Slip rates and off-fault deformation in Southern California inferred from GPS data and models

JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 8, 6 66, doi:./jgrb.6, Slip rates and off-fault deformation in Southern California inferred from GPS data and models K. M. Johnson Received 9 December ; revised September ; accepted 7 September ; published October. [] In Southern California, fault slip rate estimates along the San Andreas fault (SAF) and Garlock fault from geodetically constrained kinematic models are systematically at the low end or lower than geologic slip rate estimates. The sum of geodetic model slip rates across the eastern California shear zone is higher than the geologic sum. However, the ranges of reported model and geologic slip rate estimates in the literature are sufficiently large that it remains unclear whether these apparent discrepancies are real or attributable to epistemic uncertainties in the two types of estimates. We further examine uncertainties in geodetically derived slip rate estimates on major faults in Southern California by conducting a suite of inversions with four kinematic models. Long-term rigid elastic block models constrained by the geologic slip rates cannot fit the present-day GPS-derived velocity field. Deforming (permanent off-fault strain) elastic block models and viscoelastic earthquake cycle block models constrained by geologic slip rates can fit the present-day GPS-derived velocity field with 8 % of the total geodetic moment rate occurring as distributed deformation off of the major faults. Models incorporating viscoelastic mantle flow predict systematically higher slip rates than purely elastic models on many of the major Southern California faults with ranges of (elastic/viscoelastic) 9 / 7 mm/yr for the Carrizo SAF segment, / mm/yr for the Mojave SAF segment, 7/8 mm/yr for the Coachella SAF segment, 9/ mm/yr for the San Jacinto fault, and / mm/yr for the western Garlock fault. Citation: Johnson, K. M. (), Slip rates and off-fault deformation in Southern California inferred from GPS data and models, J. Geophys. Res. Solid Earth, 8, 6 66, doi:./jgrb.6.. Introduction [] In Southern California, slip rates derived from some geodesy-constrained kinematic models disagree with slip rate estimates from geologic techniques along major fault segments. For example, slip rates along the Garlock fault and Mojave segment of the San Andreas fault (SAF) inferred from geologic data are a factor of. to higher than slip rates inferred from GPS observations by Becker et al. [], Meade and Hager [a], and Bird [9] (Figure ). Conversely, summed right-lateral slip rates across the southern Mojave section of the eastern California shear zone (ECSZ) derived from GPS measurements is at least a factor of two higher than the summed geologic slip rate across active faults in the region [e.g., Oskin et al., 8]. Figure compares geodetically inferred fault slip rates (Figure a) with geologically inferred (Figure b) rates for some of the major faults in Southern California. The Department of Geological Sciences, Indiana University, Bloomington, Indiana, USA. Corresponding author: K. M. Johnson, Department of Geological Sciences, Indiana University, E. th St., Bloomington, IN 7, USA. (kajjohns@indiana.edu). American Geophysical Union. All Rights Reserved. 69-9//./jgrb.6 6 quoted slip rate estimates are all from traditional elastic kinematic block models which assume either long-term rigid body motion or uniform straining of blocks of crust enclosed by bounding faults. Slip rates from long-term rigid block models are labeled in black [Becker et al., ; Meade and Hager, a; Loveless and Meade, ] in Figure a. Slip rate estimates from a long-term deforming block model by McCaffrey [] are labeled in red. The reported range in geologic slip rates is a compilation from various studies (Table ). [] There are logically three possible explanations for the apparent slip rate discrepancy: () geologic slip rate estimates are incorrect for some faults, () model slip rates constrained by geodetic data are incorrect for some faults, or () the estimates from both methods are correct but yield different rates because the present-day deformation rates recorded with geodetic data are different from the Quaternary Holocene rates in the geologic record. Option has been explored in the literature by a number of investigators. It has been suggested that fault slip rates may actually vary over the long term [Bennett et al., ] due to changes of earthquake recurrence intervals [Becker and Schmeling, 998], earthquake clustering [Rockwell et al., b; Freidrich et al., ], or structural interaction [Peltzer et al., ; Dolan et al., 7]. Oskin et al. [8], for instance, suggested that the currently elevated deformation

(a) black = deforming block model, red = rigid block models (b) geologic rates 6 o N o N o N -6 Carrizo SAF Garlock Sierra Nevada - 9-6 -. Mojave SAF Sierra Madre 9 Panamint Valley - Elsinore - Helendale - Camp Rock - - Death Valley. - San Bernardino Banning - San Jacinto -8 Ludlow Coachella SAF - 6 o N o N o N Carrizo SAF 9-7 Garlock Sierra Nevada - - - Mojave SAF Sierra Madre Panamint Valley - Helendale Elsinore.- Death Valley Camp Rock 7-8 - San Jacinto -6-6 San Bernardino Banning -7 - -7 Ludlow Coachella SAF - o N o W o W 9 o W 8 o W 7 o W 6 o W o W o N o W o W 9 o W 8 o W 7 o W 6 o W o W (c) SAF-Ca model slip rates (mm/yr) GF ECSZ SAF-Mo SAF-Co SJF SAF-SB - - Gelogic fault slip rates (mm/yr) Figure. (a) Compilation of strike-slip rate estimates from three rigid block models (red) [Becker et al., ; Meade and Hager, a; Loveless and Meade, ] and one deforming block model (black) [McCaffrey, ]. Summed strike-slip rates shown across Mojave ECSZ. (b) Compilation of ranges of geologic fault slip rate estimates (see Table ). Summed strike-slip rates shown across Mojave ECSZ. (c) From Chuang and Johnson []. Geologic fault slip rates versus model slip rates inferred from geodetic data. Blue bars are slip rate comparisons from Meade and Hager [a], and red bars are from viscoelastic model of Chuang and Johnson []. ECSZ: eastern California shear zone, GF: Garlock fault, SAF: San Andreas fault, Ca: Carrizo segment, Mo: Mojave segment, SB: San Bernardino segment, Co: Coachella segment, and SJF: San Gabriel fault. rates across the southern ECSZ is a recent transient resulting from weakening and accelerate creep along lower crustal shear zones in response to a recent clustering of earthquake activity. Option is beyond the scope of this paper, but sources of epistemic uncertainties in geologic slip rate estimates have been examined in the literature [e.g., Bird, 7; Gold et al., 9; Zechar and Frankel, 9]. [] In this paper, we further explore the degree to which slip rates inferred from geodetic data are model dependent and in turn, the possibility that inferred slip rates from geodetic data are sometimes incorrect. Two previous studies suggest that the source of the discrepancy between some geologically inferred and geodetically inferred fault slip rates may be attributed to assumptions in the models used to infer slip rates from GPS-derived velocities. McCaffrey [] showed that an elastic block model for Southern California that includes long-term, inelastic distortion of tectonic blocks can explain the GPS observations with slip rates that are in better agreement with the geologic estimates than the rigid block model estimates (Figure ). Chuang and Johnson [] and HearnE.H.etal.[] showed that a viscoelastic block model that incorporates relaxing viscous flow in the mantle yields higher slip rate estimates on the Mojave San Andreas and Garlock faults than purely elastic models. Figure c is from Chuang and Johnson [] and compares geologically inferred fault slip rates along major faults in Southern California with model fault slip rates using elastic and viscoelastic models. [] It is difficult to pinpoint the reason why the model predictions of fault slip rates in Southern California are variable from study to study, and therefore, it remains unclear how to reconcile the discrepancy between geodetically inferred and geologically inferred slip rates. Each published study makes different assumptions in the model 6

Table. Summary of Geologic Slip Rate Data and Bounds a Fault Name Selected Bounds (mm/yr) References Carrizo SAF 9 7 Sieh ( 7); Liu-Zeng (); Noriega (9 6) Mojave SAF Matmon ( /7); Weldon ( ) San Bernardino SAF 7 8 McGill8 ( 8); McGill (7 ) Banning Fault 7 Yule and Sieh [] Coachella SAF Behr et al. [] San Jacinto b Rockwell (9 ); Blisniuk (7 ) Western Garlock (LL) McGill et al. [9] Central Garlock 8 (LL) Clark ( 8); McGill9 ( ) Southern ECSZ faults c..,..,.6 Oskin et al. [8] Elsinore 6 Fletcher ( ); Millman ( 6); Rockwell ( ) Newport-Inglewood Lindvall and Rockwell [99] Palos Verdes..8 Stephenson (..8); McNeilan (.7 ) Hosgri.9. Hanson et al. [99] Sierra Madre (R) Crook et al. [987] Death Valley 6 Frankel-a; Frankel-b Panamint Valley. Oswald; Burchfiel a All rates are right-lateral except noted: LL = left-lateral, R = reverse, dip-slip rate. Sieh = Sieh and Jahns [98]; Noriega = Noriega et al. [6]; Liu-Zeng = Liu-Zeng et al. [6]; Matmon = Matmon et al. []; Weldon = Weldon et al. []; McGill8 = McGill et al. [8]; McGill = McGill et al. []; Rockwell = Rockwell et al. [99]; McGill9 = McGill and Sieh [99]; Clark = Clark [98]; Fletcher = Fletcher et al. []; Millman = Millman and Rockwell [986]; Rockwell = Rockwell et al. [a]; McNeilan = McNeilan et al. [996]; Stephenson = Stephenson et al. [99]; Oswald = Oswald and Wesnousky []; Burchfiel = Burchfieletal.[987]; Frankel-a = Frankel et al. [7a]; Frankel-b = Frankel et al. [7b]. b Studies suggest slip rate tapers to the south so we select lower bound from rate at northern end near Anza. c Slip rates assigned to Helendale, Camp Rock, and Ludlow, respectively. that relates geodetic observations to fault slip rates. For example, McCaffrey [] allows fault-locking depths to vary spatially in the model, allows for internal long-term distortion of fault-bounded blocks, and uses geologically inferred fault slip rates as constraints in models. In contrast, Becker et al. [] and Meade and Hager [a] assume uniform locking depths for faults, no long-term distortion of crustal blocks, and they do no use geologically inferred fault slip rates as constraints. Chuang and Johnson [] have also fixed-locking depths and allow no internal block strain, and they also consider viscous flow in the lower crust and mantle. The purpose of this paper is to systematically examine the extent to which the discrepancy between slip rates can be reconciled with () variable-locking depths, () offfault permanent strain, and () viscous relaxation in lower crust and mantle. [6] Our approach is to conduct constrained inversions of GPS-derived velocities for fault slip rates in Southern California using kinematic deformation models with fault slip rates bounded by the minimum and maximum reported geologic slip rate on each fault segment (Figure b).. Model Construction [7] We consider four different kinematic models in this study, all of which fall in the category of kinematic block models, in which crustal blocks are entirely enclosed by boundary faults. Other non-block kinematic models have been utilized in the literature to analyze geodetic data [Liu and Bird, 8]; (Y. Zeng and Z.-K. Shen, Fault network modeling of crustal deformation in California constrained using GPS and geologic observations, submitted to Tectonophysics, ), and we do not intend to discount the validity of these approaches. However, for the purpose of making direct model comparisons, we limit this study to kinematic block models. [8] In each of these block models, the present-day deformation field is expressed as the sum of a long-term 6 component that is steady in time, and a transient, interseismic component that results from elastic distortion due to locking of faults. The first model is a traditional elastic block model [e.g., McCaffrey, ; Meade and Loveless, 9] in which fault-bounded blocks rotate undeformed over the long term about Euler poles, and interseismic elastic strain is introduced with backwards slip on dislocations in an elastic half-space. We refer to this as the rigid block model. The second model is a deforming block model which is the same as the previous model except the long-term velocity field includes a component due to steady internal straining of the blocks [also see McCaffrey, ; Meade and Loveless, 9]. The other two models are versions of the plate-block model of Johnson and Fukuda [] in which long-term, unrecoverable deformation of the blocks because geometric incompatibility is estimated. We will use elastic and viscoelastic versions of the plate-block as has been previously described in Huang et al. [], Chuang and Johnson [], and Johnson and Fukuda []. We review the main ideas of the models here... Block Models [9] The construction of the steady state, long-term velocity field is illustrated in Figures a c. Rigid body rotations of blocks about spherical Euler poles describe horizontal (tangential) motions of blocks as in, for example, McCaffrey []andmeade and Loveless [9]. In an Earth-centered Cartesian coordinate system, the velocity at coordinate X on a spherical cap rotating on the surface of Earth about a Euler pole,, is the cross product r(x) where r(x) is the vector from the center of the earth to point X. Forn blocks we define =(,,..., n ). Writing the cross product in matrix form, the Euler poles are related linearly to steady state surface velocities through a matrix, G block, V block = G block. ()

Figure. Illustration of general block model construction. (a) Block geometry with faults in elastic plate overlying layered Maxwell viscoelastic substrate. (b d) Components of the steady state velocity field. Figure b is rigid body rotation about specified Euler poles. Figure c is velocity field parameterized with cubic polynomials. Figure d illustrates fault-normal cancelation in an elastic plate for the plate-block models only. Rigid block models adopt only Figure b as steady state field. Deforming block models include Figures b and c. Plate-block models include Figures b, c, and d. (e) Illustration of back slip above specified locking depth to cancel long-term slip rate (locking of faults). Some faults creep from the surface down to km depth. (f) Illustration of elementary earthquake cycle adopted for the viscoelastic cycle model. [] In practice the Euler poles are usually specified within a fixed plate reference frame. Backwards slip (back slip) and fault-normal motions are imposed above a specified locking depth to cancel the long-term discontinuity rate and represent a fully locked fault (Figure e) as first introduced by Savage and Prescott [978] and Savage [98]. Strike-slip component of back slip is imposed on vertical faults and strike-slip and dip-slip components are imposed on dipping faults. Fault-normal back slip is imposed only 66 on vertical faults. The contribution to surface velocities due to fault-parallel back slip and fault-normal tensile motion is computed using the solution for a rectangular dislocation in an elastic half-space [Okada, 99]. For faults with known surface creep, the upper km of the fault is assumed to creep at the long-term rate. [] To allow for permanent, long-term strain in the blocks, we further parameterize the horizontal velocity field in each block with cubic polynomials. The east and north

components of this part of the velocity field in a local Cartesian (x, y) coordinate system is written as V e,n strain = a e,nx + b e,n y + c e,n xy + d e,n x + e e,n y + f e,n x y + g e,n xy + h e,n x + i e,n y, () where e and n denote east and north components. Placing the east and north components in a vector, V strain, the steady state velocity field is the sum JOHNSON: SOUTHERN CALIFORNIA SLIP RATES V ss = V block + V strain. () If V strain =, we refer to this as a rigid block model. If V strain, we refer to this as a deforming block model. Letting C be a vector of coefficients in equation () and placing the polynomial terms in a matrix, G strain, V strain = G strain C. () The long-term, steady state velocity field can then be written as V ss = G block + G strain C = [G block G strain ] = G C ss. () C.. Plate-Block Models [] The plate-block model is similar in construction to the block models described above except that the model is modified to account for dip-slip motion on dipping faults and to remove fault-normal velocity discontinuities. One further distinguishing feature of the plate-block model is that the deformation field is computed in an elastic plate overlying a viscous substrate, hence the name plate-block model. The velocity field, V ss, introduces both fault-normal and fault-parallel velocity discontinuities across faults. The fault-parallel velocity discontinuity is the fault slip rate. We want the following slip conditions on faults: () the faultnormal component of velocity discontinuities across faults is zero, () the strike component of slip rate on faults is equal to the strike component of velocity discontinuity across faults resulting from the imposed velocity field, and () the dip component of slip on faults is V? /cos where V? is the fault trace perpendicular component of the horizontal velocity discontinuity across the fault trace and is the fault dip angle. The first condition guarantees that fault surfaces do not open or inter penetrate, and the third condition guarantees that the fault trace normal component of the horizontal velocity field is continuous across dipping faults. To satisfy condition, we cancel the fault-normal velocity discontinuity with steady tensile motion on dislocations in an elastic plate under gravitational restoring forces and overlying an inviscid substrate (no shear resistance on the bottom of plate). To satisfy condition, we add the contribution from steady dip slip on faults by imposing steady slip on dislocations in the same elastic plate model. See Johnson and Fukuda [] for more discussion of this. [] We model the interseismic perturbation to the steady state as periodic locking and unlocking of faults in the elastic plate overlying a Maxwell viscoelastic asthenosphere. If the relaxation time of the Maxwell half-space is longer than the longest repeat time of earthquakes on faults, then this model is effectively an elastic model with no time dependence of the surface velocity field. Periodic earthquakes on the locked section are modeled with an infinite sequence of 67 periodic slip events on the locked sections. The slip history of a section of a fault is illustrated in Figure f. Steady back slip is modeled with steady creep on a fault in an elastic plate overlying a Maxwell viscoelastic substrate. In this model we assume uniform slip on rectangular faults. Periodic earthquakes are modeled by imposing an infinite sequence of sudden slip events on the faults such that the coseismic slip divided by the earthquake recurrence time is equal to the long-term fault slip rate. As in Johnson and Fukuda [], creep at constant resistive shear stress is approximated below the locking depth (shallow creep at the surface is modeled at the long-term slip rate). The approximation for creep at constant resistive shear stress is given in Appendix A. In short, we assume steady, uniform creep rate on the deep-creeping extension of faults below the locking depth at zero instantaneous stressing rate. The stressing rate is computed at the center of the deep-creeping rectangular fault at the current observation time. The instantaneous creep rate is assumed to be the steady rate over the entire earthquake cycle. When the rapid postseismic phase of afterslip is over (which is assumed in this model), the instantaneous creep rate is lower than the long-term fault slip rate so some coseismic slip is assumed to occur on the deep-creeping part of the fault (perhaps better interpreted as deep coseismic slip plus rapid afterslip) to make up for the creep rate deficit... Mojave Postseismic Transients [] For this study we use the Southern California Earthquake Center (SCEC) Community Motion Map (CMM ) velocity field [Shen et al., ]. It appears that the CMM velocity field contains some postseismic signal in the Mojave region due to the 99 Landers and 999 Hector Mine earthquakes in the form of accelerated right-lateral motions. To avoid biasing fault slip rate estimates in this region with the postseismic signal, we attempt to remove the accelerated transient by modeling deep afterslip with buried dislocations extending from 7 km depth along the Camp Rock fault sections. The deep- accelerated afterslip rate is bounded between and mm/yr in all of the inversions. This is intended to be a simple kinematic treatment of a likely more complicated postseismic deformation processes consisting of a combination of deep fault creep and mantle flow [e.g., Freed and Bürgmann, ]... Moment Rates On and Off Faults [] For the purpose of interpreting the off-fault strain rates in this study, we use the formula of Savage and Simpson [997] to convert the block strain rates to moment rates. Surface horizontal principal strain rates, E and E,are computed on a regular array of grid cells. The average scalar moment accumulation rate within a grid cell of area, A, is PM o =HA max ( E, E, E + E ), (6) where = Pa is the elastic shear modulus, and H = km is the assumed seismogenic thickness of the crust. We report the total moment accumulation rate as the sum of the moment accumulation rates over all grid cells. This off-fault moment accumulation rate is compared with the total on-fault moment accumulation rate which is simply the sum P i Ps ia i over i fault patches with back slip rate, Ps and area, A.

Figure. (a) Community Motion Map version (CMM ). Velocities are in fixed North America reference frame. Some major faults shown in red. (b) White lines delineate block boundaries. Block labels: TR = Transverse Ranges and ECSZ- = eastern California shear zone blocks.. Inverse Methods [6] We conduct three types of inversions in this paper: () optimizations for the best fitting model parameters using bounded least squares techniques, () mixed linear/nonlinear Markov Chain Monte Carlo (MCMC) inversions for the posterior probability density functions of model parameters, and () fully nonlinear MCMC inversions for the posterior probability distributions of model parameters. The unknown model parameters are Euler poles (from which slip rates are computed), strain rate polynomial coefficients, locking depths, a scale factor for the data uncertainties, and the deep postseismic creep rate in the Mojave region. Let be an M L -dimensional vector of unknown Euler pole coordinates and let C be a M K -dimensional vector of coefficients of cubic polynomials (equation ). Let d gps be a N gps length vector of GPS-derived surface velocities with corresponding data covariance matrix, gps gps. Here we assume the scale factor gps is unknown. Let Ps mojave be a vector of deep postseismic creep rates representing post Landers and Hector Mine postseismic deformation. Also let L be a vector of locking depths of each of fault sections. For the full inversions for posterior probability distributions of model parameters, the target distribution is the posterior probability density function, p(, C, gps, L, Ps mojave d gps ). [7] We also adopt prior bounds on-fault slip rates, locking depth, and deep postseismic Landers/Hector Mine JOHNSON: SOUTHERN CALIFORNIA SLIP RATES 68 afterslip rate. The slip rate on the ith fault segment that is bounded by the jth and kth block is, from equation (),! s i = G i ss, (7) C C where G i ss is evaluated at the center of the ith fault segment. We incorporate the bound constraints using Bayes Theorem, p(, C, gps, L, Ps mojave d gps ) /p(d gps, C, gps, L, Ps mojave) p(, C)p(L)p(Ps mojave ), where p(d gps, C, gps, L, Ps mojave) is likelihood and the prior probability distributions are, if si within bounds, for all i =,..., p(, C) = (9), otherwise, < Li <, km, for all i =,..., p(l) = (), otherwise, < Ps i p(ps mojave )= mojave <, mm/yr, for all fault sections i, otherwise, () where there are fault segments in the model. [8] Many of the inversions in this paper are actually optimizations in which we seek the best fitting model parameters by maximizing p(, C, gps, L, Ps mojave d gps ). In these inversions we assume gps is fixed, and L is set to km depth for each fault section. In this case the optimization is linear because, C, andps mojave are linearly related to the observations and we may use a bounded least squares solver (Matlab s lsqlin). In the other inversions we seek the full posterior probability density function of the model parameters using the MCMC Metropolis-Hastings algorithm [e.g., Gamerman, 997; MacKay, ]. When one of gps or L is unknown, the data-parameter relationship is nonlinear. As discussed below, the MCMC inversions are either cast as fully nonlinear and we either use the standard Metropolis- Hastings method or the efficient mixed linear/nonlinear inversion described by Fukuda and Johnson []., 9, 7.9,., geologic bounds,,,,, k.,, 6,,,.,.8.,. 7, 8.,., 7, 6,.6, j, (8) mean (mm/yr) Figure. Upper and lower bounds on strike-slip rates assumed for all inversions in this study. Positive is right lateral. Bounds are taken from a literature compilation of ranges of geologic slip rate estimates (see Table ). Colors of lines show the median value of the range for each fault section. No upper and lower bounds are specified for dashed block boundaries because of lack of data.

norm al JOHNSON: SOUTHERN CALIFORNIA SLIP RATES North (km) strike slip rates rigid block, no bounds mm/yr rigid block, bounded slip rates 7 9 9 9 9 9 9 9 6 9-6 6 - - 6 6 8 - - 9 9 6 9 8 7 7-8 9 6 8 9 6 7 6 East (km) fault-normal rates mm/yr 8 6 6 7 8 8 6 7 8 9 7 6 6 6 6 9 8 8 Figure. Best fitting strike-slip and dip-slip rates for rigid block models. Rates are shown with color and numbers. Negative is left-lateral or reverse/contraction sense of slip. r ight lateral eft latera l l n reverse/contractio. Southern California Fault System [9] The major active faults of Southern California and the GPS velocity field (Southern California Earthquake Center (SCEC) Community Motion Map (CMM)) [Shen et al., ] are shown in Figure a. The block boundaries adopted for this study are shown in Figure b, and the three-dimensional block geometry is illustrated in Figure a. The geometry is based loosely on previous block model studies [McCaffrey, ; Chuang and Johnson, ]. The fault slip rate bounds based on geologic data are shown in Figure, and the references for these bounds are summarized in Table.. Southern California Inversions [] We conduct a suite of bounded inversions of GPS data to determine the model conditions required to explain the GPS velocity field with slip rates constrained within geologic bounds. As stated in the Introduction, we seek to systematically examine the extent to which the apparent geologic/geodetic slip rate discrepancies can be reconciled with () variable-locking depths, () off-fault permanent 69 strain, () viscous relaxation in lower crust and mantle. In section.. we use a rigid block model to examine the extent to which the assumption of long-term rigidity of blocks leads to the inferred slip rate discrepancies. In section.. we examine the extent to which assumptions about locking depth can lead to biased slip rate estimates. In section. we examine the influence allowing off-fault long-term strain in blocks on slip rate estimates. Finally, in section. we investigate the extent to which the slip rate discrepancies can be reconciled with the viscoelastic plate-block model... Rigid Block Models... Fixed-Locking Depths [] We first conduct an optimization for best fitting fault slip rates using the rigid block model. We conduct one inversion without bounds on slip rates and one inversion with the slip rate bounds illustrated in Figure. The locking depth for all fault segments is fixed at km depth following Meade and Hager [a]. [] Figure shows the strike-slip and fault-normal rates. The slip rates in the unbounded inversion are quite similar to results from rigid block inversions from previous

Figure 6. Residual velocities for rigid block models (data minus model). (a, b) Residual vectors shown only for residuals exceeding. Normalized chi-square, n, is the sum of squared weighted residuals divided by the number of observations. (c, d) Color points show magnitude of all residuals. studies [e.g., Becker et al., ; Meade and Hager, a; Loveless and Meade, ]. In Particular, the model strike-slip rates on the Garlock fault and the Mojave and San Bernardino segments of the San Andreas fault are below the lower geologic bound. Also the sum of slip rates across the Mojave ECSZ is higher than the summed upper geologic bound. The fit to the GPS data (Figure 6a) is reasonably good except for a systematic pattern of residual north-south shortening in the Transverse Ranges and NE/SW extension in the southern ECSZ. Large misfits near the Coachella SAF are likely a consequence of poorly modeled shallow surface creep which has been observed here [e.g., Lyons and Sandwell, ]. Large misfits near the Imperial fault and Brawley stepover region (East coordinates to, North coordinates to ) may indicate that the idealized block and fault geometry in this region is inadequate. [] The bounded inversion fit is poor (Figure 6b) with systematic misfits in nearly every block. The normalized chi-square for the bounded and unbounded inversions are 6 n =.6 and n =.9, respectively. Here normalized reduced chi-square is the sum of weighted squared residuals divided by the number of observations (weighted by formal uncertainties times gps ). In all reported chi-square values in this paper we use the value gps =.9, which is the maximum likelihood value obtained from the plate-block inversion described in section.., and thus, the reported normalized chi-square values reflect this inflated uncertainty on the GPS-derived velocities.... Estimated Locking Depths [] The locking depth in the previous inversion was fixed at km depth. However, it is well known from inversions of GPS data with -D, antiplane strain, infinitely long strike-slip fault models that locking depth and slip rate estimates are strongly correlated [e.g., Lindsey and Fialko, ]. So now we ask, is it possible that the discrepancy between the model slip rate estimates and the geologic rates be attributed to the uniform km locking depth assumed in the previous inversion? To examine this, we estimate locking depths and slip rates by

Figure 7. Results of inversions for locking depth and slip rate. (a) Slip rates and uncertainty. (b). Locking depth and uncertainty. Thick lines delineate fault sections with statistically significant locking depth estimates. (c) Joint posterior density distributions of slip rate and locking depth for selected fault sections. conducting a mixed linear/nonlinear estimation of the posterior distribution p(, C, L, gps Ps mojave, d gps ) / p(ps mojave, d gps, C, gps, L)p(L) = p(, C gps, L, Ps mojave, d gps ) p( gps, L Ps mojave, d gps )p(l), where the expression in the second line separates the linear and nonlinear terms as in Fukuda and Johnson []. The first distribution on the right-hand side in the second line is a Gaussian distribution for which the mean and standard deviation are computed using linear least square formulas. The second distribution on the right-hand side in the second line is estimated with the MCMC Metropolis- Hastings algorithm. To make use of this simplified mixed linear/nonlinear formulation, we removed the prior bounds on-fault slip rates (prior on and C) and fixed the Mojave postseismic deep creep rates at the values determined from the previous inversion. [] Figure 7 summarizes the MCMC joint inversion for fault slip rates and locking depth. The mean and twice the standard deviation of slip rate and locking depth are shown along with selected joint posterior distributions of slip rates and locking depths. The best fitting model with optimized locking depth does fit the GPS data better ( n =.8)than the uniform locking depth inversion ( n =.9). However, the estimated slip rates in the joint inversion are not significantly different from the optimal slip rates with uniform locking depth, apart from the Coachella SAF segment. Although there is a slight correlation between slip rate and locking depth for some of the fault sections (Carrizo, San Jacinto, and Elsinore), the correlation is not as strong as one would expect from -D inversions. The reason for the weaker correlation in block models is that the fault slip rates are not as freely varying in the block models as in the -D models because of the stringent block motion constraint on slip rates. Segments with locking depths that are deemed to be significant are shown with a heavier line in Figure 7b. The inversion favors deeper locking depths for the Carrizo and Mojave SAF and shallower locking depths for the Coachella SAF, generally consistent with previous studies [e.g., Savage and Lisowski, 998; Smith-Konter et al., ]. [6] We have repeated the inversion above but with bounds on slip rates. For brevity, we do not show the results here, but this inversion fits the data poorly with n =. 6

normal JOHNSON: SOUTHERN CALIFORNIA SLIP RATES North (km) East (km) b. d. fault-normal mm/yr residual (mm/yr) 8 6.. 6 strike slip rates mm/yr 8 right lateral left lateral reverse/contraction. Figure 8. Results for inversion with deforming block model. (a) Strike-slip rates denoted by color and numbers. (b) Dip-slip rates denoted by color and numbers. (c) Residual vectors shown only for residuals exceeding. Normalized chi-square, n, is the sum of squared weighted residuals divided by the number of observations. (d) Color points show magnitude of all residuals. and the estimates of locking depth are qualitatively similar to the above inversion result. This fit is slightly better than the bounded inversion with uniform km locking depth ( n =.6), but the large systematic misfits remain. We therefore conclude that the slip rate discrepancy cannot be resolved with rigid block models with variable-locking depth because the slip rates from the joint estimate of locking depth and slip rate are similar to the fixed-locking depth inversion and the bounded inversion with optimized locking depths is unable to fit the data well... Deforming Models [7] In the previous section we showed that the rigid block with slip rates bounded by geologic rates is not consistent with the GPS data. We now consider the effect of including distributed off-fault strain in the model. We first show the results of an inversion with the deforming block model, and then we show the results for an inversion using the plate-block model. In both of these models, parameterized block strain rates are calculated only within the Transverse Ranges block (TR) and the Mojave ECSZ blocks (ECSZ- ) because we found that these are the only blocks that are needed to deform internally to fit the data and because these blocks contain known active faults within the block interiors. 6 All of the following inversions in this paper implement prior bounds on slip rates.... Deforming Block Model [8] The slip rates and fit for the deforming block model are shown in Figure 8, and the estimated maximum shear strain rates and principal directions are shown in Figure 9. The fit ( n =.9) is substantially better than the bounded rigid block model ( n =.6) suggesting that the model slip rates can be reconciled with the geologic bounds if the Transverse Range and ECSZ blocks are allowed to deform internally. [9] The strain rate pattern (Figure 9) shows roughly N/S shortening in the Transverse Ranges, consistent with reverse faulting on E/W trending faults. The southern ECSZ shows NE/SW shortening and NW/SE extension, consistent with transtensional deformation in this region. The northern ECSZ region shows N/S to NW/SE shortening and E/W to NE/SW extension. [] Although this inversion shows that the geodetic data can be largely explained with a model that is consistent with the geologic slip rate bounds, Figure a shows that the strike-slip rates on a number of fault segments are equal to the upper or lower geologic bounds. The strike-slip rate on the Mojave and San Bernardino sections of the San Andreas

normal JOHNSON: SOUTHERN CALIFORNIA SLIP RATES North (km) East (km) nano-strain/yr Figure 9. Maximum shear strain rate and principal strain rate directions for deforming block model inversion. fault and the western Garlock are equal to the lower geologic bound. Strike-slip rates on many of the ECSZ faults are equal to the upper geologic bound. This suggests that the model favors rates different from the geologic rates on these faults.... Plate-Block Model [] Some of the fault-normal velocity discontinuities are nonnegligible in the bounded inversion with block strain rates (Figure 8). This suggests that not all of the long-term 9 6 motion can be accommodated on the modeled faults. As suggested by Meade and Hager [a], one could interpret this fault-normal motion as dip slip on reverse or normal faults running near and parallel to the large strike-slip faults, although this is not explicit in the block models. In the plateblock model, we map this fault-normal motion explicitly into the long-term, steady state deformation field. [] The slip rates from the elastic plate-block model inversion are shown in Figure. The strike-slip rates are similar to the rates from the deforming block model (Figure 8) except that the Coachella SAF slip rate is mm/yr faster in the block model. [] The off-fault maximum shear strain rates and the principal directions are shown in Figure. The strain rate patterns in the Transverse and ECSZ blocks are similar to the deforming block model (Figure 9); however, the strain rates in the plate-block model are (by assumption) distributed across a large part of Southern California. Pockets of high shear strain rates exceeding 9 nanostrain/year occur at areas with complex geometry (sharp block corners and bends in faults). Lower shear strain rates of 6 nanostrain/year are distributed across much of Southern California, largely as a result of fault-normal motions required by the otherwise rigid block motion assumption... Viscoelastic Cycle Model [] The previous analyses show that the model slip rates can be reconciled with the geologic slip rate bounds if dip-slip rate (mm/yr) strike-slip rate (mm/yr) right latera left lateral reverse 9 7 6 6 9 7 7 b. d.... misfit (mm/yr) Figure. Results for inversion with elastic plate-block model. (a) Strike-slip rates denoted by color and numbers. (b) Dip-slip rates denoted by color and numbers. (c) Residual vectors shown only for residuals exceeding. Normalized chi-square, n, is the sum of squared weighted residuals divided by the number of observations. (d) Color points show magnitude of all residuals. 6

North (km) JOHNSON: SOUTHERN CALIFORNIA SLIP RATES a. nano-strain/yr elastic b. East (km) 9 6 viscoelastic Figure. Maximum shear strain rate and principal strain rates for plate-block models. (a) Elastic plateblock model. (b) Viscoelastic plate-block model. off-fault strain is included in the Transverse Ranges and Mojave ECSZ. However, the model rates on the western Garlock fault and the Mojave SAF are equal to the lower geologic bound. In fact, the model slip rates are systematically low along the entire SAF system in Southern California (Figure a). As suggested by Chuang and Johnson [] and HearnE.H.etal.[], it is possible to increase the estimated slip rate on the San Andreas fault by incorporating relaxing viscous flow in the mantle. We now explore the extent to which the GPS velocities and geologic slip rates can be reconciled through a model that incorporates flow in the lower crust and upper mantle. Before we present the results of any inversions, we first illustrate the general effect of relaxing mantle flow on surface deformation patterns.... Illustration of Mantle Flow Effect [] The fault slip rates from the elastic plate-block model inversion (Figure ) are now imposed on the viscoelastic earthquake cycle model. The model viscosity structure summarized in Figure b is based on the Thatcher and Pollitz [8] compilation of laboratory studies and geodetic estimates of lithosphere viscosity in the western U.S. (Figure a). We assume a uniform elastic thickness of km, locking depth of km, uppermost mantle viscosity of Pa s (relaxation time of years) between depths of and km, and mantle viscosity below km depth of 8 Pa s (relaxation time of. years). This viscosity structure is used for this illustration and for all cycle inversions in this study. [6] The earthquake timing is summarized in Figure c where the first number listed is the time since the last earthquake, t eq, in years, since year, and the second number is recurrence time, T, in years. The timing is assigned based on the compilation of paleoseismic studies for UCERF (Uniform California Earthquake Rupture Forecast, Version, Appendix B, USGS Open File Report 7-7). For some of the faults (typically low-slip rate faults) with no published paleoseismic data, we assign t eq = and T =. For faults with paleoseismic data, we assign the average reported repeat time as T and we use the date of the most recent earthquake in the paleoseismic record as t eq. North (km) a. (ub-lb)/ b. upper bound elastic East (km).8.6.....6.8 middle lower bound viscoelastic East (km) Figure. Color shows where model strike-slip rate estimates lie within the prior bounds (ub denotes upper bound and lb denotes lower bound). Red indicates estimate is at upper bound, green indicates estimate is in the middle, and blue indicates estimate is at the lower bound. (a) Elastic plate-block model. (b) Viscoelastic plate-block model. 6

Figure. Assumed viscosity structure and earthquake timing for all viscoelastic cycle models in this study. (a) After Thatcher and Pollitz [8]. Gray regions summarize laboratory-based estimates of lithosphere viscosity structure for different compositions at two different deformation rates. Orange boxes delineate range of Western U.S. viscosity estimates from various geodetic studies. Red lines are viscosities assumed in this study. (b) Assumed model viscosity structure for this study. (c) Assumed earthquake timing for this study. First number is time since the last earthquake, t eq, in years, and second number is recurrence time, T, in years. Data sources summarized in Table. Low-rate faults with no paleoseismic data are assigned t eq = and T =. 6 For three fault sections (Mojave SAF, San Bernardino SAF, and Coachella SAF), the time since the last earthquake is longer than the average paleoseismic repeat time. For these three sections we use the date of the last recorded event to compute t eq and we assign a T that is slightly larger than t eq. [7] Figure compares the interseismic perturbation for the elastic plate-block model with the viscoelastic earthquake cycle model. This is the total model velocity field minus the long-term,steady state velocity. In other words, Figure shows the velocities only due to back slip (plus periodic earthquakes for the viscoelastic model). The interseismic perturbation to the velocity field has a longer wavelength component in the viscoelastic calculation due to the deep-relaxing mantle flow. The perturbed viscoelastic model velocities in southwest California and off the west coast are up to mm/yr, whereas the corresponding perturbed velocities in the elastic model are very small, mm/yr or less. The perturbed velocities near faults are similar in the two models but differ significantly at greater distances. This pattern is clear from the color gradients in Figure b which show the percent differences between the elasticand viscoelastic-perturbed velocities. The percent difference is small along the SAF system and increases away from the SAF. [8] Two-dimensional earthquake cycle models with relaxing mantle flow [e.g., Savage and Prescott, 978] have been used previously in the literature to estimate fault slip rates from profiles of geodetic data [Segall, ; Dixon et al., ; Johnson and Segall, ; Johnson et al., 7]. In Appendix A we show that -D models greatly exaggerate the effect of relaxing mantle flow on predicted surface velocities because the infinitely long ruptures introduce unphysically long-wavelength patterns of flow.... Viscoelastic Inversion [9] Chuang and Johnson [] showed that an earthquake cycle model that includes viscous relaxation of the mantle and stress-driven fault creep in the lower crust is largely consistent with both the GPS velocity field and geologic fault slip rates with the exception of a systematic misfit east of the Imperial fault. However, that model did not allow additional parameterized block strain. Furthermore, that model of Chuang and Johnson [] was consistent

normal JOHNSON: SOUTHERN CALIFORNIA SLIP RATES a. b. interseismic velocity perturbations percent difference between elastic and viscoelastic V e -V v /V e x North (km) 8 6 8 6 scale, mm/yr elastic viscoelastic East (km) Figure. Comparison of interseismic perturbation in elastic and viscoelastic models. (a) Velocities due only to back slip (in elastic model) and back slip plus periodic earthquakes (in viscoelastic model). Elastic (blue) and viscoelastic (red) interseismic velocity fields. (b) Percent difference between elastic and viscoelastic interseismic velocity fields. Color saturated at %. V e is elastic velocity field and V v is viscoelastic velocity field. with geologic slip rates in the ECSZ only because the model had a significant accelerated postseismic transient in the ECSZ due to a composite earthquake that was imposed at years before present to represent known accelerated postseismic velocities due to the Landers/Hector Mine sequence, possibly the 7 Old Woman Springs earthquake [Rockwell et al., b], and any other unknown earthquakes in the ECSZ. Although we know there is some strike-slip rate (mm/yr) dip-slip rate (mm/yr) right lateral left lateral reverse 6 6 6 6 9 8 9 6 7 6 9 6 8 9 6 9 6 b. d. scale, mm/yr... misfit (mm/yr) Figure. Results for inversion with viscoelastic cycle model. (a) Strike-slip rates denoted by color and numbers. (b) Dip-slip rates denoted by color and numbers. (c) Residual vectors shown only for residuals exceeding. Normalized chi-square, n, is the sum of squared weighted residuals divided by the number of observations. (d) Color points show magnitude of all residuals. 66

Table. Compilation of Model Moment Rates in Nm/yr a Model Dip and Strike slip Fault-Normal Off Fault Total Meade and Hager [b].78 9 Deforming block.79 9 8. 8 8. 6.6 9 Elastic plate-block.88 9 N/A 9.8 8.79 9 Viscoelastic plate-block. 9 N/A 8.77 8. 9 a For the deforming block model, moment rates due to fault-parallel (dip- and strike-slip) motion are computed separately from moment rates due to fault-normal velocity discontinuities. influence on accelerated postseismic transient deformation due to the Landers/Hector Mine earthquakes, the treatment of the composite earthquake is otherwise artificial because we have no record of other recent earthquakes in the ECSZ other than the 7 Old Woman Springs earthquake. [] We now reexamine the problem using the earthquake cycle model without the composite recent ECSZ earthquake and allow for parameterized block strain. We use the same earthquake-timing parameters and viscosity structure as in the previous illustrations. Figure shows the estimated fault slip rates and fit to GPS data for the viscoelastic plate-block model inversion. The fit to the data is comparable to the elastic plate-block model inversion (Figure ) with n =. (compared to n =. for the elastic model). The strike-slip rates from the viscoelastic inversion are faster than the elastic model result along the entire SAF system ( mm/yr faster on the Carrizo segment and mm/yr faster on the Mojave and Coachella segments). The viscoelastic model also gives faster strike-slip rates on the Elsinore and eastern Garlock faults. Comparison of the two plots in Figure shows that the SAF slip rates from the viscoelastic inversion are systematically shifted up toward the middle of the geologic range compared to the elastic inversion. The viscoelastic model strike-slip rates on the Figure 6. Results of constrained slip rate inversions for selected fault segments. Solid lines show normalized chi-square as a function of fixed slip rate for elastic and viscoelastic plate-block models. Dashed lines show formal 8% confidence limits. Sides of boxes correspond with the prior upper and lower bounds on slip rate. Posterior slip rate probability density functions also shown for elastic and viscoelastic models. 67

Figure 7. Posterior probability distributions of strike-slip rates (shown as histograms of number of samples) for elastic (blue) and viscoelastic (red) plate-block models. eastern Garlock fault and the Elsinore fault are pushing up against the upper geologic bound. 6. Moment Rates [] Table compares the moment rates on and off faults for all three deforming models. The total off-fault moment rate is computed from the model strain rates using equation (6). The on-fault moment accumulation rates in the deforming block and elastic plate-block models is comparable to the moment accumulation rate computed by Meade and Hager [b]. The off-fault moment rate in the deforming block model is two orders of magnitude smaller than the off-fault moment rates in the plate-block models. However, the moment rate due to fault-normal discontinuities in the deforming block models is comparable to the off-fault moment rates in the plate-block models. Of course this is 68 not surprising because in the plate-block model, the faultnormal velocity discontinuity has been distributed as strain in the plate and so the computed off-fault moment accumulation rate should be comparable to the fault-normal moment accumulation rate. The on-fault moment accumulation rate is largest in the viscoelastic plate-block model because the slip rates on the SAF are systematically higher than in the elastic models. [] For comparison, the total year (since the 87 Fort Tejon earthquake) historical seismic moment release rate in Southern California is.6 9 Nm/yr (K. Felzer, personal communication,, based on UCERF earthquake catalog, Appendix K). This rate is only % of the total moment accumulation rate in the deforming block model and only 8% of the total moment accumulation rate in the viscoelastic plate-block model. A M8 earthquake every years is equivalent to a moment release rate of