JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. B2, PAGES 2601-2618, FEBRUARY
A constitutive law for rate of earthquake
its application to earthquake clustering
J•mes
production
10, 1994
and
Dieterich
U.S. Geological Survey, Menlo Park, California
Abstract. Seismicity is modeled as a sequenceof earthquake nucleation events in which
the distribution of initial conditions over the population of nucleation sourcesand stressing history control the timing of earthquakes.The model is implemented using solutions
for nucleation of unstable fault slip on faults with experimentally derived rate- and statedependent fault properties. This yields a general state-variable constitutive formulation
for rate of earthquake production resulting from an applied stressinghistory. To illustrate
and test the model some characteristics of seismicity following a stress step have been
explored. It is proposedthat various features of earthquake clustering arise from sensitivity of nucleation times to the stresschangesinduced by prior earthquakes. The model
givesthe characteristicOmori aftershockdecay law and interprets aftershock parameters
in terms of stresschange and stressingrate. Earthquake data appear to support a model
prediction that aftershockduration, defined as the time for rates to return to the background seismicityrate, is proportional to mainshockrecurrencetime. Observed spatial
and temporal clustering of earthquake pairs arisesas a consequenceof the spatial dependence of stresschangesof the first event of the pair and stress-sensitivetime-dependent
nucleation. Applications of the constitutive formulation are not restricted to the simple
stressstep models investigatedhere. It may be applied to stressinghistories of arbitrary
complexity. The apparent successat modeling clustering phenomenasuggeststhe possibility of using the formulation to estimate short- to intermediate-term earthquake probabilities following occurrenceof other earthquakesand for inversionof temporal variations of
earthquake rates for changesin driving stress.
Introduction
Change of the rate of production of earthquakes is a
readily observed and characteristic feature of earthquake
occurrence.
Physical causes of temporal variations of
seismic activity remain poorly understood.
However,
various attributes of earthquake activity indicate a
connection between alteration of stress and earthquake
rates.
Significant geologic events which modify stress
state are often associated with pronounced changes of
the rate of production of earthquakes. Examples include
aftershocksfollowing large earthquakes, earthquake swarms
associated with magma intrusion processes,and systematic
variation of background seismicity through the stressing
cycle of great earthquakes[e.g., Ellsworth et al., 1981].
Also, changes of effective stress related to impoundment
of reservoirs and fluid injection into deep wells are well
known to alter seismicity. Finally, at distancesgenerally not
associatedwith aftershocks,Reasenberg
and Simpson[1992]
have reported correlations between small stress changes
calculated for the 1989 Loma Prieta, California, earthquake
and small changesin the rates of production of background
seismicity.
This paper presents a general approach for obtaining rate
of earthquake activity resulting from some stressinghistory.
The concept is implemented for faults with rate- and statedependent constitutive properties which are derived from
This paper is not subject to U.S. copyright. Published in
1994 by the American Geophysical Union.
Paper number 93JB02581.
2601
laboratory fault slip experiments. Previously, some effects
of stress perturbations on rate of earthquake occurrence
and earthquake probability have been simulated for faults
with theseproperties[Dieterich, 1986, 1987, 1988b]. Those
previous applications employed numerical simulations or
limited closed-form solutions. The following presents a
general constitutive formulation for rate of earthquake
production on faults with rate- and state-dependent friction.
The formulation is amenable to exact solution for simple
stressing perturbations, and by straightforward numerical
implementation it may be applied to stressing histories of
arbitrary complexity. To illustrate and test the model
some applications to earthquake aftershocks and earthquake
clustering are presented. Table i lists symbols used in this
study and equation numbers where the symbols first appear.
Model
General
formulation.
In the following discussion
the term "earthquake nucleation" is used to describe the
processes and interactions that lead to the initiation of
an earthquake instability at some specific place and time.
Nucleation is assumed to occur over a restricted region
which
is referred
to as a nucleation
source.
The
essential
concept of the analysis is the treatment of a seismically
active volume of the Earth as having a population of sources
that nucleate successiveearthquakes to produce observed
seismicity. The objective is to find the time at which each
source in the population initiates an earthquake by following
the evolution of conditions on the sourceswhen subjected
to some stressing history.
2602
DIETERICH'
CONSTITUTIVE
LAW FOR EARTHQUAKES
Substituting the distribution of initial conditions into the
To implement this proposition requires a specific model of
the earthquake nucleation process. The general formulation
generalsolutionof (1) givesthe distributionof eart}aquake
schemedevelopedin equations(1) throu.
gh (4) can be
times for the general styessinghistory
,
employed for various models of the earthquake nucleation
process. Constitutive properties and system interactions
that provide a mechanism for the onset of unstable fault
slip must be specified, giving the time t at which a particular
source nucleates an earthquake from initial conditions and
stressing history:
t = F [C(n, r, *•), •-(t)]
(4)
Because stressing histories may be rather complicated,
involving stressjumps and irregular fluctuations, practical
Table 1.
Symbol
t - FIe, ½(t)] ,
.
Mathematical Symbols
Equation
First Used
Description
where C representsinitial conditions and r(t) is some
general stressinghistory. For example, in a model with
a,b
A,B
(•5)
Omori aftershock decay parameters
fault constitutive parameters
instantaneous nucleation at a stress threshold, C might be
expressedas the initial stressrelative to the stressthreshold.
c
C
(20)
radius
conditions
'In this work, which is implementedfor a model of timedependent nucleation by accelerating slip, C is the initial
slip speed on the nucleation source.
Changes of stressstate within the volume containing the
population of nucleation sources may arise from external
processesand possibly from earthquakes occurring within
the volume. As usedhere, stressinghistory may also include
changes of effective normal stress brought about by pore
pressurechanges. The volume is assumedto be sufficiently
small that all sourcesexperience the same changesof stress,
although conditions, which include initial stress state, will
vary over the population of sources.
This work employs the convention of redefining C at the
start of every time step in situations where time stepping is
employed to follow the evolution of conditions. In general,
the distribution
of initial
conditions
at the start
of a time
increment will be determined by the prior stressinghistory
and the prior distribution. Consequently,a distribution of
initial conditions under some reference stressinghistory is
first constructed. In setting up the reference distribution
it is assumedthat a constant steady state seismicity rate r
developsunder a condition of constant referencestressing
rate 'r.
Hence
of shear crack
general ¾epresentationof initial
on nucleation
source
Dc
(6)
G
H
(A1)
(7),(A8)
term containing model parameters
k
(7),(A1)
effectiv•stiffnessof nucelation
1
lc
(A1)
(A2)
characteristic sliding distance for
evolution
of fault
state
shear modulus
for nucleation
source
source
dimension fault patch
characteristic
nucleation
n
(2)
m
(B20)
r
_R
Ro
(2)
(11),(B24)
(16)
earthquake sequence number indicating nth source in distribution.
term used in solution for stressing
by logarithm of time
reference seismicity rate
seismicity rate
seismicity rate immediately following a stress step
characteristic time for seismicity to
return to steady state
(•8)
(19)
(B8)
where n is the sequencenumber of the earthquake source
and indicates the nth source in the distribution. Although
of
time, time of instability
t
ta
te
dimension
source
characteristic
time
for initiation
of
aftershock decay
mean earthquake recurrence time
parameters for characterization of
stressingby the logarithm of time
distance
from center
of center
of
x
(20)
c•
(6)
constitutive parameter for evolution
Poissqn
•odel of occurrence).
Thisappears
to be the "/
(7)
state variable for seismici•tyformula-
andis supported
by someobservations
[e.g.,Wyssand %s
(•0)
specify
a nonsteady
distribution
of earthquakes
overtime,
(ml)
(•)
(•)
steady state value of "/
slip and slip speed
crack geometry parameter
strictlyregularseismicity
is assumed
for construction
of
the reference distribution, the constant rate r may be
interpreted
as a statisticalmeasure
of expected
rate of
earthquake production for some magnitude interval (i.e.,
simplest assumption to make for defining the distribution
Haberman,
1988].
Alternate
assumptions
to(2)might
beto 5,5
r/
/•o
based
forexample,
onmodels
of stress
heterogeneity
or 0
variation of seismi•cpotential through the cycle of great
shear crack
of state
(5)
recurrence time of major crustal earthquakes.
C -- C (n, r, 'r)
(3)
stress
tion
nominal
state
coefficient
variable
of friction
in fault
constitutive
Are
normal
stress
shear stress and shear stressing
least
over
timeintervals
thataresmall
compared
tothemean •-''
Equating (2) with the specific solution of (1) for the
normal
formulation
earthquakes. For the specific implementation developed
below, use of steady state rate appears to be adequate, at
constant stressing rate 'r gives the distribution of initial
conditions that yields the constant reference seismicity rate
with
rate
(3)
(19)
reference stressing rate
earthquake stress change
(A2)
scaling parameter for critical
fault length l• for unstable slip
(negative)
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
2603
applicationsmay require numericaltreatment (for example, state variables [Ruina, 1983; Weeks and Tullis, 1985;
by time-marching,usingstepscomposedof simplestressing Tullis and Weeks, 1986]. Becauselittle complexity is
functions). In order to update initial conditionsfor each added, derivation of solutions for the nucleation process
time step, such applications require determination of the
(Appendix A) and derivationspertaining to earthquake
rates with populationsof sources(Appendix B) retain the
change of C arising from the stressing history and the
nucleation process.
As formulated, this approach does not predict the specific
magnitude of individual earthquakes within the magnitude
range that is implicit in the definition of r. However, because
the formulation yields rate of production of earthquakes
in response to a stressing history, it can be combined
with magnitude-frequency relations or may be used in
conjunction with stochastic models that employ seismicity
rate.
use of multiple state variables.
From experimental observations, state is inferred to
depend on sliding and normal stress history. This study
employs
for evolution of state by displacement 5 and normal stress
rr. In (6), Dc• is a characteristicdisplacementand c•i is
Implementation
using rate- and state-dependent
friction.
Using solutions for the nucleation of slip
instabilities on faults with rate- and state-dependent fault
a parameter governing normal stress dependence of Oi. At
nucleation process have been generalized somewhat from
of displacement dependence of 0, used in several studies
steadystate(rr -- const,dO/dt= O)088= Dc/•. When
not at steady state, 0 seeks 0ss over the sliding distance
strength [Dieterich, 1992] the procedure just presented D•. The displacementdependenceof (6) is that proposed
by Ruina [1983],and the normal stressdependenceis from
is followed to obtain a specific formulation for rate of
Linker and Dieterich [1992]. An alternate representation
earthquake production. In Appendix A derivations for the
Dieterich [1992]. In Appendix B the distributionsof initial
conditions and earthquake times corresponding to equations
(3) and (4), respectively,are derived. On the basis of
these results a general formulation for rate of earthquake
production is obtained. Results of those derivations are'
summarized
below.
The rate- and state-dependent representation of fault
constitutive properties provides a framework to quantify
somewhat complicated experimental observations of fault
propertiesand to unify conceptsof dynamic/static friction,
earthquake fracture energy, displacement weakening, timedependent healing, slip history dependence and slip speed
dependence[Dieterich, 1979a, 1981; Ruina, 1983].
In
addition to representing observed transient and steady state
velocity dependenciesduring stable sliding, this constitutive
relation has been applied to model the spontaneous timedependent onset of unstable slip in laboratory experiments
[Dieterich, 1979b, 1981; Ruina, 1983; Rice and Gu, 1983;
Weeks and Tullis, 1985; Blanpied and Tullis, 1986; Okubo
and Dieterich, 1986; Tullis and Weeks,1986], fault creep
[$cholz,1988],earthquakeafterslip[Maroneet al., 1991]and
the earthquakecyclealongplate boundaries[Tse and Rice,
1986; Stuart, 1988].
Several essentially equivalent rate- and state-dependent
fault constitutive representations have been discussedin the
literature. A simplifiedform usedby Ruina [1983],basedon
Dieterich [1979a,1981], and generalizedfor multiple state
variables[Rice and Gu, 1983]may be written as
[e.g., Tse and Rice, 1986; Tullis and Weeks,1986],doesnot
permit the simple derivations employed here, but numerical
simulations of the nucleation processgive comparable results
for the time dependence of earthquake nucleation for both
evolutionlaws [Dieterich, 1992].
The nucleation process on faults with these properties
is characterized by an interval of self-driven accelerating
slip that precedes instability. Over a range of stresses
the logarithm of the time to instability is found to
increasewith decreasingstress. Simulationsof faults with
heterogeneousproperties show that (1) the zone of most
rapid accelerationter}dsto shrink to a characteristicfault
length l• (Appendix"A, equation (A2)) as the time of
instability approachesand (2) nucleationsourcesof length
lc develop spontaneously in regions where the shear stress
relative to the sliding resistance averaged over lc is higher
than the surroundings. In the context of the current
model the population of sourcesis postulated to arise from
heterogeneity of fault conditions.
In general, slip speed is determined by the independent
variables 0, r, and rr. It is shown in Appendix A that in
an accelerating slip patch evolution of 0 is determined by
slip. Under this condition the dependence of the time to
instability on initial conditionsis fully specifiedby the initial
slipspee•d
•o ofthepatch(Appendix
A, equation
(A7)). Slip
speedsremain very small until a short time prior to the onset
of instability. Hence the interval of premonitory slip may be
quite long.
Following the procedure described above, the distribution
of initial slip speedsfrom Appendix B is
(•(n)
--
r=a/•o+Aln
(• -[-Slln
0-•+B21n
0-•+'"' (5)
1
Hrr"/[exP(Arrr]
5-,•n
'•_1]'
(7)
wherer andrr areshearandnormalstressrespectively,
3 where H = (B/D•) - (k/a); k is the effectivefault patch
is slip speed, and Oi are state variables. Parameters /•o,
A, and B, are experimentally determined coefficients. The
terms with asterisks are normalizing constants.
Applications to problems in nature have generally
employed only one state variable. Simulations of some
laboratory experiments has been improved by use of two
stiffness and - is a state variable that evolves with time
and stressing history. The term H does not appear again
in the
formulation.
For
the
initial
reference
distribution
correspondingto (3), ' takes the value
1
7 -- -Tr
ß
(8)
2604
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
Because slip speed increases as the nucleation process
develops,the distributionof slip speedsevolveswith time.
Appendix B shows that the distribution retains the form
of (7), independentof subsequentstressinghistory, but "/
evolves with stressing history as described by
d7= •aa dt- vdr+7 -a -.
,
(9)
wherea- a• + a• +... for applications
involvingmultiple
state variables. Recall that a is taken to be effective normal
stress;hence(9) mw be employedto modeleffectsof porefluid pressure changeson seismicity rates.
For positiveshearstressingrates with & = 0, equation(9)
has the property that 7 see• the steady state value,
%s = l/q-
(10)
0
1
2
Time to instability
with the characteristicrelaxationtime of
The distribution of earthquake times is obtained by
substituting the distribution of initial conditions into the
solutions for time to instability. In turn, seismicity rate R is
obtained by differentiating the distribution of times giving
the general result
R= 75-,-'
Ao'/• r
Figure 1. Distribution of slip speeds against time to
instability. The solid curve is the solution for time to
instability given an initial slip speedof the nucleation source
(equation (All) of Appendix A). Solid squaresgive the
distribution of slip speedsfrom equation (7) under the
(11) reference condition at
where r is the steady state seismicity rate at the reference
constant seismicity rate r. Small
circles show the effect of a stress step on the distribution
from solutionof evolutionequation (9). This solutionis
shear stressingrate ft. From (10) and (11) it is seen
givenin AppendixB (equation(Bll)). The effectof a stress
that at steady state the ratio of seismicityrates (R/r) is
step is to increaseslip speed(vertical arrows)of eachsource
simply (•/fr). Rate R may be interpretedas a statistical
as givenby (A17). This increaseof slip speeddecreases
the
representationof expected rate of earthquake production for
time to instability (horizontalarrows)and resultsin higher
some magnitude interval.
densityof points (higherseismicityrate) at short timesbut
Equations(9) and (11) are a principalresult of this study
and represent a state-dependent constitutive formulation for
rate of earthquake production. Some useful nonsteady state
solutionsof (9) are given in Appendix B. These include
evolution of 7 with time under constant stress,linear change
of shear stress with time, and change of shear stress with
the logarithm of time (equations(B7), (B17), and (B21),
respectively)and evolutionof 7 for stepof shearand normal
stress(equation(Bll)).
Figure I plots the slip speeddistributionof equation (7)
against instability time. Also shown is the effect of a stress
step on the distributionas given by (Bll).
Effects of simple stressing perturbations on seismicity
rate may be obtained by substitutingsolutionsof (9) into
(11) assumingthe seismicity is initially at steady state
(70 - 1/%). For simulationof complexstressinghistoriesa
straightforward procedure consistsof breaking the stressing
history into time steps of constant shear stressing rate
and stresssteps (of r and er) and successivelyapplying
the appropriate solutions to follow the evolution of 7 from
increment to increment. Seismicity during a time step is
then obtained by substituting the result for evolution of
with time ((B7) or (B 17)) into (11). Alternatively,numerical
solutionsof (9) may be obtained.
Parameter
values.
To implement the model, fault
does not affect density at larger times where the dependence
of time to instability against log slip speed is linear. Hence
with the passageof time, the perturbation of seismicity rates
from the stressstep evolvesout of the system.
M. L. Blanpied,personalcommunication,
1992]. Numerical
examples in the following use A = 0.01. Experimental
determination of c• is presently limited to a single study
[Linker and Dieterich,1992]whichobtainedc• = 0.23 and
establishedgenerally that 0 _• c• _• •s•, where •
state sliding friction.
Application
Seismicity
illustrate
to Earthquake
rate
following
and test the model
is steady
Clustering
a
stress
step.
some characteristics
To
of rate
of earthquake production following a stress step have
been explored. It is proposed that several types of
clustered earthquake phenomena represent a perturbation
of backgroundseismicityresultingfrom the changeof stress
state causedby previous earthquakes.
In the interest of simplicity the stress step induced
by a previous earthquake on the surrounding region is
represented as a positive step in shear stress, At, with
constitutiveparametersA and a which appear in (9) and normalstressheld constant.The solutionof (9) for a stress
(11) must be specified. From laboratory observations, step (Bll) showsthat a decreaseof effectivenormal stress
coefficientA generally has values in the range 0.005 to 0.012
for various temperature and pressure conditions. Under
hydrothermal conditions, it appears that somewhat larger
values(A •-0.02) may be appropriate[Blanpiedet al., 1991;
er produces the same qualitative effect as an increase of
shear stress in that both decrease 7 and lead to higher
rates of earthquake production. Subsequentevolution of 7
is independent of prior processesthat caused 7 to change.
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
2605
Hence time-dependentfeaturesdescribedbelow that result
from a positive step of shear stresswould also ari•sefrom
(16)
negativestepsof normal stress.The followingapplications
further assumeconstant• followinga stressstep and steady
state seismicityrate r beforethe step (i.e., % -- 1/•r).
Under these assumptions, simple expressionsmay be whereR0 is the initial seismicityrate followingthe step at
obtained for seismicity rate. First, equation (Bll) is t - 0. If stressingrate is the same before and after the
employedto evolve-/ through the stressstep to obtain -/ mainshock(• - •), then from (14), (Acr)/• -- t• and
a--(Act
b- (Act
-At
Act
r '
-•--•/)
exp(
Act
)-(-•-•/)
(•00)
immediatelyafter the step. This becomes"/0 in (B7) or
(B17) whichthengivesthe evolutionof -/for the subsequent
time interval where stressing rate is assumed constant.
Finally, the expressionsfor -/resulting from theseoperations
are substitutedinto (11) to yield seismicityrate asa function
of time after the stress step:
, +/ 0,
As shownin Figure 2, normalizedseismicityrates R/r for
differentstressstepsall decayalongthe same1/t asymptote.
The characteristictime te for the rate to mergewith the 1/t
asymptoteis obtainedby solvingfor the intersectionof R0
with the asymptoteformedby settingA•- -- oc in (13) giving
Art)- exp 1
and
(17)
a-rt•,b-t•exp Act -t• •00 '
te -- --
r•
exp
-- b.
A•
(18)
See Figure 2 for graphical interpretation of t•.
Aftershock
exp Art + t%/Arr
where % and • are the stressingrate prior to and following
the step, respectively,t is set to zero at the time of the stress
duration.
Aftershock
duration
is defined
here as ta, the characteristictime for aftershocksto return to
the backgroundseismicityrate. From (14), t• is independent
of the magnitude of the stress step but depends upon
stressingrate. Assuming the approximate relation • --
-A•-•/t•, whereA•-• is the stresschangeon the mainshock
step, and
rupture, and • is mean earthquakerecurrencetime, gives
=
Aa
t• --
Act
:t•--
Act
(19)
is the characteristic relaxation time for the perturbation of
earthquakerate. Note that for /X•- - 0 and • :/: 4-r (12)
givesthe time-dependentchangeof seismicityfor a changeof
stressing
rate from% to •. Figure1 illustratesequation(12)
by showingthe effectof a stressstep on the distributionsof
slip speeds and nucleation times.
Origin of aftershocks. It is proposedthat aftershocks
are causedby the steplikechangeof stressthat occursat the
time of the mainshock. Two general observations support
this hypothesis. First, aftershocksoften cluster in those
areas near a mainshock rupture where stress changesfavor
aftershockfault slip, including zonesoff the principal fault
The negativesign in (19) arisesbecausestressdecreaseson
the earthquake rupture surface. A similar result is given
by Dieterich [1988b]. Equation (19) constitutesa specific
prediction of this model. In this prediction, earthquake
magnitudedoesnot explicitly affectaftershockduration but
104 .
•"-•A•/Ao'
10 3
plane[e.g.,Stein andLisowski,1983].Second,suddenjumps
of stress from causes unrelated to fault slip also stimulate
aftershocklikeburstsof seismicityfollowedby the 1/t decay
characteristic of aftershock sequences. An example of the
latter are aftershocklike bursts of seismic activity on the
south flank of Kilauea volcano following dike intrusion
events[Dvoraket al., 1986].
Equations(12) and (13) are plotted in Figure 2. Note that
equation(13) has the form of Omori's law for aftershock
102
---
- 3
101
'-'•-.•.•
T2
1
'"•;;"..
..... '--'-•.;
-
100
decay:
Equation (12)
-- Equation
(13)
"
a
b+t
Similarly, equation (12) gives Omori's law for t/ta < 1,
..
o
10-1
10-4
10'3
10'2
but seismicity rate merges to the steady state background
10'1
100
101
Time, t / ta
rate for t/ta > 1. This characteristicof (12) makes it a
somewhat preferable representation of aftershock decay for
many applications.
The agreement of this model with Omori's law
permits assignment of physical interpretations to aftershock
Figure 2. Seismicity rate following a stress step from
equations(12) and (13) for the cases• • 0 and • - 0,
respectively. The family of curves gives solutions for the
indicated values of stress step A•- normalized by Act. Tho
parameters. From (13) and (15),
inset illustrates
the definition
of t•.
2606
DIETERICH:
CONSTITUTIVE
will enter indirectly if t• is a function of magnitude. As a
test of this predicted dependence of aftershock duration on
recurrence time, some preliminary data for ta and tr have
been assembled. Data sources and methods of parameter
estimation are summarized in Table 2. Figure 3 plots the
data and the predicted relationships between recurrence
time and aftershock duration. These results, though quite
Table
Event
1
2
LAW FOR EARTHQUAKES
preliminary and having large scatter, appear consistentwith•
the prediction of (19) and suggestthat more systematic
study of aftershock duration may be warranted.
Spatial effects due to nonuniform stress changes.
Because aftershock rate depends on the magnitude of the
stress step, systematic spatial variations of the magnitude
of the stresschange will result in systematic spatial variation
2. Earthquake Recurrenceand Aftershock Duration
Location
Magnitude
Bear Valley,
California, 1972
5.0
Coyote Lake,
California, 1979
5.7
years
0.2-0.5
years
14-21
Data
Sources
Recurrence
and Method
of Estimation
times are from Ellsworth
and Dietz
[1990].Aftershockdurationis obtainedby
fit to aftershockequation(12).
1.4-2.5
54-82
Upper limit on recurrence time from Reasen-
bergand Ellsworth[1982]. Minimum recurrence time is estimated assuming slip of
70 cm and Calaveras fault slip rate of
1.3 cm/yr from Matsu'ura et al. [1986].
Aftershock duration is obtained from fitting
of aftershockdata [Reasenberg
and
Ellsworth,1982]to equation(12).
3
Loma Prieta,
California, 1989
7.1
Kalapana, Hawaii,
7.5
1.7-3.5
84-100 Recurrencetime is from WorkingGroup[1990].
Aftershock duration is provided by P. Reasen-
berg (written communication,1992).
4
8.5-11.2
50-100
Recurrence time is estimated from rift opening
rate of 6-12 cm/yr and net offsetof 6 m
[Dieterich,1988a]. This is generallyconsis-
1975
tent with recurrence following the great
earthquakeof 1868 [Wyss,1988]. Aftershock
duration obtained by fitting aftershock
decayto equation (12).
5
Koaiki, Hawaii,
6.7
1-1.7
10.5-21 Recurrencetime is from Wyss[1986]. Aftershock
duration is obtained by fitting aftershock
1983
decayto equation (12).
6
Alaska, 1964
9.2
4.6-9.6
230-615
Recurrence time is estimated assuming plate slip
rate of 6.5 cm/yr and earthquakeslip of
15-40
7
Nobi, Japan, 1891
-,•8
m. Aftershock
duration
is from data
suppliedby J. Lahr (personalcommunication,
1990).
_>78 5,000-20,000Recurrencetime from Tsukuda[1983]is basedon
dating of paleoseismicevents. Aftershocksof
this event continue
to at least 1969 without
showing any indication of returning to a back-
groundrate [Utsu,1969].
8
Rat Islands, Alaska,
8.7
2.7-5.5
72-120
from Ogata and $hirnazaki[1984]to equation
(14). Recurrencetime is estimatedassuming
plate motion rate of 8.3 cm/yr and earthquake
1965
9
San Francisco,
California, 1906
Aftershock duration is obtained by fitting of data
8
5-10
129-281
slip of 6-10 m.
Recurrence time is taken to be range of estimates
for recurrence on San Francisco peninsula and
North Coast segmentsfrom Working Group
[1990]. Aftershockduration is from data of
Ellsworthet al. [1981].
10
San Juan Bautista,
California, 1979
4.8
0.5-0.8
4-8
Recurrence time is estimated from fault slip rate
of 2.5 cm/yr [Matsu'uraet al. 1986]and
estimated slip of 10-20 cm. Aftershock
duration is obtained by fitting of aftershocks
to equation(12).
DIETERICH: CONSTITUTIVE
LAW FOR EARTHQUAKES
2607
3OO
The net seismicity rate originating from a finite region
surrounding a mainshock may be obtained by numerically
IO0
integrating (21) over the region of interest. Figure 5 gives
results for finite regions assumingr, '•/%, and Act are
the same everywhere. For a finite region it is found that
earthquakerate R decaysby (l/t) p, where p = ,•0.8. This
is in contrast to the decay by 1/t arising at any point or
? lO
for finite regions where the stress change is uniform. The
decrease of decay rate derives from the dependence of te on
magnitude of the stressstep. At time t some regionswill be
at t < te (rate approachesconstant rate asymptote) while
other regionswill be at t > t, (rate decaysalong the 1/t
asymptote).
•
0.1
0.01
i i • •,•11
1
, , ,• ,1•[I
10
• • • .... I
1 O0
........
1000
I
,
10000
30000
Recurrencetime (years), t r
Figure 3. Observed and predicted aftershock duration
t• against mainshock recurrence time t•.
Curves give
solutionsof equation (19) for indicated valuesof mainshock
stress drop ATe normalized by Act. Sources of data are
summarized
in Table
2.
of seismicity in the vicinity of a previous earthquake. The
following gives a simplified representation of stress change
Figure 5 also illustrates a potential source of bias that can
arise in estimation of aftershock parameters such as t• where
aftershock rates vary in space. Estimates of aftershock
duration may be seriously underestimated if regions with
little or no increase in earthquake rate are included in the
aftershock count. For example, in Figure 5 the cumulative
rate overthe regionfrom x/c - 1.001to the large, essentially
unperturbed distance of x/c: 8 also showsan interval of
decay with p •-0.8 but has an apparent aftershock duration
that
is much
less than
the actual
duration.
Aftershock zones characteristically expand with time
[Tajima and Kanamori, 1985; Wesson, 1987]. Apparent
spread of an aftershock zone in the present model arises
from the nonuniform stress change of the mainshock.
Substitutingthe expressionfor AT of (20) into (18) yields
from slip on a planar surface in a homogeneouselastic
te -- ta exp
medium which should be common to all earthquakes:
(22)
AT--ATe 1-•
where
c is
crack
radius
-1 ,
and
x
is
x>c
radial
(20)
distance
from
Again, the result of (19) may be used to replace the'
(A%/c•A) term. Recall that te is the characteristictime that
the aftershockrate beginsto mergewith the 1It asymptote.
10 4
the center of the crack. This approximation does not
represent azimuthal dependencies of shear stress, but it
does incorporate both the square root stress singularity
time 0
at the cracktip and the stressfalloff by 1/X3 at larger
distances
which
are
characteristic
of cracks
in
an
elastic
time
..0.001
10 3
medium. Equation (20) is employedto illustrate general
-Az•/A(•=
10.0-•
•
x!
Xe••
ta
characteristics of seismicity resulting from a nonuniform
stresschange in the vicinity of an earthquake rupture. Other
stress components and stresses at locations not lying on
the projection of the mainshock fault surface will show
comparable distance effects. However, application of the
model to specific aftershock patterns would of courserequire
detailed computation of the entire stress field.
•
/•--•
10 2
--
time
=0.01
-
time_: 0.1
Substitution of (20) into (12) gives seismicityrate as a
10 1
function of time and distance following a uniform stress
change ATe on a circular crack:
{5-r
[( )({1- }-- ]-l}exp[-•-a]
lOo
1.0
1.2
1.4
1.6
1.8
2.0
Distance to mainshockrupture,x/c
x > c, -• •= 0.
Using (19), the term A%/Ao- appearingin (21) may be
replaced by -t•/t•.
Figure 4 showsrate as a function of
distanceat differenttimes using (21).
Figure 4. Earthquake rate from equation (21) against
distance at different times following a prior earthquake.
The points labeled xe indicate the apparent edge of the
aftershockzone as given by equation (22).
2608
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
(a)
105
104
Region
of
aftershock
count
= -0.8
X2
Rupture
surface
Distance
interval
a
1••-•-•-.3
b 1.001 2.0
103
•.
•
102 •
•
•
[ •
101 ?
Xl
b•
•
•
ß
d
/c
•
c
•
•
1.001
4.0
de
11:•
1
, 168:•
100
10-1
10-4
,
,
I
,
I , Ill
10-3
I
I
I
[
I I Ill
I
I
I
I
I I III
10-2
I
I
I
I
10-1
....
I
........
100
101
Time, t/t a
(b)
105
Region
of
aftershock
count"•
104
•a
103
• slope=-1.0.
1/c
I'"':••il
l
x2/c
. ';ii•??--;....;)..=•
.........
• •:.-?•
...........
--
••
Distance
interval
a
x1
102
101
100
10-4
10-3
10-2
10-1
100
101
Time, t/t a
Figure 5. Predictednet aftershockrate for finite regionssurroundinga shearcrack (seeequation(21)).
(a) Seismicityrate for cumulativedistancefrom edgeof crack. (b) Seismicityrate for annular distance
segments.
DIETERICH:
CONSTITUTIVE
At distance_• xe normalizedrates R/r begin to approach
the same 1/t asymptote and becomeindependentof the•
LAW FOR EARTHQUAKES
104
'
i
2609
!
..__ zt•/A(•= 3.0
original stress change. However, at x • xe, the normalized
rates are smaller and scale by exp(A•-/A•), where A•decreases
with distance.Hencex• in (22) marksthe location
103
of the apparent edge of the aftershock zone at time t•.
As x• -• c•,t• -• ta, and aftershocks merge with the
background rate.
The position of xe at successivetime steps is indicated
in the example of Figure 4. Figure 6 showsthe position of
Q:: 102
x• againstt• from (22). The expansioncurvesof Figure 6
appear to be generally consistent with the aftershock
• ••_/tl:/A(•=
1.5
101
observationsof Tajima and Kanamori [1985] and Wesson
[1987] for values of (tr/ta) in the range 10 to 50 which
are also suggestedby the data of Figure 3. The predicted
smooth expansion of the aftershock region shown in Figure 6
assumesboth idealized stresschangesof (20) and that
parameters r, A, and • do not vary with location. Less
systematic propagation of aftershock zones will occur if the
restrictions of these assumptionsare relaxed.
100
Stress step at t = 0,
/•? /A(• = 10.
10-1
10-3
I
10-2
I
10-1
I
100
I
101
102
Time, t/t a
Nonconstant stressing following an earthquake.
Figure 7 illustratesseismicityfor a sequenceof stresssteps Figure ?. Seismicity rate for stressing history consisting
of several stress steps. The second and third stress steps
representinga hypotheticalstressinghistory consistingof a
mainshockfollowedby two large aftershocks.This example represent large aftershocks which give rise to secondary
assumesuniform stresschangesover the region of interest aftershock sequences.
and followsthe evolutionof "7throughthe severalloading
ramps and steps by successive
applicationof (B17) and following the step again givesp = 1 with characteristic time
(Bll), respectively.Followingeachstressstep, t is set to of decay t•.
zero and the value of "7at the end of the stressstep becomes
In general, nonconstant stressing history following an
"70for the next loading ramp. Note, that the calculated
seismicityrate, when plotted against the logarithm of the
time following the mainshock, gives the appearance that
the durations of secondaryaftershocksequencesscale by
the time following the mainshock. Note also that the
effect of secondaryaftershocksis to increasethe slope of
the aftershockdecay. Both effectsare artifactsof plotting
the entire sequenceagainst logarithm of time after the
mainshock. If rates are plotted against logarithm of time
followingeach later stressstep, the slope of the decay
5
tr
4
•3
•
earthquake stress step may alter aftershock decay rates.
Figure 8 gives earthquake rate for a stressing history
consisting of an initial stress step at t = 0 followed by
stressing that varies with the logarithm of time. Evolution
of "7was found by first applyingequation(B 1l) for a stress
stepthen applyingequation(B21) for stressingby logarithm
of time. This example provides an illustration of effects
that would arise from creep processesthat add to, or relax,
an earthquake stress step. If stress increases with the
logarithm of time, the slope of the aftershock decay again
gives p: 1. Note, however, that such loading can produce
an initial buildup of earthquake rates prior to decay. If stress
decreasesby logarithm of time, rapid decay with p • 1 is
possible.
Earthquake
clustering.
Aftershocks are but one
manifestation of the pronounced tendency of earthquakes
to cluster in space and time. In their study of earthquake
clustering,Kagan and Jackson[1991] first assembleddata
from several earthquake catalogs giving the space and time
statistics on the occurrence of earthquake pairs above some
magnitude threshold. These data appear to provide a
comprehensive and robust characterization of earthquake
clustering which is well suited for a test of the present model.
The following test is based on the Kagan and Jackson
2
1
plots of all earthquakespairs (i.e., before they applied the
aleclusteringalgorithms used in their subsequentanalysis)
01
and therefore includes independent events, foreshocks,
aftershocks, and other possible types of clustered events.
Figure 6. Expansionof aftershockzonefrom equation(22).
derived from the Havard catalog and for intermediate and
deep earthquakes from the PDE catalogues are reproduced
10-5
10-4
10-3
10'2
10-1
100
Time, t ?ta
Curves give the position of the edge of the aftershock zone
x• for the indicated values of the mainshockstressdrop
normalized by Art.
Data (prior to declustering) for shallow earthquakes
from Kagan and Jackson [1991] in Figures 9a, 9b, and
9c, respectively.
These data give observed numbers
of earthquake pairs normalized by the expected number
2610
DIETERICH'
CONSTITUTIVE
LAW FOR EARTHQUAKES
10 5
•: = •:o+uln(wt+ 1)
104
•
10 3
u= .3
--o
u -- .2
--o
u = .1
u=O
ß
10 2
U= -.1
ß
U = -.201
•.
U= -.3
101
10 o
10-1
10-2
10-3
10'4
101
102
103
104
105
106
107
108
109
10lø
Time (seconds)
Figure 8. Seismici.tyrate for a stressinghistory composedof a stressstep followedby changeof stress
with logarithmof time. Parametersfor the computatiqnare A = 0.01 MPa, rr = 20 MPa, Ar = 0.5 MPa,
and w - 10 s. If stressincreaseswith the logarithm of time, seismicactivity first increasesthen decays
by 1It. If stressdecreaseswith the logarithmof time, seismicactivity decaysby rates that can exceed
the 1It decay.
assuming Poisson occurrence. The normalized number of
pairs are computed for cumulative time and cumulative
distancesfrom the first event of all possiblepairs. Each data
set showssignificant clustering of earthquakes characterized
by an intervalin whichnormalizedfrequencyde[•.ays
with
time by (l/t) p with p • 0.8. Below,it is f6findthat
apparent clustering to the very great distances shown in
Figure 9 arises as a consequenceof the convention of using
cumulative
distance intervals instead of differential
intervals
for the summing of earthquake pairs.
Simulation of these data with the present model assumes
the rate of occurrence of the second earthquake of all
possiblepairs is perturbed by the stressstep of the previous
the first event
and over the distance
interval
from
x -
c
to x - 2c, 4c, 8c, etc. Because the Kagan and Jackson
analysis began the count of pairs only after some finite
interval of time had elapsed following the first event, the
integration for the simulations also began at finite elapsed
time t 0. To obtain the model statistic equivalent to Kagan
and Jackson, the net number of events cumulative to each
time and distance was then normalized by the expected
number assumingthere was no interaction. All simulations
assumethat the stressingrate before and after the first event
are the same.
Results of simulations are plotted in Figure 9. Parameters
for the simulations ta, Are/Arr -- tr/ta,
and W were
earthquakeas givenby (21). Becausethe stressas givenby
(20) is singularat x = c, for this application(21) has been
determined by trial and error to fit the data and are listed
modified
was fixed as the minimum distance employed by Kagan
and Jackson which was judged to represent a reasonable
approximation of the source radius of the mean catalogue
magnitude. The simulations are only moderately sensitive
of--Are.
to limit
the stress increase
to a maximum
value
The simulations employ an idealized geometry
of a seismogenic
belt (Figure10) whichis represented
as
a surface of half width W. The analysis assumesthat
before the first earthquake of any pair, the unperturbed
background rate of seismicity is the same everywhere along
the surface. The first earthquake in the pair has a radius c
and is assumed to occur in the middle of the belt. Numerical
integrat!on of the modified versionof equation (21) gives
the cumulative number of ea•rthquakepairs to time t from
in Table3. To reducethe numberof freevariables,
c
to modelparametersin that nea,rlyas goodfit to the data
couldbeachieved
withsom,ewhat
differentparameter
values
(seeTable 3).
The stress-stepsolution appears to provide a rather good
representationof the data ip_cludingintermediate and deep
earthquakes which may involve mechanical processesthat
DatafromKaganand Jackson(1991)
HARVARDcatalogue,depthinterval0 - 70 kin, M_>6.0
(a)
3
•
•
/
2
I
••
"•
•
'•
I
•:•
•"•..
',• •
_
I
I
Distance
Interval
km
_
Modelparameters
ta= 3700days
x
tr/t a = 45
'¬
-
•
3",z,,_'•
'••x
W= 102kin
x
12.8
o
25.6
ß
51.2
x
tel
-
x
102.4
z
204.8
Y
409.6
n
819.2
a
1638.4
o
3276.8
•'
6553.6
*
13107.2
ø 20000.0
0
.01
I
i
.1
1.
.....
10
100
1000
10000
Time (days)
(b)
DatafromKaganandJackson(1991)
PDE catalogue,depthinterval71 - 280 km, M_>5.3
I
I
'
I
I
Distance
•
•,,
ß
Interval
km
IModel
parameters
ta= •t0days
tr/ta = 50
. c=3.2krn
W=
• 1
xx
••0
zzzz A
'"J
•x
o0
ZZ
•
.01
o
Y
.1
+1
XX
z
oo
x
1.
6.4kin
10
100
1000
o
3.2
+
6.4
x
12.8
o
25.6
,',
51.2
x
102.4
z
204.8
Y
409.6
o
819.2
•
1638.4
"
3276.8
10000
Time (days)
Data
from
Kagan
andJackson
(1991)
(c)
PDE catalogue,depthinterval281 - 700 km, M>_5.3
2
I
•
e
'•
•
I
•
•
•
•
••
'-
•,•
I
'•
•
X
c -- •.4km
ta
•,
o'•
Interval
km
ta= 4.8days
tr/ta = 1000
\
_
•
x 12.8
•
_
,• p
øoxo o
•
•')
Distance
Model
parameters
x
1
I
ooo
z
x
z
•' 51.2
x
102.4
z 204.8
xx •'AZ•
o
o25.6
x
Y409.6
o
,A
•
o,5
•>eO0
819.2
x 3276.8
+
0
.01
.1
1.
10
100
1000
10000
Time (days)
Figure 9. Clusterh•õof e•r•hqu•e p•irs for (•) shallow, (5) intermediate, •nd {c) deep e•r•hqu•es.
D•
•re from ••
•
J•c•o•
[1991]. Curves give examplesof simulationsusing •he mode] of
Figure 10.
2612
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
Table 3. Rangeof ParametersGiving SatisfactoryFit to Data of Figure 9
Data
Catalog
Depth
Harvard,
0-70 km,
PDE,
71-280 km,
PDE,
281-700 km,
*Preferred
tr/ta
Magnitude ta, years (-Ate/Act)
M >_6.0
M _>5.3
M _>5.3
Source
Radius, c
W
10.2'
45*
12.8'
102.0'
10.2
75
12.8
179.0
10.2
25
12.8
57.8
0.11'
50*
3.2*
6.4*
0.07
150
3.2
12.8
0.17
25
3.2
3.5
0.013'
1000'
6.4*
38.4*
0.012
3000
6.4
38.4
values.
are significantly different from the shallow earthquakes. In
each of the examples of Figure 9 the simulations agree to
within 10% of the data valuesfor greater that 80% of the
data points. The maximum deviations of the simulations
from the data are about 50% of a data value. Ni• attempt
was made to model the data lying within the sourceradius
of pairs at finite times following the first event, with fewer
in a later section. For the shallowearthquake
data, the
For earthquakesin southernCalifornia Jones [1985] finds
eventsat shorter times, is also seenin the simulationsand
arisesbecausethe count of earthquakepairs beganfollowing
some finite
time after the first event.
Foreshocks. Foreshockstatistics appear very similar to
that of aftershocks and to the clustering data discussed
of the first event becauseequation (21) appliesto regions above. Histograms of the frequency of foreshock-mainshock
outside of the primary rupture area. Seismicity falling pairs show rapid falloff of frequency of pairs with time and
within the source region of a prior earthquake is discussed distance[JonesandMolnar, 1979;Jones,1984;Jones,1985].
results for W appear reasonablyconsistentwith seismogenic that the probability of a mainshock following a M >_ 3.0
zone width. Also, for shallow earthquakes the estimates of
earthquakedecaysby about (l/t) p with p •00.9 which is
t• and tr/t• are consistentwith the estimatesof Figure 2 consistentwith the result p = 0.8 found for a finite region
and Table 2. However, significantly different values of
(Figure 5). In addition, Jones[1985]showsthat relative
t•, t•/t• and W are obtainedfor the intermediateand deep to foreshock magnitude, mainshock magnitudes follow a
earthquake data. In particular, ta appears to systematically normal b value distribution down to the size of the foreshock.
In the context of the model presented here, the occurrence
decreasewith increasing depth.
The characteristic shape of the data curves for variation
of mainshock following a foreshockis also interpreted as a
of number of pairs with time is controlled by the stress perturbation of backgroundseismicityby the stresschange
sensitivity of earthquake rate to elastic stress changesnear
of the foreshockas representedequation(21).
the first event. The apparentpersistenceof clusteringto
the great distances shown in Figure 9 is wholly an artifact
of summing the number of pairs cumulatively to some
distance x. Highly perturbed seismicity close to the first
event is lumped with essentially unperturbed seismicity at
greater distances. Comparable effects are seen in the results
for net aftershock
rate by cumulative
distance
(Figure5.).
Discussion
and
Conclusions
The approach presented here models seismicity as a
sequence of earthquake nucleation events. The distribution
of initial conditions over the population of nucleation
sourcesand stressinghistory determine the timing of events
in the sequence. If the nucleation process is assumed
to be instantaneous at a critical stress, then rate of
earthquake occurrence obtained from this model is simply
As previously discussedfor aftershocks,the decay of the
normalizednumber of pairs by (l/t) p with p -,,0.8 derives
from the characteristic stress change surrounding a shear proportionalto stressingrate (assumingmonotonicincrease
rupture in conjunction with the time and stress dependence
of stress). However,the model has been implemented
of seismicityrate of equation (12). The peakingof number
for time-dependent earthquake nucleation.
This yields
l•rrs•eetv
reantte
following
2W
Figure 10. Model employedfor simulationof clusteringof earthquakepairs. The downdipwidth of the
seismicbelt is W. Sourceradius of the first event of all posssiblepairs is c.
DIETERICH:
a state-variable constitutive formulation
CONSTITUTIVE
LAW FOR EARTHQUAKES
2613
for earthquake
estimate should probably be viewed with some caution until
rate (equations(9)-(11)) where state dependson stressing greater experiencewith this modeling approach is acquired.
history. It is found that seismicity rate seeksa steady state
value that is proportional to the instantaneous stressingrate
over the characteristic time ta. An important consequence
of this history dependence is that relatively modest jumps
in stress result in very large perturbations of earthquake
activity as shown, for example, by Figure 3.
Clustered earthquake phenomena have been modeled
assumingearthquake rate is perturbed by the elastic stress
change associated with prior earthquakes. The model
(equations (12) and (13)) yields Omori's law, and in
combination with the characteristic elastic stress change in
the vicinity of a shear rupture it reproduces the time and
distance statistics of earthquake pairs. These results suggest
that the formulation contains a reasonable representation
of time dependenceand stress sensitivity of the earthquake
nucleation processesin nature. Because deep earthquakes
appear to cluster in a way that is simulated by the present
model, but may involve nucleation processesvery different
from shallow earthquakes[Kirby et al., 1991], more than
one time-dependent nucleation mechanism may yield similar
results. In particular, if the various types of clustering
considered here are driven by characteristic elastic stress
changes in the vicinity of an earthquake, then an essential
nucleation characteristic needed to model the clustering data
is a dependence of the time to instability on an exponential
functionof the stresschange(i.e., t c• exp[(const)(r)]). This
is the sensitivity of the time to instability to stress obtained
from equations(A13) and (A14) when the relation of initial
slip speedto stressis used (equation(A7)).
It is also possible that alternative models of earthquake
occurrencemight be found that satisfy the various clustering
Clustering of intermediate and deep earthquakes have
been
modeled.
The
direct
relevance
of the
constitutive
formulation to intermediate and deep earthquakes may
be questioned, since mechanical processesand earthquake
instability mechanisms at those depths may be quite
different from those represented by the nucleation model.
Nonetheless, the simulations appear to provide a rather
goodrepresentationof the limited intermediate(70-280 km)
and deep (281-700 km) earthquake pair data presented
by Kagan and Jackson[1991]. From those simulations
(Table 3) it is evident that t• decreaseswith depth.
This is generally consistent with observations that deep
earthquakeshave fewer aftershocks[Frolich, 1987]. Also
the simulations clearly indicate that, compared to shallow
earthquakes,t•/t•
is larger by about order 103 for
deep earthquakes, with possible intermediate values for
intermediate earthquakes. Assuming -A•-•/rr
does not
exceed order of unity, these results indicate that fault
constitutive parameter A decreasesto very small values for
deep earthquakes.
Spatial dependence of earthquake clustering appears to
derive from the spatial dependence of the stress changes
around a prior earthquake source. Results have been
presented for an approximate representation of shear stress
change in the plane of a shear crack in an elastic
medium. Hence the model specifically applies to alteration
of earthquake rates along the plane of the prior earthquake
and outside of its rupture area. However, it is reasoned
that all stress changes resulting from an earthquake will
generally have similar spatial dependencebecausefalloff of
stresses
by 1Ix3 is characteristic
of finitesources
in a three-
data and do not involve time-dependent nucleation (e.g.,
dimensionalbody includingthe shear stressequation (20).
viscoelastic or poro-elastic stress transfer following a
Hence distance effects found for the specific model will be
generally applicable for perturbation of earthquakes from
any stress change resulting from a prior earthquake.
Thus far, only aftershocks lying outside of the zone of the
mainshock rupture have been discussed. Many aftershocks
originate within the mainshock source region, either on or
near to the mainshock rupture surface. Two explanations
for these aftershocksappear plausible. First, the aftershocks
may occur on patches within the mainshock source region
that did not slip during the mainshock. Such patches would
experiencelarge stressincreasesat the time of the mainshock
and would therefore have a high potential for aftershocks.
The second and favored explanation is that aftershocks
within the region of a mainshock may represent adjustments
on secondary faults to stressesinduced by mainshock slip.
In particular, no fault is perfectly planar and consequently
mainshock slip will give rise to large local changes of shear
mainshock).
Time-dependent nucleation is a direct
consequence of rate and state dependence of fault
constitutive properties. To the extent that those properties
exist on faults in nature, the nucleation process will
be time dependent. An independent argument for the
time-dependent nucleation model used here is based on
corroboration of these constitutive properties in other
applications. There is considerable experimental evidence
indicating these properties are characteristic of sliding
surfacesin nonmetallic materials including laboratory faults
with or without gouge layers. Additionally, various other
studies [Tse and Rice, 1986; Scholz, 1988; Stuart, 1988;
Okubo, 1989; Marone et al., 1991] have shown that
this constitutive representation is effective in modeling
fault creep, earthquake afterslip, and various earthquake
processes.
and normal stress as nonmatching geometry on opposite
An outcome of the approach presented here is that
parameters describing clustered phenomena have physical
interpretations.
For example, a predicted relationship
between aftershock duration and stressing rate or recurrence
stresseswill increase with each successiveearthquake unless
there is bulk flow or slip on secondary faults to relax the
time (equation (19)) has been obtained. In Figure 3 the
stress.
curves passing through the data for aftershock duration
mainshockand originate near to but not on the mainshock
and recurrencetime give tr/ta = -Are/Aa
= 10-40.
Similar values (tr/t• = 25-75) were obtained with the
plane [Beroza and Zoback,1993] may be evidenceof this
sidesof the fault interfere [Saucier et al., 1992]. Those
Aftershock
mechanisms
that
differ from that
of the
process.
Aftershockdecay is generallywell representedby (l/t) p
models of the data for shallow earthquake pairs. Assuming
average earthquake stress drops -A•-e : 2.5 MPa, with
with p ordinarily in the range 0.5-1.5.
A = 0.01 and A•-•/Aa = 25, these results suggesta rather
developed here, equations (12) and (13) give p =
low effective normal stress of about 10 MPa. However, this
if stress change and other model parameters are uniform
For the formulation
1.0
2614
DIETERICH:
CONSTITUTIVE
over the volume considered. Deviations from p = 1 and
variability between aftershock sequences will arise from
several causes. These include spatial variability of various
model parameters over the volume consideredand stressing
histories not satisfying the simple conditionsof (12) and
(13). Spatial variability of model parametersrequiresan
LAW FOR EARTHQUAKES
invert for changes in driving stress. Also, the formulation
presented here could be employed to estimate short- to
intermediate-term earthquake probabilities following prior
earthquakes. Such probability estimates may serve as an
alternative or adjunct to estimates that employ empirically
derived relations for earthquake potential analogous to that
appropriately weighted average of the family of curves
plotted in Figure 2 and tends to decrease p relative to the
homogeneouscase. With the exception of stress change it is
obtainedfor this model [e.g., Jones,1985; Reasenberg
and
Jones,1989; WorkingGroup,1992].
expected that model parameters (A, cr,r, ?r, and 2) would
Appendix
probably not vary systematically with respect to distance
from the mainshock. Consequently, such effects might tend
to cancel and any noncancelingeffectswould require analysis
specificto a particular mainshock(i.e., the effectswould not
be systematicto all aftershocksequences).
Use of nonuniform stress changes, characteristic of the
elastic stresses around a shear crack, yield p • 0.8
(Figure 5).
Comparable decay rates are found for
earthquakepair data compiledby Kagan and Jackson[1991]
and for foreshock-mainshock
pairs of Jones [1985] which
suggest that rates of decay shown in those composite data
sets show the systematic effect of the nonuniform elastic
stress field
near
a shear
crack.
Another source of variability of p relates to nonconstant
stress or stressing rate following an earthquake stress step.
Simulations with multiple stress steps, in which the later
stress steps represent large aftershocks with secondary
aftershock sequences(Figure 7) yield an apparent p > 1
during secondary aftershock sequences. Recall that this
effect is entirely an artifact of plotting the entire sequenceby
the logarithm of the time since the mainshock. As shown
by Figure 8 relaxation of stresses with time can result in
various rates of decay including cases where p > 1.
The formulation developed here assumesthat seismicity
rate in the absence of stress perturbations is independent
of time.
This steady state assumption appears to be
adequate for the applications explored here, at least over
time intervals comparable to ta which is small compared
to the mean earthquake recurrence time. Hence for those
time intervals the reference seismicity rate is consistent
with a Poisson model of earthquake occurrence, and the
perturbed seismicity is analogousto that of a nonstationary
Poisson
model.
in
formulation
the
The
introduction
relates
to
of the stochastic
the
distribution
element
of
initial
conditions over the population of sources or, equivalently,
to the uncertainty in initial conditions on a single source
that nucleates a specific earthquake. It is emphasized
that the nucleation process is not intrinsically uncertain.
The underlying model is of fully deterministic nucleation
of earthquake sources in which the stress on each source
begins at some low value and increases with time until the
conditions for onset of an earthquake are satisfied. Hence
conditions
on an individual
source
are consistent
with
the
A:
Earthquake
Nucleation
Nucleation model. The following summarizes results
obtained for a simplified model of a nucleation source that
is represented as single spring-slider system. Details of
the model and comparison of the simple model with plane
strain numerical solutions of a fault with spatially variable
conditionsare givenby Dieterich [1992]. Here the springand
slider resultsof Dieterich [1992]are generalizedsomewhat
by incorporation of the multiple state representation of fault
friction (equation(5)). Additionally,resultsare obtainedfor
the change of conditions on a source arising from a step in
both
shear and normal
stress.
Spring stiffnessper unit area, k, may be scaled to crack
length by
•-
Ar
Gr/
e = • '
where 1 is half-length or radius of the fault segment, G is
shear modulus (Poisson'sratio of 0.25), and r/ is a crack
geometry parameter with values near i [Dieterich, 1986,
1992]. For an instabilityto develop,k must be lessthan the
critical stiffnessfor unstable slip [Dieterich, 1981; Ruina,
1983; Rice and Ruina, 1983; Dieterich and Linker, 1992].
Hence from (A1), critical spring stiffnesscan be related to
a minimum dimension for unstable slip giving
l• = G•7D•
•rr '
In conclusion,it is noted that applications of the seismicity
rate constitutive formulation are not restricted to the simple
stress step models investigated here. It may be applied to
stressing histories of arbitrary complexity such as Earth
tides and passage of seismic waves. The demonstrated
capability for modeling the space and time characteristics
of clustering phenomena suggests the feasibility of using
observations of temporal variations of earthquake rate to
(A2)
where •c is a parameter that depends on constitutive
properties and loading conditions. It is found that the
nucleation process is characterized by an interval of selfdriven accelerating slip that precedes instability.
The
duration of the nucleation process is highly sensitive to
initial
stress
and
initial
0.
The
numerical
simulations
demonstrate that the segment on which instability nucleates
also has a characteristiclength lc that scales by (A2).
Empirical evaluation of the size of the nucleation zone from
the numerical models gives •c ,.•0.4B. If accelerating slip
initiates on a patch that exceedslc, the zone of most rapid
acceleration tends to shrink to the characteristic length as
the time of instability approaches.
Equating constitutivelaw (5) for fault strengthwith fault
stress gives
stressing cycle often assumed for quasiperiodic earthquake
occurrence.
(A•)
n
rr =/tø,+ AIn• +
i:1
BIn0i,
(A3)
where T(t) is the remotely applied stressacting on the fault
in the absence of slip and -k6
is the decrease in stress
dueto faultslip. In (A3) the constant
terms/z0,Alum*,
and Bi In 0• havebeengroupedinto /z0'. Normal stressis
assumed to be constant. Recall that 0 evolves with sliding
history as given by equation (6).
The problem may be
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
2615
introduces additional terms into the equations and does
not result in solutions that are numerically distinguishable
exceeds
thesteady
statespeed
forall 0i (i.e.,0i >> Dc•/3). from the simpler solutions provided 6i is much greater than
Asthishappens,
the1/• termin (6)willnolonger
contribute the average long term fault slip rate. This condition is
solved numerically. However, when the nucleation processis
underway and slip is accelerating, the slip speedsoon greatly
ß
wellsatisfied
for earthquake
processes
where•i variesfrom
to the evolution,and (6) can be well approximatedby
centimeters to meters per secondand long-term slip rates are
in millimeters to centimeters per year. Time of instability is
00i
) _ Oi Oi
- 0o
e-e/r%(A4)
-• cr=const
-- DC{ '
•
t=-•A(Tln(
• ø+1 ,
H(T•
Hencestateissimplydependent
ondisplacement,
where00•
is state
at 6 -
0.
For slip at constant normal stress,substitution of (A4)
into (A3) yields
n
-
•0,
(A13)
(A14)
ø , ,-0.
The
n
evolution
of state
!+Aln•+ZB•lnOo,-6Z•-•
Zi• . (A5)
=/•o
solutionand (A2).
on a nucleation
source
after
some
time interval is readily obtained from the displacement
z=l
The resultsof (A13) and (A14) are in
i:1
surprisingly good agreement with the times to instability
For a variety of functionsfor the remotestressingterm, (A5)
obtained
yields a separable differential equation that may be solved
directly.
Constant stressing rate. For the case of constant rate
spatial variation of conditions[Dieterich, 1992].
The
from detailed
above
solutions
numerical
have
simulations
the
useful
of a fault
characteristic
with
that
they may be applied repeatedly, in a step-by-step manner,
to approximate complicated stressing histories as a series of
constant stressing rate increments. Recall that initial slip
speed fully describesthe initial conditions. Hence the initial
of remote loading, r(t) - r0 + •t, (A5) may be rearranged
by solvingfor 6 -- d6/dt giving
ß
slipspeed
30,at thestartofeachtimeintervalin a loading
•0
exp
A }d6
j•0
t •t}dr- /oe
{-H6
, (A6)
history, is obtained from the solutionfor slip speed ((A13)
and (A12)) at the end of the previoustime interval.
exp<•
The effects of jumps of normal stress and shear stress on
where•0 andH contain
termsfortheinitialconditions
and slipspeedmay be foundfrom the constitutiverelations((A5)
model constants, respectively
•o-
0o•
and (A6)). Before a stressjump, slip speed dependson
initial stressand state as given by (A7). Similarly, the slip
speed following the stressjump depends on •-, (T, and 0 after
... exp•ø/• "ø (A7) the
step. Jumpsof •- do not affectstate, but from (6) change
0o2
of normal stress from (T0to (T results in the change of state
n
H-- k (
(AS)
- 0o(
•--1
•
ß
(TO
Notein (A7) that the initialslipspeed• fully describesHence,from (A7) and (A15), after the stressjump
the initial conditions
•o and 00•. Equation(A6) hasthe
following solutions:
(0o)
• -- -•-In
1-exp
A
'
X
(A16)
+ 1 * • 0,(A9)
-(To
6---•-ln 1
...
*-0.
(A10) By using (A7)
-(To
... exp A(T
A
'
the state terms in (A16) may be defined in
termsof initialstresses
and3ogiving
The slip speeds are
n
• •0((T)
r A(T
r0
(To "/A (A(T
o)
-
,* •: 0(All)
-[ro
(--•-tH(T}
{ 1+__•]
[exp
-1
-1
i
•o
Ht
A
* -0
(A12)
exp
,
c• --
Z.
t--1
(A17)
Appendix B: Formulation for Effect of
Stressing History on Earthquake Rate
Distribution
of conditions for steady state seismicity.
Taking dn/dt - r as the referencerate of seismicity,the
The timeof instability,
t, is obtainedby substituting
$i,
the slip speed at instability, into (All)
--
and (A12). The
following
employs
1/• - 0. Useoffinite•i whiletractable,
time of an earthquake at source n is
n
t= -
2616
DIETERICH:
CONSTITUTIVE
In Appendix A, initial slip speed is shown to fully describe
the initial conditions for subsequent acceleration of slip
LAW FOR EARTHQUAKES
which gives
o,) -
to instability (equation (A7)). The distribution of initial
conditions over the steady state population of patches is
obtainedby equatingthe result for time of instability (A13)
with (B1) and rearrangingto give initial slip speedas the
A similar procedureis used to evaluate (07/Or)t,• and
(07/O(T)t,•. From (B3) and (A17) the distributionof slip
dependent variable:
speedsfollowing a step in shear stress and normal stress is
1
0
3(n)--•oo exp
A(TA(T
o .
Ha7
exp
(• ]
(e2)
where 5.r is the reference stressing rate. Hence the time of
instability at the nth source may be found by evaluating
(B9)
Figure 1 illustrates the effect of a stress step on the
distribution. A stressstep that increasesslip speed (an
(B2) at n to obtain•0 andthensubstituting
that value increaseof shear stressor decreaseof normal stress)will
into the appropriate solution for time to instability, e.g.,
equation(A14). For later convenience,
(B2) is rewritten as
causesome number of patches, I, to nucleate earthquakes if
theslipspeed
•i at instability
isfinite.Setting
• - •i in (B9)
and solving for the number sources,I, that have nucleated
an earthquake gives
H(T7
[exp
(5.rn
1]' (B3)
I __
A(Tr
Hence for the referencesteady state distribution, 7 = 1/%.
In 1+•oo
(H(T•i)-i
-(T0
(B10)
Below, the effects of stressing history on the distribution
(B3) are examined. It is shownthat the distribution of slip
speedsis alwaysof the form of (B3) but with differentvalues
of 7 that evolve with stressinghistory.
The new distribution of conditions following the stress step
Evolution
equation for distribution.
The general
equation for evolution of 7 is found by evaluating the partial
derivatives
expA(T A(T
o
in
is obtained by the substitution,I + n- noid, in (B9) and
using (B10) to evaluate I. Use of either finite or infinite
& at instability again yields an evolved distribution of the
form of (B3) with
d7-(-•)....
dr+(
07 dr+(•)
07t,,_d(T
(B4)
Tofind(07/Ot)•,•.
thedistribution
•(n) ofequation
(B3)
7 - 70 -(To
(
)
exp rø r
A(T
0 A(T '
(Bll)
where 70 denotes the value of 7 immediately prior to the
stress step. From this result
is substitutedfor 6o in (A12) giving the distribution of slip
speeds after some elapsed time t at 5. -- 0, & - 0,
3(n)--
i
(B5) and
(07)
_-7
• t,•--A(T
(B12)
'
07)
_A(T
7•7c•'
t,½2 A(T
•
The use of 70 indicates the value of 7 at t = 0. After time t
has elapsed, some number of sources, I, will have nucleated
(B13)
In summary, the distribution of slip speeds retains the
earthquakes
because
thespeedforonsetofinstability
•i, has form of (B3) throughtime (with 5.= 0, & = 0) and through
been reached. Consequently,those sourceswill no longer be
a partof thepopulation.
Setting•(n) = ocandn = I in
(B5) and solvingfor I gives
(B6)
steps in shear and normal stress. Because any loading
history may be represented as a sequence of sufficiently
small stressjumps and time steps(with 5- = 0, & = 0), the
distribution of slip speedswill retain the form of (B3) for
any stressinghistory. From (B4), (BS), (B12), and (B13)
the general evolution equation for 7 is
The updated distribution of slip speeds,after time increment
1I
c•)I
(B14)
t, is obtainedby the substitutionI+n:
holdin (B5), where
holdnow refersto n appearingin (B5), prior to the time step
and using (B6) to evaluate I. Followingsomealgebra,the
new evolveddistribution is found to have the form of (B3)
tions employing friction formulations with multiple state
as previously asserted with
variables.
Recall from (A17) that c• = c•x+ c•2+ ... for applica-
Solutions
t
7--7o+A(T '
(B7)
for linear
and logarithmic
shear stress-
ing. Solutionof (B14) for changeof 7 for increaseof shear
stress at a constant rate and by logarithm of time are easy
DIETERICH:
CONSTITUTIVE
LAW FOR EARTHQUAKES
to obtain and of use for modeling earthquake processes.If
shear stressis given by
r -- r 0 q- 5.t,
1 [1- •/5.]dt,
which
stressingconditions(5. - 0, & - 0).
This suggests
(B15)
and & - 0, then from (B14), evolutionof -),is givenby
R=
as a general formulation for any stressing history. Hence,
it is proposed that at any instant seismicity rate depends
on the current value of 7 and is independent of current
stressingrate.
(B17) by
(BXS)
and &- 0, the evolutionequation(B14) becomes
?uw
d•/- •1 [I bt
q-1]dt
'
The validity of (B25) for any stressing
history may be understood by considering the nucleation
processduring unconstrained loading histories. As implied
For logarithmic stressing,
r -- r 0 + u ln(wt + 1),
(B25)
(B16)
has the solution
"/-- ["/0-•1]exp
•
+-.1
[-ts.]
2617
(B4) and as discussedpreviously,the evolution of
for any loading history may be found by breaking the
loading history into a seriesof sufficientlysmall stressjumps
and time increments with 5. = 0,& = 0. Because
determinesthe initial conditionsfor the time to instability,
(B25) will be generally valid if the time to instability
becomes independent of stressing rate as time decreases
to zero. The solutionsfor time of instability for 5. = 0
(B19)
(equation(A13)) and for 5- :/: 0 (equation(A14)) converge
and become independent of stressing rate as slip speed
approaches the critical speed for instability demonstrating
Using the integrating factor
u '
(wt+ 1)m,where
ra-- Art
(B20)
B(19) has the solution
that this conditionis satisfied.Indeed, (B25) may obtained
by the following the above procedureswith solutions for time
to instability and evolution of ? under conditions where
stressingrate is not zero (i.e., r = r o q- 5.,& = 0 and
•-=r o +uln(wt+l),&=0).
Equations (B14) and (B25) compose the general
' --'o(wtq-1)-mq-(wtq1)- (wt+ 1)-m . (B21)formulation for stressing history dependence of seismicity
Arrw(m + 1)
rates.
These solutionsmay also be obtained using the procedure of
the previous section which evolves the distribution of initial
slip speeds by employing the specific solutions for increase
of slip speed with time of a source. Conversely, the earlier
solutionsfor evolution of ? for constant stressand for stress
Acknowledgments:
This study profited from discussions
with numerous people over a period of several years. I
especially wish to acknowledgethe assistanceprovided by
Mike Blanpied, John Dvorak, John Lahr, Paul Okubo,
steps,equations(B7) and (Bll), respectively,may of course Paul Reasenberg, Bob Simpson, and Will Tanigawa. Joe
Andrews, Mark Linker, Jim Savage, George Sutton, John
be obtainedby direct solutionof (B14).
Seismicity rate. Under the condition 5. -- 0, • - 0, the Weeks, and Max Wyss provided reviews and several useful
distribution of earthquakestimes beginning at t - 0 is found suggestions.
from (A14) by substitutingthe distribution of slip speeds
References
(B3) for initial slip speed50. In making this substitution,
ß
note that the distribution of initial speeds is fixed at t - 0
and, consequently,? is fixed at 70. The distribution of times
of earthquakes is
,5. - 0, • = 0
(B22)
Rearranging for n,
n----In
r•
r
the Loma
Prieta
Zoback, Mechanism diversity in
aftershocks
and mechanics
of mainshock-
aftershock interactions, Science, 259, 210-213, 1993.
Blanpied, M. L., and T. E. Tullis, The stability and behavior
of a frictional system with a two state variable constitutive
law, Pure Appl. Geophys.,12•, 415-444, 1986.
Blanpied, M. L., D. A. Lockner, and J. D. Byerlee, Fault
stability inferred from granite sliding experiments at
hydrothermal conditions, Geophys.Res. Left., 18, 609-612,
1991.
A•ffo
q-1
'
5.--0 •--0,
'
(B23)
and differentiatinggivesseismicityrate R - dn/dt normalized by the referencerate r
R_
Beroza, G. C., and M.D.
1/5.r
70 +
t
_
_
' 5. 0,& 0 .
(B24)
Aa
Note that the denominator of the solution for seismicity rate
is simplythe specificsolutionof if(t) givenin (B7) for these
Dieterich, J. H., Modeling of rock friction, 1, Experimental
results and constitutive equations, J. Geophys. Res., 8•,
2161-2168, 1979a.
Dieterich, J. H., Modeling of rock friction, 2, Simulation of
preseismic slip, J. Geophys.Res., 8•, 2169-2175, 1979b.
Dieterich, J. H., Constitutive properties of faults with
simulated gouge, in Mechanical Behavior of Crustal Rocks,
Geophys.Monogr. Ser., vol. 24, edited by N. L. Carter, M.
Friedman, J. M. Logan, and D. W. Stearns, pp. 103-120,
AGU, Washington, D.C., 1981.
Dieterich, J. H., A model for the nucleation of earthquake
slip, in Earthquake Source Mechanics, Geophys. Monogr.
Ser., vol. 37, edited by S. Das, J. Boatwright, and C. H.
Scholz, pp. 37-47, AGU, Washington, D.C., 1986.
2618
DIETERICH:
CONSTITUTIVE
Dieterich, J. H., Nucleation and triggering of earthquake
slip: Effect of periodic stresses,Tectonophysics,1•, 127139, 1987.
Dieterich, J. H., Growth and persistence of Hawaiian
volcanic rift zones, J. Geophys.Res., 93, 4258-4270, 1988a.
Dieterich, J. H., Probability of earthquake recurrence with
non-uniform stressrates and time-dependent failure, Pure
Appl. Geophys., 126, 589-617, 1988b.
Dieterich, J. H., Earthquake nucleation on faults with rate
and state-dependent friction, Tectonophysics,211, 115134, 1992.
Dieterich, J. H., and M. F. Linker, Fault stability under
conditions of variable normal stress, Geophys. Res. Left.,
19, 1691-1694, 1992.
LAW FOR EARTHQUAKES
Reasenberg,P. A., and L. M. Jones,Earthquake hazard after
a mainshock in California, Science, 2•3, 1173-1176, 1989.
Reasenberg,P. A., and R. W. Simpson,Responseof.regional
seismicity to the static stress change produced by the
Loma Prieta earthquake, Science, 255, 1687-1690, 1992.
Rice•,.J.R.,
and J.-C. (3u, Earthquake aftereffectsand
triggered seismic phenomena, Pure Appl. Geophys., 121,
187-219, 1983.
Rice, J. R., and A. L. Ruina, Stability of steady frictional
slipping, J. Appl. Mech., 50, 343-349, 1983.
Ruina, A. L., Slip instability and state variable friction laws,
J. Geophys. Res., 88, 10,359-10,370, 1983.
Saucier, F., E. Humphreys, and R. Weldon, Stress near
geometrically
complex
strike-slip
faults:Application
to the
Dvorak, J. J., A. T. Okamura, T. T. English, R. Y.
Koyanagi, J. S. Nakata, M. R. Sako, W. T. Tanigawa, and
K. M. Yamashita, Mechanical responseof the south flank
San Andreas fault at Cajon Pass, southern California, J.
Geophys. Res., 97, 5081-5094, 1992.
Scholz, C. H., The critical slip distance for seismicfaulting,
of Kilauea volcano, Hawaii, to intrusive events along the
rift systems, Tectonophysics,12•, 193-209, 1986.
Ellsworth, W. L., and L. D. Dietz, Repeating earthquakes:
Characteristics and implications, in Proceedings of
Workshop XLVI, The 7th U. S.-Japan Seminar on
Nature, 336, 761-763, 1988.
Stein, R. S., and M. Lisowski, The 1979 Homestead Valley
earthquake sequence, California: Control of aftershocks
and postseismicdeformation, J. Geophys.Res., 88, 64776490, 1983.
Stuart, W. D., Forecast model for great earthquakes at the
Nankai trough subduction zone, Pure Appl. Geophys.,126,
Earthquake Prediction, U.S. Geol. Surv. Open File Rep.,
90-98, 226-245, 1990.
Ellsworth, W. L., A. (3. Lindh, W. H. Prescott, and D. G.
Herd, The 1906 San Francisco earthquake and the seismic
cycle, in EarthquakePrediction: An International Review,
Maurice Ewing Volume•, edited by D. W. Simpsonand P.
(3. Richards, pp. 126-140, A(3U, Washington, D.C., 1981.
Frolich, C., Aftershocks and temporal clustering of deep
earthquakes, J. Geophys.Res., 92, 13,944-13,956, 1987.
Jones,L. M., Foreshocks(1966-1980) in the San Andreas
system, California, Bull. Seismol. Soc. Am., 7J, 13611380, 1984.
Jones, L. M., Foreshocks and time-dependent earthquake
hazard assessment in southern California, Bull. $eismol.
$oc. Am., 75, 1669-1679, 1985.
Jones, L. M., and P. Molnar, Some characteristics of
foreshocks and their possible relation to earthquake
prediction and premonitory slip, J. Geophys. Res., 8•,
5709-5723, 1979.
Kagan, Y. Y., and D. D. Jackson, Long-term earthquake
clustering, Geophys. J. Int., 10•, 117-133, 1991.
Kirby, S. H., W. B. Durham, and L. A. Stern, Mantle
619-641, 1988.
Tajima, F., and H. Kanamori, Global survey of aftershock
area expansion patterns, Phys. Earth Planet. Inter., •0,
77-134, 1985.
Tse, S. T., and J. R. Rice, Crustal earthquake instability
in relationship to the depth variation of frictional slip
properties, J. Geophys. Res., 91, 9452-9472, 1986.
Tsukuda, T., Long-term seismic activity and present
microseismicity on active faults in southwest Japan,
paper presented at U.S.-Japan Seminar on Practical
Approachesto Earthquake Prediction and Warning, Tokyo
and Tsakuba, Japan, November 7-11, 1983.
Tullis, T. E., and J. D. Weeks,C'0nstitutivebehavior
and stability of frictional sliding of granite, Pure Appl.
Geophys., 12•, 383-314, 1986.
Utsu,T., Aftershocks
andearthquake
statistics
(I), J. Fac.
Sci., Hokkaido Univ., Set. 7, 3, 129-195, 1969.
Weeks,J. D., and T. E. Tulli•, Frictional slidingof dolomite:
A variation in constitutive behavior, J. Geophys.Res., 90,
7821-7826, 1985.
R. L., Modelling
aftershock
mig;•ation
andafterslip
phasechangesand deep-earthquake
faultingin subducting Wesson,
lithosphere, Science, 252, 216-225, 1991.
Linker,M. F., andJ. H. Dieterich,
Effects
ofvariable
normal
stress
on
rock
friction:
Observations
and
constitutive
equations, J. Geophys.Res., 97, 4923-•4940, 1992.
Marone,C. J., C. H. Scholz,and R. Bilham,•)n the
mechanics of earthquake afterslip, J. Geophys. Res., 96,
8441-8452, 1991.
Matsu'ura, M., D. D. Jackson, and A. Cheng, Dislocation
model for aseismic crustal deformation at Hollister,
California, J. Geophys. Res., 91, 12,661-12,674, !986.
Ogata, Y., and K. Shimazaki, Transition from aftershocks
to normal activity: The 1965 Rat Islands earthquake
aftershock sequence, Bull. Seismol. Soc. Am., 7J, 1757-
1705,1984.
Okubo, P. (3., Dynamic rupture modeling with laboratory
derived constitutive relations, J. Geophys. Res., 9•,
12,321-12,336, 1989.
Okubo, P. G., and J. H. Dieterich, State variable fault
constitutive relations for dynamic slip, in 'Earthquake
Source Mechanics, Geophys.Monogr. Set., vol. 37, edited
by S. Das, J. Boatwright, and C. H. Scholz, pp. 25-35,
AGU, Washington, D.C., 1986.
Reasenberg, P., and W. L. Ellsworth, Aftershocks of the
of the San Juan Bautista, California, earthquake of
October 3, 1972, Tectonophysics,1•, 214-229, 1987.
Working Group, Probabilitiesof large earthquakesin the
San Francisco Bay region, Califorrlia, U.S. Geol. $urv.
Circ., 1053, 51 pp.•t.•,1990.
Working Group, Future seismic hazards in southern
California, Phase I: Implications of the 1992 Landers
earthquake sequence,report, Cal. Div. of Mines and Geol.,
Sacramento, 1992.
Wyss,
M., Regular
recurrence
intervals
between
Hawaiian
earthquakes: Impl!cations for predicting the next event,
Science, 23•, 726-728, 1986.
Wyss, M., A proposed model for the great Kau, Hawaii,
earthquake of 1868, Bull. $eismol. $oc. Am., 78, 14501462, 1988.
Wyss, M., and R. E. Haberman, Precursory seismic
quiescence,Pure Appl. Geophys.,126, 319-332, 1988.
J.H. Dieterich, U.S. Geological Survey, 345 Middlefield
Road, MS/977, Menlo Park, CA 94025.
Coyote Lake, California, earthquakeof August 6, 1979: A
(Received
March
i;ri,1993;
revised
September
2 1993;
detailed study, J. Geophys. Res., 87, 10,637-10,655, 1982.
acceptedSeptember8, 1993.)