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Bulletin of the Seismological Society of America, Vol. 84, No. 3, pp. 935-953, June 1994
Static Stress Changes and the Triggering of Earthquakes
by Geoffrey C. P. King, Ross S. Stein, and Jian Lin
Abstract
To understand whether the 1992 M = 7.4 Landers earthquake
changed the proximity to failure on the San Andreas fault system, we examine
the general problem of how one earthquake might trigger another. The tendency
of rocks to fail in a brittle manner is thought to be a function of both shear and
confining stresses, commonly formulated as the Coulomb failure criterion. Here
we explore how changes in Coulomb conditions associated with one or more
earthquakes may trigger subsequent events. We first consider a Coulomb cri-
terion appropriate for the production of aftershocks, where faults most likely to
slip are those optimally orientated for failure as a result of the prevailing re-
gional stress field and the stress change caused by the mainshock. We find that
the distribution of aftershocks for the Landers earthquake, as well as for several
other moderate events in its vicinity, can be explained by the Coulomb criterion
as follows: aftershocks are abundant where the Coulomb stress on optimally
orientated faults rose by more than one-half bar, and aftershocks are sparse
where the Coulomb stress dropped by a similar amount. Further, we find that
several moderate shocks raised the stress at the future Landers epicenter and
along much of the Landers rupture zone by about a bar, advancing the Landers
shock by 1 to 3 centuries. The Landers rupture, in turn, raised the stress at site
of the future M = 6.5 Big Bear aftershock site by 3 bars. The Coulomb stress
change on a specified fault is independent of regional stress but depends on the
fault geometry, sense of slip, and the coefficient of friction. We use this method
to resolve stress changes on the San Andreas and San Jacinto faults imposed
by the Landers sequence. Together the Landers and Big Bear earthquakes raised
the stress along the San Bernardino segment of the southern San Andreas fault
by 2 to 6 bars, hastening the next great earthquake there by about a decade.
Introduction
It is generally agreed that the greatest earthquakes
in California are associated with the San Andreas or
closely related faults, and thus when the Landers earth-
quake struck within 25 km of the San Andreas a major
question concerned its potential influence on San An-
dreas behavior. In particular, were stresses redistributed
in such a way as to increase the likelihood of future San
Andreas earthquakes? In this article we examine this
question within the broad context of seeking to under-
stand more generally the causal relations between earth-
quakes. Briefly, we ask under what conditions does one
earthquake trigger another.
It has long been recognized that while each event
produces a net reduction of regional stress, events also
result in stress increases. With further tectonic loading
it seems logical that such sites of stress rise should be
the foci of future events and therefore such events should
be readily predictable from preceding ones. Despite the
apparent simplicity of this mechanical argument, earth-
quake triggering has not been observed as widely as might
be expected. Phenomena such as the steady migration of
epicenters along the North Anatolian (Ambraseys, 1970)
and San Jacinto (Sanders, 1993) faults, for example, oc-
cur, but are rare.
Recent ideas of self-organized criticality (e.g., Bak
and Tang, 1989; Cowie
et al.,
1993) help to explain this
result. If the Earth behaves in the way these authors sug-
gest, all parts of the brittle crust are at the point of failure
and, as a result of long-range elastic correlations, an
earthquake can be followed by a nearby or a distant event.
Thus, strong correlations between neighboring events need
not dominate the physics of earthquakes. Nonetheless,
in this article we show that local triggering effects can
be clearly observed. We adopt classical concepts of stress
transfer to explore interaction effects. The Coulomb fail-
ure stress changes caused by mainshock rupture effec-
tively explain the aftershock distributions for the earth-
quakes we study, with some of the more distant events
935
Static Stress Changes and the Triggering of Earthquakes
937
(containing the z direction), stress on a plane at an angle
qt from the x axis (Fig. 1) is given by
fill = O'xx C0820 -~
2c% sin 0 cos 0 + Cryy sin20
cr33 = or= sin20 - 2Cr.y sin 0 cos 0 + Cryy
C0S20
1
rz3 = ~ (~,y - Crxx) sin 20 + r~y cos 20. (7 )
We can now write the change of Coulomb stress for right-
lateral cr~ and left-lateral @ motion on planes orientated
at 0 with respect to the x axis in the following way:
= r,=3 + (8)
L
@ -~- TI3 "q- ~'~'0"33" (9)
The sign of r13 from equation (7) is unchanged for right-
lateral slip (rf3) in equation (8) and reverses in sign for
left-lateral slip (rL3) in equation (9).
Equation (9) is illustrated in Figure 2a. An elliptical
slip distribution is imposed on a master fault in a uni-
form, stress-free, elastic half-space. The contributions of
the shear and normal components to the failure condi-
tion, and the resulting Coulomb stresses, for infinitesi-
ma l faults parallel to the master fault, are shown in sep-
arate panels. Such a calculation represents the change of
Coulomb stress on these planes resulting only from slip
on the master fault. The calculation is appropriate to de-
termine, for example, the effect of Landers on a nearby
segment of th e San And reas fault. One need only know
the relative location of the San Andreas and Landers faults,
(53 Y
/ E
Figure 1. The axis system used for calcula-
tions of Coulomb stresses on optimum failure
planes. Compression and right-lateral shear stress
on the failure plane are taken as positive. The sign
of r e is reversed for calculations of right-lateral
Coulomb failure on specified failure planes.
the slip on the Landers fault, and the sense of slip on
the San Andreas to determine whether the San Andreas
fault has been brought closer to, or further from, failure.
Such calculations are independent of any knowledge of
the prevailing regional stresses or any preexisting stress
fields from other events. The signs in the calculation are
chosen such that a positive Coulomb stress indicates a
tendency for slip in the same right-lateral sense as the
fault of interest. Negative Coulomb stresses indicate a
reduction of this tendency. It is important to appreciate
that because r13 changes sign between equations (8) and
(9), a negative Coulomb stress for right-lateral fault m o -
tion is not the same as a tendency for left-lateral slip.
The distribution of increases and decreases of Cou-
lomb stress shows features common to all subsequent
figures. Lobes of increased shear stress appear at the fault
ends, corresponding to the stress concentrations that tend
to extend the fault. Off-fault lobes also appear, separated
from the fault by a region where the Coulomb stresses
have not been increased, as discussed by Das and Scholz
(1983). If the master fault were infinitesimal in length,
the off-fault lobes would be equal in amplitude to the
fault-end lobes at all distances. For a finite length fault
they are absent near the fault and reduced in amplitude
at moderate distances. The normal stress change field is
similar to the more familiar dilatational field with max-
ima and minima distributed antisymmetrically across the
fault, but here we consider only the component of ten-
sion normal to the fault. The influence of the normal
stress on the Coulomb stress distribution is to reduce the
symmetry of the final distribution and to increase the
tendency for off-fault failure.
Two-Dimensional Case: Change of Coulomb
Stress on Optimally Orientated Faults
Coulomb stress changes on optimally orientated planes
can also be calculated as a result of slip on the master
fault, and these are the planes on which aftershocks might
be expected to occur. We presume that a sufficient num-
ber of small faults exist with all orientations and that the
faults optimally orientated for failure will be most likely
to slip in small earthquakes. After an earthquake, the
optimum directions are determined not only by the stress
change due to that earthquake o -q but also by pre-existing
r t
regional stresses o-,j to give a total stress oij
l r
o-o = +
o%
(10)
The orientation of the principal axes resulting from the
total stress are therefore derived using
0 = - tan- 1
t t °
(11)
Where 0 is the orientation of one principal axis to the x
938 G.C.P. King, R. S. Stein, and J. Lin
axis as shown in Figure 1 and the other is at 0 +-- 90 °.
From these two directions, the angle of greatest
compression 0~ must be chosen. Thus the optimum fail-
ure angle q'o is given by 01 -+ ft. Whereas the optimum
planes are determined from o-~., the normal and shear
stress changes on these planes are determined only by
the earthquake stress changes 0-q. Thus the changes in
stress on the optimum planes become
0"33 = 0"q
sin2~Oo - 2o -q sin ~b o cos ~bo + o'qy cos2~bo
1
r13 = 2
(0-qy --
O'q) sin 2~o + r q cos 200 (12)
A. Coulomb stress change for
right-lateral faults parallel to master fault
Stress • Rise • Drop
+
right-lateral
shear effective
friction x right-lateral Coulomb
st re ss change
+ normal stress change =
st re ss change
+ _-
B. Coulomb stress change
for faults optimally oriented for failure
in a N7°E regional
compressive stress
(O r) of 100
bars
O r
Oplimum ~ left-lateral
Slip Planes / right-lateral
+
m
m
shear st ress effective
friction x
Coulomb s tr es s
change
+ normal
stress change = change
"c s + //' (-O n) = O~ pt
At/~J r= I.O A'~ I ~r =0.1
Figure 2. Illustration of the Coulomb
stress change. The panels show a map view
of a vertical strike-slip fault embedded in an
elastic half-space, with imposed slip that ta-
pers toward the fault ends. Stress changes
are depicted by graded colors; green rep-
resents no change in stress. (a) Graphical
presentation of equation (9). (b) Graphical
presentation of equation (13).
Change in Coulomb Failure Stress (bars) t i ~ ~ ~ ~ t ~ t i i
on optimal right-lateral faults (black) , i i i L~'~i ' ~
i I i
(~rorientedN7OE, if=0.4 -I.0 `0.8 `0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Figure 3. Dependence of the Coulomb
stress change on the regional stress mag-
nitude. If the earthquake relieves all of the
regional stress
(left panel),
resulting opti-
mum slip planes rotate near the fault. If the
regional deviatoric stress is much larger than
the earthquake stress drop
(right panel),
the
orientations of the optimum slip planes are
more limited, and regions of increased Cou-
lomb stress diminish in size and become
more isolated from the master fault. In this
and subsequent plo ts, the maximum and
minimum stress changes exceed the plotted
color bar range.
Static Stress Changes and the Triggering of Earthquakes 939
and the Coulomb stress changes
O')~ pt = "1"13 -- ~[-LtO'33 .
(13)
The two optimum planes correspond to left-lateral and
right-lateral shear with expression (13) applying to both.
It is important to emphasize that we calculate the change
of Coulomb stress on planes that are optimum after the
earthquake. The optimum orientations are calculated from
the total stress after the earthquake, and the Coulomb
stress changes caused by the earthquake stress changes
are resolved onto these planes. In general, the earth-
quake rotates the principal axis. It is possible to calculate
the change of maximum Coulomb stress at some point
before and after the earthquake, together with the change
of angle of the infinitesimal plane upon which it oper-
ates. However, we do not regard such effects as having
significance for earthquake triggering.
The results of a calculation to find optimum orien-
tations and magnitudes of Coulomb stress changes are
shown in Figure 2b. The calculations are again in a half-
space and the slip on the master fault is the same as
before. A uniform 100-bar compressional stress is intro-
duced with the orientation shown. White lines indicate
optimum left-lateral orientations and black lines, right-
lateral orientations. The shear and normal stress contri-
butions to the Coulomb stress change are again shown
in separate panels. It can be seen from expression (10)
that only the deviatoric part of the regional stress deter-
mines the orientation of principal axes, and hence the
optimum stress orientations. Thus, it is sufficient to ap-
ply the regional stress as a simple uniaxial compression
or extension.
The relative amplitude of the regional stress o -r to
the earthquake stress drop Ar might to be expected to
have an effect. This is explored in Figure 3, which shows
the Coulomb stress change
O'~ pt on
optimally orientated
right-lateral planes in which the regional field is equal
to the stress drop AT (left panel) and 10 times A~- (right
panel). These examples span likely conditions. It is ev-
ident that, except close to the master fault, the orienta-
tions of the optimal planes and Coulomb stress changes
on these planes are little altered. The optimal orienta-
tions are essentially fixed by the regional stress, except
very close to the fault where the stress change caused by
slip on the master fault is comparable to the regional
stress. If the regional stress o-~ were zero, then the Cou-
lomb stress change on optimally oriented planes 0-~P~ -
cr 7 or @. If, however, the regional stress o-~ is large
relative to the earthquake stress drop, 0-7 or @ may lo-
cally exceed o'7 pt. In all calculations that follow, we use
a uniform regional stress field o-~. However, this as-
sumption is not required, and spatially variable stress fields
can be incorporated into the calculation of 0-ffpt.
The effects of varying the orientation of regional
stresses and changing the coefficient of friction/z' are
shown in Figure 4. Possible changes of regional stress
orientation are limited since the main fault must move
as a result of the regional stress; the 30 ° range covers
the likely range. Similarly, values of friction between
0.0 and 0.75 span the range of plausible values. All of
the panels show the same general features, fault-end and
off-fault Coulomb stress lobes. Thus, our modeling is
most sensitive to the regional stress direction, modestly
sensitive to the coefficient of effective friction, and in-
sensitive to the regional stress amplitude.
Three-Dimensional Case: Strike-Slip
and Dip-Slip Conditions
In the foregoing discussion we have assumed that
only vertical faults are present and thus stress compo-
nents o'zz, O'xz, and o-y z could be neglected. In dip-slip
faulting environments, however, these stress compo-
nents cannot be ignored. If all of the regional stress com-
ponents are known, one can calculate the orientation and
magnitudes of the principal stresses. These can then used
to calculate the orientation of the plane containing o1 and
0-3, and hence the optimum orientations of slip planes
and the change of Coulomb failure stress on them can
be found.
In practice this is not straightforward. While the ver-
r
tical shear components of regional stress o'~z and O'rz can
be ignored, the magnitude of the vertical stress o-~ rel-
r r
ative to the horizontal regional stresses
O'yy,
0-xx, and
r
0-~ cannot. The relative magnitude of the horizontal to
vertical stresses determines whether events are strike slip
or dip slip, and this can alter the form of the predicted
Coulomb stress change distribution. Whereas in the two-
dimensional case only the orientation of the regional stress
is of consequence, for the three-dimensional case the ra-
tio of the vertical to horizontal stresses becomes impor-
tant, since this determines whether strike-slip or dip-slip
faulting occurs. Direct information on relative stress am-
plitudes is not generally available and varies with depth,
so alternative strategies must be adopted. One possibility
is to select relative stresses such that the calculations pre-
dict the earthquake mechanisms observed. The second
is determine probable fault orientations a priori and di-
rectly determine the Coulomb changes on them. Since
focal mechanisms are the best guide to relative stresses
and fault orientations, these two possibilities are differ-
ent technically, but not in practice. Finally, where two
principal stresses are nearly the same, the distributions
of Coulomb stress changes for dip-slip and strike-slip
faulting are similar.
Coulomb Stress Changes and Aftershocks
The methods outlined above can be applied to the
aftershock distributions of two events that preceded the
Landers event, the 1979 Homestead Valley sequence and
the 1992 Joshua Tree earthquake (Figs. 5 and 6). Neither
940 G.C.P. King, R. S. Stein, and J. Lin
event produced surface rupture directly attributable to the
mainshock, but seismic and geodetic observations fur-
nish evidence for the geometry of fault slip. The cal-
culations are carried out in a half-space with the values
of Coulomb stress plotted in the figures being calculated
at half the depth to which the faults extend.
Both Figures 5 and 6 show the four characteristic
lobes of increased Coulomb stress rise and four lobes of
Coulomb stress drop. The lobes at the ends of the fault
extend into the fault zone, while the off-fault lobes are
separated from the fault over most of its length by a zone
where the Coulomb stress is reduced. The distributions
of af t ersh o cks are consistent with theSe patterns. In-
creases of Coulomb stress of less than 1 bar appear to
be sufficient to trigger events, while reductions of the
same amount effectively suppress them. Relatively few
events fall in the regions of lowered Coulomb s tress , and
the clusters of off-fault aft e r shoc k s are separated from
the fault itself by a region of diminished activity. The
distributions of Coulomb stresses can be modified as de-
scribed earlier by adjusting the regional stress direction
and changing /x'. However, any improvements in the
correlation between stress changes and aftershock oc-
currence are modest. Consequently, we have chosen to
show examples with an average/x' of 0.4. Whatever val-
ues we adopt, we find that the best correlations of Cou-
lomb stress change to aftershock distribution are at dis-
tances greater than a few kilometers from the fault. Closer
to the fault, unknown details of fault geometry and slip
distribution influence stress changes. At distances larger
than about three fault lengths, the correlations are less
clear because there are fewer aftershocks.
N-S Regional Compression N3(I°E Regional Compression
~
lt--
0.0
I1;=
0.75
Coulomb Stre Optimum ~ Left-lateral
on Optimally Slip Planes ~ Right-lateral
Vertical Planes (bars) .0.5 0.0 0.5
Figure 4. The effect of changing the regional stress o z orientation (compare
right and left panels) and the effective coefficient of friction /z' (compare top
and bottom panels). The example is for a simplified Landers rupture (5 m of
tapered slip on a 70-km-long, 12.5-km-deep fault). Regional stress magnitude is
100 bars. Friction controls the internal angle between right- and left-lateral slip
planes, and the influence of the normal stress change on failure. The regional
stress orientation controls the size of off-fault to fault-end lobes.
Static Stress Changes and the Triggering of Earthquakes
941
An examination of the distribution of Coulomb stress
changes with depth is also instructive. Figure 7 shows a
cross section perpe ndi cul ar to the Homestead Valley fault
(L/W ~
1). Increased Coulomb stresses can be seen in
the off-fault lobes and at the base of the fault. Because
the stress concentration beneath the fault lies at the depth
at which stress is accommodated aseismically, this is
typically free from after s hock s . The off-fault lobes,
however, are seen in the well-located hypocentral dis-
tributions of the Homestead Valley (Hutton
et al.,
198 0;
Stein and Lisowski, 1983). The predicted Coulomb stress
increases diminish with depth away from the fault (Fig.
7), a feature that is also seen in the off-fault aftershock
distributions, which extend to 60 to 80% of the depth of
the mainshock and its deepest associated aftershocks. A
similar accord between the Coulomb s tre ss chang e s at
depth and well-located af t e rsho c k s is seen for the Joshua
Tree earthquake (Hauksson
et al.,
1993).
Although events such as Homestead Valley and
Joshua Tree have readily identifiable off-fault aftershock
clusters, such features have not been observed on a large
scale for great earthquakes on transcurrent faults with
rupture lengths of many tens or even hundreds of kilo-
meters. Scholz (19 82) p o inted out that fault slip scales
differently for faults that are much longer than the thick-
ness of the brit tle crust. In Figure 8, stress changes caused
by 1 of slip on a short fault
(left panel)
and a long fault
(right panel)
are compared. The lobes at the fault ends
are similar in strength and size. For the short fault, the
Figure 5. Coulomb stress changes associated
with the 15 March 1979 Homestead Valley earth-
quake sequence (ML = 4.9, 5.2, 4.5, and 4.8).
The fault (enclosed white line) is 5.5-km long by
6-km deep, with 0.5 m of tapered slip and a mo-
ment of 4.2 × 1024 dyne-cm, following Stein and
Lisowski (1983) and King
et al.
(1988). Stress is
sampled halfway down the fault.
Figure 6. Coulomb stress changes calculated
for the 23 April 1992 ML = 6.1 Joshua Tree earth-
quake. The mainshock is indicated by the star. The
model fault is 8-km long and 12.5-kin deep with
0.5 m of right-lateral slip, for a moment of 2 ×
1025 dyne-cm, following Savage
et al.
(1993) and
Ammon
et al.
(1993). Stress is sampled halfway
down the fault.
942 G.C.P. King, R. S. Stein, and J. Lin
0
Fault
¥
5
10
-10 -5 0 5 10
Distance (km)
Coulomb Stress
Change (bars) .... ~
.......... 'r
-3 -2 -1 0 1 2 3
Figure 7. A dep th cro ss section of Coulomb stress changes for the 1979
Homestead Valley fault (essentially the
L/W
= 1 fau lt in Fig. 8 made along the
distance = 0 km line), with aftershocks from Hutton
et al.
(1980). Because the
section does not pass through the centers of the off-fault lobes, the lobes appear
shallower and smaller than at their maxima. The stress concentration at base of
fault is exaggerated because fault slip is not tapered with depth.
E
eo
¢0
°~
20
0
-20
20
0
-60
-40 -20 0 20 40
-60
2O 20
20
Distance
(km) 0
Coulomb Stress
Change (bars)
0
-20 -20
-20 0 20
Distance (kin)
-1.6 -0.8 0.0 0.8 1.6
-20
1 m tapered fault slip
100-bar N45°W
compression
Figure 8. Coulomb stress changes as a function of fault length, L. Off-fault
stress lobes diminish as the fa ult lengthens relative to its down-dip dimension,
W. Both faults are 12.5-km deep with a stress drop of --45 bars. Stress is sampled
halfway down the fault.
Static Stress Changes and the Triggering of Earthquakes
943
off-fault lobes can be seen at distances from the fault of
the order of one fault length, again with comparable
strength. For the long faults the off-fault lobes have es-
sentially disappeared.
Stress Changes Associated with
the Landers Earthquake
Regional Stress Field Driving Rupture
For all subsequent calculations, we take t h e reg i ona l
stress to be a simple compression of 100 bars, orientated
at N7°E. As we demonstrate earlier, only the deviatoric
part of the stress tensor is important and the amplitude
hardly matters, provided that we can assume primarily
strike-slip mechanisms, a reasonable assumption for t he
Landers region. Figure 4 shows that Coulomb stress
changes are modestly sensitive to the orientation of t he
principal axes, and hence our choice of N7°E needs to
be justified.
The principal strain axes can be used as an indica-
tion of stress orientation, with the direction of maximum
shortening being taken to be the same as the axis of max-
imum compressive stress. Using geodetic data, Lisowski
et al.
(1991) found maximum shortening orientated at
N7 --- I°E during the pre-earthquake period 1979 to 1991
for the Joshua geodetic network, which includes most of
the Landers rupture. They also found the same direction
during the period 1934 to 1991 for the Landers and
southern San Andreas regions. Across th e nor th half of
the Landers rupture, Sauber
et al.
(1986) found max i-
mum shortening between 1934 and 1982 to lie at N4 -+
5°E. These values are all clo s e to the max i m u m short-
ening axis predicted for simple shear between the Pacific
and North America plates N9°E, given a relative plate
motion direction in central California of N36°W (DeMets
et al.,
1990).
Se ismi c foca l mechanisms also supply information
on the principal stress. The mean principal stress direc-
tion derived from small shocks along the 50 to 150 km
of the S an A ndre as faul t nearest to the Landers region
(Banning and Indio segments) is N6 -+ 2°E (Jones, 1988).
Williams
et al.
(1990) found that the average principal
stress direction for the 50-km stretch of the San Andreas
fault adjacent to Landers (San Gregorio Pass and Eastern
Transverse Range-I regions) to be N 8 - 5°E. Thus, sev-
eral independent techniques yield a stress direction within
a few degrees of our adopted value. Only data from the
borehole at Cajon Pass (Zoback and Lachenbruch, 1992)
gives a different orientation (N57 --- 19°E), but this could
not drive local or regional right-lateral m o t ion on the Sa n
Andreas fault and may instead be attributable to local
effects (Shamir and Zoback, 1992).
Coulomb Stress Changes Preceding
the Landers Rupture
In Figure 9 we show the Coulomb stress changes
caused by the four M > 5 earthquakes within 50 km of
100
80
60
40
20
0 20 4 0 60 km
Figure 9. Coulomb stress changes calculated
for the four M > 5 earthquakes in the Caltech-
USGS catalog within 50 km of the future Landers
epicenter. Each earthquake raised the stress at the
future Landers epicenter (star). All ruptures (en-
closed white lines) except the North Palm Springs
shock are modeled as vertical right-lateral rup-
tures. The ML = 5.2 Galway Lake earthquake is
modeled with 0.07 m of slip on a 6-kin-long fault,
for a moment of 6.3 × 1023 dyne-cm (Hill and
Beeby, 1977; Lindh
et al.,
1978). The North Palm
Springs fault dips 45 ° NE and has 0.42 m of right-
lateral and 0.27 m of reverse slip, following Jones
et al.,
(1986), Pacheco and Nfibelek (1988), and
Savage
et al.
(1993).
Landers that preceded the Landers earthquake. The 1975
ML = 5 . 2 Galway Lake, 1979 ML = 5.2 Homestead Val-
ley, 1986 ML = 6 North Palm Springs, and 1992 Mc =
6.1 Joshua Tree earthquakes progressively increased
Coulomb stresses by a bout 1 bar at the future Landers
epicenter. Together they also produced a narrow zone of
Coulomb stress increase of 0.7 to 1 bars, whi ch the fu-
ture 70-km-long Landers rupture followed for 70% of its
length. The Landers fault is also nearly optimally ori-
ented for failure along most of its length. The four mod-
erate earthquakes may themselves have been part of a
larger process of earthquake preparation within the
944 G.C.P. King, R. S. Stein, and J. Lin
earthquake cycle, as suggested by Nu t
et al.
(1993). It
is noteworthy that the three largest events are roughly
equidistant from t h e future Landers epicenter; the right-
lateral Homestead Valley and Joshua Tree events en-
hanced stress as a result of the lobes beyond the ends of
their ruptures, whereas the North Palm Springs event en-
hanced rupture as a result of an off-fault lobe. Increasing
the effective friction from/x' from 0.4 (Fig. 9) to 0.75
slightly enhances the effects, and dropping the friction
to zero reduces them.
Paleoseismic trench excavations across the Landers
rupture suggest that the 1992 Landers fault last slipped
about 6000 to 9000 yr ago (Hecker
et al.,
1993; Rock-
well
et al.,
1993; Rubin and Sieh, 1993). The mean static
stress drop of the 1992 Landers earthquake was about
35 bars , usi ng a sh ear modulus of 3.3 x 10 ~ dyne-cm,
and mean fault slip of 2.6 m and a fault width of 15 km
from Wald and Heaton (1994). Thus, the 1-bar stress
rise contributed by the neighboring earthquakes may rep-
resent about 150 to 250 yr of typical stress accumulation.
The four moderate earthquakes can therefore be thought
of as advancing the occurrence of the Landers earth-
quake by 1 to 3 centuries.
Ru ptur e Mo d e l fo r the Landers Earthquake
Unlike the earthquake sources modeled so far, which
we approximated by tapered slip on single planes, there
is vastly more information about the M = 7 .4 L ande r s
source. Here we use Wald and Heaton's (1994) model
of fault slip, which they derive from joint inversion of
broadband teleseismic waveforms, near-field and re-
gional strong motions, geodetic displacements from
Murray
et al.
(1993), and sur face fault slip measure-
ments from Sieh
et al.
(1993). We smooth the i r 2 by 2
km variable slip model to 5 by 5 km, and retain their
three planar fault segments (Fig. 10). Cohee and Beroza
(1994) found a similar slip distribution from near-source
low-gain seismograms. Using a she ar modulus of 3.3 x
l0 II
dyne-cm -2, in the range typically employed to de-
rive seismic moment, this slip distribution gives a total
Landers seismic moment of 0.9 × 1027 dyne-cm
Stress Changes following the Landers Rupture
but before the Big Bear Earthquake
The stress changes caused by the Landers event are
shown in Figure 11. At first glance the off-fault stress
lobe to the west of the fault appears surprisingly large,
considering the 70-km length of the surface rupture. In-
spection of Figure 10 reveals, however, that most of the
fault slip is confined to a 40-km-long, 15-km-deep cen-
tral section. Thus, the source has an effective
L/W
ratio
of 2 to 3, rather than 6, as it would appear from the
length of the surface rupture, which results in large off-
fault lobes (Fig. 8). The western off-fault lobe is large
at the expense of the eastern lobe because of the fault
curvature, with the off-fault stresses adding on the con-
cave side of the fault. These two factors account for the
concentration of stress 20 km west of the Landers rup-
ture, where the Big Bear aftershock would occur.
The largest lobe of increased Coulomb stress is cen-
tered on t he epicenter of the future
ML
= 6.5 Big Bear
event, where stresses were raised 2 to 3 bars. The Big
Bear earthquake was apparently initiated by this stress
rise 3 hr 26 min after the Landers mainshock. The Cou-
lomb stress change at the epicentre is greatest for high
effective friction but remains more than 1.5 bars for/~
= 0. There is no surface rupture or Quaternary fault trace
associated with the Big Bear earthquake. Judging from
its epicenter and focal mechanism, Hauksson
et al.
(1993)
suggest left-lateral rupture on the plane that is seen to
be optimally aligned for failure, with the rupture appar-
ently propagating northeast and terminating where the
Landers stress change became negative. Jones and Hough
(1994), however, argue for a multiple event with both
right- and left-lateral rupture on orthogonal faults bi-
secting at the epicenter. In this case as well, rupture on
each plane terminates where the stress changes become
negative.
In addition to calculating the stress changes caused
by the Landers rupture, we estimate the slip on the Big
Bear fault needed to relieve the shear stress imposed by
the Landers rupture. This is achieved by introducing a
freely slipping boundary element along the future Big
Bear rupture. The potential-slip along the Big Bear fault
is 60 mm (left lateral), about 5 to 10% of the slip that
occurred several hours later. These calculations suggest
that the Big Bear slip needed to relieve the stress im-
posed by Landers was a significant fraction of the total
slip that later occurred. Thus, from consideration of the
stress changes and the kinematic response to those
changes, it is reasonable to propose that stresses from
the Landers event played a major role in triggering the
Big Bear shock.
Stress Changes Caused by the Landers, Big Bear,
and Joshua Tree Ruptures
The Big Bear earthquake was the largest of more
than 20,000 aftershocks located after the Landers earth-
quake, large enough to result in significant stress redis-
tribution at the southwestern part of the Landers rupture
zone. Consequently, the distribution of later events can-
not be examined without considering its effect. Although
smaller, a similar argument can be applied to the Joshua
Tree event, whose aftershock sequence was not com-
plete at the time of the Landers rupture. In Figure 12 we
therefore plot th e combined Coulomb stress changes fo r
the Joshua Tree, Landers, and Big Bear earthquakes. This
distribution is shown together with all well-located ML
--> 1 earthquakes that occurred in the box shown, within
0 20 40 60 Distance (km)
E
E 5
¢-
15
I I I I I I I I
Modeled Fault Slip 011)
JLIIfl~Ogt2,t~14 ¢ l.q~lGl,j
Landers-Johnson Valley
tli
15
6- 5 4 3 2 1 0
Figure 10. Distribution of modeled fault slip for the Landers earthquake from
Wald and Heaton (1994), derived from joint inversion of strong motion, telese-
ismic, geodetic, and surface slip data . Three planar fault segments are used, which
correspond approximately to the mapped fault trace. Although the surface rupture
is 70-kin long, the fault slip is concentrated over a strike length of just 40 km.
Coulomb stress change caused by l.atlders
and Joshua Tree Earthquakes before
occurrence of the Big Bear shock (bars)
-1.0 -0.5 0.0 0.5 1.0
Figure 11. Coulomb stress change caused by the Landers rupture. The left-
lateral ML = 6.5 Big Bear rupture occurred along dotted line 3 hr 26 min after
the Landers mainshock. The Coulomb stress increase at the future Big Bear ep-
icenter is 2.2 to 2.9 bars.
946 G.C.P. King, R. S. Stein, and J. Lin
about 250 km of the 28 June mainshock, during t he fol-
lowing 25 days.
Most ML > 1 aftershocks occur in regions where the
failure stress is calculated to have increased by --- 0.1
bar, and few events are found where the stress is pre-
dicted to have dropped (Fig. 12). Even when all seis-
micity within 5 k m of the Landers, Big Bear, and Joshua
Tree faults is excluded, more than 75% of the after-
shocks occur where the stress is predicted to have risen
by >0.3 bar. In c o ntr a st, less than 25% of the after-
shocks occur where the stress dropped by >0.3 bar. The
same correspondence can be found among ML --> 4 earth-
quakes that took place during the 9-month period April
through December 1992 from Hauksson
et al.
(1993).
The largest shock to fall on or near the San An dre a s, the
29 June 1992 ML = 4.7 Yucaipa event (due east of San
Bernardino in Fig. 13), occurred where the failure stress
change on the San A ndre as is calculated to have risen
by 5 bars. Aftershocks to Landers also occurred as far
as 1250 km north of the mainshock, largely in geother-
mal areas (Hill
et al.,
1993). At these distances, the static
Coulomb stress changes are much smaller than the tidal
Cou lomb Stress Change caused by the
Landers, Big Bear, and Joshua Tree ~
Earthquakes (bars} -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
Figure 12. Coulomb stress changes at a depth of 6.25 km caused by the Lan-
ders, Big Bear, and Joshua Tree earthquakes. The Big Bear earthquake, which
lacked surface rupture, is modeled as an 18-km-long by 12.5-km-deep vertical
fault with 0.83 m of left-lateral slip, for a moment of 5,5
x 10 25
dyne-era, fol-
lowing limited seismic (Hanksson
et al.,
1993) and geodetic (Murray
et al.,
1993;
Massonnet
et al.,
1993) evidence.
Static Stress Changes and the Triggering of Earthquakes 947
stress changes, and thus we do not attempt to include
them in our modeling.
Few aftershocks are seen near Indio in the Coachella
Valley where the San Andreas was loaded by the Lan-
ders earthquake and, to a lesser extent, by the Imperial
Valley (Hanks and Allen, 1989), Elmore Ranch, and Su-
perstition Hills (Hudnut et al., 1989) events [the effect
of these earthquakes is shown in Stein et al. (1992)].
The lack of earthquakes near Indio appears to be an ex-
ception to the observation that Coulomb stress rises are
accompanied by at least some activity, although trig-
gered surface slip was seen at several points between
Indio and the southeastern end of the Sa n Andre as (D.
Ponti, personal comm., 1992). Landers aftershocks could
be absent in the Coachella Valley because the total stress
there is lower, because the fault is locally tougher, or
because modulus contrasts modify the Coulomb stresses.
We examined the effect of a low-modulus Mojave tec-
Figure 13. The largest Coulomb stress changes at depths between 0 and 12.5
km caused by the Landers, Big Bear, and Joshua Tree earthquakes, shown with
the first 25 days of seismicity from Hauksson et al. (1993). Also shown are the
largest two aftershocks to occur during the following 8 months, the 17 November
1992 ML ----- 5.3 and 4 December 1992 ML = 5.1 shocks.
948 G.C.P. King, R. S. Stein, and J. Lin
tonic block surrounded by a stiffer crust, consistent with
the
Pg
velocity contours of Hearn and Clayton (1986).
The average velocity contrast between the 100- by 300-
km Mojave region with
Pg
velocity <6.2 km/sec and
the surrounding medium is 0.45 km/sec. We gave the
Mojave block a Young's modulus, E, of 6.2 × 1011 dyne-
cm -z. In this plane stress calculation, the Coulomb fail-
ure stress rise is halved in the Coachella Valley, but is
nearly unchanged elsewhere. Thus, although we may
overestimate the stress change for the Coachella Valley
in Figure 12, a stress increase did apparently occur, but
has not been expressed in seismicity.
In Figure 13 we plot the most positive Coulomb stress
change at depths between 0.5 and 12.5 km. Thus, this
is the Coulomb stress change on optimally oriented faults
at the optimum depth. This function gives perhaps the
best correlation between stress changes and aftershocks,
since aftershocks will likely occur at the depth and lo-
cation where the stress change is greatest. Note that two
of the largest earthquakes in southern California during
the 9-month period, the 27 November 1992
Mr
= 5.3
and the 4 December 1992 ML = 5.1 events, occurred
north of the Big Bear epicenter in a region where stress
was increased as a result of the Big Bear earthquake.
Coulomb Stress Changes along the San Andreas
and San Jacinto Faults
Thus far we have been calculating Coulomb stress
changes on opt!mally oriented planes Ao-2 pt. This is be-
cause for small events, sufficient small faults exist that
those optimally orientated will be activated. The only
possible exceptions have been the Landers and Big Bear
earthquakes, but in both cases, the difference between
optimum and actual orientations were small. The San
Andreas and San Jacinto faults are not optimally orien-
tated, so to examine Coulomb changes we resolve the
right-lateral Coulomb stress changes on these faults
Ao-f, rather then calculate optimum changes at the fault
locations Ao'~ pt.
Resolved Coulomb changes for the San Andreas are
shown in Figure 14. Because the failure stress change
on a particular fault is independent of the regional stress,
as illustrated in Figure 2a, the calculation is a function
only of the Landers source and the San Andreas fault
geometry, sense of slip, and friction coefficient. In the
top panel
of Figure 14, Coulomb stress changes are shown
for/z = 0 and/x = 0.75. San Andreas segment bound-
aries inferred by the Working Group on California Earth-
quake Probabilities (1988) are shown below and accord
roughly to sign changes in the failure stress change im-
posed by the Landers event. Regardless of the value of
friction, the stress change is positive along the San Ber-
nardino Mountain segment, generally positive along the
Coachella Valley segment, and negative along the Mo-
jave segment. The failure stress changes are greatest for
high values of effective friction, reflecting the role that
the normal stress plays in the total Coulomb stress change.
The stress changes for an intermediate value of friction
(/~ = 0.4) are shown in Figure 14
(middle panel).
The
stress change on the northern San Jacinto fault, which is
farther from Landers but more favorably oriented than
the San Andreas, is positive along parts of the San Ber-
nardino Valley and San Jacinto Valley segments (Fig.
15,
upper panel).
These results are in substantial agree-
ment with those of Harris and Simpson (1992) and Jaum6
and Sykes (1992).
Potential Slip on the San Andreas and San Jacinto
Faults Caused by the Landers Rupture
The correspondence between seismicity and the
Coulomb failure stress changes produced by the Landers
and earlier events suggests that regions of predicted in-
crease are candidates for future major events. If the
earthquakes are in some sense time-predictable, with
rupture occurring when a failure threshold is exceeded,
then the stress increase will hasten the time to the next
earthquake. To predict how the Landers earthquakes have
advanced or delayed the next great southern San Andreas
earthquake, we let a frictionless San Andreas (Fig. 14,
bottom panel)
or San Jacinto fault (Fig. 15,
bottom panel)
slip to relieve the stress imposed by the Landers, Big
Bear, and Joshua Tree earthquakes. This is accom-
plished by introducing freely slipping boundary elements
along the faults to a depth of 12.5 km.
The response on the San Andreas is slip of 20 cm
along the central San Bernardino segment (equivalent to
an M = 6.2 event if it occurred seismically), and 7 cm
in the northern Coachella Valley segment (equivalent to
M = 5.7). The calculated slip does not depend on the
number of San Andreas segments allowed to slip at once,
or on the coefficient of friction. Slip on the San Andreas
fault with a moment equivalent to two moderate events
are therefore needed simply to relieve the stresses added
by the Landers sequence. In contrast, potential slip com-
parable to an M = 6.2 event is removed from the Mo-
jave segment, and slip equivalent to an M = 6 event is
removed north of Palm Springs [site of the 1948 M =
6 Desert Hot Spring earthquake; see Sykes and Seeber
(1985)], taking these portions of the fault farther from
failure. On the San Jacinto fault, the San Bernardino
Valley and San Jacinto Valley segments have an added
potential for slip of 5 cm along the northernmost 50 km
of the fault (Fig. 15,
lower panel).
So far creep has not
been detected and no M > 5 earthquakes have occurred
on the San Andreas or San Jacinto faults since the Lan-
ders event. If these events do not take place, the like-
lihood of great earthquakes on the San Andreas must in-
crease as well.
Static Stress Changes and the Triggering of Earthquakes
949
Time Change to the next Large Earthquakes on
the San Andreas and San Jacinto Faults
Because the southern San Andreas fault is late in the
earthquake cycle, the long-term probability of a great
earthquake on any of its three southern segments was
high before the Landers earthquake took place (Working
Group on California Earthquake Probabilities, 1988). The
San Bernardino Mountain segment last ruptured in 1812
(Fumal
et al.,
1993); given its 24 --- 3 mm/yr slip rate
(Weldon and Sieh, 1985), a ->4.3-m slip deficit has since
accumulated, which could yield an M -> 7.5 event. The
Coachella Valley segment last ruptured in 1680, has a
L
0
g
-4
-40
12 i
° F
r$)
-4
60
50
"~ 40
30
20
10
W
t i | i i i • i ii . i i . i ii . | . i .
Coulomb Stress Change /~12 = 0.75
...............................
i E
i i i i ! i i i i ! i
-20 0 20 40 60 80 100 120 140 160 180 200
Coulomb Stress
Change
on the San Andreas
Fault (for 12 = 0.4)
Long~I~rm
Immediate
(halfspace)
E-
l
=iI=~
• ~ " ~:!i Coachella Valley
i~i: !i!~! Mtn. Segment ~! Segment ~;~
35 mm/yr, last
event in
1857 i~i~
24+-3 mm/yr,
1812 ~:~
25-30 mm/yr,
1680
Slip Needed to Relieve ~ .~,
~ ShearStressChanges ! ~ ~ ~ ~ "~ ~.
0 ....
-40 -20 0 20 40 60 80 100 120 140 160 180 200
Distance East of Palmdale (kin)
Immediate
LRemoves load
Long-Term
Stress Changes
1 Adds load
~ Stress Changes
Figure 14. Coulomb stress change caused by the Landers, Big Bear, and Joshua
Tree earthquakes resolved on the San Andreas fault
(top two panels).
The San
Andreas is assumed to be vertical, purely right-lateral, and 12.5-km deep. The
fault is traced along the Mission Creekbranch (the northern strand between San
Bernardino and Indio in Fig. 13) and stress is sampled every 5 km along the
fault at a depth of 6.25 km. The
bottom panel
depicts the slip required to relieve
the imposed stress increase.
~
Removes load
Adds load
950 G.C.P. King, R. S. Stein, and J. Lin
slip rate of 25 to 30 mm/yr, and thus has accumulated
a -> 6-m deficit (M -> 7.5). Its prehistoric repeat time
is ---235 yr (Lindh, 1988). The Mojave segment last rup-
tured in 1857, has a slip rate of -35 mm/yr (Weldon
and Sieh, 1985), and thus has accumulated a 4.7-m def-
icit (M --- 7.7); its repeat time is 100 to 130 yr (Fumal
et al.,
1993; Jacoby
et al.,
1988). The San Beruardino
Valley segment of the San Jacinto fault may have last
ruptured in 1890; it has a slip rate of 8 --- 3 m m / yr
(Working Group on California Earthquake Probabilities,
1988), and thus has a slip deficit of ->0.8 m (M >- 6.8).
We estimate the advance and delay times of great
earthquakes by dividing the slip required to relieve the
applied stress by the local San Andreas or San Jacinto
slip rates. Alternatively, one could divide the calculated
Coulomb or shear stress change by the assumed stress
drop A~- of the last earthquake, and then multiply this
ratio by the next earthquake repeat time. Our calcula-
tion, however, benefits from being independent of the
earthquake repeat time or earthquake stress drop, for which
there is both uncertainty and variability. Our estimate
probably supplies a lower bound on the earthquake time
change because we neglect changes in normal stress act-
ing on the fault, which tended to increase the Coulomb
stress changes on the San Andreas.
We thus find that the next great San Andreas earth-
quake along the San Bernardino Mountain segment will
strike 8 to 10 yr sooner than it would have in the absence
of the Landers shock. Similarly, the next great San An-
dreas earthquake along the Coachella Valley segment is
advanced by 2 yr; and the next large earthquake on the
San Bernardino Valley segment of the San Jacinto fault
is advanced 8 yr. In contrast, we estimate a delay in the
next great Mojave shock by 2 yr.
Long-Term Stress Changes Caused by
the Landers Earthquake Sequence
All of the calculations so far have been carded out
in an elastic half-space on the assumption that for short
periods of time, creep processes at depth can be ignored.
Results from measuring and modeling earthquake-re-
lated geodetic data suggest that over periods of months
to a year or two this is a reasonable approximation.
However, stresses at depth will in due course relax and
modify the stress distributions that we calculate. For ex-
ample, after relaxation of the viscous substrate, the stress
concentration below the fault shown in Figure 7 is trans-
ferred back to the elastic part of the crust. Relaxation of
the lower crust reloads the upper crust, regardless of
whether relaxation takes place by creep on the down-dip
continuation of the fault, or by viscous flow in the lower
crust or asthenosphere (Thatcher, 1990). We can there-
fore approximate complete relaxation of the lower crust
by considering a 12.5-km-thick elastic plate over an in-
viscid fluid, with the material below the plate transmit-
ting only vertical buoyancy forces to the plate. At the
depths shown in our figures, the stress change on the
San Andreas and surrounding faults roughly doubles (Fig.
14,
middle panel,
and Fig. 15,
upper panel)
and the slip
required to relieve the stresses likewise grows (Fig. 14,
lower panel,
and Fig. 15,
lower panel).
The time needed
for substantial relaxation depends on the viscosity of the
lower crust, or on the rate at which creep propagates
down the fault, which is perhaps in the range of 30 to
100 yr. Thus, the stress changes caused by major events
such as Landers do not diminish with time; rather they
grow and diffuse outward from the source. If one were
also to include the secular rate of stressing caused by the
g~
O
0 15 30 45 60
Distance (kin)
Immediate
Removes load
Stress Changes
Adds load
Long-Term
~Removes load
Stress Changes ~'~
Adds load
Figure 15. Coulomb' stress change caused by
the Landers, Big Bear, and Joshua Tree earth-
quakes resolved on the San Jacinto fault. The San
Jacinto is assumed to be vertical, right-lateral, and
12.5-kin deep; stress is sampled every 5 km along
the fault at a depth of 6.25 kin. The
lower panel
depicts the slip required to relieve the imposed
stress increase.
Static Stress Changes and the Triggering of Earthquakes
951
plate tractions, the long-term stress changes would be
larger still.
Conclusions
Earthquake slip causes stresses to change. The stress
increases result in further earthquakes. Aftershocks are
the most readily studied of such events because of their
large number. The aftershocks of the Joshua Tree,
Homestead Valley, Big Bear, and Landers earthquakes
all have epicenter distributions that may be predicted on
the basis of the Coulomb failure criterion; events occur
where Coulomb stresses have risen. In the case of Joshua
Tree and Homestead Valley, where the depth distribu-
tions of aftershocks are reliably determined, these are
also effectively predicted by increases in Coulomb
stresses. The exact locations of off-fault stress changes
are modestly sensitive to assumptions about regional stress
direction and, to a lesser extent, the effective friction
coefficient. Since a range of plausible values can repro-
duce observed aftershock distributions, neither the ef-
fective friction coefficient nor the regional stress field is
constrained by our results. Conversely, the predictive
power of the method that we use does not depend on
having a detailed knowledge of these parameters.
Stress increases of less than one-half bar appear suf-
ficient to trigger earthquakes, and stress decreases of a
similar amount are sufficient to suppress them. The for-
mer, in agreement with current ideas of self-organized
criticality, suggests that some parts of the brittle crust
are always on the threshold of failure. This indicates that
over periods of aftershock sequences, other processes do
not change stresses by even modest amounts.
Over long time periods Coulomb stress changes in
the upper crust will increase as a result of stress relax-
ation processes in the lower crust. Thus, our calculations
may understate the amplitude of the triggering stress for
delayed events. Jaum6 and Sykes (1992) and Simpson
and Reasenberg (1994) have also argued that postseismic
fluid flow will, under some circumstances, raise the ef-
fective coefficient of friction, causing long-term in-
creases in the static stress changes. The time constant
for these effects is subject to speculation, and thus we
leave the evolution of the Coulomb stress changes to fur-
ther study.
Coulomb stress changes do not only predict after-
shock distributions. The Landers earthquake rupture oc-
curred within a narrow zone where a series of previous
events had enhanced Coulomb stresses. The Big Bear
earthquake that followed Landers was also apparently
controlled both in its initiation point (near the maximum
Coulomb stress increase due to the Landers rupture) and
in its termination point (where the Coulomb stress change
was negative) by stresses due to the main event, These
observations suggest that regions of enhanced Coulomb
stress should be regarded as candidates for future events.
The stresses transmitted by the Landers sequence to the
nearby San Bernardino segment of the San Andreas fault
are substantial, between 2 and 6 bars, and these could
grow to 10 bars as the lower crust beneath Landers re-
laxes. The rate of small earthquakes has risen on this
portion of the San Andreas since the Landers earth-
quake. But unless these stresses are relieved by the oc-
currence of an M - 6.5 event on the San Andreas, the
next great earthquake on the San Bernardino segment
may be advanced by a decade or more.
Acknowledgments
We are grateful for illuminating discussions with Albert Tarantola,
Paul Tapponnier, and Ruth Harris. We thank Robert Simpson for con-
ducting an extensive joint calibration of our respective programs. The
calculations we report were carried out using the boundary element
program VARC 0.9, written by King. The program incorporates half-
space code modified from Okada (1992), and employs the methods
outlined by Crouch and Starfield (1983) as modified by Bilham and
King (1989) for freely slipping fault elements.
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Institut de Physique du Globe
67084 Strasbourg France
(G.C.P.K.)
U.S. Geological Survey
Menlo Park, California 94025
(R.S.S.)
Woods Hole Oceanographic Institution
Woods Hole, Massachusetts 02543
(J.L.)
Manuscript received 20 August 1993.
