. Gagnon and R. P. Young (1995), Microcracking during stressrelief of polycrystalline ice formed at high pressure, Geophys. Res. Lett., 22 (16), Meltzer, A. (1989), Crustal structure and tectonic evolution; Central California, Ph. D. dissertation, , Rice Uni. at Houston, Texas. Myers, D.J., J.L. Nabelek, and R. S. Yeats (2003), Dislocation modeling of blind thrusts in the eastern Los Angeles basin, California, Journal of Geophysical Research, 108, B9, 2443, doi: /2002jb Mynatt, I., G. E. Hilley and D. D. Pollard (2007), Inferring fault characteristics using fold geometry constrained by Airborne Laser Swath Mapping at Raplee Ridge, UT, Geophysical Research Letters, 34, L16315, doi: /2007gl Patton, T. L. and R. C. Fletcher (1995), Mathematical block motion for deformation layer above a buried fault of arbitrary dip and sense of slip, J. Struct. Geol., 17, Petersen, F. A. (1983), Foreland detachment structures, in Rocky Mountain foreland basins and uplifts, edited by J. D. Lowell, 65-78, Denver, Colorado, Rocky Mountain Association of Geologists. Pfaff, V. J. (1986), On the Forms of Folds. Formation of kink and box folds in viscous multilayers, Ph.D. dissertation, 563pp, Uni. of Cincinnati at Cincinnati, Ohio. Pfaff, V. J. and A. M. Johnson (1989), Opposite senses of fold asymmetry, Engineering Geology, 27, Reches, Z. and A. M. Johnson (1978), Theoretical analysis of monoclines, in Laramide folding associated with basement block fault in the western United States, edited by V. III Mattews, Geological Society of America Memoir, 151, Rich, J. L. (1934), Mechanisms of low-angle overthrust faulting as illustrated by Cumberland thrust block, Virginia, Kentucky, and Tennessee, American Association of Petroleum Geologists Bulletin, 18,

33 Sanford, A. R. (1959) Analytical and experimental study of simple geologic structures, Geol. Soc. Am. Bull., 70, Savage, H. M. and L. C. Cooke (2003), Can flat-ramp fault geometry be inferred from fold shape? A comparison of kinematic and mechanical folds, J. Struct. Geol., 25, Shackleton, J. R. and M. L., Cooke (2007), Is plane strain a valid assumption in noncylindrical fault-cored folds? J. Struct. Geol., 29, Shaw, J. H. and J. Suppe (1996), Earthquake hazards of active blind-thrust faults under the central Los Angeles basin, California, J. Geophys. Res., 101(B4), Shaw, J. H., A. Plesch, J. F. Dolan, T. L. Prattt, and P. Fiore (2002), Puente Hills blind-thrust system, Los Angeles, California, Bull. Seismol. Soc. Am., 92(8), Shaw, J. H., and P. M. Shearer (1999), An elusive blind-thrust fault beneath metropolitan Los Angeles, Science, 283, Sibson, R. H. (1986), Earthquakes and rock deformation in crustal fault zones. Annual Review of Earth and Planetary Science, 14, Stein, R. S. and G. Ekström (1992), Seismicity and Geometry of a 110-km-long blind thrust fault 2. Synthesis of the California earthquake sequence, J. Geophys. Res., 97(B4), Stein, R., and G. C. P. King (1984), Seismic potential revealed by surface folding: 1983 Coalinga, California, earthquake, Science, 224, Stone, D. S. (1983a), The Greybull Sandstone pool (Lower Cretaceous) on the Elk Basin thrust-fold complex, Wyoming and Montana, in Rocky Mountain foreland basins and uplifts, edited by J. D. Lowell, Rocky Mountain Association of Geologists, Stone, D. S. (1983b), Seismic profile: South Elk Basin area, Big Horn Basin, Wyoming, in seismic expression of structural styles, 3, edited by A. W. Bally, American Association of Petroleum Geologists, Stone, D. S. (1993), Basement-involved thrust-generated folds as seismically imaged in the subsurface of the central Rocky Mountain foreland, in Laramide Basement Deformation in the Rocky Mountain Foreland of the Western United States, edited by C. J. Schmidt, R. B. Chase, and E. A. Erslev, Geological Society of America Special Paper 280, Strayer, L. M. and J. Suppe (2002), Out-of-plane motion of a thrust sheet during along-strike propagation of a thrust ramp: a distinct-element approach, J. Struct. Geol., 24, Suppe, J. (1983), Geometry and kinematics of fault-bend folding, American Journal of Science, 283, Suppe, J. and D. A. Medwedeff (1994), Fault-propagation folding, Geol. Soc. Am. Abs. w. Prog. 16, 670. Wentworth, C. M. and M. D. Zoback (1989), The style of late Cenozonic deformation at the eastern front the California Coast ranges, Tectonics, 8,

34 Wiltschko, D. V. (1979), A mechanical model for thrust sheet deformation at a ramp, J. Geophys. Res., 84, Yamashita, Tadayoshi and K. Ojima (1968), Deformation twin tips with flat surfaces in pure iron crystals, Journal of Election Microscopy, 17(4),

35 Figure 1. Models of folding of passive markers or mechanical layers in an elastic medium. Layers are horizontal at onset. The layers slip freely at contacts (a)-(c) and are bonded (no slip, passive markers) in (d)-(f). W is initial fault width. (a) Fold formed by fault slip and buckling after shortening of 21.5%. (b) Fold formed by imposed slip on the fault with no horizontal shortening. The fold is produced by imposing the same slip on the fault as in (a). (c) Buckle fold after 21.5% shortening. The small deflection of the layers after two increments of deformation in (a) is introduced as an initial perturbation. (d) Fold in passive markers after shortening of 30.5% with the same maximum fault slip as in (a). (e) Fold in passive markers due to the same imposed fault slip as in (b). (f) Passive markers are initially assigned the same small perturbation as in (c) and then subjected to 21.5% shortening Figure 2. (a) Plots of amplification factor for periodic folds in viscous layers as a function of wavelength, L, normalized by the thickness of a single layer, h. N indicates number of layers in multilayer. Layers slip freely at contacts. Layers and surrounding medium have same viscosity. The plots are produce by using the folding theory developed by Johnson and Pfaff [1989]. (b) Illustration of a multilayer bounded above and below by semi-infinite media. The number of layers in the multilayer, N, is Figure 3. Geometry and boundary conditions of two models of a fault embedded in an elastic medium with mechanical layering.. Notation is σ : stress, σ n : normal traction, σ s : shear traction, ff ε : remote strain and S: uniform slip. Wiggly edges indicate that 35

36 899 the medium extends to infinity. (a) Embedded fault case. The loading condition is 900 horizontal shortening, i.e. ff ε xx. (b) Ramp fault case. The loading condition is a 901 uniform slip applied to the detachment on the far right side of the model domain Figure 4. Models of fault in an elastic half space underlying a stack of elastic layers. Entire medium is subjected to far-field horizontal shortening. W is initial fault width. Fault dips 25 at onset. (a) Fault-cored passive fold (layers are passive markers.) (b) Fault-cored buckle fold (layers slip freely at contacts) Figure 5. Models of a fault embedded in layers. W is initial fault width. Fault dips 25 at onset. (a) Fault-cored passive fold (layer interfaces are bonded). (b) Fault-cored buckle fold (Layers slip freely at contacts) Figure 6. Models of a ramp anticline. W is the width of ramp. Ramp initially dips 25. The maximum fault slip is the amount of slip applied to the detachment at the far right side of the model domain. (a) Ramp-cored passive fold (layer interfaces are bonded). (b) Ramp-cored buckle fold (layers slip freely at contacts) Figure 7. Comparison of amplitudes of folds in Figure 4 and amplitudes of fault-cored passive and buckle folds in a full space (no free surface) Figure 8. Models of a fault embedded in stacks of mechanical layers with different layer thicknesses. The faults in all models have the same maximum fault slip of 0.18 W. W is initial fault width. n is number of layer interfaces

37 Figure 9. Comparisons of fold features with different numbers of layer interfaces. (a) Ratio of amplitude of fault-cored buckle folds in Figure 8 to amplitude of folds in passive markers under the same amount of shortening plotted with number of layer interfaces. (b) Fold wavelengths from Figure 8 normalized by the initial fault width plotted with number of layer interfaces Figure 10. Geologic map of the west flank of the Bighorn Basin, showing the location of Pitchfork Anticline. A-A is the location of profile in Figure Figure 11. Reconstruction of Pitchfork anticline by Durdella (2001) based on the combination of surface data, well logs and a seismic profile, where Pc is Precambrian, Pa is Paleozoic, Me is Mesozoic and Te is Tertiary. Numbers above each well indicate the distance it was projected to the line of transect. The location of this profile is shown in Figure 10. Formations are projected above the ground surface to show extent of strata incorporated in the folding of Pitchfork anticline Figure 12. Comparison of Pitchfork Anticline and results from passive marker model and mechanical layer model. (a) Pitchfork Anticline based on the combination of surface data, well logs and a seismic profile. (b) Passive layer model. A broad asymmetric anticline forms above the upper tip of a curved fault under horizontal shortening of 21.5%. (c) Mechanical layer model. A tight asymmetric anticline forms above the upper tip of a curved fault under a horizontal shortening of 21.5%

38 Figure 13. Geological map and cross sections of Coalinga and Kettleman Hills (modified after Stein and Ekström, 1992). Geometry is based on seismic reflection images. (a) Coalinga profile with moment tensor for 1983 earthquake. Hypocenters of small earthquakes are shown with small circles. (b) Kettleman Hills North Dome with moment tensor for 1985 earthquake. Hypocenters of small earthquakes are shown with small circles. (c) Kettleman Hills South Dome Figure14. Predicted elevation change of 1985 Kettleman Hills earthquake. (a) Vertical deformation field predicted from a rectangular dislocation with reverse-slip by Ekström at el. [1992]. (b) Model of a fault embedded in mechanical layers like in Figure 5. W is initial width of the fault and U y is vertical displacement. The plot of coseismic uplift attributed to only fault slip in the upper part of (b) is centered above the forelimb of Kettleman Hills Anticline, much like the observed pattern in (a) Figure 15. Comparison of South Dome anticline and simulations. (a) Profile of Kettleman Hills South Dome (profile in Figure 13c). (b) Result from mechanical layer model. (c) Result from passive marker model. The anticline in (b) resembles the South Dome anticline better than the anticline in (c) Figure A1. Plots of amplification factor of periodic folds using analytical viscous folding theory and boundary element theory Figure A2. Evaluation of results from the mechanical layer model with various incremental steps. W is initial fault width. (a) Average fold amplitude plotted with number of incremental steps. (b) Fold forms from models of a fault embedded in 38

39 basement underlying sedimentary layering. The fold form does not change appreciably as the number of steps is increased above Figure A3. Evaluation of results from the mechanical layer model with various lengths of layer interfaces. R is initial length of layer interfaces. W is initial fault width. There is no appreciable difference in final fold forms after 21.4% shortening

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