Inverse and forward numerical modeling of trishear faultpropagation

Transcription

1 TECTONICS, VOL. 17, NO. 4, PAGES , AUGUST 1998 Inverse and forward numerical modeling of trishear faultpropagation folds Richard W. Allmendinger Department of Geological Sciences and Institute for the Study of the Continents, Snee Hall, Cornell University, Ithaca, New York Abstract. Fault-propagation folds commonly display footwall synclines as well as changes in stratigraphic thickness and dip on their forelimbs, features that cannot easily be explained by simple parallel kink fold kinematics. An alternative kinematic model, trishear, can explain these observations, as well as a variety of other features which have long intrigued structural geologists. Trishear has received little attention until recently, in part because it must be applied numerically rather than graphically. A new computer program has been developed to analyze trishear and hybrid trishear-fault-bend fold deformation. Trishear fold shape can vary considerably by changing the apical angle of the trishear zone and/or the propagation to slip ratio (P/S) during the evolution of the structure. Breakouts, anticlinal and synclinal ramps, and inversion structures can also be modeled, tracking the kinematics with growth strata. Strain within trishear zones can be used to predict fracture orientations throughouthe structures as demonstrated by comparison with analog clay models. Also presented is a method for inverting data on real structures for a best fit trishear model by performing a grid search over a sixparameter space (ramp angle, trishear apical angle, displacement, P/S, and X and Y positions of the fault tip line). The inversion is performed by restoring a key bed to a planar orientation by least squares regression. Because trishear provides a bulk kinematic description of a deforming zone, it is complementary to, rather than competing with, other kinematic models. 1. Introduction The fault-related folding in thick-skinned tectonic provinces such as the Laramide Rocky Mountain foreland or the Sierras Pampeanas of western Argentina has long challenged structural geologists [Erslev, 1991; Erslev and Rogers, 1993; Matthews and Work, 1978; Mitra and Mount, 1998; Narr and Suppe, 1994; Reches, 1978]. The basement rocks in these provinces commonly do not display the layered, stratigraphic anisotropies that are thought to control fold kinematics in thin-skinned thrust belts, where layer-parallel shear and parallel folding is considered the norm. In thick-skinned provinces, basement and the overlying strata are commonly folded over the tips of propagating faults. The now classical model fault-propagation folding based on kink geometries [Suppe, 1983; Suppe and Medwedeff, 1990] does not explain very well the broad crested anticlines and monoclines in these provinces. Furthermore, it has long been recognized that, Copyright 1998 by the American Geophysical Union. Paper number 98TC /98/98TC $12.00 even in thin-skinned provinces, fault-propagation folds with changes in forelimb bedding thickness and dip are common (Figure la). Early attempts to model such cases were purely geometric exercises [Jamison, 1987; Mitra, 1990] or kinematic exercises but with extremely restrictive assumptions [Suppe et al., 1992; Suppe and Medwedeff, 1990]. Erslev [ 1991 ] proposed a strikingly different, kinematically explicit model for fault-propagation folds, the "trishear" model, in which many geometries can be reproduced. This model has received relatively little attention, perhaps because it must be implemented numerically and there were no generally available forward modeling programs. Recently, Hardy and Ford [1997] expanded Erslev's [1991] initial trishear model. They present a clear mathematical formulation of the problem, have analyzed the effect of variable propagation to slip ratios, and have illustrated growth strata geometries associated with trishear fault-propagation folds. Their computer program represented a first step in a general trishear forward modeling program. I have applied Hardy and Ford's [1997] mathematical analysis in a completely new computer program which allows great flexibility in the description of the starting parameters, variations in parameters during the analysis, strata of variable initial thicknesses and dips, and strata added during the growth of the structure (although surface transport and base level changes are not included). The program is used to produce a series of simple, multistage forward models to demonstrate the array of ideal geometries that can be produced by trishear. The basic processes modeled, a combination of fault-bend and fault-propagation folding, breakouts, inversion structures, and progressive and instantaneous rotation in growth strata, are already well known in the literature, but the use of a trishear approach puts them in a new and different light. Then, a new inverse modeling approach for analyzing real structures is introduced. One of the cases to which the inverse method is applied shows that trishear is not restricted to thick-skinned tectonic provinces but also occurs in thin-skinned regions. 2. Kinematics of Trishear Deformation In the trishear model, a single fault in "basement" expands outward into a triangular zone of distributed shear (Figure lb). The reason for the triangular shape of the shear zone must ultimately lie in the still largely unexplored mechanics of trishear. Blind faults like those modeled here are essentially large mode II cracks. Theoretical studies of the stress field around mode II cracks show that there is a triangularegion of high shear stress concentration around the tip (Figure l c) [Pollard and Segall, 1987]. Erslev [ 1991 ] and Erslev and Rogers [ 1993] showed that 64O

2 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 641 A. downward steepening dips rn Bearpaw Gardiu ' ---" " footwall synclines B. Trishear Kinematics C. maximum shear stress, mode II crack Figure 1. (a) Much simplified cross-section of the Turner Valley anticline, foothills of the Canadian Rocky Mountains [modified from Gallup, 1951](AAPG 1951, reprinted by permission of the American Association of Petroleum Geologists). Section highlights several long standing problems in balancing fault-propagation folds. (b) Basic trishear geometry as described by Erslev [1991] and Hardy and Ford [1997]. (c) Contour plot of maximum shear stresses at the tip of a model II crack. Note symmetric triangular region of high stresses at crack tip. Crack model is based on linear elasticity fracture mechanics as described in Pollard and Segall [1987]; plot was produced using notebooks of the computer program Mathematica described by Crider et al. [ 1996]. to conserve cross-sectional area the triangular zone must be symmetric with respecto the fault. At the top of the trishear zone, slip vectors are equal to that of the hanging wall: they are parallel and equal in magnitude to the master fault. At the base of the trishear zone, the slip is zero. Within the trishear zone, the slip vector varies linearly in magnitude and orientation from top to bottom [Hardy and Ford, 1997]. Thus the direction of shear varies from the dip of the fault to the dip minus the half apical angle of the trishear zone. Although the displacement field is easy to calculate, it must be done iteratively, and therefore the method cannot be applied graphically or analytically. The apex of the trishear zone is located either on the tip line of the fault (attached to the hanging wall in Erslev's [1991] terminology), or it is attached to the footwall. Hardy and Ford [ 1997] show that these two conditions are precisely described in terms of the propagation-to-slip ratio (P/S), which determines how rapidly the tip line propagates relative to the slip on the fault itself. Footwall-attached trishear zones have a P/S ratio of zero, whereas in Erslev's [ 1991 ] hanging wall attached trishear zones, P/S = 1. However, there is no need to restrict P/S to 0 or 1 [Erslev and Mayborn, 1997; Hardy and Ford, 1997] (Figure 2). Low values of P/S result in pronounced forelimb thickening and tight folding

3 642 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 3. Variable Trishear Forward Modeling In variable trishear deformation, various parameters can be changed at any time during a model run. With respecto the trishear zone itself, either P/S or the apical angle of the triangular zone can be varied during growth of the structure. Factors which might produce a change in the trishear angle or P/S during deformation are unknown. It seems likely that mechanical properties of the lithologic sequence, strain rate, and perhaps variable fluid pressure may play a role. A ramp in the fault may also form during growth of the structure, producing a fault-bend fold. Finally, beds can be added during the formation of the structure, simulating growth strata. Several well-known types of structural interactions can be modeled from a trishear perspective. Sections demonstrate the effects of varying model parameters through time, emphasizing the final geometry. Though growth strata are shown in all models, an explicit discussion of the growth geometries is saved until the end of this section Changing P/S Ratio Through Time D. P/S = 2.0 / I Figure 2. Illustration of the effects of varying propagation to slip ratio: (a) P/S - 0, (b) P/S = 1, (c) P/S = 1.5, and (d) P/S = 2.0. All models have the same slip; only the propagation of the tip line varies. Strain ellipses document variation with kinematics. in the trishear zone, whereas P/S > 1 results in less thickening, more open folding, and in folding of the hanging wall, even though the trishear zone is attached to the tip line [Hardy and Ford, 1997]. This occurs because the hanging wall boundary of the trishear zone must migrate through the material of the hanging wall as the tip line propagates. This migration has significant consequences for growth strata geometries, as discussed in section 3.6. The strain field within the trishear zone is heterogeneous but continuous (Figure 2). Because the shear planes are oblique to layering, the folding within the trishear zone involves changes in thickness of the layers. In general, beds thicken during the early stages of deformation but then thin as they steepen and overturn later on. Because trishear has not been studied exten- sively, the physical conditions which determine whether or not a trishear zone occurs as well as the specific apical angle are not well understood. Hardy and Ford [ 1997] showed that the style of folding depends on P/S. If P/S is large, then any material point spends less time within the trishear zone than if P/S is small, and thus it is less deformed, and the folding is more open. If this ratio varies through time, the macroscopic effect will be that beds at different stratigraphic levels will display different degrees of folding. The change in P/S during deformation seems likely to be a common scenario. It may happen, for example, when the tip line enters a unit of different mechanical properties, one that is overpressured, etc. Molinero et al. [ 1996] have suggested that variable P/S occurred during the development of one of the structures in the Ebro Basin. In Figure 3a, two episodes of low P/S were separated by an episode of high P/S. This produces overturned folds at low stratigraphic levels, more open folds at intermediate levels, and overturned folds again at higher levels. The rapid propagation produced a fold geometry with relatively straight limbs and a narrow rounded axial zone which could be interpreted as a kink surface. Within the growth strata, a switch from high to low P/S produces a distinct kink, whereas the reverse switch, from low to high P/S, does not. Because there is little folding at high P/S, the previous form of the fold, produced during low P/S, rapidly becomes part of the hanging wall and is simply transported along. In Figure 3b, two episodes of high P/S are separated by a period of low P/S. Predictably, this has just the opposite effect of the previous model: open folds at low and high stratigraphic levels with overturned folds along the thrust in the middle (produce during the time of low P/S) Variable Trishear Angles In trishear zones with small apical angles, intense strain is concentrated in a narrow wedge of rock, whereas broader angles result in more diffuse, less intense strain. Changing the apical angle during thrusting produces geometric effects, which are particularly striking when the angle is suddenly reduced (Figure 4a). This case results in an instantaneous incorporation of trishear zone material into the hanging wall, "freezing" its geometry as well as the focusing of strain into a smaller zone. Two pseudokinks are produced; the first is a more rounded fold hinge which

4 ß ß ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 643 A. Variable Propagation to Slip Ratio (P/S) - 7 Slip = 210 Propagation = 420 Ramp angle = 30 ø irishear angle = 50 ø ß ' :: -: -': ::: -:'-' : : :'. :: ' :- : :i i, --:- / :x.'- : '? ¾! iiiii?iii :.-.-', 4....:..:,, :::... :, :?':': ' ':::::: ::: * % :* :..:::::..:. '"':%,. ":iii :: :: : ": '"" :....? : ::.. i.". : i/..i:..'...i'.':i... '.' ::::::::::::::::::::::::::::: e. 5 Slip = 150 Propagation = 435 Ramp angle = 30 ø irishear angle = 50 ø ß ' A",: 2 1 :,.,.,.,.., "..,.,..., -,-,..,:. },..,.! ' '""...'" :g :.: ;?:* : ". i :: :i }!:. :?,,.. :,..,,..-, ¾. ::.:. : : i..., :. :.,:...-', :. ::: ::, ::,-...,'-::.. i,-.:.--: ::... :.:: -.-,,..,-. :.: :.:.... :: :-2!...! i; ::i!:., : ß ' '.....,.::;,,.: <. :.:... ::... Figure 3. Variable P/S through time. (a) The top model was produced with P/S = 1.5 until the deposition of bed 4, P/S = 5 until the deposition of bed 5, and P/S = 1.5 between the times of beds 5 and 7. (b) The bottom model was formed by P/S = 5 until bed 1 time, P/S = 1.5 between beds 1 and 4, and P/S = 5 between beds 4 and 5. The half circles on the fault trace show the position of the tip line of the fault (in the hanging wall and footwall) when P/S was changed. Note that beds overturn during times of low P/S and are upright during high P/S. corresponds to the initial position of the hanging wall boundary of the trishear zone. Because this boundary was oriented at a higher angle to the fault zone, it migrated a greater distance through the rock as the tip line propagated up section, producing the broadly rounded hinge. The dip panel to the right of the rounded hinge (but to the left of the current trishear boundary) are rocks which were within the trishear zone during the initial open angle but then suddenly became part of the hanging wall. This dip p el narrows down section, as both the broad and the narrow trishear zone must have had the same vertex (i.e., the tip line). The second (right hand) kink (Figure 4a) is more pronounced for two reasons. First, because the new trishear boundary is oriented at a small acute angle to the slip vector for the hanging wall, it migrates little through the material as the tip line propa- gates. Second, the strains are concentrated in a smaller crosssectional area. The kink due to the second hanging wall boundary of the trishear zone, as well as the switch from open to tight trishear angle is clearly marked in the growth strata. The opposite change, an initial narrow and later open apical angle, produces quite a different anticlinal form (Figure 4b). The right dipping panel of strata is much more subtle in both growth and particularly pregrowth strata. The opening of the apical angle at a later stage in the deformation has a smoothing effect, smearing out sudden changes in dips Anticlinal and Synclinal Ramps A change in the dip of the fault during movement produces a fault bend fold, either an anticline or a syncline, which trails trishear zone (Figure 5). Because the bend in the fault is modeled as sharp change in dip, the fault-bend fold is a kink fold which conforms to the geometry and kinematics described by Suppe [ 1983]. Synclines which preserve bedding thickness on both flanks can be produced by virtually any change in ramp angle but anticlines which preserve bedding thickness and have no angular shear in horizontal beds have a very limited range of changes in fault dip (the angle { of Suppe [1983]). Because the trishear zone is symmetric about the fault, a new ramp at a different angle will produce a change in the orientation of the trishear zone, suddenly involving rocks that were previously part of the footwall or the hanging wall. In the case of a bend which produces an anticline (Figure 5a), the forelimb displays two prominent steps: The higher of the two is produced by the fault-bend fold, and the lower, which is somewhat more rounded, was produced by the hanging wall boundary of the trishear zone prior to the formation of the second

5 644 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR Variable Trishear Apical Angle A. 8 Slip= 240 I... ]: "";' I 4 Ramp angle= 30 ø I..,... /':' ;, 3 P/S ratio = I 5 I /"'" ' E 2..-, z::z:.:. ::.. i...; Figure 4. Variation in apical angle through time. (a) Model wa started with an apical angle of 50 ø until the tip line reached the position of the half circle (in hanging wall and footwall) at the time of deposition of bed 5. Then the apical angle was reduced to 20 ø between beds 5 and 8 times. (b) Apical angle was 20 ø until bed 5 time and then was increased to 50 ø between beds 5 and 8. ramp. The step related to the trishear boundary would be sharper if a P/S ratio of 1 were used rather than the value of 1.5 that was used to make Figure 5. The model also shows a pronounced zone of forelimb thickening and local, more subtle forelimb dip changes related to the hanging wall boundary of the trishear zone at the end of the second ramp. The forelimb displays growth geometries reflecting both progressive rotation during trishear and instantaneous rotation at higher stratigraphic levels formed by kink band migration in the fault-bend fold. These same strata (labeled 4-7 in Figure 5a) would also show progressive rotation had the model been plotted farther to the right. Where the second ramp is steeper than the first, a synclinal fault-bend fold results (Figur. e 5B). The forelimb geometry is produced solely by the variation in trishear angle. The upper part of the forelimb is relatively planar even though it lies within the trishear zone; the lower part is notably steeper. The change in dip between the two parts of the structure records the position of the hanging wall boundary of the trishear zone before the second ramp formed. The back limb reflects only the fault-bend fold geometry; it is planar, the kinks are angular, and it follows Suppe's [1983] geometric relations. The growth strata reflect the domi- nant kinematics on the two limbs: the back limb shows a typical growth triangle [Suppe et al., 1992], whereas the forelimb displays fanning of strata characteristic of composite progressive unconformities [Anaddn et al., 1986; Hardy and Ford, 1997; Riba, 1976]. The forelimb growth strata deposited during movement on the second ramp migrate toward the crest of the anticline because the steepening of the fault rotates the trishear zone to a higher angle, "focusing" the uplift farther to the left Breakouts A break-out forms when the fault associated with a fault- propagation fold cuts rapidly across the stratigraphic section, abandoning the fold as a relic of the former position of the tip line. In terms of modified trishear, this phenomenon is precisely modeled as a large, sudden increase the P/S ratio; the tip line moves rapidly away from the structure, which can only happen if the propagation is much larger than the slip. There are three general types of breakouts [Mitra, 1990; Suppe and Medwedeff, 1990]: along the axis of the tip line syncline, as a decollement at the stratigraphic level of the tip line, or by cutting

6 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 645 V=riable Ramp Angles t x 7 I fault-bend told I /'- -- growth triangle 6 Slip= 210 '! Propagation = 316 trishearangle= 40 ø I, ' ::?;i.:_ ' " "'":':':'""'"":...:....-, '" /... ii - -.: I /" :::: '.' iii:.-."::ii.':-":: - :: ::. ':' :: i... '... / i I -: : :... :.:.:.... :..:....:.....::..:::::: ' '"= ":'"'" '" -' "':'" ':"- fault-bend fold_ L. _ -. '_ - growth tdangle "' _ 7 ½ /// :... / 6, 4 Propagation = t sheara le= o 1 P/S ratio = 1.5.,.: -:, :: -, - :=:,,,, : -<.-.. :<<<.:... s: :.. :.:..,, :.. :.., ::::-. :: :-,-.,: ::: :::..../'"'" :...,:,,.., : :.,....,..., ,. Figure 5. Geometries produced by combining trishear fault-propagation folding with fault-bend folds produced by a change in the ramp angle. (a) An anticline produced by lowering the ramp angle from 30 ø to 5 ø at the time of deposition of bed 4. (b) A synclinal bend produced by increasing the ramp angle from 30 ø to 55 ø at bed 4 time. across the forelimb of the anticline. These are modeled as rapid propagation along the same ramp, as an anticlinal bend in the fault to a near horizontal position, and as a steepening of the fault producing a synclinal bend cutting across the anticlinal forelimb, respectively (Figure 6a, b, and c). The synclinal breakout produces the simplest geometry; the fold geometry which formed up to the point of rapid propagation is simply translated up the ramp without any further modification. The time of the breakout is clearly marked in the growth strata as the point where fanning of the strata ceases (bed 5 in Figure 6a); the equivalent strata are, of course, thicker in the footwall but are also unfolded. Not surprisingly, the geometry for the decollement breakout (Figure 6b) is quite similar to that of the anticlinal bend (Figure 5a). The main difference is that total forelimb thickening is less in the breakout case because there is no thickening and only translation related to the second and younger ramp. In the anticlinal breakout (Figure 6c), there is no further steepening of the forelimb after the formation of the second ramp (unlike in Figure 6b); the hanging wall is simply translated up the ramp, producing the classic "snake head" anticline. The pattern of fanning growth strata deposite during movement of the first ramp (prebed 5) is readily apparent only in the footwall Inversion Structures Erslev [ 1991 ] showed that trishear can be applied equally well to normal faults as to reverse faults. I take the next step of showing the geometry that results when a trishear normal fault is reactivated as a reverse fault (Figure 7a, b) and then the fault dip flattens as it enters the growth strata sequence (Figure 7c). In the rift stage, the tip line propagates upward as the hanging wall lowers, matching the behavior observed in experimental normal drape folds [Withjack et al., 1990]. Growth strata thin and onlap the footwall as they are folded with a typical "drag fold" geometry. As the direction of slip reverses to a thrust sense and the total slip returns to zero at the base of the model, a curious thing happens: there is upward increasing displacement in the pregrowth strata (i.e., with zero displacement on the base of the model, there is a pronounced anticline above the tip line at the top of the prenormal growth strata). This marked contrast to existing reactivation models [e.g., Mitra, 1993] is due to the folding produced in the trishear zone which propagates upward during both normal and reverse movement. If the trishear zone were to propagate downward with the hanging wall, no anticline would result. At this point in the model (Figure 7b), the normal fault-

7 646 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR A. Synclinal Breakout ' : :--- -,.. ' '"' "" '. "": ':"' "'" ' -:'.' "'"'"' '"''"'' '"" ':' ":" '-'' '" ':'" :' ' "': ' " ""- ' ' " '" "'"' '" '' ' --': -.:..'ii:i..-'...' '"'"'"'" ' '"' '" ' :"-'<... %.->?...-, ",.-, '... :.:.:.:,.:.:.:...:.:...:...:o: :.,.:...:...::.,...>.:::: o / B. Decollement Breakout A",. 7,%,...?'"" '"'" :: : :,,,..,.... ii... i' ':T'?ii'ii'"i' "'ig i ggi'" "!i":i!!?:g':i' "'.'.:i.'g.'... '.' C. Anticlinal Breakout Figure 6. Breakouts produced by rapid tip line propagation along (a) the leading synclinal axis at bed 5 time, (b) as a bedding-parallel decollement at bed 4 time, and (c) by cutting across the anticlinal forelimb at bed 5 time. related growth strata are completely inverted, the postrift/prethrust strata have folded and thickened in the trishear zone, and the thrust-related growth strata show a typical fanning composite progressive unconformities (CPU' s). Several interesting complications occur when the thrust flattens into the growth strata (Figure 7c). A fault-bend fold anticline forms with an active kink and a typical growth triangle related to a passive or fixed kink axis in the synorogenic strata. The early formed, thrust-related CPU is folded by the newly oriented trishear zone. Because of the shallowing of the fault ramp, the fanning geometry in the later growth strata steps out away from the locus of uplift. Note that the prenormal growth strata in the foot- wall preserve the normal-fault related folding; that is, they appear to have a "reverse drag" with respecto the thrusting. The same strata in the hanging wall also preserve a gentle syncline (located between the second ramp and the active kink axis) as a relic of the normal motion on the forelimb of the main anticline Growth Strata In general, as shown by Hardy and Ford [1997], these growth strata mimic composite progressive unconformities (CPUs) such as those described in the Pyrenees [Anad6n et al., 1986; Riba, 1976]. Many detailed aspects of growth strata geometries have al-

8 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 647 "... '"'""" ' ' : 'i ii"'"'--'"" " " ' ' : -...,,... ili Propagation Ramp angle = ø trishear angle 40.0 ø P/S ratio* ':"" %--"... i i?!dg:..;. g Slip '"'" thrust-related growth strata post-rift, pre-thrust strata rift-related growth strata,growth triangle --Total Shortening---), Propagation il Ramp angle 25.0ø I "" ' 40.0or - ' in the footwall are PIS ratio 1.5 these dips a relict of the normal faulting I trishear angle Figure 7. Model of rift inversion and fault reactivation. (a) The stippled layer was deposited postrift, prethrust reactivation. Rift synorogenic strata are immediately below this layer on the left side of the diagram and onlap the pre-rift strata draped over the footwall. (b) Fault has been reactivated in a reverse sense so that the net slip is now zero. Note, however, the formation of a tight anticline near the fault. Thrust growth strata have been added above the stippled bed. (c) Fault ramp reduced in the synorogenic strata forming a broad anticline. Note the fanning geometry of the trishear zone and, farther to the left, the growth triangle in strata of the same age. "Drag" left over from the rift phase is preserved in the pregrowth strata of the footwall. ready been described in sections Here I address a more general topic: the formation of "growth triangles" related to propagating trishear zones. The hanging wall boundary of a trishear zone is kinematically similar to a kink axis in parallel folding in that both separate a domain characterized by no shear, just translation, from a domain in which bedding is sheared. Clearly, the nature of the shear is different for the'parallel case than for the trishear cases. Where P/S > 1, the hanging wall trishear boundary propagates faster than the hanging wall slips, and therefore the boundary migrates through the material. Under these conditions, the hanging wall trishear boundary acts like an active kink axis, with its initial position in the rock equivalent to a fixed axis (Figure 8a). The gently inclined part of the passive kink axis must connect the steeply inclined passive axis in the pregrowth strata with the active axis at the top of the growth strata, producing the equivalent of a parallel fold growth triangle. In the trishear case, this growth triangle is a measure of the rate of propagation of the tip line of the fault. The more rapid the propagation of the tip line is, the lower the angle is that the fixed axis makes in the growth strata (see dashed lines in Figures 3 and 8). Note that, unlike the parallel fold case, the "active axis" is virtually undetectable in the growth strata beneath the depositional surface. Because the boundary is propagating through the material, the strain varies heterogeneously and continuously from the deformed strata now located in the hanging wall and the strata within the current trishear zone. When P/S = 1, the hanging wall boundary of the trishear zone remains fixed in the material, and no growth triangle forms (Fig-

9 _ 648 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR A. P/S = 2.0 growth "triangle." bounda throu. h the rock '" " :'.: : : hanging tdshe bounda wa ;:> ;. : :.. j *:..... ================================================ :..=.>::., :.:.:.:.: Figure 8. (a) Detail of the growth triangle produced by propagating trishear zones (P/S > l) in growth strata. The slope of the fixed axis in the growth strata is directly related to the rate of propagation of the tip line. (b) With a P/S = 1, the hanging wall boundary of the trishear zone is fixed in the material, and no growth triangle occurs. ure 8b). There is an abrupt increase in strain across the hanging wall boundary of the trishear zone and a distinct kink forms in both growth and pregrowth strata. 4. Strain and Fracturing One of the most useful aspects of the forward modeling is that the strain throughout the structure can be predicted (Figure 2). Initial circles are described as an array of points and are spaced evenly along each bed in the model. Each time the model is iterated, the displacements of these points are calculated. True distortion only occurs within the trishear zone; in the hanging wall, the points are only translated, and in the footwall they are fixed (Figure 1). Where P/S > 1, material deformed in the trishear zone becomes incorporated into the hanging wall (Figure 2b, c). Because the strain is very heterogeneous, the deformed array of initially circular points is only approximately elliptical and represents an average strain over the region covered by the points. The larger the initial circle is, the less truly elliptical are the points in the deformed state. Nonetheless, they yield a good first approximation to the strain distribution throughouthe structure. The strain in the trishear zone can be accommodated in a vari- ety of ways, depending on the lithologies present. Weak units such as shale or evaporites may flow or experience intricate small-scale duplexing. More competent units surrounded by weaker ones may experience tight folding. Massive units, including basement, may fracture intricately. Trishear kinematics does not dictate which of these processes will occur; it provides nothing more than a bulk kinematic description of the deformed region and the strain path by which it arrived at its present configuration. Comparison with analog clay models of extensional forced folds [Withjack et al., 1990] demonstrates the utility of the trishear model strain predictions for line length balancing and for understanding fracture orientation and distribution. Withjack et al. [ 1990] and S. Hardy and K. McClay (Kinematic modelling of extensional forced folds, submitted to Journal of Structural Geology, 1998) showed that such folds form in triangular regions above the tip lines of planar normal faults (Figure 9a). In the clay models, much of the deformation is concentrated in the hanging wall of the normal fault (Figure 9a). This pattern is mimicked in the trishear forward models: the largest strain magnitudes are observed in the hanging wall of the normal fault and

10 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 649 in the hanging wall of the projection of the normal fault into the trishearegion above the tip line (Figure 9a). This occurs because there is a net transfer of material from footwall to hanging wall in the normal fault case (the opposite occurs in thrust faults), as originally recognized by Erslev [ 1991]. There is an exceptionally good fit, in both magnitude and orientation, between the trishear predicted stretch and the line length stretch measured from offset markers in the experiments of Withjack et al. [ 1990] (Figure 9c). Shear planes commonly occur along lines of no finite elongation (LNFEs), as in the classic card deck shearing experiment known to all structural geology students. This is also why bed length balancing works in parallel folding: the beds are LNFEs that do not change length and must have shear parallel to them. There are two such lines in any area-conserving, two-dimensional strain. In the case of Withjack et al.'s [1990] extensional forced fold, the LNFEs from the trishear model fit remarkably well with A. Original clay model of Withjack et al. [1990] B. Trishear model with predicted strain magnitude and orientation C. Trishear predicted strain & bed length balance in D. Lines of no finite elongation and fractures in clay model clay model Trishear Strain Ellipse... I... :,._..,A..,...-A.., N :--..,a z'- -, X... ::::::::::::::::::::::::::::: -.. :.. ::::...,..,.../....;,, principal stretch = 1...!i" ' '"': '"": :ii? " " ',// -/. X i!12., ' '" " :i" ' 't,... X... '?" '. ß.... $ = - i = 1.9' r' ' l' 'x' Figure 9. Trishear modeling of extensional forced folds in analog models from Withjack et al. [1990](AAPG!990, reprinted by permission of the American Association of Petroleum Geologists). In all illustrations, the shaded lineshow the actual beds, and the irregular solid lineshow the distribution of macroscopic fractures in the clay model. (a) Sketch of the original clay model, (b) best fitting inverse model and forward modeled strain magnitude and orientation superimposed on clay model. The stretch contoured is that along the greatest principal axis of the finite strain ellipse. Note concentration of strain magnitudes in the hanging wall. (c) Detail showing close match of trishear predicted stretch and line length balance. (d) Comparison of fracture pattern in clay model with lines of no finite elongation (short ticks in cross pattern) in trishear model. Note close coincidence orientations. Horizontal ruled area is where reverse faults are observed in Withjack et al.'s model; shaded region is where trishear model predicts reverse faulting.

11 650 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR the observed fracture patterns in the clay (Figure 9c), even to the point of predicting where high-angle reverse faults will form in this extensional system. The second set of LNFEs, antithetic to the main fault zone, does not correspond to fracture planes and simply rotates passively during the deformation. by trial and error and tedious comparison with existing deformed sections, converge the final geometry matching the structure of interest. There is, however, a better way Inverse Procedure 5. Inverse Modeling of Real Structures The inverse method takes advantage of the fact that the trishear kinematics is reversible; one can run models backward to In general, one would like to fit trishear models to real struc- unfold beds to their original, approximately planar orientations. tures. In the parallel kink fold case, one can measure panels Although it would seem to make no difference, it is far easier, in where the strata have coherent dip and, using the geometric rela- practice, to evaluate the goodness of fit of a model by how well it tions of Suppe [1983] and Suppe and Medwedeff[1990], make restores the beds rather than by how well it deforms them. This is predictions abouthe geometry and magnitude of slip on a fault, because the initial state (approximately planar beds) is much the primary variables. This is not possible in the trishear case be- simpler than the final state (complexly deformed beds), and there cause of the continuously varying nature of the bed orientations. are simple statistical descriptions of that initial state. The inverse Furthermore, there are more unknown parameters in trishear method finds a best fit initial geometry, and then a forward model kinematics: (1) fault ramp, (2) slip, (3) propagation-to-slip ratio, of a smoothed version of the initial geometry can be used to (4) trishear apical angle, and (5) tip line position, which is actu- model the strain in a structure. ally two parameters (X and Y coordinates, or a vector magnitude One can invert for all six parameters mentioned above by perand angle). These can produce a broad array of possible fold forming a grid search across a prespecified parameter space. The geometries, not even counting the hybrid structures described in statistic used to evaluate goodness of fit is the simple least section 3. One could generate a series of forward models which, squares linear regression, carried out by minimizing Z 2 as de- A ß 74 ø B 112 ø 108 ø 104 ø I I I I '"l anticline ) \ -' :atskill Mountains o o o 38 ø 42 ø 34 ø I I I Figure 10. Geologic sketch map showing the locations of the two structures used to demonstrate the inverse method. (a) Hudson Valley fold-thrust belt in eastern New York state, simplified from Marshak [1986]. Horizontal hatch pattern shows the outcrop belt of Silurian through lower Middle Devonian in which the thrust belt is developed. Western edge of the Taconic allochthon is shown with the barbed line. Inset map shows location in New York State. (b) Laramide Rocky Mountain foreland province of the western United States. Barbs are on the upper plates of the thrust faults; arrowshow the vergence of the monoclines. Inset map shows location in the western United States.

12 ALLMENDINGER' INVERSE AND FORWARD MODELS OF TRISHEAR 651 scribed by Press et al. [1986]. Because the geometry of one bed in a cross section is commonly known much better (i.e., is better constrained by real data) than the rest, the program currently finds the best fitting model for that bed only and then evaluates how that model applies to the other beds in the section. Grid searching is a brute force method that, when used to find all six parameters over a broad range of values with small step size, can be extremely time consuming. For example, searching for the best fit tip line in a 200 by 200 unit area with a one unit step means calculating and evaluating 40,000 models for each unique combination of the other parameters. If, in the same run, one also specifies 20 different ramp angles, 20 different P/S values, 20 different trishear angles, each run out to 500 displacement units, the 11 program. will test 1.6'10 individual models. Fortunately, in many cases, the tip line position and ramp angle are well constrained by outcrop or seismic data, and thus the number of models can be reduced substantially. For the above example, the number of individual models tested would be reduced to 2* 10, which can be carried out on a modem desktop computer in less than 5 min. If more than one bed is well known, the grid search can simply be repeated for that bed, and the best average model can be used. For the reasons discussed in the section 5.3 this approach is justified at present. both have P/S = 2.5, trishear angles between 30 ø and 35 ø, and displacements in the mid-300 unit range. The minima are slightly better defined using bed 3 as the key bed (Figure 12b), but the differences are minimal. Several consistencies among the various models are striking. The best models for beds 1 and 4 require consistently less shortening ( units less) that those for the other beds. This may have to do with the fact that those wo beds have sharper local curvature than the others. Also, in all three cases, the curves for beds 4, 5, and 6 "plateau" at small displacements beyond their best models. This is related to the fact that they are located farther from the final position of the tip line. Strain diminishes at distances away from the tip line, meaning that those beds rapidly reach a point where they are as linear as they are going to get; subsequent displacement will do little to change their geometry Application I show the application of the inverse modeling approach by applying it to two previously published cross sections. It is not my intent to prove that the trishear modeling approach s superior to that described in the original articles. Indeed, there are many reasons, some discussed in section 5.3, why indiscriminate application of the method to published cross sections could be misleading at best. These examples stand solely as a demonstration of the modeling procedure. Note that, although both examples are thrust faults, the exact same procedure can be applied to trishear normal faults, as in the case of the extensional forced folds, discussed in section 4 (Figure 9) Hudson Valley Fold and Thrust Belt. The Hudson Valley fold and thrust belt of eastern New York (Figure 10a) [Marshak, 1986] has some splendid outcrop-scale examples of fault-propagation folds. Even though no basement is involved in this deformation, many of these structures display upward shallowing dips on the forelimb due to thickening in the core of the tip line syncline. Figure 11 shows the results of this procedure applied to one of these structures (a tracing of Figure 16 in the work of Mitra, [1990]). In this example, the position of the tip line and the dip of the fault are known; thus the search is for just three parameters: P/S, trishear apical angle, and displacement (Figure 12). This grid search was repeated with beds 1, 3, and 6 as key beds. At present, one can make only qualitative comparisons among the results from using each of the three beds as the key bed for restoration. Bed 1 clearly yields an inferior solution (Figure 12a): not only is the least squares fit for that bed relatively poor compared to the others, but the best fitting model for bed 1 produces broad, poorly defined minima in the chi-squared plots for the other beds, particularly beds 4, 5, and 6. The poor fit is due, at least in part, to the fact that bed 1 is faulted. Points on the bed near the fault may be poorly restored, as can be seen in Figure 1 lb, resulting in strong local deviations from the linear model. The best models for beds 3 and 6 produce very similar solutions: B. / / Figure 11. Inverse model of a fault-propagation fold from the Hudson Valley fold and thrust belt. (a) Simplified tracing of a photograph which was published as Figure 16 in the work of Mitra [1990]. Bed tops are labeled as referred to in the text. (b) Best fit restoration to a planar state for bed 3. Dashed line shows starting position of hanging wall boundary of the trishear zone. Diagonaline across beds 1 and 2 is the restored position of the final fault cutoffs across those units. (c) Forward model using best fit parameters. Original data are shown dashed beneath the model. Bold ellipses occur every fifth ellipse; at the start of the deformation, they were circles aligned vertically and perpendicular to bedding.

13 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR A ' ' ' Key bed = 1 = 3.0 Apical angle = 30 ø B,,'1 f B. 5oooo Bedd Bed/ Be, d o 2oooo Bed 2 ø20000 C= \ \ k,, %./'.,.-/j "-Bed-5 ed-4 Bed ,...,... i...,... i...,...,...,... i Displacement D= y be 0 Apical angle = 34 ø 0... '... : )U... -'1000 Displacement 40000' 30000' : Be 10000: i Bed Bedd =6 _- = 3 o...!...!... a...,... a...,--.!,.! Displacement Grid Search Parameters minimum maximum step ramp angle 36 ø 36 ø -- displacement P/S trishear angle 20 ø 60 ø 1 (tip line is also fixed) Figure 12. Summary of best model statistics for the Hudson Valley fold and thrust belt example shown in Figure 11. (a) Curves of chi-squared versus displacement for all beds for the combination of trishear parameters yielding a best fit for bed 1; the best model for each bed occurs at the minimum chi-squared. (b) Same as Figure 12a but with best fit parameters for bed 3, (c) same as Figure 12a but with best fit parameters for bed 6. (d) Grid search parameters. Ideally, the best fit model for each individual bed should be similar to that for every other bed. Visually, one looks for tight curves with well-defined minima near the same value of displacement. Similar curves can be constructed for any combination of two parameters. Restoration with bed 3 as the key bed appears to be the best overall restoration. Forward modeling using the parameters of the best model produces a geometry remarkably similar to that of the initial deformed section (Figure 11 c). The only areas of serious discrepancy between model and section occur where the real rocks are in kink-like areas of tight curvature, particularly in bed tops 4, 5, and 6. The trishear model provides a simple explanation for the folding observed in the footwall below the stratigraphic level of the tip line. Such features cannot be explained in kink faultpropagation folding without resorting to a ramp breakout across the anticlinal forelimb [Mitra, 1990; Suppe and Medwedeff, 1990]. Despite the excellent fit between model and observation, one curious result, the 9 ø difference in dip between the highest and lowest beds in the restored state across less than 10 m hori- zontal distance (Figure 1 lb), suggests that an artifact rather then a true view of a real structure has been modeled (see discussion in section 5.3) Rangely Anticline. Mitra and Mount [1998] have recently described a method for analyzing fault-propagation folds in basement faulted anticlines with triangular-shaped zones of deformation in the overlying sedimentary cover. I apply the inverse method to one of the structures tha they analyzed: the Rangely anticline (Figures 10b and 13). Their crossection of the Rangely anticline has a small anticlinal ramp near the tip; the main part of the fault dips 38 ø but changes to 28 ø within a few hundred meters of the tip. As shown in section 3.3, such changes can be forward modeled, but the inverse model works only for a single fault dip, so 38 ø was used. The position of the tip line is not precisely known (at leasto me) but is limited to a small range of values.

14 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 653 O' -2OOO 5 - B. / starting position of the hang- wedge due ramp in original section,ng wa!l trish. boundaty,._ Pm tnsnear angle = 76,o[ o Figure 13. Inverse model of the Rangely anticline. (a) Rangely cross section modified from Mitr and Mount [1998](AAPG 1998, reprinted by permission of the American Association of Petroleum Geologists). Vertical lines on the section are petroleum exploration wells. (b) Best fit restoration using bed 9 as the key bed. (c) Forward model using best fit parameters. Original datare shown dashed beneath the model. Bold ellipses occur every fifth ellipse; at the start of the deformation, they were circles aligned vertically and perpendicular to bedding. Therefore the grid search involves a larger number of parameters smaller (with the exception, of course, of bed 1), indicating a than previously: trishear angle, P/S, displacement, and the X and better overall fit than in the previous case (Figure 14b). However, the minima for some of the beds are stretched out, and best fit Y positions of the tip line. As before, three different bed tops as key beds (1, 5, 9) are analyzed as key beds. Because I do not displacements range from-160 to -220 model units. The overall have access to the original data from which the crossection was best combination of parameters for all beds is obtained when bed constructed, the constraints each of these beds is unclear. top 9 is used as the key (Figure 14c). With those parameters, the Clearly, bed top 9 has the best control in depth, as it is pierced by curves for all beds are tight with well-defined minima, and the all of the petroleum exploration wells. Top 5 was drilled in only best fit displacements vary between just-180 and -200 model one area, and top 1 is pierced nowhere. units. Even the Z 2 minimum value for bed top 1 is only twice that for the best fit for bed 1. The results of the statistical analysis for each of the key beds (Figure 14) also indicate that bed top 9 is the most reliable. The The best fit parameters for bed top 9 (trishear angle of 76 ø, P/S curves of Z 2 where bed 1 was the key show poor linear fits to all of 2.3, slip of-4200 m) are put into a forward model the Rangely of the other beds (Figure 14a). This occurs because the program anticline to compare to the original crossection (Figure 13c). finds the best combination of values which unfolds the sharp cur- Overall, the fit is remarkably good, particularly where there is vature in bed 1 near the fault, particularly in the footwall. Be- well control. The 4200 m of displacement exactly what one cause this curvature is greater than for any other bed, the best pa- would predict by projecting the straight (i.e., unfolded) parts of rameters for bed 1 produce a broad syncline all the other beds. bed top 1 to the fault (see dashed line in Figure 13c). The actual The minima in curves that result when bed 5 is the key bed are slip marked by the cutoffs bed 1 against the fault reflects the

15 , ß 654 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR A B ' 40000' : : i Key bed = _1 Slip P/S 2.8 Apical angle 4,6 ø O... i... i... i... i... i... i... i... i ' 000 Displacement C D= P/S = 2.8 Slip -218 Ap, ical angle= 5,4 ø... i... i... i... i... i... i... i Displacement 40000' i 10000, 0 AKey bed = 9 Slip P/S = 2.4 pical angle = 76 ø... i... i... t... i... i... i... i... i... i Displacement Grid Search Param.ete: S minimum maximum step ramp angle 38 ø 38 ø m displacement P/S trishear angle 20 ø 85 ø 1 tip line 60 tip line positions tested (note that each model displacement unit is 21.6 m) Figure 14. Summary of best model statistics for the Rangely anticline. See Figur t2 for explanation. (a) Curves of Z 2 versus displacement for all beds for the combination of trishear parameters yielding a best fit for bed 1, (b) Same as Figure 14a but with best fit parameters for bed 5, (c) same as Figure 14a but with hest fit parameters for bed 9. (d) Grid search parameters. Bed 9 as the key bed provides the best overall fit. up-section decrease in slip due to the fault-propagation fold the fault. This thickening occurred in the trishear zone above the kinematics. The fit of the forward model to the original cross tip line of the fault. Once fully incorporated into the footwall, no section is worst near the fault, as might be predicted from the further strain occurs. restoration in Figure 13b). The trishear forward model predicts much less tight folding near the fault. It is likely thathe original 5.3. Caveats data on bed dips are least reliable near the fault, and thus the mismatchere could be due to poor constraints the original A good statistical fit does not guarantee a good, or even a reasection and not to an incorrect trishear model. The total thickness sonable, model af reality. The two examples illustrate not only between bed tops 1 and 10 is well matched in the hanging wall, the application of the inverse modeling, but they also highlight but the trishear forward model predicts mild thickening of beds in differentypes of potential pitfalls which may await a user the footwall, particularly within 7 or 8 km of the fault. This oc- blindly using the program. The Hudson Valley fold and thrust curs because the trishear angle used is quite open (76ø). The belt example provides a rather complete geometric description of strain ellipses predict a component of extension perpendicular to the structure because it is all contained within a single outcrop. the layers throughout the footwall but particularly approaching The ramp angle, tip line position, and tops of the beds can all be

16 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 655 seen in the original photograph [see Mitra, 1990, Figure 20]. However, a photograph may be subject to problems of depth of field (different parts of the photo being at different distances from the lens), perspective, and the flattening of a three dimensional morphology into two dimensions. Furthermore, one must take care to be sure that the photo provides a true down-plunge view of the fold to be modeled. I suspecthat the extreme wedging of the stratigraphic units in the restoration in Figure l lb probably occurs because the right-hand side of the cross section (left hand side in the original photo) was more distant from the camera lens, and therefore the beds look thinner there. This does not negate the modeling procedure described above but suggests that it is just as easy to model artifact as real data. One could argue that, in this case, the modeling has actually helped to identify a potential artifact. The model of the Rangely anticline is subject to a different problem: in this case an interpretation rather than data has been modeled. Without access to the original seismic data, it is difficult to evaluate independently the goodness of fit of the model, which of the bed tops is really the best "key bed" to use, etc. In fact, whenever one models anything but outcrop data, the model will mostly be inverting interpretation of limited data with a broad range of confidence limits. Seismic reflection data are subject to errors inherent in the conversion from time to depth, picks in areas without good well control, processing artifacts, and areas of poor data quality. Again, the trishear modeling can help improve the interpretation in several of these instances if one knows independently that trishear is the appropriate kinematics. For these reasons, any attempt at more sophisticated inverse modeling, such as finding the model which minimizes the misfit for all beds simultaneously, while technologically possible (if computationally intensive), may be unwarranted. One would have to assign quantitative confidence limits to all of the observations and interpretations. In an extreme example, bed top 10 of the Rangely anticline is mostly interpreted because the real bed has been eroded away. Though the interpretation of where it should be (based on entirely different kinematic assumptions of Mitra and Mount [1998]) is included in the model, it makes no sense to attempt to minimize its misfit. Finally, the inverse procedures described here can only be applied to simple trishear structures in which there is no change in P/S, trishear apical angle, or ramp angle through time. The hybrid structures described earlier in section 3 can only be forward modeled. In both of the examples above, some of the misfits between model and data may be due to the fact that both data (in the case of the Hudson Valley example) and interpretation (Rangely) suggest a flattening of the thrust near the tip line, producing an anticlinal ramp. 6. Discussion Trishear is an undeniably powerful way to model structures and provides a reasonably simple explanation for a variety of structural complications which are much more cumbersome to explain with other kinematic approaches. The strikingly different kinematics of trishear would seem to put this model in direct competition with now classical parallel kink fold models, but this is not the case. Because trishear provides a bulk description of a deforming zone, it is complementary to those models, rather than an incompatible alternative. Trishear explains the gross thicken- ing or thinning between units but dictates nothing about the specific structural geometry and processes by which those strains occur. Duplex thickening in an anticlinal stack or passive roof duplex can be modeled with trishear, even though the detailed structural geometries obey all of the Dahlstrom and Suppe rules. Thus it is not just a technique for modeling basement cored structures (as the Hudson Valley example in section demonstrates), but neither is it a replacement for existing crosssection balancing techniques. Because strain can be accommodated in a variety of ways, no one strain indicator will, necessarily, match the strain ellipses predicted in trishear forward models. To do so, a single indicator would have to accrue strain throughout the history of the structure. For example, it may be that much of the late strain in a structure is produced by out-of-sequence thrust faults, after faults within forward breaking duplexes have locked up and ceased movement. Likewise, intracrystalline strain may only be recorded during specific steps or events in the formation of the structure such as during early layer-parallel shortening or late stage locking of folds. The most reliable measure of strain in the structure is the change in bedding thickness itself. If a trishear forward model matches the thickness changes throughout a real structure, then the strain predicted by the forward model must be at least consistent with, if not a unique description of, the bulk strain in the real structure. One area in which the trishear technique is particularly useful is in the prediction of gross structural geometries in areas of poor (or nonexistent) subsurface data. For simple structures, one can invert the existing data for a best fit model. The model allows one to answer such questions as the following: how tightly are the deeper units folded? What is the most likely displacement for the structure? What were the apical angle and the P/S ratio? Where are fractures most likely to be concentrated? How much of the unit thickening or thinning is accommodated by lateral flow of material and from where? How deep was the tip line of the fault (i.e., the nucleation point) at the start of folding? Once these broader-scale issues have been addressed, one can then concentrate on the detailed structural geometry and the processes by which the deformation occurred. 7. Conclusions Trishear kinematics provides an excellent description of the bulk geometry and deformation associated with fault-propagation folds. By allowing changes in the basic parameters (trishear apical angle, propagation-to-slip ratio) and by permitting the formation of second ramps and trailing fault-bend folds through time, many different geometries can be modeled. For simple trishear fault-propagation folds, the inverse method presented here provides, for the first time, a rational statistically valid basis for selecting the appropriate trishear parameters that most closely duplicate the real structure of interest. The method can be applied either to thrust or normal fault-related folds and either thick- or thin-skinned deformation. Perhaps the greatest remaining unknowns concerning trishear structures are the physical factors that control the apical angle and propagation-to-slip ratio. Because the inverse method gives us an objective method for determining best fit to real structures, we can now address these un- knowns.

17 656 ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR Acknowledgments. I am indebted to Ren6 Manceda of YPF, S.A., for showing me an outcrop in southern Mendoza Province, Argentina, that sparked my interest in trishear. Ren6's and Tom s Zapata's (also YPF, S.A.) enthusiasm for my initial modeling attempts inspired the current effort. YPF, S.A., provided support for my visit, and I am grateful to Rafil Gorrofio and Ricardo Manoni for arranging all the details. That I was thinking of trishear at all at the time is thanks to Stuart Hardy and to the AGU Editor who sent the Hardy and Ford paper to review for Tectonics. It was their simple and precise mathematical description of tris- hear that enabled me to write so quickly the modeling program described in this paper. I am further indebted to Stuart Hardy for suggesting that I model Withjack et al.'s [1990] extensional forced folds. Ben Brooks constantly encouraged me to pursue a rigorous inverse model and advised me on how to implement it, although we both know that more can be done in this area. Reviews of the manuscript by Hardy, Brooks, Zapata, Francisco Gomez, and Tectonics reviewers Eric Erslev and Don Medwedeff are much appreciated. Thanks also to Editor Dave Scholl, who suggested that I make the paper dirtier. This work was supported in part by National Science Foundation grant EAR References Anad6n, P., L. Cabrera, F. Colombo, M. Marzo, and O. Riva, Syntectonic intraformational unconformities in alluvial fan deposits, eastern Ebro Basin margins (NE Spain), in Foreland Basins, vol. 8, Spec. Publ., vol. 8, edited by P. Allen and P. Homewood, pp , International Association of Sedimentologists, Crider, J.C., M.L. Cooke, E.J.M. Willemse, and J.R. 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