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TECTONICS, VOL. 17, NO. 4, PAGES 640-656, AUGUST 1998 Inverse and forward numerical modeling of trishear faultpropagation folds RichardW. Allmendinger Departmentof GeologicalSciencesandInstitutefor the Studyof the Continents, SneeHall, CornellUniversity, Ithaca, New York Abstract. Fault-propagationfolds commonly display footwall synclinesas well as changesin stratigraphic thicknessanddip on their forelimbs,featuresthat cannoteasilybe explainedby simple parallel kink fold kinematics.An alternativekinematicmodel, trishear,can explain theseobservations, as well as a variety of other features which have long intrigued structuralgeologists. Trishear has received little attentionuntil recently, in part becauseit must be appliednumericallyratherthan graphically.A new computerprogramhas been developedto analyzetrishear and hybrid trishear-fault-bendfold deformation.Trishear fold shapecan vary considerablyby changingthe apical angleof the trishearzone and/orthe propagationto slip ratio (P/S) duringthe evolution of the structure. Breakouts, anticlinal and synclinal ramps,andinversionstructures canalsobe modeled,trackingthe kinematicswith growth strata.Strainwithin trishearzonescanbe usedto predictfractureorientationsthroughoutthe structuresas demonstrated by comparisonwith analogclay models.Also presentedis a methodfor invertingdataon real structuresfor a best fit trishear model by performing a grid search over a sixparameterspace(rampangle,trishearapicalangle,displacement, P/S, and X and Y positionsof the fault tip line). The inversionis performedby restoringa key bed to a planarorientationby least squaresregression.Becausetrishearprovidesa bulk kinematic descriptionof a deformingzone, it is complementary to, rather thancompetingwith, otherkinematicmodels. 1. Introduction The fault-relatedfolding in thick-skinnedtectonicprovinces suchas the Laramide Rocky Mountain forelandor the Sierras Pampeanasof westernArgentinahaslong challengedstructural geologists[Erslev,1991;Erslevand Rogers,1993;Matthewsand Work, 1978; Mitra and Mount, 1998;Narr and Suppe,1994; Reches,1978]. The basementrocksin theseprovincescommonlydo not displaythe layered,stratigraphic anisotropies thatarethought to control fold kinematics in thin-skinned thrust belts, where layer-parallelshearand parallelfoldingis considered the norm. In thick-skinned provinces,basementandthe overlyingstrataare commonlyfolded over the tips of propagatingfaults.The now classicalmodel fault-propagationfolding based on kink geometries[Suppe,1983;Suppeand Medwedeff,1990] doesnot explain very well the broadcrestedanticlinesand monoclinesin theseprovinces.Furthermore,it haslongbeenrecognizedthat, even in thin-skinned provinces,fault-propagationfolds with changesin forelimbbeddingthicknessanddip are common(Figure la). Early attemptsto model suchcaseswere purely geometric exercises [Jamison, 1987; Mitra, 1990] or kinematic exercises but with extremely restrictive assumptions[Suppeet al., 1992; Suppeand Medwedeff,1990]. Erslev [ 1991] proposeda strikingly different, kinematically explicit model for fault-propagation folds, the "trishear"model, in which many geometriescan be reproduced.This model has receivedrelativelylittle attention,perhapsbecauseit mustbe implementednumericallyand therewere no generallyavailableforwardmodelingprograms. Recently, Hardy and Ford [1997] expandedErslev's [1991] initial trishear model. They presenta clear mathematicalformulation of the problem,have analyzedthe effect of variablepropagationto slip ratios,and have illustratedgrowthstratageometries associatedwith trishearfault-propagationfolds. Their computer program representeda first step in a general trishear forward modelingprogram. I have applied Hardy and Ford's [1997] mathematicalanalysis in a completely new computerprogramwhich allows great flexibility in the descriptionof the startingparameters,variations in parametersduringthe analysis,strataof variableinitial thicknessesand dips, and strataaddedduringthe growthof the structure (although surfacetransportand base level changesare not included). The program is used to producea seriesof simple, multistage forward models to demonstratethe array of ideal geometriesthat can be producedby trishear.The basicprocesses modeled, a combination of fault-bend and fault-propagation folding, breakouts,inversionstructures,and progressiveand instantaneousrotation in growth strata,are alreadywell known in the literature, but the use of a trishearapproachputs them in a new and different light. Then, a new inversemodelingapproach for analyzing real structuresis introduced.One of the casesto whichthe inversemethodis appliedshowsthat trishearis not restrictedto thick-skinnedtectonicprovincesbut also occursin thin-skinnedregions. 2. Kinematics of Trishear In the trishearmodel, a singlefault in "basement"expands outwardinto a triangularzone of distributedshear(Figure lb). The reasonfor the triangularshapeof the shearzone mustultimately lie in the still largely unexploredmechanicsof trishear. Blind faultslike thosemodeledhereareessentially largemodeII cracks. Theoretical Copyright1998by the AmericanGeophysical Union. studies of the stress field around mode II cracksshowthat thereis a triangularregionof high shearstress concentrationaround the tip (Figure l c) [Pollard and Segall, 1987]. Erslev [ 1991] and Erslevand Rogers[ 1993] showedthat Papernumber98TC01907. 0278-7407/98/98TC-01907 Deformation $12.00 64O downward steepening dips A. 0 641 Bearpaw 1000 rn Gardiu • '•---"•"---.--•• • footwall synclines B.Trishear Kinematics C.maximum shearstress,modeII crack Figure 1. (a) Much simplifiedcross-section of the Turner Valley anticline,foothillsof the CanadianRocky Mountains [modifiedfrom Gallup, 1951](AAPG ¸ 1951, reprintedby permissionof the AmericanAssociationof Petroleum Geologists).Sectionhighlightsseverallong standingproblemsin balancingfault-propagationfolds. (b) Basic trisheargeometryas describedby Erslev [1991] and Hardy and Ford [1997]. (c) Contourplot of maximum shear stresses at the tip of a model II crack.Note symmetrictriangularregionof high stresses at cracktip. Crack model is basedon linear elasticityfracturemechanicsas describedin Pollard and Segall [1987]; plot was producedusing notebooksof the computerprogramMathematicadescribedby Crider et al. [ 1996]. to conservecross-sectionalarea the triangular zone must be symmetric with respectto the fault. At the top of the trishear zone,slipvectorsareequalto thatof thehangingwall: theyare parallelandequalin magnitude to themaster fault.At thebaseof the trishearzone,the slip is zero. Within the trishearzone,the slipvectorvarieslinearlyin magnitude andorientation fromtop to bottom[Hardyand Ford, 1997].Thusthe directionof shear variesfromthe dip of the fault to the dip minusthe half apical angleof the trishearzone.Althoughthe displacement field is easyto calculate, it mustbe doneiteratively,andthereforethe methodcannotbe appliedgraphicallyor analytically. The apex of the trishearzoneis locatedeitheron the tip line of the fault (attachedto the hangingwall in Erslev's [1991] terminology), or it is attachedto the footwall. Hardy and Ford [ 1997] showthat thesetwo conditionsare preciselydescribedin termsof the propagation-to-slip ratio (P/S), which determineshow rapidly the tip line propagatesrelative to the slip on the fault itself. Footwall-attached trishear zones have a P/S ratio of zero, whereas in Erslev's [ 1991] hangingwall attachedtrishearzones,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 resultin pronounced forelimbthickening andtightfolding 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR 3. Variable Trishear Forward Modeling In variable trisheardeformation,variousparameterscan be changedat any time duringa modelrun.With respectto the trishear zone itself, either P/S or the apicalangleof the triangular zone can be varied during growth of the structure.Factorswhich might producea changein the trishearangle or P/S during deformationare unknown.It seemslikely that mechanicalproperties of the lithologicsequence,strainrate, and perhapsvariable fluid pressuremay play a role. A ramp in the fault may alsoform duringgrowthof the structure, producinga fault-bend fold. Finally, beds can be added duringthe formationof the structure,simulatinggrowthstrata. Severalwell-known typesof structuralinteractionscan be modeledfrom a trishearperspective.Sections3.1-3.6 demonstrate the effectsof varyingmodel parametersthroughtime, emphasizing the final geometry.Though growth strataare shownin all models, an explicitdiscussion of the growthgeometries is saveduntil the end of this section 3.6. 3.1. ChangingP/S Ratio Through Time Hardy and Ford [ 1997] showedthat the style of folding dependson P/S. If P/S is large,thenany materialpointspendsless time within the trishear zone than if P/S is small, and thus it is D. P/S = 2.0 / I lessdeformed,and the folding is more open.If this ratio varies throughtime, the macroscopic effectwill be thatbedsat different stratigraphiclevelswill displaydifferentdegreesof folding.The changein P/S during deformationseemslikely to be a common scenario.It may happen,for example,when the tip line entersa unit of differentmechanicalproperties,onethatis overpressured, etc. Molinero et al. [ 1996] have suggestedthat variableP/S occurred during the developmentof one of the structuresin the Ebro Basin. Figure 2. Illustrationof the effectsof varyingpropagation to slip ratio: (a) P/S - 0, (b) P/S = 1, (c) P/S = 1.5, and (d) P/S = 2.0. All modelshave the sameslip; only the propagationof the tip line varies.Strainellipsesdocumentvariationwith kinematics. in the trishear zone, whereasP/S > 1 resultsin less thickening, more open folding, and in folding of the hanging wall, even though the trishear zone is attachedto the tip line [Hardy and Ford, 1997]. This occursbecausethe hangingwall boundaryof the trishear zone must migrate through the material of the hanging wall as the tip line propagates.This migrationhassignificant consequences for growth stratageometries,as discussedin section 3.6. The strainfield within the trishearzoneis heterogeneous but continuous(Figure 2). Becausethe shearplanesare oblique to layering,the folding within the trishearzoneinvolveschanges in thicknessof the layers. In general,beds thicken during the early stagesof deformationbut then thin as they steepenand overturn later on. Because trishear has not been studied exten- sively, the physicalconditionswhich determinewhetheror not a trishearzone occursas well as the specificapical angle are not well understood. In Figure 3a, two episodesof low P/S were separatedby an episodeof high P/S. This producesoverturnedfolds at low stratigraphiclevels, more open folds at intermediatelevels,and overturnedfolds again at higher levels. The rapid propagationproduceda fold geometrywith relativelystraightlimbsanda narrow roundedaxial zonewhich couldbe interpretedas a kink surface. Within the growthstrata,a switchfrom high to low P/S produces a distinctkink, whereasthe reverseswitch,from low to highP/S, doesnot. Becausethereis little foldingat highP/S, the previous form of the fold, producedduringlow P/S, rapidlybecomespart of the hangingwall and is simplytransported along.In Figure3b, two episodesof high P/S are separatedby a period of low P/S. Predictably,this has just the oppositeeffect of the previous model: openfolds at low and high stratigraphiclevelswith overturnedfolds alongthe thrustin the middle (producedduringthe time of low P/S). 3.2. Variable Trishear Angles In trishear zones with small apical angles, intensestrain is concentratedin a narrow wedge of rock, whereasbroaderangles resultin more diffuse,lessintensestrain.Changingthe apicalangle during thrustingproducesgeometriceffects,which are particularly strikingwhen the angleis suddenlyreduced(Figure4a). This case results in an instantaneousincorporationof trishear zone material into the hanging wall, "freezing" its geometryas well as the focusingof straininto a smallerzone. Two pseudokinks are produced;the first is a more roundedfold hinge which 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 642 INVERSE AND FORWARD MODELS OF TRISHEAR 643 Variable Propagation to Slip Ratio (P/S) A. - 7 '•::•-:•-':•:::•-:'-'•:•-•-•-•:•:'•.•::•'•:-•:•:i•i,•--:-•/•:x.'-•:•.5 ................................. '?...... •¾!•iiiii?iii•:.-.-', 4 Slip = 210 Propagation = 420 •:?':':•'•'::::::•:::•*•%•:*•:..:::::•..:. '"':%,. ":iii ::•::•: • 2 ..................... .:..:,, .... ::: :, ........... Ramp angle = 30 ø irishear angle = 50ø ................ .................................... .............................................................. ß ":•'""• ......................... :................................. .?•:•::•..• ......... i.". ...... :........ i/..i:..'...i'.':i ............ '.'........ ::::::::::::::::::::::::::::: 0 250 5 e. Slip = 150 ß Propagation= 435 '•A"•,: • 2 1 Ramp angle = 30 ø irishear angle = 50 ø :•,.,.,.,.., ,•.•. ..,.,...,•-,,:.•},..,•.•!• ...... ::•:i•}!:.•:?,,..•:•,..,,..-•,•¾•.•::•.:.•:•:•i ................... :.::•-.-,,•..,-.•:.:•:.:.• ....................... • •::•:-2•!• ............................. •!•i;•::i!:.,• •" •'•'""•....'"•:g•:.: ................ ;?:*•:•".•i ............... ,•:.•:•.,:...-',•:.•:::•::•,•::•,-•..•.•,'-::..•i•,-.:.--:•::• :• ß ß'•' 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: ................................................................... ..,.::;,,.: ........... <.......... .•:.:....• ................................................... ::....................................... ß Figure 3. Variable P/S throughtime. (a) The top model was producedwith P/S = 1.5 until the depositionof bed 4, P/S = 5 until the depositionof bed 5, and P/S = 1.5 betweenthe timesof beds5 and 7. (b) The bottommodelwas formedby P/S = 5 until bed 1 time, P/S = 1.5 betweenbeds 1 and4, andP/S = 5 betweenbeds4 and 5. The half circleson the fault traceshowthe positionof the tip line of the fault (in the hangingwall andfootwall) when P/S was changed.Note thatbedsoverturnduringtimesof low P/S andareuprightduringhighP/S. corresponds to the initial positionof the hangingwall boundary of the trishear zone. Because this boundary was oriented at a higher angle to the fault zone, it migrated a greater distance throughthe rock as the tip line propagatedup section,producing the broadly rounded hinge. The dip panel to the right of the roundedhinge (but to the left of the currenttrishearboundary) are rocks which were within the trishearzone during the initial open angle but then suddenlybecamepart of the hangingwall. Thisdipp•el narrows downsection, asboththebroad andthe narrow trishearzone must have had the samevertex (i.e., the tip line). The second(right hand) kink (Figure4a) is more pronounced for two reasons.First, becausethe new trishearboundaryis oriented at a small acute angle to the slip vector for the hanging wall, it migrateslittle throughthe material as the tip line propagates. Second, the strains are concentratedin a smaller crosssectionalarea. The kink due to the secondhangingwall boundary of the trishearzone, as well as the switchfrom opento tight trishear angleis clearly markedin the growthstrata. The oppositechange,an initial narrowand later openapical angle,producesquite a differentanticlinalform (Figure4b). The right dippingpanel of stratais muchmore subtlein both growth and particularlypregrowthstrata.The openingof the apicalangle at a later stagein the deformationhas a smoothingeffect, smearing out suddenchangesin dips. 3.3. Anticlinal and SynclinalRamps A changein the dip of the fault duringmovementproducesa fault bendfold, eitheran anticlineor a syncline,which trails trishear zone (Figure 5). Becausethe bendin the fault is modeledas sharpchangein dip, the fault-bendfold is a kink fold which conformsto the geometryandkinematicsdescribedby Suppe[ 1983]. Synclineswhich preservebedding thicknesson both flanks can be producedby virtually any changein ramp anglebut anticlines which preservebeddingthicknessand have no angularshearin horizontalbedshave a very limited rangeof changesin fault dip (the angle { of Suppe[1983]). Becausethe trishearzone is symmetric aboutthe fault, a new ramp at a differentanglewill producea changein the orientationof the trishearzone, suddenlyinvolving rocks that were previously part of the footwall or the hangingwall. In the caseof a bend which producesan anticline(Figure 5a), the forelimb displaystwo prominentsteps:The higherof the two is produced by the fault-bend fold, and the lower, which is somewhat more rounded, was produced by the hanging wall boundaryof the trishearzone prior to the formationof the second ALLMENDINGER:INVERSEAND FORWARDMODELSOFTRISHEAR VariableTrishearApicalAngle A. 8 Slip= 240 I ..... .................................................................... ]:•"";'• I • • 4 Ramp angle= 30 øI P/Sratio= I 5 I .•.•,•... /':'•;••, /"'"•'•E • • 3 2 ..-•,•z::z:.:.•::•.. i.• ..; Figure 4. Variation inapical angle through time.(a)Model wasstarted withanapical angle of50øuntilthetipline reached theposition of thehalfcircle(inhanging wallandfootwall) atthetimeof deposition of bed5. Thenthe apicalanglewasreduced to 20øbetween beds5 and8 times.(b)Apicalanglewas20øuntilbed5 timeandthenwas increased to 50 ø between beds 5 and 8. ramp.The steprelatedto thetrishearboundary wouldbe sharper if a P/S ratio of 1 were used rather than the value of 1.5 that was usedto makeFigure5. Themodelalsoshowsa pronounced zone of forelimbthickeningand local, more subtleforelimbdip changes relatedto thehangingwall boundary of thetrishearzone at the end of the secondramp.The forelimbdisplaysgrowth geometriesreflectingboth progressive rotationduringtrishear andinstantaneous rotationat higherstratigraphic levelsformed nantkinematics onthetwolimbs:thebacklimbshows a typical growthtriangle[Suppeet al., 1992], whereasthe forelimbdis- playsfanning of stratacharacteristic of composite progressive unconformities [Anaddnet al., 1986;Hardy and Ford, 1997; Riba,1976].Theforelimb growth strata deposited during movementon thesecondrampmigratetowardthecrestof theanticline because the steepening of the fault rotatesthe trishearzoneto a higherangle,"focusing" theupliftfarthertotheleft. by kink bandmigrationin the fault-bendfold. Thesesamestrata (labeled4-7 in Figure5a) wouldalsoshowprogressive rotation hadthemodelbeenplottedfartherto theright. Wherethe secondrampis steeperthanthe first, a synclinal fault-bend fold results(Figur. e 5B). The forelimbgeometry is produced solelyby thevariationin trishearangle.Theupperpart of the forelimbis relativelyplanareventhoughit lieswithinthe trishear zone;thelowerpartis notablysteeper. Thechange in dip betweenthetwo partsof the structure records thepositionof the hangingwall boundaryof the trishearzone before the second rampformed.The backlimb reflectsonlythefault-bend foldgeometry;it is planar,the kinksare angular,andit followsSuppe's [1983] geometricrelations.The growthstratareflectthe domi- 3.4. Breakouts A break-out forms when the fault associatedwith a fault- propagation fold cutsrapidlyacross the stratigraphic section, abandoning thefoldasa relicof theformer position of thetip line.In terms of modified trishear, thisphenomenon isprecisely modeled asa large,sudden increase in theP/Sratio;thetipline moves rapidly away fromthestructure, which canonlyhappen if thepropagation ismuch larger thantheslip. There arethree general types ofbreakouts [Mitra, 1990; Suppe andMedwedeff, 1990]: along theaxisofthetiplinesyncline, asa decollement atthestratigraphic levelofthetipline,orbycutting 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 644 INVERSE AND FORWARD MODELS OF TRISHEAR 645 V=riable Ramp Angles tx I • I Slip=210 '•! trishearangle= 40ø I Propagation = 316 / '-•-- , 7 fault-bend told growth triangle '•::•?;i•.:_ ' '" • / 6 "•"'":':':'""'"":•....:.•...-•,.• • ------2 ............................ ii•-•-•.:• .... I /"•::::•'.'•iii:.-."::ii.':-"::•-•::•::. • "•':'•::•i• .... '........ / iI -:•:•:• ......................... :.:.:.•....... :..: ........ .:......... ............... ..::..::::::. .............................................................................. •..• 0 ..................................... '-' "':'"•':"-• '•'"=•":'"'"•'" fault-bend fold_ L.•_-.•'__-• growth tdangle•"'•• _• • ½ ///•:• ............................ • .................... • • 7 / 6 ,• 4 Propagation = 318• t•sheara•le= •o • 2 1 P/S ratio =1.5 • . ,.:•-:,•::•-• •,•-•:=:,,,, ...•-•. :•-<•.-..•:<<<.:• ................................ •s:•:..•:.:..,•,•:..•:..•,•::::-.•::•:-,-.,:•:::•:::• ............ ./'"'" :•••• ...... ••••,:•,•••••,.•.,•:•:.•,• ................... •.,....,.•.............•.•......•,.•••••••••• Figure5. Geometries produced by combining trishearfault-propagation foldingwithfault-bend foldsproduced by a changein therampangle.(a) An anticlineproduced by loweringtherampanglefrom30ø to 5ø at thetimeof depositionof bed4. (b) A synclinalbendproduced by increasing therampanglefrom30øto 55ø at bed4 time. acrossthe forelimb of the anticline.These are modeledas rapid propagationalong the same ramp, as an anticlinal bend in the fault to a nearhorizontalposition,and as a steepening of the fault producinga synclinalbendcuttingacrossthe anticlinalforelimb, respectively(Figure 6a, b, and c). The synclinalbreakoutproducesthe simplestgeometry;the fold geometrywhich formedup to the point of rapid propagationis simplytranslatedup the ramp without any further modification. The time of the breakout is clearlymarkedin the growth strataas the point where fanningof the strataceases(bed 5 in Figure 6a); the equivalentstrataare, of course,thicker in the footwall but are also unfolded. Not surprisingly,the geometryfor the decollementbreakout (Figure 6b) is quite similar to that of the anticlinalbend (Figure 5a). The main difference is that total forelimb thickeningis less in the breakout case becausethere is no thickening and only translationrelatedto the secondandyoungerramp.In the anticlinal breakout (Figure 6c), there is no further steepeningof the forelimbafter the formationof the secondramp (unlike in Figure 6b); the hangingwall is simplytranslatedup the ramp,producing the classic"snakehead" anticline.The patternof fanninggrowth stratadepositedduringmovementof the first ramp (prebed5) is readilyapparentonly in the footwall. 3.5. Inversion Structures Erslev[ 1991] showedthat trishearcanbe appliedequallywell to normalfaultsas to reversefaults.I take the next stepof showing the geometrythat resultswhen a trishearnormalfault is reactivated as a reversefault (Figure 7a, b) and then the fault dip flattensas it entersthe growthstratasequence (Figure7c). In the rift stage,the tip line propagates upwardasthe hangingwall lowers, matching the behavior observedin experimentalnormal drapefolds [Withjacket al., 1990]. Growthstratathin andonlap the footwall as they are foldedwith a typical"dragfold" geometry. As the directionof slip reversesto a thrustsenseandthe total slip returnsto zero at the baseof the model,a curiousthinghappens:there is upward increasingdisplacementin the pregrowth strata(i.e., with zero displacementon the baseof the model,there is a pronouncedanticline above the tip line at the top of the prenormal growth strata). This marked contrast to existing reactivationmodels[e.g., Mitra, 1993] is due to the foldingproducedin the trishearzone which propagatesupwardduringboth normal and reverse movement. If the trishear zone were to propagatedownwardwith the hangingwall, no anticlinewould result. At this point in the model (Figure 7b), the normal fault- 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: 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• Figure6. Breakouts produced by rapidtip linepropagation along(a) theleadingsynclinal axisat bed5 time,(b) as a bedding-parallel decollement at bed4 time,and(c) by cuttingacross theanticlinal forelimbatbed5 time. related growth strata are completely inverted, the postrift/prethrust strata have folded and thickened in the trishear zone, and the thrust-relatedgrowth stratashowa typical fanning compositeprogressive unconformities (CPU' s). Several interestingcomplicationsoccur when the thrust flattensinto the growth strata(Figure 7c). A fault-bendfold anticline formswith an activekink and a typicalgrowthtrianglerelatedto a passiveor fixed kink axis in the synorogenicstrata.The early formed, thrust-relatedCPU is folded by the newly orientedtrishear zone. Becauseof the shallowingof the fault ramp, the fanning geometryin the later growth stratastepsout away from the locusof uplift. Note that the prenormalgrowthstratain the foot- wall preservethe normal-faultrelatedfolding;thatis, theyappear to havea "reversedrag" with respectto the thrusting.The same stratain the hangingwall alsopreservea gentlesyncline(located betweenthe secondramp and the active kink axis) as a relic of the normal motion on the forelimb of the main anticline. 3.6. Growth Strata In general,as shownby HardyandFord [1997],thesegrowth stratamimic compositeprogressive unconformities (CPUs) such as thosedescribedin the Pyrenees[Anad6net al., 1986;Riba, 1976].Manydetailedaspects of growthstratageometries haveal- 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 646 .... •" ..... 647 '"'"""•'•'•:•'i•••••ii"'"'--'""•"•"•' •'"'" •'•:••••-........... ,,....•••ili Propagation 90.0 ......... ----':"" %--"• ........................................... •i................ i?!dg:.• ................................... •.;.•g ...... Slip =-60.0 0 Ramp angle = trishearangle 40.0ø 40.0ø P/S ratio* -1.5 250 ,growthtriangle thrust-relatedgrowth strata post-rift,pre-thruststrata rift-related growth strata --Total 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: INVERSE ANDFORWARD MODELS OFTRISHEAR Shortening---), • Propagation 361.7.•..•. I ' ilRamp angle • inthefootwall are these dips a relict of the normal faulting ""•' 25.0ø • 40.0or -• PIS ratio Itrishear angle1.5• Figure 7.Model ofriftinversion andfaultreactivation. (a)Thestippled layer wasdeposited postrift, prethrust reactivation. Riftsynorogenic strata areimmediately below thislayer ontheleftside ofthediagram andonlap the pre-rift strata draped over thefootwall. (b)Fault has been reactivated inareverse sense sothat thenetslipisnow zero. Note,however, theformation ofatightanticline near thefault. Thrust growth strata have been added above thestippled bed. (c)Fault ramp reduced inthesynorogenic strata forming abroad anticline. Note thefanning geometry ofthetrishear zone and, farther totheleft,thegrowth triangle instrata ofthesame age. "Drag" leftover fromtheriftphase ispreserved inthepregrowth strata ofthefootwall. readybeendescribed in sections 3.1-3.5.HereI address a more general topic:theformation of "growth triangles" related to propagating trishear zones. Thehanging wallboundary ofa trishearzoneis kinematically similarto a kinkaxisin parallelfold- inginthatbothseparate a domain characterized bynoshear, just translation, froma domainin whichbedding is sheared. Clearly, thenatureof theshearis different forthe'parallel casethanfor the trishearcases.Where P/S > 1, the hangingwall trishear stratawith the activeaxisat thetopof thegrowthstrata,produc- ingtheequivalent of a parallelfoldgrowthtriangle. In thetrishearcase,thisgrowthtriangleis a measure of therateof propagationof thetiplineof thefault.Themorerapidthepropagation of thetip lineis,thelowertheangleis thatthefixedaxismakes in thegrowthstrata(seedashed linesin Figures 3 and8). Note that,unliketheparallelfold case,the"activeaxis"is virtually undetectable in the growthstratabeneaththedepositional sur- throughthematerial, boundary propagates faster thanthehanging wallslips, andthere- face.Becausetheboundaryis propagating andcontinuously fromthedeforetheboundary migrates through thematerial. Underthese the strainvariesheterogeneously formed strata now located in the hanging wall and the strata conditions, thehanging walltrishear boundary actslikeanactive kinkaxis,withitsinitialposition in therockequivalent to a fixed axis(Figure 8a).Thegently inclined partofthepassive kinkaxis within the current trishear zone. WhenP/S= 1, thehangingwallboundary of thetrishearzone mustconnect thesteeply inclined passive axisin thepregrowth remainsfixedin the material,andno growthtriangleforms(Fig- ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR _ A. P/S = 2.0 growth "triangle." bounda• throu. htherock tdshe• bounda• hanging wa. .. • '"•"•:'•.:•:•••:• ..... ........ -.......... ;:>•;.•:•:..•j•*:•..... •..• 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 648 ================================================ ................................................................................................................................. ..................... :..=.>::.,.. ............•............... :.:.:.:.:.• .................... ..................... .• ...... Figure 8. (a) Detail of the growthtriangleproducedby propagating trishearzones(P/S > l) in growthstrata.The slopeof thefixedaxisin thegrowthstratais directlyrelatedto therateof propagation of thetip line.(b) With a P/S = 1, thehangingwall boundary of thetrishearzoneis fixedin thematerial,andnogrowthtriangleoccurs. ure 8b). There is an abruptincreasein strainacrossthe hanging wall boundaryof the trishearzone and a distinctkink forms in bothgrowthandpregrowthstrata. 4. Strain and Fracturing One of the mostusefulaspectsof the forwardmodelingis that the strain throughoutthe structurecan be predicted(Figure 2). Initial circlesare describedas an array of pointsand are spaced evenly alongeachbed in the model.Each time the modelis iterated, the displacementsof thesepointsare calculated.True distortion only occurswithin the trishearzone; in the hangingwall, the pointsare only translated,and in the footwall they are fixed (Figure 1). Where P/S > 1, materialdeformedin the trishearzone becomesincorporatedinto the hangingwall (Figure 2b, c). Becausethe strainis very heterogeneous, the deformedarrayof initially circularpointsis only approximatelyelliptical and representsan averagestrainoverthe regioncoveredby the points.The largerthe initial circle is, the lesstruly ellipticalare the pointsin the deformedstate.Nonetheless,they yield a goodfirst approximationto the straindistributionthroughoutthe structure. The strain in the trishear zone can be accommodated in a vari- ety of ways, dependingon the lithologiespresent.Weak units such as shale or evaporitesmay flow or experienceintricate small-scale duplexing. More competentunits surroundedby weaker ones may experiencetight folding. Massiveunits, includingbasement,may fracture intricately.Trishearkinematics doesnot dictatewhich of theseprocesses will occur;it provides nothingmore than a bulk kinematicdescriptionof the deformed regionand the strainpath by which it arrivedat its presentconfiguration.Comparisonwith analogclay modelsof extensional forcedfolds [Withjacket al., 1990] demonstrates the utility of the trishearmodelstrainpredictionsfor line lengthbalancingandfor understandingfracture orientationand distribution.Withjack et al. [ 1990] and S. Hardy and K. McClay (Kinematicmodellingof extensional forcedfolds,submitted to Journalof StructuralGeology, 1998) showedthat such folds form in triangular regions abovethetip linesof planarnormalfaults(Figure9a). In the clay models,much of the deformationis concentratedin the hangingwall of the normal fault (Figure 9a). This patternis mimickedin the trishearforward models:the largeststrainmagnitudesare observedin the hangingwall of the normal fault and INVERSE AND FORWARD in the hangingwall of the projectionof the normalfault into the trishearregionabovethe tip line (Figure9a). This occursbecause thereis a net transferof material from footwall to hangingwall in the normal fault case (the oppositeoccursin thrustfaults), as originallyrecognizedby Erslev[ 1991].Thereis an exceptionally goodfit, in bothmagnitudeand orientation,betweenthe trishear predictedstretchandthe line lengthstretchmeasured from offset markersin the experimentsof Withjacket al. [ 1990] (Figure9c). A. Original clay model of Withjack et al. [1990] MODELS OF TRISHEAR 649 Shearplanescommonlyoccuralonglines of no finite elongation (LNFEs), as in the classiccard deck shearingexperiment known to all structuralgeology students.This is also why bed length balancingworks in parallel folding: the beds are LNFEs that do not changelengthand musthave shearparallelto them. There are two suchlines in any area-conserving, two-dimensional strain. In the case of Withjack et al.'s [1990] extensionalforced fold, the LNFEs from the trishearmodel fit remarkablywell with B. Trishear model with predicted strain magnitude and orientation C. Trishearpredictedstrain& bed lengthbalancein clay model D. Linesof no finite elongationand fracturesin clay model ............ I ............................. •:,._.•.,A..,.......-A..,...-.•.-. •..N.. ........................... ...... ................... -,• X ............ -•!i"•'•'"':•'"":•:ii? .•. :.•.•::::•..... ,..•,.../• ::::::::::::::::::::::::::::: -..................................................... "••"•',// -/.X...•.;,•,...•.-•. Trishear Strain Ellipse principal stretch =1 ............. :--•..,a z'- ..................................... i!12., '................... .......................................................... " • '•'t, X '" .•-•-. :i"• .................................... ...................................... -.......... .r'•'•• l'•'x' ...... ß .• ......................................... '?"•'. ......... ................. $=-•i=1.9'1 Figure9. Trishear modeling of extensional forcedfoldsin analog models fromWithjack et al. [1990](AAPG ¸ !990,reprinted by permission of theAmerican Association of Petroleum Geologists). In all illustrations, theshaded linesshow theactual beds, andtheirregular solid lines show thedistribution ofmacroscopic fractures intheclay model. (a)Sketch of theoriginal claymodel, (b)bestfittinginverse model andforward modeled strain magnitude andorientation superimposed onclaymodel. Thestretch contoured isthatalongthegreatest principal axisof thefinitestrainellipse. Noteconcentration of strainmagnitudes in thehanging wall.(c)Detailshowing closematchof trishear predicted stretch andlinelengthbalance. (d) Comparison of fracture pattern in claymodelwithlinesof no finite elongation(shortticksin crosspattern)in trishearmodel.Noteclosecoincidence in orientations. Horizontal ruledareais wherereverse faultsareobserved in Withjacket al.'smodel;shaded regionis wheretrishear model predictsreversefaulting. 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: ALLMENDINGER: INVERSE ANDFORWARD MODELS OFTRISHEAR theobserved fracture patterns in theclay(Figure9c),evento the pointof predictingwherehigh-anglereversefaultswill form in this extensional system.The secondsetof LNFEs,antitheticto themainfaultzone,doesnotcorrespond to fracture planesand bytrialanderrorandtedious comparison withexisting deformed sections, converge onthefinalgeometry matching thestructure ofinterest. Thereis,however, a betterway. simplyrotatespassively duringthedeformation. 5.1. Inverse Procedure The inversemethodtakesadvantage of thefactthatthetris- 5. InverseModeling of Real Structures hear kinematicsis reversible;one can run modelsbackwardto In general, one would like to fit trishearmodelsto real struc- tures.In the parallelkink fold case,onecanmeasure panels wherethestratahavecoherent dipand,usingthegeometric relationsof Suppe[1983]andSuppeandMedwedeff[1990], make predictions aboutthegeometry andmagnitude of slipona fault, theprimaryvariables. Thisis notpossible in thetrishear casebe- unfoldbedsto theiroriginal,approximately planarorientations. Although it wouldseemto makeno difference, it is far easier,in practice, to evaluate thegoodness of fit of a modelbyhowwellit restores thebedsratherthanby howwell it deformsthem.Thisis because the initial state(approximately planarbeds)is much simplerthanthefinalstate(complexly deformed beds),andthere causeof the continuously varyingnatureof thebedorientations. are simplestatistical descriptions of thatinitial state.The inverse Furthermore, thereare moreunknownparameters in trishear methodfindsa bestfit initialgeometry, andthena forwardmodel kinematics: (1) faultramp,(2) slip,(3) propagation-to-slip ratio, of a smoothed versionof the initialgeometry canbe usedto (4) trishearapicalangle,and(5) tip lineposition, whichis actu- model the strain in a structure. allytwoparameters (X andY coordinates, ora vector magnitude Onecaninvertforall sixparameters mentioned above byperandangle).Thesecanproducea broadarrayof possible fold forming a gridsearch across a prespecified parameter space. The geometries, not evencountingthe hybridstructures described in statisticusedto evaluategoodness of fit is the simpleleast section3. Onecouldgenerate a seriesof forwardmodelswhich, squares linearregression, carriedoutby minimizing Z2asde- A ß B 74 ø 112 ø 108 ø 104 ø I I I I anticline '"l :atskill Mountains ) \ -' o 38 ø o o 42 ø 34 ø I I I Figure10.Geologic sketch mapsshowing thelocations of thetwostructures usedto demonstrate theinverse method. (a)Hudson Valleyfold-thrust beltineastern NewYorkstate, simplified fromMarshak [1986]. Horizontal hatch pattern shows theoutcrop beltof Silurian through lowerMiddleDevonian in whichthethrust beltisdevel- oped. Western edge oftheTaconic allochthon isshown withthebarbed line.Inset mapshows location inNewYork State. (b)Laramide Rocky Mountain foreland province ofthewestern United States. Barbs areontheupper plates ofthethrust faults; arrows show thevergence ofthemonoclines. Insetmapshows location in thewestern United States. 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 650 INVERSE AND FORWARD MODELS OF TRISHEAR 651 scribedby Presset al. [1986]. Becausethe geometryof one bed in a crosssectionis commonlyknown much better (i.e., is better constrainedby real data) than the rest, the programcurrently finds the best fitting model for that bed only and then evaluates how that model appliesto the other beds in the section.Grid searchingis a brute force method that, when usedto find all six parametersover a broadrangeof valueswith small stepsize,can both have P/S = 2.5, trishearanglesbetween30ø and 35ø, and be extremelytime consuming.For example,searchingfor the bestfit tip line in a 200 by 200 unit areawith a oneunit step meanscalculatingandevaluating40,000modelsfor eachunique have sharperlocal curvaturethan the others.Also, in all three cases,the curvesfor beds4, 5, and6 "plateau"at smalldisplacements beyond their best models.This is related to the fact that they are located farther from the final positionof the tip line. Strain diminishesat distancesaway from the tip line, meaning that thosebedsrapidly reacha point where they are as linear as they are going to get; subsequentdisplacementwill do little to changetheir geometry. combinationof the otherparameters. If, in the samerun, onealso specifies20 differentrampangles,20 differentP/S values,20 differenttrishearangles,each11run out to 500 displacement units,the program.will test 1.6'10 individual models. Fortunately, in many cases,the tip line position and ramp angle are well constrainedby outcropor seismicdata,andthusthe numberof models can be reduced substantially.For the above example, the displacements in themid-300unitrange.The minimaareslightly betterdefinedusingbed 3 as the key bed (Figure12b),but the differencesareminimal.Severalconsistencies amongthevarious modelsare striking.The bestmodelsfor beds 1 and4 require consistentlylessshortening(50-100 unitsless)that thosefor the otherbeds.This may haveto do with the factthatthose•wo beds number of individual models tested would be reduced to 2* 10, which can be carriedout on a modem desktopcomputerin less than 5 min. If more than one bed is well known, the grid search can simply be repeatedfor that bed, and the best averagemodel canbe used.For the reasonsdiscussed in the section5.3 this approachisjustifiedat present. 5.2. Application I show the applicationof the inversemodeling approachby applyingit to two previouslypublishedcrosssections.It is not my intentto provethat the trishearmodelingapproach•s superior to that describedin the original articles.Indeed, there are many reasons,somediscussed in section5.3, why indiscriminate application of the methodto publishedcrosssectionscould be misleading at best. These examplesstandsolely as a demonstration of the modelingprocedure.Note that, althoughboth examplesare thrustfaults, the exact sameprocedurecan be appliedto trishear B. / / normal faults, as in the case of the extensional forced folds, dis- cussedin section4 (Figure9). 5.2.1. Hudson Valley Fold and Thrust Belt. The Hudson Valley fold and thrust belt of eastern New York (Figure 10a) [Marshak, 1986] has some splendidoutcrop-scaleexamplesof fault-propagation folds.Even thoughno basementis involvedin this deformation,many of thesestructuresdisplayupwardshallowing dips on the forelimb due to thickeningin the core of the tip line syncline.Figure 11 showsthe resultsof this procedure appliedto one of thesestructures(a tracingof Figure 16 in the work of Mitra, [1990]). In this example,the positionof the tip line and the dip of the fault are known;thusthe searchis for just three parameters:P/S, trishearapical angle, and displacement (Figure 12). This grid searchwas repeatedwith beds1, 3, and6 as key beds. At present,one canmakeonly qualitativecomparisons among the resultsfrom usingeachof the threebedsas the key bed for restoration.Bed 1 clearlyyieldsan inferiorsolution(Figure12a): not only is the leastsquaresfit for that bed relativelypoor comparedto the others,but the bestfitting modelfor bed 1 produces broad, poorly defined minima in the chi-squaredplots for the otherbeds,particularlybeds4, 5, and 6. The poor fit is due, at leastin part, to the fact that bed 1 is faulted.Pointson the bed near the fault may be poorly restored,as can be seenin Figure 1lb, resultingin stronglocal deviationsfrom the linear model. The bestmodelsfor beds3 and 6 producevery similar solutions: Figure 11. Inversemodelof a fault-propagation fold fromthe HudsonValleyfold andthrustbelt.(a) Simplified tracingof a photograph whichwaspublished asFigure16 in theworkof Mitra [1990].Bedtopsarelabeledasreferredto in thetext.(b) Best fit restorationto a planar state for bed 3. Dashedline shows startingpositionof hangingwall boundaryof the trishearzone. Diagonalline across beds1 and2 is therestored positionof the final faultcutoffsacross thoseunits.(c) Forwardmodelusing bestfit parameters.Originaldata are showndashedbeneaththe model.Boldellipsesoccureveryfifth ellipse;at thestartof the deformation, they werecirclesalignedverticallyandperpendicularto bedding. 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER' ALLMENDINGER: A. 50000• INVERSE AND FORWARD MODELS OF TRISHEAR B. 5oooo Key bed =1 Bedd = 3.0 Apicalangle = 30ø 40000' Bed/•Be, d 40000 . '13 • 30000'• •30000 B ,,'1 f ø20000 o 2oooo• Bed 2 10000• \ \ k,•,%./'• .,.-/j ••ed-4• 0 .... , .... , .... i .... , .... 0 10000- "-Bed-5 y be Bed-6 i .... , .... , .... , .... 0 i .... -500 0......... ' ............. -1000 Displacement 50000 Apical angle = 34ø :•)U .................... -'1000 Displacement C= D= 40000' Be Bed Bedd •30000' Grid Search Parameters ramp angle displacement : 020000• P/S . minimum maximum step 36 ø 0 36 ø - 1000 --2 1.5 3.5 0.1 trishear angle 20ø (tip line is also fixed) 10000: i .... ! .... ! .... 0 a .... , .... a ........ -500 60ø 1 =6 _,--.! ,. = 3• o ! .... -1000 Displacement Figure12.Summary of bestmodelstatistics fortheHudson Valleyfoldandthrust beltexample shown in Figure 11.(a)Curves of chi-squared versus displacement forallbedsforthecombination of trishear parameters yielding a bestfit forbed1; thebestmodelforeachbedoccurs at theminimum in chi-squared. (b) SameasFigure12abut withbestfit parameters forbed3, (c)same asFigure12abutwithbestfit parameters forbed6. (d)Gridsearch parameters. Ideally,thebestfit modelforeachindividual bedshould besimilarto thatforeveryotherbed.Visually, onelooksfor tightcurveswith well-defined minimanearthe samevalueof displacement. Similarcurvescanbe constructed for anycombination of twoparameters. Restoration withbed3 asthekeybedappears to be thebest overall restoration. Forwardmodelingusingtheparameters of thebestmodelproducesa geometryremarkablysimilar to that of the initial de- a trueview of a realstructure hasbeenmodeled(seediscussion in section5.3). formedsection(Figure11c). The only areasof seriousdiscrep- 5.2.2. RangelyAnticline.Mitra andMount[1998]haverecentlydescribed a method for analyzing fault-propagation folds kink-likeareasof tight curvature,particularlyin bedtops4, 5, in basement faultedanticlines withtriangular-shaped zones of deand6. The trishearmodelprovidesa simpleexplanation for the formation in theoverlying sedimentary cover.I applytheinverse foldingobserved in the footwallbelowthe stratigraphic levelof method to oneof thestructures thattheyanalyzed: theRangely the tip line. Such featurescannotbe explainedin kink fault- anticline (Figures 10band13).Theircross section oftheRangely propagation foldingwithoutresortingto a rampbreakoutacross anticline hasa smallanticlinal rampnearthetip;themainpartof ancybetweenmodel and sectionoccurwherethe real rocksare in to 28øwithina few hundred meters the anticlinalforelimb [Mitra, 1990;Suppeand Medwedeff, thefaultdips38øbutchanges 1990]. Despitethe excellentfit betweenmodel and observation, of thetip.As shown in section 3.3,suchchanges canbeforward onecuriousresult,the 9ø differencein dip betweenthe highest modeled, buttheinverse model works onlyfora single faultdip, and lowest beds in the restored state across less than 10 m hori- zontaldistance(Figure 1lb), suggests that an artifactratherthen so38ø wasused.Theposition of thetip lineis notprecisely known(at leastto me)butis limitedto a smallrangeof values. 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 652 653 O' -2OOO __ B. 5- / startingpositionof the hang- • wedgedue ramp in originalsection ,ng wa!l trish. boundaty •,•__._ • Pm ..... 7- tnsnear angle =76,o[•o Figure 13.Inverse model oftheRangely anticline. (a)Rangely cross section modified from MitraandMount [1998](AAPG ¸ 1998, reprinted bypermission oftheAmerican Association ofPetroleum Geologists). Vertical lines onthesection arepetroleum exploration wells. (b)Best fitrestoration using bed9 asthekeybed. (c)Forward model using best fitparameters. Original data areshown dashed beneath themodel. Bold ellipses occur every fifth ellipse; atthestart ofthedeformation, they were circles aligned vertically and perpendicular tobedding. of course,of bed 1), indicatinga Therefore thegridsearch involves a largernumber of parameters smaller(with the exception, than previously: trishear angle, P/S,displacement, andtheX and betteroverallfit thanin thepreviouscase(Figure14b).However, Y positions ofthetipline.Asbefore, three different bedtopsas the minima for someof the beds are stretchedout, and best fit range from-160to-220model units. Theoverall keybeds(1, 5, 9) areanalyzed askeybeds. Because I donot displacements of parameters forallbedsis obtained whenbed haveaccess totheoriginaldatafromwhichthecross section was bestcombination 14c).Withthose parameters, the constructed, the constraints on eachof thesebedsis unclear. top9 isusedasthekey(Figure minima,andthe Clearly, bedtop9 hasthebestcontrol indepth, asit ispierced by curvesfor all bedsaretightwithwell-defined varybetween just-180and-200 model allofthepetroleum exploration wells. Top5 wasdrilled inonly bestfit displacements units.EventheZ2minimum valueforbedtop1isonlytwicethat onearea,andtop 1 is piercednowhere. The resultsof the statistical analysis for eachof thekey beds for the best fit for bed 1. Thebestfit parameters forbedtop9 (trishear angle of76ø,P/S of2.3,slipof-4200m) areputintoa forward model theRangely to compare to theoriginalcrosssection (Figure13c). of theotherbeds(Figure14a).Thisoccursbecause theprogram anticline (Figure 14)alsoindicate thatbedtop9 isthemostreliable. The curves of Z2where bed1 wasthekeyshowpoorlinearfitsto all findsthe bestcombinationof valueswhich unfoldsthe sharpcur- vaturein bed 1 nearthe fault,particularly in the footwall.Because thiscurvature is greaterthanfor anyotherbed,thebestpa- Overall,the fit is remarkably good,particularly wherethereis well control.The 4200 m of displacement is exactlywhatone wouldpredict byprojecting thestraight (i.e.,unfolded) partsof bed top 1 to the fault (see dashed line in Figure 13c). The actual rameters forbed1 produce a broadsyncline in all theotherbeds. slipmarked bythecutoffs of bed1 against thefaultreflects the The minima in curvesthat resultwhenbed 5 is the key bed are 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: INVERSEANDFORWARD MODELSOFTRISHEAR ALLMENDINGER:INVERSE AND FORWARDMODELS OF TRISHEAR A. 50000 B. 50000 40000' 40000' ß 30000• 0 20000: 30000 : •20000 10000 i Key bed = _1 162 Slip P/S 2 O• .... 0 C. i .... i .... i .... i .... 10000 • 2.8 Apical angle 4,6 ø-' 000 i .... -500 i .... i .... i ...... .... P/S i .... i .... i .... 0 Displacement 50000 i .... Slip =-218 D= = 2.8 Ap, ical angle= 5,4 ø i ........ i .... i ........ -500 - 1000 Displacement 40000' GridSearchParam.ete:•S minimum maximum , 30000. . • 20000i • AKeybed Slip P/S 10000, 0 .... 0 = 9 -- -194 = 2.4 pical angle =76 ø i .... i .... t .... i .... i .... i .... -500 i .... i .... step ramp angle 38 ø 38 ø m displacement P/S trishearangle 0 1.5 20ø -1000 3.5 85ø -2 0.1 1 tip line 60 tiplinepositions tested (note thateachmodel displacement unitis21.6m) i .... -1000 Displacement Figure 14.Summary ofbest model statistics fortheRangely anticline. SeeFigur• t2forexplanation. (a)Curves of Z2versus displacement forallbeds forthecombination oftrishear parameters yielding abest fitforbed1,(b)Same asFigure 14abutwithbest fit parameters forbed5,(c)same asFigure 14abutwithhest fitparameters forbed9. (d)Gridsearch parameters. Bed9 asthekeybedprovides thebestoverallfit. up-sectiondecreasein slip due to the fault-propagation fold thefault.This thickeningoccurred in thetrishearzoneabovethe kinematics. The fit of the forwardmodelto the originalcross tip lineof thefault.Oncefullyincorporated intothefootwall,no sectionis worstnearthefault,asmightbe predicted fromthe further strain occurs. restoration in Figure13b).Thetrishear forwardmodelpredicts muchlesstightfoldingnearthefault.It islikelythattheoriginal dataon bed dips are leastreliablenearthe fault, and thusthe mismatch herecouldbe dueto poorconstraints in theoriginal sectionand not to an incorrecttrishearmodel.The total thickness 5.3. Caveats A goodstatistical fit doesnotguarantee a good,orevena rea- sonable, model af reality. Thetwoexamples illustrate notonly between bedtops1 and10is wellmatched in thehanging wall, theapplication of theinverse modeling, buttheyalsohighlight butthetrishear forward modelpredicts mildthickening of bedsin differenttypesof potential pitfallswhichmayawaita user the footwall,particularlywithin 7 or 8 km of the fault. This ocblindlyusingtheprogram. TheHudson Valleyfoldandthrust cursbecause the trishearangleusedis quiteopen(76ø).The beltexample provides a rather complete geometric description of strainellipses predicta component of extension perpendicular to thestructure because it is allcontained withina single outcrop. thelayersthroughout thefootwallbutparticularly approaching Therampangle,tip lineposition, andtopsof thebedscanall be 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 654 seen in the original photograph[see Mitra, 1990, Figure 20]. However,a photographmay be subjectto problemsof depthof field (differentpartsof the photobeingat differentdistances from the lens), perspective,and the flatteningof a three dimensional morphologyinto two dimensions.Furthermore,one must take careto be surethat the photoprovidesa true down-plungeview of the fold to be modeled.I suspectthat the extremewedgingof the stratigraphic unitsin the restorationin Figure l lb probably occursbecausethe right-handside of the crosssection(left hand sidein the originalphoto)was moredistantfrom the cameralens, and thereforethe bedslook thinnerthere.This doesnot negate the modeling proceduredescribedabovebut suggeststhat it is just aseasyto modelartifactasreal data.One couldarguethat,in thiscase,the modelinghasactuallyhelpedto identifya potential artifact. The model of the Rangely anticlineis subjectto a different problem:in this casean interpretationratherthan data hasbeen modeled.Without accessto the original seismicdata, it is difficult to evaluateindependentlythe goodnessof fit of the model, which of the bed topsis really the best"key bed" to use,etc. In fact, wheneverone modelsanythingbut outcropdata,the model will mostly be inverting interpretationof limited data with a broad range of confidence limits. Seismic reflection data are subjectto errorsinherentin the conversionfrom time to depth, picksin areaswithoutgoodwell control,processing artifacts,and areasof poordataquality.Again, the trishearmodelingcanhelp improve the interpretationin severalof these instancesif one knowsindependently thattrishearis theappropriate kinematics. For thesereasons,any attemptat more sophisticated inverse modeling,suchas finding the model which minimizesthe misfit for all beds simultaneously,while technologicallypossible(if computationally intensive), may be unwarranted. One would have to assignquantitativeconfidencelimits to all of the observationsandinterpretations. In an extremeexample,bedtop 10 of the Rangely anticlineis mostly interpretedbecausethe real bed has been eroded away. Though the interpretationof where it shouldbe (basedon entirelydifferentkinematicassumptions of Mitra and Mount [1998]) is included in the model, it makes no senseto attemptto minimize its misfit. Finally, the inverseproceduresdescribedherecanonly be applied to simpletrishearstructuresin which there is no changein P/S, trishearapical angle,or ramp anglethroughtime. The hybrid structuresdescribedearlier in section3 can only be forwardmodeled. In both of the examplesabove,someof the misfitsbetween model and data may be due to the fact that both data (in the case of the HudsonValley example)andinterpretation(Rangely)suggesta flatteningof the thrustnearthe tip line, producingan anticlinal ramp. 6. Discussion Trishear is an undeniablypowerful way to model structures and providesa reasonablysimple explanationfor a variety of structuralcomplicationswhich are much more cumbersometo explainwith otherkinematicapproaches. The strikinglydifferent kinematicsof trishear would seem to put this model in direct competitionwith now classicalparallelkink fold models,but this is not the case.Becausetrishearprovidesa bulk descriptionof a deformingzone, it is complementary to thosemodels,ratherthan an incompatiblealternative.Trishear explainsthe grossthicken- 655 ing or thinningbetweenunits but dictatesnothingaboutthe specific structuralgeometry and processesby which those strains occur. Duplex thickeningin an anticlinal stackor passiveroof duplex can be modeledwith trishear,even thoughthe detailed structuralgeometriesobey all of the Dahlstromand Supperules. Thus it is not just a technique for modeling basementcored structures(as the HudsonValley examplein section5.2.1 demonstrates), but neither is it a replacement for existing crosssectionbalancingtechniques. Becausestrain can be accommodatedin a variety of ways, no one strain indicator will, necessarily,match the strain ellipses predictedin trishearforward models.To do so, a singleindicator would have to accrue strain throughoutthe history of the structure. For example, it may be that much of the late strain in a structureis producedby out-of-sequence thrustfaults, after faults within forward breaking duplexeshave locked up and ceased movement.Likewise, intracrystallinestrainmay only be recorded during specific stepsor eventsin the formationof the structure suchas during early layer-parallelshorteningor late stagelocking of folds. The most reliable measureof strainin the structure is the change in bedding thicknessitself. If a trishear forward model matchesthe thicknesschangesthroughouta real structure, then the strain predictedby the forward model must be at least consistentwith, if not a unique descriptionof, the bulk strainin the real structure. One area in which the trisheartechniqueis particularlyuseful is in the predictionof grossstructuralgeometriesin areasof poor (or nonexistent)subsurfacedata. For simple structures,one can invert the existingdatafor a bestfit model.The modelallowsone to answersuchquestionsas the following: how tightly are the deeperunitsfolded?What is themostlikely displacement for the structure?What were the apical angle and the P/S ratio?Where are fracturesmost likely to be concentrated? How much of the unit thickeningor thinningis accommodated by lateral flow of materialandfrom where?How deepwas the tip line of the fault (i.e., the nucleationpoint) at the start of folding? Once these broader-scaleissueshave been addressed,one can then concen- trate on the detailedstructuralgeometryand the processes by which the deformation occurred. 7. Conclusions Trishearkinematicsprovidesan excellentdescription of the bulkgeometryanddeformation associated with fault-propagation folds.By allowingchanges in thebasicparameters (trishearapical angle,propagation-to-slip ratio) andby permittingthe formation of secondrampsand trailing fault-bendfoldsthroughtime, manydifferentgeometriescan be modeled.For simpletrishear fault-propagation folds,the inversemethodpresented hereprovides,for the first time, a rationalstatisticallyvalid basisfor selectingthe appropriatetrishearparameters that mostcloselyduplicatethe real structureof interest.The methodcanbe applied either to thrust or normal fault-related folds and either thick- or thin-skinneddeformation.Perhapsthe greatestremainingunknownsconcerningtrishearstructuresare the physicalfactors that controlthe apical angle and propagation-to-slip ratio. Becausethe inversemethodgivesus an objectivemethodfor determining best fit to real structures,we can now addresstheseunknowns. 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR ALLMENDINGER: INVERSE AND FORWARD MODELS OF TRISHEAR Acknowledgments. I am indebtedto Ren6 Mancedaof YPF, S.A., for showing me an outcropin southernMendoza Province, Argentina, that sparkedmy interestin trishear. Ren6's and Tom•s Zapata's (also YPF, S.A.) enthusiasmfor my initial modelingattemptsinspiredthe current effort. YPF, S.A., providedsupportfor my visit, and I am gratefulto Rafil Gorrofio and Ricardo Manoni for arrangingall the details. That I was thinking of trishearat all at the time is thanksto StuartHardy and to the AGU Editor who sentthe Hardy and Ford paperto review for Tectonics.It was their simple and precisemathematicaldescriptionof tris- hearthat enabledme to write so quickly the modelingprogramdescribed in this paper.I am further indebtedto StuartHardy for suggesting that I model Withjacket al.'s [1990] extensionalforcedfolds. Ben Brooksconstantlyencouragedme to pursuea rigorousinversemodelandadvisedme on how to implementit, althoughwe both know that more canbe donein this area.Reviewsof the manuscriptby Hardy,Brooks,Zapata,Francisco Gomez, and Tectonics reviewers Eric Erslev and Don Medwedeff are muchappreciated. Thanksalsoto Editor Dave Scholl,who suggested that I make the paperdirtier. This work was supportedin part by National ScienceFoundationgrantEAR-9614759. 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(Received March 9, 1998; Vetterling, Numerical Recipes:The Art of Scientific Computing,818 pp., CambridgeUniv. Press, revisedMay 21, 1998; New York, 1986. acceptedJune5, 1998) 19449194, 1998, 4, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98TC01907 by Indian Institute Of Science Education And Research Bhopal, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 656