Geodetic strain across the San Andreas fault reflects...

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tters 269 (2008) 352–365www.elsevier.com/locate/epsl

Earth and Planetary Science Le

Geodetic strain across the San Andreas fault reflects elastic plate thicknessvariations (rather than fault slip rate)

Jean Chéry

UNIVERSITE MONTPELLIER 2, Géosciences Montpellier, CNRS (ou CNRS/INSU), UMR 5243, Place Eugène Bataillon, CC 60 34095 Montpellier, France

Received 11 January 2007; received in revised form 15 October 2007; accepted 27 January 2008

Editor: CAvailable online

.P. Jaupart16 February 2008

Abstract

The interseismic velocity field provided by geodetic methods is generally interpreted in the framework of a thick elastic lithosphere with aslipping fault at depth. Because lateral variations of lithospheric rheology play a key role in determining the geological strain distribution, Iexamine the idea that interseismic strain rate variations also occur in response to lateral variations in the elastic thickness of the lithosphere. Usinga stress balance principle and some simplifying assumptions, I show using a 1D model that elastic thickness is inversely proportional to strain ratefor the simple case of pure strike-slip faulting. Elastic thickness computed on three profiles crossing the San Andreas fault system (SAFS) suggeststhat the distribution of interseismic strain rate is compatible with a thick elastic lithosphere in the Great Basin-Sierra Nevada province and on thePacific plate. Conversely, a thin plate with a shallow asthenosphere is needed on the SAFS to explain its high strain rate. A 2.5D Finite Elementmodel of interseismic strain in the Carrizo Plain region in Central California shows how known vertical and horizontal variations of elasticproperties refine 1D model predictions. In a case of a multiple fault system, I point out that the interseismic velocity is not causally tied to faultsslip rate. Therefore, analysing the velocity field across the SAFS cannot reliably provide faults slip rate distribution as previously claimed. Rather,the apparent correlation between geologic slip rate and interseismic strain may only indicate that the elastic thickness plays a dominant role incontrolling fault strength. Finally, I suggest that interseismic geodetic strain could be a new way to infer effective elastic plate thickness on thecontinents.© 2008 Elsevier B.V. All rights reserved.

Keywords: geodesy; interseismic strain; elastic thickness; GPS; lithosphere; rheology; fault; San Andreas fault; stress

1. Introduction

Seismic hazard assessment strongly relies on the measure-ment of fault slip rates. For times longer than 10–100 kyrs,repeated earthquakes offset the geomorphic features crossed bya fault (gullies and moraines). Offsets measurements and datedfeatures determine an average fault slip rate (Sieh and Jahns,1984). This geological approach is thought to be accurate due toits direct relation to fault slip observations. However, thismethod is not always straightforward as it requires unambigu-ously datable offset features. Another way to compute fault sliprate is to measure the interseismic strain by geodetic means suchas the GPS technique. Due to the global GPS coverage, thismethod is potentially applicable worldwide on land and is

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becoming a major tool to provide a global strain pattern at theEarth's surface (Kreemer et al., 2003). However, switchingfrom interseismic strain to fault slip rate remains challenging.Because GPS observations are made on a small fraction of theseismic cycle, computing the fault slip results from a huge timeextrapolation using a physical model of repeated earthquakes.Also, one must be confident that the surface strain arerepresentative of the bulk deformation of the continentallithosphere, especially in the case of weak stratified plate. Forstrike-slip faulting, a widely used concept is based on a thicklithosphere with an embedded fault (Savage and Burford,1973), referred in this paper as SB73 model or thick lithospheremodel.

During the interseismic phase, the fault is locked from thesurface to a depth d (locking depth, see Fig. 1a). The fault belowthis depth slips at a constant rate s. This model corresponds to a

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Fig. 1. a) The thick lithosphere model in a vertical cross-section. The verticalfault is infinite perpendicular to the cross-section. During interseismic period,the fault above d is locked while the deep part slips at a rate s; b) adjustment ofthe GPS velocity data (solid dots) parallel to the SAF in the Carrizo segment inCentral California (see Fig. 2 for precise location) using the screw dislocationmodel (solid curve and Eq. (1)). Values of 12 km and 34 mm/yr are used for dand s respectively.

353J. Chéry / Earth and Planetary Science Letters 269 (2008) 352–365

screw dislocation and the velocity variation at the surface isgiven by:

v ¼ s=pð Þ arctan x=dð Þ ð1Þimplying that interseismic velocity v reaches ~90% to thegeological fault slip rate swhen the distance to the fault |x| is largerthan ~2πd. Such a formulation is highly attractive as an entire setof velocities may be fitted by adjusting the locking depth and thefault slip rate. Velocity profile across the central segment of theSan Andreas fault (SAF) illustrates well this aspect (Fig. 1b).Using a slip rate of 34 mm/yr and a locking depth of 12 km leadsto a RMSmisfit between the SCEC 3.0 velocity field (Shen et al.,2003) and the model of 2.25 mm/yr, about twice as higher as the1-σ formal data uncertainty of ~1 mm/yr (Schmalzle et al., 2006).It is remarkable that the geodetic slip rate matches well with thegeological slip rate of 34 mm/yr (Sieh and Jahns, 1984; Brown,1990) and that the locking depth corresponds to the maximumseismicity depth in this zone (Miller and Furlong, 1988). Thearctangent shape of the velocity profiles across many large strike-

slip faults encouraged scientists to extend the concept developedby Savage and Burford for a single strike-slip fault to multiplefault settings in which vertical faults delimit elastic blocks. Again,interseismic velocities are well fit by least square inversion forfault slip rate and locking depth. Based on its conceptualsimplicity and on successful slip rate predictions of the SB73model, its extension to the block model is about to become aroutine tool to estimate fault slip rates in Northern California(Freymueller et al., 1999; Savage et al., 2004; D'Alessio et al.,2005) Southern California (Lisowski et al., 1991; Bennett et al.,1996; Becker et al., 2005; Fay et al., 2005; Meade and Hager,2005) and other continental areas (McClusky et al., 2000;Wallaceet al., 2004).

To summarize, the use of the thick lithosphere model ismostly due 1) its conceptual simplicity 2) its ability to model thegeodetic strain field 3) the determination of desirable para-meters such as the long term fault slip rate. Because the thicklithosphere model provides a long term slip rate prediction, thelink between short term and long term time scales is mandatoryto assess this prediction reliability. Therefore, the motivation ofthis paper is to examine how this model is compatible with along term mechanical model of lithospheric strain. Indeed, aninterseismic strain model should be viewed— in principle— asa time fraction of a seismic cycle model. Also, a seismic cyclemodel spans another time fraction embedded in a long termevolution of the geological strain. Using the example of theSAFS, I start with describing the mechanical and rheologicalaspects of a long term model for parallel strike-slip faults. Then,I attempt to extract the model behaviour for the different phasesof the seismic cycle (coseismic, postseismic, interseismic). Thisanalysis leads me to propose that the interseismic strain patternmay be so much influenced by variations in lithosphere elasticthickness that it becomes difficult to obtain unambiguously anestimation of the slip rate on this sole basis. I test this hypothesisusing a mechanical analysis of the geodetic interseismic strainon the SAF using a variable thickness mechanical model.

2. Geological strain, lithosphere rheology and the seismiccycle on the San Andreas fault

Plate reconstruction and geological analysis have shown thatmost of the Quaternary strain between the Pacific plate and theSierra Nevada is concentrated on a few faults spaced by a fewtenths of km (Brown, 1990). In zones where the SAF orientationis aligned with the Pacific plate motion, these faults are nearlypurely strike-slip. Therefore, the sum of the geological slip rate ofthese faults is believed to be (at 10% uncertainty) equal to theSierra Nevada–Pacific plate differential motion (Dixon et al.,2000). In this context, two kinds of active fault settings occur (Fig.2). First, the slip rate can be distributed over a few faults. Innorthern California at the latitude 38–39°N, the slip occurs fromwest to east on the San Andreas fault (20–25 mm/yr) theMaacama–Rodger Creek faults (6–10 mm/yr) and the BartlettSpring–Green Valley fault (~5mm/yr). A similar situation occursin southern California at a latitude of 33°N but with an inversedistributionwith respect to the northernCalifornia setting. Indeed,the slip rate is small on the Elsinore fault on the coast (~3 mm/yr),

Fig. 2. Surface velocity (blue arrows) across the SAF system given by permanentand campaign modeGPS data (USGS and SCEC public data) in a North Americanreference frame. This velocity field (black arrows) mostly represents interseismicstrain accumulation. Major faults are given in red (SAF = San Andreas fault; RC =Rodgers Creek fault; GV = Green Valley fault; SJ = San Jacinto fault; ELS =Elsinore fault). The three profiles across the fault system are drawn with a blackline. The surface velocity field used in the computations and in Figs. 6, 7 and 8 aremarked with blue dots (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.).

354 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352–365

while it is larger on the San Jacinto fault (8 mm/yr) and maximumon the SAF to the east (22 mm/yr). In contrast, in centralCalifornia between the San Francisco Bay and the Big bend, mostof the strain occurs on the SAF only at a rate of 34 mm/yr.

Geophysical evidence may explain this pronounced strainlocalization in Central California. First, a high heat flow of~80 mW/m2 has been measured in a zone of 100 km around the

Fig. 3. Typical stress envelops with depth for a) cold continental lithosphere, b) hot con(the dashed curve represents the frictional stress associated to a high friction as in a) astrain rate does not vary with depth. Horizontal and vertical scales are indicative. T

SAF (Lachenbruch and Sass, 1980; Williams et al., 2004),suggesting high temperature in the lower crust and the upper-most mantle induces low viscosities. Also, stress measurementsin deep boreholes indicate that the SAF supports a shear stress aslow as 10–20MPa in the seismogenic zone (Zoback et al., 1987;Rice, 1992) contrasting with high values of ~150 MPa expectedfrom laboratory measurements of fault friction (Byerlee, 1967).The combination of high heat flow in a 100 kmwide zone aroundthe SAF and the low resolved shear stress on the fault itselfprovides a physical explanation for slip localization on the SAFin a thermally weakened lithosphere (Furlong, 1993). Further-more, the low compliance of the lithosphere seems to occur at asmaller scale. For example, magnetotelluric experiments atParkfield show a low resistivity area beneath the seismogenicpart of the SAF (Unsworth et al., 1997), probably due to a highfluid concentration related to intense shear at depth. Geomecha-nical modelling supports this view in requiring an effectivefriction coefficient of 0.05–0.17 for the Central SAF (Bird andKong, 1994) (Chéry et al., 2001). In zones of multiple parallelfaults such as northern and southern California, the measuredfault slip rates can be explained by a combination of lateral heatflow variations with low effective friction coefficients on faults(Provost and Chéry, 2006). In general, the contribution of thebasal stress inmechanical modelling is neglected, although someauthors have provided arguments that it may significantly affectthe stress balance of the SAFS (Lachenbruch and Sass, 1973).

Geological strain across active strike-slip faults results from acombination of the rheological stratification of the lithosphereversus depth and low resisting stress in fault zones (Gilbert et al.,1994). Depth stratification is mainly temperature dependent, andthe depth of the 350 °C isotherm represents the transition betweenfrictional faulting and thermally activated viscous strain (Sibson,1982). Another transition occurs at the crust–mantle transitionthat marks a strength increase due to olivine rheology (Brace andKohlstedt, 1980). As both crustal and mantle viscous laws aretemperature dependent, an easy way to estimate differential stressvariation with depth is to assume a constant strain rate through the

tinental lithosphere c) weak fault zone embedded in a hot continental lithospherend b). τ represents the magnitude of deviatoric stress and :e ¼ ct means that thehe crust–mantle transition occurs at ~30 km depth.


lithosphere for different geotherms. Although the constant strainrate assumption is generally incorrect in actively deforming zonesas demonstrated by mechanical modelling (Chéry et al., 2001),three end member models emerge for the lithospheric strength(Fig. 3). At low heat flow (40 mW/m2), the 350 °C isotherm isclose to Moho depth, causing the crust to be mostly brittle (Fig.3a). Low temperature in the subcrustal mantle should imply veryhigh viscosities (Strehlau and Meissner, 1987). However, stresscontrolled plastic behaviour is likely to limit the maximumsustainable stress to about 600 MPa (Tsenn and Carter, 1987). Athigh heat flow (80 mW/m2), the 350 °C isotherm is shallow (7–15 km) and the Moho temperature is high (N700 °C), leading to astrength profile mostly controlled by upper crustal friction andmiddle crust viscosity (Fig. 3b). However, this model has to bemodified for a fault zone, with the constrain of low effectivefriction within the seismogenic layer (Zoback et al., 1987; Wanget al., 1995; Hassani et al., 1997) (Fig. 3c). In addition, strain rateweakening and metamorphic reactions may alter the deformationprocesses in the middle crust, and some authors have proposed

Fig. 4. a) Strain rate invariant associated with parallel strike-slip faults of northern Caintrinsically weak material and display high strain rate (Maa = Maacama fault; BS =surface (solid line) and at 25 km depth (dashed line); c) fault-parallel stress profilescorresponds to the maximum fault strength for this slip rate. The profile to the right insto the dashed curve. Stress integrals with depth must be equal on the two profile to

that a drastic strength reduction occurs at the brittle–ductiletransition (Gueydan et al., 2001).

Lithospheric strength profiles are useful to study the linkbetween geophysical variables such as P and T and the effectiverheology of the lithosphere. However, they represent themaximum sustainable stress (yield stress) of the lithospherefor a given strain rate, not the actual lithospheric stress. As anexample, I consider the behaviour of the northern SAF asmodelled by Provost and Chéry (Fig. 4a,b). In their study, theauthors account with both strike-slip and shortening betweenthe Pacific plate and the Sierra Nevada. The small shorteningstrain, which is accommodated by plastic strain inside the crustand also by dip-slip fault motion, is not considered in thepresent paper to keep a simple mechanical analysis. In the caseof pure strike-slip motion, the stress magnitude along faultsdepends on the effective friction coefficient of the seismogeniczone and the viscosity of the middle crust. Because bothfrictional and viscous parameters in fault zones have low values,this limits the stress of the surrounding lithosphere that cannot

lifornia (adapted from Provost and Chéry, 2006); major faults are modelled withBartlett Springs fault); b) long term velocity in the fault-parallel direction at thefor the fault zone located on the SAF and for the crust. The profile to the leftide the crust remains much below the maximum sustainable stress correspondingensure stress equilibrium.

Fig. 5. Interseismic stress accumulation and corresponding velocity. a) Cross-section of a lithosphere with lateral variation along x-axis of its geodetic elasticthickness Tg. The position of the 350 °C isotherm is given by the dashed line andmarks the upper limit of the viscoelastic zone. A low deviatoric stress is assumedto occur below corresponding to effective viscosities lower than 1019–1020 Pa s.A fault-parallel stress increase Δτxy occurring a time interval Δt (see Eq. (2)) isshown on two profiles. For each profile, the stress increase is assumed to beconstant with depth accordingly and is materialized with a solid rectangle. Dueto stress equilibrium the stress increase integrals on the two profiles are equal. b)Corresponding interseismic velocity field (y-axis component) across thelithosphere computed using Eq. (3).


reach its maximum value. Therefore, the lithosphere does notdeform and the viscous stress is zero due to the lack of strain. Inthis case, the lithospheric stress of this zone corresponds toelastic strain accumulation without involving frictional pro-cesses (right profile on Fig. 4c).

Let us now consider the behaviour of such a long term modelduring the seismic cycle. In contrast with the SB73 model whichis driven directly by imposing fault motion on the fault plane,this long termmodel is kinematically driven from the sides of theblock at the prescribed side plate velocities. In this context,coseismic motion occurs in response to a step in effective faultfriction, initially proposed by Brace and Byerlee (1966). Faultrupture during large continental earthquakes takes place betweenthe surface and 10–20 km depth and produces elastic strain in thefault vicinity. Postseismic motion occurs minutes to yearsfollowing the earthquake as a result of a variety of processes likeafterslip of the deep fault plane, viscoelastic processes in themiddle crust and the mantle and poroelastic effects in the crust. IfI ignore the poroelastic effect for this discussion, both afterslipand viscoelastic strain act to release stress buildup at depth andreload the fault plane above. Due to the thermal stratification ofthe lithosphere, a large viscosity spectrum [1017–1019 Pa s] islikely to control the strain following large earthquakes as shownby postseismic modelling studies (Freed et al., 2006). Once thisstress transfer is complete, a steady interseismic strain build upoccurs in response to plate motion. A chief difference of thisphase with respect to the long term behaviour is that the fault islocked, implying that the whole seismogenic layer behaveselastically or viscoelastically with viscosities higher than 1021 Pas. At depth, a low viscosity stress occurs in response tointerseismic strain beneath the seismogenic zone.

If one attempts to interpret the interseismic strain across theSAFS with the model of Fig. 4c, it becomes clear that thesurface strain has to be influenced by the thickness of the elasticand viscoelastic layers. As suggested by Fig. 5, thin parts of thelayer should display strain accumulation while thick partsshould not accumulate much strain. Although this two-layermodel has been invoked for decades to explain postseismicstrain (Nur and Mavko, 1974; Pollitz et al., 2000) (Kenner andSegall, 1999), it is less used to explain the interseismic straindistribution across fault systems (Bourne et al., 1998; Cohenand Darby, 2003; Schmalzle et al., 2006). In such a case, elasticstrength is likely to depend on the product of the thickness of theseismogenic zone and the average shear modulus of this layer asdetailed later. The viscoelastic strength determination followsthe same scheme and depends on the product of the thickness ofthe viscous layer and its average viscosity. Two lines ofevidence suggest that the viscous contribution to the litho-spheric strength is relatively small. First, a large stress reductionwith depth occurs in only a few km only due to the highsensitivity of power law flow to temperature. For example, atypical granite-type power law (Kirby, 1985) loaded at a strainrate of 10−14 s−1 implies a deviatoric stress of 100 MPa at350 °C and only 1–10 MPa at 450 °C. Assuming a surfacethermal gradient of 20 °C/km, this suggests that the thickness ofthe layer hosting significant viscous stress should not exceed 5–10 km. Second, effective viscosities determined in the crust and

the mantle by postseismic modelling are on the order of 1019 Pas, suggesting that these zones produce a stress contribution of~1 MPa if an interseismic strain rate of 10−14 s−1 is applied.

To summarize this analysis, interseismic strain measured bygeodetic tools at the Earth's surface may reflect the elastic strainaccumulation of the seismogenic zone (i.e., grossly the part ofthe crust above 350 °C and possibly the cold uppermost mantle)and the viscoelastic behaviour of a thin zone below theseismogenic layer in the crust and possibly in the mantle. Tosignificantly contribute to interseismic stress accumulation, thecorresponding viscosities must be higher than 1021 Pa s. Thisway to interpret interseismic strain has been proposed for theSAFS and for the Alpine fault of New Zealand (Bourne et al.,1998) but assuming that viscous strain in the lower crust and inthe mantle drives and controls the entire fault system. Rather, Ipropose that interseismic strain reflects lateral variations in thethickness and elastic modulus of a lithospheric stress guide. Thisview is obviously close to the elastic plate model used to explainthe flexural behaviour of the lithosphere (Watts, 2001). Themaindifference comes from the kind of forces applied to the plate.Classical plate theory aims to explain the relationship betweenvertical motion (plate bending) and plate thickness, while theconcept I propose here considers the relation between horizontalstrain and plate thickness. Because this thickness is evaluatedby analysing horizontal geodetic strain, I name it geodetic elastic

Fig. 6. Interpretation of the GPS velocity field on a profile perpendicular to thenorthern SAFS. a) Fault-parallel GPS interseismic velocity fields (see Fig. 2 forlocation). A continuous version of the velocity measurements is given by the redcurve. The RMS (mm/yr) of the adjustment between the curve and the data is givenby the blue solid line with the scale to the right; b) fault-parallel horizontal strainrate corresponding to the continuous velocity field slope c) elastic thickness acrossthe SAFS based on Eq. (3) (black curve) and on the assumptions given in the text.Minimum and maximum thickness associated to the strain rate formal error arerepresented by red lines (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.).


thickness (GET) or Tg in order to differentiate it to the flexuralelastic thickness (Te) provided by plate bending analysis.

3. A simple model between strain rate and elastic thickness

To build a simple model explaining the relation between theinterseismic strain and the lithospheric strength, I assume herethat the crust and the mantle are elastic for temperature lowerthan respectively 350 °C and 750 °C. Also, I neglectviscoelastic effects that is to say that the stress at temperaturegreater than 350 °C and 750 °C for the crust and the mantlerespectively is negligible. The contribution of interseismicstrain corresponds therefore to the deformation of a plate havingan effective thickness Tg. as presented in Fig. 5a. Because themechanical system obeys stress equilibrium and assuming thatonly the stress component τxy contributes to the total force,stress increase Δτxy during a time Δt on two profiles ofthickness Tg

1 and Tg2 satisfies the relation:

Z T1g

0Ds1xy zð Þdz ¼

Z T2g

0Ds2xy zð Þdz ¼ DF ð2Þ

where ΔF represents the shear force change applied to thelithosphere. I assume a linear shear stress–shear strain relationgiven by Δτxy=G·Δεxy where G is the average shear moduluson the layer. Assuming that Δεxy is constant along a verticalprofile and dividing the previous equation by Δt leads to:

PG xð Þ � Tg xð Þ � :exy xð Þ ¼ C ð3Þwhere

PG is the average shear modulus on the profile and C is a

constant, meaning that fault-parallel strain rate :exy is inverselyproportional to the integrated elastic strength

PG � Tg. With the

simplifying assumptions above, knowledge of the interseismicstrain rate should teach us how elastic thickness may vary. I testthis hypothesis on three profiles across the SAFS.

4. Analysis of velocity profiles crossing the SanAndreas fault

GPS profiles shown on Fig. 2 cross the SAFS north of SanFrancisco Bay (North Bay profile, see Fig. 6), in the Carrizoplain south of Parkfield (Carrizo profile, see Fig. 7) and close tothe Salton Sea south of Los Angeles (Salton profile, see Fig. 8).Faults are straight around these three locations so the deforma-tion is nearly two-dimensional. Also, no large earthquakes haveaffected the SAF since the 1906 San Francisco earthquake forthe North Bay profile and the 1857 Ft. Tejon earthquake for theCarrizo profile. Also, there is no evidence that the 1857 ruptureextended down into the Salton Trough region. Therefore, thepostseismic signal on the three profiles is expected to be smallcompared to the interseismic strain. I use the public dataavailable on USGS and SCEC web sites corresponding to GPSsurface velocities with respect to the North American plate.

Because the RMS associated to the data are of the order1 mm/yr, the direct derivative of the linear velocity fieldinterpolation with respect to the profile direction is meaninglessand not of a practical use in our case. In order to obtain a smooth

space derivative of the velocity field, I compute the strain rateusing a least square adjustment of the interseismic velocity overa moving window of 20 km minimum half-width. Therefore,most of the long and short wavelength features of the profiles arepreserved (Fig. 6a) and interseismic strain rate is given by theslope of the least square adjustment (Fig. 6b). The given formalerror on the strain rate corresponds to the slope uncertaintyassociated to the least square adjustment. This leads to obtain asmall error when all the points are aligned in the movingwindow, therefore corresponding to a constant strain rate model.Amore complete formulation that should also include individualRMS associated to the data has not been developed here. Thecomputation of the elastic thickness using Eq. (3) requires apriori information on Tg because a trivial solution is given byTg=0, corresponding to the lack of information on the shearforce change of Eq. (2). Because earthquake seismicity near theSAF does not occur below 12–15 km, I assume a minimum


elastic thickness on each profile of 13 km. Elastic thicknessdistribution is computed according to this value, also assumingthat the average shear modulus does not vary along the profile. Ifcomputations were done with a different minimum elasticthickness, it would have been affected elastic thickness on theprofile only by a multiplicative constant (see Eq. (3)). Fromnorth to south, interseismic strain rate patterns and theircorresponding elastic thickness are markedly different.

Along the North Bay profile, strain rate displays a markedasymmetry with a maximum along the Pacific plate to the westand a slow decrease towards the east up to the Central Valley andthe Sierra Nevada. Consequently, the elastic thickness is higherclose to the Pacific plate (40–70 km) and gently decreases (13 to20 km) across the Coast Range from the SAF to the Green Valleyfault. Because of the small strain rate of the Central Valley, theinverse relation between strain rate and thickness leads to anelastic thickness larger than 100 km (Fig. 6c).

Interseismic velocity in central California leads to a differentinterseismic strain variation (Fig. 7b). Strain rate gently increasesfrom southwest (the Pacific coast) to northeast up to a maximum10 km east to the SAF trace as noticed by previous work(Schmalzle et al., 2006). Despite a limited GPS data set in theCentral Valley and Sierra Nevada, the strain rate 20 km east to theSAF is virtually zero according to other geodetic studies of theSierra Nevada block (Dixon et al., 2000). According to this strain

Fig. 7. Same as for Fig. 6 for the central SAFS (see Fig. 2 for location).

Fig. 8. Same as for Fig. 6 for the southern SAFS (see Fig. 2 for location).

variation, elastic thickness gradually decreases from values higherthan 80 km to 13 kmacross thewesternCoast Ranges (Fig. 7c), andjumps back to large values east to the SAF in the Central Valley.

The southern California profile near the Salton Sea reveals areversed strain pattern compared to the one obtained in theNorth Bay area. Elastic thickness progressively decreases fromthe Pacific coast when crossing the Elsinore fault, reaches itsminimum value between the San Jacinto and SAF and increasestowards ~80 km on the north American plate to the east (Fig. 8c).

5. Finite Element model of the interseismic strain

The simple formulation used to compute the elastic thicknessof the SAFS provides a first order relation between interseismicstrain and plate thickness. However, stress components arelikely to vary vertically within the plate, implying that anaccurate strain solution requires solving a more complete stressbalance equation. Also, the assumption of constant elasticmodulus through the entire lithosphere is questionable, asvertical and lateral elastic properties variations of the litho-sphere are known to be significant (Meissner, 1986). Involvingthese effects with solving stress balance equation with complexgeometry needs to be done numerically. I design one experimentto determine if the simple concept developed above about thestrain rate–elastic thickness relation still holds for a finitethickness lithosphere. Because errors induced by the 1D model

Fig. 9. a) Horizontal and vertical Young's modulus variation used in the Finite Element model; b) Geometry of the FEM and stress rate accumulation:sxy corresponding

to a 34 mm/yr loading velocity between the lateral sides (at 0 and 350 km). Zone of low stress rate to the right corresponds to the Great Valley and the Sierra Nevada.Zone 20 km east to the SAF has low elastic modulus based on seismological evidence and loads at a high stress rate of 0.01 MPa/yr. Note that this value predicts aninterseismic stress loading of 2.5 MPa for a recurrence time of 250 yr, which is compatible with the average static stress drop for a large earthquake (Hanks, 1977).Zone west to the SAF corresponds to a progressive plate thickening increase.

Fig. 10. Interseismic velocity field provided by the Finite Element model (thickblack line) compared with the smooth data fit (dashed thin line) and with thethick lithosphere model (purple curve). Discrete GPS velocity values across thecentral SAFS are given by the solid black circles.


are likely to be higher when large geometrical variations occur, Iuse the example of the central SAF for which high thicknessgradients are expected.

The geometry of the model represents a cross-section of theelastic part of lithosphere perpendicular to the fault direction. Theloading corresponds to a motion of the Pacific plate with respectto a fixed Sierra Nevada at a rate of 34 mm/yr parallel to the faultdirection. Assuming no velocity variation in the direction parallelto the fault direction (y), non-zero stress components are τxy andτyz. Both horizontal and vertical bulk elastic variations are takeninto account (Fig. 9a). I incorporate Young's modulus increasewith depth as observedworldwide in continents (Meissner, 1986).Based on a seismic velocitymodel of the Parkfield area (Eberhart-Philips and Michael, 1993) I also account for a lateral decrease ofYoung's modulus in a 20 km width zone east of the SAF. I alsoadjust elastic thickness along the Carrizo profile. Elastic thicknessof the Great Valley–Sierra Nevada is set to 200 km in order toaccount for both its low geodetic deformation (Dixon et al., 2000)and its high flexural rigidity (Kennelly and Chase, 1989). Elasticthickness of the Pacific plate west to the SAF is set to 40 km tomake it compatible with currently estimated elastic thickness(Watts, 2001).

The interseismic profile velocity for the model of Fig. 9bdisplays a data-model RMS of 1.77 mm/yr (Fig. 10). Due tointrinsic velocity errors of about 1 mm/yr on the Carrizo profile,the smooth interseismic curve (1D model of Fig. 6) fits velocitydata with a RMS of 1.34mm/yr. The data fit of the FEMmodel isnot as good as the one provided by the 1D model. This isunderstandable as the 1D strain model is obtained by direct

smoothing of the discrete velocity data, therefore representingthe best possible data fit. Despite its lower accuracy, I argue thatthe data adjustment provided by the FEM has a greater physicalmeaning than the 1D model presented before. This opinion isbased on three factors. First, it corresponds to a finite thicknesslithosphere, implying that the hypothesis of having :sxy constant


on a vertical profile is no longer needed. Clearly, this 1Dassumption is violated when high thickness gradients are presentas shown by Fig. 9b. Second, the FEM allows us to include ourbest a priori knowledge of the elastic properties of thelithosphere based on seismology. Interestingly, a reliableestimate of bulk elastic properties requires the thickness onlyto be adjusted. Third, interseismic strain provided by the FEMresults from a trial-and-error procedure. It is likely that a suitableinversion technique based on a grid search for an optimal elasticthickness would lead to a fit as good as the one provided by the1D smooth fit. In comparison to the 1D and the FEM data fit, theadjustment of fault slip rate and locking depth of the SB73modelleads to a data fit of 2.25 mm/yr. This is mostly because theasymmetry of the elastic profile resulting from the variableelastic thickness model fits the GPS data better compared to thesymmetric solution provided by the SB73 model. However, thefit of the thick lithosphere model is quite acceptable given thesmall number of free parameters and may be still improved ifelastic modulus contrast across the SAF is taken into account.

6. Discussion

Choosing among different mechanical models of interseismicstrain on the sole basis of geodetic data fitting has been shown tobe meaningless because of the non-uniqueness of the problemeven in the case of a single fault (Savage, 1990). In other words, abad data fit can allow us to discard a model but a good data fit isnot proof of a model's relevance. Therefore, in chasing amongpossible models I need to consider the problem from a broaderpoint of view than the one of finding the best data. Rather, itrequires consideration of each model's relevance from ageophysical, rheological and geodynamical point of view. Havingin mind the current view of the thermomechanic state of thelithosphere beneath the SAF (Lachenbruch and Sass, 1980;Furlong, 1993; Chéry et al., 2001), the use of a thick elastic modelfor the SAFS seems at odds to this knowledge. For example, thenorthern SAFS is very juvenile as it results from the northwardprogression of the Mendocino triple junction (Furlong, 1984).There, the disappearance of the subducting plate is thought to haveopened an asthenospheric window in contact with the upper plate.As a consequence, a sharp contrast is likely to exists between athin lithosphere (10–20 km) surrounding the SAFS and a thicklithosphere (50–100 km) to the east in the Great Valley and in theSierra Nevada. In other words, an interpretation of the whole faultsystem using a thick elastic lithosphere is therefore unlikelybecause of the presence of a hot low viscosity zone in the uppermantle in the region of the slab window. By contrast, the variableelastic thickness model predict a low rigidity of the SAFS, whichis compatible with a shallow low viscosity zone. As already noted(Le Pichon et al., 2005; Schmalzle et al., 2006), the data fitprovided by the Savage and Burford model is limited by theasymmetrical character of the velocity field. In the simple case of asingle vertical strike-slip fault, one expects a perfect symmetrywith respect to the fault axis. This is not always the case, as shownby Le Pichon et al. for different strike-slip faults. A plausibleexplanation is that lateral variations of elastic properties of a thicklithosphere cause a symmetry break (Rybicki and Kasahara,

1977). Indeed, large lateral variations of bulk rigidity are likely tooccur and could explain the asymmetry of the deformation.However, a bulk rigidity contrast of 10 that is required to explainthe strain asymmetry around the Sumatra fault (Le Pichon et al.,2005) probably exceeds known variations of shear moduluswithin the crust. An alternative explanation for strain asymmetryis a variation of the elastic thickness in conjunction with a bulkrigidity contrast (Melbourne and Helmberger, 2001; Cohen andDarby, 2003; Schmalzle et al., 2006).

6.1. Interseismic strain, elastic plate thickness and fault slip rate

As shown by the analysis of the central SAF profiles, SB73and variable thickness models both provide a good fit to theinterseismic velocity field. In contrast, different behaviours canbe expected between these models when fault slip rates aresearched. In the case of the SB73 model, the long term fault sliprate is equal to the far-field velocity (the differential platevelocity). The long term slip rate for the variable thicknessmodel is less straightforward to define because of the remotefault drive. As discussed in the introduction, the effective faultfriction should be considered here. If the fault friction is low, thefault will slip at the differential plate velocity because thelithosphere does not deform anelastically. If the fault friction ishigh, two cases have to be considered. If one assumes that thelithosphere always behaves elastically (meaning that it cansustain arbitrarily high strain) the fault slip rate must still beequal to the differential plate velocity. However, both in-situ andlaboratory rock strength estimates indicate that the differentialstress of the seismogenic lithosphere is limited by a frictioncoefficient of [0.6–0.8] (Townend and Zoback, 2000). There-fore, a high fault friction is likely to pervasively deform thelithosphere around, especially where the plate is the thinnest. Forthis case only, the fault slip rate is expected to be different to thedifferential plate velocity. Considering that large faults separat-ing thick lithospheric plates are often thought to sustain lowshear stress, the fault slip rate determined by the variablethickness model must therefore be in fair agreement with the bestfit SB73 model.

6.2. Slip rate computation of a parallel fault system

From a mechanical viewpoint (see Appendix for a discus-sion), the SB73 model mimics the mechanical behaviour of twothick lithospheric blocks separated by a low strength fault zone.Interestingly, most fault slip rates inferred from SB73 and blockmodels analysis around continental plate boundaries are in goodagreement with the corresponding geologic slip rates (Reilingeret al., 2006). Therefore, it is logical to conclude that the geodeticfault slip determination using the SB73 model is valid only if itcorresponds to a setting of a weak fault between two strongplates. However, many other fault settings occur in nature asmultiple fault system, faults embedded in wide orogens or highplateaus. To discuss this point in the framework of the SAFS, Iattempt here to explain how the variable thickness modelbehaves when two faults or more are embedded (Fig. 11). Thismodel is directly adapted from the 1-fault model of Fig. A1c but

Fig. 11. Mechanical model of a variable thickness lithosphere model with two embedded fault. Each fault strength i is the sum of a frictional stress occurring on a widthhfi and of a viscous stress occurring on a width hv

i .

Fig. 12. Normalized relation between slip rate and mechanical thickness usingEq. (8) for different values of the viscosity parameter (red lines). Black circlesgive the relation between slip rate and thickness as computed in Table 1. Blacklines represents slip rate uncertainties.


adding another frictional discontinuity and a viscoelastic zoneaccording to Fig. 4c.

As shown by the previous mechanical modeling, frictionaland viscous fault strength both control slip rate distribution in amultiple fault system (Bird and Baumgardner, 1984; Provostand Chéry, 2006). Using the example of Fig. 11 and using a 1Dbalance assumption similar of Eq. (2), it can easily be shownthat the effective strength of both faults are equal. Also, thestrength of the fault i is equal to the sum of the frictionalstrength Ff

i and of the viscous strength Fvi . For each fault, the

frictional strength is equal to the stress integral along thefrictional thickness hf

i. Assuming a frictional shear stressconstant with depth leads the frictional strength to be equal to:

Fif ¼ sihif ð4Þ

The viscous strength is equal to the integral of the viscousstress along the viscous layer which has a thickness hv

i . Thisintegral depends on the average strain rate inside this layeraround the fault plane and is also affected by the large viscosityvariations that likely occur with depth. Assuming that theviscous strain occurs on an average width w, the viscousstrength is therefore defined by the relation

Fiv ¼ giwh

iv � si ð5Þ

where is the fault viscosity ηi divided by w and si the fault sliprate. The total fault strength is equal to

Fi ¼ si 1þ hisi� �

hif ð6Þ

with

hi ¼ giwhim

sihifð7Þ

Therefore, θi represents the viscous component of the faultstrength. Assuming for purposes of discussion that both θi and τi

are equal for both faults, the slip rate ratio is given by:

s2

s1¼ max 0; 1þ 1

h1s1

� �h1fh2f

� 1

h1s1

" #ð8Þ

This relation is represented on Fig. 12 for different values ofh1s1 in the case or h2Nh1. As the respective contribution offrictional and viscous fault strength is still a matter of debate,this formulation is well suited as it does not require an a-priorichoice. In order to see how southern and northern SAFS behavewith respect to Eq. (8), let us assume that viscous to frictionalratio is the same for each fault. Thus the ratio between frictionalthicknesses is equal to the ratio given by geodetic elasticthicknesses. Slip rate and frictional thickness ratios are given inTable 1 and plotted in Fig. 12. Using preferred values for sliprates, only pairs including the Elsinore fault display θ1s1 valueslarger than 3. For other pairs, elastic thickness and slip rateratios match curves computed with between 1/3 and 3,suggesting that viscous and frictional strengths may have anequal influence on the SAFS. In order to see how this parametercompares with an a priori calculation, I use the following set ofparameter: a viscosity of 1019 Pa s; a fault zone width of 1 km; aratio hivh

if=0.25; a frictional shear stress of 10 MPa. In such a

Table 1Geodetic elastic thickness (provided by this study) and known fault slip rates fornorthern (lines 1 to 3) and southern (lines 4 to 6) SAFS (Shen-Tu, Holt et al., 1999)

Fault1–fault2 Tg1

(km)Tg2

(km)s1

(mm/yr)s2

(mm/yr)hf1/hf

2 s2/s1

SAF–RC (a) 13 18 20 [12–30] 9 [7–10] 0.72 0.45 [0.30–0.75]SAF–GV (b) 13 20 20 [12–30] 6 [4–8] 0.65 0.30 [0.20–0.50]RC–GV (c) 18 20 9 [7–10] 6 [4–8] 0.90 0.66 [0.44–0.88]SAF–SJ (d) 13 17 29 [10–35] 12 [8–24] 0.76 0.41 [0.27–1.00]SAF–ELS (e) 13 33 29 [10–35] 6 [2–9] 0.39 0.21 [0.06–0.6]SJ–ELS (f) 17 33 12 [8–24] 6 [2–9] 0.51 0.50 [0.16–0.75]

Numbers in columns 5, 6, 8 represent preferred values while brackets indicatemin/max estimates. SAF = San Andreas fault; RC = Rodgers Creek fault; GV =Green Valley fault; SJ = San Jacinto fault; ELS = Elsinore fault.


case, θi=2.5 108 s. Using a velocity of 25 mm/yr for the SAF(7.92 10−10 ms−1), is close to 0.2.

Given the large uncertainty in rheological parameters such asthe fault zone viscosity and width, the agreement between thisa-priori computation and the value deduced from the elasticthickness and slip rate plot is encouraging.

This analysis suggests that several geometrical and rheologicalparameters control fault strength and therefore geological faultslip rates. Frictional strength is controlled by the fault friction andthe seismogenic thickness, while viscous strength is controlled bythe viscosity and the size of the viscous domain. Any variation ofthese rheological parameters on a fault will cause the slip rate tovary as already demonstrated by some studies (Roy and Royden,2000; Provost and Chéry, 2006). The present analysis suggests acomplex link between geologic slip rate and interseismic strain ifmore than one fault are active. Despite that the SB73 model isbased on a direct and causal relation between interseismic strainand fault slip rate, these two parameters may only be positivelycorrelated in nature. In order to properly interpret their relationone must understand their causal relation. Indeed, fault slip ratechiefly depends on effective fault friction, fault zone viscosity andthe mechanical thickness of all these process zones. Because agreater thickness induces a strength increase if other rheologicalparameters remains unchanged, such an increase in thicknesscauses both the slip rate and the interseismic strain to decrease.But a change of effective fault friction modifies fault slip ratewithout changing the interseismic strain. Therefore, the apparentcorrelation between geologic slip rate and interseismic strain mayonly indicate that the elastic thickness plays a dominant role incontrolling fault strength.

6.3. A global relation between elastic thickness and interseismicstrain?

The goal of this paper is to study the interseismic strain ratedistribution across the SAFS and its relation to lateral rigidityvariations. If such a relation holds for this plate boundary,perhaps a similar relation can be expected for other deformationareas such as subduction zones, mountain belts, continental oroceanic rifts. If a relation between interseismic strain and elasticthickness is assumed, then plate boundaries that displays highstrain rate concentration would need to be interpreted like thinelastic zones. Is this mechanically understandable? Current

knowledge of the SAFS suggests that a thin elastic lithosphereindicates a locally weakened plate. This kind of weakness can bethermally induced, as high temperature gradients reduce thethickness of both brittle and viscous layers. This is likely to occurin mountain belts, rifts or extending plateaus as shown bygeophysical measurements and mechanical modelling (Gaude-mer et al., 1988; Buck 1991; Cattin et al., 2001). Also, a largecontribution to the lithospheric weakness could be related to lowstrength faults that decouple (in a stress meaning) adjacentplates. Subduction faults and large intracontinental faults areprobably a good example of this kind of weakness (Wang et al.,1995; Hassani et al., 1997; Cattin et al., 2001).

Up to now, elastic plate thickness has been computed forcontinental and oceanic plates using the idea that vertical loadsapplied to the lithosphere produce vertical motions by platebending (Watts, 2001). Using stress equilibrium of the plate andisostatic assumption, knowledge of the horizontal distribution ofthe loading function (topography, internal loads, mantle buoy-ancy, glaciations, etc) permits determination of the flexuralrigidity that best explains topographic and gravimetric signals.Knowledge of the elastic parameters of the lithosphere permitsconversion of flexural rigidity to equivalent plate thickness Te.However, one needs to be careful when plate thickness areobtained with different kinds of loads. Because the modelassumes an elastic plate over an inviscid fluid, the determinationof the flexural rigidity in nature is correctly done if the load isapplied for a time scale long enough to allow a complete stressrelaxation at depth. Short time scale loading does not allow suchstress relaxation, therefore leading to larger values of equivalentplate thickness. For example, mechanical modelling of the rapidfilling of the lake Mead in the Basin and Range leads to a platethickness estimate of 20–30 km (Kaufmann and Amelung,1995) much higher than the 5–15 km thickness found using thesurface topography as a loading function (Lowrie and Smith,1995). On both oceanic and continental plates, a clear correlationis found between elastic thickness and the thermal state of thelithosphere if a long term loading is considered. The 500–550 °Cisotherm matches the lower limit of the elastic plate in oceans,while no strong correlation has been found between effectiveelastic thickness and some particular isotherm for continentalcrust (Watts, 2001). The current interpretation for oceaniclithosphere is that most of the elastic stress is stored between thesurface and the 500–550 °C isotherm. In the case where elasticstresses reach the frictional strength of the lithosphere (generallydue to high plate curvature as it occurs in subduction zones), anelastoplastic plate bending model has to be used (Judge and McNutt, 1991) to account for a reduced elastic thickness. Thesituation is more complex in a moderately cold continentallithosphere for which a large part of the strength is probablystored in the uppermost mantle. As I show using the example ofthe SAF, the use of horizontal GPS velocity gradients at theEarth's surface through a stress equilibrium principle is anotherway to estimate effective plate thickness using its transverserigidity as a parameter to invert. However, such a use of GPSvelocities relies on two key assumptions. First, the strain mustreally represent the interseismic stage during which faults arelocked. The second assumption is that the plate strain is the result

Fig. A1. Differences and similarities between the thick lithosphere model andthe variable thickness lithosphere model; a) thick elastic lithosphere model(SB73 model); b) modified SB73 model with remote boundary conditions c)variable thickness elastic lithosphere.


of stress equilibrium inside the plate without coupling with themantle other than hydrostatic forces. For example, horizontalforces applied at the base of the elastic plate such as deepconvection or slab traction can induce strain on the surface,therefore leading to an incorrect plate thickness estimate.Keeping in mind that the relation between interseismic strainand plate thickness may therefore breakdown, the applicabilityof this theory on continents is worldwide. Indeed, the rapidgrowth of GPS and InSARmapping is already providing a denseand accurate velocity fields in many active region. Suitableinterpolation of this velocity field makes possible the strain ratecomputation (Kreemer et al., 2003). Using an inversionprocedure, this would potentially allow computation of rigiditymaps on continents.

7. Conclusion

The idea that interseismic strain is linked to variable elasticplate thickness is markedly different than the usual geodeticdata interpretation using the block model driven by fault slip atdepth. I have argued that the former model is rheologically moreplausible as it is directly linked to brittle and ductile lithosphereproperties. In such a model, faults do not play a direct roleduring the interseismic phase because they are locked.Interseismic strain is therefore chiefly controlled by elasticplate properties (mostly bulk elastic modulus and platethickness). Two main implications can be drawn:

– the use of the block model to infer fault slip rate must berestricted to single faults embedded in high rigidity domainslike for example the central north Anatolian fault or thecentral SAF. Incorrect slip rates are likely to be inferred ifthis condition is not fulfilled. This case occurs when multiplefaults are present as in southern and northern California andsuggests that such geodetically based fault slip rates could beunreliable.

– In a simple tectonic setting such as pure strike-slip faulting,high interseismic strain can be interpreted in the same way as alow rigidity (small elastic thickness) zone, while low strainwould correspond to a higher rigidity (large thickness). If this1D analysis can be extended to 2D based on stress equilibriumprinciple, the variable plate rigidity model could be applied tocompute elastic thickness on continents with a suitableinversion procedure using interseismic strain as input data.

Acknowledgements

This paper benefited from several discussions with mycolleagues. I particularly thank Xavier Le Pichon who forcedme to clarify my interpretations. Philippe Vernant read an earlyversion of the paper. Careful reviews of Wayne Thatcher and oftwo anonymous reviewers allowed to significantly improve themanuscript. I thank also Jessica Murray for helping me ingathering USGS geodetic data. SCEC and USGS are acknowl-edged for providing geodetic data of high quality. GenericMapping Tool (GMT) software has been used to prepare mostof the figures.

Appendix A

The variable elastic thickness model has markedly differentseismic hazard implications compared to those deduced fromSB73 and block models. For the latter, fault slip rates are a freeparameter to adjust. For the former, elastic thickness can becomputed, but no direct link can bemade with the long term faultslip rate. This lack of information is clearly related to the model'sassumption: interseismic strain is interpreted on the time scaleduring which geodetic measurements are made. By contrast withthe SB73 model, no hypothesis is made about the relationbetween short term interseismic strain and long term slip rate.Does this mean that interseismic strain cannot be used to predictfault slip rate? To discuss this important issue, let us consideragain the formal differences between these two models. Themost significant is the way to apply the boundary conditions tothe model. In the case of SB73 model, the differential velocity isapplied on the deep part of the fault (near field boundarycondition, see Fig. A1a). In such a case, the fault slip rate is akinematical parameter and has no relation to the fault strength. Inthe case of the variable elastic thickness model, the velocityboundary condition is remotely applied according to differentialplate motion. Because the source of plate motion is thought to becontrolled by large-scale body forces, this way to setup theboundary condition is probably wiser than a local drive. Let us


now imagine how to reproduce the interseismic velocity field ofthe SB73 model using a remote boundary condition (Fig. A1b).

Fig. A1 Differences and similarities between the thicklithosphere model and the variable thickness lithosphere model;a) thick elastic lithosphere model (SB73 model); b) modifiedSB73 model with remote boundary conditions c) variablethickness elastic lithosphere.

If the fault strength is high, the slip will be not allowed and thestrain would be homogeneous throughout the model. Conversely,the way to simulate the velocity jump on the deep fault is to assigna low strength for this part of the fault. Therefore, the low strengthof the deep part of the fault allows to interpret the SB73 model asthe juxtaposition of two thick blocks only coupled by the lockedzone above the velocity jump. This mechanical view is equivalentto a variable thickness model in which two thick blocks areseparated by an asthenospheric channel (Fig. A1c), and thesethree models should display similar interseismic surface velocityfield (although they are not identical).

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Geodetic strain across the San Andreas fault reflects...

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