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.
References
Anad6n, P., L. Cabrera, F. Colombo, M. Marzo, and
O. Riva, Syntectonicintraformationalunconformities in alluvial fan deposits,easternEbro Basin
margins(NE Spain), in Foreland Basins,vol. 8,
Spec. Publ., vol. 8, edited by P. Allen and P.
Homewood,pp. 259-271, InternationalAssociation of Sedimentologists,
1986.
Crider, J.C., M.L. Cooke, E.J.M. Willemse, and J.R.
Arrowsmith,Linear-elasticcrackmodelsof jointing and faulting, in StructuralGeologyand PersonalComputers,editedby D.G. de Paor,pp. 359388, Elsevier, New York, 1996.
Erslev,E.A., Trishearfault-propagation
folding,Geology,19, 617-620, 1991.
Erslev,E.A., and K.R. Mayborn, Multiple geometries
andmodesof fault-propagation
foldingin the Canadian thrust belt, J. Struct. Geol., 19, 321-335,
1997.
Erslev, E.A., and J.L. Rogers, Basement-covergeometry of Laramide fault-propagationfolds, in
Laramide basement deformation in the Rocky
Mountain foreland of the Western United States,
editedby C.J. Schmidt,R.B. Chase,and E.A. Erslev, Spec. Pap. Geol. Soc. Am., 280, 125-146,
1993.
Gallup, W.B., Geologyof Turner Valley oil and gas
field, Alberta, Canada, Am. Assoc.Pet. Geol. Bull.,
34, 797-821, 1951.
Hardy, S., and M. Ford, Numericalmodelingof trishear fault-propagationfolding and associated
growthstrata,Tectonics,16, 841-854,1997.
Jamison,W.R., Geometricanalysisof fold development in overthrust terranes, J. Struct. Geol., 9,
207-219, 1987.
Marshak, S., Structure and tectonics of the Hudson
Reches, Z., Development of monoclines:Part I.
Valley fold-thrust belt, easternNew York State,
Structure of the Palisades Creek branch of the East
Geol. Soc. Am. Bull., 97, 354-368, 1986.
Kaibab monocline, Grand Canyon, Arizona, in
Laramide Folding Associated With Basement
BlockFaulting in the WesternUnitedStates,edited
by V. Matthews III, Mere. Geol. Soc. Am. 151,
Matthews, V., III, and D.F. Work, Laramidefolding
associatedwith basementblock faulting along the
northeastern
flank of the FrontRange,Colorado,in
Laramide Folding Associated With Basement
BlockFaultingin the WesternUnitedStates,edited
by V. Matthews III, Mem. Geol. Soc. Am. 151,
101-123, 1978.
Mitra, S., Fault-propagationfolds: Geometry, kinematic evolution, and hydrocarbontraps, AAPG
Bull., 74, 921-945, 1990.
Mitra, S., Geometryand kinematicevolutionof inversion structures,AAPG Bull., 77, 1159-1191, 1993.
Mitra, S., and V.S. Mount, Foreland basementinvolved structures,AAPG Bull., 82, 70-109, 1998.
Molinero, J., F. Colombo,and S. Hardy, Disposici6n
geom6trica
profundade losmateriales
terciarios
en
el corte del rfo Najerilla (Sector Riojano de la
cuencadel Ebro),Geogaceta,20, 792-795, 1996.
Narr, W., and J. Suppe, Kinematics of basementinvolvedcompressivestructures,
Am. J. Sci., 294,
235-271, 1978.
Riba, O., Syntectonic unconformities of the Alto
Cardener,SpanishPyrenees:A geneticinterpretation, Sediment. Geol., 15, 213-233, 1976.
Suppe, J., Geometry and kinematics of fault-bend
folding,Am. J. Sci., 283,684-721, 1983.
Suppe,J., and D. Medwedeff,Geometryand kinemat-
ics of fault-propagationfolding, Eclogae Geol.
Helv., 83, 409-454, 1990.
Suppe,J., G.T. Chou,andS.C. Hook,Ratesof folding
and faulting determinedfrom growth strata,in
ThrustTectonics,editedby K.R. McClay,pp. 105121, Chapmanand Hall, New York, 1992.
Withjack, M.O., J. Olson, and E. Peterson,Experimental models of extensionalforced folds, AAPG
Bull., 74, 1038-1054, 1990.
802-860, 1994.
Pollard,D.D., andP. Segall,Theoreticaldisplacements
and stressesnear fracturesin rock: With applications to faults, joints, veins, dikes, and solution
surfaces,in Fracture Mechanicsof Rock,editedby
R. W. Allmendinger,Departmentof Geological
B.K. Atkinson,pp. 277-349, Academic,San DiSciences,Cornell University, Snee Hall, Ithaca, NY
ego,Calif., 1987.
14853-1504. (e-mail: rwal @cornell.edu)
Press,W.H., B.P. Flannery,S.A. Teukolsky,andW.T.
(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