A model of characteristic earthquakes and its implications for regional seismicity

Abstract

Regional seismicity (i.e. that averaged over large enough areas over long enough periods of time) has a size–frequency relationship, the Gutenberg–Richter law, which differs from that found for some seismic faults, the Characteristic Earthquake relationship. But all seismicity comes in the end from active faults, so the question arises of how one seismicity pattern could emerge from the other. The recently introduced Minimalist Model of Vázquez-Prada et al. of characteristic earthquakes provides a simple representation of the seismicity originating from a single fault. Here, we show that a Characteristic Earthquake relationship together with a fractal distribution of fault lengths can accurately describe the total seismicity produced in a region. The resulting earthquake catalogue accounts for the addition of both all the characteristic and all the non-characteristic events triggered in the faults. The global accumulated size–frequency relationship strongly depends on the fault length fractal exponent and, for fractal exponents close to 2, correctly describes a Gutenberg–Richter distribution with a b exponent compatible with real seismicity.

Full text

Page 1

A model of characteristic earthquakes and its implications
for regional seismicity
Ricardo Lo ´pez-Ruiz,1Miguel Va ´zquez-Prada,2Javier B. Go ´mez3and Amalio F. Pacheco2
1Department of Computer Sciences and BIFI,
Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
2Department of Theoretical Physics, and
3Department of Earth Sciences, University of
Introduction
In regional and global seismicity,
there is a well established fractal
relationship, the Gutenberg–Richter
law, that can be expressed in the form
_ n / S?b;
ð1Þ
where _ n is the number of observed
earthquakes in a year with rupture
area greater than S, and b is in the
range 0.5–1.5. The Gutenberg–Richter
law implies that earthquakes are, on a
regional or global scale, self-similar
phenomena – without any character-
istic length scale.
It is important to note, however,
that the Gutenberg–Richter law is a
property of the regional seismicity,
appearing when we average seismicity
over large enough areas and over long
enough time intervals (e.g. Kosso-
bokov and Mazhkenov, 1994; Mol-
chan et al., 1997). In the last 10 years,
a wealth of data have been collected to
extract statistics on individual systems
ofearthquakefaults
1994; Petersenet al.,
1996; Stirling et al., 1996; McCalpin
and Slemmons, 1998; Working Group
on California Earthquake Probabili-
ties, 2003). Interestingly, it has been
(Wesnousky,
1996;Sieh,
found that the distribution of earth-
quake magnitudes may vary substan-
tially from one fault to another and
that, in general, this type of size–
frequencyrelationship
fromthe Gutenberg–Richter
Many single faults or fault zones
display power-law distributions only
for small events (small compared with
the maximum earthquake size a fault
can support, given its area), which
occur in the intervals between roughly
quasi-periodic earthquakes of much
larger size and that rupture the entire
fault.These large
earthquakes are termed ?characteristic?
(Schwartz and Coppersmith, 1984),
and the resulting size–frequency rela-
tionship is the Characteristic Earth-
quake distribution. The characteristic
earthquake hypothesis is not univer-
sally supported by the seismological
community and alternative scenarios
do exist (e.g. Kagan and Jackson,
1991, 1995; Kagan, 1993, 1996; Sav-
age, 1993). Ellsworth (1995) reviews
evidence for and against the notion of
the characteristic earthquake.
With the purpose of representing
the seismicity originating from a single
fault, we have recently introduced a
simple model called ?Minimalist? (Va ´ z-
quez-Prada et al., 2002, 2003). The
main goal of this paper is to study the
total seismicity produced in a region,
using the Minimalist Model (hereafter
MM) for the description of the seis-
micity originating from each individ-
isdifferent
law.
quasi-periodic
ual fault. In the sections below we
briefly review some properties of the
MM, and identify its asymptotic
(N fi ¥) behaviour. This informa-
tion, combined with the observed
fractal distribution of fault sizes at a
regional level (Barton et al., 1986;
Hirata, 1989; Ouillon et al., 1996;
Bonnet et al., 2001), will give us a
size–frequency distribution for the
earthquake population, which will
bediscussed and
Eq. (1).
comparedwith
Rules and properties of the
Minimalist Model
The MM was devised in a manner akin
to the sandpile model of Self-Organ-
ized Criticality (Bak and Tang, 1989):
the physics are not apparent, but
the model is able to grasp the routine
of a fault’s dynamics.
This model is explained as follows.
Consider a one-dimensional vertical
array of length N. The ordered posi-
tions, or levels, in the array will be
labelled by an integer index i varying
from 1 to N. This system performs
two functions, it is loaded by receiving
individual stress particles in the var-
ious positions of the array, and unloa-
ded by emitting groups of particles
through the first level, i ¼ 1, which are
called relaxations
(Fig. 1).
These two functions proceed with
the following four rules:
orearthquakes
ABSTRACT
Regional seismicity (i.e. that averaged over large enough areas
over long enough periods of time) has a size–frequency
relationship, the Gutenberg–Richter law, which differs from
that found for some seismic faults, the Characteristic Earth-
quake relationship. But all seismicity comes in the end from
active faults, so the question arises of how one seismicity
pattern could emerge from the other. The recently introduced
Minimalist Model of Va ´zquez-Prada et al. of characteristic
earthquakes provides a simple representation of the seismicity
originating from a single fault. Here, we show that a Charac-
teristic Earthquakerelationship together with a fractal
distribution of fault lengths can accurately describe the total
seismicity produced in a region. The resulting earthquake
catalogue accounts for the addition of both all the character-
istic and all the non-characteristic events triggered in the faults.
The global accumulated size–frequency relationship strongly
depends on the fault length fractal exponent and, for fractal
exponents close to 2, correctly describes a Gutenberg–Richter
distribution with a b exponent compatible with real seismicity.
Terra Nova, 16, 116–120, 2004
Correspondence: Javier B. Go ´ mez, Depart-
ment of Earth Sciences, University of
Zaragoza, Pedro Cerbuna 12, 50009 Zar-
agoza, Spain. Tel.: +34 976762124; fax:
+34 976761106; e-mail: jgomez@unizar.es
116
? 2004 Blackwell Publishing Ltd
doi: 10.1111/j.1365-3121.2004.00538.x

Page 2

1 The incoming particles arrive at the
system at a constant rate. Thus, the
time interval between each two
successive particles will be consid-
ered the basic time unit in the
evolution of the system.
2 All the positions in the array, from
i ¼ 1 to i ¼ N, have the same prob-
ability of receiving a new particle.
When a position receives a particle
we say that it is occupied.
3 If a new particle arrives at a level
that is already occupied, this parti-
cle is dissipated. Thus, a given
position i can only be either non-
occupied when no particle has
arrived at it, or occupied when one
or more particles have arrived at it.
4 Level i ¼ 1 is special. When a par-
ticle arrives at this first position a
relaxation event occurs. Then, if all
the successive levels from i ¼ 1 up
to i ¼ k are occupied, and position
k + 1 is empty, the effect of the
relaxation – or earthquake – is to
unload all the levels from i ¼ 1 up
to i ¼ k. Hence, the size of this
relaxation is k, and the remaining
levels i > k remain unaltered in
their occupancy.
Thus, from what has been men-
tioned above, this model has no
parameter; the size N is the unique
specification to be made, and the
spatial correlation is induced by rule
4. The state of the system is given by
stating which of the (i > 1) N ) 1 lev-
els are occupied. Each of these states
corresponds to a stable configuration,
and therefore the total number of
possible configurations is 2N)1.
The prominent role given to the
level i ¼ 1 is equivalent to considering
that the asperity of the fault is located
there. Each relaxation empties the
lower levels of the system as explained
in rule 4, and the system is left in
another stable configuration. The size
of the earthquakes can thus range
from 1 up to N and the earthquake of
maximum size, k ¼ N, is termed the
characteristic earthquake.
Because the model is one dimen-
sional, extensive Monte Carlo simula-
tions can be performed to explore its
properties accurately. It can also be
studied, for small system sizes, from
the perspective of Markov chains
(Va ´ zquez-Prada et al., 2002).
The size–frequency relationship of
the MM is shown in Fig. 2. There, the
probability of occurrence of an event
of size k in a system of size N, pN(k), is
shown for N ¼ 10, N ¼ 100 and
N ¼ 1000. Note that this spectrum
has a distribution of the characteristic-
earthquake type: it exhibits a power-
law relationship for small events, an
excessofmaximal
events and very few of intermediate
size. The size–frequency relationships
obtained by Wesnousky (1994) for
the Southern San Andreas, Garlok,
Whittier–ElsinoreandNewport–Ingle-
wood faults are consistent with the
basic shape shown in Fig. 2. The gap
that can be seen in the size–frequency
relationship of most faults between the
small, non-characteristic earthquakes,
and the characteristic earthquakes ap-
pears in our model when considering
that the probabilities of intermediate-
size events are so low that they would
be absent when accumulating the seis-
micity over a few consecutive cycles
(as is done in real faults).
Besides, we note that in this model
(characteristic)
pNðkÞ ¼ pN0ðkÞif k < N andk < N0;
ð2Þ
i.e. the probability of having non-
characteristic earthquakes of size k is
Fig. 1 Layout of the Minimalist Model.
The vertical array of N cells is closed at
the top and open at the bottom. For
each time step a stress particle is added
randomly to the array and a relaxation
of size k occurs when the added stress
particle hits the lowermost level. k is the
number of consecutive levels, starting
from the bottom, that contain a stress
particle.
Fig. 2 Probability of occurrence of earthquakes of size k. Note that the simulations
corresponding to N equal to 10, 100 and 1000 are superimposed.
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a constant, independent of the size of
the system. This is a consequence of
the scale-invariance of the model for
non-characteristic earthquakes, which
can be derived from a Markov-chain
approach to the model (Va ´ zquez-
Prada et al., 2002). The property sum-
marized by Eq. (2) is apparent in
Fig. 2 by the perfect coincidence of
the probability curves for systems of
different size N.
Asymptotic behaviour of the
Minimalist Model
A direct consequence of Eq. (2) is the
following property of the MM:
pNðNÞ ¼ pNþ1ðN þ 1Þ þ pNþ1ðNÞ:
ð3Þ
The meaning of Eq. (3) is as follows:
the probability of occurrence of a
characteristic event in a system of size
N is equal to the sum of the probab-
ility of a characteristic event in a
system of size N + 1 plus the prob-
ability of a non-characteristic event of
size k ¼ N in a system of size N + 1.
It is easily derived from the property
that the sum of the probabilities of
earthquakes of all sizes is unity. For a
system of size N we have:
1 ¼ pNð1Þ þ pNð2Þ þ ::: þ pNðN ? 1Þ
þ pNðNÞ;
and for a system of size N + 1,
1¼pNþ1ð1ÞþpNþ1ð2Þþ:::þpNþ1ðN ?1Þ
þpNþ1ðNÞþpNþ1ðN þ1Þ:
Because the first N ) 1 terms are
identical in both sums, subtracting
one equation from the other gives
Eq. (3).
The most important consequence of
Eq. (3) is that if one knows pN(k) for
all 1 £ k < N, then one knows pM(M)
for 1 £ M < N. And vice versa, using
Eq. (3), the knowledge of pM(M) leads
to the knowledge of pN(k < N). In the
MM, the value of these two sets of
probabilities can be easily obtained,
for small N, by means of diagonaliza-
tions of the corresponding Markov
matrices, and in general by Monte
Carlo simulations. For large systems
one would expect regular behaviour in
the decay rate of these probabilities.
This is shown in Fig. 3. In panel
Fig. 3(a), pN(N) is fitted to the func-
tion:
pNðNÞ ¼ cðlogNÞ?b;
ð4Þ
where b¼ 0.95 and c is a constant.
Equation (4)impliesthattheprobability
of occurrence of a characteristic earth-
quake tends to zero as the system size
tends to infinity, although very slowly.
As a verification of the goodness of
this fit, in Fig. 3(b) we have plotted
the ratio pN(N)⁄pN¢(N¢) of the prob-
ability of a characteristic earthquake
for two different system sizes (N and
N¢) against the ratio (log N¢⁄log N)b
for b¼ 0.95. The points are colour-
(a)
(b)
Fig. 3 Asymptotic behaviour of the MM. (a) Logarithmic-decaying fit to the
probability of occurrence of the characteristic earthquake in a system of size N as a
function of N. (b) Plot of the ratio pN(N)⁄pN¢(N¢) against the ratio (log N¢⁄log N)b, for
b¼ 0.95, where N and N¢ are two different system sizes. If the logarithmic-decaying fit
is correct, all the data points in this graph should plot on the diagonal straight line of
slope 1, as is the case. In the legend, an offset of 1 means that the ratio pN(N)⁄pN¢(N¢)
is calculated for two consecutive data points. An offset of 2 is for ratios calculated for
next-nearest neighbour data points, and so on.
A model of characteristic earthquakes • R. Lo ´pez et al.
.............................................................................................................................................................
Terra Nova, Vol 16, No. 3, 116–120
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Page 4

coded with respect to the offset
between the values of N and N¢ used
to calculate the ratios pN(N)⁄pN¢(N¢)
and (log N¢⁄log N)b. As can be seen,
all the data points, regardless of the
offset, fall on the straight line of
gradient 1, indicating that Eq. (4) is
a good description of the asymptotic
behaviour.
Now, using Eqs. (3) and (4), always
for large N, we have
cðlogNÞ?b¼ cðlogðN þ 1ÞÞ?b
þ pNþ1ðNÞ;
and therefore
ð5Þ
pNþ1ðNÞ ¼
cb
NðlogNÞbþ1:
ð6Þ
Because of the property expressed by
Eq. (2), Eq. (6) gives the probability
of occurrence of all non-characteristic
earthquakes of size N in a system of
size larger than N, not just the prob-
ability of the second-largest earth-
quake in a system of size N + 1. So,
we can say that Eq. (6), together with
Eq. (4) for the probability of the
characteristic earthquakes, completely
defines the size–frequency relation of
the MM for large N. We make use
of both equations in the following
section.
Implications for regional
seismicity and discussion
Observational data seem to show that
the frequency–size
faults is fractal (Barton et al., 1986;
Hirata, 1989; Ouillon et al., 1996;
Bonnet et al., 2001). Specifically, the
results of Barton et al. (1986) in
Nevadaindicate
dimension of fault lengths has a mean
value of D ¼ 1.7. That means that the
number of faults of size N is propor-
tional to N–D. Using this information,
our result for the probability of occur-
rence of an earthquake of size k
originating in any of the faults of the
region, P(k), is
distribution of
thatthefractal
PðkÞ ? k?D
c
ðlogkÞb
N?D
þ
X
N¼kþ1
equation,
1
cb
kðlogkÞbþ1:
the
ð7Þ
In
which derives from Eq. (4), accounts
for the contribution of the charac-
thisfirst term,
teristic earthquakes originated in the
faults, and the second term corres-
ponds to the contribution of only the
non-characteristic earthquakes. This
second term, which derives from
Eq. (6), is the sum of the contribu-
tions to P(k) of all the faults whose
size is greater than k, i.e. the sum
extends to all fault sizes larger than
k.
After computing the sum in the
second term, we obtain
PðkÞ ? k?D
c
ðlogkÞbþ
cb
kðlogkÞbþ1
k1?D
D?1:
ð8Þ
Rearranging this equation, we obtain
PðkÞ ? k?D
c
ðlogkÞb
1 þ
b
D ? 1
1
logk
??
:
ð9Þ
When k is large, the second term
inside the brackets is small compared
with 1 (the 1⁄log k constant prefactor
b⁄(D ) 1) is of the order of unity),
and so Eq. (9) can be simplified to
PðkÞ ? k?D
c
ðlogkÞb:
ð10Þ
In other words, the net result is equiv-
alent to the product of a first factor
originating from the fault size distribu-
tion times a second factor originating
from the seismicity of the individual
faults. Note that the impact of the
second factor (the logarithmic term) in
the decay rate of P(k) is very small in
comparison with the first power-law
factor coming from the assumed fault
distribution. To check the validity of
Eq. (10), we have carried out Monte
Carlo simulations on a large number of
minimalist systems, acting simulta-
neously and distributed in size in the
fractal form N–Dwith D ¼ 1.7. The
resultisshowninFig. 4.Thegreylineis
the resulting non-cumulative size–fre-
quency distribution, and the black
curve the accumulated distribution (to
facilitate comparison with Eq. (1),
which is a cumulative distribution). As
shown in this figure, our result corres-
ponds to a Gutenberg–Richter rela-
tionship with an exponent of c. 0.75,
lowbut compatible
b-values. Note that the simulations do
take into account both the non-charac-
teristic and the characteristic earth-
quakes and, in that respect, they
represent the result of Eq. (9) and not
its simplified version, Eq. (10).
Hence, our main conclusions are:
(a) the Gutenberg–Richter law can
be obtained from the interplay of
with seismic
Fig. 4 Non-cumulative (grey line) and cumulative (black line) Gutenberg–Richter
relationship for earthquakes occurring in minimalist-model faults with a power-law
(fractal) size–frequency distribution with fractal dimension D ¼ 1.7. The slope of the
cumulative straight line is )0.75.
Terra Nova, Vol 16, No. 3, 116–120
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R. Lo ´pez et al. • A model of characteristic earthquakes
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Page 5

fractally size-distributed faults each
obeying a characteristic earthquake
relationship(which
maximalearthquakes
non-characteristic
each fault); and (b) if we assume that
earthquakes on a single fault follow a
Characteristic Earthquake distribu-
tion, and the faults do not interact,
then the slope b appearing in the
Gutenberg–Richter
regional seismicity generally reflects
the fractal distribution of fault sizes.
We aim, however, to study further
the properties of the model in the case,
more realistic in some respects, of a set
of interacting faults. The results pre-
sented here should serve as a baseline
to compare the non-interacting case
with any other interacting version, as
one can devise more than one way in
which the faults are made to interact.
We want to stress that the Minim-
alist Model is able to reproduce
several properties of real seismicity,
apart from the size–frequency rela-
tionship commented on here. These
are: (1) the model has a seismic-cycle
behaviour with foreshocks (see, for
example, Va ´ zquez-Prada et al., 2002,
2003); (2) the model has a stress
shadow (no characteristic earthquake
is possible when the number of time
steps, n, is lower than N); (3) the
model hasquiescence
quakes occur during a period of
average length N prior to the char-
acteristic earthquakes); (4) the model
can be used as a renewal model for
thetime-dependent
strong earthquakes in an earthquake
sequence (J. B. Go ´ mez and A. F.
Pacheco, unpublished observations);
and (5)themodel
importance of high-resistance patches
(asperities) in the fault as a necessary
ingredient forthe
Earthquake distribution.
includes
and
earthquakes
both
smaller
for
graphofthe
(no earth-
forecasting of
stressesthe
Characteristic
Acknowledgement
This work was supported in part by the
Spanish DGICYT
01798).
(ProjectBFM2002-
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