Simulation of Ground Motion Using the Stochastic Method
DAVID M. BOORE
1
Abstract — A simple and powerful method for simulating ground motions is to combine parametric or
functional descriptions of the ground motion’s amplitude spectrum with a random phase spectrum
modified such that the motion is distributed over a duration related to the earthquake magnitude and to
the distance from the source. This method of simulating ground motions often goes by the name ‘‘the
stochastic method.’’ It is particularly useful for simulating the higher-frequency ground motions of most
interest to engineers (generally, f > 0:1 Hz), and it is widely used to predict ground motions for regions of
the world in which recordings of motion from potentially damaging earthquakes are not available. This
simple method has been successful in matching a variety of ground-motion measures for earthquakes with
seismic moments spanning more than 12 orders of magnitude and in diverse tectonic environments. One of
the essential characteristics of the method is that it distills what is known about the various factors
affecting ground motions (source, path, and site) into simple functional forms. This provides a means by
which the results of the rigorous studies reported in other papers in this volume can be incorporated into
practical predictions of ground motion.
Key words: Stochastic, simulation, ground motion, random vibration, earthquake, strong motion, site
amplification.
Introduction
Keiiti Aki was one of the first seismologists to derive an expression for the
spectrum of seismic waves radiated from complex faulting. In a 1967 paper (A
KI,
1967) he used assumptions about the form of the autocorrelation function of slip as a
function of space and time to derive an x-square model of the spectrum (and he
coined the term ‘‘x-square model’’ in that paper). He then used the assumption of
similarity to derive a source-scaling law, showing that the spectral amplitude at the
corner frequency goes as the inverse-cube power of the corner frequency. He
explicitly recognized that this is a constant-stress-drop model. His work has been
used knowingly and unknowingly by several generations of seismologists to predict
ground motions for earthquakes, particularly at high frequencies where the space-
and time-distribution of fault slip is complicated enough to warrant a stochastic
description of the source. Usually these predictions are for a specified seismic
1
U.S. Geological Survey, Mail Stop 977, 345 Middlefield Road, Menlo Park, California 94025, U.S.A.
E-mail: boore@usgs.gov
Pure appl. geophys. 160 (2003) 635–676
0033 – 4553/03/040635 – 42
Birkha
¨
user Verlag, Basel, 2003
Pure and Applied Geophysics
moment, and this is another place in which Kei’s work had a long-term impact: in
1966 (A
KI, 1966) he determined the seismic moment of an earthquake for the first
time and also explicitly related the seismic moment to the product of rigidity, slip,
and fault area. His research on the shape and scaling of source spectra and on seismic
moment form the basis for the method for simulating ground motions discussed in
this paper. In recognition of its use of a partially stochastic, rather than a completely
deterministic, description of the source and path, this method is often referred to as
‘‘the stochastic model’’ or ‘‘the stochastic method.’’ A word about terminology may
be in order here: I refer to the means of simulating ground motions as the ‘‘stochastic
method,’’ whereas a particular application of the method results in a ‘‘stochastic
model’’ of the ground motion (often associated with a particular study, such as the
F
RANKEL et al. (1996) model). The terminology is not standardized, however, and
more usually (and loosely) people refer to any application of the stochastic method as
the stochastic model; the distinction between the two is important, because the
ground motions for different applications of the method (different models) might be
very different.
There are several methods that use stochastic representations of some or all of
the physical processes responsible for ground shaking (e.g., P
APAGEORGIOU and AKI,
1983a; Z
ENG et al., 1994). In this paper I review the particular stochastic method that
I and a number of others developed in the last several decades. The paper includes a
few new figures and an improvement in the calculation of random vibration results
that previously appeared only in an USGS open-file report (B
OORE, 1996), Other
authors have published papers applying the stochastic method and extending the
method in various ways. Table 1 contains a partial list of papers primarily concerned
with development of the method; a table of references applying the method is given
later.
Most of the discussion assumes that the motions to be simulated are S waves—
these are the most important motions for seismic hazard. The method can be
modified to simulate P -wave motions, as was done in B
OORE (1986).
The Essence of the Method
The stochastic method described in this paper has its basis in the work of Hanks
and McGuire, who combined seismological models of the spectral amplitude of
ground motion with the engineering notion that high-frequency motions are basically
random (H
ANKS, 1979; MCGUIRE and HANKS, 1980; HANKS and MCGUIRE, 1981).
Assuming that the far-field accelerations on an elastic half space are band-limited,
finite-duration, white Gaussian noise, and that the source spectra are described by a
single corner-frequency model whose corner frequency depends on earthquake size
according to the B
RUNE (1970, 1971) scaling, they derived a remarkably simple
relationship for peak acceleration that was in good agreement with data from 16
636 David M. Boore Pure appl. geophys.,
earthquakes. I generalized their work to allow for arbitrarily complex models,
extended it to the simulation of time series, and considered many measures of ground
motions, the most important of which are response spectra (B
OORE, 1983). The
underlying simplicity of the method, however, remains unchanged. The essence of the
method is shown in Figure 1: The top of the figure shows the spectrum of the ground
motion at a particular distance and site condition for magnitude 5 and 7 earthquakes,
based on a standard seismological model; by assuming that this motion is distributed
with random phase over a time duration related to earthquake size and propagation
distance, the time series shown in the bottom of the figure are produced.
The essential ingredient for the stochastic method is the spectrum of the ground
motion—this is where the physics of the earthquake process and wave propagation
are contained, usually encapsulated and put into the form of simple equations. Most
of the effort in developing a model is in describing the spectrum of ground motion.
As is traditional, I find it convenient to break the total spectrum of the motion at a
site (Y ðM
0
; R; f Þ) into contributions from earthquake source (E), path (P ), site (G),
and instrument or type of motion (I), so that
Y ðM
0
; R; f Þ¼EðM
0
; f ÞP ðR; f ÞGðf ÞIðf Þ ; ð1Þ
where M
0
is the seismic moment, introduced into seismology in 1966 by K. AKI (AKI,
1966). I usually use moment magnitude M rather than seismic moment as a more
familiar measure of earthquake size; there is a unique mapping between the two:
M ¼
2
3
log M
0
10: 7 ð2Þ
(H
ANKS and KANAMORI, 1979).
Seismic moment has a number of advantages as the predictor variable for
earthquake size in applications:
It is the best single measure of overall size of an earthquake and is not subject to
saturation.
It can be determined from ground deformation or from seismic waves.
Table 1
Some references on methodology
B
ERESNEV and ATKINSON (1997, 1998a), BOORE (1983, 1984, 1989b, 1996, 2000),
B
OORE and JOYNER (1984), CAMPBELL (1999), CARTWRIGHT and LONGUET-HIGGINS (1956),
CORREIG (1996), ERDIK and DURUKAL (2001), GHOSH (1992), HANKS and MCGUIRE (1981),
HERRMANN (1985), JOYNER (1984, 1995), JOYNER and BOORE (1988), KAMAE and IRIKURA (1992), KAMAE
et al. (1998), KOYAMA (1997), LAM et al. (2000), LIAO and JIN (1995), LIU and PEZESHK (1998, 1999), LOH
and YEH (1988), MILES and HO (1999), O
´
LAFSSON and SIGBJO
¨
RNSSON (1999), OU and HERRMANN
(1990a, 1990b), PAPAGEORGIOU and AKI (1983a), PEZESHK et al. (2001), RATHJE et al. (1998), SABETTA and
PUGLIESE (1996), S¸ AFAK and BOORE (1988), SCHNEIDER et al. (1991), SHAPIRA and VAN ECK (1993),
SILVA (1992), SILVA and LEE (1987), SILVA et al. (1988, 1990, 1997), TAMURA et al. (1991),
W
ENNERBERG (1990), YU et al.
Vol. 160, 2003 Simulation of Ground Motion 637
It can be estimated from paleoseismological studies.
It can be related to slip rates on faults.
It is the variable of choice for empirically and theoretically based equations for the
prediction of ground motions.
Figure 1
Basis for stochastic method. Radiated energy described by the spectra in the upper part of the figure is
assumed to be distributed randomly over a duration equal to the inverse of the lower corner frequency (f
0
).
Each time series is one realization of the random process for the actual spectrum shown. When plotted on a
log scale, the levels of the low-frequency part of the spectra are directly proportional to the logarithm of
the seismic moment and thus to the moment magnitude. Various peak ground-motion parameters (such as
response spectra, instrument response, and velocity and acceleration) can be obtained by averaging the
parameters computed from each member of a suite of acceleration time series or more simply by using
random vibration theory, working directly with the spectra. The examples in this figure came from an
actual simulation and are not sketched in by hand.
638 David M. Boore Pure appl. geophys.,
By separating the spectrum of ground motion into source, path, and site
components, the models based on the stochastic method can be easily modified to
account for specific situations or to account for improved information about
particular aspects of the model.
The Source ðEðM
0
; f ÞÞ
Both the shape and the amplitude of the source spectrum must be specified as a
function of earthquake size. This is the most critical part of any application of the
method. References given later should be consulted to see how various authors have
approached this issue. The most commonly used model of the earthquake source
spectrum is the x-square model, a term coined by A
KI (1967). Figure 2 shows this
spectrum for earthquakes of moment magnitude 6.5 and 7.5. The scaling of the
spectra from one magnitude to another is determined by specifying the dependence of
the corner frequency f
0
on seismic moment. AKI (1967) recognized that assuming
similarity in the earthquake source implies that
M
0
f
3
0
¼ constant ; ð3Þ
Figure 2
Source scaling for single-corner-frequency x-square spectral shape. For constant stress drop M
0
f
3
0
is a
constant (AKI, 1967), and this dependence of the corner frequency f
0
on the moment M
0
(given by the
shaded line) determines the scaling of the spectral shapes.
Vol. 160, 2003 Simulation of Ground Motion 639
