Yehuda Ben-Zion
Department of Earth Sciences,
University of Southern California,
Los Angeles, CA 90089-0740
Thomas K. Rockwell
Zheqiang Shi
Department of Geological Sciences,
San Diego State University,
San Diego, CA 92182-1020
Shiqing Xu
Department of Earth Sciences,
University of Southern California,
Los Angeles, CA 90089-0740
1
Reversed-Polarity Secondary
Deformation Structures Near
Fault Stepovers
We study volumetric deformation structures in stepover regions using numerical simulations 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
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
back into the rupture segment. Dynamic rupture propagating as a symmetric bilateral
crack produces similar (anti-symmetric) structures at both rupture ends. In contrast, rupture in the form of a predominantly unidirectional pulse produces pronounced reversedpolarity 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 system in southern California illustrate the existence of reversed-polarity secondary deformation structures. In the examples shown, relatively-small pressure-ridges are seen only
on one side of relatively-large extensional stepovers. This suggests frequent predominantly unidirectional ruptures in at least some of those cases, although multisignal observations 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]
Introduction
Earthquake faults are geometrically complex and consist of collections of segments having various sizes [1–3]. The ends of fault
segments form partial or complete barriers for rupture propagation
and are generally regions of stress concentration and generation of
related deformation structures. In the present paper, we focus on
stepover regions of large strike-slip faults where neighboring segments are separated by a lateral offset. Burchfiel and Stewart [4]
and Crowell [5] pointed out that right- and left-stepping of rightlateral faults produce, respectively, regions with extensional and
compressive stress fields (Fig. 1). The regions between the rightand left-stepping segments of right-lateral faults are expected to
be associated generally with structures referred to as pull-apart
basins and pressure-ridges, respectively. The expected stress fields
and deformation features are reversed for corresponding leftlateral faults. These expectations should be applicable over a wide
range of scales because the equations of elasticity governing the
amplitudes and signs of the stress field are scale-free. There is a
long and productive history of using the forgoing ideas in numerous field and laboratory studies involving different stress regimes
and scales [6–9]. Various studies attempted to estimate the likelihoods that earthquake ruptures can propagate across stepovers as
functions of geometrical and rheological parameters [10–12] and
tested these estimates with field observations [13–16].
If a fault stepover acts as a barrier for earthquake ruptures, the
co-seismic slip distribution will decrease to zero near the stepover.
As illustrated in Fig. 2 for a right-lateral case (top plot), this can
be represented as a superposition of right-lateral motion with high
slip over essentially the entire fault (bottom left), combined with
left-lateral slip that is concentrated near the barriers (bottom
right). The slip profile in Fig. 2 top is similar to the slip distribution simulated in Sec. 2 for a crack-like rupture. While the particular superposition in Fig. 2 bottom is nonunique, it provides an
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 shortManuscript received August 11, 2011; final manuscript received October 21,
2011; accepted manuscript posted February 21, 2012; published online April 6,
2012. Assoc. Editor: Huajian Gao.
Journal of Applied Mechanics
wavelength (high-frequency) wavefield generated by earthquakes
[17,18]. The relative length-scales, amplitudes, and signs of the
opposite slip components in the schematic representation of Fig. 2
suggest that stepovers acting as strong barriers should produce
relatively-large “classical” pull-apart basins and pressure-ridges
within the stepover regions, combined with relatively-small
reversed-polarity superposed structures near the barriers.
The existence and amplitude of the reversed-polarity secondary
deformation structures near the different segment-ends in stepover
regions should depend on the abruptness of the stopping process
and directivity of the ruptures. Abrupt stopping processes are naturally expected to produce more localized and clearer features
than gradual processes. Approximately symmetric ruptures that
propagate bilaterally and generate considerable arrest phases at
both segment ends are expected to produce significant reversedpolarity features near both barriers. In contrast, ruptures that are
predominately unidirectional are likely to produce significant
reversed-polarity deformation feature only near the barrier in the
dominant propagation direction. Clarifying the conditions leading
to different types of reversed-polarity deformation features near
segment ends can improve the ability of deriving from field observations information on properties of earthquake ruptures and
expected seismic motion in different locations.
In Sec. 2, we demonstrate with numerical simulations the basic
properties of reversed-polarity small-scale deformation structures
near stepovers generated by symmetric and predominantly unidirectional dynamic ruptures. In Sec. 3, we present several observational examples of such features from the San Andreas, San
Jacinto, Elsinore, and Rose Canyon faults in southern California.
The results contribute to the development of better links between
models of earthquake ruptures and fault geomorphology. The
reversed-polarity deformation features in stepover regions augment the list of signals that may be used to infer on likely propagation directions of earthquake ruptures on given fault sections.
2
Numerical Modeling
The essential behavior associated with propagation and arrest
of dynamic ruptures on the main fault of Fig. 1 can be studied
using the 2D plane-strain model of Fig. 3 for Mode-II ruptures on
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Fig. 1 Large-scale volumetric stresses expected in fault stepover regions
a finite fault segment bounded by strong barriers at both ends.
We perform numerical simulations for this model using the
Support Operator Rupture Dynamics (SORD) code [19,20],
with further addition of rate-and-state friction on the fault and
Drucker-Prager viscoplasticity in the bulk. To illustrate the interaction of different types of dynamic ruptures with strong barriers,
we simulate two end-member cases: (1) a symmetric bilateral
crack-like rupture with slip-weakening friction and (2) a predominantly unilateral pulse-like rupture with strongly rate-weakening
friction.
To produce bounded elastic stresses near the barriers and
reduce numerical noise, we incorporate inelastic yielding in the
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 dilatancy to shear localization. This framework was used in several
recent studies of earthquake ruptures with off-fault yielding. In
particular, Templeton and Rice [25] demonstrated that a realistic
amount of dilatancy helps to suppress shear localization, in agreement with the Rudnicki and Rice theory, and Viesca et al. [26]
showed that in the presence of pore fluids the incorporation of
dilatancy can affect significantly the elastic-plastic response.
Since our calculations employ the standard Drucker-Prager plasticity, the volumetric deformation generated by dynamic ruptures
in the results below is purely elastic.
As shown in Fig. 3, the fault line is located at y ¼ 0. The shear
strength of the fault is set to be sufficiently large for j xj 25 km
so that sliding occurs only within the confined segment of –25
km < x < 25 km. In the crack-like rupture case, the fault is governed by a slip-weakening friction where the friction coefficient f
linearly decreases [27–29] with slip Du from static value fs to
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 barriers (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 jx j 25 km confine the ruptures to the finite segment between the yellow stars. A 3 km wide nucleation zone is centered at x 5 0 and x 5 215 km for cases producing bilateral crack
and unidirectional pulse, respectively.
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f ¼
fs ðfs fd ÞDu=Dc
fd
for Du < Dc
for Du Dc
(1)
antee numerical convergence and resolve properly the dynamical
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-andstate dependent friction with strong rate weakening. The friction
coefficient f ðV; wÞ is a function of slip velocity V and state variable w [30–34]
rffiffiffiffiffiffiffiffiffiffiffi
1
1
sij sij rkk sin / þ c cos /
2
3
f ðV; wÞ ¼ a lnðV=V0 Þ þ w
where sij ¼ rij 13dij rkk are the deviatoric stress components, rkk
is the first stress invariant, / is the angle of internal coefficient of
friction, and c is the cohesion. A Maxwellian viscoplasticity
scheme is used, so during yielding the return of the stress state to
the yield surface occurs smoothly over a relaxation time Tv set to
be the time for S wave to traverse one grid size [39]. As discussed
by Templeton and Rice [25], the Drucker-Prager yield envelop
approximates the Mohr-Coulomb yield criterion in certain 2D
stress conditions. We set the initial out-of-plane normal stress r0zz
to be the average of the other two initial in-plane normal stresses
r0xx and r0yy , such that these two yield criteria match initially but
then generally deviate as plastic yielding occurs. For simplicity,
we use in this study r0xx ¼ r0yy ¼ r0zz ¼ 100 MPa and cohesion
c ¼ 0. The values for shear stress r0xy , which is applied in a rightlateral 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.
The ruptures are initiated by gradually increasing the shear traction along a 3 km nucleation zone to slightly above the static shear
strength. In the simulation associated with a symmetric bilateral
crack rupture, the nucleation zone is centered at x ¼ 0 (red zone in
Fig. 3). In the case designed to produce predominantly unilateral
pulse (in the þ x direction), the nucleation zone is centered at
x ¼ 15 km (cyan zone in Fig. 3) and the friction properties to the
left of x ¼ 20 km change gradually from velocity-weakening to
velocity-strengthening to suppress the left-propagating rupture
front.
Figure 4 shows the temporal evolution of slip-rate and slip profiles along the fault during the propagation and arrest of a dynamic
crack-like rupture. Starting from the nucleation zone around x ¼ 0,
the rupture expands symmetrically in both directions with increasing slip rates at the propagating tips. The barriers at x ¼ 625 km
produce sudden arrest of right-lateral slip at the arrivals of the rupture fronts with peak slip rates of 5.8 m/s. This generates strong
reflected stress waves that trigger arrest phases, which propagate
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
stress, in a case similar to that leading to Fig. 4 but with elastic
off-fault response and smaller grid size of 20 m, which illustrates
the discussed stages of the rupture process.
To study the interaction between the dynamic rupture and
strong barriers on the deformation field in the vicinity of the barriers, we examine the differential volumetric strain DeV around the
barriers. This is obtained by subtracting a reference volumetric
strain field at time t ¼ tref from the volumetric strain fields at
t > tref, where tref is the time when the rupture front tip nominally
reaches the barrier. The use of differential rather than total strain
helps to identify small local changes that may be difficult to see
on the background of larger regional variations. This is analogous
to plotting a small topographic feature on a high mountain (e.g.,
the Tibetan plateau) with respect to the local environment rather
than the sea level.
Figure 5 shows contours of DeV near the right barrier (top plots)
and left barriers (bottom plots) at two different times (left and
right plots) for the bilateral crack rupture of Fig. 4. The red and
blue colors denote relative compression and tension, respectively.
As expected, the differential fields at the left and right barriers are
anti-symmetric. The relatively large volumetric strain fields that
extend ahead of the segment ends are the classical features of
stepover regions, i.e., pressure-ridge and pull-apart-basin above
and below the right barrier and the other way around for the left
(2a)
with evolution of state variable governed by the slip law
V
w_ ¼ ½w wss ðVÞ
L
and
wss ðVÞ ¼ a ln
2V0
fss ðVÞ
sinh
a
V
(2b)
In Eq. (2b), fss ðVÞ is the steady-state friction coefficient at slip
velocity V given [35,36] by
(
fss ðVÞ ¼
f0 ðb aÞlnðV=V0 Þ
for V Vw
fw þ ½f0 ðb aÞlnðV=V0 Þ fw Vw =V
for V > Vw
(2c)
The definitions and values of the various friction parameters are
summarized in Table 1. The parameter values are chosen based on
previous studies [37,38] to produce subsonic crack- and pulsetype ruptures.
The size of the process zone for strength reduction on a fault
governed by slip-weakening friction can be estimated using the
formula of Palmer and Rice [27]. A similar estimate can be made
for the rate-and-state friction by approximating the frictional
behavior with equivalent slip-weakening parameters [38]. Using
the employed parameters (Table 1), we estimate the process zone
to be 1500 m in the crack-like rupture case with slip-weakening
friction and 700 m in the pulse-like rupture case with rate-andstate friction. The simulations are done with a grid spacing of 50
m, which is sufficiently small compared to both estimates to guar-
Table 1 Employed physical and numerical parameters in simulations with rate-and-state dependent (RSD) or slip-weakening
(SW) friction and Drucker-Prager (DP) viscoplasticity
Parameter
Shear wave speed Cs
Shear modulus G
Poisson’s ratio Model grid spacing (uniform) dx
Nucleation zone size Lnucl
Initial shear stress r0xy
Direct effect parameter (RSD) a
Evolution effect parameter (RSD) b
Steady-state friction coefficient at V0 (RSD) f0
Fully weakened friction coefficient (RSD) fw
State evolution distance (RSD) L
Reference slip rate (RSD) V0
Weakening slip rate (RSD) Vw
Static friction coefficient (SW) fs
Dynamic friction coefficient (SW) fd
Characteristic slip distance (SW) Dc
Internal friction coefficient (DP) tanð/Þ
Viscoplasticity relaxation time (DP) Tv
Journal of Applied Mechanics
Value
3464 m=s
32.04 GPa
0.25
50 m
3000 m
45 MPa (crack),
30 MPa (pulse)
0.01
0.014
0.6
0.2
0.4 m
1 lm=s
0.2 m=s
0.6
0.4
0.4 m
0.8
0.0144 s
(3)
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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 5 625 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.
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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 5 625 km indicate strong barriers.
barrier (compare with Fig. 1). The smaller-scale opposite-polarity
volumetric strain fields within and near the rupture ends are new
features of this study, which may be associated with localized
reversed slip (as in the bottom right plot of Fig. 2). Comparing
Fig. 5(a) with Fig. 5(b), it is seen that the spatial extent of significant volumetric strain continues to increase for t > tref. This is
especially significant in the dilatational quadrant at the bottom
right area near the right barrier at (þ25, 0). In the bottom halfplane (y < 0) immediately to the left of the area with dilatational
DeV, there is a corresponding increase of the reversed-polarity secondary deformation feature producing a prominent localized
region of compressive DeV. In the top half-plane (y > 0) to the left
of the right barrier, a localized region of dilatational strain merges
with the larger scale dilatational DeV to the bottom right of the
barrier. Similar features with reversed signs are exhibited in Figs.
5(c) and 5(d) near the vicinity of the left barrier.
Figure 6 shows the temporal evolution of slip-rate and slip profiles along the fault for the case producing a predominately unidirectional pulse. The left rupture front gradually loses its strength as
it propagates from the nucleation zone (centered at x ¼ 15 km)
and traverses the velocity-strengthening transition zone (x < 20
km). The slip is approximately constant along the fault (with the
exception of the nucleation zone and the near-barrier regions), but
the slip velocity is strongly asymmetric. When the left rupture tip
reaches the barrier, the peak slip rate is 2.5 m/s. The barrier at
x ¼ 25 km arrests the left rupture front and generates a healing
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,
the peak slip rate is 7.5 m/s. This generates a strong healing
phase that arrests the rupture abruptly.
Figure 7 shows contours of the differential volumetric strain
DeV in the pulse case around the barriers at different times. Due to
the asymmetric nature of the rupture propagation, the time of the
reference volumetric strain field tref in this case has different values for the left and right barriers. The contours in Figs. 7(a) and
7(b) for the deformation field near the right barrier exhibit similar
distributions and trends of DeV as in the crack rupture but with
higher intensity since the arrest phase in this case is stronger. The
fields of DeV at the left barrier are (as expected) much weaker
(Figs. 7(c) and 7(d)) than those associated with the right barrier or
the crack-like case (Fig. 5). In particular, the reversed-polarity
Journal of Applied Mechanics
secondary features near the left barrier can be easily masked by
subsequent deformation, erosion processes, etc. We assume that
similar differences of small-scale reversed-polarity features will
characterize ruptures that propagate from the left through the left
barrier and arrest at the right barrier. Such and other cases will be
examined in a follow up study.
3 Observations of Small-Scale Compressional
Structures on the Margins of Larger-Scale
Extensional Steps Along Strike-Slip Faults
There are numerous examples of transtensive steps along
strike-slip faults, with pull-apart basins, sags, and other structures
that accommodate extension, and superposed smaller transpressive features within the stepover on one of the bounding faults. In
this section, we discuss observations of five such features at different scales from the system of sub-parallel strike-slip faults that
form the plate boundary in southern California. The examples are
taken from the San Andreas, San Jacinto, Elsinore, and Rose Canyon faults, which collectively account for most of the motion
between the Pacific and North American plates in the region. We
assume that the observed structures reflect statistical tendencies of
multiple earthquake ruptures. We do not attempt to present observations associated with individual ruptures, as this requires careful
mappings before and after the ruptures, which are generally not
available.
3.1 The San Andreas Fault in the Salton Trough. At the
southern end of the San Andreas fault in the vicinity of the Salton
Trough, slip steps southward to the Imperial fault via the Brawley
Seismic Zone (Fig. 8). The width of this step is on the order of 10
km, which exceeds the largest known stepover width that has
been ruptured through by an historical earthquake [15]. Paleoseismic data for the Imperial fault [41], the Brawley fault [42], and
the southernmost San Andreas fault [43] show a large earthquake
on each fault at about AD 1700. However, the events occurred at
different times during the last highstand of Lake Cahuilla (of
which presently only the Salton sea is covered by water), and are
thus not the same earthquake [42], consistent with the assessment
of Wesnousky [44] that this step is too large to rupture through in
a single earthquake.
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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
within the region of the step, there are a series of transpressive
structures including, from NW to SE, the Indio Hills, Mecca Hills,
and Durmid Hill. The latter is very close to the southernmost end of
the San Andreas fault, and active uplift is exhibited at Bat Cave
Buttes, where the late Holocene shorelines of Lake Cahuilla have
been elevated, presumably during large earthquakes on the southern
San Andreas fault. The scales and polarities of the Salton Trough
and discussed pressure ridges are consistent with the calculated differential volumetric strain fields in the region y < 0 of Fig. 7(b).
To the south of the step, the Imperial fault is marked by a very
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.
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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
transtension with oblique normal displacement in Mesquite Basin
[42]. Within the step, the Brawley seismic zone marks the active
area of deformation, with both strike-slip and normal fault epicenters, although most earthquakes appear to occur on northeaststriking cross faults [45]. There are no known zones of uplift
within the Imperial or Brawley fault zones that would indicate
local transpression, so the observations of transpressive structures
along the southernmost San Andreas fault appear to be asymmetric in reference to the stepover itself.
3.2 The Mt. Soledad Uplift on the Rose Canyon Fault. San
Diego Bay represents a large releasing stepover between the Rose
Canyon fault to the north and the Descanso fault to the south. Farther south, the Descanso fault feeds southward into the Agua
Blanca fault, where it comes on shore south of Ensenada. The
stepover at San Diego Bay is nearly 10 km in width (Fig. 9),
which is considered large enough to terminate any large earthquake along the Rose Canyon fault [44]. Displacement at Rose
Creek in the most recent earthquake is about 3 m, as determined
by 3D trenching [46]. Of interest are a series of small pressure
ridges along the fault [46,47] between the Mt. Soledad uplift and
the San Diego Bay step (Fig. 9 inset), one of which was trenched
for determination of fault activity in the 1970s and shown to be
active in the late Quaternary [48].
To the south of the San Diego Bay step, there are no equivalent
transpressive features that have been recognized along the Descanso fault, again suggesting asymmetry of the secondary
reversed-polarity deformation features. However, the Descanso
fault lies offshore and small transpressive features may not be
well-expressed or preserved.
3.3 The Elsinore Lake Stepover. Hull [49] documented the
presence of three major releasing steps along the Elsinore fault in
the Elsinore Trough between Wolf Valley and the Santa Ana
River. Two of these steps have apparently been bypassed and are
now inactive, whereas Lake Elsinore itself represents the third
step and continues to exhibit active subsidence with about 1 km of
accumulated Quaternary sediments.
Figure 10 shows the Lake Elsinore step, with the Wildomar
fault on the southeast and the Glen Ivy fault to the northwest. Adjacent to the lake itself along the northernmost part of the WildoJournal of Applied Mechanics
mar fault, there are small areas of active uplift, or pressure ridges,
that represent zones of transpression immediately next to the large
zone of transtension. We observe no similar features along the
Glen Ivy fault, which itself exhibits oblique normal displacement
along most of its length [50,51]. As in the previous examples,
there is an apparent asymmetry in the presence of small transpressive structural features adjacent to and within a larger transtensive
step. This situation corresponds to an opposite case to that calculated in Figs. 6 and 7.
3.4 The Hemet Stepover on the San Jacinto Fault. The
Hemet stepover along the San Jacinto fault is one of the bestdescribed steps in a major strike-slip fault zone. The step separates
the Clark-Casa Loma fault to the southeast from the Claremont
strand to the northwest (Fig. 11) and has a stepover width of several kilometers. The shallow and deep structures in the area have
been investigated with seismic reflection profiling [52,53], microseismicity [54], and CPT soundings [55]. Paleoseismic observations are known for the Clark fault to the south [56] and are
currently being collected to the north along the Claremont fault
[57]. Based on these studies, the Hemet stepover appears to be
evolving towards a straighter course [55] and some large events
may rupture through the step [57].
South of the step, there is a series of small transpressive structures, including Park Hill in eastern Hemet, and several small
unnamed uplifts closer to Mystic Lake (Fig. 11). The latter represents the active locus of sedimentation and is presumably the part
of the step which is subsiding most rapidly [52]. To the north of
Mystic Lake, there is no evidence for any small transpressive
structures in the vicinity of the step. A small sag feature on the
northeast margin of Mystic Lake [57] appears to be an additional
transtensive structural feature superposed on the larger Hemet
stepover. The linear uplift of the San Timateo badlands to the
north of the step likely also represents transpressive deformation
associated with the southern bend in the San Andreas fault at Banning Pass. In any case, we observe small transpressive features
only to the south of Mystic Lake and within the overall “Hemet
step”, so at the scale of the step there again appears to be asymmetry in the presence of small transpressive structures within and adjacent to the large transtensive step. This situation corresponds to
a case that is opposite of that calculated in Figs. 6 and 7 (i.e., a
predominantly left-propagating rupture pulse).
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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 southeast of Mystic Lake, the locus of deep sedimentary fill.
3.5 The Hog Lake Small-Scale Step. The final example is a
small pressure ridge located within a hundred meter scale stepover
at Hog Lake on the San Jacinto fault near Anza [56]. Figure 12
shows this small transpressive structure relative to the releasing
step. Extensive trenching within and south of the step has not
exposed any similar evidence of transpression, so this feature is
exclusively to the north of the step. From the paleoseismic observations, it appears that transpression is not observed in all surface
ruptures: the small pressure ridge grew during events 3, 4, and 5
but subsided during the most recent two ruptures. This behavior is
interesting and may reflect the characteristics of the individual
ruptures, including their direction of rupture.
031025-8 / Vol. 79, MAY 2012
4
Discussion
We used numerical simulations to study basic types of deformation structures produced by interactions of dynamic ruptures on
strike-slip faults with barriers at segment ends representing stepover regions. The results show that, in addition to the classical primary structures associated with the overall sense of volumetric
stress change near the rupture ends, there are reversed-polarity
secondary features that reflect the stopping phases of the ruptures.
A symmetric bilateral rupture with abrupt stopping at both ends
produces primary and secondary structures that are pronounced
near both barriers (Fig. 5). In contrast, a predominantly
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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 propagation direction that is arrested abruptly at one barrier, and low sliprate in the opposite direction with gradual arrest, produces deformation structures that are pronounced only near the rupture end in
the main propagation direction (Fig. 7). In particular, the
reversed-polarity secondary features in the opposite direction are
unlikely to be preserved and observed, leading to strongly asymmetric field configurations as in Figs. 8–12.
The two simulations presented in this work (Figs. 4–7) are simple end-member 2D cases designed to highlight the different deformation structures that are expected for symmetric and strongly
asymmetric ruptures. There are many possible variations that can
influence the generated structures including bilateral ruptures with
abrupt stopping at one end and gradual stopping at the other, overlapping fault segments and other geometrical configurations, ruptures that are subshear in one direction and supershear in the
other, etc. In addition, 3D cases that account for free surface
effects, and off-fault yielding rheologies that allow for inelastic
volumetric changes [24,58], are expected to modify the nearbarrier strain fields. While the results associated with such cases
should be clarified by future studies, they are unlikely to change
our interpretation that the reversed-polarity secondary features are
associated with reversed transient stress field near the barriers produced by arrest phases of ruptures. Field observations of reversedpolarity secondary structures that exist only at one side of a stepover region provide a possible indicator of frequent predominately
unidirectional ruptures. This adds to the list of signals that may be
used to infer on a statistically preferred propagation direction of
earthquakes on given fault sections. Other signals include asymJournal of Applied Mechanics
metric rock damage across the fault [59–64], asymmetric alongstrike aftershock sequences [65,66], asymmetric branching structures with respect to the main fault [67,68], and direct imaging of
many small ruptures on a fault [69–71].
We described five examples at different scales from strike-slip
faults of the San Andreas system in southern California, with
overall transtensive steps and related classical extensional structures [6,7,72], and superposed transpressive structures that have
much smaller scales than the releasing steps. The secondary
reversed-polarity features in each of the presented examples exist
on only one side of the step, so there is a clear asymmetry of the
type shown in Fig. 7 for a predominantly unidirectional rupture.
We note that the asymmetry observed for the southern San
Andreas fault (Fig. 8) may be produced, at least in part, by a gradual arrest of earthquakes that propagate toward the step from the
south. This may be expected given the relatively high heat flow
and associated low seismic coupling coefficient in the Brawley
Seismic Zone and Imperial fault [73–75]. However, Zaliapin and
Ben-Zion [66] observed asymmetric seismicity on the southern
San Andreas fault, which is consistent with a preferred direction
of earthquake ruptures to the SE and predicted behavior of bimaterial ruptures [29,76–80] associated with the local velocity contrast in the area [81,82]. In this and other examples, careful
measurements of slip gradients on the opposite sides of the stepover, mapping of rock damage across the faults, examination of
symmetry properties of seismicity, and analysis of other signals
can identify the key mechanism(s) responsible for the strong
asymmetry of the reversed-polarity secondary structures. In
Sec. 3, we also noted that the observed asymmetry for the Rose
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Canyon fault (Fig. 9) may reflect reduced resolution on the offshore structures. Sites with comparable good exposures and similar conditions at both sides of the stepover, as in Figs. 10–12, are
clearly preferable.
There are no historical observations of large earthquakes to
indicate the likely direction of ruptures on the southern San
Andreas, Elsinore, or Rose Canyon faults. With the above caveats
we can make the following tentative inferences, assuming that the
reversed-polarity small-scale structural features reflect at least
partially statistically preferred rupture directions. For the southern
San Andreas fault, the presence of transpressive structures to the
northwest of the Salton Sea step (Fig. 8) is consistent with common NW to SE ruptures that terminate rapidly at the Salton
Trough. See also Table 2 and Fig. 9 of Zaliapin and Ben-Zion
[66]. If this is correct, the ground motion in Calexico and Mexicali
would often be amplified by rupture directivity effects. For the
Rose Canyon Fault, the observed configuration (Fig. 9) is consistent with common southward propagating ruptures. If correct, the
shaking in San Diego would likely be amplified by directivity
effects. For the Elsinore step, the presence of the small transpressive structures only to the southwest of the releasing step (Fig. 10)
implies common SE to NW ruptures that are arrested quickly at
Lake Elsinore.
For the examples from the San Jacinto fault, the sense of asymmetry at the Hemet stepover (Fig. 11) is consistent with the inferences of Dor et al. [61] from rock damage asymmetry and
Zaliapin and Ben-Zion [66] from along-strike asymmetry of seismicity on a preferred direction of ruptures from SE to NW. However, the small pressure ridge at Hog Lake (Fig. 12) is consistent
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 transtension in the same trenches and adjacent to the areas that previously experienced transpression. Earlier transpressive deformation
is also locally evident in some trenches for much older events.
This is particularly interesting when we assess what is known of
these events.
Events 1 and 2 have clear geomorphic expression and are interpreted to be associated with rupture of the entire Clark fault from
Clark Valley to Hemet [83], with maximum displacements near
Anza of 3-4 m. Their time separation is consistent with the
observed recurrence interval of the San Jacinto fault. In contrast,
events 3, 4, and 5 occurred as a cluster within about a hundred
year interval. This is somewhat an enigma but may be explained
if each of these events represents rupture of the segment to the
NW of Hog Lake and that displacement terminated in the Anza
seismicity gap. The 1918 earthquake apparently did this [83],
although the 1918 rupture probably did not reach quite as far
south as Hog Lake, terminating with a km or two to the NW. It is
possible that two (or maybe all three) of events 3-5 terminated
close to Hog Lake and that the small pressure ridge represents the
southward-dying ruptures from these earthquakes.
A possible scenario is that earthquakes in the area tend to nucleate just NW of Hog Lake, and have a predominant propagation
direction to the NW and a small dying component to the SE as in
the case shown in Fig. 6. This can reconcile the observations of
Dor et al. [61] and Zaliapin and Ben-Zion [66] on asymmetry of
rock damage and seismicity, tomographic images of velocity contrast across the San Jacinto [82,84], and the observed small
pressure-ridge at Hog Lake. Events 1 and 2, which ruptured the
entire Clark fault, may have nucleated farther south and simply
ruptured through Hog Lake, and therefore did not affect the small
transpressive feature within Hog Lake. We note that there are no
similar transpressive features immediately SE of the small Hog
step. It exists only to the NW. We also note that the three events
that show clear transpression in the sediments of Hog Lake all
occurred within a cluster in a short time interval. This suggests
that they belong to a different class of events. We will test further
and refine our inferences using results from ongoing detailed
observational studies in the area.
031025-10 / Vol. 79, MAY 2012
In summary, we presented results from numerical simulations
and field observations that small reversed-polarity deformation
features, superposed in stepover regions on larger classical structures, may indicate common direction of earthquake ruptures on
the master faults. Recognition of this signal may increase the
number and type of features that can be used to infer the probable
rupture propagation direction for large earthquakes on a given
fault section. As usual, interpreting from observations the generating mechanism is subjected to nonuniqueness and other processes
(e.g., a gradual arrest process on one side of a stepover) may also
produce asymmetry of the discussed features. Multisignal observations associated with different geological and seismological
data can reduce from the nonuniqueness and help to identify the
key operating mechanism(s). Because rupture directivity can
amplify the ground motion in the propagation direction by a factor
3 or more, it is important to continue to develop improved understanding of signals that reflect ruptures directions. More detailed
theoretical parameter-space study and field observations on
reversed-polarity secondary deformation structures will be presented in follow-up works.
Acknowledgment
It is a pleasure and privilege to contribute this paper to a special
volume in honor of Jim Rice who pioneered, among other things,
the development of bridges between mechanics and geological deformation. The study was supported by the National Science
Foundation (Grant Nos. EAR-0908903, EAR-0944066, and EAR0908515).
Nomenclature
x¼
y¼
t¼
Cs ¼
G¼
¼
dx ¼
Lnucl ¼
r0ij ¼
Du ¼
V¼
w¼
f(V, w) ¼
a¼
b¼
V0 ¼
Vw ¼
f0 ¼
fw ¼
L¼
wss(V) ¼
fss(V) ¼
f¼
fs ¼
fd ¼
Dc ¼
/¼
c¼
Tv ¼
sij ¼
rij ¼
dij ¼
tref ¼
DeV ¼
fault-parallel position
fault-normal position
time
shear wave speed
shear modulus
Poisson’s ratio
model grid spacing
nucleation zone size
initial stress components
slip along fault
slip velocity on a frictional fault
state variable in rate-state friction
rate-state friction coefficient
direct effect parameter in rate-state friction
evolution effect parameter in rate-state friction
reference slip rate in rate-state friction
weakening slip rate in rate-state friction
steady-state friction coefficient at V0 in rate-state
friction
fully weakened friction coefficient in rate-state friction
state evolution distance in rate-state friction
steady-state friction state variable at slip velocity V in
rate-state friction
steady-state friction coefficient at slip velocity V in ratestate friction
friction coefficient in slip-weakening friction
static friction coefficient in slip-weakening friction
dynamic friction coefficient in slip-weakening friction
characteristic slip distance in slip-weakening friction
angle of internal friction coefficient in Drucker-Prager
yielding
cohesion in Drucker-Prager yielding
viscoplasticity relaxation time
deviatoric stress components
stress components
Kronecker delta
reference time of rupture front arrival
differential volumetric strain around barriers
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