Underthrusting-accr etion cycle: W ork budget as revealed ... · 2 Underthrusting-accr etion cycle: W ork budget as revealed 3 by the boundary element method ... 78 etry of the forearc

2 Underthrusting-accretion cycle: Work budget as revealed

3 by the boundary element method

4 Mario Del Castello1 and Michele L. Cooke1

5 Received 16 February 2007; revised 3 August 2007; accepted 20 September 2007; published XX Month 2008.

6 [1] Sandbox models of accretionary wedges have demonstrated that fault systems grow7 episodically via cycles of alternating wedge thickening, which is accommodated by slip8 along faults within the wedge (underthrusting), and wedge lengthening, which is9 accommodated by growth of new faults at the wedge toe (accretion). The transition10 between these two modes of deformation is controlled by the interplay of work against11 gravity, frictional heating, the work of deformation around faults, and the work of12 fault propagation and seismic/acoustic energy. Using numerical mechanical models based13 on the boundary element method, we have simulated the deformation observed in sandbox14 experiments, providing a mechanical analysis of the underthrusting/accretion15 transition. Our results show that the total work done by the contracting wedge increases16 during the underthrusting stage up to a critical value when the propagation of a new frontal17 thrust significantly reduces the work required for further deformation. The numerical18 models also predict the location of the maximum shear along the basal decollement during19 underthrusting as well as the energetically most viable position and vergence for the20 nucleation of a new thrust. These locations do not coincide, and the match of the21 energetically most favorable position with the experimental results suggests that the new22 thrust ramps develop first ahead and then link down and backward to the propagating23 basal decollement. The shear localization producing a new thrust ramp will occur24 where the energy spent by the deforming wedge is minimized due to an optimal25 combination of gravitational, frictional, internal, and propagation work terms.

26 Citation: Del Castello, M., and M. L. Cooke (2008), Underthrusting-accretion cycle: Work budget as revealed by the boundary27 element method, J. Geophys. Res., 113, XXXXXX, doi:10.1029/2007JB004997.

29 1. Introduction30 1.1. Critical Taper Theory

31 [2] Since the proposition of the critical taper theory32 [Dahlen et al., 1984; Dahlen, 1990], fold and thrust belts33 have been successfully analyzed with the assumption that34 they deform as a self-similar wedge of noncohesive brittle35 material such as in front of an advancing bulldozer. The36 theory states that the wedge deformation is ultimately37 controlled by the ratio between internal coefficient of38 friction (m) and the basal coefficient of friction (mb) and39 that the wedge must be on the verge of Coulomb failure40 everywhere and at every time. Only in this case can the41 taper angle be calculated as a function of m and mb [Dahlen,42 1990]. The requirements set in the analytical treatment43 produce a constant taper angle during wedge growth, i.e.,44 the wedge preserves a fixed length to height ratio through-45 out deformation. However, this reflects observations only to46 a first degree.47 [3] The critical taper theory has been largely applied to48 the evolution of fold and thrust belts since its proposition

49and systematically tested against both natural analogues50[Lallemand et al., 1994; Ford, 2004; Bangs et al., 2004;51Vannucchi et al., 2003; Saffer and Bekins, 2006; Wang and52Hu, 2006] and experimental techniques [Bombolakis, 1994;53Lohrmann et al., 2003; Adam et al., 2005]. The wide54application of the theory of critical taper owes to its55intuitively simple approach to interpret wedge mechanics56in terms of m/mb ratio, despite the complexity of the57analytical treatment. By simply reducing or increasing the58aforementioned ratio, the equations derived for this theory59successfully predict the complete sequence of faulting60processes occurring at contractional margins (accretion,61underthrusting, underplating and basal erosion). Neverthe-62less, the current form of the critical taper theory provides an63end-member scenario that neglects the importance of elastic64processes and strain hardening/softening that might affect65the temporal distribution of the state of stress and ultimately66the wedge geometry [Wang and Hu, 2006]. For instance, the67theory correctly envisages transition between accretion and68underthrusting, but does not address why a wedge should69activate a deeper decollement level nor where shear stresses70would localize in order to produce a new fore-thrust ramp.71Furthermore, the predicted constant taper is seldom ob-72served in natural case studies. Rather than growing self-73similarly, new material is generally seen to be incorporated

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74 into the wedge episodically via processes that occur at75 distinct times.76 [4] Elastic effects related to subduction megathrust faults77 locking are well known to influence the large-scale geom-78 etry of the forearc. Zones of interseismic locking may act as79 boundaries between subsiding basins and accreting regions80 [Wells et al., 2003] and control variations in trench-parallel81 forearc topography [Song and Simons, 2003], such as82 steepening of the forearc slope [Bangs et al., 2004]. Also83 rotations of large-scale sectors of orogens have been attrib-84 uted to plate locking [McCaffrey et al., 2000]. However,85 how the role of local elastic processes that operate on single86 thrust fault might influence the wedge mechanics has not87 yet been investigated in detail.

88 1.2. Overview of Sandbox Experiments of Wedge89 Growth

90 [5] Sandbox models are a common technique to study91 wedge kinematics. Analysis of analogue experiments sug-92 gest that the wedge development is characterized by dis-93 continuous phases of alternating wedge thickening,94 accommodated by slip along thrust faults within the wedge,95 and wedge lengthening, accommodated by nucleation of96 new thrust faults in front of the wedge [e.g., Mulugeta and97 Koyi, 1992; Storti and McClay, 1995; Gutscher et al., 1996;98 McClay and Whitehouse, 2004; Konstantinovskaia and99 Malavieille, 2005]. These two processes are referred to as100 underthrusting and accretion. After an early phase of101 accretion (Figure 1a), an underthrusting stage begins, when102 a coherent sheet of undeformed material slides underneath

103the wedge via activation of a roof thrust. The roof thrust104allows for the wedge to vertically thicken, increasing the105taper angle (Figure 1b). By contrast, the accretionary stage106predicts nucleation of new thrust ramps in front of the toe of107the wedge by concentrating the slip along the basal decolle-108ment. New fore thrusts accrete at the wedge front and the109wedge taper decreases accordingly (Figure 1c). Recent110analyses showed the cyclic nature of this transition111[Gutscher et al., 1998; Del Castello et al., 2004; Adam et112al., 2005; Naylor et al., 2005]. As a result of these cyclic113processes, the wedge may display a variety of slope angles114during growth and does not obey the self-similar growth115requirement of critical taper theory. What controls the116duration in time and the persistence of the accretion and117underthrusting stages? Observation of wedge growth in118sand-box models suggests that the predominance of one119mode over the other can be controlled by such factors as120syntectonic erosion/sedimentation or temporarily unbal-121anced topography [Storti and McClay, 1995; Del Castello122et al., 2004], high basal friction [Gutscher et al., 1996;123Kukowski et al., 2002; Konstantinovskaia and Malavieille,1242005], wedge thickening and basal friction [Burbidge and125Braun, 2002], mechanical interaction between prowedge126and retrowedge [Del Castello et al., 2004; Naylor et al.,1272005]. Yet, all these approaches envisage static m/mb ratio128with faults in constant state of creep. In contrast, natural129faulting processes incur friction drop during slip (i.e., slip130weakening) that can change the mechanics of the system.131Consequently, previous studies underestimate or totally132neglect the effect of the phases of elastic loading predating133episodes of frictional slip on faults.134[6] Stick-slip behavior [e.g., Rabinowicz, 1958] has been,135until recently, disregarded in analogue modeling of accre-136tionary wedges, because of the general assumption that sand137cannot preserve a ‘‘mechanical’’ memory on preexisting138fractures. Because of the poor measurements of analog139material properties, the elastic interval of deformation and140strain hardening processes have been traditionally consid-141ered as of secondary importance for noncohesive wedge142growth. Recent refined laboratory analyses demonstrate that143granular analogue materials show a significant departure144from the Coulomb elastoperfectly plastic behavior, with a145marked decrease in m at the transition between material peak146strength and residual strength [Lohrmann et al., 2003; Ellis147et al., 2004; Panien et al., 2006]. The strain hardening prior148to peak strength and the strain softening post peak strength149has been related to decompaction/compaction cycles affect-150ing the sand during the compressional stress loading and151subsequent rupturing [Lohrmann et al., 2003]. The stress-152strain curves for the sand materials tested agree well with153characteristic curves of rocks and soils [Jaeger and Cook,1541969; Lambe andWhitman, 1968;Marone, 1998; Barnhoorn155et al., 2004]. Adam et al. [2005] used particle imaging156velocimetry (PIV) techniques within sandbox experiments157of wedge growth to document how accretion and under-158thrusting processes are not synchronous but cyclic and might159be related to strain hardening/softening cycles occurring at160the roof thrust and basal decollement.161[7] In this paper we use a boundary element method162(BEM) code to numerically reproduce the sandbox exper-163iment of wedge deformation published by Adam et al.164[2005]. Analogue models based on noncohesive, Coulomb

Figure 1. Schematic diagram illustrating two sequentialcycles of underthrusting-accretion. (a) Initial accretion. (b)Episodic deactivation of the basal decollement causesunderthrusting under the roof thrust (white arrows). (c)Nucleation of a new fore thrust that steps the wedge toeforward. Continuous thick lines indicate active faultsegments. Insets in Figures 1b and 1c show transitoryincrease/decrease of taper angle during underthrusting andaccretion phases, respectively.

XXXXXX DEL CASTELLO AND COOKE: UNDERTHRUSTING-ACCRETION WORK BUDGET

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165 material are excellent tools to test the applicability of the166 critical taper theory, yet their information is limited to167 kinematics rather than a full mechanical deformation. We168 try to bridge this gap by constructing a numerical model that169 utilizes recently revisited strain hardening/softening proper-170 ties of sand within a fully elastic BEM code. The investi-171 gation is restricted to a single underthrusting/accretion172 transition for which we calculate the work balance and173 discriminate between different work terms, trying to assess174 both the trade-off mechanisms between the energy terms175 and the impact of each of them at the transition. The setup176 of our numerical models is based on the fault configurations177 and sequence of faulting revealed by PIV analysis of178 sandbox experiments by Adam et al. [2005]. This paper179 presents a systemic view of wedge growth using work180 budget assessments that are consistent with natural and181 sandbox observations, encompasses existing ideas of wedge182 growth and does not rely on temporal nor spatial variations183 in material properties to achieve cyclic transitions between184 underthrusting and accretion modes.

186 2. Work Analysis by Boundary Element Method187 2.1. Description of Work Terms

188 [8] Transitions between underthrusting and accretion189 modes may reflect adjustments to the balance between work190 against frictional slip along the basal and inner wedge faults,191 work of deformation within the host material and uplift192 work against gravity. Despite the lack of consensus on the193 applicability of the principle of work minimization to large-194 scale systems in nature [e.g., Bird and Yuen, 1979; Sleep et195 al., 1979], the systemic analysis of the interaction between196 various work terms has yielded insights to the study of197 deformation mechanics of geological structures [Mitra and198 Boyer, 1986; Platt, 1988; Melosh and Williams, 1989;199 Gutscher et al., 1998; Masek and Duncan, 1998; Burbidge200 and Braun, 2002]. However, studies to date have not201 revealed the trade-off between the various work components202 as the system shifts between accretion and underthrusting203 modes. Within accretionary systems, faults are constantly204 growing, shutting down and initiating via a complex inter-205 play of the different work terms as the system persistently206 seeks to minimize the total work. The total work in the207 fault system, Wtot, is the sum of work against gravity, Wgrav,208 work against frictional sliding, Wfric, work of internal209 deformation in the material around the fault, Wint, work of210 propagating new faults, Wprop, and work of seismic radiated211 energy, Wseis [Cooke and Murphy, 2004].

Wtot ! Wgrav "Wfric "Wint "Wprop "Wseis #1$

Each of these work terms is calculated as the product of214 appropriate force times displacement. The work against215 gravity is positive for contractional systems and negative for216 extensional systems. Wgrav depends on the vertical uplift217 and vertical normal stress due to overburden weight218 integrated over the area of the system. In a two-dimensional219 system with horizontal position, x, and depth of burial, d, as220 a function of vertical position, z,

Wgrav !ZZ

rgd z# $dxdz #2$

221where g is gravitational acceleration and r is the average223density of the overburden. When a fault slips, work is done224resisting friction along the fault surface; this work is225expressed as frictional heating. For example, weak faults226(m = 0) produce no work against friction and do no exhibit227frictional heating.Wfric depends on fault slip and shear stress228integrated along the length of each fault in the system. Wfric

229is one of several forms of work that is path-dependent.230Faults do not necessarily unslip when they are unloaded so231that different loading paths may produce different Wfric.232Consequently, this work is integrated over the loading path;233in this case the path of applied horizontal strain, "hor. Wfric is234also integrated over the fault length, l, along which normal235stress, sn, slip, s, and frictional coefficient, m, may vary.

Wfric !ZZ

sn "hor;l! "

m l# $s "hor;l! "

d"hor;dl #3$

236The internal work is the stored strain energy density, a238measure of mechanical work within the system. Wint is239calculated as strain times stress at a point, integrated over the240area of the system [e.g., Timoshenko and Goodier, 1934].241Using Hooke’s law, the strains can be formulated as stresses242to get

Wint !ZZ

1% n2# $2E

s2xx " s2

zz

! "

" 1" n# $E

s2xz " nsxxszz

! "

# $

dzdx

#4$

244where E and v are the elastic properties Young’s modulus245and Poisson’s ratio, respectively. The work of fault246propagation, based on Gibb’s free energy, is the energy247required to break intact rock and create new fault surface248energy integrated over the length of new fault surfaces249[Scholz, 2002]. Fault surfaces are often accompanied by250abundant damage within the host rock so that the total new251surface area created is far greater than the area of new fault252surface. The seismic radiated energy, expressed as ground253shaking or acoustic energy, is the energy released from the254surrounding rock during fault slip. Wseis is path-dependent255and depends on the shear stress drop, Dt, and slip along the256sliding faults [Scholz, 2002].

Wseis !ZZ

Dt "hor;l! "

s "hor;l! "

d"hor;dl #5$

257For finite models, such as that considered in this study, we259can also calculate external work on the system, which should260equal the sum of the individual work terms. This external261work is the product of displacement along the boundaries of262the system times the forces along that boundary.Wext is path-263dependent and is integrated over the loading path. Within264this study we are able to compare the Wext from our models265to Wtot in order estimate the errors in our work calculations.266[9] Whereas some forms of work are nonconservative267and lost to the system such as Wfric, Wprop and Wseis, others268serve to store work that can be later used to drive faulting,269Wgrav and Wint. For example, thrust faulting may release270stored internal strain within contracting systems, but this271occurs at the expense of producing greater work against272gravity. Cooke and Murphy [2004] expanded the theoretical


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273 formulations of Mitra and Boyer [1986] to include a more274 rigorous internal work term (equation (4)) so that a com-275 pletely balanced work budget can be evaluated within276 numerical models to simulate the growth fault systems.277 We use this formulation (1) to evaluate the principle of278 work minimization theory for an accretion/underthrusting279 cycle, (2) to quantify the role of each work component280 within a cycle, and (3) to test the similarity of analog281 experiments to results obtained by the boundary element282 method (BEM) technique.

283 2.2. BEM

284 [10] The BEM uses the principle of continuum mechanics285 to calculate stress and strain throughout a faulted rock body286 [e.g., Crouch and Starfield, 1990]. All the data necessary287 for analysis of the work terms, including slip, traction,288 internal stress and strain and vertical displacements, are289 constrained by the governing differential equations of290 continuum mechanics. Unlike the Finite Element Method,291 which requires discretization of the entire body, the BEM292 only requires discretization of model boundaries and dis-293 continuities (i.e., faults). This is advantageous for modeling294 multiple interacting faults because BEM requires less effort295 for discretization, and errors due to discretization and296 approximation arise only on the boundaries and along fault297 surfaces [Crouch and Starfield, 1990]. The BEM code used298 in this study, FRIC2D, computes the two-dimensional299 elastic and inelastic deformation associated with frictional300 slip along faults using the displacement discontinuity for-301 mulation of Crouch and Starfield [1990] with special302 constitutive frictional-slip elements [Cooke and Pollard,303 1997]. The model boundaries and fault surfaces are dis-304 cretized into linear elements each with uniform shear and305 normal displacement discontinuities. Additionally, along306 fault surfaces FRIC2D requires prescription of constitutive307 properties such as cohesion and friction coefficient.308 FRIC2D has been used to investigate faulting [Cooke and309 Pollard, 1997; Roering et al., 1997; Cooke and Kameda,310 2002; Cooke and Murphy, 2004], bedding plane slip within311 folds [Cooke et al., 2000], joint propagation [Cooke and312 Underwood, 2001] and sediment compaction associated313 with buried craters [Buczkowski and Cooke, 2004]. FRIC2D314 assumes linear elastic, isotropic and homogeneous rheology,315 which may not appropriately simulate sand deformation at

316all scales or at all time intervals. Specifically, the code does317not capture the details of individual grain movement and318instead treats the entire sandbox as elastic for the short time319periods simulated. Because loose sand accumulates a large320amount inelastic deformation over the course of an analog321experiment, infinitesimal snap shots of quasi-elastic defor-322mation are simulated within each numerical experiments of323this study. These snapshots allow us to investigate the first-324order effects of fault geometry and growth on the work325budget of the accretionary system.

3273. Model Setup

328[11] The shape and properties of the wedge in the model329was designed to mimic the sandbox analog experiments330analyzed by the particle image velocimetry technique (PIV)331[Adam et al., 2005]. By comparing sequential high-resolu-332tion images, PIV is able to track grain-by-grain dislocation333and thus produce maps of shear strain within the wedge. In334this way, it is possible to obtain a continuous recording of335the activity of faults and their geometry. In order to simulate336an underthrusting stage in our numerical model, we sequen-337tially lengthen the roof thrust (Figure 2).338[12] The BEM model simulates a 0.6 m length of the339sandbox experiment performed by Adam et al. [2005]. We340follow the observations of Adam et al. [2005] and focus our341investigation on the deformation within the toe of the wedge342at the transition from underthrusting to accretion. The total343length of the model is 1 m to ensure that the lateral sides of344the model are sufficiently far from the region of investiga-345tion to minimize boundary effects. The topography of the346top of the model resembles the laboratory experiment and in347the model the upper surface is prescribed to be stress free348(Figure 2). A series of numerical models investigates small349increments of deformation during the transition from un-350derthrusting to accretion; in this manner, the models simu-351late snapshots through this transition. Within each snapshot,352only fault surfaces with activity detected by PIVof Adam et353al. [2005] and the basal decollement are included in the354model. For each snapshot, the left side and base of the355model are heaved to the right 0.5 cm (0.5% contraction)356while the upper portion of the right side of the mode is357horizontally fixed. The lower two elements of the right side358heave to the right creating a discontinuity along a horizontal

Figure 2. Conceptual diagram of model set up and boundary conditions. The deformation is modeledby heaving the left, bottom elements and bottommost two elements of the right end side toward the right.The residual and peak coefficients of friction are chosen for active and inactive segments of the basaldecollement, respectively. The underthrusting stage is modeled by progressively lengthen the roof thruststo the right. Because of wedge geometry, the roof thrust will be loaded by increasing overburden as itlengthens.


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359 frictional interface that simulates the base of the laboratory360 experiment. Where the shear stress is low (i.e., less than361 frictional strength) along this base, the overlying sand is362 transported to the right; where shear stresses are high, the363 sand may not be transported to the right creating a decolle-364 ment surface. In addition to the frictional interface simulat-365 ing the basal decollement, various roof thrust lengths and366 accretion fore thrusts are prescribed as frictional interfaces367 within the models.368 [13] The frictional properties assigned to the fault surfa-369 ces depend on whether they are slipping or not. For sliding370 surfaces (faults within the wedge and portions of the basal371 decollement to the right of these faults) a friction coefficient372 of 0.5 is assigned to mimic frictional properties under373 dynamic slip conditions measured in laboratory tests of374 Adam et al. [2005]. The portion of the basal decollement to375 the left of the thrust faults (in front of the thrust faults) is376 assigned higher friction (m = 0.7). This higher value377 corresponds to the coefficient of friction measured for the378 sand at peak strength, i.e., at the verge of failure [Adam et379 al., 2005]. The roof thrust and fore thrusts added in380 sequential models are assigned the lower-valued dynamic381 coefficient of friction (m = 0.5). The effective normal and382 shear stiffness are estimated from the material stiffness and383 the width of the fault zone observed in the experiments of384 Adam et al. [2005]. To establish the values of Young’s385 modulus and Poisson’s ratio for our models, we compared386 the frictional strength of the fine sand used by Adam et al.387 [2005] to physical properties compiled from soil engineer-388 ing studies of a uniform fine to medium sand [Lambe and389 Whitman, 1969]. Average values for the Poisson’s ratio390 (0.25–0.40) and Young’s modulus (10.35–17.25 MPa)391 were prescribed for our models (Turner-Fairbank Highway392 Research Center, http://www.tfhrc.gov/safety/pubs/04094/

39304.htm). The sand density was set at 1.732 g/cm3 [Adam394et al., 2005].395[14] Using the prescribed boundary conditions and mate-396rial properties, the BEM solves for the displacements and397stresses along external boundaries and faults within the398model so that that Wfric and external work Wext can be399calculated. For example, the only boundaries that produce400external work are the left side and bottom two elements of401the right side. The product of force times displacement is402zero along the other model boundaries. Additionally, dis-403placements and stresses can be sampled at specific points404throughout the fault system to calculate Wgrav and Wint. The405seismic and fault propagation energy are not directly406assessed within the quasi-static models because they reflect407a change in conditions over time. The seismic energy is the408energy released during slip. This can be calculated by409comparison of fault stresses within two models before and410after the fault has slipped. Similarly, the propagation energy411can be estimated by the length of new fault surface times the412energy needed to create new fault surface area in models413before and after the development of the new fault. The414energy needed to create new fault surface has been mea-415sured in the lab. Depending on different confining pressures416and rock type, the fracture energy can vary greatly (between417101 and 104 J/m2 [Wong, 1982, 1986; Cox and Scholz,4181988]). Recent studies suggest that these values can be419decreased by a factor of 3 to 5 if the dilation process before420rupture is properly taken into account [Labuz and Dai,4212000]. The sum of the seismic energy and the fault422propagation should equal the change in external energy423between models before and after fault slip or a new fault424develops.

4254. Model Results

426[15] Progressive underthrusting is simulated by lengthen-427ing the roof thrust. We examine five snapshots of under-428thrusting at intervals or 2 cm thrust length measured along429the upper horizontal portion of the roof thrust. The cumu-430lative error of all the work terms can be assessed from the431difference between the sum of all the work components432(Wtot) and the work done on the external boundary of the433system (Wext). Most models produce cumulative errors less434than 1.5% of the Wext (Figure 3). Much of this error arises435from the internal and gravitational work calculations, which436are integrated over the area of the model. Because the437stresses and displacements are numerically unreliable near438the edges of individual fault elements, some critical points439within regions of stress concentration near the faults are left440out of the integrations, resulting in underestimation of Wint

441and Wgrav. Consequently, Wtot is consistently less than Wext

442and the external work, which is not hampered by data gaps,443provides a reliable estimate of the total work in the system.

4444.1. Work Increase During the Underthrusting Mode

445[16] Numerical results show a systematic increase in Wext

446as the sheet is progressively underthrust (Figure 3). This447suggests that with increasing underthrusting, continued448deformation of the wedge requires greater work applied to449the system. The largest portion of work goes into frictional450heating (Wfric > 0.5 Wext), while uplift against gravity451consumes only about 2% of Wext.

Figure 3. Increase in Wgrav, Wfric, and Wint for each stageof roof thrust lengthening. This increase leads to systematicincrease of total work during progressive underthrusting.The new fore thrust produces significantly less frictionalheating and less total work in spite of increased Wint. Thetransition from underthrusting to accretion via the develop-ment of the new fore thrust follows total work minimizationof the fault system.


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452 [17] Within the underthrusting system, uplift against453 gravity only occurs near the dipping ramp section of the454 roof thrust (Figure 4). Although small, Wgrav increases with455 thrust sheet length due to increasing thickness of material456 uplifted above the ramp. The increase in Wfric with progres-457 sive underthrusting can be attributed to both increasing roof458 thrust length and increased weight of overburden, which459 resists frictional sliding and subsequently increases friction-460 al heating within the system. For similar reasons, Wint

461 systematically increases as the roof thrust grows longer.462 As the active ramp is under deeper portions of the wedge,463 fault slip is inhibited and contraction instead is accommo-464 dated by greater internal deformation of the surrounding465 material. We see in the models a trade-off between the466 contraction accommodated by fault slip and that accommo-467 dated by deformation of the surrounding material.

468 4.2. Work Decrease During Transition

469 [18] We simulate the beginning of the accretionary stage470 documented in sandbox experiments by removing the471 inactive roof thrust and adding an active fault ramp at the472 front of the deforming wedge in the position of fore473 thrusting observed by Adam et al. [2005]. The transition474 from underthrusting to accretionary mode is associated with475 a significant decrease in the total work on the system476 (Figure 3). This suggests that deforming a wedge system477 with a ramp in front of the wedge takes less overall tectonic478 work than deforming a system with a mature underthrust479 sheet. The transition produces a net decrease in Wext via an480 increase in Wint and a correspondingly larger decrease in481 Wfric.482 [19] The work against gravity decreases upon the transi-483 tion to accretion because the active fault ramp has now less484 overburden to uplift than the mature roof thrust (Figure 2).485 The reduction of overburden weight on the active fault ramp486 also decreases the resistance of the ramp to frictional487 sliding. Thus, to accommodate the same amount of con-488 traction, the accretionary fault system produces less fric-489 tional heating than the long and mature underthrust fault490 system. In apparent contrast to the lesser resistance of the491 fore thrust to frictional slip, this fault has less slip than the492 roof thrust in the underthrust models just prior to transition493 to underthrusting (Figure 5). This is demonstrated by the494 significant increase in internal work within the accretionary495 model as the surrounding material accommodates deforma-496 tion in lieu of fore-thrust slip (Figure 3). Consequently, the497 accretion model that minimizes the overall contributions of498 Wgrav, Wint and Wfric is not the fault configuration that499 accommodates greatest slip.

5004.3. Position of the Maximum Shear Stress Along the501Basal Decollement

502[20] The position of the maximum shear stress on the503basal decollement was calculated in the model with the504longest roof thrust just before the transition to the accretion505stage (based on the sandbox models of Adam et al. [2005]).506The numerical model gives a left-lateral maximum shear507stress located at 26 cm along the basal decollement508(Figure 6b) that is also associated with a high distortional509strain energy (white arrow, Figure 6a) along the base of the510numerical wedge. High levels of distortional strain energy511density have been suggested to characterize the propagation512path of deformation bands [Okubo and Schultz, 2005].513Accordingly, we hypothesize that the localization of distor-514tional strain energy in our model may mimic the incipient515propagation of a new fore-thrust ramp at the wedge toe. A516striking fit can be observed between the position of the517maximum basal shear stress within the numerical model and518the location of accumulating shear strain at the base of the

Figure 4. Plot of vertical displacement within the wedge during shortening. The vertical uplift isconcentrated on the ramp portion of the roof thrust. The total work against gravity integrated throughoutthe model is much smaller than other work terms.

Figure 5. Slip distribution shown by the (1) roof thrustramp and (2) fore-thrust ramp, respectively. The fore-thrustramp has less slip despite the lesser vertical overburdenacting on the fault compared to on the roof thrust ramp. Theroof thrust is at its maximum length, prior to transition toaccretion observed by Adam et al. [2005].


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519 sandbox model of Adam et al. [2005] (black arrow pointing520 at yellow red patches in Figures 6d and 6e).

521 4.4. Efficient New Ramp Location and Vergence

522 [21] In the sandbox experiment, before the roof thrust is523 totally abandoned (turning pale green from red) an area of524 diffuse deformation can be observed ahead of the wedge toe525 (Figure 6d). Adam et al. [2005] interpreted this area as a526 dilational domain that produces strain hardening of the sand527 just prior to nucleation of a new fault, which eventually528 occurs toward the leftmost part of the dilational domain529 (Figure 6e). The new fore-thrust ramp location is expected530 to breach the dilational domain. On this premise, we tested531 the most efficient position for a new ramp to nucleate based532 on the principle of energy minimization. In our numerical533 model, we explored two likely locations for fore-thrust and534 back-thrust ramps for the accretionary stage: (1) the location535 of maximum shear stress along the basal decollement536 (location 1, fore thrust 1 (FT1) and back thrust 1 (BT1);537 Figure 7) and (2) beyond the wedge toe in the location of538 faulting observed by Adam et al. [2005] (location 2, FT2539 and BT2; Figure 7). FT 2, in front of the wedge, requires the540 least external work to accommodate the same contraction as541 the other ramps. The Wfric and Wgrav show nearly equal542 values for both FT 1 and FT 2 (Figure 7); however, the Wint

543 within the FT 1 model exceeds the Wint within the FT 2544 model. This reflects the resistance of FT 1 to slip (Figure 8)545 due to increased overburden weight. The reduced slip on the

546ramp increases the amount of local deformation (i.e., Wint)547ahead of the wedge toe, as displayed by plots of Strain548Energy Density (SED) (Figure 9b). Conversely, FT 2 more549efficiently accommodates contraction via a greater slip550(Figure 9a), resulting in a net decrease of the Wtot.551[22] The mechanical efficiency of fore thrusts and back552thrusts can also be compared. The wedge needs to expend a553larger amount of energy in order to develop back thrusts554rather than fore thrusts (Figure 7). Both BT1 and BT2 are555characterized by a lesser slip than the fore thrusts (Figure 8),556so that the Wfric is smaller than along the FTs, even where557BT1 has greater overburden than FT1. However, the lesser558slip yields disproportionably large increase in Wint. The559distribution of strain energy density within the wedge560indicates a higher level of wedge internal strain for back561thrusts compared to the model with fore thrusts (Figures 9c562and 9d). The overall effect is that of a larger Wtot that makes563the wedge less efficient to deform with a back thrust rather564than with a fore thrust. This is consistent with many565analogue experimental observations in that back thrusts do566not display prolonged activity. Consequently, the most567efficient location for a new ramp is not at the point of568maximum basal shear stress, as one would expect for a ramp569branching off the basal decollement. Instead new faults570develop where the gravitational, frictional and internal571mechanical work terms are optimally combined. As valida-572tion of the work minimization principle, the fault configu-

Figure 6. (a) Distortional strain energy (DSE) density within the wedge. (b) Greatest shear stress(negative is left lateral), which occurs at x = 26 cm along the base of the numerical sandbox, where themodel also shows an area of relatively high distortional strain energy. (c) PIV image of sandboxexperiment of Adam et al. [2005]. (d) Decrease of slip on the roof thrust that follows stress buildup at thebase of the wedge and frontal dilation of sand. (e) Shear localization in the form of a fore-thrust–back-thrust couple. The numerical model is superimposed to show coincidence of numerically calculatedmaximum shear stress with area of maximum shear strain at the base of the sand-box experiment (blackarrows in Figures 6d and 6e).


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573 ration that minimizes the overall work of the system574 correlates with sandbox observations.

576 5. Discussion577 5.1. Friction and Slip Evolution in Sandbox Models

578 [23] We have implemented a numerical model in order to579 quantitatively study the work components of an underthrust-580 ing/accretion transition. The analysis was based on the idea581 that faults nucleate within sandbox models of wedge growth582 to minimize the system’s overall work. Our results indicate583 that gravity plays a fundamental role during deformation by584 controlling the balance between the shear resistance on fault585 surfaces (Wfric) and the overall strain around the fault (Wint).586 While the critical taper theory predicts uniform deformation587 within a growing wedge of constant taper angle, locally588 changing wedge slope has been proposed to alter the normal589 stress distribution along the basal decollement to promote590 the development of distinct under thusting and accretion591 modes [Gutscher et al., 1998; Burbidge and Braun, 2002;592 Del Castello et al., 2004]. Our numerical experiments593 further verify that cycles of episodic cycles of underthrust-594 ing and accretion should be expected within growing595 accretionary wedges rather than monotonic growth pre-596 dicted by critical taper theory. The analysis in this paper597 offers an explanation of the trade-off mechanisms between598 different energy terms that force slip to jump from fault to599 fault across a sand wedge when a new fault propagates600 along the basal decollement, which was fully disclosed by601 recent advances in visualization techniques of analogue602 models of Coulomb wedges (PIV [Adam et al., 2005]).

6035.2. Work Minimization

604[24] The principle of work minimization predicts that the605fault system will evolve to a configuration that requires the606least total work so that we can compare the energetic costs607of various fault geometries to predict wedge system evolu-608tion. If we neglect the acoustic energy released by slip609(Wseis) and energy consumed in the formation of new fault610surface (Wprop), we can predict that the accretionary mode is611favored once the underthrust sheet is longer than 4.5 cm612(Figure 10). At this length the total work of accommodating613contraction by roof thrusting equals the work of accommo-614dating contraction along a new fore thrust in front of the615wedge. As the thrust sheet grows beyond 4.5 cm, the Wext

616(Wint + Wgrav + Wfric) of continued underthrusting requires617greater energy than accretion (9.04 J). However, the system618persists in the underthrusting stage. Why? The energy619required to produce the new frontal ramp from the basal620decollement must also be taken into account. Underthrust-621ing will be favored as long as the Wext is less than that622necessary for accretion plus Wseis and Wprop for the accre-623tionary model (equation (1)). The experiment of Adam et al.624[2005] demonstrates that the transition from underthrusting625to accretion occurs when the roof thrust is 10.5 cm long,626when the model is expending 9.44 J to continue under-627thrusting (Figure 10). The difference between theWext of the628underthrust model at the transition and Wext of the new629accretion model (0.4 J) reflects the energy lost during this630transition to acoustic energy and development of new fault631surface (Wseis + Wprop; Figure 10).

Figure 8. Slip along different fault locations and rampvergencemodeled. Fore thrust 2 shows the greatest amount ofslip. Positive sign is for left-lateral motion (FT), and negativesign is for right-lateral motion along the ramp (BT).

Figure 7. Weight of different work terms for under-thrusting phase (roof thrust length just prior to transition)and accretion. Fore thrust 2 is the most energeticallyefficient faulting geometry for this system setup as itdisplays the least amount of Wtot required for wedgedeformation. In all other cases, the Wint increase overrunsWfric decrease, making these faults not favorably positionedin terms of work minimization.


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632 [25] Although the energy required to propagate faults in633 sand is not available due to the difficulty of these measure-634 ments, laboratory tests on soft cemented sedimentary rocks635 at low confining pressure have shownWprop around 140 J/m

2

636 [Labuz and Dai, 2000]. If such sandstone grew a fault as637 long as the fore thrust in our accretionary models, the rock638 would consume &9.7 J. Of course, we expect loose sand to639 require much less energy than soft cemented sandstone640 because the unconfined compressive strength of soft,641 cemented sandstone is 100 times greater than that of sand.642 The order of magnitude of Wprop estimated by the model643 (<0.4 J) agrees with our expectations from available labora-644 tory data.645 [26] The values of the various work terms reported here646 reflect the specific fault geometry of and material properties647 within the numerical experiment. Although the values of the648 work components will vary with different fault configura-649 tions, the tradeoffs between the different work components650 within the overall work budget will remain the same. The651 transition from underthrusting to accretion occurs when the

652energetic benefit of slip along the new fore thrust exceeds653the energetic cost of producing this surface.

6545.3. Is Work Against Gravity Not Important in655Sandboxes?

656[27] The progressive increase of vertical overburden due657to thickening by no doubts constitutes one of the main658drivers to induce the wedge to switch from underthrusting to659accretion during cycles observed in analogue experiments660[Mulugeta and Koyi, 1992; Gutscher et al., 1996; Del661Castello et al., 2004]. The BEM models have relatively662small components of work against gravity (Wgrav '; Wint

663and Wfric). To some degree, this is due to the small scale of664the sandbox, which produces small overburden pressures;665larger scaled numerical models display a higher portion of666work against gravity [Cooke and Murphy, 2004]. Adding a667dip to the base of the sandbox model would increase the668work against gravity slightly but not to the level of the other669work components. Our gravitational work values are in the670same order as the analytical estimates of Gutscher et al.

Figure 9. Plots of strain energy density for different fore-thrust and back-thrust location. A progressiveenlargement of wedge regions affected by high levels of strain (yellow, see scale for reference) can beobserved from Figure 9a (fore thrust 2) to Figure 9d (back thrust 1). These cross sections confirm thatfore thrust 2 accumulates the least internal strain within the wedge. The box in Figure 9a indicates thelocation of Figure 11.


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671 [1998]. Does this mean that gravity is insignificant in672 sandbox experiments of accreting systems? No. While work673 against gravity is small (uplift against gravity occurs only674 on the ramp portion of each fault, Figure 4), overburden675 weight due to gravity critically influences changes in Wfric

676 and Wint that govern the transition from underthursting to677 accretion. With increasing vertical overburden, the ramp678 will tend to slip less, decreasing the Wfric and eventually679 result in the clamping of the fault. However, at the same680 time, decreasing the slip means a larger fraction of energy is681 consumed in the form of internal strain of the host rock682 (Wint). The variation of Wint can be smaller or larger than683 that of Wfric (Figure 7). This trade-off will determine684 whether or not a fault is efficiently located such that the685 wedge minimizes the energy necessary for deformation.686 Yet, gravity will always be the driving variable. Neverthe-687 less, within natural systems that operate on larger scales, the688 work against gravity can also become significant compared689 to other work terms [Cooke and Murphy, 2004]. Conse-690 quently, the work budget for natural systems may differ691 from sandbox models, which are quite sensitive to changes692 in Wfric. Future work should explore the scales of models so693 that consideration can be given in extrapolating from694 sandbox interpretations.

695 5.4. Fore Thrusts Are Energetically More Efficient696 Than Back Thrusts

697 [28] The numerical model can also be used to evaluate698 whether the site for spontaneous shear localization in a new699 ramp observed in the models of Adam et al. [2005] responds

700to the work minimization principle. In principle, the new701fore-thrust ramp could nucleate within the width of the702dilation zone along the weakest grain boundaries or largest703grains. A new imbricate splaying off at the basal decolle-704ment tip may develop where shear stress is maximum along705the basal decollement. The BTs consistently show a higher706Wtot than FTs, suggesting that back thrusts are less efficient707and may only be secondary features of the accretion process708that allow for a partial accommodation of contraction during709the folding process above the fault ramp. Even if our710numerical experiments simulate a very small amount of711contraction, we believe that work minimization can also712explain why the back thrusts are generally observed to die713out quite rapidly in sandbox models, immediately after their714nucleation [Mulugeta and Koyi, 1992]. This may occur715because these faults are steeper than fore thrusts and716therefore are not optimally oriented to the horizontal con-717traction [Cooke and Murphy, 2004]. This produces very718high level of internal strain, possibly leading to strain719hardening and clamping of the fault [Lohrmann et al.,7202003]. Other mechanisms must be active to produce a721long-lived BT, as in the case of weaker layers in the722deforming section [Liu et al., 1992; Cotton and Koyi,7232000; Bonini, 2001]. Another mechanism that may favor724back thrusting is exceptionally high levels of fluid over-725pressure, which produces micro fracturing, deformation726bands and cataclastic. Our model results show that the727stored internal work, Wint, controls the preference of fore728thrusts over back thrusts. This suggests that weak layers729and/or damage from fluid overpressures may serve to730reduce Wint locally in order to favor back-thrust develop-731ment. In nature, very few systems have back thrusts with732larger slip than fore thrusts. In the Cascadia subduction zone733accretionary complex back thrusts seem to be favored where734conspicuous and rapid sedimentation promotes fluid pres-735sure build up [MacKay, 1995; Gutscher et al., 2001; Adam736et al., 2004].737[29] Panian and Wiltschko [2004, 2007] have suggested738that back thrusts play a key role in forward propagation of739fore thrusts. In their model, the initial back thrust develops740at the toe of the wedge and propagates downdip to the741decollement. During propagation of back thrust a fore thrust742develops that intersects the back thrust at the decollement.743Within our model prior to fore thrusting, the area of greatest744strain energy density and Navier-Coulomb stress, measures745of propensity for fault development, occurs at the tip of the746wedge (Figure 11) and at the same location of back-thrust747initiation of Panian and Wiltschko [2004, 2007]. While the748stress distribution may promote the development of a back749thrust, our models results show that for the overall system, a750fore thrust requires less work than the back thrust to751accommodate contraction; the fore thrust is more efficient752and takes over.

7535.5. Downdip Propagation of the Fore-Thrust Ramp

754[30] The observation that the energetically most efficient755fore thrust is located away from the position of the maxi-756mum shear along the basal decollement may raise some757further questions. The position of maximum shear stress758calculated by our model just beneath the wedge toe is759consistent with previous mechanical analyses of frontal760imbrication [Platt, 1988]. In the absence of the results

Figure 10. (a) Models simulating successive thrust sheetgrowth. In this setup, the topography remains constant foreach underthrust model. (b) External work (Wext) requiredfor contraction increases with length of the underthrustsheet. The development of a fore thrust in front of thewedge is energetically favored when the Wext required forcontinued underthrusting exceeds the Wext for fore thrustingplus the work required to develop the new fault surface ofthe fore thrust (Wprop) and acoustic energy associated withslip (Wseis).


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761 offered by our numerical model, one would normally expect762 that the branching would occur at the point of maximum763 shear along the basal decollement or at the location of764 maximum distortional strain or positions of maximum765 Coulomb stress. The PIV experimental data of Adam et al.766 [2005] show an area of diffuse dilatation within the not yet767 faulted sand pack ahead of the maximum shear strain768 (equivalent to our maximum shear stress point) location769 on the basal decollement. Within the dilatational domain,770 material may be on the verge of failure. Our numerical771 results show that it is mechanically more efficient to772 nucleate a new ramp at the outer frontal limit of the773 dilatational domain, which might then propagate downward774 to the basal decollement, rather than propagating a new775 ramp from the basal decollement at the point of maximum776 shear strain/stress. The nucleation of the new ramp at777 shallow depths is also supported by the shallow region of778 high Navier-Coulomb stress displayed by the numerical779 model in the area corresponding to the dilatational zone780 (Figure 11). Our results contrast the generally accepted781 paradigm that fault ramps propagate upward from the basal782 thrust decollement tip [e.g., Suppe and Medwedeff, 1990]783 (Figure 12a) and concur with alternative models of ramp784 nucleation that evoke failure within the layer and propaga-785 tion of both fault tips toward upper and lower bedding plane786 decollements. These models propose that, under horizontal787 compression, the more rigid layer fails first, creating the788 ramp, which later propagates into the less competent bed-789 ding planes [Eisenstadt and De Paor, 1987; McConnell et790 al., 1997; Morley, 1994] (Figure 12b). This outcrop-scale791 mechanical stratigraphy is analogous in the sandbox models

792to the sand layer, which acts as a single homogeneous layer793bounded by a weaker basal decollement and in the accre-794tionary wedge to deforming rock sequence. In principle, the795scale-invariant interaction between the various work terms796analyzed in this study at the laboratory scale can be applied797to the wedge scale to predict the nucleation of a shallow,798large-scale ramp at the foot of a deforming accreting wedge.799For example, initiation of a new frontal ramp was recently800proposed at the outer Western Alps to explain unexpectedly801high rates of local uplift [Niviere et al., 2006]. Unfortunate-802ly, the initiation of a new thrust ramp is difficult to detect in803nature as it is likely to be short-lived. Once the thrust ramp804localizes, the concentrated stresses at the fault tips will805facilitate propagation downward and the ramp will rapidly806link to the basal decollement tip.

8075.6. Coincidence Between BEM and Sandbox Model808for Small Amounts of Contraction

809[31] A remarkable outcome of this paper is the coinci-810dence between the position of the maximum shear stress on811the basal decollement calculated by our numerical model812with the maximum shear strain observed in the sandbox813experiment of Adam et al. [2005] (Figures 6d and 6e). We814emphasize that the location of the maximum shear stress815was neither set as a model boundary condition nor was it816prescribed at the beginning of the modeling procedure. This817is significant considering that although input parameters of818the numerical wedge were the same as the mechanical819properties of the noncohesive analogue material used in820the sand-box experiment, the BEM code was implemented821with the linear elastic continuum mechanics. It is certainly

Figure 11. Plots of (a) strain energy density and (b) Navier-Coulomb stress. High values of strain/stressare visible at shallow depth in the vicinity of the wedge toe (outlined by the dashed line).


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822 surprising that sand, which is generally thought as deform-823 ing plastically, can be modeled so well by a purely elastic824 numerical code along short timescales. In the recent past825 different models (FEM, DEM [Ellis et al., 2004; Burbidge826 and Braun, 2002; Yamada and Matsuoka, 2005; Naylor et827 al., 2005]) were used to numerically simulate deformation828 observed in experimental Coulomb wedges. Our BEM829 model, which underestimates the effects of plasticity, might830 not be able yet to capture the full continuous deformation831 process within a Coulomb wedge. Certainly, more detailed832 models are needed to further assess whether elasticity can833 be effective on a continuous timescale or to appraise the834 importance of elasticity over plasticity as the key rheology835 driving the wedge deformation. However, our model dem-836 onstrates that at least for small temporal scales elasticity837 contributes to the wedge mechanics. Indeed, if the wedge838 deformation is discretized into several snapshots, we can839 observe that important wedge characteristics such as (1) size840 and spacing of imbricates, (2) switching between different841 deformation modes, and (3) activation/deactivation of single842 fault due to strain hardening/weakening processes can be843 analyzed in the light of elastic models and that these844 respond to mechanical concepts of work minimization.

846 6. Conclusions

847 [32] Our BEM models have successfully reproduced the848 transitory deformation of accreting wedges during an accre-849 tion-underthrusting cycle observed in analog experiments.850 The modeling results confirm that the underthrusting to851 accretion transition is driven by a complex interaction852 between gravitational forces and shear resistance along853 fault surfaces such that a trade-off exists between the854 amount of energy dissipated as internal strain and that855 absorbed via frictional heating on the faults. Consideration856 of individual and total work of the accreting system857 demonstrates that the development of new fore thrusts858 reflects the overall tendency of the system to minimize859 total work even while individual work terms may increase.860 Because every new fore thrust will accumulate slip and861 mature to a progressively less efficient underthrusting stage,862 a repetitive cycle of underthrusting and accretion arises.863 The transition from underthrusting to accretion occurs when864 the increase in system efficiency associated with the devel-865 opment of a new fore thrust exceeds the energetic cost of

866growing the fore thrust (Wprop + Wseis). Our models results867show distinct phases of underthrusting and accretion during868wedge growth, such as observed in different segments of869the Alaskan margin [Gutscher et al., 1998; Moore et al.,8701991] that are not explained by critical taper theory. In the871models, fore thrusts are characterized by higher efficiency872compared to back thrusts. This is consistent with observa-873tion from previous analogue and numerical models that slip874along back thrusts rapidly lessens with increasing displace-875ment. Furthermore, the position and vergence of a ramp that876is most energetically efficient matches that produced in the877analog experiments.878[33] The work budget analysis offers results consistent879with both observations of physical models [Adam et al.,8802005] and analytical treatments of accretion [Platt, 1988].881Our numerical approach fills in knowledge gaps from882previous analyses of accreting systems that were limited883to work against gravity and frictional sliding [Gutscher et884al., 1998; Burbidge and Braun, 2002]. By considering the885significant component of work accumulating as internal886deformation within the wedge (Wint), we are able to paint887a more complete picture of the dynamics of underthrusting-888accretion transitions. During the transition from underthrust-889ing to accretion, the decrease of work against friction890accounts for all of the increase in fault system efficiency891and overcomes associated increases in internal work.892Consequently, we expect that even small changes in wedge893slope and thickness, such as investigated by Gutscher et894al. [1998] and Burbidge and Braun [2002], will have895significant impact on slip distribution and mode of wedge896deformation.

897[34] Acknowledgments. The work was funded by a grant from the898National Science Foundation (EAR-0349070). N. Kukowski, M. A.899Gutscher, and G. Moore are acknowledged for their fruitful comments.900D. Wiltschko kindly offered suggestions and unpublished material as a901useful comparison.

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Figure 12. Two alternative models for ramp generation. (a) Ramps growing upward from the tip of thepropagating decollement. (b) Ramps nucleating first within the rigid layer and then propagatingdownward to the tip of the sole thrust. (c) Two scenarios producing almost identical styles of ramp-relatedfolding (see McConnell et al. [1997] and Morley [1994] for more details).


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%%%%%%%%%%%%%%%%%%%%%%%1103M. L. Cooke and M. Del Castello, Geosciences Department, University1104of Massachusetts, 611 North Pleasant Street, Amherst, MA 01003, USA.1105([email protected])


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