Reversed-Polarity Secondary Yehuda Ben-Zion Deformation Structures Near Department of Earth Sciences, University of Southern California, Los Angeles, CA 90089-0740 Fault Stepovers

We study volumetric deformation structures in stepover regions using numerical simula- Thomas K. Rockwell tions and field observations, with a focus on small-scale features near the ends of rupture segments that have opposite-polarity from the larger-scale structures that characterize Zheqiang Shi the overall stepover region. The reversed-polarity small-scale structures are interpreted to be generated by arrest phases that start at the barriers and propagate some distance Department of Geological Sciences, back into the rupture segment. Dynamic rupture propagating as a symmetric bilateral San Diego State University, crack produces similar (anti-symmetric) structures at both rupture ends. In contrast, rup- San Diego, CA 92182-1020 ture in the form of a predominantly unidirectional pulse produces pronounced reversed- polarity structures only at the fault end in the dominant propagation direction. Several observational examples at different scales from strike-slip faults of the San Andreas sys- Shiqing Xu tem in southern California illustrate the existence of reversed-polarity secondary defor- Department of Earth Sciences, mation structures. In the examples shown, relatively-small pressure-ridges are seen only University of Southern California, on one side of relatively-large extensional stepovers. This suggests frequent predomi- Los Angeles, CA 90089-0740 nantly unidirectional ruptures in at least some of those cases, although multisignal obser- vations are needed to distinguish between different possible mechanisms. The results contribute to the ability of inferring from field observations on persistent behavior of earthquake ruptures associated with individual fault sections. [DOI: 10.1115/1.4006154]

1 Introduction wavelength (high-frequency) wavefield generated by earthquakes [17,18]. The relative length-scales, amplitudes, and signs of the Earthquake faults are geometrically complex and consist of col- opposite slip components in the schematic representation of Fig. 2 lections of segments having various sizes [1–3]. The ends of fault suggest that stepovers acting as strong barriers should produce segments form partial or complete barriers for rupture propagation relatively-large “classical” pull-apart basins and pressure-ridges and are generally regions of stress concentration and generation of within the stepover regions, combined with relatively-small related deformation structures. In the present paper, we focus on reversed-polarity superposed structures near the barriers. stepover regions of large strike-slip faults where neighboring seg- The existence and amplitude of the reversed-polarity secondary ments are separated by a lateral offset. Burchfiel and Stewart [4] deformation structures near the different segment-ends in stepover and Crowell [5] pointed out that right- and left-stepping of right- regions should depend on the abruptness of the stopping process lateral faults produce, respectively, regions with extensional and and directivity of the ruptures. Abrupt stopping processes are nat- compressive stress fields (Fig. 1). The regions between the right- urally expected to produce more localized and clearer features and left-stepping segments of right-lateral faults are expected to than gradual processes. Approximately symmetric ruptures that be associated generally with structures referred to as pull-apart propagate bilaterally and generate considerable arrest phases at basins and pressure-ridges, respectively. The expected stress fields both segment ends are expected to produce significant reversed- and deformation features are reversed for corresponding left- polarity features near both barriers. In contrast, ruptures that are lateral faults. These expectations should be applicable over a wide predominately unidirectional are likely to produce significant range of scales because the equations of elasticity governing the reversed-polarity deformation feature only near the barrier in the amplitudes and signs of the stress field are scale-free. There is a dominant propagation direction. Clarifying the conditions leading long and productive history of using the forgoing ideas in numer- to different types of reversed-polarity deformation features near ous field and laboratory studies involving different stress regimes segment ends can improve the ability of deriving from field obser- and scales [6–9]. Various studies attempted to estimate the likeli- vations information on properties of earthquake ruptures and hoods that earthquake ruptures can propagate across stepovers as expected seismic motion in different locations. functions of geometrical and rheological parameters [10–12] and In Sec. 2, we demonstrate with numerical simulations the basic tested these estimates with field observations [13–16]. properties of reversed-polarity small-scale deformation structures If a fault stepover acts as a barrier for earthquake ruptures, the near stepovers generated by symmetric and predominantly unidir- co-seismic slip distribution will decrease to zero near the stepover. ectional dynamic ruptures. In Sec. 3, we present several observa- As illustrated in Fig. 2 for a right-lateral case (top plot), this can tional examples of such features from the San Andreas, San be represented as a superposition of right-lateral motion with high Jacinto, Elsinore, and Rose Canyon faults in southern California. slip over essentially the entire fault (bottom left), combined with The results contribute to the development of better links between left-lateral slip that is concentrated near the barriers (bottom models of earthquake ruptures and fault geomorphology. The right). The slip profile in Fig. 2 top is similar to the slip distribu- reversed-polarity deformation features in stepover regions aug- tion simulated in Sec. 2 for a crack-like rupture. While the particu- ment the list of signals that may be used to infer on likely propa- lar superposition in Fig. 2 bottom is nonunique, it provides an gation directions of earthquake ruptures on given fault sections. indication of features expected to exist near the barriers. The localized reversed slip may be interpreted as producing the arrest phases of ruptures, which are known to dominate the short- 2 Numerical Modeling The essential behavior associated with propagation and arrest Manuscript received August 11, 2011; final manuscript received October 21, 2011; accepted manuscript posted February 21, 2012; published online April 6, of dynamic ruptures on the main fault of Fig. 1 can be studied 2012. Assoc. Editor: Huajian Gao. using the 2D plane-strain model of Fig. 3 for Mode-II ruptures on

Journal of Applied Mechanics Copyright VC 2012 by ASME MAY 2012, Vol. 79 / 031025-1

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm material off the fault based on the Drucker-Prager plasticity [21]. The yielding surface is a function of the mean normal stress and zero inelastic volumetric deformation is assumed. The latter is not fully realistic for brittle deformation of rocks that includes dilation [22,23] but is adopted nevertheless for simplicity. Rudnicki and Rice [24] generalized the Drucker-Prager plasticity to include dilatancy and discussed, among other things, the relevance of dila- tancy to shear localization. This framework was used in several recent studies of earthquake ruptures with off-fault yielding. In Fig. 1 Large-scale volumetric stresses expected in fault step- particular, Templeton and Rice [25] demonstrated that a realistic over regions amount of dilatancy helps to suppress shear localization, in agree- ment with the Rudnicki and Rice theory, and Viesca et al. [26] a finite fault segment bounded by strong barriers at both ends. showed that in the presence of pore fluids the incorporation of We perform numerical simulations for this model using the dilatancy can affect significantly the elastic-plastic response. Support Operator Rupture Dynamics (SORD) code [19,20], Since our calculations employ the standard Drucker-Prager plas- with further addition of rate-and-state friction on the fault and ticity, the volumetric deformation generated by dynamic ruptures Drucker-Prager viscoplasticity in the bulk. To illustrate the inter- in the results below is purely elastic. action of different types of dynamic ruptures with strong barriers, As shown in Fig. 3, the fault line is located at y ¼ 0. The shear we simulate two end-member cases: (1) a symmetric bilateral strength of the fault is set to be sufficiently large for jjx 25 km crack-like rupture with slip-weakening friction and (2) a predomi- so that sliding occurs only within the confined segment of –25 nantly unilateral pulse-like rupture with strongly rate-weakening km < x < 25 km. In the crack-like rupture case, the fault is gov- friction. erned by a slip-weakening friction where the friction coefficient f To produce bounded elastic stresses near the barriers and linearly decreases [27–29] with slip Du from static value fs to reduce numerical noise, we incorporate inelastic yielding in the dynamic level fd over a characteristic slip distance Dc

Fig. 2 Schematic representation of right-lateral slip (top purple curve) on a finite fault segment bounded by barriers (yellow stars) as superposition of near constant right-lateral slip (bottom left blue curve) and reversed left-lateral slip near the bar- riers (bottom right red curve)

Fig. 3 A model for 2D in-plane dynamic rupture on a frictional fault at y 5 0 with initial stress loading at the remote boundaries. Strong barriers at jjx 25 km con- fine the ruptures to the finite segment between the yellow stars. A 3 km wide nucle- ation zone is centered at x 5 0 and x 5215 km for cases producing bilateral crack and unidirectional pulse, respectively.

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Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm f ðf f ÞDu=D for Du < D antee numerical convergence and resolve properly the dynamical f ¼ s s d c c (1) fd for Du Dc features associated with the rupture process. The yield criterion of Drucker and Prager [21] is given by In the pulse-type rupture case, the fault is governed by a rate-and- rffiffiffiffiffiffiffiffiffiffiffi state dependent friction with strong rate weakening. The friction 1 1 s s r sin / þ c cos / (3) coefficient f ðV; wÞ is a function of slip velocity V and state vari- 2 ij ij 3 kk able w [30–34] where r 1d r are the deviatoric stress components, r f ðV; wÞ¼a lnðV=V0Þþw (2a) sij ¼ ij 3 ij kk kk is the first stress invariant, / is the angle of internal coefficient of with evolution of state variable governed by the slip law friction, and c is the cohesion. A Maxwellian viscoplasticity scheme is used, so during yielding the return of the stress state to V the yield surface occurs smoothly over a relaxation time Tv set to w_ ¼ ½w w ðVÞ be the time for S wave to traverse one grid size [39]. As discussed L ss by Templeton and Rice [25], the Drucker-Prager yield envelop and approximates the Mohr-Coulomb yield criterion in certain 2D stress conditions. We set the initial out-of-plane normal stress r0 zz to be the average of the other two initial in-plane normal stresses 2V0 fssðVÞ 0 0 wssðVÞ¼a ln sinh (2b) rxx and ryy, such that these two yield criteria match initially but V a then generally deviate as plastic yielding occurs. For simplicity, 0 0 0 we use in this study rxx ¼ ryy ¼ rzz ¼100 MPa and cohesion In Eq. (2b), fssðVÞ is the steady-state friction coefficient at slip 0 c ¼ 0. The values for shear stress rxy, which is applied in a right- velocity V given [35,36]by lateral fashion, and all other model parameters are listed in Table 1. The employed values are chosen to represent conditions near active ( faults and to produce smooth numerical results. f0 ðb aÞlnðV=V0Þ for V Vw The ruptures are initiated by gradually increasing the shear trac- fssðVÞ¼ fw þ ½f0 ðb aÞlnðV=V0Þfw Vw=V for V > Vw tion along a 3 km nucleation zone to slightly above the static shear strength. In the simulation associated with a symmetric bilateral (2c) crack rupture, the nucleation zone is centered at x ¼ 0 (red zone in Fig. 3). In the case designed to produce predominantly unilateral The definitions and values of the various friction parameters are pulse (in the þ x direction), the nucleation zone is centered at summarized in Table 1. The parameter values are chosen based on x ¼15 km (cyan zone in Fig. 3) and the friction properties to the previous studies [37,38] to produce subsonic crack- and pulse- left of x ¼20 km change gradually from velocity-weakening to type ruptures. velocity-strengthening to suppress the left-propagating rupture The size of the process zone for strength reduction on a fault front. governed by slip-weakening friction can be estimated using the Figure 4 shows the temporal evolution of slip-rate and slip pro- formula of Palmer and Rice [27]. A similar estimate can be made files along the fault during the propagation and arrest of a dynamic for the rate-and-state friction by approximating the frictional crack-like rupture. Starting from the nucleation zone around x ¼ 0, behavior with equivalent slip-weakening parameters [38]. Using the rupture expands symmetrically in both directions with increas- the employed parameters (Table 1), we estimate the process zone ing slip rates at the propagating tips. The barriers at x ¼ 625 km to be 1500 m in the crack-like rupture case with slip-weakening produce sudden arrest of right-lateral slip at the arrivals of the rup- friction and 700 m in the pulse-like rupture case with rate-and- ture fronts with peak slip rates of 5.8 m/s. This generates strong state friction. The simulations are done with a grid spacing of 50 reflected stress waves that trigger arrest phases, which propagate m, which is sufficiently small compared to both estimates to guar- back towards the nucleation region. See http://earth.usc.edu/~ybz/ pubs_recent/BZ_etal_JAM12/ShearStressCrackRupture.mov in the supplementary material [40] for animation of evolving shear Table 1 Employed physical and numerical parameters in simu- stress, in a case similar to that leading to Fig. 4 but with elastic lations with rate-and-state dependent (RSD) or slip-weakening off-fault response and smaller grid size of 20 m, which illustrates (SW) friction and Drucker-Prager (DP) viscoplasticity the discussed stages of the rupture process. To study the interaction between the dynamic rupture and Parameter Value strong barriers on the deformation field in the vicinity of the bar- riers, we examine the differential volumetric strain DeV around the Shear wave speed Cs 3464 m=s barriers. This is obtained by subtracting a reference volumetric Shear modulus G 32.04 GPa strain field at time t ¼ tref from the volumetric strain fields at Poisson’s ratio 0.25 t > tref, where tref is the time when the rupture front tip nominally Model grid spacing (uniform) dx 50 m reaches the barrier. The use of differential rather than total strain Nucleation zone size Lnucl 3000 m 0 helps to identify small local changes that may be difficult to see Initial shear stress rxy 45 MPa (crack), 30 MPa (pulse) on the background of larger regional variations. This is analogous Direct effect parameter (RSD) a 0.01 to plotting a small topographic feature on a high mountain (e.g., Evolution effect parameter (RSD) b 0.014 the Tibetan plateau) with respect to the local environment rather Steady-state friction coefficient at V0 (RSD) f0 0.6 than the sea level. Fully weakened friction coefficient (RSD) fw 0.2 Figure 5 shows contours of DeV near the right barrier (top plots) State evolution distance (RSD) L 0.4 m and left barriers (bottom plots) at two different times (left and Reference slip rate (RSD) V0 1 lm=s right plots) for the bilateral crack rupture of Fig. 4. The red and Weakening slip rate (RSD) Vw 0.2 m=s blue colors denote relative compression and tension, respectively. Static friction coefficient (SW) fs 0.6 Dynamic friction coefficient (SW) f 0.4 As expected, the differential fields at the left and right barriers are d anti-symmetric. The relatively large volumetric strain fields that Characteristic slip distance (SW) Dc 0.4 m Internal friction coefficient (DP) tanðÞ/ 0.8 extend ahead of the segment ends are the classical features of Viscoplasticity relaxation time (DP) Tv 0.0144 s stepover regions, i.e., pressure-ridge and pull-apart-basin above and below the right barrier and the other way around for the left

Journal of Applied Mechanics MAY 2012, Vol. 79 / 031025-3

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 4 Profiles of along-fault slip velocity (top) and slip (bottom) at various times (indicated by the color scale) during propagation of symmetric bilateral crack-type rupture. The star symbols at x 5625 km indicate locations of strong barriers.

Fig. 5 Contours of differential volumetric strain DeV (t > tref 5 9.380 s) in the case of symmetric bilateral crack-like rupture at (a) tref 1 0.16 s around the right barrier, (b) tref 1 2.32 s around the right barrier, (c) tref 1 0.16 s around the left barrier, and (d) tref 1 2.32 s around the left barrier. The black line at y 5 0 indicates the slipping portion of the fault. See text for additional explanations.

031025-4 / Vol. 79, MAY 2012 Transactions of the ASME

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 6 Profiles of along-fault slip velocity (top) and slip (bottom) at various times (indicated by the color scale) during propagation of predominately unilateral pulse-like rupture. The star symbols at x 5625 km indicate strong barriers.

barrier (compare with Fig. 1). The smaller-scale opposite-polarity secondary features near the left barrier can be easily masked by volumetric strain fields within and near the rupture ends are new subsequent deformation, erosion processes, etc. We assume that features of this study, which may be associated with localized similar differences of small-scale reversed-polarity features will reversed slip (as in the bottom right plot of Fig. 2). Comparing characterize ruptures that propagate from the left through the left Fig. 5(a) with Fig. 5(b), it is seen that the spatial extent of signifi- barrier and arrest at the right barrier. Such and other cases will be cant volumetric strain continues to increase for t > tref. This is examined in a follow up study. especially significant in the dilatational quadrant at the bottom right area near the right barrier at (þ25, 0). In the bottom half- plane (y < 0) immediately to the left of the area with dilatational 3 Observations of Small-Scale Compressional DeV, there is a corresponding increase of the reversed-polarity sec- Structures on the Margins of Larger-Scale ondary deformation feature producing a prominent localized Extensional Steps Along Strike-Slip Faults region of compressive DeV. In the top half-plane (y > 0) to the left There are numerous examples of transtensive steps along of the right barrier, a localized region of dilatational strain merges strike-slip faults, with pull-apart basins, sags, and other structures with the larger scale dilatational DeV to the bottom right of the that accommodate extension, and superposed smaller transpres- barrier. Similar features with reversed signs are exhibited in Figs. sive features within the stepover on one of the bounding faults. In 5(c) and 5(d) near the vicinity of the left barrier. this section, we discuss observations of five such features at dif- Figure 6 shows the temporal evolution of slip-rate and slip pro- ferent scales from the system of sub-parallel strike-slip faults that files along the fault for the case producing a predominately unidir- form the plate boundary in southern California. The examples are ectional pulse. The left rupture front gradually loses its strength as taken from the San Andreas, San Jacinto, Elsinore, and Rose Can- it propagates from the nucleation zone (centered at x ¼15 km) yon faults, which collectively account for most of the motion and traverses the velocity-strengthening transition zone (x < 20 between the Pacific and North American plates in the region. We km). The slip is approximately constant along the fault (with the assume that the observed structures reflect statistical tendencies of exception of the nucleation zone and the near-barrier regions), but multiple earthquake ruptures. We do not attempt to present obser- the slip velocity is strongly asymmetric. When the left rupture tip vations associated with individual ruptures, as this requires careful reaches the barrier, the peak slip rate is 2.5 m/s. The barrier at mappings before and after the ruptures, which are generally not x ¼25 km arrests the left rupture front and generates a healing available. wave that trails the right rupture front, transforming the initially crack-like rupture to a pulse that continues to propagate in the þ x direction. When the pulse reaches the right barrier at x = þ 25 km, 3.1 The San Andreas Fault in the Salton Trough. At the the peak slip rate is 7.5 m/s. This generates a strong healing southern end of the San Andreas fault in the vicinity of the Salton phase that arrests the rupture abruptly. Trough, slip steps southward to the Imperial fault via the Brawley Figure 7 shows contours of the differential volumetric strain Seismic Zone (Fig. 8). The width of this step is on the order of 10 DeV in the pulse case around the barriers at different times. Due to km, which exceeds the largest known stepover width that has the asymmetric nature of the rupture propagation, the time of the been ruptured through by an historical earthquake [15]. Paleoseis- reference volumetric strain field tref in this case has different val- mic data for the Imperial fault [41], the Brawley fault [42], and ues for the left and right barriers. The contours in Figs. 7(a) and the southernmost San Andreas fault [43] show a large earthquake 7(b) for the deformation field near the right barrier exhibit similar on each fault at about AD 1700. However, the events occurred at distributions and trends of DeV as in the crack rupture but with different times during the last highstand of Lake Cahuilla (of higher intensity since the arrest phase in this case is stronger. The which presently only the Salton sea is covered by water), and are fields of DeV at the left barrier are (as expected) much weaker thus not the same earthquake [42], consistent with the assessment (Figs. 7(c) and 7(d)) than those associated with the right barrier or of Wesnousky [44] that this step is too large to rupture through in the crack-like case (Fig. 5). In particular, the reversed-polarity a single earthquake.

Journal of Applied Mechanics MAY 2012, Vol. 79 / 031025-5

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 7 Contours of differential volumetric strain DeV (t > tref) in the case of unilateral pulse-like rupture at (a) tref 1 0.16 s around the right barrier, (b) 17.0 s around the right barrier, (c) tref 1 0.16 s around the left barrier, and (d) 17.0 s around the left barrier. The black line at y 5 0 indicates the slipping portion of the fault. Here tref 5 16.256 s for the differential fields around the right barrier ((a), (b)) and tref 5 5.552 s for the left barrier ((c), (d)). See text for additional explanations.

Along the San Andreas fault adjacent to the Salton Sea, and well been elevated, presumably during large earthquakes on the southern within the region of the step, there are a series of transpressive San Andreas fault. The scales and polarities of the Salton Trough structures including, from NW to SE, the Indio Hills, Mecca Hills, and discussed pressure ridges are consistent with the calculated dif- and Durmid Hill. The latter is very close to the southernmost end of ferential volumetric strain fields in the region y < 0ofFig.7(b). the San Andreas fault, and active uplift is exhibited at Bat Cave To the south of the step, the Imperial fault is marked by a very Buttes, where the late Holocene shorelines of Lake Cahuilla have linear, localized zone that has ruptured at least two times

Fig. 8 The southern end of the San Andreas fault with a large transtensive step across the Brawley Seismic Zone to the Imperial fault that continues right lateral shear southward into Mexico. A series of transpressive steps are present to the northwest of the ~10 km step and include Durmid Hill, the Mecca Hills, and the Indio Hills.

031025-6 / Vol. 79, MAY 2012 Transactions of the ASME

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 9 The Rose Canyon fault in southern California terminates in the vicinity of San Diego Bay, a depression cause by the ~10 km right step to the Descanso fault. Mt. Soledad, which is a large transpressive uplift, and several small pressure ridges exist in the Rose Canyon fault to the northwest of the step. At least one small transpressive step is present just north of the San Diego River. Paleoseismic investigations demonstrate that it is a late Quaternary feature [48].

historically (1979, 1940). The north end is marked by a region of mar fault, there are small areas of active uplift, or pressure ridges, transtension with oblique normal displacement in Mesquite Basin that represent zones of transpression immediately next to the large [42]. Within the step, the Brawley seismic zone marks the active zone of transtension. We observe no similar features along the area of deformation, with both strike-slip and normal fault epicen- Glen Ivy fault, which itself exhibits oblique normal displacement ters, although most earthquakes appear to occur on northeast- along most of its length [50,51]. As in the previous examples, striking cross faults [45]. There are no known zones of uplift there is an apparent asymmetry in the presence of small transpres- within the Imperial or Brawley fault zones that would indicate sive structural features adjacent to and within a larger transtensive local transpression, so the observations of transpressive structures step. This situation corresponds to an opposite case to that calcu- along the southernmost San Andreas fault appear to be asymmet- lated in Figs. 6 and 7. ric in reference to the stepover itself. 3.4 The Hemet Stepover on the San Jacinto Fault. The 3.2 The Mt. Soledad Uplift on the Rose Canyon Fault. San Hemet stepover along the San Jacinto fault is one of the best- Diego Bay represents a large releasing stepover between the Rose described steps in a major strike-slip fault zone. The step separates Canyon fault to the north and the Descanso fault to the south. Far- the Clark-Casa Loma fault to the southeast from the Claremont ther south, the Descanso fault feeds southward into the Agua strand to the northwest (Fig. 11) and has a stepover width of sev- Blanca fault, where it comes on shore south of Ensenada. The eral kilometers. The shallow and deep structures in the area have stepover at San Diego Bay is nearly 10 km in width (Fig. 9), been investigated with seismic reflection profiling [52,53], micro- which is considered large enough to terminate any large earth- seismicity [54], and CPT soundings [55]. Paleoseismic observa- quake along the Rose Canyon fault [44]. Displacement at Rose tions are known for the Clark fault to the south [56] and are Creek in the most recent earthquake is about 3 m, as determined currently being collected to the north along the Claremont fault by 3D trenching [46]. Of interest are a series of small pressure [57]. Based on these studies, the Hemet stepover appears to be ridges along the fault [46,47] between the Mt. Soledad uplift and evolving towards a straighter course [55] and some large events the San Diego Bay step (Fig. 9 inset), one of which was trenched may rupture through the step [57]. for determination of fault activity in the 1970s and shown to be South of the step, there is a series of small transpressive struc- active in the late Quaternary [48]. tures, including Park Hill in eastern Hemet, and several small To the south of the San Diego Bay step, there are no equivalent unnamed uplifts closer to Mystic Lake (Fig. 11). The latter repre- transpressive features that have been recognized along the Des- sents the active locus of sedimentation and is presumably the part canso fault, again suggesting asymmetry of the secondary of the step which is subsiding most rapidly [52]. To the north of reversed-polarity deformation features. However, the Descanso Mystic Lake, there is no evidence for any small transpressive fault lies offshore and small transpressive features may not be structures in the vicinity of the step. A small sag feature on the well-expressed or preserved. northeast margin of Mystic Lake [57] appears to be an additional transtensive structural feature superposed on the larger Hemet 3.3 The Elsinore Lake Stepover. Hull [49] documented the stepover. The linear uplift of the San Timateo badlands to the presence of three major releasing steps along the Elsinore fault in north of the step likely also represents transpressive deformation the Elsinore Trough between Wolf Valley and the Santa Ana associated with the southern bend in the San Andreas fault at Ban- River. Two of these steps have apparently been bypassed and are ning Pass. In any case, we observe small transpressive features now inactive, whereas Lake Elsinore itself represents the third only to the south of Mystic Lake and within the overall “Hemet step and continues to exhibit active subsidence with about 1 km of step”, so at the scale of the step there again appears to be asymme- accumulated Quaternary sediments. try in the presence of small transpressive structures within and ad- Figure 10 shows the Lake Elsinore step, with the Wildomar jacent to the large transtensive step. This situation corresponds to fault on the southeast and the Glen Ivy fault to the northwest. Ad- a case that is opposite of that calculated in Figs. 6 and 7 (i.e., a jacent to the lake itself along the northernmost part of the Wildo- predominantly left-propagating rupture pulse).

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Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 10 The Elsinore fault with ~2.5 km releasing step at Lake Elsinore. Several small pressure ridges are present along the Wildomar fault on the southeast side of the step.

Fig. 11 The San Jacinto fault zone near Hemet with a large releasing step referred to as the Hemet stepover. Along the Casa Loma strand, several small pressure ridges are present south- east of Mystic Lake, the locus of deep sedimentary fill.

3.5 The Hog Lake Small-Scale Step. The final example is a 4 Discussion small pressure ridge located within a hundred meter scale stepover at Hog Lake on the San Jacinto fault near Anza [56]. Figure 12 We used numerical simulations to study basic types of deforma- shows this small transpressive structure relative to the releasing tion structures produced by interactions of dynamic ruptures on step. Extensive trenching within and south of the step has not strike-slip faults with barriers at segment ends representing step- exposed any similar evidence of transpression, so this feature is over regions. The results show that, in addition to the classical pri- exclusively to the north of the step. From the paleoseismic obser- mary structures associated with the overall sense of volumetric vations, it appears that transpression is not observed in all surface stress change near the rupture ends, there are reversed-polarity ruptures: the small pressure ridge grew during events 3, 4, and 5 secondary features that reflect the stopping phases of the ruptures. but subsided during the most recent two ruptures. This behavior is A symmetric bilateral rupture with abrupt stopping at both ends interesting and may reflect the characteristics of the individual produces primary and secondary structures that are pronounced ruptures, including their direction of rupture. near both barriers (Fig. 5). In contrast, a predominantly

031025-8 / Vol. 79, MAY 2012 Transactions of the ASME

Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Fig. 12 The Hog Lake paleoseismic site near Anza associated with a 100 m scale releasing step. A small pressure ridge to the north of the step has produced folding and off-fault thrusting during a subset of the events observed at the site.

unidirectional rupture having high slip-rate in the main propaga- metric rock damage across the fault [59–64], asymmetric along- tion direction that is arrested abruptly at one barrier, and low slip- strike aftershock sequences [65,66], asymmetric branching struc- rate in the opposite direction with gradual arrest, produces defor- tures with respect to the main fault [67,68], and direct imaging of mation structures that are pronounced only near the rupture end in many small ruptures on a fault [69–71]. the main propagation direction (Fig. 7). In particular, the We described five examples at different scales from strike-slip reversed-polarity secondary features in the opposite direction are faults of the San Andreas system in southern California, with unlikely to be preserved and observed, leading to strongly asym- overall transtensive steps and related classical extensional struc- metric field configurations as in Figs. 8–12. tures [6,7,72], and superposed transpressive structures that have The two simulations presented in this work (Figs. 4–7) are sim- much smaller scales than the releasing steps. The secondary ple end-member 2D cases designed to highlight the different de- reversed-polarity features in each of the presented examples exist formation structures that are expected for symmetric and strongly on only one side of the step, so there is a clear asymmetry of the asymmetric ruptures. There are many possible variations that can type shown in Fig. 7 for a predominantly unidirectional rupture. influence the generated structures including bilateral ruptures with We note that the asymmetry observed for the southern San abrupt stopping at one end and gradual stopping at the other, over- Andreas fault (Fig. 8) may be produced, at least in part, by a grad- lapping fault segments and other geometrical configurations, rup- ual arrest of earthquakes that propagate toward the step from the tures that are subshear in one direction and supershear in the south. This may be expected given the relatively high heat flow other, etc. In addition, 3D cases that account for free surface and associated low seismic coupling coefficient in the Brawley effects, and off-fault yielding rheologies that allow for inelastic Seismic Zone and Imperial fault [73–75]. However, Zaliapin and volumetric changes [24,58], are expected to modify the near- Ben-Zion [66] observed asymmetric seismicity on the southern barrier strain fields. While the results associated with such cases San Andreas fault, which is consistent with a preferred direction should be clarified by future studies, they are unlikely to change of earthquake ruptures to the SE and predicted behavior of bima- our interpretation that the reversed-polarity secondary features are terial ruptures [29,76–80] associated with the local velocity con- associated with reversed transient stress field near the barriers pro- trast in the area [81,82]. In this and other examples, careful duced by arrest phases of ruptures. Field observations of reversed- measurements of slip gradients on the opposite sides of the step- polarity secondary structures that exist only at one side of a step- over, mapping of rock damage across the faults, examination of over region provide a possible indicator of frequent predominately symmetry properties of seismicity, and analysis of other signals unidirectional ruptures. This adds to the list of signals that may be can identify the key mechanism(s) responsible for the strong used to infer on a statistically preferred propagation direction of asymmetry of the reversed-polarity secondary structures. In earthquakes on given fault sections. Other signals include asym- Sec. 3, we also noted that the observed asymmetry for the Rose

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Downloaded 28 Apr 2012 to 128.125.230.113. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Canyon fault (Fig. 9) may reflect reduced resolution on the off- In summary, we presented results from numerical simulations shore structures. Sites with comparable good exposures and simi- and field observations that small reversed-polarity deformation lar conditions at both sides of the stepover, as in Figs. 10–12, are features, superposed in stepover regions on larger classical struc- clearly preferable. tures, may indicate common direction of earthquake ruptures on There are no historical observations of large earthquakes to the master faults. Recognition of this signal may increase the indicate the likely direction of ruptures on the southern San number and type of features that can be used to infer the probable Andreas, Elsinore, or Rose Canyon faults. With the above caveats rupture propagation direction for large earthquakes on a given we can make the following tentative inferences, assuming that the fault section. As usual, interpreting from observations the generat- reversed-polarity small-scale structural features reflect at least ing mechanism is subjected to nonuniqueness and other processes partially statistically preferred rupture directions. For the southern (e.g., a gradual arrest process on one side of a stepover) may also San Andreas fault, the presence of transpressive structures to the produce asymmetry of the discussed features. Multisignal obser- northwest of the Salton Sea step (Fig. 8) is consistent with com- vations associated with different geological and seismological mon NW to SE ruptures that terminate rapidly at the Salton data can reduce from the nonuniqueness and help to identify the Trough. See also Table 2 and Fig. 9 of Zaliapin and Ben-Zion key operating mechanism(s). Because rupture directivity can [66]. If this is correct, the ground motion in Calexico and Mexicali amplify the ground motion in the propagation direction by a factor would often be amplified by rupture directivity effects. For the 3 or more, it is important to continue to develop improved under- Rose Canyon Fault, the observed configuration (Fig. 9) is consist- standing of signals that reflect ruptures directions. More detailed ent with common southward propagating ruptures. If correct, the theoretical parameter-space study and field observations on shaking in San Diego would likely be amplified by directivity reversed-polarity secondary deformation structures will be pre- effects. For the Elsinore step, the presence of the small transpres- sented in follow-up works. sive structures only to the southwest of the releasing step (Fig. 10) implies common SE to NW ruptures that are arrested quickly at Acknowledgment Lake Elsinore. For the examples from the San Jacinto fault, the sense of asym- It is a pleasure and privilege to contribute this paper to a special metry at the Hemet stepover (Fig. 11) is consistent with the infer- volume in honor of Jim Rice who pioneered, among other things, ences of Dor et al. [61] from rock damage asymmetry and the development of bridges between mechanics and geological de- Zaliapin and Ben-Zion [66] from along-strike asymmetry of seis- formation. The study was supported by the National Science micity on a preferred direction of ruptures from SE to NW. How- Foundation (Grant Nos. EAR-0908903, EAR-0944066, and EAR- ever, the small pressure ridge at Hog Lake (Fig. 12) is consistent 0908515). with opposite direction of ruptures. Paleoseismic studies at Hog Lake have shown that transpression was produced only during events 3, 4, and 5, whereas events 2 and 1 are expressed as trans- Nomenclature tension in the same trenches and adjacent to the areas that previ- x ¼ fault-parallel position ously experienced transpression. Earlier transpressive deformation y ¼ fault-normal position is also locally evident in some trenches for much older events. t ¼ time This is particularly interesting when we assess what is known of Cs ¼ shear wave speed these events. G ¼ shear modulus Events 1 and 2 have clear geomorphic expression and are inter- ¼ Poisson’s ratio preted to be associated with rupture of the entire Clark fault from dx ¼ model grid spacing Clark Valley to Hemet [83], with maximum displacements near Lnucl ¼ nucleation zone size 0 Anza of 3-4 m. Their time separation is consistent with the rij ¼ initial stress components observed recurrence interval of the San Jacinto fault. In contrast, Du ¼ slip along fault events 3, 4, and 5 occurred as a cluster within about a hundred V ¼ slip velocity on a frictional fault year interval. This is somewhat an enigma but may be explained w ¼ state variable in rate-state friction if each of these events represents rupture of the segment to the f(V, w) ¼ rate-state friction coefficient NW of Hog Lake and that displacement terminated in the Anza a ¼ direct effect parameter in rate-state friction seismicity gap. The 1918 earthquake apparently did this [83], b ¼ evolution effect parameter in rate-state friction although the 1918 rupture probably did not reach quite as far V0 ¼ reference slip rate in rate-state friction south as Hog Lake, terminating with a km or two to the NW. It is Vw ¼ weakening slip rate in rate-state friction possible that two (or maybe all three) of events 3-5 terminated f0 ¼ steady-state friction coefficient at V0 in rate-state close to Hog Lake and that the small pressure ridge represents the friction southward-dying ruptures from these earthquakes. fw ¼ fully weakened friction coefficient in rate-state friction A possible scenario is that earthquakes in the area tend to nucle- L ¼ state evolution distance in rate-state friction ate just NW of Hog Lake, and have a predominant propagation wss(V) ¼ steady-state friction state variable at slip velocity V in direction to the NW and a small dying component to the SE as in rate-state friction the case shown in Fig. 6. This can reconcile the observations of fss(V) ¼ steady-state friction coefficient at slip velocity V in rate- Dor et al. [61] and Zaliapin and Ben-Zion [66] on asymmetry of state friction rock damage and seismicity, tomographic images of velocity con- f ¼ friction coefficient in slip-weakening friction trast across the San Jacinto [82,84], and the observed small fs ¼ static friction coefficient in slip-weakening friction pressure-ridge at Hog Lake. Events 1 and 2, which ruptured the fd ¼ dynamic friction coefficient in slip-weakening friction entire Clark fault, may have nucleated farther south and simply Dc ¼ characteristic slip distance in slip-weakening friction ruptured through Hog Lake, and therefore did not affect the small / ¼ angle of internal friction coefficient in Drucker-Prager transpressive feature within Hog Lake. We note that there are no yielding similar transpressive features immediately SE of the small Hog c ¼ cohesion in Drucker-Prager yielding step. It exists only to the NW. We also note that the three events Tv ¼ viscoplasticity relaxation time that show clear transpression in the sediments of Hog Lake all sij ¼ deviatoric stress components occurred within a cluster in a short time interval. This suggests rij ¼ stress components that they belong to a different class of events. We will test further dij ¼ Kronecker delta and refine our inferences using results from ongoing detailed tref ¼ reference time of rupture front arrival observational studies in the area. DeV ¼ differential volumetric strain around barriers

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