Influence of anelastic surface layers on postseismic

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. B2, PAGES , FEBRUARY 10, 2000 Influence of anelastic surface layers on postseismic thrust fault deformation Gregory A. Lyzenga Department of Physics, Harvey Mudd College, Claremont, California Wendy R. Panero Department of Geology and Geophysics, University of California, Berkeley Andrea Donnellan Jet Propulsion Laboratory, Pasadena, California Abstract. We presenthe results of a systematic modeling study of postseismic deformation following blind thrust earthquakes. The results include qualitative and quantitative predictions of the surface movement caused by relaxation in viscoelastic near-surface layers. Finite element forward models are used in conjunction with elastic dislocation inversions to characterize the postseismic deformation. A viscoelastic surface layer overlying a blind thrust fault in an elastic basement shows characteristic signatures of postseismic surface movement. Simple equivalent elastic dislocations located in the hanging wall wedge are found to provide an effective proxy for nearsurface postseismic relaxation in two-dimensional numerical simulations. A model survey of a range of fault dip angles and layer geometrieshows the time evolution and geometry of the proxy fault to be simply related to fault dip and sedimenthickness. The results are of significance in the interpretation of postseismic Global Positioning System (GPS) strain data from the 1994 Northridge, California, earthquake and other similar events in regions characterized by poorly consolidated or otherwise anelastic layers overlying the brittle seismogenic zone. 1. Introduction Transient crustal deformation in the weeks to years following a large earthquake has been the subject of considerable observational and theoretical study over the past two decades. Postseismic deformation has been observed with geodetic techniques following shallow strike-slip and dip-slip events that were large enough to rupture all or much of the thickness of the brittle seismogenic crust. Theoretical work has ascribed this deformation to various combinations of fault creep (occurring on one or more discrete planes) and bulk viscoelastic relaxation off the fault. Observational evidence suggests that in many cases, more than a single mechanism and timescale for postseismic deformation are active. While long-term geodetic measurements of the earthquake cycle appear to demonstrate lower crustal viscoelasticity with decadal timescales [e.g., Thatcher, 1983], recent studies including those of the 1992 Landers [Shen et al., 1994], 1994 Northridge [Heflin et al., 1998], and 1994 Sanriku-Haruka-Oki, Japan [Heki et al., 1997], events show transient motions on timescales of several months. In particular, Donnellan and Lyzenga [1998] have found that the M W 6.7 Northridge thrust earthquake was followed by -1 year timescale relaxation, associated with the combined effects of fault plane afterslip and bulk relaxation in the uppermost shallow crust. While earlier preearthquake observations and models of the Ventura Basin region [Hager et al., 1999] near the Northridge event provide con- Copyright 2000 by the American Geophysical Union. Paper number 1999JB JB straints on the longer-term viscoelasticity of the lower crust, the immediate postseismic geodesy is uniquely valuable in elucidating the behavior of poorly consolidated and anelastic sedimentary near-surface layers treated as a low-viscosity viscoelastic layer. An important deformation mechanism which is potentially active along with or instead of the bulk processes described here is transient afterslip on one or more discrete planes [Marone et al., 1991]. While spatial separation is probably adequate to distinguish main rupture afterslip from distributed upper crustal shear, it is more problematic to unambiguously distinguish off-rupture triggered slip. A possible discriminant is the time dependence, which for transient slip is expected to have a logarithmic character, while Newtonian bulk shear relaxation should follow an exponential decay law. However, in this work we recognize that the actual rheology of shallow crustal materials is likely to be complex, and the distinction between true afterslip and true bulk relaxation is likely to be gradational. Savage and Prescott [1978] and Savage [1990] discussed the equivalence between vertical strike-slip postseismic viscoelastic relaxation and strain in an elastic half-space due to proxy faults. In a related vein, Savage [1987] found distributions of elastic half-space faulting that are equivalent to vertical strike-slip faulting in layered elastic structures. The present study examines a similar but distinct problem, equivalent elastic dislocation proxies for viscoelastic relaxation following shallow thrust earthquakes. 2. Model Formulation and Approach A simple two-layer, two-dimensional (2-D) elasticviscoelastic structure is used in a finite element simulation (Figure 1). Vis- 3151

2 3152 LYZENGA ET AL.: ANELASTIC POSTSEISMIC SURFACE DEFORMATION surface Maxwell viscoelastic "proxy" fault ('c = 1.0 yr) IIII IIII IIII ljll IIII I I III IIII -" to 900 km to +_450 km surface Figure 1. (top) Schematic representation of the 2-D finite element model used in the forward modeling part of the study. The seismogenic fault is introduced in the layered elastic-viscoelastic half-space through the use of split nodes. Dashed line indicates the approximate apparent location of the postseismic deformation source. (bottom) Finite element grid at the same scale as in Figure 1 (top), an indication of model resolution and geometry. coelastic properties are restricted to the uppermost layer, which is intended to represent a near-surface sedimentary layer or otherwise poorly consolidated material with relatively low Maxwell relaxation time 'c M. (The Maxwell time is defined as 'cm = BIt, where B is the viscosity and g is the elastic rigidity of the medium.) While in a broad sense, it is also important to consider viscoelastic response of the lower crest, the effect of viscoelastic deformation in the lower crest is not considered in this work. Here, we examine only shallow relaxation occurring over timescales short in comparison with typical lower crustal Maxwell times. The seismogenic thrust fault in these models breaks the elastic crust from the base of the sedimentary lid, down to a maximum depth of 16 km. The fault dimensions approximate typical values for blind thrust faulting in western North America [e.g., Stein et al., 1988]. However, the results of this study are presented in a nondimensionalized form which permits their application to different assumed layer thicknesses. The fault is represented in the finite element calculation by "split nodes" [Melosh and Raefsky, 1981] which apply a specified constant dislocation across the fault surface. While the technique allows the dislocation amplitude to vary from point to point on the grid, for simplicity, this study employs spatially uniform slip on the fault. The finite element grid resolution along the fault varies from 0.5 to 1.0 km, and in the far field the grid spacing is gradually increased stepwise to several tens of kilometers. As an approximation of an infinite half-space, the fixed boundaries of the grid form a roughly square box of dimension -900 km. The elastic and rheological properties of the model are simplified to provide a suite of earthquake models as generic as possible. The elastic part of the crust and upper mantle is a Poisson isotropic solid, with an assumed rigidity of 36 GPa. The uppermost ductile crust, with a thickness h, varying from 3 to 8 km, is assumed to be a linear Maxwell viscoelastic material. The rigidity is 18 GPa (half that of the elastic crust), and the viscosity is chosen to yield a Maxwell relaxation time of 'cm--1.0 year. Another simplification is that the models do not include the effects of buoyancy or isostatic forces in driving postseismic rebound. The applicability of a simple Newtonian viscosity model to sediments (or other ductile upper crustal materials) is debatable, as is the 1-year timescale; however, the intent of the approach is to provide a simplified indicative model, which by virtue of its linearity is easily scaled to different effective viscosities. In this way, the models discussed here may provide broad (though perhaps less detailed) generalizations to be drawn regarding this style and geometry of crustal faulting and deformation. The choice of an upper crustal rigidity lower than that of the underlying half-space approximates the elastic properties of real sediments, but it is not the only choice that could be made. While this study focuses on the layered rigidity structure, some forward models were also run using a half-space of homogeneous elastic properties. These models yielded proxy dislocation deformation sources closely analogous to the principal results described below. So while variations in assumed elastic structure change the numerical details of the results (typically by some tens of percent in the sensitive parameters), the qualitative character of the solutions is unchanged. The toregoing description covers the forward modeling aspects of this work. These results are used in turn, as input to elastic half-space inversion as a means of finding equivalent dislocation proxies for the postseismic deformation process. As discussed above, this approachas precedent in the work of Savage [ 1990] on vertical strike slip faults. An important difference from Savage's work, in addition to the depth of the viscous layer, is that in the case of thrust faulting, no rigorous mathematical equivalence between the viscoelastic problem and its elastic dislocation proxy is known. The philosophy of this study is to identify and characterize approximatequivalences for the deformation fields, where the working definition of "approximate equivalence" is governed by the smallest easily resolvable differences in motion as measured by modem geodetic methods. A numerical residual-minimization procedure is used to search for the best fitting elastic dislocation solutions. A Z 2 goodnessof-fit objective function is formed from the horizontal and vertical components of displacement at each surface grid point in the finite element forward model. In keeping with our approach of relating quality of fit to practical geodetic measurement precision, the horizontal component "pseudomeasurements" are more heavily weighted than the vertical, reflecting approximately the relative levels of Global Positioning System (GPS) campaign survey precision. On the basis of this precision, each horizontal pseudomeasurement is assigned a conservative formal uncertainty for weighting of 2.0 mm yr -1, while the vertical components are assigned an uncertainty of 10 mm yr -1. The residual-minimization program developed by the authors for this study (SIMPLEX 4.0) is based upon the downhill simplex simulated annealing algorithm described by Press et al. [1988].

3 , LYZENGA ET AL.' ANELASTIC POSTSEISMIC SURFACE DEFORMATION 3153 A trial solution is represented as a single point in an N- dimensional space, where N is the number of fault parameters to be determined. N+I distinct trial guesses define the vertices of a simplex in this abstract space, and the algorithm seeks out a "downhill" path to contract the locus of vertices about a minimum in the objective function. Function evaluations are based upon the elastostatic solutions for rectangular dislocations given by Okada [1992]. Since the function may have a very complex configuration of local minima, the simulated annealing strategy is employed as a means of locating a global minimum. In addition to its present application to the postseismic deformation problem, this technique is useful in general for inversion of coseismic and interseismic deformation for fault parameters [e.g., Donnellan and Lyzenga, 1998]. In the present study, all fault parameters (depth, dip, horizontal position, width, and slip) were free estimated parameters, and none were constrained. In addition to the explicit weighting of horizontal versus vertical movements, a less obvious implicit weighting is given to points located nearest the fault. This arises because the finite element grid is more closely spaced near the fault, so there are more points per unit area in the near field and more relative importance given to this region in the resulting fit. A typical comparison between forward model pseudomeasurements (derived from the viscoelastic model calculation) and the best fitting elastic dislocation solution (Figure 2) yields a qualitatively good fit, as emphasized in this example by a 2 per (a) vertical velocities 2.5 I 2... I -- FE model : i : -::;ii ;..... :,.... : :... "::::: : i : :: i : x (km) ' (b) horizontal velocities ]... = FE model... _. i..]... dislocation fit i, x (km) Figure 2. Typical postseismic results for a rupture with 8 = 40 ø and h = 5 km. (a) Vertical and (b) horizontal surface velocities are plotted as a function of position (acros strike), at time t=l.0 year following the model earthquake The dashed line indicates the position where the extension of the coseismic fault plane intersects the surface. Solid points show the results of the finite element (FE) forward model of postseismic deformation; shaded line shows the best fitting dislocation proxy source.

4 LYZENGA ET AL.' ANELASTIC POSTSEISMIC SURFACE DEFORMATION degree of freedom objective function value of Visual examination of the correspondence shows that the elastic equivalent solution represents most of the important qualitative features and scales of the deformation field. Some short-wavelength discrepancies exist between the curves, perhaps most noticeably in the footwall near the surface projection of the fault surface. This misfit may be related to strong permanent deformation concentrated near the upper termination of the seismogenic fault, which is poorly represented by a simple dislocation source proxy. In this regard, it seems significant that analogous efforts to fit visoelastic relaxation of the lower crustupper mantle following a thrust earthquake with a simple dislocation model have been unsuccessful. Typically, modeled deep relaxation following thrust earthquakes produces surface deformations quite different in character from the coseismic pattern. The result is a deformation field inconsistent with any simple equivalent single dislocation. We speculate that the character of motions in the comparatively unconstrained half-space of viscoelastic material (in the deep case) is dominated by non-double-couple motions which render the proxy fault approach ineffective. In contrast, the thin-layer geometry of the relaxing material in the present study leads to deformation patterns that, while not purely fault-like in character, do more closely approximate such motions. 3. Study Parameters and Results As a means of exploring the applicability and generality of these models, a systematic model study was carried out allowing two principal parameters to vary, the fault dip5 and the "sediment" thickness h. The resulting dislocation fits serve a dual purpose. First, they demonstrate the practical range of earthquake parameters for which this postseismic model might be expected to be valid. Second, they provide a numerical prescription for approximately modeling this type of postseismic deformation by interpolating and appropriately scaling the current results to any similar case of interest. Forward models and dislocation fits were run for cases with dip angles of5 = 30 ø, 40 ø, 50 ø, and 60 ø and sedimentary layer thicknesses of h = 3, 4, 5, 6, 7, and 8 km. Each model holds the depth limit (H+h) of the coseismic rupture fixed at 16 km. Thus we may regard the variable parameter in this suite of models as the ratio of faulted crustal thickness to ductile layer thickness. The ratio Hh therefore covers the range 1.0 to 4.3 in this work. For consistency, the forward models all assume the same value of 1.0 m of coseismic slip on the rupture plane. However, since the rupture width varies from case to case, comparison of results between cases should take account of the different seismic moments where relevant. We present here in detail the results of the single baseline model already considered (Figure 2). The sedimenthickness is 5 km (thickness ratio Hh = 2.2) and5 = 40 ø, closely approximating the geometry of the Northridge earthquake [Hauksson et al., 1995]. The slip rate of the equivalent dislocation as well as its geometric parameters are variable functions of the time after the earthquake. The location and migration over time of the postseismic equivalent dislocation are presented in Figure 3. The proxy fault generally lies above the coseismic fault plane, about halfway between the free surface and the upper coseismic rupture termination. This geometry almost seems suggestive of an "image fault" solution, although it is evidenthat the best fitting dislocation is much narrower in downdip extent than the actual rupture. The dip of the proxy fault is, within fitting uncertainty, constant in time and approximately equal to that of the coseismic fault. During the first few Maxwell relaxation times, the location and size of the fault remain roughly constant, although the amplitude of the dislocation rate is decaying exponentially with time. For times greater than a few multiples of M = 1.0 year, the amplitudes of the motions everywhere become small compared with the assumed pseudomeasurement error, so that the fit becomes noisy and the results lose significance. Figure 4 shows more clearly the time evolution of equivalent moment release by the postseismic proxy fault. Because the approximate fitting procedure used here yields some mild time dependence in the fault width as well as the dislocation amplitude, the best quantity to analyze is their product. In the case of a twodimensional fault, this product amounts to the seismic moment (per unit length in the out-of-plane direction), divided by the elastic rigidity. Adapting seismological terminology, we will refer to this quantity as the specific potency of the 2-D dislocation. (Potency is also sometimes preferred over moment in working with actual observational data, since it is more directly related to seismological observables [Heaton and Heaton, 1989].) Because viscoelastic h - 5 kmf ': 40ø dips ' Id x= 1.Oyr - f ili'st ';... i':.o :i': :'::'..: H=11 km 0.1yr,, ' Figure 3. Detail of geometry and time evolution of the best fitting proxy source from the same model depicted in Figure 2. The apparent dislocation source lies above the main fault plane and exhibits weak time dependence in position and extent. Position of the proxy fault is parameterized in terms of the variables x and d, as defined here.

5 LYZENGA ET AL.: ANELASTIC POSTSEISMIC SURFACE DEFORMATION 3155 '7, 0.06 >, c- (D (1) 0.02 N E 0.01 O 0.05 :: :: I i e- t I 1.7 i...i... i.i..i !... i... :: ,0 time (yr) Figure 4. Time evolution of the moment release rate from the same model depicted in Figure 2. The potency rate is plotted, thus eliminating the crustal rigidity as a variable. The plotted results are normalized by the mainshock potency. The curve shows the exponential time function fit to the proxy fault results. Plate 2a shows the difference in dips between the coseismic rupture and the equivalent relaxation fault plane. In all cases the two dip angles differ by no more than a few degrees. The dip deviation A5 varies principally as a function of the dip 5, with weak dependence on thickness. The proxy fault systematically dips at a more shallow angle than the causative fault when 5 exceeds -45 ø, while for a shallow causative fault, the proxy is slightly steeper. Plate 2b plots the downdip width of the proxy fault. The width varies appreciably with both dip and thickness, and the narrowest equivalent fault solutions correspond to large sediment thickness and steep dip. On the other extreme, thin sedimentary layers produce an equivalent fault width that is two or more times the layer thickness h. Plates 3a and 3b show the vertical and horizontal locations re- spectively, of the proxy fault plane. Both of these parameters vary as functions of both dip and thickness. In Plate 3a the depth d to the top of the equivalent fault varies from nearly the full thickness h of the sediment layer (for large 5 and h), down to somewhat less than half of that relative depth for shallow dips and thin sediments. The horizontal position of the proxy fault, plotted in Plate 3b, varies over a narrower range, shifting slightly negative (toward the hanging wall) for large 5 and h, and shifting the proxy fault represents deformation by continuouslippage, it positive in the opposite limits. is the specific potency rate that describes the deformation process at any given moment in time. The specific potency rate is seen to decay exponentially in time with a decay time constant similar (though not identical) to the Maxwell time M of the viscoelastic upper crust. The time dependence is well fit by the simple decay law 4. Conclusions The results presented above can be used to compute expected rates and distributions of short-term postseismic deformation following blind thrust earthquakes. Consider the hypothetical example of a 45 ø dipping fault that ruptures brittle crust from 16 W: W o exp(-t x). (1) to 4 km depth. Interpolating from the plots, we predict proxy parameters W o = 0.045, 'c = 1.55 X M, A5 = -1.0 ø, wh = 2.0, dh = The initial rate W o corresponds to 5.3% of the mainshock potency 0.6, and xh = 0.0. When relaxation of the upper ductile layer is per year, and the rate subsequently follows this simple exponential history. These results suggest that a useful presentation of the whole suite of examined models may be presented in terms of just a few output variables. We can approximately characterize the behavior of a given model in the dip versus thickness parameter space in terms of the proxy fault's: (1) initial potency rate, (2) decay time constant, (3) deviation in dip A5 from the causative fault's dip, (4) initial width, (5) initial depth, and (6) horizontal position relative to the causative fault. The graphical summaries which follow in this paper give the derived values tbr these variables. The systematics of the proxy dislocation results are most clearly and efficiently presented in graphical form. Accordingly, Plates 1, 2, and 3 contain plots showing the variation of these six output parameters as functions of the two model parameters, thickness ratio (Hh), and fault dip (5). Plate 1 summarizes the time dependence of the equivalent dislocation in terms of the exponential function of equation (1). Plate l a gives W o the best fitting potency rate at t=0, while Plate lb gives the best fit decay complete, deformation equivalent to W o 'c-- 7% of the mainshock moment will be released in postseismic strain. The geodetic signature will mimic that of a 44 ø dip creeping thrust fault that reaches within -2.4 km of the free surface. The upper terminus of this proxy fault is directly above that of the primary rupture, and it has a downdip extent of (2.0)(4.0) -- 8 km. Although these predictions are approximate, particularly if there exist crustal structural inhomogeneities, they provide a useful means of quantifying the effects of this ductile deformation process. Apart from their potential utility in predicting blind thrust deformation fields, these results may shed new light on the interpretation of past postseismic deformation observations. Biirgmann et al. [1997] have analyzed postseismic geodetic observations following the 1989 Loma Prieta earthquake. Direct comparison of that event with these models is problematic because rather than a pure thrust event, it involved oblique slip with comparable components of reverse and right-lateral strike slip. Furthermore, the dip of the mainshock fault plane was somewhat steeper (70 ø ) than the range of dip angles considered in this time constant 4. This time constant is measured in units of the study. However, the fact that the coeismic rupture terminated ~8 sedimentary Maxwell time x M, which in these models was taken km below the surface suggests that the general picture of brittle to be 1 year. W o showstrong dependence on the layer thickness rupture underlying a more ductile layer may have relevance. ratio, with the largest equivalent slip rates occurring for the cases with the thickest sedimentary layers (Hh small). The results show relatively weak dependence of W o on dip angle. The time constant x shows rather weak dependence on either thickness or dip. As a result, no strong trend stands out in the plot of Plate lb, with the time constant tending to be close to Significantly, that study finds a preferred model in which two creeping dislocation fault sources are active in the immediate postseismic period. The first fault plane is fairly well colocated with the mainshock rupture, while the second plane resides in the upper crustal region not ruptured by the mainshock. These findings are themselves somewhat uncertain in their 1.6 x M, except for very thick or very thin relaxing sedimentary interpretation but lend weight to a picture of anelastic deformalayers. tion in the uppermost crust, perhaps acting in concert with true

6 O o I ' I ' I I '! ' I.,. I '!! h specific potency rate, Wo decay constant, 'c Plate 1. (a) Graphical summary of proxy potency rate results for the range of investigated clip angles and layer thickness ratios. Larger values of Hh correspond to thinner sediment layer thickness. Plotted quantity is normalized specific potency rate (yr-1) at time t=0, immediately following the earthquake. (b) Summary of potency decay rate constant (year), as a function of the same range of dip angles and layer ratios.,-,, 50 I \ ; 40 " _..._. ' 40 L,,,,,, :,,,,,,; O :2.0 ß 30 I O I dip deviation (degrees) relative width, wh Plate 2. (a) Graphical summary of difference between proxy fault dip angle and main fault dip angle, as a function of dip angles and layer ratios. (b) Summary of along-dip width of best fitting proxy fault, normalized by the sediment thickness h O I-Ih relative depth, dh -o. z -o. x -o. o o. t horizontal translation, xh Plate 3. (a) Summary of the depth d to the uppermost edge of best fitting proxy fault, normalized by the sediment thickness h. (b) Summary of horizontal offset x of best-fitting proxy fault, normalized by the sediment thickness h.

7 LYZENGA ET AL.: ANELASTIC POSTSEISMIC SURFACE DEFORMATION 3157 fault afterslip to produce a complex composite pattern of postseismic uplift and strain. This speculation becomes more concrete in examination of the previously mentioned 1994 Northridge earthquake. Donnellan and Lyzenga [1998] present a suite of inversions of postseismic GPS data in the Northridge area. The study uses a single fault plane to fit the data, obtaining a shallow thrust fault lying above the principal fault plane in the hanging wall block, similar to the proxy fault results of this study. A better fitting (though less constrained) solution was obtained by allowing two fault planes in the inversion. In that case, it was found that one plane fell directly on the main coseismic rupture plane, while the second remained in the uppermost hanging wall block. This result suggest strongly that both main fault afterslip and shallow ductile deformation were active the first year after the Northridge earthquake, a conclusion that is consistent with the observed spatial distribution of aftershocks [Hauksson et al., 1995]. Although the similarity of the Northridge geodetic results to this generic modeling study are encouraging, it is possible that the agreement is illusory. Outstanding questions following this work concern the possible influence of three-dimensional structural complexities or more realistic plastic rheological descriptions for sedimentary materials. As mentioned previously, transient fault slip models may occupy one place in a continuum of complex models which also include bulk deformation. Also, there is a considerable body of current research [e.g., Peltzer et al., 1996] that suggests that poroelastic effects involving the relaxation of fluid pore pressure may play an important role in the short-term postseismic period. Preliminary calculations (G. Peltzer, personal communication, 1999) suggesthat such a mechanism may produce qualitatively similar deformation signatures in the Northridge geometry. These and related processes will remain topics of ongoing investigation. The secure conclusion of this work, however, remains that within the contraints of simple geometry and rheology, blind thrust faulting can be expected to give rise to relatively simple, dislocation-like postseismic deformation fields. The parameterized proxy dislocations given in this work provide a convenient and broadly applicable tool for describing and predicting such deformation. Acknowledgments. We gratefully acknowledge the helpful advice and discussions with Jay Parker, Donald Argus, Gilles Peltzer, Chris Marone, and Brad Hager in preparing this work. We acknowledge the early inversion code development work done by Floyd H. Ross. Reviews and suggestions by Gerald Bawden, Jishu Deng, and Ruth Harris were extremely helpful. Parts of this research were carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. Support was also provided by the U.S. Geological Survey and the National Science Foundation through the National Earthquake Hazards Reduction Program (NEHRP) and the Southern California Earthquake Center (SCEC). SCEC is funded by NSF Cooperative Agreement EAR and USGS Cooperative Agreements A0899 and 1434-HQ-97AG This is SCEC contribution 477. References Btirgmann, R., et al., Postseismic strain following the 1989 Loma Prieta earthquake frown GPS and leveling neasurements, J. Geophys. Res., 102, , Donnellan, A., and G. A. Lyzenga, GPS observations of fault afterslip and upper crustal deformation following the Northridge earthquake, J. Geophys. Res., ,285-21,297, Hager, B. H., G. A. Lyzenga, A. Donnellan and D. Dong, Reconciling rapid strain accumulation with deep seismogenic fault planes in the Ventura basin, California, J. Geophys. 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Rundle, The growth of geological structures by repeated earthquakes, 2, Field examples of continental dip-slip faults, J. Geophys. Res., 93, 13,319-13,331, Thatcher, W., Nonlinear strain buildup and the earthquake cycle on the San Andreas fault, J. Geophys. Res., 88, , A. Donnellan, Jet Propulsion Laboratory, MS , Pasadena, CA (andrea@cobra.jpl.nasa.gov) G. A. Lyzenga, Department of Physics, Harvey Mudd College, Claremont, CA (lyzengacg hmc.edu) W. R. Panero, Department of Geology and Geophysics, University of California, Berkeley, CA (panero@uclink4.berkeley.edu) (Received April 12, 1999; revised July 23, 1999; accepted August 9, 1999.)