of 9
Geophys. J. Int.
(2005)
162,
841–849
doi: 10.1111/j.1365-246X.2005.02679.x
GJI Seismology
Th
e
effect of slip variability on earthquake slip-length scaling
Jing Liu-Zeng,
Thomas Heaton and Christopher DiCaprio
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena,
CA 91125
,
USA.
E-mails: liu@ipgp.jussieu.fr; heaton
t@caltech.edu; dicaprio@gps.caltech.edu
Accepted 2005 May 9. Received 2004 November 26; in original form 2004 January 7
SUMMARY
There has been debate on whether average slip
D
in long ruptures should scale with rupture
length
L
,o
r
with rupture width
W
.
This scaling discussion is equivalent to asking whether
av
erage stress drop
,w
hich is sometimes considered an intrinsic frictional property of
af
ault, is approximately constant over a wide range of earthquake sizes. In this paper, we
e
xamine slip-length scaling relations using a simplified 1-D model of spatially heterogeneous
slip. The spatially heterogeneous slip is characterized by a stochastic function with a Fourier
spectrum that decays as
k
α
,w
here
k
is the wavenumber and
α
is a parameter that describes
the spatial smoothness of slip. We adopt the simple rule that an individual earthquake rupture
consists of only one spatially continuous segment of slip (i.e. earthquakes are not generally
separable into multiple disconnected segments of slip). In this model, the slip-length scaling
relation is intimately related to the spatial heterogeneity of the slip; linear scaling of average
slip with rupture length only occurs when
α
is about 1.5, which is a relatively smooth spatial
distribution of slip. We investigate suites of simulated ruptures with different smoothness, and
we
show that faults with large slip heterogeneity tend to have higher
D
/
L
ratios than those
with spatially smooth slip. The model also predicts that rougher faults tend to generate larger
numbers of small earthquakes, whereas smooth faults may have a uniform size distribution of
earthquakes. This simple 1-D fault model suggests that some aspects of stress drop scaling are
a
consequence of whatever is responsible for the spatial heterogeneity of slip in earthquakes.
Key w
ords:
av
erage stress drop, earthquake slip-length scaling, self-affine fractals, slip het-
erogeneity, 1-D stochastic model.
INTRODUCTION
Studying the scaling relationships between earthquake rupture pa-
rameters provides insight into the physics of the rupture process.
One of the most fundamental of these scaling relationships, which
is that between average slip and rupture dimensions, has been dis-
cussed intensively over the last decade. Does average slip
D
scale
with the rupture length
L
or with the rupture width
W
?A
ke
y
ele-
ment in this debate, however, is the hypothesis that earthquakes are
crack-like processes with scale-invariant average stress drops.
The ratios of average slip to rupture dimensions provide direct
estimates of the strain change in the vicinity of earthquakes, and
hence the average stress drops on rupture surfaces. The average slip
D
,
from a uniform stress drop
,o
na
r
upture surface of area
S
,in
an elastic whole space of rigidity
μ
,
can be written as (e.g. Kanamori
&
Anderson 1975)
D
=
S
1
/
2
σ/
C
μ
(1)
Corresponding author: Laboratoire de Tectonique, Institut de Physique du
Globe de Paris, 4 Place Jussieu, 75252 Paris, cedex 05, France.
w
here
C
is a dimensionless number whose value depends on the ge-
ometry of the rupture and the orientation of the shear stress. Equa-
tion (1) can also be shown to apply to a class of half-space problems
with sufficient symmetry. It is convenient to characterize rupture
surfaces as rectangular, having length
L
and width
W
.
The relation-
ship between average slip and stress drop for rectangular faults in
both whole- and half spaces was first reported by Boore & Dunbar
(1977). Parsons
et al.
(1988) found some inconsistencies with these
solutions and reported revised results, which can be summarized for
av
ertical strike-slip fault (Table 1).
Das (1988) demonstrated that these relationships are still approx-
imately true even when the stress drop is spatially heterogeneous,
but stress drop is replaced by spatially averaged stress drop. Aver-
age stress drop
wa
s
initially suggested to be scale independent
from the observation that seismic moment
M
0
scales with
S
3
/
2
over
a
wide range of earthquake sizes (Aki 1972; Abe 1975; Kanamori
&
Anderson 1975; Hanks 1977).
A
common and seemingly logical interpretation of
S
3
/
2
scaling
of moment is that
is a material property of faults. For small
earthquakes, which are essentially equal dimensional,
S
1
/
2
could be
either
L
or
W
in eq. (1), thus constant stress drop means average slip
scales linearly with rupture length, as well as with rupture width.
C

2005 RAS
841
842
J.
Liu-Zeng, T. Heaton and C. DiCaprio
Ta bl e 1 .
Summary of Parsons
et al.
’s
(1988) formulae of average stress drop for strike-slip faults.
Whole space
Half space with surface rupture
Square fault (
L
=
W
)
=
2
.
55
μ
D
/
L
=
2
.
04
μ
D
/
L
Long, narrow fault (
L

W
)
1
.
28
μ
D
/
W
0
.
65
μ
D
/
W
(3)
However, if
is a material property of the fault, then we should
e
xpect an upper bound on the average slip for strike-slip earthquakes
with very long ruptures. These earthquakes have rupture dimen-
sions that are confined by the thickness of the seismogenic zone in
downdip width, but unbounded in rupture length. In this case, av-
erage stress drop is determined by the ratio of average slip versus
width, and constant stress drop means that average slip will reach a
constant value related to the maximum fault width
W
max
.
Thus we
might expect average slip to depart from linear growth with rupture
length, when the rupture length becomes much larger than the fault
width. This is sometimes referred to as the
W
-model (
W
represents
f
ault width). The
W
-model assumes that average stress drops are
independent of rupture dimensions. Therefore, the
W
-model pre-
dicts that
D
L
W
w
hen
L
<
W
max
,but
D
W
max
w
hen
L

W
max
.
Equivalently, the
W
-model predicts seismic moment
(
M
0
=
μ
LW
D
)
scales with
L
3
for small events and with
L
for large
ev
ents.
Despite this expectation, Scholz (1982) reported that average slip
scales linearly with rupture length for both small and large earth-
quakes. To explain linear slip-length scaling, Scholz (1982) intro-
duced the
L
model in which stress drop and average slip are de-
termined by rupture length
L
.
Unlike the
W
-model, the
L
-model
predicts that
M
0
scales with
L
2
(
D
L
and
W
=
constant) for
large earthquakes and with
L
3
for small events. Shimazaki’s (1986)
observations on a set of Japanese intraplate earthquakes are also con-
sistent with an
L
model. More recently, Romanowicz (1992) argued
that existing data for large earthquakes are compatible with
M
0
L
scaling, which is evidence for the
W
model. This spurred fur-
ther debate about whether existing data favoured the
W
model (Ro-
manowicz 1994; Romanowicz & Ruff 2002), or the
L
-model (Scholz
1994a; Pelger & Das 1996; Wang & Ou 1998). Recent modelling
and compilations of data (Scholz 1994b; Bodin & Brune 1996; Mai
&
Beroza 2000; Shaw & Scholz 2001), however, show that neither
the
L
model nor the
W
model may be adequate to explain slip-length
scaling observations. Current compilations seem to indicate that av-
erage slip continues to increase with
L
w
hen
L

W
max
,but
at a
slower rate than implied by the
L
model. Furthermore, the transition
from linear to non-linear length scaling seems to occur at rupture
lengths that are relatively large compared to the maximum rup-
ture width. This observation suggests the limitation of static
L
and
W
models, in which fault slip is determined by the average stress
drop and the final dimension of fault rupture.
Dynamic rupture models, such as partial stress drop, abrupt lock-
ing and self-healing models (Housner 1955; Brune 1970, 1976;
Heaton 1990), may help to explain the messy slip-length scaling
observations. For example, in the slip-pulse model, Heaton (1990)
speculated that the most natural explanation for a correlation be-
tween average slip and rupture length (regardless of the rupture
width) is that slip pulses with very large slip tend to propagate
larger distance. If earthquake rupture grows like a narrow ‘slip-
pulse’ propagating along the fault, then there would be a stochastic
element in rupture propagation, and this stochastic element would
have consequences in how far the rupture can go, thus affecting the
scaling of earthquake rupture parameters.
In this paper, we introduce a simple 1-D stochastic model of fault
slip as a function of along-strike distance. This model is purely geo-
metric in nature and is not intended to satisfy equations of motion or
specific boundary conditions. It does, however, share features that
we
e
xpect to occur in slip-pulse type models of dynamic rupture.
The purpose of this paper is to argue that constant stress drop is not
necessarily required to explain slip-length scaling. Specifically, we
show that the spatial heterogeneity of fault slip can play a key role
in slip-length scaling. We find that our model does not generally
result in linear scaling of slip with rupture length, although linear
scaling is a special case for a particular degree of rupture smooth-
ness. However, when a suite of faults with different smoothness is
considered, the overall pattern of average slip with length is simi-
lar to that contained in the data compiled by Wells & Coppersmith
(1994).
We
restrict our discussion of slip-length scaling for individual
earthquake ruptures, not the maximum displacement—length scal-
ing for faults. The latter scaling describes the relationship between
the maximum cumulative displacements and the total lengths of
f
aults (e.g. Walsh & Watterson 1988; Cowie & Scholz 1992; Clark
&C
ox
1996; Manighette
et al.
2001). These fault parameters rep-
resent summed effect of either numerous seismic ruptures or non-
seismic fault growth. It is unclear whether the scaling for individual
earthquakes also applies to faults.
1-D SLIP VARIATION MODEL
The basic motivation for our 1-D model is the notion that a pulse
of slip propagates along a fault. At any given point,
x
,i
t
has slip
of amplitude
D
(
x
). As the rupture propagates to an adjacent point,
x
+

x
,
the amplitude of the slip changes by some random amount,
w
hich can be either positive or negative. In this way, the earthquake
increases or decreases in intensity in some stochastic way as rupture
propagates along a fault. If the slip amplitude decreases to zero, then
the earthquake is over.
We
demonstrate that the relationship between average slip and
r
upture length for this class of models is closely related to the as-
sumed spatial heterogeneity of slip. While the physical origin of
spatial slip heterogeneity may be a profoundly important issue, we
do not address it here. We are simply emphasizing the role that slip
heterogeneity plays in determining stress drop.
Our model consists of two simple rules.
Rule 1: Slip as a function of position,
D
(
x
), can be approx-
imated by a convolution in the Fourier space between a random
function of position and some function with power law dependence.
Mathematically,
D
(
x
)
=
D
0
|
FT
1
[
ˆ
R
(
k
)
k
α
]
|
(2)
w
here
D
o
is a constant, FT
1
refers to taking inverse Fourier trans-
form,
ˆ
R
(
k
)i
s
the Fourier transform of
R
(
x
), which is a Guassian ran-
dom function of
x
with zero mean and variance of 1,
k
is wavenum-
ber, and
α
is a filtering parameter that determines the smoothness
or roughness of the slip function.
Rule 2: Any earthquake consists of a spatially contiguous segment
of positive slip. That is, any point at which
D
(
x
)
=
0, defines the
end of an individual rupture.
We
use the procedure described by Turcotte (1997, p. 149–155) to
generate a fractal series
D
(
x
). The series is then rescaled to obtain the
C

2005 RAS,
GJI
,
162,
841–849
Earthquake slip-length scaling
843
desired mean and standard deviation, and we then call it the parent
series. The parent series consists of alternative segments of positive
and negative values. We then take the absolute value of the parent
series, so that both positive and negative segments are retained as
analogues of earthquake slip functions, which are assumed to be non-
negative functions. Accordingly, the length of a rupture is thus taken
to be the distance between zeroes of the parent series (rule 2). That
is, an earthquake rupture consists of only one spatially continuous
segment of slip. In this way, each parent series is split into a set of
earthquake slip functions of varying length with approximately the
same
α
.S
e
gments that are represented by fewer than 10 data points
are not used in our statistical analysis, because they may be too
short to be statistically stable. Indeed, statistics show more scatter
in average displacement for short segments. However, this choice of
truncation is arbitrary, as we do not see a sharp change in statistics
for segments of various sizes. In this article, we define
α
,
the slope of
its Fourier spectral amplitude with wave number, as the smoothness
of the slip function. Fig. 1 shows
D
(
x
)
and
ˆ
D
(
k
)
for typical simulated
ev
ents with smoothness
α
of 1.0, 1.25 and 1.5. The combination of
r
ules 1 and 2 implies that a model with a rougher slip distribution
(lower
α
)
will produce more short events because those functions
are more likely to pass through zero.
0
20
40
60
80
0
2
4
10
0
10
2
10
4
10
-5
10
0
10
5
0
20
40
60
80
0
2
4
10
0
10
2
10
4
10
-5
10
0
10
5
0
20
40
60
80
0
2
4
10
0
10
2
10
4
10
-5
10
0
10
5
Amplitude spectrum
(a)
(b)
(c)
(d)
W
avenumber (1/100km)
Amplitude
Amplitude
Amplitude
(e)
(f)
Synthetic slip functions
Distance along strike (km)
Displacement (m)
Displacement (m)
Displacement (m)
slope=1.25
slope=1.0
slope=1.5
Figure 1.
Examples of synthetic self-affine slip functions of different variability (a,c,e) and their Fourier amplitude spectra (b,d,f). As the slip function
gets
smoother from (a) to (c) to (e), the slope of the amplitude spectrum grows from 1.0 (b) to 1.25 (d) to 1.5 (f).
The parent series described by the first rule is known as 1-D frac-
tional Brownian motion (fBm) (Mandelbrot 1985). A fBm function
has the property that
D
(
x
)i
s
statistically similar to
D
(
rx
)
r
H
,w
here
H
is the Hausdorff measure and
r
is any scale length. The theoretical
relationship between
H
and
α
is (Turcotte 1997)
H
=
α
0
.
50
.
5
<α<
1
.
5
.
(3)
While there is no conclusive evidence that
D
(
x
)i
s
fractal for real
earthquakes, the use of fractal slip functions is motivated by numer-
ous observations suggesting fractal fault traces, fractural surfaces,
internal structure of fault zones and fault networks (e.g. Brown &
Scholz 1985; Scholz & Aviles 1986; Power
et al.
1987; Chester
et al.
1993; Schmittbuhl
et al.
1995). Furthermore, fractal slip dis-
tributions have been adopted in earthquake models (e.g. Andrews
1980; Herrero & Bernard 1994; Mai & Beroza 2002; Lavall ́
ee &
Archuleta 2003). However, previous studies have focused mostly
on self-similar fractals. In contrast, we have considered the more
general self-affine fractals and exploited the implication of the as-
sumption of anisotropic fractal slip functions in the relationship
between average slip and length.
Our spatially heterogeneous slip functions imply that stress and
strain changes are even more spatially heterogeneous. That is, strain
C

2005 RAS,
GJI
,
162,
841–849
844
J.
Liu-Zeng, T. Heaton and C. DiCaprio
changes on the fault are a function of first derivatives of slip with
respect to distance. If slip can be described by spatially filtered
random functions with a spectral decay of
k
α
,
then stress and strain
change functions will have a spectral decay of
k
α
+
1
.
Thus
α
=
1i
s
consistent with Gaussian random stress changes and
α<
1
corresponds to stress change functions that are deficient in long-
wa
v
elength variations compared to random.
MODELLED RESULTS
We
have considered seven cases with slip smoothness
α
correspond-
ing to 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 and 2.0. For each case, 10 par-
ent series consisting of
N
=
65 536 points were constructed, and
the distance between adjacent data points is assumed to be

L
=
10 m. This value was adopted such that the longest synthetic rupture
lengths are comparable to those in the real world.
F
ig. 2 shows the modelled relationship (log–log) between aver-
age displacement
D
,
and length
L
for earthquakes with intermediate
v
alues of slip smoothness,
α
=
1.0, 1.25 and 1.5. The lengths of the
earthquakes range from 1 to 300 km. Several features can be seen
in this plot. (1) For a given average displacement, earthquakes with
smoother slip functions tend to have longer ruptures. (2) Ruptures
with a slip smoothness of
α
=
1.5 produce average slips that are lin-
early proportional to
L
.H
ow
ev
er, linearity breaks down for rougher
slip functions (
α<
1.5). That is,
D
L
0
.
8
for ruptures of
α
=
1.25,
and
D
L
0
.
6
for
α
=
1.0.
The scaling exponent
γ
in the relationship of
D
L
γ
depends
nonlinearly on the slip smoothness,
α
,a
s
shown in Fig. 3. Theoret-
ically, the exponent
γ
in the slip-length scaling is equivalent to the
Hausdorff measure (
H
), and linear scaling of
D
and
L
implies
H
=
1, corresponding to smooth or nearly differentiable slip functions.
Note that the value of
γ
deviates systematically from the theoretical
line of
H
=
α
0.5. Similar discrepancy also exists between power-
law spectral exponent and fractal dimension (Fox 1989; Pickering
et al.
1999). The reason for the deviation is two-fold. One, the sim-
ple relation of theoretical prediction can only be an approximation.
Hausdorff measure must lie between 0 and 1 by definition. This
corresponds to 0.5
α
1.5. However, it is possible to construct
10
-2
10
-1
10
0
10
1
1
10
100
1000
L (km)
D (m)
=1.0
=1.25
=1.5
D ~ L
0.8
D ~ L
0.6
D ~ L
α
α
α
Figure 2.
Al
o
g–log plot of average slip
D
v
ersus rupture length
L
for
modelled ruptures with slip smoothness
α
=
1.0, 1.25 and 1.5. The data can
be fitted by power scaling relations whose exponents depend on
α
.
D
scales
linearly with
L
only for earthquakes with smooth slip (
α
=
1.5).
Smoothness
Scaling exponent in D
γ
γ
L
0
0.5
1.0
2.5
2.0
1.5
0
0.4
0.8
1.2
α
Figure 3.
The dependence of slip-length scaling on slip heterogeneity
α
.
It is not well constrained for
α<
0.5 or
α>
1.5. Large values of
α
tend to
make series in which D diverges with
L
.
The line indicates the theoretical
prediction of
γ
=
α
0.5, if
γ
is the Hausdorff exponent.
ap
ow
er-law function with any value of
α
.F
or example, a random
noise of
α
=
0
implies
H
=−
0.5 if
H
=
α
0.5. This is beyond
the possible range of Hausdorff measure. It is equally possible to
construct a series
α
=
2, which implies a Hausdorff exponent of 1.5,
larger than the possible upper bound of 1. Another factor for the dis-
crepancy is that spectral analysis has its limitations, even though it
is the most common technique applied in literature to describe the
scaling properties of series. For example, Fourier analysis assumes
an infinite length, stationary function. In practice, the data series is
finite and may be non-stationary.
We
also ran the model with
α
larger than 1.5. As
α
becomes
g
reater than 1.5, the slip becomes so smooth that distances between
zero crossings become a significant fraction of the total length of the
parent series and the statistical analysis becomes unreliable. When
α>
2, the parent series becomes so smooth that it tends to diverge
with distance. That is, zero crossings disappear and this modelling
procedure is inappropriate.
In Fig. 4, we plot
D
/
L
ratios as a function of length
L
.
There are
two noteworthy features seen in Fig. 4. (1) rougher slip distributions
10
-6
10
-5
10
-4
10
-3
=1.0
=1.25
=1.5
1
10
100
1000
L
(km)
Modeled D/L ratio
α
α
α
Figure 4.
Modelled slip-length ratio
D
/
L
as a function of rupture length
L
.
Spatially rougher slip (smaller
α
)
produce higher
D
/
L
ratios.
D
/
L
ratios
generally decrease with increasing rupture length, except when
α
=
1.5, in
w
hich case synthetic earthquakes have constant
D
/
L
ratios.
C

2005 RAS,
GJI
,
162,
841–849
Earthquake slip-length scaling
845
(smaller
α
)
produce higher
D
/
L
ratios, and (2)
D
/
L
ratio decreases
with increasing rupture length, except when
α
=
1.5.
The first feature, higher
D
/
L
ratios for rougher slip distributions,
is a simple result of the fact that, for a given slip, a rupture is more
likely to terminate in a short distance if the distribution is rough.
The second feature,
D
/
L
ratios decrease with rupture length if
α<
1.5, is a result of the fact that there is always the chance that some
percentage of the slip distributions will have a long rupture length.
These features of our modelled
D
/
L
ratios are the results of a kind
of statistical game of chance, which is determined by the variability
of our stochastic slip function.
COMPARISON TO OBSERVATIONS
Can our simple rupture simulation help us to interpret observations
of slip-length scaling for real earthquakes? To answer this question,
we
e
xamine Wells and Coppersmith’s (1994) data set, the most com-
plete compilation of earthquake rupture parameters to date. We only
use their strike-slip events for which the seismic moment, subsur-
f
ace rupture length and rupture width are simultaneously given. We
have supplemented the Wells & Coppersmith (1994) compilation
with additional events, which have occurred since their compilation
(see Table 2).
We
take the rupture length to be the estimated subsurface rupture
length, since some events have little or no surface rupture. However,
we
do assume that the lengths of surface rupture of several very long
pre-instrumental earthquakes (e.g. 1857 Fort Tejon, or 1872 Owens
V
alley) are close to their subsurface rupture lengths. For shorter
r
uptures, Wells & Coppersmith (1994) use the distribution of early
aftershocks to estimate subsurface rupture length. Although this is
not a direct measurement of rupture length, Mogi (1968), Wyss
(1979) and Kanamori & Given (1981) demonstrated that it seems
to provide an adequate estimate. We determine average slip using
D
=
M
0
/
(
μ
LW
), where
μ
is assumed to be 3.0
×
10
10
Pa,
L
is
the subsurface rupture length and
W
is the down-dip width of the
r
upture plane.
W
is also estimated from the aftershock distributions.
When the aftershock distribution is unknown (e.g. pre-instrumental
large ruptures),
W
is assumed to be the estimated thickness of the
seismogenic zone (Wells & Coppersmith 1994).
Since the importance of tectonic environment in scaling relations
is still a matter of debate (Scholz
et al.
1986; Kanamori & Allen
1986; Kanamori & Anderson 1975; Wells & Coppersmith 1994;
Romanowicz 1992), we further separate the data into interplate and
non-interplate events. Earthquakes that occurred on the San Andreas
f
ault, California, the Fairweather fault, Alaska, the North Anatolian
Table 2.
Major strike-slip earthquakes that occurred since Wells & Coppersmith’s (1994) compilation.
Event
Date
M
0
(Nm)
L
(km)
Downdip width
W
(km)
Average slip
D
(m)
D
/
L
ratio (
×
10
5
)E
v
ent type
Ref.
K
obe, Japan
01/16/1995
2
×
10
19
45
20
0.74
1.65
a
1
Manyi, China
11/08/1997
2.23
×
10
20
170
20
2.19
1.29
a
2
Izmit, Turkey
08/17/1999
1.47
×
10
20
100
20
2.45
2.45
b
3
Duzce,Turkey
11/12/1999
4.7
×
10
19
55
20
1.42
2.59
b
3
Hector Mine, California
10/16/1999
6.2
×
10
19
84
16
1.56
1.85
a
4
Ko
ko
xili, China
11/14/2001
5.27
×
10
20
450
20
1.95
0.43
a
5,6
Denali, Alaska
11/03/2002
7.5
×
10
20
340
15
4.9
1.44
b
7
Notes: a. Non-interplate type; b. Interplate type.
Av
erage slip is determine using
D
=
M
0
μ
L
W), where
μ
is assumed to be 3.0
×
10
10
Pa,
L
is the rupture length and
W
is the down-dip width of the rupture
plane,
W
is generally assumed to be the estimated thickness of the seismogenic zone.
References: 1. Ide
et al.
(1996); 2. Peltzer
et al.
(1999); 3. Tibi
et al.
(2001); 4. Ji
et al.
(2002); 5. Antolik
et al.
(2004); 6. Klinger
et al.
(2005);
7. Eberhart-Phillips
et al.
(2003).
0.01
0.1
1
10
11
0
100
1000
Interplate
Non-interplate
L (km)
D (m)
D = 2.5 x 10
-5
L
_
Figure 5.
Slip-length relation of real earthquakes in a log–log plot. The
solid line indicates a linear relationship. Data are modified from Wells &
Coppersmith (1994), plus several earthquakes occurred since their compi-
lation. Open symbols represent interplate earthquakes, and solid ones are
non-interplate events.
f
ault in Turkey and the Motagua fault in Guatemala are classified
under the interplate group. Earthquakes that clearly occurred within
a
plate, and those that occurred in a diffuse zone surrounding plate
boundaries, are classified as non-interplate earthquakes. This qual-
itative classification is often ambiguous, and varies somewhat with
different authors. Furthermore, Scholz
et al.
(1986) and Kanamori
&
Allen (1986) argued that slip rates and repeat times provide a
more meaningful categorization. Unfortunately, we cannot use this
approach since slip rates and repeat time data are unknown for most
of the earthquakes in Wells and Coppersmith’s (1994) data set. Nev-
ertheless, faults that we categorize as interplate (open symbols) all
have high slip rates and large total geologic offsets.
F
ig. 5 shows the relationship between
D
and
L
obtained from
data on a log–log scale. At first impression, a linear relationship
D
=
2
.
5
×
10
5
L
,
seems to fit this data (Fig. 5). This seems com-
patible with constant stress drop scaling. But when we plot
D
/
L
as a function of
L
(Fig. 6a), we see that the data seem to fall into
aw
edge-shaped region with a flat bottom at 6
×
10
6
and a top
that decreases from 2
×
10
4
at
L
=
1kmto2
×
10
5
at
L
=
1000 km. Fig. 6(b) shows the suite of
D
/
L
v
alues obtained for our
1-D simulation and it includes only the points where
α
=
1.25 and
C

2005 RAS,
GJI
,
162,
841–849
846
J.
Liu-Zeng, T. Heaton and C. DiCaprio
10
-6
10
-5
10
-4
10
-3
α
=1.25
α
=1.5
1
10
100
1000
L
(km)
Modeled D/L ratio
(b)
11
0
100
1000
L (km)
(a)
Observed D/L ratio
10
-6
10
-5
10
-4
10
-3
Interplate
Non-interplate
Figure 6.
(a) Slip-length ratios (
D
/
L
)v
ersus lengths (
L
)o
f
real earthquakes. Data seem to fall into a wedge-shaped region, as outlined by two solid lines.
There is a larger variation in the slip-length ratio among small earthquakes than among large events. (b) A suite of modelled
D
/
L
v
alues with
α
=
1.25 and
1.5 show a similar pattern as in (a).
α
=
1.5. Viewed in this way, it appears that the actual data (Fig. 6b)
is similar to our 1-D model assuming that there is a combination of
f
aults having slip smoothness between
α
=
1.25 and
α
=
1.5.
DISCUSSION
There has been a lot of discussion about whether scaling of slip
with rupture dimensions should be interpreted with an ‘
L
model’
or a ‘
W
model’. A key issue is whether the average stress drop
increases with rupture length for very large earthquakes. The decay
of geodetically determined ground displacement with distance from
the rupture surface of large California strike-slip earthquakes clearly
indicates that the bottom of the rupture surface is in the vicinity of
15 km (Thatcher 1975). In this sense, it seems clear that the stress
drops of long strike-slip ruptures are determined by the average slip
and the width of the seismogenic zone. Observations indicate that
av
erage slip increases even for very long ruptures whose widths
are saturated. Therefore, average stress drop increases perhaps by a
f
actor of 2 or 3 as rupture gets longer.
Our simple 1-D model of slip heterogeneity offers a different in-
terpretation of this scaling data. Determining the slip and the length
of an event becomes a simple game of chance. The stress drop is
only determined by how the ‘dice roll’. In this model,
L
and
W
are
statistically linked by a game of chance to D. For earthquakes in
w
hich
L
is less than the width of the seismogenic zone,
W
and
L
are
more or less comparable; both
L
and
W
are statistically linked to D.
Fo
r
long ruptures,
L
is still linked to D by a game of chance even
though
W
becomes fixed at the width of the seismogenic zone. Thus,
although
W
is of fundamental importance in determining the stress
drop, it may not be important for determining the average fault slip
w
hen
L

W
.
There are several key consequences of the type of model that we
introduced. We expect that faults with more heterogeneous slip will
have higher
D
/
L
ratios. This implies that, for a given earthquake
moment, we expect to see higher
D
/
L
ratios on faults with more
heterogeneous slip. This is a simple robust result of rule 2; ruptures
are contiguous. If there is variability in the roughness of slip from
one fault to another, then it seems clear that the faults with rougher
slip distributions will tend to have higher
D
/
L
ratios.
The model also predicts that
D
/
L
for a particular fault will, in
general, decrease with magnitude unless it has
α
1
.
5
D
/
L
ratio is
sometimes taken as a rough measure of average stress drop. In this
notion, our model suggests decreasing stress drop with increasing
r
upture size unless
α
1.5. When
α
1
.
5
.
D
/
L
ratio is invariant
ov
er all sizes of length, thus average stress drop is constant.
α
1.5 can be translated to Hausdorff measure
H
1
and fractal di-
mension
D
1
for 1-D slip functions. Thus the constant stress drop
model requires that slip functions are essentially Euclidean. This
is incompatible with evidence of many highly irregular slip func-
tions (e.g. McGill & Rubin 1999; Mai & Beroza 2002; Rockwell
et al.
2002). Our results are in general agreement with Mai & Beroza
(2002). They characterized earthquake slip complexity of published
finite-source rupture models and found that stress drop is inversely
proportional to source size. The constant stress drop model may not
apply to the published slip maps.
Unlike Mai & Beroza (2002), we emphasize that in real world
there may be a combination of faults having different slip smooth-
ness. If the Earth is considered to have a suite of faults with different
α
,
then we expect to observe larger variability of
D
/
L
for small
earthquakes than we do for large ones. In fact, there is compelling
e
vidence that the stress drops of some small earthquakes are in the
range of 100 MPa (e.g. Abercrombie & Leary 1993). However, we
sincerely hope that we do not encounter such a stress drop for a large
earthquake, since the average slip would have to be in the vicinity
of 100 m.
We
base our modelling mainly in 1-D. It should be possible to
e
xtend our simple model into 2-D, but implementing the model
in 2-D has not been straight forward, as one has to make ad hoc
assumptions as to how slip tapers to zero at the bottom for a long
and narrow rupture. In literature, people have used composite 2-
D
models consisting of several 1-D horizontal layers (Lavall ́
ee &
Archuleta 2003), or finite 2-D fault planes of small widths or square
lattices, which cannot be scaled realistically to represent shallow and
long ruptures (e.g. Andrews 1980; Herrero & Bernard 1994; Mai &
Beroza 2002). The 1-D model, as over-simplified as it is, captures
the basic features of slip-pulse type models of dynamic rupture. In
the slip-pulse models, the width of slip pulse is smaller than that
of the seismogenic zone, thus the propagation of the pulse in the
horizontal direction may be independent from that in the vertical
C

2005 RAS,
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,
162,
841–849
Earthquake slip-length scaling
847
direction. Pulses with large slip tend to propagate larger distance,
regardless of the final width of the rupture.
Nonetheless, we have tested a 2-D model of a square element
matrix, and the results from this model are similar to those in one
dimension. That is, a wedge-shaped
D
/
L
ratios as a function of
L
,
for a suite of smoothness. For our 2-D simulations,
D
(
x
,
y
)
=
D
0
FT
1
2
d
R
(
kx
,
ky
)
α


kx
2
+
ky
2
(4)
w
here
α

is the measure of smoothness of slip in two dimensions.
α

is different from, but is mathematically linked to that in 1-D,
α

=
2
α
.
While we have emphasized the relationship between
D
/
L
and
L
for this simple model, we can also investigate the statistical be-
haviour of the frequency of occurrence of different size events. Fig. 2
shows that when
α<
1.5, there are more small events than there
are large ones. This is slightly suggestive of real earthquake phe-
nomena. However, we cannot directly compare the statistics of our
1-D model with actual earthquake statistics since they involve fault
systems that are at least 2-D. We ran our preliminary 2-D model
multiple times with different values of
α
,
and magnitudes of events
w
ere calculated using
M
0
=
μ

S
Ddxdy
(5)
and
M
log
M
0
1
.
5
6
.
033
,
w
here
M
0
in N m
(6)
The number frequency statistics is shown in Fig. 7. The num-
ber frequency statistics with
α
=
1.5 and 1.25 are closest to the
Gutenberg–Richter relationship. But none of the statistic with a sin-
gle value of
α
fit the Gutenberg–Richter relationship over the entire
range of earthquake size. While
α
=
1.5 seem to fit at high moment
range, but does not fit at low moment range. On the other hand,
α
=
1.25 seems to produce a fit of the G-R relationship in the low moment
range, but has misfit at high moment range. Perhaps a combination
of earthquakes with
α
between 1.5 and 1.25 would produce close to
10
20
10
21
10
22
10
23
10
0
10
1
10
2
10
3
Seismic moment (Nm)
Number of events
0.5
0.75
1
1.25
1.5
symbol
Gutenberg - Richter
α
Figure 7.
F
requency-magnitude statistics for a 2-D stochastic model with
a
set of smoothness
α
.
Solid lines represent Gutenberg-Richter law. A com-
bination of earthquakes with
α
=
1.5 and 1.25 would produce close to
Gutenberg-Richter statistics.
Gutenberg–Richter statistics. If it is true, then the combination of
α
between 1.25 and 1.5 indicated by the frequency-size distribution is
coincident with that by the pattern of
D
/
L
ratio with
L
that seems
appropriate for slip-length scaling.
Although we have hypothesized that slip is spatially heteroge-
neous at all length scales, we have not provided any physical ex-
planation for such behaviour. One hypothesis is that the slip hetero-
geneity is related to complexities in the geometry of the fault zone
or in the frictional properties of the fault zone (Rice 1993; Andrews
1994; Aki 1995); that is, spatially heterogeneous slip is the result of
spatially heterogeneous media. However, others have demonstrated
that heterogeneous slip can be spontaneously generated by nonlinear
processes that arise from positive feedback between slip and fric-
tion in some classes of dynamic rupture simulations (Shaw 1995;
Cochard & Madariaga 1996; Aagaard
et al.
2001).
We
chose a simple fractal form for our slip heterogeneity, but
we
are unable to provide compelling evidence that this particular
form is a close approximation of real earthquake slip heterogeneity.
Fo
r
instance, our slip distributions are constructed from random
number sequences; changing this assumption is likely to change the
specifics of our relationship between
D
and
L
.N
ev
ertheless, it does
seem clear that slip is an irregular function of position on a fault
(McGill & Rubin 1999; Rockwell
et al.
2002). Furthermore, it has
been shown that finite-fault source inversions of earthquake slip can
be well described with stochastic models with power law decay (e.g.
Mai & Beroza 2002; Lavall ́
ee & Archuleta 2003). We speculate that
the existence of heterogeneity, together with the notion that ruptures
are spatially contiguous, are sufficient conditions for average stress
drop to increase with increasing heterogeneity; in geography, islands
with rough topography have higher average elevations than similar
sized islands with smooth topography.
CONCLUSIONS
We
introduced a simple 1-D model of stochastic slip variation along
f
ault to explore the effect of slip heterogeneity on slip-length scal-
ing and average stress drop. The key assumptions are that (1) slip
heterogeneity has a power law decay
α
with respect to wave number,
and (2) individual events are spatially connected regions.
Our model shows that for a given average displacement, earth-
quakes with smoother slip tend to have longer ruptures. Using only
these simple assumptions, we find that the relationship of average
slip,
D
,
and rupture length
L
depends on the degree of slip hetero-
geneity. Slip-length scaling is approximately linear only for rela-
tively smooth slip heterogeneity (e.g. smoothness
α
=
1.5). This
type of model is motivated by slip pulse models, in which rupture
length is determined by the average slip and conditions on a fault
(e.g. fault roughness), and static stress drop may not be a controlling
f
actor in rupture dynamics.
The modelled average slip to length ratio
D
/
L
and therefore
av
erage stress drop decreases with increasing rupture length, except
w
hen
α
1.5, in which case
D
/
L
ratio is constant. This general
pattern is consistent with the real data. Our model also implies that
slip heterogeneity affects average stress drop; average stress drops
are generally higher on faults with rougher slips than on those with
smoother slips.
ACKNOW
LEDGMENT
We
thank Jon Pelletier for providing the code to generate syn-
thetic self-affine series. We thank Hiroo Kanamori, Anupama
V
enkataraman, Edwin Schauble, Yann Klinger and Lingsen Zeng
for suggestions on earlier versions of the manuscript. Reviews from
C

2005 RAS,
GJI
,
162,
841–849
848
J.
Liu-Zeng, T. Heaton and C. DiCaprio
D.
Lavallee and an anonymous reviewer were helpful in improv-
ing the manuscript. This research was supported by the Southern
California Earthquake Center. SCEC is funded by NSF coopera-
tive agreement EAR-8920136 and USGS cooperative agreements
14-08-0001-A0899 and 1434-HQ-97AG01718. The SCEC contri-
bution number for this paper is 686.
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