Bulletin of the Seismological Society of America, Vol. 83, No. 1, pp. 7-24, February 1993
STATISTICS OF CHARACTERISTIC EARTHQUAKES
BY Y. Y. KAGAN
ABSTRACT
Statistical methods are used to test the characteristic earthquake hypothesis. Several distributions of earthquake size (seismic moment-frequency relations) are described. Based on the results of other researchers as well as my
own tests, evidence of the characteristic earthquake hypothesis can be explained either by statistical bias or statistical artifact. Since other distributions
of earthquake size provide a simpler explanation for available information, the
hypothesis cannot be regarded as proven.
INTRODUCTION
Size distribution of earthquakes has important implications both for practical
applications in engineering seismology (earthquake hazard analysis) and for
construction of a consistent physical theory of earthquake occurrence. Since the
1940s, we have discovered (see more in Kagan, 1991) that magnitude distribution when observed over broad areas follows the Gutenberg-Richter (G-R)
relation. This relation can be transformed into a power-law distribution for the
scalar seismic moment. (In this note, as a measure of earthquake size, I
consistently use the scalar seismic moment of an earthquake, which I denote by
symbol M. Occasionally, for illustration purposes, I also use an empirical
magnitude of earthquakes denoted by m.)
Considerations of finiteness of the total seismic moment or deformational
energy available for an earthquake generation, and the finite size of tectonic
plates, require that the power-law relation be modified at the high end of the
moment scale. This upper limit is usually treated by the introduction of an
additional parameter, maximum magnitude or maximum moment (McGarr,
1976; Anderson, 1979; Molnar, 1979; Anderson and Luco, 1983; Kagan, 1991).
The behavior of the size distribution in the neighborhood of the maximum
moment is not well known due to an insufficient number of great earthquakes
in available catalogs. Although evidence exists that the moment-frequency
relation bends down at the large size end of the distribution, at least, for the
world wide data (see discussion in section on Earthquake Size Distribution), the
data on the size distribution in particular regions are scarce. However, it is very
important, especially for engineering purposes, to evaluate the maximum credible size of an earthquake in a region and the probability of its occurrence. Thus,
the problem can be formulated as follows: is it possible to estimate the maximum size earthquake in a region, and is its occurrence rate higher or lower
than the extrapolation of the G-R relation for small and moderate shocks?
Recently a more complex model of earthquake size distribution has been
proposed: the characteristic earthquake distribution (Singh et al., 1983;
Wesnousky et al., 1983; Schwarz and Coppersmith, 1984; Davison and Scholz,
1985; see also Scholz, 1990). Generally, the characteristic distribution presupposes the existence of two separate and independent earthquake populations:
(1) that of regular earthquakes and (2) that of characteristic earthquakes.
Regular earthquakes, which might include foreshocks and aftershocks of characteristic events, follow the G-R relation characterized by three parameters: a,
'7
8
Y . Y . KAGAN
fl, and Me, where (1) a = the number of earthquakes in the unit of time with
seismic moment higher t h a n a cutoff level, (2) fi = the slope of the momentfrequency relation in the loglog plot, and (3) M a = the upper m o m e n t limit for
regular earthquakes. The characteristic events correspond to the largest earthquakes in a region. In order to characterize these events occurrence we need at
least two additional parameters: the size and the rate of occurrence of the
characteristic earthquakes, Mma x and ama x.
From the point of view of general statistical and continuum-mechanical
considerations, there should be no serious objection to the idea of "characteristic" earthquakes, if the hypothesis signifies t h a t the distribution of earthquakes
in a region that has a particular geometry and stress pattern, m a y differ from
the standard G-R relation. An earthquake occurrence is dependent on stress
distributions and geometry of faults, therefore the power-law moment-frequency
law is only a first approximation. However, questions remain as to the degree of
difference between the moment distribution for a large region and the distribution for a small area centered on an earthquake fault segment, even to the form
of such postulated difference. Proponents of the "characteristic earthquake"
hypothesis assert t h a t this difference is substantial (see Wesnousky et al., 1983,
their Fig. 1) and has a particular form of sharp probability density peak at the
high magnitude end of the size distribution; i.e., the frequency of these earthquakes is significantly higher t h a n an extrapolation based on the G-R law would
predict. This model is proposed on the basis of geologic evidence for similarity of
large earthquakes on fault segments, as well as on some considerations of the
geometrical structure of earthquake faults, their barriers, and the rupture
process on these complex fault systems. The determination of characteristic
earthquakes for an extended fault system involves subdividing the system into
individual faults and their segments. This procedure of segment selection is
based on geological, seismic, and other information, and it is of a qualitative
nature, since no general algorithm exists for this selection. Each of the segments is assumed to be ruptured completely by nearly identical (characteristic)
earthquakes.
Recently the characteristic earthquake hypothesis has been used to calculate
the seismic risk (see, for example, Youngs and Coppersmith, 1985; Wesnousky,
1986; Nishenko, 1991) and to model earthquake rupture propagation (Brown et
al., 1991; Carlson, 1991). Because of the importance of this hypothesis it
deserves rigorous testing. Kagan and Jackson (1991b) tested whether the
seismic gap hypothesis predicts time of occurrence for large earthquakes in the
circum-Pacific seismic zones better t h a n a random guess (the Poisson distribution) and found t h a t it does not. The negative results of the gap hypothesis
verification demonstrate t h a t the plausible and even seemingly compelling
physical and geological arguments do not guarantee t h a t the hypothesis is
correct. Thus, the arguments proposed for the justification of the characteristic
hypothesis are not sufficient for its acceptance as a valid scientific model; the
hypothesis needs to be tested and validated. This paper addresses only the
subject of the size distribution of earthquakes; their temporal relations are
discussed only if they are relevant for the testing of the characteristic earthquake hypothesis. Such testing encounters several difficulties:
1. The major justification for the characteristic earthquake hypothesis comes
from general geological, physical, and mechanical considerations, which are
STATISTICS OF CHARACTERISTIC EARTHQUAKES
9
of a qualitative nature and subject to various, sometimes contradictory
interpretations.
2. Since the above considerations are non-unique, the quantitative verification
of the hypothesis is based almost exclusively upon the statistical analysis of
earthquake size distributions in various seismogenic regions (see references
above).
3. Unfortunately, the seismic record in most seismic zones is too short to
estimate the distribution of event sizes directly from the data. Many additional assumptions about large earthquakes, the stationarity of earthquake
occurrence and relation between various magnitude scales, for example, are
needed before testing is possible. These assumptions make verification of
results inconclusive.
4. Moreover, the characteristic hypothesis is not specified to the point where it
can be subjected to a formal testing; some basic parameters of the model are
known only in broad qualitative terms.
The above considerations make it almost impossible to verify or refute the
characteristic hypothesis on the basis of current evidence. The best outcome we
can hope to obtain is to show that the quantitative proofs proposed to validate it
are not sufficient; simpler models can explain available data as successfully as
the characteristic hypothesis. In this paper, I do not challenge the physical,
geological, and geometrical arguments that are used to support the characteristic hypothesis, although in the discussion section I provide some reasons that
cast doubts on the possibility of objective segmentation of earthquake faults,
and therefore on the unique definition of characteristic earthquakes. However,
the main point of this paper is to review the quantitative statistical arguments
used to verify the characteristic hypothesis. These arguments constitute a
backbone for the validation of the hypothesis. In particular, the results of this
paper indicate that earlier published validation attempts contain some serious
defects, at least from a statistical point of view.
Howell (1985) observes that certain statistical evidence used to prove the
characteristic earthquake hypothesis m a y be the result of statistical fluctuations and thus selective reporting of "successful" samples. Another possible
biased selection involves, for example, taking a relatively small area around a
large known earthquake and comparing the seismicity of w e a k and strong
earthquakes in the area. Such a selection uses earthquake catalog information
to choose the sample that, in general, follows statistical laws that differ from a
sample chosen by an unbiased procedure, j u s t as the maximum member of a
population has a different distribution than a randomly selected member. Thus,
the catalog is used twice: first to select a sample and then to test it. As an
example of such selection, suppose a fault system of length L is enclosed in a
rectangle L x W. If we reduce the width W so that the largest earthquake is
kept inside the rectangle, the final sample (when W -~ 0) will contain a single
earthquake.
Anderson and Luco (1983) show that m a n y geologic and paleo-seismic studies
compare earthquake size distributions belonging to different distribution classes:
those of earthquake ruptures at a specific site and those events occurring in a
specific area. It is obvious, for example, that a large earthquake has a much
higher chance to rupture the surface, thus its statistics follows a different
distribution than the statistics of events that are registered by a seismographic
10
Y.Y.
KAGAN
network (Anderson and Luco, 1983). Moreover, since these data are usually
collected by different methods, their systematic errors may differ. For example,
large surface ruptures are easier to discover by paleo-seismological surveys.
Comparison of seismicity levels obtained from local instrumental catalogs and
from catalogs of strong earthquakes of longer time duration is potentially an
error-prone procedure. Local catalog data are usually collected over quiet periods of earthquake activity; even if a strong earthquake occurs in an area
covered by a local network, a threshold level of detection and reporting usually
rises abruptly immediately following the main event, thus introducing a negative bias in the seismicity levels of small earthquakes. Hence, incomplete
reporting of these events causes significant bias in seismicity levels. Moreover,
there is substantial evidence that seismicity undergoes both short- and long-term
variations (Kagan and Jackson, 1991a, and references therein). Whereas shortterm changes of seismicity, due mostly to aftershock sequences, could reasonably be taken into account, no such models yet exist for the long-term variations. These variations m a y span decades, centuries, or even milleniums (Kagan
and Jackson, 1991a).
Another source of bias is the saturation of all magnitude scales. This saturation is explained by the finite frequency of a seismographic network (Kagan,
1991). The saturation causes a loss of important information that cannot fully
be restored by any transformation rule. For example, an evaluation of the recent
seismicity levels by Singh e t a l . (1983) and Davison and Scholz (1985) is based
on body-wave magnitude ( m b) data. m b saturates at magnitudes as low as 6.0;
moreover, in order to compare m b with strong earthquake data and fault slip
data, m b is first converted into m s ; m s is, in turn, converted into the seismic
moment. Clearly, significant systematic errors m a y accumulate during such
conversions.
Since m b data are usually limited to earthquakes of lower magnitude limits,
their extrapolation to the domain of m a x i m u m events depends strongly upon
accepted b values of the G-R law: even insignificant modification of the b value
might bring these data into agreement with earthquake data in the upper
magnitude range. In the publications cited above, the standard errors in the
b-value estimates have not been evaluated; these errors are significant if the
total number of events in a sample ( N ) is small: the coefficient of variation for
the b value is 1 / ~ N (Aki, 1965). For example, if we consider that the discrepancies in rates of occurrence of strong and w e a k earthquakes reported in Singh e t
a l . (1983) are due to saturation, b-value uncertainties, and possible long-term
fluctuations of seismicity, then only one region out of four (Oaxaca) exhibits a
significant difference between extrapolated seismicity for m s events and registered "maximum" earthquakes. Such a discrepancy m a y readily be attributed to
random fluctuations of the data (Howell, 1985).
Singh e t a l . (1983) often compare actually observed earthquake numbers with
expected numbers according to extrapolation of the G-R law. If the ratio of these
numbers is high, it is considered as evidence in favor of the characteristic
earthquake hypothesis. However, if the expected number is small, the relatively
high ratio values might be the result of random fluctuations. For example, for
the Poisson variable with the expected value of 0.5, in 9% of cases two and more
events are observed (Gnedenko, 1962, pp. 431-434). If the expected number
is 1.0, there is still 1.9% probability that the actual number of events is equal
to or exceeds 4. Similarly, one should not make significant conclusions when
STATISTICS OF CHARACTERISTIC EARTHQUAKES
11
observing two or three events in a certain magnitude interval versus zero or one
event in another interval (Singh et al., 1983); such variations can be explained
again by random fluctuations.
A single work among those cited escapes criticism: Davison and Scholz (1985)
investigate a large portion of the circum-Pacific belt (more than 3000 km), hence
there is little possibility for the selection bias or statistical fluctuations explaining their results. Except for use of the m b data, the study analyzes a relatively
homogeneous m s catalog to infer suitability of the characteristic earthquake
hypothesis. Davison and Scholz (1985) first normalize m s data to the recurrence
times of characteristic earthquakes; then they extrapolate the normalized seismicity levels to magnitudes of the maximum events.
For five out of nine zones where great earthquakes occurred during this
century, Davison and Scholz (1985) find that extrapolation of frequency-moment
data is smaller by a factor of 10 than the moment of earthquakes that actually
originated on these segments. However, these tests are critically dependent on
the estimate of the return time for the m a x i m u m magnitude events. Davison
and Scholz (1985) evaluate the recurrence time by dividing the earthquake
moment by the moment accumulation rate. This procedure assumes that most
elastic deformation is effected by characteristic events. However, we need to test
whether another model, for example, a regular G-R relation, can explain these
data with the same efficiency.
To accomplish this test, we need first to describe statistical distributions of
earthquake size. Although most of the distributions in this paper have been
published earlier (see Anderson, 1979; Molnar, 1979; Anderson and Luco, 1983;
Kagan, 1991), consistent use of the seismic moment allows us to simplify the
formulas significantly. Thus this paper has two objectives: (1) to compile formulas for distributions of the seismic moment and related quantities and (2) to
review available evidence and statistical tests for the characteristic earthquake
hypothesis.
EARTHQUAKE SIZE DISTRIBUTIONS
The distribution density ¢ of the seismic moment is usually assumed to follow
the power-law or a Pareto distribution, which is an appropriate transformation
of the G-R relation (see more in Kagan, 1991):
¢(M)=fiMc~.M
-1 ~
forMc <M<~.
(1)
M c is a lower threshold seismic moment. This lower limit is introduced for two
reasons: (1) since any seismographic network has a detection threshold and (2)
since otherwise the distribution would diverge at M ~ 0 (there is an infinite
number of events with M -~ 0). The distribution (1) requires only one degree of
freedom for its characterization. In the traditional form of the G-R relation
another p a r a m e t e r a or a (Kagan, 1991) is added; this p a r a m e t e r is the rate of
occurrence of earthquakes with M > M c. In later considerations, in addition to
the density function, I often use the cumulative distribution function F ( M ) and
the complementary function q)(M) = 1 - F ( M ) .
Statistical analysis of magnitude and seismic moment distributions yields the
value of fi in (1) between ~1 and 1 for small and medium-size earthquakes
(Kagan, 1991). Simple considerations of finiteness of seismic moment or deformational energy, available for an earthquake generation, require that the
12
Y.Y. KAGAN
power-law relation be modified at the large size end of the moment scale. At the
minimum, the distribution tail must have a decay stronger than M -1 -~ with
fi > 1. This problem is generally solved by the introduction of an additional
parameter, called a "maximum moment" (M~), to the distribution. This new
parameterization of the modified G-R relation has several forms (Anderson and
Luco, 1983; Kagan, 1991). In their simplest modifications, they require at least
three parameters to characterize an earthquake occurrence.
I consider four possible distributions of earthquake size that all satisfy the
requirement of finiteness of the moment flux: (a) characteristic earthquake
distribution; (b) maximum moment distribution; (c) truncated Pareto distribution; and (d) g a m m a distribution. Below I refer to cases (b), (c), and (d) as to
a l t e r n a t i v e models.
In this paper the term "characteristic earthquake distribution" refers to a
distribution for which the major part of total seismic moment flux is carried out
by characteristic earthquakes, so that we essentially ignore t h e contribution of
regular earthquakes in the total budget of seismic moment rate. Thus, the
seismic deformation is characterized by two parameters M~c and ax~ (moment
and rate of characteristic earthquakes), which are either both evaluated independently using the available geological/seismological data or, if only one of
them can be evaluated, the other can be computed from the relation
(2)
= a~cMxc,
where M is the moment flux.
Here I understand the "maximum moment distribution" as a truncated
power-law distribution of the complementary cumulative function (Molnar,
1979)
q)(M) =
for M c < M < M x m ,
and
(I)(M) = 0
for M ~
< M,
(3)
where Mx~ is the m a x i m u m moment for this model. For the cumulative
function, we obtain
F( M)
M s _ Me t3
M~
-
and
F(M)
for M e < M < M x m ,
(4)
= 1 forMxm < M .
For the maximum moment distribution density,
~(M)
= fiMc~'M -1-~ + 6(M-
~b(M) = 0
Mxm ) •
Mc]8 f o r M c
forMxm<M,
<=M~Mxm,
(5)
where 6 is the Dirac delta function.
The two first distributions can be regarded as limiting cases of the general
characteristic hypothesis. In the former distribution (a), the regular earthquakes do not play a significant role in the total balance of fault slip rate and
13
STATISTICS OF CHARACTERISTIC EARTHQUAKES
therefore can be ignored. In the latter distribution (b), characteristic earthquakes are considered as an extrapolation of the regular events distribution, or
as a density spike at the high m o m e n t end of the distribution. The size
distributions referred to in most of the publications on characteristic earthquakes can easily be obtained as a mixture of (a) and (b).
Another model does not truncate the complementary cumulative distribution
but the density function: for the Pareto distribution truncated on both ends (c),
the distribution density is
¢(M)
=
fi M x~p• M c~
M-1-8
Mx~p- M f
forM~ < M < M x p ,
=
(6)
=
where Mzp is the m a x i m u m m o m e n t for the Pareto model, which m a y differ
fromMxm of (3). For cumulative function, we obtain
F(M) =
[_M__~_]~ M~ - M f
Mx~p - M f
for Mc < M < Mxp,
=
=
(7)
and the complementary distribution function has a form
¢p( M ) = [ _ ~ ] ~ M~p - M~
Mx~p:Mcc ~
(s)
f°rMc<M<Mxp"
For the g a m m a distribution (d),
¢(M) =C-1×
~
exp
~/xl
f°rMc<M<%
(9a)
where Mxg is the m a x i m u m m o m e n t p a r a m e t e r for the g a m m a distribution.
The p a r a m e t e r Mxg has a different meaning from the p a r a m e t e r Mxp or Mxm:
whereas the latter two represent a "hard" limit, the former is a "soft" limit;
hence in the g a m m a distribution some earthquakes m a y have the m o m e n t
M~g < M. The normalizing coefficient C is (Bateman and Erdelyi, 1953)
( )
C= 1-
~Mc
n
~exp([Mxg
Mc
( M c / M xf ig+ ~)
F(1 - fi) - n~O ( - 1 )ni(1----
I
, (9b)
where F is the g a m m a function. For M~ << M~g, C = 1. The cumulative function
is
F(M)=C
=
l×fi
[ ]expl
[Mxg]JMo
~
exp
[ ~
dM
7( - f i , M / M x e) - 7( - f i , Mc/Mxg))
forM~ < M < %
(10)
14
Y.Y. KAGAN
where ~ is the incomplete g a m m a function (Bateman and Erdelyi, 1953). The
complementary distribution function has a form
¢p(M) = 1 - F ( M )
forM c <M<~.
(11)
To illustrate the gamma distribution fit to actual data, Figure l a displays a
cumulative histogram for the scalar seismic moment of earthquakes in the
H a r v a r d catalog (Dziewonski et al., 1992, and references therein), for shallow
earthquakes (depth interval 0 to 70 km). The available catalog covers the period
from 1 J a n u a r y 1977 to 31 December 1991. To ensure the data uniformity
events with M >__10177 Nm (m w >=5.8) are used. The distribution of the worldwide data for the largest earthquakes differs from the G-R law, b u t this
difference has an opposite sign than the characteristic hypothesis predicts: the
number of great events is smaller than the extrapolation of the G-R curve.
Thus, our null hypothesis for the size distribution of the largest events should
be the gamma or truncated Pareto distribution (see below).
The maximum likelihood procedure (Kagan, 1991) allows us to retrieve the
values of distribution parameters. The values of fi and Mxg are listed in Table 1
for the maximum of the likelihood function and for four extreme points of an
approximate ellipse corresponding to the 95% confidence area (see Fig. 2 in
Kagan, 1991). The five curves in Figure l a are calculated using the above
values, they form a 95% envelope of possible approximations for the momentfrequency relation by the gamma distribution. The curves in Figure l a demonstrate that worldwide earthquake size data are reasonably well-approximated
by the gamma distribution.
For comparison, Figure lb displays the curves for the truncated Pareto
distribution (8). (We have not tried distributions (a) and (b) since they are
clearly inappropriate for approximation of the worldwide data.) To obtain the
values of parameters that are also listed in Table 1, I again apply the likelihood
method. From (6) it is clear that in this case the function should reach the
m a x i m u m at the M value of the largest earthquake in the catalog ( M m a x = 3.57
• 1021 Nm), thus the m a x i m u m likelihood estimate and the lower bound of the
95% confidence area coincide. Even a visual inspection shows that the gamma
distribution gives a better approximation for the experimental curve. The
disadvantages of the hard limit used in the truncated Pareto distribution are
more obvious if we compare the H a r v a r d catalog with the history of the seismic
moment release during the 20th century (McGarr, 1976; Kanamori, 1983). The
largest earthquake (Chilean of 1960) had a moment 2 • 1023 Nm, thus we would
need to significantly modify the Mxp parameter of the Pareto distribution,
whereas the g a m m a distribution approximates these data with only a slight
change of parameters: since earthquakes with M > Mxg are allowed in the
gamma distribution, the Chilean earthquake can be accommodated by using the
"upper" curve of the 95% envelope (see Table 1).
Figure 2 displays complementary functions of distributions (3), (8), and (11)
for t~ = 2 (Davison and Scholz, 1985). The value of Mxm = 1023 is set, approximately corresponding to the largest observed earthquake in the Aleutian islands and Alaska (Kanamori, 1983). Other Mx's are adjusted so that the
distributions would yield the same seismic moment release as (b) (see also
below, equation 15). The characteristic model (a) would represent a step function in the plot at the abscissa value 1023 Nm. For the truncated Pareto
STATISTICS
10 4
, , ,,,,,,,
OF
CHARACTERISTIC
, , ,,,,,,,
, , ,,,,,,,
EARTHQUAKES
, , ,,,,,,,
15
, , ,,,,,,,
, , ,,,,,
10 3
"
'~siii"..
i;, .. i~i;!ii!!!!!!iiiii.
"
10 2
Y.
E
'-,..~
101
%,
l00
•.
10-1
i
i
i iiiiit
10 17
i
llll,,i
i
1018
,
i
lllliH
10 19
i
i
,llll,,
10 20
~ .i
-.. \
"..
,
I.IHH
10 21
i ".i
i lilt
10 22
1023
L o g Seismic M o m e n t ( N m )
(a)
10 4
, , ,,,,,,,
, , ,,,,,,,
, • ,,,,,,,
, , ,,,,,,,
,
, ,,,,,,,
~,
, ,,,,,~
I
~llll
i
i
103
E
Z
102
7
E
10 I
~9
10 o
10_
1
1017
,
i
,
J ,,liH
1018
i
, ill,zt
t
i
, ,,Llll
1019
1020
_t
r
i itttt~
1021
tit
1022
L,,
1023
Log Seismic M o m e n t (Nm)
(b)
FIG. 1. L o g o f s c a l a r s e i s m i c m o m e n t v e r s u s c u m u l a t i v e f r e q u e n c y f o r t h e 1 9 7 7 to 1 9 9 1 H a r v a r d
c a t a l o g . T h e c u r v e s s h o w t h e n u m b e r s o f s h a l l o w e a r t h q u a k e s w i t h t h e m o m e n t l a r g e r o r e q u a l to
M . T h e t o t a l n u m b e r o f e v e n t s w i t h M > 10177 is 2 3 5 8 . D a s h e d a n d d o t t e d c u r v e s c o r r e s p o n d to t h e
approximation by the gamma (la) and-Pareto (lb) distributions (maximum likelihood shown by a
d a s h e d line; 9 5 % e n v e l o p e s h o w n by d o t t e d lines).
16
Y. Y. KAGAN
TABLE 1
PARAMETERVALUES(PARETO AND GAMMADISTRIBUTIONS)
FOR THE HARVARDCATALOG1977-1991
Likelihood
Function
_+95% Limits
/3-value
M x x
Gamma distribution
max
0.667
left
0.627
down
0.650
right
0.706
upper
0.683
Pareto distribution
max, down
0.684
left
0.645
right
0.715
upper
0.687
102] Nm
M x 1021 Nm/yr
2.51
1.58
1.00
4.60
22.0
2.49
2.46
1.92
2.57
4.65
3.57
3.57
3.57
11.0
2.91
3.36
2.56
4.08
10o
ii
10-1
,
10-2
o
10 -3!
~
10 4
e~
~o
1@5
10-6
10_ 7
1018
T i I1.1.
1019
, i IIH,.
1020
, ~ iiitl,
1021
i i iiiiiit
1022
1 0 23
1 0 24
Seismic Moment (Nm)
FIG. 2. Complementary distribution functions for several models of earthquake size distribution:
(1) Pareto distribution with unlimited maximum seismic moment (dotted line); (2) maximum
moment model (solid line); (3) truncated Pareto distribution (dashdot line); (4) gamma distribution
(dashed line). We take seismic moment cutoff at 10 18 Nm; M x _ = 10 2 3 ; other maximum moment
quantities are adjusted to yield the same moment rate, given num]aer of earthquakes with M > 10 is :
Mxp = 3.38 x 10-23 and Mxg = 4.74 x 1023 (see equation 15).
distribution, the return times for earthquakes with M ~ Mxp are approaching
infinity since the probability of occurrence of such events obtained by integration of (6) is close to zero. The gamma distribution displays similar behavior
(Fig. 2).
Using this figure, one can calculate the number of expected earthquakes with
the moment greater or equal to M, by multiplying the value of alp(M) by the
S T A T I S T I C S OF C H A R A C T E R I S T I C
EARTHQUAKES
17
total number of events with M >=M c. The difference between the alternative
distributions is small for most of the moment range, only at M > 5 • 1021 the
difference exceeds 10%. To distinguish between these models we need information on great earthquakes, which are extremely rare according to (c) and (d).
Such events might correspond to simultaneous breaking of several segments of
the subduction belt.
Using models (a) through (d) one can calculate the seismic moment release
rate (moment flux), taking the seismic activity level a o for earthquakes with
moment M o and greater. To avoid awkward algebra, M x >> M 0. In most cases,
M 0 can be chosen to correspond to M e in (1). For the characteristic distribution
the moment flux is given by (2). The total seismic moment rate according to the
m a x i m u m moment distribution is (Molnar, 1979; Anderson and Luco, 1983)
/I//1- ~/I// f i - - "
M=ao,,,xm
,,~o 1 -
1
(12)
fi'
for the Pareto distribution we calculate (McGarr, 1976; Anderson, 1979; Anderson and Luco, 1983)
M
/3
= ~ nzl-~t
~
.
-0-,-x,
-,-o 1 - 13'
(13)
and for the g a m m a distribution
/3
: a°Mx~g-¢M°¢F(2 - / 3 ) -1- - - ~ "
(14)
Table 1 displays the moment rate per year for the H a r v a r d catalog. The
estimates are relatively stable, even though the parameter values display
significant variations. The rate value obtained by a direct summation of the
moment is 2.33 • 1 0 21 N m / y r ; i.e,. this value is close to the values corresponding
to the m a x i m u m of likelihood functions. Parenthetically I note t h a t the direct
summation of Kanamori (1983) data for 1921 to 1976 yields 8.53 • 1021 N m / y r ,
whereas McGarr (1976) found the M value of 3.4- 1022 N m / y r on the basis of
plate motion calculations. F u r t h e r discussion of this problem is outside the
scope of this article.
By equating expressions (12) to (14) for total moment rate, we can find the
ratios between m a x i m u m moments in all of these distributions:
Mxg[r(2 -/3)] 1/(1-~) Mxp =
---
Mxm/3- 1/(1-~).
(15)
For values of/3 equal to ~2 or to ~1 the g a m m a function is F(2 - 2) = 0.893 or
F(2 - 1)
2 = 0.886. Using formulas (5), (6), and (9), I calculate the distribution of
the total seismic moment t h a t is released by an earthquake with the moment
M. In this case, the distributions converge for M ~ 0, thus we do not need the
truncation on the left-hand side of the distributions, and we can define the
distributions on the interval 0 < M < ~. For the characteristic distribution (a),
the distribution is a step function at Mxc; for other models, the total moment
18
Y.Y.
KAGAN
release distributions can be calculated as
(16)
F(M) -- fjch(M)MdM.
In particular, for the maximum moment distribution (b) the cumulative function
is
F(M)
=fl
, for0<=M<__Mxm,
and
F=
1 forMxm < M ;
(17)
and
F=
1 forMxm < M ;
(18)
for the Pareto distribution
F(3~) =
, for0<M<M~m,
and for the gamma distribution
1
F(]~/) -
- fi, M/Mxg),
y(1
F(1
for 0 < M < ~.
(19)
Figure 3 displays these distributions for fi = ~.
2 The plots can be compared to
similar results by Anderson and Luco (1983, their Table 1). According to all of
the distributions, the major part of seismic moment release is carried out by
great earthquakes (i.e., events with ( M x / 1 0 ) < M < Mx): for case (b) it is
1
J
E lll,tll
i
i iil,l,i
i
i ii,ll~r
i
i ii~lll
!
0.9
/
[ /
0.8
i/
};
0.7
0.6
0.5
0.4
0.3
¢,
0.2
0.1
0
1019
.
, ,111,.
1020
~ J ii1.,,,
1021
,
i t,lllll
1022
i
, , i,l,i,
1023
i
, .,,,,,~
1024
i
i ,qll
1025
Seismic Moment (Nm)
FIG. 3. Distribution of total seismic m o m e n t released by e a r t h q u a k e s with seismic m o m e n t M.
The line types a n d values of p a r a m e t e r s correspond to t h a t of Figure 2.
STATISTICS OF CHARACTERISTIC EARTHQUAKES
19
69.1%, for case (c) 53.6%, and for (d) 39.7%. In the last case (the g a m m a
distribution), earthquakes with M > Mx contribute an additional 9.6% to he
total. The above values are obtained for fi = ~.
2 The m a x i m u m magnitude
1
distribution (b) implies t h a t y of the seismic moment release is due to m a x i m u m
moment earthquakes. Alternative distributions differ little in their total mom e n t release until the earthquake moment approaches m a x i m u m size. This
m a y signify t h a t it would be difficult to infer the proper form of a distribution on
the basis of the relatively short histories of earthquake occurrences available for
most of the Earth's regions.
RESULTS
Next we will examine whether the discrepancy between the rates of event
occurrence in the zones selected by Davison and Scholz (1985) and the rates of
great events which originate in the same zones might have an explanation other
t h a n characteristic distribution (a). In case of a characteristic distribution, the
moment release is concentrated in a sharp, delta-function-like spike, whereas
other models envision such a release in a more continuous manner. Davison and
Scholz (1985) find t h a t the strongest earthquakes have an occurrence rate with
a factor 5 to 30 higher t h a n the expected rate for those events using the
extrapolation from weak events by the G-R law.
Let us suppose t h a t in reality earthquakes obey any of the alternative
distributions (b to d) but we process the largest events through the same
procedure as applied by Davison and Scholz (1985): we normalize the occurrence
rate of medium-size earthquakes according to the recurrence time of characteristic events calculated using (2) and extrapolate the obtained rate to great
events. In particular, let us assume t h a t all earthquakes with moment larger
t h a n Mf are t a k e n to be characteristic events. Then the recurrence time for an
event with moment M is
T= Mf
A~ILf
(20)
where Lf is the rupture length. M and a in this section are normalized per
unit of time as well as per unit of fault length. The expected total number of
earthquakes with Mf < M during this time
I- M r 11-~
Nr= TLfaf= ( 1 - fi)[~xm J
,
(21)
where in the right-hand part of (21) we calculate Nf for model (b). Let us call
the inverse of Nf the discrepancy ratio p. This ratio shows how much the
procedure employed by Davison and Scholz (1985) under-counts the expected
number of earthquakes. For example, even if an earthquake has a moment
equal to the m a x i m u m (Mxm) and fi = y,
2 then p = 3; i.e., this procedure
reduces the expected number of regular events by a factor of 3, which is the
consequence of the distribution of the total moment displayed in Figure 3; only
one third of the moment is released by earthquakes with Mxm (cf. Anderson and
Luco, 1983).
20
Y.Y. KAGAN
One needs to remember t h a t the "true" value of M~ is unknown, so the
difference obtained by using (2) is an average over possible values of M and Mx.
Since we do not know the value of M~m, for model (b) I assume t h a t any
earthquake with M f < M is a characteristic event. These earthquakes occur
with probabilities (5), hence we calculate the average value of p as
,
= fMx~p4~(M) d M = Mxm ] + - JMr
1--fl
Mxm ]
"
(23)
For example, assuming again t h a t fi = -~, and MW = 1021, which corresponds to
earthquakes with m s = 8.0, we obtain ~ = 9.3. This means t h a t the average
discrepancy for model (b) is of one order of magnitude; i.e., it is about the same
as was observed by Davison and Scholz (1985). Due to random fluctuations, this
ratio m a y be larger or smaller for certain areas.
Similar expressions for other models can be obtained, for example, for the
Pareto distribution (c)
~ = l f i 2 f i M f -1 Mxp - M f
M~p -- M--~"
(24)
The values of ~ are only slightly different from t h a t of (23): for the same
example ~ = 9.5. Even if we t a k e n M f = 1022 (m s -~ 8.7), the ratio is still large
--4.6.
Combined data from the Aleutian Arc (Fig. 11 in Davison and Scholz, 1985)
follow one of the "noncharacteristic" alternative distributions. From a statistical
point of view this is to be expected, since the data set is extensive and its
selection is largely unbiased, except for the 1964 Alaskan earthquake, which
was one of the two largest worldwide earthquakes in this century. Hence we
should expect t h a t the point corresponding to this event would be an outlier in
the moment-frequency plot.
Davison and Scholz (1985) identify four out of nine zones where no strong
earthquakes have occurred in this century, as potential seismic gaps. They
normalize the seismicity levels measured in these zones to the r e t u r n times of
m a x i m u m moment earthquakes. These m a x i m u m events are assumed to rupture the whole extent of the appropriate segment. The arguments above are
applicable to those segments of the Aleutian Arc: if the distribution of earthquake size follows model (c) or (d), we will obtain similar differences between
the probabilities of the strongest events and extrapolation of seismicity, using
the characteristic hypothesis (2). Parenthetically, I mention t h a t during the 6
years t h a t have elapsed since the publication of the Davison and Scholz' (1985)
paper, strong earthquakes have not followed the seismic gap hypothesis: according to the Harvard catalog of seismic moment tensor solutions (Dziewonski et
al., 1992), two earthquakes out of the six events with M > 1019"5 in the Aleutian
Arc occurred in the previously identified zones (M = 1021 in 1957 zone, M =
10137 in 1938 zone), three events near, but outside the Yakataga gap, and only
one (M = 1019's) occurred in the Kommandorski gap.
If we subscribe to the position t h a t the size distribution of earthquakes in the
Aleutian Arc follows the truncated Pareto or g a m m a distribution, then we
STATISTICS OF CHARACTERISTIC EARTHQUAKES
21
calculate from Figure 11 in Davison and Scholz (1985) that about eight events
with moment larger than 3.5 • 1021 occurred in the 85 years prior to 1985. The
probability of eight earthquakes falling into five fault zones out of a total of nine
is 0.34; thus the observed earthquake pattern can easily be explained by a
simpler assumption of the Pareto or g a m m a distribution.
DISCUSSION
The existence of several competing earthquake size distributions suggests
that we do not yet know the exact seismic moment-frequency relation. In
general, by introducing delta-functions or sharp corners in the distributions, we
impose rather strong constraints on the size of an earthquake. These constraints are not w a r r a n t e d by the available evidence, since they contradict the
known behavior of dissipative physical systems. Therefore, we should introduce
some degree of smoothness in the distributions which would require additional
parameter(s), thus such distributions as "characteristic earthquake," "maxim u m moment," or the truncated Pareto distributions will lose much appeal due
to their simplicity. Only the g a m m a distribution satisfies both conditions:
simplicity and satisfactory approximation of available data.
Although alternative distributions are similar at most of the moment range
(see Fig. 2) where one has empirical evidence and all of them can be used for an
approximation of moment-frequency histograms, we should be concerned with
the simplicity and logical consistency of the models. For example, the characteristic earthquake model (a) cannot by itself explain the size distribution of all
earthquakes making it necessary to invoke a general characteristic model, as
discussed in the introduction. The latter model, in turn, requires five to six
parameters for characterization, and the analysis above shows that there is no
direct quantitative evidence to justify its acceptance. E a r t h q u a k e occurrence
models with three degrees of freedom approximate available data as satisfactorily as a general characteristic distribution. We m a y invoke the Ockham razor
rule, thus choosing the simplest explanation and the most economical conceptual formulation of a model.
I interpret the Ockham rule to select among models consistent with data and
other available information, the model that has a minimum n u m b e r of degrees
of freedom; we should choose the model with m a x i m u m entropy, i.e., the model
that, under known constraints, maximizes the uncertainty of our knowledge. In
other words, unless there is compelling evidence to the contrary, we should
accept the simplest distribution of earthquake size.
In conclusion, let us examine whether the characteristic hypothesis can be
effectively tested. Unless formulated as a formal quantitative algorithm to
specify segmentation of a fault zone and values of distribution parameters
expected from earthquakes occurring on these segments, it would be difficult to
validate this hypothesis statistically. At present, fault segment selection is
based on qualitative criteria and hence is largely subjective.
Thatcher (1990) demonstrates that even very large earthquakes do not break
the same segment of subduction belts; thus subdivision of a fault into segments
does not assure that earthquakes will obey these boundaries. The Working
Group (1988, 1990) provided two segmentation variants of the San Francisco
peninsula part of the San Andreas fault, the Hayward, and the Rodgers Creek
faults. Although the same scientists served on both panels (8 out of 12), the
results of fault segmentation are significantly different (cf. Fig. 5 of the 1988
22
Y.Y. KAGAN
report and Figs. 5 and 6 of the 1990 report). As another example, Nishenko
(1991) proposes a subdivision of the circum-Pacific earthquake zone into segments that are significantly different from that of the earlier work by McCann
et al. (1979).
I believe that this disagreement is not accidental: if, as accumulating evidence
suggests (Kagan, 1992), the fault pattern is scale-invariant (fractal), there are
no separate scales for "individual" faults or their segments. Moreover, the fault
geometry analysis (Kagan, 1992, p. 10) indicates that earthquakes do not occur
on a single (possibly wrinkled or even fractal) surface, b u t on a fractal structure
of many closely correlated faults. This would indicate that we cannot meaningfully define an individual fault surface. Thus, the objective selection of fault
segments is as impossible as it is infeasible to devise a computer algorithm that
would subdivide a mountain range into individual mountains or a cloudy sky
into individual clouds. Suppose we try to subdivide fault patterns as shown in
Figures 10 to 12 of Tchalenko (1970) without considering the scale of the plot. In
principle each segment can be subdivided further into subsegments and so on;
each fault trace (surface) could again be subdivided into m a n y quasi-parallel
traces. Therefore any segmentation rule would be arbitrary, and the results
obtained through such subdivision would be strongly dependent on the algorithm used.
Although it is difficult to formalize the segmentation procedure, the geological
insight of such studies m a y still yield important information on the size
distribution of very large earthquakes. We cannot test this possibility statistically through a retrospective forecasting, since current segmentation models are
descriptive and hence the number of degrees of freedom in the models is
comparable to the number of data. However, it is possible to test the actual
forecast of characteristic earthquakes (such as made by Nishenko, 1991), where
such considerations do not apply. However, in order for the prediction to be
testable, it should be forecast for a large number of fault segments, so that after
several years there will be a sufficient dataset to make statistically valid
conclusions.
Moreover, a meaningful forecasting method m u s t be testable against a random occurrence. Suppose an earthquake satisfying characteristic event criteria
occurs in a certain zone: it does not automatically validate the hypothesis; such
an earthquake could occur purely by chance. Thus we need to compare the
forecast data set with that expected from the standard worldwide relation such
as displayed in Figure 1. In other words, the characteristic earthquake hypothesis, as any meaningful scientific model, must be falsifiable: accumulated evidence should either validate or reject the hypothesis.
Thus, to be verifiable, not only characteristic events have to be described
completely in the forecast, b u t the distribution of regular earthquakes also
needs to be specified; i.e., all five or six parameters of the general characteristic
distribution m u s t be specified. At a preliminary stage, the general characteristic
distribution might be defined as a worldwide generic model; thus, only one or
two parameters need be adjusted for a particular fault segment. Only when it
can be shown that the forecast distribution is a significantly superior approximation of the experimental data than relations (b) to (d) can we conclude that
the hypothesis has been verified. Before such validation is accomplished, in
practical evaluations of seismic risk a range of results should be presented to
display possible alternatives. These calculations should be performed using
STATISTICS OF CHARACTERISTIC EARTHQUAKES
23
v a r i o u s m o d e l s of size d i s t r i b u t i o n w i t h i n d i c a t i o n of p o s s i b l e v a r i a t i o n s o f
parameter estimates.
ACKNOWLEDGMENTS
I appreciate support from the National Science Foundation through Cooperative Agreement
EAR-8920136 and USGS Cooperative Agreement 14-08-0001-A0899 to the Southern California
Earthquake Center (SCEC). I am grateful for useful discussions held with D. Vere-Jones of
Wellington University, D. D. Jackson of UCLA, and P. Bak of Brookhaven National Laboratory. I
thank Associate Editor S. G. Wesnousky, reviewer P. A. Reasenberg (USGS), and an anonymous
referee for their valuable remarks, which significantly improved this manuscript. Publication 5,
SCEC. Publication 3808, Institute of Geophysics and Planetary Physics, University of California,
Los Angeles.
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24
Y, Y. KAGAN
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Note Added in Proof--More complete discussion of the truncated Pareto distribution and evaluation of the maximum earthquake magnitude can be found in the paper by V. F. Pisarenko (1991)
"Statistical evaluation of maximum possible earthquakes," Phys. Solid Earth 27, 757-763, (English
translation).
INSTITUTEOF GEOPHYSICSANDPLANETARYPHYSICS
UNIVERSITYOF CALIFORNIA
LOS ANGELES,CALIFORNIA90024-1567
Manuscript received 10 December 1991