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

Available online at www.sciencedirect.com Earth and Planetary Science Letters 269 (2008) 352 365 www.elsevier.com/locate/epsl Geodetic strain across the San Andreas fault reflects elastic plate thickness variations (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 Available online 16 February 2008 Editor: C.P. Jaupart Abstract The interseismic velocity field provided by geodetic methods is generally interpreted in the framework of a thick elastic lithosphere with a slipping fault at depth. Because lateral variations of lithospheric rheology play a key role in determining the geological strain distribution, I examine the idea that interseismic strain rate variations also occur in response to lateral variations in the elastic thickness of the lithosphere. Using a stress balance principle and some simplifying assumptions, I show using a 1D model that elastic thickness is inversely proportional to strain rate for the simple case of pure strike-slip faulting. Elastic thickness computed on three profiles crossing the San Andreas fault system (SAFS) suggests that the distribution of interseismic strain rate is compatible with a thick elastic lithosphere in the Great Basin-Sierra Nevada province and on the Pacific plate. Conversely, a thin plate with a shallow asthenosphere is needed on the SAFS to explain its high strain rate. A 2.5D Finite Element model of interseismic strain in the Carrizo Plain region in Central California shows how known vertical and horizontal variations of elastic properties refine 1D model predictions. In a case of a multiple fault system, I point out that the interseismic velocity is not causally tied to faults slip 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 in controlling fault strength. Finally, I suggest that interseismic geodetic strain could be a new way to infer effective elastic plate thickness on the continents. 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 measurement of fault slip rates. For times longer than 10 100 kyrs, repeated earthquakes offset the geomorphic features crossed by a fault (gullies and moraines). Offsets measurements and dated features determine an average fault slip rate (Sieh and Jahns, 1984). This geological approach is thought to be accurate due to its direct relation to fault slip observations. However, this method is not always straightforward as it requires unambiguously datable offset features. Another way to compute fault slip rate is to measure the interseismic strain by geodetic means such as the GPS technique. Due to the global GPS coverage, this method is potentially applicable worldwide on land and is E-mail address: jean@dstu.univ-montp2.fr. becoming a major tool to provide a global strain pattern at the Earth's surface (Kreemer et al., 2003). However, switching from interseismic strain to fault slip rate remains challenging. Because GPS observations are made on a small fraction of the seismic cycle, computing the fault slip results from a huge time extrapolation using a physical model of repeated earthquakes. Also, one must be confident that the surface strain are representative of the bulk deformation of the continental lithosphere, especially in the case of weak stratified plate. For strike-slip faulting, a widely used concept is based on a thick lithosphere with an embedded fault (Savage and Burford, 1973), referred in this paper as SB73 model or thick lithosphere model. During the interseismic phase, the fault is locked from the surface to a depth d (locking depth, see Fig. 1a). The fault below this depth slips at a constant rate s. This model corresponds to a 0012-821X/$ - see front matter 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2008.01.046

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 353 Fig. 1. a) The thick lithosphere model in a vertical cross-section. The vertical fault 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 of the GPS velocity data (solid dots) parallel to the SAF in the Carrizo segment in Central California (see Fig. 2 for precise location) using the screw dislocation model (solid curve and Eq. (1)). Values of 12 km and 34 mm/yr are used for d and s respectively. screw dislocation and the velocity variation at the surface is given by: v ¼ ðs=pþ arctan ðx=dþ ð1þ implying that interseismic velocity v reaches ~90% to the geological fault slip rate s when the distance to the fault x is larger than ~2πd. Such a formulation is highly attractive as an entire set of velocities may be fitted by adjusting the locking depth and the fault slip rate. Velocity profile across the central segment of the San Andreas fault (SAF) illustrates well this aspect (Fig. 1b). Using a slip rate of 34 mm/yr and a locking depth of 12 km leads to a RMS misfit between the SCEC 3.0 velocity field (Shen et al., 2003) and the model of 2.25 mm/yr, about twice as higher as the 1-σ formal data uncertainty of ~1 mm/yr (Schmalzle et al., 2006). It is remarkable that the geodetic slip rate matches well with the geological slip rate of 34 mm/yr (Sieh and Jahns, 1984; Brown, 1990) and that the locking depth corresponds to the maximum seismicity depth in this zone (Miller and Furlong, 1988). The arctangent shape of the velocity profiles across many large strikeslip faults encouraged scientists to extend the concept developed by Savage and Burford for a single strike-slip fault to multiple fault settings in which vertical faults delimit elastic blocks. Again, interseismic velocities are well fit by least square inversion for fault slip rate and locking depth. Based on its conceptual simplicity and on successful slip rate predictions of the SB73 model, its extension to the block model is about to become a routine 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; Wallace et al., 2004). To summarize, the use of the thick lithosphere model is mostly due 1) its conceptual simplicity 2) its ability to model the geodetic strain field 3) the determination of desirable parameters such as the long term fault slip rate. Because the thick lithosphere model provides a long term slip rate prediction, the link between short term and long term time scales is mandatory to assess this prediction reliability. Therefore, the motivation of this paper is to examine how this model is compatible with a long term mechanical model of lithospheric strain. Indeed, an interseismic strain model should be viewed in principle as a time fraction of a seismic cycle model. Also, a seismic cycle model spans another time fraction embedded in a long term evolution of the geological strain. Using the example of the SAFS, I start with describing the mechanical and rheological aspects of a long term model for parallel strike-slip faults. Then, I attempt to extract the model behaviour for the different phases of the seismic cycle (coseismic, postseismic, interseismic). This analysis leads me to propose that the interseismic strain pattern may be so much influenced by variations in lithosphere elastic thickness that it becomes difficult to obtain unambiguously an estimation of the slip rate on this sole basis. I test this hypothesis using a mechanical analysis of the geodetic interseismic strain on the SAF using a variable thickness mechanical model. 2. Geological strain, lithosphere rheology and the seismic cycle on the San Andreas fault Plate reconstruction and geological analysis have shown that most of the Quaternary strain between the Pacific plate and the Sierra Nevada is concentrated on a few faults spaced by a few tenths of km (Brown, 1990). In zones where the SAF orientation is aligned with the Pacific plate motion, these faults are nearly purely strike-slip. Therefore, the sum of the geological slip rate of these faults is believed to be (at 10% uncertainty) equal to the Sierra 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. In northern California at the latitude 38 39 N, the slip occurs from west to east on the San Andreas fault (20 25 mm/yr) the Maacama Rodger Creek faults (6 10 mm/yr) and the Bartlett Spring Green Valley fault (~5 mm/yr). A similar situation occurs in southern California at a latitude of 33 N but with an inverse distribution with respect to the northern California setting. Indeed, the slip rate is small on the Elsinore fault on the coast (~3 mm/yr),

354 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 Fig. 2. Surface velocity (blue arrows) across the SAF system given by permanent and campaign mode GPS data (USGS and SCEC public data) in a North American reference frame. This velocity field (black arrows) mostly represents interseismic strain 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 black line. The surface velocity field used in the computations and in Figs. 6, 7 and 8 are marked with blue dots (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.). while it is larger on the San Jacinto fault (8 mm/yr) and maximum on the SAF to the east (22 mm/yr). In contrast, in central California between the San Francisco Bay and the Big bend, most of the strain occurs on the SAF only at a rate of 34 mm/yr. Geophysical evidence may explain this pronounced strain localization in Central California. First, a high heat flow of ~80 mw/m 2 has been measured in a zone of 100 km around the SAF (Lachenbruch and Sass, 1980; Williams et al., 2004), suggesting high temperature in the lower crust and the uppermost mantle induces low viscosities. Also, stress measurements in deep boreholes indicate that the SAF supports a shear stress as low as 10 20 MPa in the seismogenic zone (Zoback et al., 1987; Rice, 1992) contrasting with high values of ~ 150 MPa expected from laboratory measurements of fault friction (Byerlee, 1967). The combination of high heat flow in a 100 km wide zone around the SAF and the low resolved shear stress on the fault itself provides a physical explanation for slip localization on the SAF in a thermally weakened lithosphere (Furlong, 1993). Furthermore, the low compliance of the lithosphere seems to occur at a smaller scale. For example, magnetotelluric experiments at Parkfield show a low resistivity area beneath the seismogenic part of the SAF (Unsworth et al., 1997), probably due to a high fluid concentration related to intense shear at depth. Geomechanical modelling supports this view in requiring an effective friction coefficient of 0.05 0.17 for the Central SAF (Bird and Kong, 1994) (Chéry et al., 2001). In zones of multiple parallel faults such as northern and southern California, the measured fault slip rates can be explained by a combination of lateral heat flow variations with low effective friction coefficients on faults (Provost and Chéry, 2006). In general, the contribution of the basal stress in mechanical modelling is neglected, although some authors have provided arguments that it may significantly affect the stress balance of the SAFS (Lachenbruch and Sass, 1973). Geological strain across active strike-slip faults results from a combination of the rheological stratification of the lithosphere versus depth and low resisting stress in fault zones (Gilbert et al., 1994). Depth stratification is mainly temperature dependent, and the depth of the 350 C isotherm represents the transition between frictional faulting and thermally activated viscous strain (Sibson, 1982). Another transition occurs at the crust mantle transition that marks a strength increase due to olivine rheology (Brace and Kohlstedt, 1980). As both crustal and mantle viscous laws are temperature dependent, an easy way to estimate differential stress variation with depth is to assume a constant strain rate through the Fig. 3. Typical stress envelops with depth for a) cold continental lithosphere, b) hot continental lithosphere c) weak fault zone embedded in a hot continental lithosphere (the dashed curve represents the frictional stress associated to a high friction as in a) and b). τ represents the magnitude of deviatoric stress and : e ¼ ct means that the strain rate does not vary with depth. Horizontal and vertical scales are indicative. The crust mantle transition occurs at ~30 km depth.

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 355 lithosphere for different geotherms. Although the constant strain rate assumption is generally incorrect in actively deforming zones as 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/m 2 ), the 350 C isotherm is close to Moho depth, causing the crust to be mostly brittle (Fig. 3a). Low temperature in the subcrustal mantle should imply very high viscosities (Strehlau and Meissner, 1987). However, stress controlled plastic behaviour is likely to limit the maximum sustainable stress to about 600 MPa (Tsenn and Carter, 1987). At high heat flow (80 mw/m 2 ), the 350 C isotherm is shallow (7 15 km) and the Moho temperature is high (N700 C), leading to a strength profile mostly controlled by upper crustal friction and middle crust viscosity (Fig. 3b). However, this model has to be modified for a fault zone, with the constrain of low effective friction within the seismogenic layer (Zoback et al., 1987; Wang et al., 1995; Hassani et al., 1997)(Fig. 3c). In addition, strain rate weakening and metamorphic reactions may alter the deformation processes in the middle crust, and some authors have proposed that a drastic strength reduction occurs at the brittle ductile transition (Gueydan et al., 2001). Lithospheric strength profiles are useful to study the link between geophysical variables such as P and T and the effective rheology of the lithosphere. However, they represent the maximum sustainable stress (yield stress) of the lithosphere for a given strain rate, not the actual lithospheric stress. As an example, I consider the behaviour of the northern SAF as modelled by Provost and Chéry (Fig. 4a,b). In their study, the authors account with both strike-slip and shortening between the Pacific plate and the Sierra Nevada. The small shortening strain, which is accommodated by plastic strain inside the crust and also by dip-slip fault motion, is not considered in the present paper to keep a simple mechanical analysis. In the case of pure strike-slip motion, the stress magnitude along faults depends on the effective friction coefficient of the seismogenic zone and the viscosity of the middle crust. Because both frictional and viscous parameters in fault zones have low values, this limits the stress of the surrounding lithosphere that cannot Fig. 4. a) Strain rate invariant associated with parallel strike-slip faults of northern California (adapted from Provost and Chéry, 2006); major faults are modelled with intrinsically weak material and display high strain rate (Maa = Maacama fault; BS = Bartlett Springs fault); b) long term velocity in the fault-parallel direction at the surface (solid line) and at 25 km depth (dashed line); c) fault-parallel stress profiles for the fault zone located on the SAF and for the crust. The profile to the left corresponds to the maximum fault strength for this slip rate. The profile to the right inside the crust remains much below the maximum sustainable stress corresponding to the dashed curve. Stress integrals with depth must be equal on the two profile to ensure stress equilibrium.

356 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 reach its maximum value. Therefore, the lithosphere does not deform and the viscous stress is zero due to the lack of strain. In this case, the lithospheric stress of this zone corresponds to elastic strain accumulation without involving frictional processes (right profile on Fig. 4c). Let us now consider the behaviour of such a long term model during the seismic cycle. In contrast with the SB73 model which is driven directly by imposing fault motion on the fault plane, this long term model is kinematically driven from the sides of the block at the prescribed side plate velocities. In this context, coseismic motion occurs in response to a step in effective fault friction, initially proposed by Brace and Byerlee (1966). Fault rupture during large continental earthquakes takes place between the surface and 10 20 km depth and produces elastic strain in the fault vicinity. Postseismic motion occurs minutes to years following the earthquake as a result of a variety of processes like afterslip of the deep fault plane, viscoelastic processes in the middle crust and the mantle and poroelastic effects in the crust. If I ignore the poroelastic effect for this discussion, both afterslip and viscoelastic strain act to release stress buildup at depth and reload the fault plane above. Due to the thermal stratification of the lithosphere, a large viscosity spectrum [10 17 10 19 Pa s] is likely to control the strain following large earthquakes as shown by postseismic modelling studies (Freed et al., 2006). Once this stress transfer is complete, a steady interseismic strain build up occurs in response to plate motion. A chief difference of this phase with respect to the long term behaviour is that the fault is locked, implying that the whole seismogenic layer behaves elastically or viscoelastically with viscosities higher than 10 21 Pa s. At depth, a low viscosity stress occurs in response to interseismic strain beneath the seismogenic zone. If one attempts to interpret the interseismic strain across the SAFS with the model of Fig. 4c, it becomes clear that the surface strain has to be influenced by the thickness of the elastic and viscoelastic layers. As suggested by Fig. 5, thin parts of the layer should display strain accumulation while thick parts should not accumulate much strain. Although this two-layer model has been invoked for decades to explain postseismic strain (Nur and Mavko, 1974; Pollitz et al., 2000) (Kenner and Segall, 1999), it is less used to explain the interseismic strain distribution across fault systems (Bourne et al., 1998; Cohen and Darby, 2003; Schmalzle et al., 2006). In such a case, elastic strength is likely to depend on the product of the thickness of the seismogenic zone and the average shear modulus of this layer as detailed later. The viscoelastic strength determination follows the same scheme and depends on the product of the thickness of the viscous layer and its average viscosity. Two lines of evidence suggest that the viscous contribution to the lithospheric strength is relatively small. First, a large stress reduction with depth occurs in only a few km only due to the high sensitivity of power law flow to temperature. For example, a typical granite-type power law (Kirby, 1985) loaded at a strain rate of 10 14 s 1 implies a deviatoric stress of 100 MPa at 350 C and only 1 10 MPa at 450 C. Assuming a surface thermal gradient of 20 C/km, this suggests that the thickness of the layer hosting significant viscous stress should not exceed 5 10 km. Second, effective viscosities determined in the crust and Fig. 5. Interseismic stress accumulation and corresponding velocity. a) Crosssection of a lithosphere with lateral variation along x-axis of its geodetic elastic thickness T g. The position of the 350 C isotherm is given by the dashed line and marks the upper limit of the viscoelastic zone. A low deviatoric stress is assumed to occur below corresponding to effective viscosities lower than 10 19 10 20 Pa s. A fault-parallel stress increase Δτ xy occurring a time interval Δt (see Eq. (2)) is shown on two profiles. For each profile, the stress increase is assumed to be constant with depth accordingly and is materialized with a solid rectangle. Due to stress equilibrium the stress increase integrals on the two profiles are equal. b) Corresponding interseismic velocity field (y-axis component) across the lithosphere computed using Eq. (3). the mantle by postseismic modelling are on the order of 10 19 Pa s, 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 by geodetic tools at the Earth's surface may reflect the elastic strain accumulation of the seismogenic zone (i.e., grossly the part of the crust above 350 C and possibly the cold uppermost mantle) and the viscoelastic behaviour of a thin zone below the seismogenic layer in the crust and possibly in the mantle. To significantly contribute to interseismic stress accumulation, the corresponding viscosities must be higher than 10 21 Pa s. This way to interpret interseismic strain has been proposed for the SAFS and for the Alpine fault of New Zealand (Bourne et al., 1998) but assuming that viscous strain in the lower crust and in the mantle drives and controls the entire fault system. Rather, I propose that interseismic strain reflects lateral variations in the thickness and elastic modulus of a lithospheric stress guide. This view is obviously close to the elastic plate model used to explain the flexural behaviour of the lithosphere (Watts, 2001). The main difference comes from the kind of forces applied to the plate. Classical plate theory aims to explain the relationship between vertical motion (plate bending) and plate thickness, while the concept I propose here considers the relation between horizontal strain and plate thickness. Because this thickness is evaluated by analysing horizontal geodetic strain, I name it geodetic elastic

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 357 thickness (GET) or T g in order to differentiate it to the flexural elastic thickness (T e ) provided by plate bending analysis. 3. A simple model between strain rate and elastic thickness To build a simple model explaining the relation between the interseismic strain and the lithospheric strength, I assume here that the crust and the mantle are elastic for temperature lower than respectively 350 C and 750 C. Also, I neglect viscoelastic effects that is to say that the stress at temperature greater than 350 C and 750 C for the crust and the mantle respectively is negligible. The contribution of interseismic strain corresponds therefore to the deformation of a plate having an effective thickness T g. as presented in Fig. 5a. Because the mechanical system obeys stress equilibrium and assuming that only the stress component τ xy contributes to the total force, stress increase Δτ xy during a time Δt on two profiles of thickness T 1 g and T 2 g satisfies the relation: space derivative of the velocity field, I compute the strain rate using a least square adjustment of the interseismic velocity over a moving window of 20 km minimum half-width. Therefore, most of the long and short wavelength features of the profiles are preserved (Fig. 6a) and interseismic strain rate is given by the slope of the least square adjustment (Fig. 6b). The given formal error on the strain rate corresponds to the slope uncertainty associated to the least square adjustment. This leads to obtain a small error when all the points are aligned in the moving window, therefore corresponding to a constant strain rate model. A more complete formulation that should also include individual RMS associated to the data has not been developed here. The computation of the elastic thickness using Eq. (3) requires a priori information on T g because a trivial solution is given by T g =0, corresponding to the lack of information on the shear force change of Eq. (2). Because earthquake seismicity near the SAF does not occur below 12 15 km, I assume a minimum Z T 1 g 0 Z T 2 Ds 1 xy ðþdz z ¼ g 0 Ds 2 xyðþdz z ¼ DF ð2þ where ΔF represents the shear force change applied to the lithosphere. I assume a linear shear stress shear strain relation given by Δτ xy =G Δε xy where G is the average shear modulus on the layer. Assuming that Δε xy is constant along a vertical profile and dividing the previous equation by Δt leads to: P Gx ðþt g ðþ x e : xy ðþ¼c x where P G is the average shear modulus on the profile and C is a constant, meaning that fault-parallel strain rate : e xy is inversely proportional to the integrated elastic strength P G T g. With the simplifying assumptions above, knowledge of the interseismic strain rate should teach us how elastic thickness may vary. I test this hypothesis on three profiles across the SAFS. 4. Analysis of velocity profiles crossing the San Andreas fault ð3þ GPS profiles shown on Fig. 2 cross the SAFS north of San Francisco Bay (North Bay profile, see Fig. 6), in the Carrizo plain south of Parkfield (Carrizo profile, see Fig. 7) and close to the Salton Sea south of Los Angeles (Salton profile, see Fig. 8). Faults are straight around these three locations so the deformation is nearly two-dimensional. Also, no large earthquakes have affected the SAF since the 1906 San Francisco earthquake for the North Bay profile and the 1857 Ft. Tejon earthquake for the Carrizo profile. Also, there is no evidence that the 1857 rupture extended down into the Salton Trough region. Therefore, the postseismic signal on the three profiles is expected to be small compared to the interseismic strain. I use the public data available on USGS and SCEC web sites corresponding to GPS surface velocities with respect to the North American plate. Because the RMS associated to the data are of the order 1 mm/yr, the direct derivative of the linear velocity field interpolation with respect to the profile direction is meaningless and not of a practical use in our case. In order to obtain a smooth Fig. 6. Interpretation of the GPS velocity field on a profile perpendicular to the northern SAFS. a) Fault-parallel GPS interseismic velocity fields (see Fig. 2 for location). A continuous version of the velocity measurements is given by the red curve. The RMS (mm/yr) of the adjustment between the curve and the data is given by the blue solid line with the scale to the right; b) fault-parallel horizontal strain rate corresponding to the continuous velocity field slope c) elastic thickness across the 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 are represented by red lines (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

358 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 elastic thickness on each profile of 13 km. Elastic thickness distribution is computed according to this value, also assuming that the average shear modulus does not vary along the profile. If computations were done with a different minimum elastic thickness, it would have been affected elastic thickness on the profile only by a multiplicative constant (see Eq. (3)). From north to south, interseismic strain rate patterns and their corresponding elastic thickness are markedly different. Along the North Bay profile, strain rate displays a marked asymmetry with a maximum along the Pacific plate to the west and a slow decrease towards the east up to the Central Valley and the Sierra Nevada. Consequently, the elastic thickness is higher close to the Pacific plate (40 70 km) and gently decreases (13 to 20 km) across the Coast Range from the SAF to the Green Valley fault. Because of the small strain rate of the Central Valley, the inverse relation between strain rate and thickness leads to an elastic thickness larger than 100 km (Fig. 6c). Interseismic velocity in central California leads to a different interseismic strain variation (Fig. 7b). Strain rate gently increases from southwest (the Pacific coast) to northeast up to a maximum 10 km east to the SAF trace as noticed by previous work (Schmalzle et al., 2006). Despite a limited GPS data set in the Central Valley and Sierra Nevada, the strain rate 20 km east to the SAF is virtually zero according to other geodetic studies of the Sierra Nevada block (Dixon et al., 2000). According to this strain Fig. 8. Same as for Fig. 6 for the southern SAFS (see Fig. 2 for location). variation, elastic thickness gradually decreases from values higher than 80 km to 13 km across the western Coast Ranges (Fig. 7c), and jumps back to large values east to the SAF in the Central Valley. The southern California profile near the Salton Sea reveals a reversed strain pattern compared to the one obtained in the North Bay area. Elastic thickness progressively decreases from the Pacific coast when crossing the Elsinore fault, reaches its minimum value between the San Jacinto and SAF and increases towards ~80 km on the north American plate to the east (Fig. 8c). 5. Finite Element model of the interseismic strain Fig. 7. Same as for Fig. 6 for the central SAFS (see Fig. 2 for location). The simple formulation used to compute the elastic thickness of the SAFS provides a first order relation between interseismic strain and plate thickness. However, stress components are likely to vary vertically within the plate, implying that an accurate strain solution requires solving a more complete stress balance equation. Also, the assumption of constant elastic modulus through the entire lithosphere is questionable, as vertical and lateral elastic properties variations of the lithosphere are known to be significant (Meissner, 1986). Involving these effects with solving stress balance equation with complex geometry needs to be done numerically. I design one experiment to determine if the simple concept developed above about the strain rate elastic thickness relation still holds for a finite thickness lithosphere. Because errors induced by the 1D model

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 359 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 : s xy 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 an interseismic 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. are likely to be higher when large geometrical variations occur, I use the example of the central SAF for which high thickness gradients are expected. The geometry of the model represents a cross-section of the elastic part of lithosphere perpendicular to the fault direction. The loading corresponds to a motion of the Pacific plate with respect to a fixed Sierra Nevada at a rate of 34 mm/yr parallel to the fault direction. Assuming no velocity variation in the direction parallel to the fault direction (y), non-zero stress components are τ xy and τ yz. Both horizontal and vertical bulk elastic variations are taken into account (Fig. 9a). I incorporate Young's modulus increase with depth as observed worldwide in continents (Meissner, 1986). Based on a seismic velocity model of the Parkfield area (Eberhart- Philips and Michael, 1993) I also account for a lateral decrease of Young's modulus in a 20 km width zone east of the SAF. I also adjust elastic thickness along the Carrizo profile. Elastic thickness of the Great Valley Sierra Nevada is set to 200 km in order to account for both its low geodetic deformation (Dixon et al., 2000) and its high flexural rigidity (Kennelly and Chase, 1989). Elastic thickness of the Pacific plate west to the SAF is set to 40 km to make it compatible with currently estimated elastic thickness (Watts, 2001). The interseismic profile velocity for the model of Fig. 9b displays a data-model RMS of 1.77 mm/yr (Fig. 10). Due to intrinsic velocity errors of about 1 mm/yr on the Carrizo profile, the smooth interseismic curve (1D model of Fig. 6) fits velocity data with a RMS of 1.34 mm/yr. The data fit of the FEM model is not as good as the one provided by the 1D model. This is understandable as the 1D strain model is obtained by direct smoothing of the discrete velocity data, therefore representing the best possible data fit. Despite its lower accuracy, I argue that the data adjustment provided by the FEM has a greater physical meaning than the 1D model presented before. This opinion is based on three factors. First, it corresponds to a finite thickness lithosphere, implying that the hypothesis of having : s xy constant Fig. 10. Interseismic velocity field provided by the Finite Element model (thick black line) compared with the smooth data fit (dashed thin line) and with the thick lithosphere model (purple curve). Discrete GPS velocity values across the central SAFS are given by the solid black circles.

360 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 on a vertical profile is no longer needed. Clearly, this 1D assumption is violated when high thickness gradients are present as shown by Fig. 9b. Second, the FEM allows us to include our best a priori knowledge of the elastic properties of the lithosphere based on seismology. Interestingly, a reliable estimate of bulk elastic properties requires the thickness only to be adjusted. Third, interseismic strain provided by the FEM results from a trial-and-error procedure. It is likely that a suitable inversion technique based on a grid search for an optimal elastic thickness would lead to a fit as good as the one provided by the 1D smooth fit. In comparison to the 1D and the FEM data fit, the adjustment of fault slip rate and locking depth of the SB73 model leads to a data fit of 2.25 mm/yr. This is mostly because the asymmetry of the elastic profile resulting from the variable elastic thickness model fits the GPS data better compared to the symmetric solution provided by the SB73 model. However, the fit of the thick lithosphere model is quite acceptable given the small number of free parameters and may be still improved if elastic modulus contrast across the SAF is taken into account. 6. Discussion Choosing among different mechanical models of interseismic strain on the sole basis of geodetic data fitting has been shown to be meaningless because of the non-uniqueness of the problem even in the case of a single fault (Savage, 1990). In other words, a bad data fit can allow us to discard a model but a good data fit is not proof of a model's relevance. Therefore, in chasing among possible models I need to consider the problem from a broader point of view than the one of finding the best data. Rather, it requires consideration of each model's relevance from a geophysical, rheological and geodynamical point of view. Having in mind the current view of the thermomechanic state of the lithosphere beneath the SAF (Lachenbruch and Sass, 1980; Furlong, 1993; Chéry et al., 2001), the use of a thick elastic model for the SAFS seems at odds to this knowledge. For example, the northern SAFS is very juvenile as it results from the northward progression of the Mendocino triple junction (Furlong, 1984). There, the disappearance of the subducting plate is thought to have opened an asthenospheric window in contact with the upper plate. As a consequence, a sharp contrast is likely to exists between a thin lithosphere (10 20 km) surrounding the SAFS and a thick lithosphere (50 100 km) to the east in the Great Valley and in the Sierra Nevada. In other words, an interpretation of the whole fault system using a thick elastic lithosphere is therefore unlikely because of the presence of a hot low viscosity zone in the upper mantle in the region of the slab window. By contrast, the variable elastic thickness model predict a low rigidity of the SAFS, which is compatible with a shallow low viscosity zone. As already noted (Le Pichon et al., 2005; Schmalzle et al., 2006), the data fit provided by the Savage and Burford model is limited by the asymmetrical character of the velocity field. In the simple case of a single vertical strike-slip fault, one expects a perfect symmetry with respect to the fault axis. This is not always the case, as shown by Le Pichon et al. for different strike-slip faults. A plausible explanation is that lateral variations of elastic properties of a thick lithosphere cause a symmetry break (Rybicki and Kasahara, 1977). Indeed, large lateral variations of bulk rigidity are likely to occur and could explain the asymmetry of the deformation. However, a bulk rigidity contrast of 10 that is required to explain the strain asymmetry around the Sumatra fault (Le Pichon et al., 2005) probably exceeds known variations of shear modulus within the crust. An alternative explanation for strain asymmetry is a variation of the elastic thickness in conjunction with a bulk rigidity contrast (Melbourne and Helmberger, 2001; Cohen and Darby, 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, SB73 and variable thickness models both provide a good fit to the interseismic velocity field. In contrast, different behaviours can be expected between these models when fault slip rates are searched. In the case of the SB73 model, the long term fault slip rate is equal to the far-field velocity (the differential plate velocity). The long term slip rate for the variable thickness model is less straightforward to define because of the remote fault drive. As discussed in the introduction, the effective fault friction should be considered here. If the fault friction is low, the fault will slip at the differential plate velocity because the lithosphere does not deform anelastically. If the fault friction is high, two cases have to be considered. If one assumes that the lithosphere always behaves elastically (meaning that it can sustain arbitrarily high strain) the fault slip rate must still be equal to the differential plate velocity. However, both in-situ and laboratory rock strength estimates indicate that the differential stress of the seismogenic lithosphere is limited by a friction coefficient of [0.6 0.8] (Townend and Zoback, 2000). Therefore, a high fault friction is likely to pervasively deform the lithosphere around, especially where the plate is the thinnest. For this case only, the fault slip rate is expected to be different to the differential plate velocity. Considering that large faults separating thick lithospheric plates are often thought to sustain low shear stress, the fault slip rate determined by the variable thickness model must therefore be in fair agreement with the best fit SB73 model. 6.2. Slip rate computation of a parallel fault system From a mechanical viewpoint (see Appendix for a discussion), the SB73 model mimics the mechanical behaviour of two thick lithospheric blocks separated by a low strength fault zone. Interestingly, most fault slip rates inferred from SB73 and block models analysis around continental plate boundaries are in good agreement with the corresponding geologic slip rates (Reilinger et al., 2006). Therefore, it is logical to conclude that the geodetic fault slip determination using the SB73 model is valid only if it corresponds to a setting of a weak fault between two strong plates. However, many other fault settings occur in nature as multiple fault system, faults embedded in wide orogens or high plateaus. To discuss this point in the framework of the SAFS, I attempt here to explain how the variable thickness model behaves when two faults or more are embedded (Fig. 11). This model is directly adapted from the 1-fault model of Fig. A1c but

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 361 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 width h f i and of a viscous stress occurring on a width h v i. adding another frictional discontinuity and a viscoelastic zone according to Fig. 4c. As shown by the previous mechanical modeling, frictional and viscous fault strength both control slip rate distribution in a multiple fault system (Bird and Baumgardner, 1984; Provost and Chéry, 2006). Using the example of Fig. 11 and using a 1D balance assumption similar of Eq. (2), it can easily be shown that the effective strength of both faults are equal. Also, the strength of the fault i is equal to the sum of the frictional strength F f i and of the viscous strength F v i. For each fault, the frictional strength is equal to the stress integral along the frictional thickness h f i. Assuming a frictional shear stress constant with depth leads the frictional strength to be equal to: F i f ¼ s i h i f The viscous strength is equal to the integral of the viscous stress along the viscous layer which has a thickness h v i. This integral depends on the average strain rate inside this layer around the fault plane and is also affected by the large viscosity variations that likely occur with depth. Assuming that the viscous strain occurs on an average width w, the viscous strength is therefore defined by the relation ð4þ This relation is represented on Fig. 12 for different values of h 1 s 1 in the case or h 2 Nh 1. As the respective contribution of frictional and viscous fault strength is still a matter of debate, this formulation is well suited as it does not require an a-priori choice. In order to see how southern and northern SAFS behave with respect to Eq. (8), let us assume that viscous to frictional ratio is the same for each fault. Thus the ratio between frictional thicknesses is equal to the ratio given by geodetic elastic thicknesses. Slip rate and frictional thickness ratios are given in Table 1 and plotted in Fig. 12. Using preferred values for slip rates, only pairs including the Elsinore fault display θ 1 s 1 values larger than 3. For other pairs, elastic thickness and slip rate ratios match curves computed with between 1/3 and 3, suggesting that viscous and frictional strengths may have an equal influence on the SAFS. In order to see how this parameter compares with an a priori calculation, I use the following set of parameter: a viscosity of 10 19 Pa s; a fault zone width of 1 km; a ratio h i vh i f=0.25; a frictional shear stress of 10 MPa. In such a F i v ¼ gi w hi v si ð5þ where is the fault viscosity η i divided by w and s i the fault slip rate. The total fault strength is equal to F i ¼ s i 1 þ h i s i h i f ð6þ with h i ¼ gi w hi m s i h i f ð7þ Therefore, θ i represents the viscous component of the fault strength. Assuming for purposes of discussion that both θ i and τ i are equal for both faults, the slip rate ratio is given by: " # 1 h 1 s 1 s 2 1 h 1 f ¼ max 0; 1 þ s1 h 1 s 1 h 2 f ð8þ Fig. 12. Normalized relation between slip rate and mechanical thickness using Eq. (8) for different values of the viscosity parameter (red lines). Black circles give the relation between slip rate and thickness as computed in Table 1. Black lines represents slip rate uncertainties.

362 J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 Table 1 Geodetic elastic thickness (provided by this study) and known fault slip rates for northern (lines 1 to 3) and southern (lines 4 to 6) SAFS (Shen-Tu, Holt et al., 1999) Fault1 fault2 T g 1 (km) T g 2 (km) s 1 (mm/yr) s 2 (mm/yr) h f 1 /h f 2 s 2 /s 1 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 indicate min/max estimates. SAF = San Andreas fault; RC = Rodgers Creek fault; GV = Green Valley fault; SJ = San Jacinto fault; ELS = Elsinore fault. case, θ i =2.5 10 8 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 as the fault zone viscosity and width, the agreement between this a-priori computation and the value deduced from the elastic thickness and slip rate plot is encouraging. This analysis suggests that several geometrical and rheological parameters control fault strength and therefore geological fault slip rates. Frictional strength is controlled by the fault friction and the seismogenic thickness, while viscous strength is controlled by the viscosity and the size of the viscous domain. Any variation of these rheological parameters on a fault will cause the slip rate to vary as already demonstrated by some studies (Roy and Royden, 2000; Provost and Chéry, 2006). The present analysis suggests a complex link between geologic slip rate and interseismic strain if more than one fault are active. Despite that the SB73 model is based on a direct and causal relation between interseismic strain and fault slip rate, these two parameters may only be positively correlated in nature. In order to properly interpret their relation one must understand their causal relation. Indeed, fault slip rate chiefly depends on effective fault friction, fault zone viscosity and the mechanical thickness of all these process zones. Because a greater thickness induces a strength increase if other rheological parameters remains unchanged, such an increase in thickness causes both the slip rate and the interseismic strain to decrease. But a change of effective fault friction modifies fault slip rate without changing the interseismic strain. Therefore, the apparent correlation between geologic slip rate and interseismic strain may only indicate that the elastic thickness plays a dominant role in controlling fault strength. 6.3. A global relation between elastic thickness and interseismic strain? The goal of this paper is to study the interseismic strain rate distribution across the SAFS and its relation to lateral rigidity variations. If such a relation holds for this plate boundary, perhaps a similar relation can be expected for other deformation areas such as subduction zones, mountain belts, continental or oceanic rifts. If a relation between interseismic strain and elastic thickness is assumed, then plate boundaries that displays high strain rate concentration would need to be interpreted like thin elastic zones. Is this mechanically understandable? Current knowledge of the SAFS suggests that a thin elastic lithosphere indicates a locally weakened plate. This kind of weakness can be thermally induced, as high temperature gradients reduce the thickness of both brittle and viscous layers. This is likely to occur in mountain belts, rifts or extending plateaus as shown by geophysical measurements and mechanical modelling (Gaudemer et al., 1988; Buck 1991; Cattin et al., 2001). Also, a large contribution to the lithospheric weakness could be related to low strength faults that decouple (in a stress meaning) adjacent plates. Subduction faults and large intracontinental faults are probably 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 for continental and oceanic plates using the idea that vertical loads applied to the lithosphere produce vertical motions by plate bending (Watts, 2001). Using stress equilibrium of the plate and isostatic assumption, knowledge of the horizontal distribution of the loading function (topography, internal loads, mantle buoyancy, glaciations, etc) permits determination of the flexural rigidity that best explains topographic and gravimetric signals. Knowledge of the elastic parameters of the lithosphere permits conversion of flexural rigidity to equivalent plate thickness T e. However, one needs to be careful when plate thickness are obtained with different kinds of loads. Because the model assumes an elastic plate over an inviscid fluid, the determination of the flexural rigidity in nature is correctly done if the load is applied for a time scale long enough to allow a complete stress relaxation at depth. Short time scale loading does not allow such stress relaxation, therefore leading to larger values of equivalent plate thickness. For example, mechanical modelling of the rapid filling of the lake Mead in the Basin and Range leads to a plate thickness estimate of 20 30 km (Kaufmann and Amelung, 1995) much higher than the 5 15 km thickness found using the surface topography as a loading function (Lowrie and Smith, 1995). On both oceanic and continental plates, a clear correlation is found between elastic thickness and the thermal state of the lithosphere if a long term loading is considered. The 500 550 C isotherm matches the lower limit of the elastic plate in oceans, while no strong correlation has been found between effective elastic thickness and some particular isotherm for continental crust (Watts, 2001). The current interpretation for oceanic lithosphere is that most of the elastic stress is stored between the surface and the 500 550 C isotherm. In the case where elastic stresses reach the frictional strength of the lithosphere (generally due to high plate curvature as it occurs in subduction zones), an elastoplastic plate bending model has to be used (Judge and Mc Nutt, 1991) to account for a reduced elastic thickness. The situation is more complex in a moderately cold continental lithosphere for which a large part of the strength is probably stored in the uppermost mantle. As I show using the example of the SAF, the use of horizontal GPS velocity gradients at the Earth's surface through a stress equilibrium principle is another way to estimate effective plate thickness using its transverse rigidity as a parameter to invert. However, such a use of GPS velocities relies on two key assumptions. First, the strain must really represent the interseismic stage during which faults are locked. The second assumption is that the plate strain is the result

J. Chéry / Earth and Planetary Science Letters 269 (2008) 352 365 363 of stress equilibrium inside the plate without coupling with the mantle other than hydrostatic forces. For example, horizontal forces applied at the base of the elastic plate such as deep convection 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 strain and plate thickness may therefore breakdown, the applicability of this theory on continents is worldwide. Indeed, the rapid growth of GPS and InSAR mapping is already providing a dense and accurate velocity fields in many active region. Suitable interpolation of this velocity field makes possible the strain rate computation (Kreemer et al., 2003). Using an inversion procedure, this would potentially allow computation of rigidity maps on continents. 7. Conclusion The idea that interseismic strain is linked to variable elastic plate thickness is markedly different than the usual geodetic data interpretation using the block model driven by fault slip at depth. I have argued that the former model is rheologically more plausible as it is directly linked to brittle and ductile lithosphere properties. In such a model, faults do not play a direct role during the interseismic phase because they are locked. Interseismic strain is therefore chiefly controlled by elastic plate properties (mostly bulk elastic modulus and plate thickness). Two main implications can be drawn: Appendix A The variable elastic thickness model has markedly different seismic hazard implications compared to those deduced from SB73 and block models. For the latter, fault slip rates are a free parameter to adjust. For the former, elastic thickness can be computed, but no direct link can be made with the long term fault slip rate. This lack of information is clearly related to the model's assumption: interseismic strain is interpreted on the time scale during which geodetic measurements are made. By contrast with the SB73 model, no hypothesis is made about the relation between short term interseismic strain and long term slip rate. Does this mean that interseismic strain cannot be used to predict fault slip rate? To discuss this important issue, let us consider again the formal differences between these two models. The most significant is the way to apply the boundary conditions to the model. In the case of SB73 model, the differential velocity is applied on the deep part of the fault (near field boundary condition, see Fig. A1a). In such a case, the fault slip rate is a kinematical parameter and has no relation to the fault strength. In the case of the variable elastic thickness model, the velocity boundary condition is remotely applied according to differential plate motion. Because the source of plate motion is thought to be controlled by large-scale body forces, this way to setup the boundary condition is probably wiser than a local drive. Let us the use of the block model to infer fault slip rate must be restricted to single faults embedded in high rigidity domains like for example the central north Anatolian fault or the central SAF. Incorrect slip rates are likely to be inferred if this condition is not fulfilled. This case occurs when multiple faults are present as in southern and northern California and suggests that such geodetically based fault slip rates could be unreliable. In a simple tectonic setting such as pure strike-slip faulting, high interseismic strain can be interpreted in the same way as a low rigidity (small elastic thickness) zone, while low strain would correspond to a higher rigidity (large thickness). If this 1D analysis can be extended to 2D based on stress equilibrium principle, the variable plate rigidity model could be applied to compute elastic thickness on continents with a suitable inversion procedure using interseismic strain as input data. Acknowledgements This paper benefited from several discussions with my colleagues. I particularly thank Xavier Le Pichon who forced me to clarify my interpretations. Philippe Vernant read an early version of the paper. Careful reviews of Wayne Thatcher and of two anonymous reviewers allowed to significantly improve the manuscript. I thank also Jessica Murray for helping me in gathering USGS geodetic data. SCEC and USGS are acknowledged for providing geodetic data of high quality. Generic Mapping Tool (GMT) software has been used to prepare most of the figures. Fig. A1. Differences and similarities between the thick lithosphere model and the variable thickness lithosphere model; a) thick elastic lithosphere model (SB73 model); b) modified SB73 model with remote boundary conditions c) variable thickness elastic lithosphere.