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Physics of the Earth and Planetary Interiors 163 (2007) 35–51

Parallel computing of multi-scale continental deformationin the Western United States: Preliminary results

Mian Liu a, Youqing Yang a,∗, Qingsong Li b, Huai Zhang a,c

a Department of Geological Sciences, University of Missouri-Columbia, Columbia, MO 65211, USAb Lunar and Planetary Institute, Houston 77058, USA

c Computational Geodynamics Lab, Graduate University of Chinese Academy of Sciences, Beijing, China

Received 18 January 2007; received in revised form 16 June 2007; accepted 16 June 2007

bstract

Lithospheric deformation in the western United States is one of the best examples of diffuse continental tectonics that deviate fromhe plate tectonics paradigm. Conceptually, diffuse continental deformation is known to result from (1) weak and heterogeneousheology of continents and (2) driving forces that arise from plate boundaries as well as within the continental lithosphere. However,he dynamic interplay of continental rheology and driving forces, hence the geodynamics of continental tectonics, remains poorlynderstood. The heterogeneous rheology and multiple driving forces cause continents to deform over different spatiotemporal scalesith different physical processes, yet most geodynamic models for continental tectonic avoid dealing with such multiphysics partlyecause of (1) the limited observational constraints of lithospheric structure and deformation, and (2) high demands on computinglgorithms and resources. These constraints, however, have relaxed significantly in recent years to permit exploration of some ofhe multi-scale physics governing continental tectonics. Here we present preliminary results of modeling multi-scale tectonics inhe western United States using parallel finite element computation. In a 3D subcontinental-scale model, we used fine numerical

eshes to incorporate all major tectonic boundaries and rheological heterogeneities in the model to explore their interplay with

ectonic driving forces in controlling active tectonics in the western US. In another model for the entire San Andreas Fault system,e explored strain localization and simulated fault behavior at multi-timescales ranging from rupture in seconds to secular fault

reep in tens of thousands of years. These models can help to integrate data grids with distributed high-performance computingesources in the emerging geosciences cyberinfrastructure.

2007 Elsevier B.V. All rights reserved.

Sa

eywords: Parallel computing; Continental tectonics; Finite elements;

. Introduction

In the plate tectonics paradigm, the outer shell of thearth consists of a dozen or so rigid plates that move

elative to each other. The relative motion between aair of plates can be entirely determined by a simple

∗ Corresponding author.E-mail address: yangyo@missouri.edu (Y. Yang).

031-9201/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.pepi.2007.06.008

n Andreas Fault; Western US; Cyberinfrastructure

Euler vector (DeMets et al., 1990, 1994), and deforma-tion of the plates is limited to narrowly defined plateboundaries. Although the rigid-plate approximation issatisfactory in explaining many geological observations,broadly diffuse deformation away from plate boundaries,and significant deformation in plate interior, are common

(Gordon and Stein, 1992; Molnar and Tapponnier, 1975).Such non-plate behavior is particularly conspicuous incontinents; examples include the broad crustal deforma-tion in the Tibetan plateau and central Asia, and diffuse

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36 M. Liu et al. / Physics of the Earth a

continental deformation in the western United States.The diffuse continental tectonics have helped propellingmodern geosciences research to advance from rigid-platekinematics toward a more holistic understanding of litho-spheric deformation in the context of the Earth’s dynamicsystems (Molnar, 1988).

The cause of diffuse continental deformation isconceptually clear: the relatively weak and laterally het-erogeneous rheology of the continental lithosphere, andthe driving forces that arise not only from plate bound-aries but also from within and at the base of continentallithosphere (Artyushkov, 1973; Forsyth and Uyeda,1975; Molnar, 1988). However, the dynamic interplaybetween lithospheric rheology and driving forces, hencethe geodynamics of continental tectonics, remain poorlyunderstood, partly because numerical modeling of conti-nental tectonics is difficult. One of the major challengesis that continental deformation varies at different spatialand temporal scales. For example, present crustal defor-mation measured by the Global Positioning Systems(GPS) and other space-based geodetic techniques reflectmainly short-term deformation. Given the knowledgeof timescale-dependent lithospheric rheology (Jeanlozand Morris, 1986; Liu et al., 2000; Pollitz, 1997, 2003a;Ranalli, 1995), and the fact that tectonic processes andboundary conditions can change significantly throughgeological history, it is not surprising that short-termdeformation derived from space-geodesy often differsignificantly from long-term crustal deformation indi-cated by geological records (Dixon et al., 2003; Friedrichet al., 2003; He et al., 2003; Liu et al., 2000; Pollitz,2003a; Shen et al., 1999). Continental deformationalso varies strongly with spatial scales. At continentalscales, most geological discontinuities, including faultsand structural boundaries, may be ignored in first-orderapproximations (Bird, 1999; England and McKenzie,1982). However, at smaller scales these features oftendominate the tectonic processes.

Current numerical models are usually limited to cer-tain spatial and temporal scales. Whereas there are soundscientific rationales for doing so, this practice oftenrepresents a compromise to two major obstacles: (1)the lack of detailed observational constraints, especiallyfor large-scale models, and (2) the limitation of com-puting power and algorithms for solving non-linear,time-dependent, and coupled physics arising from multi-scale continental tectonics. The situation has improveddramatically in recent years, and will get much better

in the near future. Many recent and ongoing researches,including the EarthScope (www.earthscope.org) project,will dramatically improve the volume and quality ofobservational constraints of crustal deformation and

anetary Interiors 163 (2007) 35–51

lithospheric structure in the western United States. Fur-thermore, rapid advancement of computer hardware inrecent years, marked by the affordable Beowulf PCclusters, and new computational algorithms, have madehigh-performance computing no longer a luxury.

In this paper we introduce our ongoing effort of mod-eling multi-scale tectonics in the western US. Followinga brief overview of the tectonic background, we present ascheme of parallel computation of lithospheric dynamicsusing an automated finite element modeling system. Wethen show two preliminary models: (1) a subcontinental-scale model that explores the roles of various forcesdriving active tectonics in the western US, and (2) amodel of the entire San Andreas Fault (SAF) systemthat explores multi-timescale slips on the SAF and theresulting stress distribution and strain localization in theSAF plate boundary zone.

2. Multi-scale tectonics in the western US

The western US is one of the best examples of diffusecontinental deformation (Fig. 1). Its tectonic history hasbeen extensively reviewed (Atwater, 1970; Atwater andStock, 1998; Axen et al., 1993; Burchfiel et al., 1992;Burchfiel and Davis, 1975; Coney, 1978; Dickinson,2002; Lipman et al., 1972; Sonder and Jones, 1999;Stewart, 1978; Wernicke, 1992; Zoback et al., 1981).During the Mesozoic and the early Cenozoic, subductionof the oceanic Farallon plate under the North Americaplate produced the Sevier fold-and-thrust belts in theNorth American Cordillera. Along the coastal area, acontinuous, narrow belt of arc magmatism was estab-lished (Cross and Pilger, 1982; Lipman et al., 1971).During the Late Cretaceous the arc magmatism was sig-nificantly disrupted and shifted eastward as far inlandas Colorado (Cross and Pilger, 1982). Accompanyingthe shifting magmatism was the Laramide deformation,which features crustal bulking and giant fault-bounded,basement-cored uplifts (Brown, 1988; Dickinson andSnyder, 1978).

Since Tertiary the Farallon-North American plateconvergence was gradually replaced by shear motionalong the San Andreas transform boundary (Atwater,1970). In the western Cordillera, compressive tec-tonics was replaced by widespread crustal extension,which reaches up to 300% in the Basin and Rangeprovince (Hamilton and Myers, 1966; Wernicke, 1992;Zoback et al., 1981). Extension in the southern Basin

and Range province ceased ∼10 Myr ago but remainsactive in northern Basin and Range, as indicated bythe widespread seismicity (Fig. 1). Although exten-sion in the western US is diffuse over time and space

M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51 37

tes. Epntal m

Fig. 1. Seismicity and tectonics provinces of the western United Stasimplified boundaries of tectonic provinces are used in the subcontinefigure legend, the reader is referred to the web version of the article.)

(Axen et al., 1993), most extension concentrated nearthe western and eastern margins of the northern Basinand Range province (Lowry and Smith, 1994; Wernickeet al., 1987). In the past 10 Ma extension has encroachedinto the relative rigid Great-Valley Sierra Nevada block(Boyd et al., 2004; Jones et al., 2004; Wernicke et al.,1996).

In the past two decades space-based geodetic mea-surements, especially the GPS, have greatly improvedthe kinematics of active crustal deformation in the west-ern US (Argus and Gordon, 1991; Bennett et al., 2002,1999, 2003; Davis et al., 2006; Dixon et al., 2000, 1995;Miller et al., 2001b; Minster and Jordan, 1987; Murrayand Lisowski, 2000; Thatcher et al., 1999; Ward, 1990;Wernicke et al., 2000). The results show broadly diffuseand spatially variable deformation. South of lat. 36◦Nmuch of the relative motion between the Pacific andNorth America plates is taken up by dextral shear alongthe southernmost San Andreas Fault system. At the lati-tudes of northern Basin and Range province, significantportion of the Pacific-North America relative motion isaccommodated by broad deformation across the Basin

and Range province and the Sierra Nevada-Great Valleymicroplate. Up to 25% of relative plate motion is accom-modated by the Eastern California Shear Zone (Dokkaand Travis, 1990; Gan et al., 2000; McClusky et al., 2001;

icenter data (red dots) are for 1973–2002 earthquakes (USGS). Theodel (see Fig. 2). (For interpretation of the references to color in this

Miller et al., 2001a; Savage et al., 2001). Further northin the coastal regions of Pacific Northwest, crustal defor-mation is dominated by NE compression related to thesubduction of the Juan de Fuca plate under the NorthAmerica plate.

Whereas the short-term strain measurements arebroadly consistent with late Cenozoic tectonics in thewestern US, the discrepancies are noticeable in manyplaces. For example, in the Cascadian coastal region,geological records do not show strong northeastwardcrustal shortening as indicated by the GPS data. Instead,the moment tensor of small earthquakes in this regionindicate E–W extension (Lewis et al., 2003), consis-tent with evidence of structural geology (Wells andSimpson, 2001; Wells et al., 1998). Some of the discrep-ancy may be expected, because space-geodesy measuresonly short-term strain rates, whereas lithospheric defor-mation is temporally variable and time-scale dependent(Liu et al., 2000). Furthermore, tectonic boundary con-ditions and driving forces, and lithospheric structure,may have changed through time. Through the late Ceno-zoic, the plate boundary force has changed continuously

as the convergent Farallon-North American boundarywas gradually replaced by dextral shear along the grow-ing San Andreas Fault system (Atwater and Hemphill,1997; Atwater, 1970); the gravitational buoyancy force

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38 M. Liu et al. / Physics of the Earth a

also changed as a result of continuous extension inthe western Cordillera that reduced elevation (Wolfe etal., 1997), while mantle upwelling under the Basin andRange province may have had the opposite effect (Liu,2001; Sonder and Jones, 1999). Taking a closer look,one may find that the GPS velocity field, which is heav-ily influenced by localized strain and fault activity, maybe different from those derived from neotectonic studiesbecause of secular variations of fault slip rates (Bennettet al., 2004; Meade and Hager, 2005).

3. Parallel computing of lithospheric dynamics:a new approach

Geodynamic models of lithospheric deformation aremostly based on continuum mechanics. The defor-mation field is simulated by solving the governingequations with specified initial and boundary conditionsand the given rheological structure, often using finite ele-ment (FE) method. However, traditional finite elementmodels are not suited for simulating multi-scale, three-dimensional (3D) continental deformation. Most modelsare based on various 2D approximations. For large-scalecontinental deformation, the viscous thin sheet (shell)model is commonly used (e.g., Bird, 1999; Bird andPiper, 1980; England and McKenzie, 1982; Flesch etal., 2000; Sonder and Jones, 1999; Vilotte et al., 1982).This is a 2D, plane stress model for simulating long-termdeformation of a large area, but the viscous rheologydoes not allow simulation of strain localization. Othersare 2D cross-section plane strain models (e.g., Goversand Wortel, 1993; Harry and Sawyer, 1992; Liu andShen, 1998; Willett et al., 1993). Complex rheologyassociated with multi-scale continental deformation isoften greatly simplified in these models. For example,crustal deformation in a single earthquake cycle mayinclude coseismic slip (frictional or plastic deformation),postseismic deformation (viscoelastic deformation), andanelastic/plastic creep in the fault zone and unfaultedcrust. These processes are often isolated in modelswith a simple rheology, such as the elastic dislocationmodel for coseismic motion (Okada, 1985) or viscoelaticmodel for postseismic deformation (e.g., Pollitz et al.,2001). These models are useful for addressing a largerange of lithospheric problems. However, a more holisticunderstanding of continental geodynamics requires thatinteractions between multi-scale physics be explored ina self-consistent geodynamic framework.

Taking this challenge requires high-performancecomputing power, which nowadays is available mostlyfrom parallel computers. Whereas parallel computing iscommon in some fields, such as atmospheric sciences

anetary Interiors 163 (2007) 35–51

and mantle convection, it has been rare in the studies oflithospheric dynamics. This is partly because problemsin lithospheric dynamics are more diverse, making it dif-ficult to pool community resources to develop a singlecomprehensive community model. In the western US,the tectonic problems vary from rupture of fault segmentsto extension of the entire Basin and Range province. Forthe same reason, researchers of lithospheric dynamicsoften struggle to find suitable “canned” finite elementsoftware packages for their studies.

We have been developing a parallel finite elementmodeling system as part of the GEON (Geoscience Net-work, www.geongrid.org) cyberinfrastructure building;the details are given by Zhang et al. (this volume). Thisgeneral-purpose modeling system can generate parallelfinite element source codes from user input. In the inputfiles users specify the partial differential equations (PDE)and the desired solving algorithms, the type of FE ele-ments, and other model-specific properties and options.This system is based on the fact that many parts of finiteelement programs, from preprocessing to assemblingand solving large sparse matrices, are common to dif-ferent FE models. These common parts may be pluggedinto a new program and be reused. Thus, in this modelingsystem we have built libraries of “standard” segmentsof subroutines, solvers, algorithms, and pre- and post-processing routines. Most of these routines are availablefrom public domain, and the open design of this systemallows emerging novel algorithms and solvers to be read-ily linked. The user-interface is based on an English-likemodeling language (scripts). The system use informationin the input files to call desired routines and segments ofsource codes, to plug them into a chosen program stencil,and to assemble them into a complete finite element pro-gram. The following finite element models are developedusing this system.

4. Modeling active tectonics in the western US

In this section we use a subcontinental-scale finiteelement model to explore the major forces that drive thediffuse active crustal deformation in the western US.

The main driving forces include plate boundaryforce, basal traction, and gravitational potential energy.However, the relative roles of these driving forcesremain uncertain. Sonder et al. (1986) used a viscousthin sheet model to show that shear coupling across theSan Andreas Fault cannot induce the broad deformation

in the Basin and Range province. Sonder and coworkersargued that gravitational force has been the main causeof diffuse deformation in the western US (Jones etal., 1996; Sonder and Jones, 1999). Others, using

M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51 39

Fig. 2. The 3D finite element model for active tectonics in the west-ern US. The model has 371260 FE nodes and 693522 triangle prismelements. The vertical extension is 200 (km), divided into 18 layers.Baa

sspbChUosAS

drfiUoPwa61ttWtp

mriepip

oundaries of major tectonic units are included, so is the topographys shown in the figure with vertical exaggeration. Boundary conditionsnd other parameters are given in the text.

pace-based geodetic data and viscoelastic modeling,howed that present extension in the Basin and Rangerovince reflects mainly the dextral shear impartedy the relative plate motion (Thatcher et al., 1999).hoi and Gurnis (2003) suggested that, if the lateraleterogeneity of lithospheric rheology in the westernS is considered, plate boundary force is capablef inducing broad deformation. Other workers havetudied the role of traction on the base of the Northmerican plate (Bokelmann, 2002; Liu and Bird, 2002;ilver and Holt, 2002), with different conclusions.

To explore the dynamic interplay of the variousriving forces with the 3D lithospheric structure andheology in the western US, we have developed a 3Dnite element model encompassing the entire westernS (Fig. 2). The parallel computing algorithm is basedn the domain decomposition method, using Messageassing Interface. The large distributed linear systemsere solved using preconditioned Krylov subspace iter-

tion solvers. The model has 371260 FE nodes and93522 triangle prism elements, require solution of overmillion equations at each step. It is a nearly impossible

ask for traditional single-CPU desktop computers, butakes only 145 s to solve on a 36-processor PC cluster.

e compared the results of the parallel FE codes withhose of serial FE codes using a model with 12224 nodaloints, the difference is less than 0.01%.

The fine FE mesh allows us to include in theodel all first-order tectonics boundaries and major

heological contrasts in the western US (Fig. 2). We alsoncluded the Juan de Fuca and the Gorda plates, and the

astern margin of the Pacific plate, to better simulatelate interaction and its influence on deformationn the western US. This preliminary model assumesower-law viscous rheology for the continent; the

Fig. 3. Color-filled contours of the predicted vertical stress, which islargely dependent on topographic loading. The results demonstrate thefine spatial resolution of the 3D model.

plate boundaries and major tectonic boundaries aresimulated as weak zones with relatively low viscosities.The model extends to 200 km depth, and includes thevariations of crustal thickness based the Crust 2.0 model(http://mahi.ucsd.edu/Gabi/rem.dir/crust/crust2.html).In most cases, we impose the velocities of plate motionrelative to stable North America as the boundary con-ditions: the Juan de Fuca plate moves northeastward at34 mm/year, and the Pacific plate moves northwestwardat 51 mm/year. The eastern margin of the model domainis fixed. The north and south sides are fixed in N–Sdirection and free-traction in E–W direction. The surfaceis free-traction in horizontal directions and verticallyloaded with topographic weight based on the ETOPO5 model. Fig. 3 shows the calculated vertical stresses.The results closely resemble the topography, becausetopographic loading is the major contributor to thevertical stresses. The modeling is an ongoing effort; weshow here some preliminary results to demonstrate thepotential of parallel computing for modeling large-scalecontinental tectonics.

4.1. Effects of gravitational potential energy

Gravitational potential energy has been suggested asthe primary driving forces in the western US (Jones etal., 1996; Sonder and Jones, 1999) and other regions ofdiffuse continental deformation (England and Molnar,1997). To isolate the effects of gravitational potentialenergy, we conducted a series of experiments with alledges of our model domain fixed in horizontal direc-tions, and no basal traction is applied, so the deformationis driving solely by the gravitational buoyancy forces ofthe continent. Fig. 4 shows the steady-state maximumshear stresses and the stress tensor in the upper crust

plotted using the lower hemisphere stenographicprojection. It shows widespread extensional stresses,driving by gravitational spreading. The highest stressesoccur in the Rockies, because in this preliminary model

40 M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51

ss states (shown as lower hemisphere stenographic projections) in the uppert for details.

Fig. 4. The predicted maximum shear stress (background color) and strecrust resulting solely from excess gravitational potential energy. See tex

the gravitational potential energy derives mainly fromtopography. The general pattern is comparable withdeformation styles and earthquake-indicated stress statein the western US. The predicted orientation of the max-imum principal stress compares well with those fromthe World Stress Map (http://www-wsm.physik.uni-karlsruhe.de/). The averaged misfit is 32◦, close to thebest fit achieved by Liu and Bird (2002).

On the other hand, gravitational potential energyalone cannot reproduce the surface velocity as observedby the GPS. Whereas the magnitude of the predictedsurface velocity is sensitive to the viscosity of the modellithosphere, the fits to the GPS data are poor for typi-cal lithospheric viscosities (1023–1021 Pa s, Fig. 5). This

is not surprising, as many have argued that the current

Fig. 5. Predicted surface velocity (curves) driven solely by gravita-tional spreading. The values are along a profile from (230◦, 40◦) to(270◦, 36◦). The model assumes a homogeneous viscosity, and thevalues are from models with different viscosities: purple: 1023 Pa s;

crustal motion in western US is strongly influenced byshear coupling across the plate boundary in the SanAndreas fault (Bennett et al., 1999; Thatcher et al., 1999).

4.2. Effects of plate boundary forces

To isolate the effects of plate boundary forces, in thefollowing experiments we removed the topography fromthe model. The result show that shear coupling across

blue: 2 × 1022 Pa s, and green: 5 × 1021 Pa s. The dots with error barsare the GPS data from a 2◦ swath projected onto the profile. SAF: thelocation of the San Andreas Fault. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web versionof the article.)

M. Liu et al. / Physics of the Earth and Pl

Fig. 6. Predicted surface velocity (curves) driven solely by the plateboundary forces, which in the model varies with the viscosity of theplate boundaries. Blue curves: 1021 Pa s; purple curves: 1018 Pa s. SeeFig. 5 for the location of the profile and the GPS data (dots with errorbars). The green curves results of a model with heterogeneous rheo-logical structure that includes the Eastern California Shear Zone andoto

pmdrf

weak zones (the Eastern California Shear Zone, the

Fi

ther major faults as weak zones. (For interpretation of the referenceso color in this figure legend, the reader is referred to the web versionf the article.)

late boundaries can have significant impact on crustalotion in the western US, but the magnitudes are depen-

ant on the strength of coupling at the plate interface,epresented in the model by the effective viscosity in theault zones. Some of the results are shown in Fig. 6. With

ig. 7. Comparison of the predicted surface velocities (black) with the averan this figure legend, the reader is referred to the web version of the article.)

anetary Interiors 163 (2007) 35–51 41

moderate viscosity (1021–1018 Pa s) in plate boundaries,the model predict a surface velocity field in the west-ern US generally comparable to the GPS data. Fig. 6also shows that the dextral shear coupling across the SanAndreas Fault can significantly impact crustal motion asfar inland as lat. 244–248◦, the eastern side of the Basinand Range province.

On the other hand, the predicted stress field fits poorlyto the observations. The predicted maximum principalstresses have a 42–44◦ average misfit to those from theWorld Stress Map. Part of the reason is that when weremoved topography from the model, much of the verti-cal stress, which derives from the topography and is themaximum stress in most part in the western US, is lost.

In summary, the gravitational potential energy is pri-marily responsible for the steady-state stress fields andlong-term deformation in the western US. Conversely,the present-day short-term crustal motion is controlledby relative plate motions through shear coupling at theplate interfaces. A model includes both gravitationalbuoyancy force and the plate boundary shear force pro-duce a better fit to both stress orientations (Fig. 7) andsurface velocity (Fig. 8). By considering major tectonic

Wasatch fault zone, and the Rio Grand rift) and the stiff-ness in the Colorado plateau and the Great-Valley SierraNevada microplate, we can reduce the average misfit of

ged GPS velocities (red). (For interpretation of the references to color

42 M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51

Fig. 8. Comparison of the predicted directions of the maximum horizontal compressive stresses (black) with those from the World Stress Map (red).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 9. Predicted strain energy in the upper crust. Grey dots are epicenters of earthquakes with magnitude ≥3.5 from 1973 to 2006 (data from NEICcatalog).

nd Pla

coupling across the San Andreas Fault. Because of therepeated buildup and release of stress during earthquakecycles, the long-term, averaged shear stress on the SanAndreas Fault must be lower than that during inter-

netary Interiors 163 (2007) 35–51 43

seismic locking. Viscoelstic models that simulate thestick-slip behavior of the San Andreas Fault would bebetter suited for replicating GPS velocities in this region(see below). Thus, different rheology and plate cou-pling are needed to reproduce deformation at differenttimescales, because continental deformation is timescaledependent, and so is lithospheric rheology (Liu et al.,2000).

5. Modeling multi-timescale slips along the SanAndreas Fault

The viscous model discussed above is not suited forsimulating strain localization and faulting, which areimportant aspects of continental tectonics. In this sec-tion, we present a regional-scale model for the SanAndreas Fault (SAF) system. Parallel computing is usedin this model to explore multi-timescale slips on the SAFand the resulting stress evolution and strain localization.

The SAF is a complex fault system including manyactive fault branches (Fig. 10). About ∼70% of the rel-ative plate motion between the Pacific and the NorthAmerican plates (49 mm/year, DeMets et al., 1994) isaccommodated by the SAF and its closely subparallel

fault strands (Becker et al., 2005; Bennett et al., 2004;Sieh and Jahns, 1984), with significant along-strike vari-ations. In northern California, the SAF and the closelysubparallel fault branches accommodate ∼35 mm/year

M. Liu et al. / Physics of the Earth a

stress orientations to those of the World Stress Map to∼31◦, and the average surface velocity misfit to the GPSdata to 5–8 mm/year. Fig. 9 shows the predicted strainenergy in the upper crust (given by the product of themaximum strain rate and the maximum shear stress).The high strain energy is found along plate boundariesand near the Eastern California Shear Zone, generallyconsistent with modern seismicity.

4.3. Effects of basal shear

We also explored the effects of shear traction onthe bottom of the western US lithosphere. We find thatthe average basal shear should be less than 2 MPa. Theoverall role of basal shear, however, is secondary to grav-itational potential energy and plate boundary forces. Ourresults prefer an eastward shear under much of the west-ern US, consistent with previous results of Bokelmann(2002) but differ from those of Liu and Bird (2002). With2 MPa eastward shear stress, we were able to reduce sur-face velocity misfit to 5–6 mm/year, and misfit of stressorientation to 30◦. Westward basal shear always worsenthe misfits. This may indicate that the westward motionof the North American plate is resisted by the sublitho-spheric mantle, but this statement cannot be conclusivebecause of the small role of basal shear in our model.

4.4. Short-term versus long-term deformation

By fitting the GPS velocity near the plate boundaries,we may overestimate the plate coupling in the Cascadiansubduction zone, because the GPS data reflect mainlyinterseismic coupling, whereas the long-term, averagedcoupling is expected lower, because much of the short-term strain near the trench is transient and recoversduring trench earthquakes or aseismic slips (Dragert etal., 2001; Liu et al., 2000). This explains the predictedE–W compressive stress in the Cascadian coastal region(Fig. 4), which is inconsistent with earthquake (Lewis etal., 2003) and structural geological data that show E–Wextension (Wells and Simpson, 2001; Wells et al., 1998).A better fit to the long-term stress field can be producedby assuming a weaker coupling along the Cascadiansubduction zone.

The same can be said for the San Andreas Faultplate boundary zone and the adjacent Basin and Rangeprovince. Using the long-term viscous model to fit theGPS velocity, we may have overestimated the shear

Fig. 10. Topographic relief and seismicity in California and surround-ing regions. Data of seismicity (includes M > 5.0 earthquakes from1800 to present) are from the NEIC catalog.

nd Planetary Interiors 163 (2007) 35–51

Liu, 2006, in press). Fig. 11 shows the model for theSAF with its first-order geometry of the surface-trace.A 300-km wide extra model domain is added to bothends of the SAF to minimize artificial boundary effects.

Fig. 11. Numerical mesh and boundary conditions of the finite element

model for the SAF system. The entire San Andreas Fault (black line)is explicitly included in the model.

The eastern side of the model domain is fixed, while thewestern side is loaded by a shear velocity of 49 mm/year,representing relative motion between the Pacific and theNorth American plates. The northern and southern sidesare free boundary in the direction of plate motion butfixed in the other two directions within the plane.

The model includes a 20-km thick upper crust withan elasto-plastic rheology, and a 40-km thick viscoelasticlayer representing both the lower crust and the uppermostmantle. Viscosity for the lower crust and upper mantlebetween 1019 and 1021 Pa s (Hager, 1991; Kenner andSegall, 2000; Pollitz et al., 2001) are explored. Young’sModulus is taken to be 8.75 × 1010 N/m2, and Poisson’sratio to be 0.25 for both crust and mantle. The cohe-sion for the SAF fault is set to be 10 MPa, which isclose to the upper bound permitted by heat flow data(Lachenbruch and Sass, 1980). The surrounding uppercrust is assumed to be relatively strong with cohesion of50 MPa. The effective internal frictional coefficients forthe SAF fault and surrounding upper crust are set to be0 and 0.4, respectively.

For simulation of long-term fault slip and strainlocalization in the crust, we used the Drucker–Pragerplastic model (Li and Liu, 2006), which is similar tothe Mohr–Coulomb model but is better suited for 3Danalysis. Different plastic yield criteria are assigned tothe fault zone and unfaulted crust. For this rheologicalsystem, stain increments can be written as

{dε} = {dεv} + {dεe} + {dεp} (1)

where the superscripts v, e and p denote viscous,

44 M. Liu et al. / Physics of the Earth a

fault slip. In southern California, slip rates on the SAFdrops significantly over the Big Bend, and roughly12 mm/year fault slip is accommodated by the San Jac-into fault (SJF). In the Eastern California Shear Zone(ECSZ), the slip rates are up to ∼12 mm/year (Dixonet al., 2003; Dokka and Travis, 1990; Gan et al., 2000;McClusky et al., 2001; Miller et al., 2001a; Peltzer etal., 2001; Reheis and Dixon, 1996; Sauber et al., 1994;Savage et al., 2001).

The establishment of dense, high precision geode-tic network in California since the 1970s has producedrich datasets that permit regional inversion of fault sliprates (Becker et al., 2005; Bennett et al., 1996; Meadeand Hager, 2005). The slip rates derived from the short-term geodetic measurements differ significantly fromthose derived from geological data (Becker et al., 2005;Li and Liu, 2006). Part of the reason is the differenttimescales represented in these data sets. Space-geodeticmeasurements are dominated by interseismic strains.Theoretically, interseismic strain plus coseismic dis-placement would give the total slip as in geologicalrecords (Savage, 1983). However, given the transientsassociated with earthquakes and other factors, such assecular slip rate variations, integrating short-term geode-tic and long-term geological data is not straightforward(Pollitz, 2003b; Segall, 2002).

5.1. A 3D visco-elasto-plastic model

Geodynamic models have not kept up with the fastgrowth of geodetic and geological measurements of faultslip rates. Most models of fault slips are based on half-space elastic dislocation model (Savage and Burford,1973). Various simplifications are used in viscoelas-tic thin-shell model (Bird and Kong, 1994) and 3Delasto-frictional model (Parsons, 2002). Some modelsuse prescribed slip rates (Smith and Sandwell, 2003;Williams and Richardson, 1991), thus the effect of faultgeometry on slip rates cannot be directly tested. Insome dynamic models (Du and Aydin, 1996; Duan andOglesby, 2005; Fitzenz and Miller, 2004), fault slip rateswere not explicitly calculated. Furthermore, most previ-ous studies have focused on the fault zone; the plasticdeformation in the surrounding crust, hence the impacton strain localization, remain to be explored.

We have developed a preliminary 3D visco-elasto-plastic finite element model for simulating crustalmotion and fault evolution in multi-timescales (Li and

elastic and plastic, respectively. Here {} contains allsix components of strain increments in the strain tensor.Viscous component is associated with stress, and elastic

nd Planetary Interiors 163 (2007) 35–51 45

Fig. 12 shows the predicted long-term fault slip rateson the SAF. They are higher in northern and central SAF(∼35 mm/year) than in southern SAF (∼20 mm/year),where fault strike bends ∼25◦ counterclockwise over

Fig. 12. Predicted long-term slip rates (numbers beside the

fault) along the San Andreas Fault. Numbers in parenthesisare geological slip rates from the California Geological Survey(http://www.consrv.ca.gov/CGS/rghm/psha/index.htm).

the Big Bend. The along-strike variation of slip rates isconsistent with geological slip rates. The predicted faultslip rates depend on the viscosity of the lower crust andupper mantle, but the general pattern stays unchanged (Liand Liu, 2006). Because less of the relative plate motionis accommodated over the Big Bend, more has to beabsorbed by deformation in the surrounding crust. This isconsistent with diffuse seismicity in southern California.

The Big Bend also causes strain localization in thesurrounding crust. Fig. 13 shows the predicted steady-state rates of plastic energy release outside the SAFproper. The most conspicuous feature is the two high-energy belts caused by the Big Bend. One is to thesouthwest of the SAF, coincides with the Palos Verdes-Coronado Bank fault zone and off the coast; the othercoincides with the ECSZ. In southern California, up to40% of the relative plate motion is taken up by the SanJacinto fault (Becker et al., 2005). When the San Jac-into fault is included in the model, strain localizationalong the ECSZ is enhanced, at the expense of strainenergy along the coast (Li and Liu, in press). Theseresults differ from previous models (Bird and Kong,1994; Parsons, 2002) where deformation in the unfaulted

M. Liu et al. / Physics of the Earth a

component is associated with stress increments:

{dεv} = [Q]−1{σt} dt

{dεe} = [D]−1{dσ} (2)

where {σt} are stresses at time t, dt time increment, {dσ}the stress increments, [Q] the viscous material propertymatrix, and [D] is the elastic material property matrix.The yield function for the Drucker–Prager model is:

F = αI1 +√

J ′2 − k = 0 (3)

where I1 and J ′2 are first and second invariants of

the deviatoric stress tensor, respectively, and α andk are parameters related to cohesion c and effectivecoefficient of friction μ. Different strengths (in terms ofc and μ) are assigned to fault zones and outside crust.

The stress and displacement of the lithosphere underloading are solved in a parallel FE model generated usingour automated modeling system described in Section 3.The general procedure for determining displacement andstress at time t includes the following steps: (1) calcu-late stress increments for the incremental displacementsfrom the previous iteration step; (2) update the trial vis-coelastic stress; (3) calculate the plastic yield function;(4) if no plastic yield is expected, accept trial stresses, andcalculate displacement increments; (5) otherwise deter-mine the fraction that is plastic; (6) calculate stresses thatresult from both viscoelastic and plastic portions of theincrement; (7) calculate displacement increments withplastic component. Return to step 1 until convergence isreached.

5.2. Long-term fault slip rates and strainlocalization

To simulate long-term fault slip rates, we run themodel till it reaches a steady state. At that point theinfluence of artificial initial conditions are largely gone,and the deformation reflects the long-term slip on theSAF and strain partitioning in the surrounding crustdriven by secular tectonic loading from relative platemotion. We then calculated stress evolution over aperiod of tens of thousands of years. Over such a timescale, the model SAF experiences continuous creeping,which is a long-term approximation of stick-slip processover shorter timescales. In the surrounding crust, excessstress over the yield strength is released by plasticdeformation.

crust is continuous and diffuse, because of elastic or vis-cous rheology used in these models. Our model doesnot include slip-weakening, which may further enhancestrain localization.

46 M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51

Fig. 13. The predicted plastic energy release off the SAF main trace,vertically integrated through the upper crust. The areas of high-energyrelease coincide with many active faults in California, including theMaacama-Garberville Fault (MGF), the Rodgers Creek Fault (RC),

Fig. 14. Predicted long-term surface velocities (arrows) and GPSvelocities (arrows with confidence ellipses). The GPS data are

the Hayward Fault (HF), the Calaveras Fault (CF), the Garlock Fault(GF), the East California Shear Zone, the San Jacinto Fault (SJF), theElsinore Fault (EF), the Palos Verdes Fault (PVF), and the CoronadoBank Fault (CBF). Cycles are seismicity from 1800 to 2004 (NEIC).

The predicted long-term surface velocities are com-parable with the GPS velocity field in general (Fig. 14),but they drop more sharply across the SAF than theGPS velocities. This discrepancy results from the differ-ent time scales: most GPS measurements occur duringinterseismic periods when the fault is locked and platecoupling is strong, whereas over seismic cycles, mosttransients are averaged out and fault motion is domi-nated by cumulative coseismic slips, resulting in sharpervelocity gradient across the fault.

5.3. Multi-timescale fault behavior: from rupturesto earthquake cycles

The difference between stick-slip fault behaviorduring earthquake cycles and long-term creeping iswell known, but modeling these processes over multi-timescales have been numerically difficult. Thus mostprevious models focus on a single physical process overa specific timescale: fault rupture over seconds, postseis-

mic deformation over months to decades, and long-termsteady-state fault motion over many seismic cycles.However, separating these fault physics is not alwayseasy. For example, the initial stress state for dynamic

from USGS and the SCEC Crustal Motion Map Version 3.0(http://epicenter.usc.edu/cmm3/). AA′ shows the location of the profilein Fig. 16.

ruptures depends on stress evolution following previousearthquakes, and fault rupture pattern, the main outcomeof studies of dynamic rupture processes, is a critical inputfor modeling postseismic stress evolution.

The parallel computing algorithms and visco-elasto-plastic rheology in our model allow exploration of someof the multi-timescale fault behavior. The results shownhere are from a model similar to that for long-term faultmotion discussed above, the only difference is that weadded a simplified fault rupture process. When stressreaches the yield strength at any element within the faultzone, that part of the fault zone ruptures, simulated byplastic deformation to lower the stress in the element by10 MPa. This stress drop may cause stress in surroundingelements to increase, perhaps leading to rupture in moreelements. In our model the time steps are reduced to1 s when the first element fails, and the rupture processis iterated until no more fault elements fail. Then thepostseismic process starts again with fault strength fullyrestored, and the model resumes with 1-year time steps.After ∼20,000 years of simulation, the system entersa statistically steady state. The following results reflect

this state of the fault system.

Fig. 15 shows one of the predicted short-term(instantaneous) surface velocities taken at a randompoint of time. Because in this model the fault behavior

M. Liu et al. / Physics of the Earth and Planetary Interiors 163 (2007) 35–51 47

Fv

iavwatdbtvi

t

Fv

could be tested in the future when longer and morecomplete earthquake history along the SAF becomesavailable.

ig. 15. Predicted short-term surface velocities (arrows) and GPSelocities (arrows with confidence ellipses).

s time-depend, with stick-slip fault motion and rupturest various fault segments and time (see below), the exactalues of the predicted surface velocity slightly varyith time. In essence, the surface velocity at any location

nd time is influenced by the previous fault motion athis location, as well as the ruptures and postseismiceformation in the surrounding regions, as suggestedy Pollitz (2003a). Fig. 16 compares the predicted long-erm and instantaneous surface velocities with the GPS

elocities along a profile across the SAF. The predictednstantaneous surface velocities better fit the GPS data.

The cyclic stress changes, from coseismic stress dropo postseismic stress buildup, are shown in Fig. 17 for

ig. 16. Comparison of the predicted long- and short-term surfaceelocity with the GPS data along AA′ profile shown in Figs. 14 and 15.

Fig. 17. Stress evolution at three sample points in the central segmentof the SAF.

three randomly chosen locations along the central seg-ment of the SAF. The gradual postseismic stress increaseis due to tectonic loading and viscous relaxation in thelower crust and upper mantle, and the stress jump-upsare caused by fault ruptures in neighboring fault seg-ments. The fault ruptures, shown by stress drops at eachlocation, are more regular along central SAF (Fig. 18a)where the fault trace is relatively straight. Conversely,the rupture pattern in southern SAF over the Big Bendis more chaotic (Fig. 18b). These results suggest thatlarge earthquakes along central SAF segments may bemore periodic and regular than in southern SAF wherethe fault geometry is more complex. This prediction

Fig. 18. Stress evolution of groups of sample points at central SAF (a)and southern SAF (b).

d Pl

48 M. Liu et al. / Physics of the Earth an

6. Discussion

The common practice of geodynamic modeling is toisolate a geological process for a narrowly specified spa-tial and time scale, to simplify the process so it canbe represented by a relatively simple set of partial dif-ferential equations, and then to solve the PDEs usingnumerical methods, often the finite element method.This approach is partly based on the desire to isolateand hence better understand the physics of certain geo-logical processes, but it often represents a compromiseto the numerical challenges and limited computationalresources. Although generally it has worked well, insome fields, such as continental dynamics where dif-ferent physical processes operating over large spectrumof spatiotemporal scales are closely coupled with eachother, the traditional approach becomes inadequate. Wehave discussed the coupled multiphysical processes offaulting at different timescales. Another example is thelasting debate of the nature of large-scale continentaldeformation. Results based on viscous thin-sheet mod-els argues for continuously diffuse deformation (Englandand Houseman, 1988; England and Molnar, 1997; Fleschet al., 2001), while geological studies show deforma-tion highly concentrated along boundaries of discretecrustal blocks (Taponnier et al., 1982; Tapponnier et al.,2001). While there are merits in both end-member mod-els, integrating them for a more holistic understandingof continental deformation would require more realisticgeodynamic models with enough spatial resolution torepresent the critical 3D lithospheric structures and rhe-ological heterogeneities, and more sophisticated modelrheology to simulate strain localization and fault evolu-tion.

Recent development of computing algorithms andhardware, marked by the affordable Beowulf class PCclusters, provides an opportunity to attack some ofthe multi-scale, multiphysical problems in continentaltectonics. However, parallel finite element packagesfor lithospheric dynamics are few, and the cannedpackages are often difficult to meet the demands formodeling diverse lithospheric dynamics. Zhang etal. (this volume) introduce an parallel finite elementmodeling environment where complete source codescan be automatically generated for specified sets ofPDEs and model-specific properties, though a friendlyuser-interface. The flexibility of this system is idealfor modeling the diverse problems in lithospheric

dynamics, and with its open design, it can be readilyintegrated in the emerging cyberinfrastructures ofdistributed data grids and high-performance computingresources.

anetary Interiors 163 (2007) 35–51

In this paper we showed two models generated usingthis modeling system to simulate multi-scale continentaldeformation in the western US. These models are rudi-mentary with many features yet to be developed. Thesubcontinental-scale model shows the potential of incor-porating detailed 3D lithospheric structures that willbe provided by EarthScope and other ongoing researchprojects. However, the current model does not includeplastic deformation, thus cannot fully simulate strainlocalization. The model for the San Andreas Fault sys-tem made the initial attempt to simulate multi-timescalefault motion and crustal deformation, from rupture inseconds to steady-state creeping over thousands of years.The amount of computation is prohibitive for traditionalsingle-processor desktop workstations. Even with large-size PC clusters, the computations are intensive, thusnumerous simplifications have been made, includingreplacing the full dynamic rupture processes with a sim-ple plastic deformation on the failing element. Theseaspects of the model will be improved in future.

7. Conclusions

Our preliminary results indicate that active tectonicsin the western US is mainly driven by the excess gravi-tational potential energy, and shear coupling at the plateboundaries along the SAF and the Cascadian subduc-tion zone. Basal traction on the bottom of the westernUS lithosphere is likely small (<2 MPa) and dominatedby eastward shear. The GPS velocities, dominated byinterseismic deformation, are largely controlled by plateboundary forces. Conversely, the long-term stress states,as indicated by earthquakes and faulting patterns, is pri-marily controlled by the gravitational spreading force,which has been driving widespread extension in thewestern US since early Tertiary. The lateral variationsof lithospheric rheology in the western US facilitatedeformation to diffuse away from the plate boundaries.Further viscoelastic–plastic modeling is needed to bet-ter simulate strain localization and explore its role inshaping continental deformation in the western US.

Our regional-scale model for the SAF system showsthat the long-term slip rates on the SAF and strain par-titioning in the surrounding crust is strongly influencedby the geometry of the SAF. The Big Bend reduces faultslip on the SAF and causes a greater portion of the rel-ative plate motion to be absorbed in the surroundingcrust, consistent with the diffuse seismicity in south-

ern California. Interestingly, the Big Bend is shown tocause high concentration of plastic strain energy alongthe Eastern California Shear Zone and thus may havecontributed to its initiation. Our results show that stress

nd P

M. Liu et al. / Physics of the Earth a

evolution and fault behavior at any segment of the SAFare closely related to the earthquakes history of thenearby regions. Hence the GPS velocity at a given placemay be influenced not only by tectonic loading but alsoby postseismic deformation in the surrounding regions.Again, the fault geometry may affect earthquake behav-iors. Along central segments of the SAF where the faulttrace is relatively straight and subparellel to the direc-tion of relative plate motion, large earthquakes tend tobe periodic and regular, whereas along southern SAFover the Big Bend, earthquake patterns tend to be moreirregular.

Acknowledgements

We thank two anonymous reviewers for their help-ful comments. This work was supported by NSF grantsEAR 0225546 as part of the GEON (Geoscience Net-work) project, and USGS grant 04HQGR0046. Liu’scollaboration with the Computational Geodynamics Labof the Chinese Academy of Sciences (CAS) is partiallysupported by NSF China grant 40228005 and the CAS,and the Research Council of the University of Missouri-Columbia.

References

Argus, D.F., Gordon, R.G., 1991. Current Sierra Nevada-North Amer-ica motion from very long baseline interferometry; implications forthe kinematics of the Western United States. Geology (Boulder) 19(11), 1085–1088.

Artyushkov, E.V., 1973. Stresses in the lithosphere caused by crustalthickness inhomogeneities. J. Geophys. Res. 78, 7675–7690.

Atwater, B.F., Hemphill, H.E., 1997. Recurrence intervals for greatearthquakes of the past 3,500 years at northeastern Willapa Bay,Washington. U.S. Geological Survey Professional Paper: 108.

Atwater, T., 1970. Implications of plate tectonics for the Cenozoictectonic evolution of western North America. Geol. Soc. Am. Bull.81, 3513–3536.

Atwater, T., Stock, J., 1998. Pacific North America plate tectonics ofthe Neogene southwestern United States: an update. Int. Geol. Rev.40 (5), 375–402.

Axen, G.J., Taylor, W., Bartley, J.M., 1993. Space-time patterns andtectonic controls of Tertiary extension and magmatism in the GreatBasin of the western United States. Geol. Soc. Am. Bull. 105,56–72.

Becker, T.W., Hardebeck, J.L., Anderson, G., 2005. Constraints onfault slip rates of the southern California plate boundary from GPSvelocity and stress inversions. Geophys. J. Int. 160, 634–650.

Bennett, R.A., Davis, J.L., Normandeau, J.E., Wernicke, B.P., 2002.Space Geodetic Measurements of Plate Boundary Deformation

in the Western U.S. Cordillera, Plate Boundary Zones. AmericanGeophysical Union, Washington, DC, United States, pp. 27–55.

Bennett, R.A., Davis, J.L., Wernicke, B.P., 1999. The present-daypattern of Werstern U.S. Cordillera deformation. Geology 27,371–374.

lanetary Interiors 163 (2007) 35–51 49

Bennett, R.A., Friedrich, A.M., Furlong, K.P., 2004. Codependent his-tories of the San Andreas and San Jacinto fault zones from inversionof fault displacement rates. Geology 32, 961–964.

Bennett, R.A., Rodi, W., Reilinger, R.E., 1996. Global positioningsystem constraints on fault slip rates in southern California andnorthern Baja, Mexico. J. Geophys. Res. 101 (B10), 21943–21960.

Bennett, R.A., Wernicke, B.P., Niemi, N.A., Friedrich, A.M., Davis,J.L., 2003. Contemporary strain rates in the northern Basinand Range Province from GPS data. Tectonics 22 (2), 1008,doi:10.1029/2001TC001355.

Bird, P., 1999. Thin-plate and thin-shell finite-element programs forforward dynamic modeling of plate deformation and faulting. Com-put. Geosci. 25 (4), 383–394.

Bird, P., Kong, X., 1994. Computer simulations of California tecton-ics confirm very low strength of major faults. GSA Bull. 106 (2),159–174.

Bird, P., Piper, K., 1980. Plane-stress finite element models of tectonicflow in southern California. Phys. Earth Planet. Int. 21, 158–175.

Bokelmann, G.H.R., 2002. Convection-driven motion of the NorthAmerican Craton; evidence from P-wave anisotropy. Geophys. J.Int. 148, 278–287.

Boyd, O.S., Jones, C.H., Sheehan, A.F., 2004. Foundering lithosphereimaged beneath the Southern Sierra Nevada, California, USA. Sci-ence 305 (5684), 660–662.

Brown, W.G., 1988. Deformational style of Laramide uplifts in theWyoming foreland. Geol. Soc. Am. Mem. 171, 1–25.

Burchfiel, B.C., Cowan, D.S., Davis, G.A., 1992. Tectonic overviewof the Cordilleran orogen in western United States. In: Burchfiel,B.C., Zoback, M.L., Lipman, P.W. (Eds.), The Cordilleran Orogen:Conterminous. U.S. Geol. Soc. Am., Boulder, Colorado, USA, pp.407–480.

Burchfiel, C.B., Davis, G.A., 1975. Nature and controls of cordilleranorogenesis, western United States: extensions of an earlier synthe-sis. Am. J. Sci. 275-A, 363–396.

Choi, E.s., Gurnis, M., 2003. Deformation in transcurrent and exten-sional environments with widely spaced weak zones. Geophys.Res. Lett. 30 (2), 1076, doi:10.1029/2002GL016129.

Coney, P.J., 1978. Mesozoic-Cenozoic Cordilleran plate tectonics.Mem. Geol. Soc. Am. 152, 33–50.

Cross, T.A., Pilger Jr., R.H., 1982. Controls of subduction geome-try, location of magmatic arcs, and tectonics of arc and back-arcregions. In: Symposium on Subduction of Oceanic Plates. Geolog-ical Society of America (GSA), Boulder, CO, United States, pp.545–562.

Davis, J.L., Wernicke, B.P., Bisnath, S., Niemi, N.A., Elosegui, P.,2006. Subcontinental-scale crustal velocity changes along thePacific-North America plate boundary. 441 (7097) 1131–1134.

DeMets, C., Gordon, R.G., Argus, D.F., Stein, S., 1990. Current platemotions. Int. J. Geophys. 101, 425–478.

DeMets, C., Gordon, R.G., Argus, D.F., Stein, S., 1994. Effect of recentrevisions to the geomagnetic reversal time scale on estimates ofcurrent plate motion. Geophys. Res. Lett. 21, 2191–2194.

Dickinson, W.R., 2002. The basin and range province as a compositeextensional domain. Int. Geol. Rev. 44, 1–38.

Dickinson, W.R., Snyder, W.S., 1978. Plate tectonics of the Laramideorogeny. In: V., Mattews (Ed.), Laramide Folding Assocatedwith Basement Block Faulting in the western United States, pp.

355–366.

Dixon, T.H., Miller, M., Farina, F., Wang, H., Johnson, D., 2000.Present-day motion of the Sierra Nevada block and some tectonicimplications for the Basin and Range Province, North AmericanCordillera. Tectonics 19 (1), 1–24.

nd Pla

50 M. Liu et al. / Physics of the Earth a

Dixon, T.H., Norabuena, E., Hotaling, L., 2003. Paleoseismology andGlobal Positioning System; earthquake-cycle effects and geodeticversus geologic fault slip rates in the Eastern California shear zone.Geology (Boulder) 31 (1), 55–58.

Dixon, T.H., Robaudo, S., Lee, J., Reheis, M.C., 1995. Constraintson present-day Basin and Range deformation from space geodesy.Tectonics 14 (4), 755–772.

Dokka, R.K., Travis, C.J., 1990. Role of the eastern California shearzone in accommodating Pacific-North American plate motion.Geophys. Res. Lett. 17 (9), 1323–1326.

Dragert, H., Wang, K., James, T.S., 2001. A silent slip event onthe deeper Cascadia subduction interface. Science 292 (5521),1525–1528.

Du, Y., Aydin, A., 1996. Is the San Andreas big bend responsible for thelanders earthquake and the Eastern California shear zone? Geology24 (3), 219–222.

Duan, B., Oglesby, D.D., 2005. Multicycle dynamics of nonplanarstrike-slip faults. J. Geophys. Res. 110, B03304.

England, P., Houseman, G., 1988. The mechanics of the Tibetanplateau. Philos. Trans. R. Soc. Lond. A327, 379–413.

England, P., Molnar, P., 1997. Active deformation of Asia: from kine-matics to dynamics. Science 278, 647–650.

England, P.C., McKenzie, D.P., 1982. A thin viscous sheet model forcontinental deformation. Geophys. J. R. Astron. Soc. 70, 295–321.

Fitzenz, D.D., Miller, S.A., 2004. New insights on stress rotations froma forward regional model of the San Andreas fault system near itsBig Bend in southern California. J. Geophys. Res. 109, B08404.

Flesch, L.M., Haines, A.J., Holt, W.E., 2001. Dynamics of the India-Eurasia collision zone. J. Geophys. Res. 106, 16435–16460.

Flesch, L.M., Holt, W.E., Haines, A.J., Shen, T.B., 2000. Dynamics ofthe Pacific-North American plate boundary in the Western UnitedStates. Science 287 (5454), 834–836.

Forsyth, D., Uyeda, S., 1975. On the relative importance of the drivingforces of plate motion. Geophys. J. R. Astron. Soc. 43, 163–200.

Friedrich, A.M., Wernicke, B.P., Niemi, N.A., Bennett, R.A., Davis,J.L., 2003. Comparison of geodetic and geological data from theWasatch region, Utah, and implications for the spectral character-istics of Earth deformation at periods of 10 to 10 million years. J.Geophys. Res. 108 (B4), 2199, doi:10.1029/2001JB000682.

Gan, W.J., Svarc, J.L., Savage, J.C., Prescott, W.H., 2000. Strain accu-mulation across the Eastern California Shear Zone at latitude 36degrees 30′ N. J. Geophys. Res. 105 (B7), 16229–16236.

Gordon, R.G., Stein, S., 1992. Global tectonics and space geodesy.Science 256 (5055), 333–342.

Govers, R., Wortel, M.J.R., 1993. Initiation of asymmetric extensionin continental lithosphere. Tectonophysics 223, 75–96.

Hager, B.H. (Ed.), 1991. Glacial Isostacy, Sea Level and Mantle Rhe-ology. Kluwer Academic Publishers, London, pp. 493–513.

Hamilton, W., Myers, W.B., 1966. Cenozoic tectonics of the westernUnited States. Rev. Geophys. 5, 509–549.

Harry, D.L., Sawyer, D.S., 1992. A dynamic model of extension in theBaltimore canyon trough region. Tectonics 11 (2), 420–436.

He, J., Liu, M., Li, Y., 2003. Is the Shanxi rift of northern China extend-ing? Geophys. Res. Lett. 30, 2213, doi:10.1029/2003GL018764.

Jeanloz, R., Morris, S., 1986. Temperature distribution in the crust andmantle. Ann. Rev. Earth Planet. Sci. 14, 377–415.

Jones, C.H., Farmer, G.L., Unruh, J., 2004. Tectonics of Pliocene

removal of lithosphere of the Sierra Nevada, California. Geol. Soc.Am. Bull. 116 (11/12), 1408–1422.

Jones, C.H., Unruh, J.R., Sonder, L.J., 1996. The role of gravitationalpotential energy in active deformation in the southwestern UnitedStates. Nature 381, 37–41.

netary Interiors 163 (2007) 35–51

Kenner, S.J., Segall, P., 2000. Postseismic deformation following the1906 San Francisco earthquake. J. Geophys. Res. B Solid EarthPlanets 105 (6), 13, 195–13, 209.

Lachenbruch, A.H., Sass, J.H., 1980. Heat flow and energetics of theSan Andreas fault zone. J. Geophys. Res. 85 (B11), 3222–6185.

Lewis, J.C., Unruh, J.R., Twiss, R.J., 2003. Seismogenic strain andmotion of the Oregon coast block. Geology 31 (2), 183–186.

Li, Q., Liu, M., 2006. Geometrical impact of the San Andreas Fault onstress and seismicity in California. Geophys. Res. Lett. 33, L08302,doi:10.1029/2005GL025661.

Li, Q., Liu, M. Initiation of the San Jacinto Fault and its interactionwith the San Anreas Fault. Pure Appl. Geophys., in press.

Lipman, P.W., Prostka, H.J., Christiansen, R.L., 1971. Evolving sub-duction zones in the western United States, as interpreted fromigneous rocks. Science 174, 821–825.

Lipman, P.W., Prostka, W.J., Christiansen, R.L., 1972. Cenozoic vol-canism and plate tectonic evolution of the western United States.Part I. Early and Middle Cenozoic. R. Soc. Lond. Philos. Trans.271-A, 217–248.

Liu, M., 2001. Cenozoic extension and magmatism in the North Amer-ica Cordillera: the role of gravitational collapse. Tectonophysics342, 407–433.

Liu, M., Shen, Y., 1998. Sierra Nevada uplift; a ductile link to mantleupwelling under the Basin and Range Province. Geology (Boulder)26 (4), 299–302.

Liu, M., Yang, Y., Stein, S., Zhu, Y., Engeln, J., 2000. Crustal shorten-ing in the Andes: why do GPS rates differ from geological rates?Geophys. Res. Lett. 27, 3005–3008.

Liu, Z., Bird, P., 2002. North America Plate is driven westwardby lower mantle flow. Geophys. Res. Lett. 29 (24), 2164,doi:10.1029/2002GL016002.

Lowry, A.R., Smith, R.B., 1994. Flexural rigidity of the Basin andRange-Colorado Plateau-Rocky Mountain transition from coher-ence analysis of gravity and topography. J. Geophys. Res. 99 (B10),20, 123–20, 140.

McClusky, S.C., et al., 2001. Present day kinematics of the Eastern Cal-ifornia Shear Zone from a geodetically constrained block model.Geophys. Res. Lett. 28 (17), 3369–3372.

Meade, B.J., Hager, B.H., 2005. Block models of crustal motion insouthern California constrained by GPS measurements. J. Geo-phys. Res. 110 (B3), B03403.

Miller, M.M., Johnson, D.J., Dixon, T.H., Dokka, R.K., 2001a. Refinedkinematics of the Eastern California shear zone from GPS obser-vations, 1993–1998. J. Geophys. Res. 106 (B2), 2245–2263.

Miller, M.M., et al., 2001b. GPS-determination of along-strike varia-tion in Cascadia margin kinematics; implications for relative platemotion, subduction zone coupling, and permanent deformation.Tectonics 20 (2), 161–176.

Minster, J.B., Jordan, T.H., 1987. Vector constraints on Western U.S.deformation from space geodesy, neotectonics, and plate motions.J. Geophys. Res. B Solid Earth Planets 92 (6), 4798–4804.

Molnar, P., 1988. Continental tectonics in the aftermath of plate tec-tonics. Nature (London) 335 (6186), 131–137.

Molnar, P., Tapponnier, P., 1975. Cenozoic tectonics of Asia: effectsof a continental collision. Science 189, 419–426.

Murray, M.H., Lisowski, M., 2000. Strain accumulation along the Cas-cadia subduction zone. Geophys. Res. Lett. 27 (22), 3631–3634.

Okada, Y., 1985. Surface deformation due to shear and tensile faultsin a half-space. Bull. Seismol. Soc. Am. 75, 1135–1154.

Parsons, T., 2002. Post-1906 stress recovery of the San Andreas faultsystem calculated from three-dimensional finite element analysis.J. Geophys. Res. Solid Earth 107 (B8).

nd Pl

M. Liu et al. / Physics of the Earth a

Peltzer, G., Crampe, F., Hensley, S., Rosen, P., 2001. Transient strainaccumulation and fault interaction in the Eastern California shearzone. Geology 29 (11), 975–978.

Pollitz, F.F., 1997. Gravitational viscoelastic postseismic relaxation ona layered spherical Earth. J. Geophys. Res. B Solid Earth Planets102 (8), 17, 921–17, 941.

Pollitz, F.F., 2003a. The relationship between the instantaneous veloc-ity field and the rate of moment release in the lithosphere. Geophys.J. Int. 153, 596–608.

Pollitz, F.F., 2003b. Transient rheology of the uppermost mantlebeneath the Mojave Desert, California. Earth Planet. Sci. Lett. 215(89–104).

Pollitz, F.F., Wicks, C., Thatcher, W., 2001. Mantle flow beneath acontinental strike-slip fault; postseismic deformation after the 1999Hector Mine earthquake. Science 293 (5536), 1814–1818.

Ranalli, G., 1995. Rheology of the Earth. Chapman & Hall, London,p. 413.

Reheis, M.C., Dixon, T.H., 1996. Kinematics of the Eastern Californiashear zone: evidence for slip transfer from Owens and Saline Valleyfault zones to Fish Lake Valley fault zone. Geology 24 (4), 339–342.

Sauber, J., Thatcher, W., Solomon, S.C., Lisowski, M., 1994. Geodeticslip rate for the Eastern California Shear Zone and the recurrencetime of Mojave desert earthquakes. Nature 367 (6460), 264–266.

Savage, J.C., 1983. A dislocation model of strain accumulation andrelease at a subduction zone. J. Geophys. Res. 88 (6), 4984–4996.

Savage, J.C., Burford, R., 1973. Geodetic determination of relativeplate motion in central California. J. Geophys. Res. 78, 832–845.

Savage, J.C., Gan, W.J., Svarc, J.L., 2001. Strain accumulation androtation in the Eastern California Shear Zone. J. Geophys. Res.106 (B10), 21995–22007.

Segall, P., 2002. Integrating geologic and geodetic estimates of sliprate on the San Andreas fault system. Int. Geol. Rev. 44 (1), 62–82.

Shen, T.B., Holt, W.E., Haines, A.J., 1999. Deformation kinematicsin the Western United States determined from Quaternary faultslip rates and recent geodetic data. J. Geophys. Res. B Solid EarthPlanets 104 (12), 28, 927–28, 956.

Sieh, K.E., Jahns, R.H., 1984. Holocene activity of the San AndreasFault at Wallace Creek, California. Geol. Soc. Am. Bull. 95 (8),883–896.

Silver, P.G., Holt, W.E., 2002. The mantle flow field beneath westernNorth America. Science 295 (5557), 1054–1058.

Smith, B., Sandwell, D., 2003. Coulomb stress accumulation along theSan Andreas Fault system. J. Geophys. Res. Solid Earth 108 (B6).

Sonder, L.J., England, P.C., Houseman, G.A., 1986. Continuum cal-culations of continental deformation in transcurrent environments.

J. Geophys. Res. 91, 4797–4818.

Sonder, L.J., Jones, C.H., 1999. Western United States extension: howthe west was widened. Annu. Rev. Earth Planet. Sci. 27, 417–462.

Stewart, J.H., 1978. Basin and Range structure in western NorthAmerica: a review. In: Smith, R.B., Eaton, G.L. (Eds.), Ceno-

anetary Interiors 163 (2007) 35–51 51

zoic Tectonics and Regional Geophysics of the Western Cordillera.Mem. Geol. Soc. Am., pp. 1–31.

Taponnier, P., Pelzer, G., LeDain, A.Y., Armijo, R., Cobbold, P., 1982.Propagating extrusion tectonics in Asia: new insights from simpleplasticine experiments. Geology 10, 611–616.

Tapponnier, P., et al., 2001. Oblique stepwise rise and growth of theTibet Plateau. Science 294 (5547), 1671–1677.

Thatcher, W., et al., 1999. Present-day deformation across the Basinand Range Province, Western United States. Science 283 (5408),1714–1718.

Vilotte, J.P., Daignieres, M., Madariaga, R., 1982. Numerical modelingof intraplate deformaion: simple mechanical models of continentalcollision. J. Geophys. Res. 87, 10709–10728.

Ward, S.N., 1990. Pacific-North America plate motions; new resultsfrom very long baseline interferometry. J. Geophys. Res. B SolidEarth Planets 95 (13), 21, 965–21, 981.

Wells, R.E., Simpson, R.W., 2001. Northward migration of the Casca-dia forearc in the northwestern US and implications for subductiondeformation. Earth Planets Space 53 (4), 275–283.

Wells, R.E., Weaver, C.S., Blakely, R.J., 1998. Fore-arc migration inCascadia and its neotectonic significance. Geology (Boulder) 26(8), 759–762.

Wernicke, B., Christiansen, R.L., England, P.C., Sonder, L.J., 1987.Tectonomagmatic evolution of Cenozoic extension in the NorthAmerican Cordillera. In: Coward, M.P., Dewey, J.F., Hancock, P.L.(Eds.), Continental Extensional Tectonics. Geol. Soc. Spec. Pub.,pp. 203–221.

Wernicke, B., et al., 1996. Origin of high mountains in the continents:the southern Sierra Nevada. Science 271 (5246), 190–193.

Wernicke, B., Friedrich, A.M., Niemi, N.A., Bennett, R.A., Davis, J.L.,2000. Dynamics of plate boundary fault systems from Basin andRange Geodetic Network (BARGEN) and geological data. GSAToday 10 (11), 1–7.

Wernicke, B.P., 1992. Cenozoic extensional tectonics of the U.S.Cordillera. In: Burchfiel, B.C., Zoback, M.L., Lipman, P.W. (Eds.),The Cordilleran Orogen: Conterminous. U.S. Geol. Soc. Am.,Boulder, Colorado, pp. 553–581.

Willett, S., Beaumont, C., Fullsack, P., 1993. Mechanical model forthe tectonics of double vergent compressional orogens. Geology21, 371–374.

Williams, C.A., Richardson, R.M., 1991. A rheological layered three-dimensional model of the San Andreas Fault in central and southernCalifornia. J. Geophys. Res. 96, 16597–16623.

Wolfe, J.A., Schorn, H.E., Forest, C.E., Molnar, P., 1997. Paleobotan-ical evidence for high altitudes in Nevada during the Miocene.

Science 276, 1672–1675.

Zoback, M.D., Anderson, R.E., Thompson, G.A., 1981. Cenozoic evo-lution of the state of stress and style of tectonism of the Basin andRanges Province of the western United States. Philos. Trans. R.Soc. Lond. Ser. A 300, 407–434.