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1841
Bulletin of the Seismological Society of America, Vol. 95, No. 5, pp. 1841–1855, October 2005, doi: 10.1785/0120040181
Global Observational Properties of the Critical Earthquake Model
by C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Abstract The preshock (critical) regions of 20 mainshocks with magnitudes be-
tween 6.4 and 8.3, which occurred recently (since 1980) in a variety of seismotectonic
regimes (Greece, Anatolia, Himalayas, Japan, California), were identified and inves-
tigated. All these strong earthquakes were preceded by accelerating time-to-main-
shock seismic crustal deformation (Benioff strain). The time variation of the cumu-
lative Benioff strain follows a power law with a power value (m ⳱ 0.3) in very good
agreement with theoretical considerations. We observed that the dimension of the
critical region increased with increasing mainshock magnitude and with decreasing
long-term seismicity rate of the region. An increase of the duration of this critical
(preshock) phenomenon with decreasing long-term seismicity rate was also observed.
This spatial and temporal scaling expresses characteristics of the critical earthquake
model, which are of importance for earthquake prediction research. We also showed
that the critical region of an oncoming mainshock coincides with the preparing region
of this shock, where other precursory phenomena can be observed.
Introduction
The critical earthquake model is based on principles of
statistical physics and has been proposed (Sornette and Sor-
nette, 1990; Allegre and Le Mouel, 1994; Sornette and Sam-
mis, 1995) to explain accelerating intermediate magnitude
seismicity observed before strong mainshocks (Tocher,
1959; Mogi, 1969; Varnes, 1989; Sykes and Jaume, 1990;
Knopoff et al., 1996; Tzanis et al., 2000; among others).
According to this model, the physical process of generation
of these moderate magnitude shocks (preshocks) is consid-
ered a critical phenomenon, culminating in a large event
(mainshock), which is considered as the critical point. Such
behavior has been also supported by independent observa-
tions, which suggest that rupture in heterogeneous media is
a critical phenomenon (Vanneste and Sornette, 1992; La-
maignere et al., 1996; Andersen et al., 1997). Thus, the criti-
cal earthquake model is now supported by seismological ob-
servations, by principles of statistical physics, and rock
mechanics experiments.
Several researchers have recently investigated proper-
ties of the model (Bufe and Varnes, 1993; Bowman et al.,
1998; Brehm and Braile 1999; Jaume and Sykes, 1999; Run-
dle et al., 2000). Bufe and Varnes (1993) proposed a power
law for the time variation of the cumulative Benioff strain,
S (square root of energy), released by preshocks in the criti-
cal region:
m
S ⳱ A Ⳮ B(t ⳮ t ) (1)
c
where t
c
is the failure time (origin time of the mainshock)
and A, B, and m are model parameters. Bowman et al. (1998)
quantified the degree of deviation of the time variation of S
from linearity by proposing the minimization of a curvature
parameter, C, which is defined as the ratio of the root mean
square error of the power-law fit (relation 1) to the corre-
sponding linear fit error, and showed that the size of the
accelerating (critical) region scales with the mainshock mag-
nitude. Similar results were independently obtained by
Brehm and Braile (1998, 1999) and supported by mining-
induced seismicity and laboratory experiments (Quillon and
Sornette, 2000; Johansen and Sornette, 2000). Furthermore,
Zoller et al. (2001) and Zoller and Hainzl (2002) have per-
formed systematic spatiotemporal tests in an attempt to
quantify the predictive power of accelerated deformation re-
lease and the related growing of the spatial correlation
length. Despite the fact that the critical earthquake concept
has been questioned regarding the robustness of the reported
analyses (Gross and Rundle, 1998), several theoretical ap-
proaches (Ben-Zion et al., 1999; Rundle et al., 2000; Ben-
Zion and Lyakhovsky, 2002), as well as laboratory results
(Guarino et al., 1999, 2002) have verified the existence of
critical behavior, at least under some conditions.
Papazachos and Papazachos (2000, 2001) suggested the
use of elliptical critical regions and several constraints to the
critical earthquake model expressed by relations between pa-
rameters of the accelerating Benioff strain (seismic crustal
deformation) and the long-term mean rate of the seismic
deformation (seismicity release rate). As a representative
measure of the seismicity/seismic deformation, they used the
average annual Benioff strain rate in the critical region, S
r
(in J
1/2
/yr), as this is determined from the cumulative, long-
1842 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
term Benioff strain release for events with M ⱖ5.2. The use
of this specific cutoff magnitude value was imposed by the
fact that such events are usually available for a long-time
interval (in some areas almost for the whole twentieth cen-
tury) for many seismogenic regions worldwide, allowing the
reliable determination of the average seismic release rate for
each examined critical region. The most important of these
constraint relations are: (1) A relation between the logarithm
of the radius R (in km) of the circle with area equal to the
area of the elliptical critical region and the magnitude M
(moment magnitude) of the mainshock; (2) a relation be-
tween the logarithm of the duration t
p
(in years) of the pre-
shock sequence (that is, of the duration of the accelerating
seismic deformation) and the logarithm of the long-term rate
of seismic crustal deformation per unit area, S
r
(in J
1/2
/yr and
per 10 km
2
); (3) a relation between the log(A/t
p
) and log S
r
,
where A is the constant of relation (1); and (4) a relation
between the magnitude of the mainshock, M , and the mean
magnitude of the three largest preshocks, M
13
.
Observations for 18 mainshocks of the Aegean area (Pa-
pazachos et al., 2002a, b) showed that preshock accelerating
deformation can be identified in all cases with C ⱕ 0.6 and
m ⱕ 0.35. The C cutoff value observed for the Aegean is
slightly smaller than the cutoff value used by Bowman et al.
(1998). The average observed value for m for the Aegean
area is 0.3 (Papazachos et al., 2001), in accordance with
theoretical considerations and a large number of worldwide
observations (see Ben-Zion and Lyakhovsky [2002] for an
indicative summary of results from various studies). Papa-
zachos and Papazachos (2001) defined a parameter P as the
average value of the probability that each of the four param-
eters (R, t
p
, A, M) attains a value close to its expected one.
Furthermore, Papazachos et al. (2002a) chose a quality in-
dex, q, defined by the relation
P
q ⳱ (2)
mC
in an attempt to simultaneously evaluate the compatibility
of an investigated accelerating seismic deformation with the
behavior of past real preshock sequences (large P), the de-
gree of seismic acceleration (small m), and the fit of the data
to the power law (small C). From the investigation of the
same sample of 18 preshock sequences in the Aegean area,
the cutoff values of P ⱖ 0.45 and q ⱖ 3.0 have also been
determined (Papazachos et al., 2002a,b).
The main target of the present work is to investigate
properties of critical regions in active areas with different
seismotectonic conditions (Greece, Anatolia, Himalayas, Ja-
pan, California) by using the most recent and reliable data.
Furthermore, an attempt is made to derive globally valid
relations between model parameters that can be used as ad-
ditional constraints to the critical earthquake model. It is
shown that such work supports the validity of the critical
earthquake model and can contribute to the improvement of
methodology for intermediate-term earthquake prediction,
since certain basic properties of the model have a premoni-
tory character for the oncoming mainshock.
The terms “preshocks” and “postshocks” are used in the
present article in their broad sense. Thus, while foreshocks
and aftershocks are spatially distributed in or near the fault
(rupture) zone of the mainshock, and the duration of their
sequences is of the order of months, preshocks and post-
shocks are distributed in a broader region, with dimensions
that are typically an order of magnitude larger than the fault
length of the mainshock, while the duration of preshock and
postshock sequences is of the order of years or decades.
The choice of the critical region shape is of importance
for the identification of such regions, and several shapes such
as circular (e.g., Bufe and Varnes, 1993; Bowman et al.,
1998), elliptical (e.g., Papazachos and Papazachos 2000,
2001), predicted stress-drop pattern (e.g., Bowman and
King, 2001) have been used. Circular ones, traditionally
used in most studies, have the advantage that the probability
that parts of more than one preshock (critical) region will be
included in such a shape is small. There are, however, sev-
eral elongated seismic zones for which no solutions can be
found using circular critical regions and reliable solutions
are found only for elongated shapes (e.g., elliptical). This is
either due to data limitations and critical region overlap or
to the real necessity to use ellipses to approximate the true
critical region, which some studies have suggested have
complicated shapes (e.g., Bowman and King, 2001). Unfor-
tunately, elliptical regions include two additional degrees of
freedom (ellipticity, azimuth), hence increasing the proba-
bility of finding valid solutions even in random catalogs
(e.g., Papazachos et al., 2002a). For this reason, identifica-
tion of a critical region in a broad area must be tested for
both circular and elliptical shapes. Thus, for the 20 preshock
sequences studied in the present paper, for 16, both circular
and elliptical shapes gave valid solutions, while for the re-
maining 4 cases, which concern preshock sequences in elon-
gated seismic regions, only elliptical shapes gave valid so-
lutions.
Data and Procedure
Three samples of data are used for each of the five areas
considered in the present article. The first sample concerns
the mainshocks, the second the preshocks of each main-
shock, and the third one includes the shocks that define the
long-term mean seismicity in each critical region. Each sam-
ple must be complete (including all shocks that occurred
during a certain time period with magnitudes larger than a
certain value), large enough (to be representative) and ac-
curate (a property that is better satisfied for recent earth-
quakes). These properties differ from area to area; hence,
each area of interest was examined separately.
The main target of the data processing procedure was
to obtain homogeneous catalogs exhibiting similar charac-
teristics for all examined areas. For this reason, all magni-
Global Observational Properties of the Critical Earthquake Model 1843
tudes of earthquakes included in the final catalogs are mo-
ment magnitudes or equivalent to moment magnitudes, that
is, magnitudes that have been converted to moment magni-
tudes from other scales (i.e., M
s
, m
b
published by ISC and/
or
NEIC) by using appropriate formulas and/or graphs pro-
posed by Scordilis (2005). The finally adopted magnitude
for each earthquake is the weighted mean of the values cal-
culated by the above transformations by taking as weight for
each magnitude value the inverse of the standard deviation,
r, of the corresponding transformation formula. A similar
procedure was successfully applied in order to create a ho-
mogeneous earthquake catalog, which was used in a retro-
spective study of accelerated crustal deformation in the Adri-
atic (Scordilis et al., 2004).
An example of the adopted procedure is shown in Fig-
ure 1a, where the equivalent moment magnitude, , com-
*
M
W
puted from other magnitudes (M
S
, m
b
, M
L
) is plotted against
original M
W
determinations for more than 700 events in
California for which both estimates were available. In gen-
eral a very good correlation between the two magnitudes is
observed with the standard error being less than 0.2, sug-
gesting that the equivalent moment magnitude can be reli-
ably used in the present study.
Furthermore, in all cases the data completeness has been
checked using both, the frequency–magnitude distribution
and the cumulative frequency–magnitude relation. Several
time periods have been tested to assure the data complete-
ness for each region. For instance, the data completeness for
the broader Himalayas area for the time period from 1965
until 2003 has been checked for seven subperiods. It has
been finally determined that the data are complete for earth-
quakes of M ⱖ4.7 since 1965 and M ⱖ4.2 since 1981 (Fig.
1b and 1c).
In order to ensure the reliability of the data set used,
only mainshocks that occurred during the last two decades
or so (1980–2002) were considered. The starting time was
changed to 1981 for Greece (date of full operation of the
first digital telemetric network in Greece) and 1990 for the
Himalayas, which is the only area for which a regional cat-
alog source was not available; hence we limited the search
to the last decade in order to use the most reliable main-
shock–preshock–postshock information.
The main sources of all the catalogs used in the present
work were the bulletins of the International Seismological
Centre (
ISC, 2005) and of the National Earthquake Infor-
mation Center (NEIC, 2005), as well as the CMT solutions
catalog of Harvard (Harvard Seismology, 2005), which were
used to create the master catalog for each area until the end
of 2002. Using this master catalog for the Himalayas and
surrounding area (20⬚N–45⬚N, 60⬚E–90⬚E) four mainshocks
with M ⱖ 7.0 that occurred since 1990 were considered. The
final data sample used concerns shocks with M ⱖ4.7 that
occurred since 1965.
However, for several of the examined areas, we also
used individual catalogs available mainly from reliable local
sources, which were merged with the master catalog. Spe-
cifically, for the Mediterranean region (Greece and Anatolia,
28⬚N–55⬚N, 15⬚W–46⬚E), data from the catalogs of Euro-
pean earthquakes compiled by Karnik (1996), which was
complemented by the catalog of Papazachos et al. (2003)
for earthquakes in Greece and surrounding areas, as well as
the bulletins of National Observatory of Athens and the Geo-
physical Laboratory of the University of Thessaloniki have
been used. Using the final catalog for the area of Greece and
surrounding region (34⬚N–42⬚N, 19⬚E–28⬚E), all six main-
shocks with M ⱖ6.4 that occurred between 1981 and 2002
were considered. The following three complete samples of
data have been used: 1911–2002, M ⱖ5.2 (used for calcu-
lation of the mean seismic deformation); 1950–2002, M
ⱖ5.0; 1965–2002, M ⱖ4.5 (used for estimation of the pre-
shock seismic deformation). Similarly, for Anatolia (35⬚N–
42⬚N, 28⬚E–42⬚E), three mainshocks that occurred during
the last decade with magnitude M ⱖ6.4 were considered.
The following three complete samples have been used:
1911–2002, M ⱖ5.2; 1950–2002, M ⱖ5.0; 1965–2002,
M ⱖ4.5.
For the area of Japan (30⬚N–50⬚N, 130⬚E–150⬚E), the
catalog has been enriched with data of the Japan Meteoro-
logical Agency (
JMA), in order to obtain additional infor-
mation for low-magnitude earthquakes. Four mainshocks
with M ⱖ7.3 that occurred since 1980 were considered. Ex-
amination of completeness showed that the data were com-
plete for M ⱖ5.0 since 1926.
Finally, for California (33⬚N–42⬚N, 125⬚W–114⬚W),
additional data from
NEIC regarding significant U.S. earth-
quakes, as well as moment magnitudes from the USGS
(Earthquake Hazards Program) have also been used. Fur-
thermore, additional data have been extracted from the Ad-
vanced National Seismic System (
ANSS) composite earth-
quake catalog (http://quake.geo.berkeley.edu/anss), which
includes contributions from the Southern California Seismic
Network, the Seismographic Station of the University of
California at Berkeley, and the Northern California Seismic
network, among other contributing members. Earthquake
magnitudes for California in the
ANSS catalogue are given
in various scales (M
s
, m
b
, M
L
, M
w
, M
c
, M
d
, etc.). For ex-
ample, about 65% of the M ⱖ3.0 events have M
L
or M
c
(which is essentially equal to M
L
, [Felzer et al., 2002]) mag-
nitudes. All magnitudes were also converted to equivalent
moment magnitudes using appropriate relations (Scordilis,
2005). Finally, four mainshocks with M ⱖ7.0 that occurred
since 1980 were considered, while the examination for com-
pleteness showed that the data were complete for M ⱖ4.8
since 1930.
The involved uncertainties are typically less than 30 km
for the epicenter and less than 0.3 for the equivalent moment
magnitude. Therefore, for 20 normal-depth mainshocks that
occurred since 1980 and had magnitudes between 6.4 and
8.3 (Table 1), reliable data (magnitude, epicenters of pre-
shocks, etc.) were available, and these data are used to define
parameters of their critical (preshock) regions. Although a
few preliminary results for some intermediate-depth events
1844 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Figure 1. (a) Comparison of the equivalent moment magnitudes, , determined
*
M
W
from various magnitude sources (M
S
, m
b
, M
L
) with original M
W
estimations using the
relations of Scordilis (2005) for the catalog used for the broader California area. (b, c)
Frequency–magnitude and cumulative frequency–magnitude distributions used for the
Himalayas area to define the data completeness for two time periods: 1965–1980 (b)
and 1981–2003 (c).
indicate that they also exhibit a similar accelerated defor-
mation preshock behavior, we decided to exclude them from
the present work owing to the limited number of associated
preshocks, as well as their special characteristics, and leave
their study for future work.
To identify an elliptical critical (preshock) region of an
already occurred mainshock, the algorithm of Papazachos
(2001) has been used. According to this algorithm, the broad
seismic zone (e.g., of dimensions 400 km ⳯ 400 km), where
the mainshock is located (e.g., Hokkaido zone in Japan) is
divided using a grid with the desired density (e.g., 0.2⬚ NS,
0.2⬚ EW). Each point of the grid is considered as the center
of the elliptical (or circular) critical region, and magnitudes
of shocks (preshocks) with epicenters in this region are used
to calculate parameters of relation (1) and the curvature pa-
rameter C. Calculations for each point of the grid are re-
peated for a large set of values for the azimuth, z,ofthe
large ellipse axis, its length, a, ellipticity, e, the start time,
t
s
, of the preshock sequence, the minimum preshock mag-
nitude, M
min
, and the magnitude of the mainshock (starting
Global Observational Properties of the Critical Earthquake Model 1845
Table 1
Information on the Dates, Geographic Coordinates of the
Epicenter, E, and Magnitudes of the 20 Mainshocks for which
Data Have Been Used in the Present Study
Area Date (t
c
) E(
, k)MQ(
1
, k
1
)
Greece 1 1981, 02, 24 38.2, 22.9 6.7 40.8, 21.9
2 1981, 12, 19 39.0, 25.3 7.2 41.0, 22.8
3 1983, 01, 17 38.1, 20.2 7.0 39.0, 19.8
4 1995, 05, 13 40.2, 21.7 6.6 38.8, 21.0
5 1995, 06, 15 38.4, 22.2 6.4 38.7, 21.2
6 1997, 11, 18 37.6, 20.6 6.6 39.4, 21.1
Anatolia
7 1995, 10, 01 38.1, 30.2 6.4 38.4, 28.8
8 1999, 08, 17 40.8, 30.0 7.4 39.6, 28.2
9 2002, 02, 03 38.7, 31.2 6.5 38.6, 29.6
Himalayas
10 1992, 08, 19 42.1, 73.6 7.2 41.4, 72.2
11 1997, 02, 27 30.0, 68.2 7.1 26.8, 66.2
12 1997, 11, 08 35.1, 87.4 7.5 35.0, 90.8
13 2001, 01, 26 23.4, 70.3 7.6 26.4, 72.2
Japan
14 1983, 05, 26 40.4, 139.1 7.7 40.2, 139.4
15 1993, 07, 12 42.8, 139.2 7.7 41.2, 141.0
16 1994, 10, 04 43.7, 147.7 8.3 44.6, 146.4
California
17 1980, 11, 08 41.1, 124.6 7.3 38.4, 125.2
18 1989, 10, 18 37.0, 121.9 7.0 38.0, 117.0
19 1992, 04, 25 40.3, 124.2 7.2 38.9, 122.4
20 1992, 06, 28 34.2, 116.4 7.3 34.8, 120.8
The geographic coordinates are also given for the point, Q, which cor-
responds to the best solution and is considered as the center of the critical
region.
typically from 6.2, up to the magnitude of the largest earth-
quake in the zone). Calculations were initially performed
with variable m value, allowing a wide range of m values
between 0.1 and 0.5. The final calculations were made using
a constant m value equal to 0.3, as this was determined from
the average m value, in very good agreement with theoretical
values and laboratory results ranging between 0.25–0.33
(Ben-Zion et al., 1999; Guarino et al., 1999; Rundle et al.,
2000; Ben-Zion and Lyakhovsky, 2002) and standard prac-
tice in similar studies (e.g., Zoller and Hainzl, 2002; Rundle
et al., 2003). All grid points with valid solutions (using the
cutoff determined for the Aegean area, C ⱕ 0.60, with m
fixed
⳱ 0.30, and a minimum number of 20 preshocks) are con-
sidered and the geographical point, Q, with the smallest C
value was considered as the center of the critical region,
while the solution (q, C, M, a, z, e, M
min
, n, t
s
) for this point
was adopted as the best solution. The application of this
algorithm was repeated for all 20 cases by considering cir-
cular regions, and it was found that the results are very simi-
lar for 16 cases. For this reason, the solution for the simplest
shape (circular) was adopted in these 16 cases. For the other
four cases, no valid solutions were found when circular
shapes were used. For this reason, the best elliptical solution
for each of these four cases was adopted. One of these cases
concerns a mainshock along the elongated zone of north
Anatolia (case with code number 8 in Tables 1 and 2, with
ellipticity e ⳱ 0.9 and major-axis azimuth z ⳱ 60⬚), and
three concern mainshocks in the Indian plate boundary (case
10 with e ⳱ 0.9, z ⳱ 0⬚; case 11 with e ⳱ 0.7, z ⳱ 120⬚;
and case 12 with e ⳱ 0.7, z ⳱ 0⬚).
Results
Application of the previously described algorithm for
the 20 critical regions resulted in the calculation of appro-
priate model parameters, which are listed in Tables 1 and 2.
These parameters can be used to derive several empirical
relations. Figures 2–6 show the critical regions (circular or
elliptical), while the inset figures show the corresponding
time variations of the cumulative Benioff strain.
From the estimated values of M* and the minimum pre-
shock magnitude, M
min
(Table 2) for which the best solution
(smallest C value) was obtained, the following relation can
be derived (Fig. 7):
M* ⳮ M ⳱ 0.54M* ⳮ 1.92, r ⳱ 0.08 , (3)
min
which is similar to the equation obtained by Papazachos
(2003). We made a further attempt to examine the effect of
the variation of the minimum preshock magnitude on the
results, and we observed that for each mainshock there was
a lower as well as a higher cutoff minimum preshock mag-
nitude for which valid solutions (C ⬍ 0.60, m
fixed
⳱ 0.3)
still existed. Thus, for the Japanese mainshock of 4 October
1994, M 8.3, valid solutions were obtained only for M
min
between 5.5 and 5.9. The higher minimum-magnitude limit
was clearly a result of the minimum number of preshocks
(typically 20), which we imposed in the optimization pro-
cedure in order to ensure the reliability of the obtained re-
sults (Papazachos and Papazachos, 2000, 2001). Further-
more, the lower minimum magnitude verifies the theoretical
results (Rundle et al., 2000), as well as independent obser-
vations (e.g., Jaume, 2000; Karakaisis et al., 2002) that such
a minimum magnitude exists as a slope change above M
min
in the Gutenberg–Richter distribution during the preshock
period. Equation (3) verifies that it is the accelerating gen-
eration of intermediate-magnitude preshocks that character-
izes the critical phenomenon. This observation also explains
why the time variation of the crustal deformation (Benioff
strain) better expresses the critical phenomenon (Bufe and
Varnes, 1993) than other quantities, such as the frequency
of shocks, which is dominated by a large number of small
shocks, or the seismic moment, which is dominated by the
magnitudes of the largest preshocks.
Extending the initial results of Bowman et al. (1998),
Papazachos and Papazachos (2000) have shown that the ra-
dius, R (in km), of the circle with area equal to the area of
the elliptical critical region, scales with the magnitude, M,
of the mainshock according to a relation of the form:
logR ⳱ 0.42M Ⳮ a (4)
m
1846 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Table 2
Information on the Critical Regions of the 20 Mainshocks
Area M* CqARM
13
log S
r
Log s
r
M
min
nt
s
t
i
Greece 1 7.0 0.43 4.6 0.16 247 6.4 7.18 5.90 5.1 33 1972 1980.7
2 7.4 0.41 6.0 0.40 368 6.8 7.38 5.91 5.4 41 1969 1980.7
3 7.4 0.44 6.1 0.52 348 6.8 7.50 5.87 5.4 42 1968 1980.7
4 6.6 0.35 7.9 0.21 167 6.0 7.21 6.27 4.9 48 1984 1993.1
5 6.5 0.57 4.6 0.18 139 6.0 7.13 6.35 4.9 43 1984 1993.0
6 6.9 0.45 6.5 0.34 216 6.4 7.31 6.15 5.1 60 1984 1994.8
Anatolia
7 6.4 0.24 8.8 0.13 207 5.8 7.01 5.88 4.9 39 1983 1993.0
8 7.4 0.25 7.0 0.96 621 6.8 7.88 5.80 5.2 124 1986 1996.9
9 6.6 0.49 4.7 0.21 234 6.1 6.93 5.69 4.9 62 1981 1995.9
Himalayas
10 6.9 0.42 5.0 0.39 373 6.2 7.39 5.75 5.1 69 1979 1990.4
11 7.1 0.59 5.4 0.34 447 6.5 6.91 5.11 5.2 43 1962 1990.7
12 7.2 0.44 4.6 0.31 454 6.3 6.86 5.05 5.2 37 1961 1994.9
13 7.6 0.49 5.2 0.49 735 7.0 7.09 4.89 5.5 42 1956 1992.4
Japan
14 7.7 0.52 5.3 0.67 393 7.2 8.02 6.43 5.5 76 1972 1981.6
15 7.5 0.47 5.2 0.12 336 7.1 8.03 6.48 5.1 180 1984 1990.8
16 8.1 0.46 4.9 2.60 614 7.5 8.35 6.32 5.6 121 1983 1993.2
California
17 6.9 0.36 5.0 0.69 356 6.1 6.87 5.59 5.1 32 1962 1977.1
18 7.1 0.34 8.3 0.52 458 6.4 7.22 5.48 5.1 84 1962 1983.5
19 7.6 0.48 5.9 0.98 726 7.1 7.46 5.39 5.5 86 1964 1984.3
20 7.3 0.32 7.8 0.90 529 6.7 7.35 5.50 5.3 89 1955 1988.3
M* is the predicting magnitude, C is the curvature parameter, q is the quality index, A (in 10
9
J
1/2
)isthe
parameter of the power-law relation (1), R (in km) is the radius of the circular critical regions (or the radius of
the circle with area equal to the area of the elliptical regions for cases 8,10,11,12), M
13
is the average magnitude
of the three largest preshocks, S
r
(in J
1/2
/yr) is the long-term Benioff strain rate in the critical region, s
r
is the
same quantity per 10
4
km
2
, M
min
is the smallest preshock magnitude, n is the number of preshocks, t
s
the start
year of the preshock sequence, t
i
is the identification time.
This relation fits a large sample of data (mainshock magni-
tude range 4.6–8.6), which concerns elliptical critical re-
gions in the Aegean area as well as data for California and
other areas presented by Bowman et al. (1998). The values
of R and M* presented in Table 2 have been used to calculate
the values of a
m
for each of the twenty mainshocks listed in
this table. The mean a
m
value for each of the five areas is
listed on Table 3, where the corresponding mean deforma-
tion rate, s
r
(in J
1/2
/yr and per 10
4
km
2
) is also given. The
plot of a
m
as a function of s
r
is presented in Figure 8, where
a linear relation can be identified (a
m
⳱ 1.38 ⳮ 0.32 log s
r
,
r ⳱ⳮ0.95). Thus, relation (4) becomes:
log R ⳱ 0.42M* ⳮ 0.32log s Ⳮ 1.38, r ⳱ 0.11 . (5)
r
This equation suggest that the size of a critical region scales
with both the magnitude of the mainshock, M*, and the
long-term rate of seismic crustal deformation, s
r
, in the criti-
cal region. In particular, this dimension increases with the
magnitude of the mainshock but decreases with the rate of
seismic crustal deformation. Hence, the size of a critical re-
gion, which corresponds to a mainshock of certain magni-
tude is, for example, smallest for Japan, where the crustal
deformation rate is high, larger for Greece, and much larger
for Anatolia, California, and the Himalayas (Fig. 8).
Dobrovolsky et al. (1979) adopted the model of an elas-
tic soft inclusion in a more rigid elastic space in order to
determine the region of precursory deformation for a future
earthquake. In their model, they assumed that this region is
centered at the epicenter of the oncoming mainshock and
showed that the mean radius, r (in km), of the region is
related to the magnitude, M, of the expected earthquake by
the formula:
log r ⳱ 0.43M ⳮ 0.33log e ⳮ 2.73, M ⱖ 5.0 (6)
where e is the greatest principal strain. Comparison of their
theoretical results with observations on several precursory
phenomena (geochemical, resistance, telluric, radon, light
effects) showed that most precursors were observed for dis-
tances corresponding to a strain levels, e, between 10
ⳮ6
and
10
ⳮ8
. The similarity between relation (5), which gives the
radius of the critical region, and relation (6), which gives the
size of the preparing region, is striking since in both relations
the radius scales with the magnitude of the oncoming main-
shock and with the deformation measure with almost iden-
tical coefficients. Furthermore, by using the values of the
parameters a
m
of relation (4) given in Table 3, we can obtain
the corresponding values of e (relation 6), which vary be-
Global Observational Properties of the Critical Earthquake Model 1847
Figure 2. The circular critical regions associated with the preshock accelerating
seismic sequences of six mainshocks (M ⱖ6.4) which occurred in Greece. Preshock
epicenters are shown by small circles, and the mainshock epicenter is denoted by a
star. Numbers and dates correspond to the code numbers and dates in Tables 1 and 2.
The inset figures display the time variation of seismic deformation (Benioff strain).
1848 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Figure 3. One elliptical and two circular critical regions associated with preshock
accelerating seismic sequences of three mainshocks (M ⱖ6.4), which occurred in An-
atolia (Turkey). Symbols are as in Figure 1.
tween 10
ⳮ6.3
and 10
ⳮ7.3
, in agreement with the values of e
for which precursory phenomena are observed.
The similarity of equations (5) and (6) should take into
account that, despite the fact that Benioff strain is a stable
quantity to use in similar studies, as early works (e.g., Var-
nes, 1989; Bufe and Varnes, 1993) have shown, it is difficult
to assign a clear physical meaning to Benioff strain owing
to its divergence when smaller magnitude events are grad-
ually included. However, Benioff strain is finite if only
events above a threshold value (e.g., M
min
in this work) are
considered, which are those events that contribute to the ac-
celerated deformation phenomenon examined in the present
study (equation 3). In this case and for several models, it
can be shown that the Benioff strain is proportional to the
classical strain, e, (e.g., Ben-Zion and Lyakhovsky, 2002),
hence log s
r
and log e have a simple constant shift. In any
case, the similarity of equations (5) and (6) shows that for a
specific model (Dobrovolsky et al., 1979), the scaling rela-
tions have a similar dependence not only on the mainshock
magnitude but also on the loading/seismicity rate, as is also
shown by real data. An important point is that this similarity
also suggests that the critical (preshock) region of an oncom-
ing mainshock coincides with its preparing region, where
other geophysical precursors are also observed.
Table 3 presents the mean values of the durations, t
p
(in
years), of the preshock sequence for each of the five areas.
The plot of t
p
versus s
r
in Figure (9) shows that the duration
of preshock sequences scales with the rate of seismic crustal
deformation according to the relation
log t ⳱ 3.87 ⳮ 0.45log s , r ⳱ 0.10. (7)
pr
Thus, for Japan (highest s
r
values), the duration of the pre-
shock sequence is of the order of 11 years and increases for
Global Observational Properties of the Critical Earthquake Model 1849
Figure 4. Three elliptical regions and one circular critical region associated with
the preshock accelerating seismic sequences of four mainshocks (M ⱖ7.1), which
occurred in the Himalayas area; symbols are as in Figure 1.
lower deformation rates, becoming about 13 years for
Greece, 15 years for Anatolia, 27 years for California, and
35 years for the Himalayas.
From the values of A, t
p
, and S
r
presented in Table 2,
we find that
A
log ⳱ 1.01log S , r ⳱ 0.04 , (8)
r
冢冣
t
p
which is similar to the relation derived for the Aegean area.
This relation suggests that the mean rate of deformation,
A/t
p
, during the accelerating deformation in the critical re-
gion is almost equal to the long-term rate of seismic defor-
mation, S
r
, in the same critical region.
From the values of M* and M
13
given in Table 2, the
following relation can be derived:
M* ⳱ M Ⳮ 0.60, r ⳱ 0.15. (9)
13
This simple formula constitutes an important constraint to
the critical earthquake model and can be used for an inde-
pendent estimation of the magnitude of an ensuing main-
shock. A linear relation between the magnitude of a main-
shock and the mean magnitude of the three largest shocks
of preshock swarms has also been proposed by Evison and
Rhoades (1997).
We should note that the results previously presented
were recomputed using the highest q value (equation 2),
where the smallest C value is combined with the highest P
value, as this is determined from the compatibility of the
observed accelerated deformation behavior with the prelim-
inary form of equations (5), (7), (8), and (9). This is clearly
seen in Figure 10, where the spatial distribution of the C and
Q values is presented for event 15 of Table 1. A much
sharper peak of the q value is observed close to the epicenter
than of the C value, verifying the usefulness of the q param-
eter in the search for valid accelerated deformation patterns.
1850 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Figure 5. The circular critical regions associated with the preshock accelerating
seismic sequences of three mainshocks (M ⱖ7.3), which occurred in Japan; symbols
are as in Figure 1.
Valid accelerating seismic deformation (C ⬍ 0.60, m ⳱
0.30) cannot be identified until a certain time, t
i
(before the
origin time of the mainshock), which can be considered as
the identification time. This is the earliest time up to when
the available data of a preshock sequence give a valid so-
lution, and it is usually associated with a seismic excitation
(Papazachos et al., 2001). To estimate the identification time
of a preshock sequence before the generation of the main-
shock, the following procedure is applied. Considering sev-
eral assumed values, T
c
, for the unknown origin time of the
mainshock (e.g., in steps of 1 month), we let the program
try to fit relation (1) to the data until the first valid solution
is obtained. At this time, the assumed origin time has a value,
T
i
, which corresponds to a value, t
i
, of the time up to when
preshock data have been used and which is the identification
time. In the last column of Table 2, the values of t
i
deter-
mined by this procedure are given for each one of the 20
preshock sequences. From the values of t
i
given in Table 2,
it is clear that the difference t
c
ⳮ t
i
also scales with the long-
term seismicity rate s
r
, with a scaling coefficient (slope)
similar to the total preshock duration, t
p
(relation 7). The
scaling coefficient of the corresponding linear relation is
ⳮ0.43, hence practically identical to that of equation (7),
which applies for the total preshock duration. For this rea-
Global Observational Properties of the Critical Earthquake Model 1851
Figure 6. The circular critical regions associated with the preshock accelerating seis-
mic sequences of four mainshocks (M ⱖ7.0) in California; symbols are as in Figure 1.
son, we adopted the same scaling coefficient (ⳮ0.45), and
from the listed values the following relation was obtained:
log(t ⳮ t ) ⳱ 3.08 ⳮ 0.45log s , r ⳱ 0.24. (10)
ci r
Comparing equation (10) with equation (7) shows that the
ratio of the duration of the identification period, t
c
ⳮ t
i
,is
⬃16% of the total duration, t
p
, of the preshock sequence, in
agreement with previous results concerning preshock se-
quences in the Aegean area (Papazachos et al., 2002b), as
well as with the value (0.17) determined by an independent
method and data (Yang et al., 2001). Relation (10) can take
the form
t ⳱ t Ⳮ exp(7.09 ⳮ 1.04log s ), r ⳱ 1.7 years, (11)
ci r
which can be also used to estimate the origin time, t
c
,ofan
oncoming mainshock.
It is interesting to examine the average distance of the
point where the highest q value is observed from the true
1852 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Figure 7. Variation of the difference, M ⳮ M
min
,
of the mainshock magnitude, M, and the minimum
preshock magnitude, M
min,
as a function of M, for the
M
min
value for which the best solution (highest ac-
celerating deformation) is obtained.
Table 3
Mean Values for the Logarithm of the Seismic Deformation Rate,
s
r
(in J
1/2
per year and per 10
4
km
2
); the Parameter a
m
of Relation
(5); and the Duration, t
p
(in years) of the Preshock Sequences for
Each of the Five Examined Areas
Area log s
r
a
m
t
p
Greece 6.08 Ⳳ 0.19 ⳮ0.55 Ⳳ 0.03 12.5 Ⳳ 2.0
Anatolia 5.71 Ⳳ 0.15 ⳮ0.36 Ⳳ 0.03 15.0 Ⳳ 4.3
Himalayas 5.20 Ⳳ 0.33 ⳮ0.32 Ⳳ 0.03 34.8 Ⳳ 8.2
Japan 6.41 Ⳳ 0.07 ⳮ0.68 Ⳳ 0.03 11.0 Ⳳ 1.0
California 5.49 Ⳳ 0.07 ⳮ0.34 Ⳳ 0.01 27.0 Ⳳ 5.3
Figure 8. Variation of parameter a
m
(relation 5)
with the seismic deformation rate, s
r
.
Figure 9. Variation of the duration, t
p
, of the pre-
shock sequence with the seismic deformation (Beniof
strain rate), s
r
.
epicenter. Examination of the available results showed that
this distance does not scale with the magnitude of the earth-
quake but only with the seismicity, as this is expressed by
the Benioff strain rate, s
r
, according to the relation
x ⳱ 1150 ⳮ 160log s , r ⳱ 60 km . (12)
r
Discussion
In the present work, where properties of preshock se-
quences of known mainshocks are investigated, the basic
parameter used to quantify the accelerating time variation of
Benioff strain is the curvature parameter C, which expresses
the deviation from linearity and the degree of acceleration.
Thus, the small positive values of C (ⱕ0.60) observed for
all 20 preshock sequences, which occurred in a variety of
seismotectonic regimes, indicate that this accelerating seis-
micity is a systematic premonitory pattern.
The fitting of the data by a power law with a power
value (m ⳱ 0.3) predicted by theory and the excellent agree-
ment between the empirically derived scaling relation of the
spatial dimension of the critical region (relation 5) with a
theoretically obtained relation (6) give support to the critical
earthquake model and a physical meaning to the preshock
accelerating seismicity.
Global Observational Properties of the Critical Earthquake Model 1853
Figure 10. Comparison of the spatial variation of the C (a) and q (b) values for the
1993 M 7.7 Hokkaido Nansei-Oki earthquake in Japan (event 15 in Table 1). A much
sharper high-value peak close to the epicenter (solid circle) is observed for the q dis-
tribution.
The results of the present work can be of importance
for intermediate-term earthquake prediction research, be-
cause the magnitude, the origin time, and their uncertainties
for an oncoming mainshock can be estimated by relations
that express properties of the critical earthquake model. By
such relations, useful information can also be obtained for
the epicenter of an oncoming mainshock, which can lead to
the estimation of the epicenter coordinates and their uncer-
tainties, in combination with other data.
The magnitude of an ensuing mainshock can be esti-
mated by relations (5) and (9). Comparison of the magni-
tudes estimated by this method (M* in Table 2) with ob-
served magnitudes (M in Table 1) shows differences ranging
between ⳮ0.4 and 0.4, with a mean difference about zero
and a standard deviation of 0.22. This result indicates that
the magnitude of an oncoming mainshock can probably be
estimated with a maximum error of Ⳳ 0.4 with a high con-
fidence (⬃ 95%).
Relations (10) and (11) can be used to estimate the or-
igin time of an ensuing mainshock because the identification
time, t
i
, can be estimated from properties of the preshock
sequence. The uncertainties in these relations (r ⳱ 1.7 yrs)
indicate that the origin time of the ensuing mainshock can
probably be estimated by this method with an error of Ⳳ3.4
yrs with high confidence (⬃95%).
It should be pointed out that there are a few additional
events in the specific time–magnitude–space windows pre-
viously defined for each study area, for which results are not
presented in this work. In most cases (e.g., five cases in
Greece, one in Anatolia, and one in Japan), these events are
normal aftershocks or foreshocks (one case in Greece),
where the process is controlled by the mainshock and no
accelerated deformation pattern can be identified and sepa-
rated for these events. The same is true for the 16 October
1999, M 7.1, Hector Mine event in California, which can be
considered as a late aftershock of the Landers 1992, M 7.3,
mainshock. Moreover, no identification of accelerated de-
formation was possible for the 28 December 1994, M 7.7,
event, which occurred immediately after the big 4 October
1994, M 8.3, Shikotan event in Japan (event 16 in Table 1)
within its critical region, as well as for the 1 November 1989,
M 7.4 event, which again fell in the critical region of the 26
May 1983, M 7.7 (event 14 in Table 1), and occurred within
a few years after the previous larger event (M 7.7) within
its critical region. Hence, it appears that it is impossible to
identify an accelerated deformation pattern for smaller mag-
nitude events within the critical region or for a few years
after the generation of a large mainshock such as the big
Shikotan 1994 event, as the pattern of the mainshock dom-
inates and does not allow the grid-search algorithm to iden-
tify and separate the phenomena for the later, smaller mag-
nitude event. This observation sets a limit on the possible
predictive ability of the described pattern, as the method is
blind for a few years after the generation of a large main-
shock, at least within its critical region.
The information given in the present work shows that
identification of accelerated deformation is not enough for
locating the epicenter of an oncoming mainshock, with a
reasonable accuracy, since the epicenter tends to lie at a dis-
tance (equation 12) from the best solution (point Q). Similar
results have been obtained by other researchers (Robinson,
2000), who have examined cases where the center of the
critical region (maximum q value in our grid search) does
not coincide with the mainchock epicenter. A possible physi-
cal explanation for this bias (relation 12) could be the fact
that a large number of observations suggest that decelerating
seismic deformation (quiescence) occurs in the narrower
rupture zone of an ensuing mainshock (Wyss et al., 1981;
1854 C. B. Papazachos, G. F. Karakaisis, E. M. Scordilis, and B. C. Papazachos
Figure 11. Variation of the cumulative Benioff
strain for the narrower seismogenic source region of
the 1993 M 7.7 Hokkaido Nansei-Oki earthquake in
Japan (elliptical area in Fig. 10). A clear decelerating
pattern is observed, in contrast to the accelerating de-
formation pattern of the broader critical region (cir-
cular area in the upper right part of Fig. 5).
Wyss and Habermann, 1988; Hainzl et al., 2000; Zoller et
al., 2002; Papazachos et al., 2005). Thus, the epicenter zone
of the mainshock where decelerating deformation occurs
during the critical period has the tendency to push the best
solution obtained from the grid-search algorithm away from
the center of the region where accelerating deformation oc-
curs (see Fig. 10), as this is seen by relation (12). This is
clearly seen in Figure 11, where the variation of the cumu-
lative Benioff strain is shown for the same event examined
in Figure 10, for a narrower seismogenic source area (ellip-
tical region in Fig. 10). A clearly decelerating pattern is ob-
served, and we can easily apply equation (1) with an expo-
nent m ⬎1 in an attempt to model this decelerating
(quiescence) behavior. This observation and equation (12)
suggest that the results obtained in this work should be com-
bined with independent information (decelerating seismicity
in the rupture zone, location of active faults, etc.) to locate
the epicenter of an oncoming mainshock with a higher ac-
curacy.
The procedure followed in the present work must be
applied to future mainshocks in order to test its capability
more objectively. It must be noticed that the identification
of the critical region of an ensuing mainshock has to be
based on the spatial distribution of the values of the quality
parameter q(⳱ P/[C*m]), because it expresses the degree of
accelerating seismicity (through C), the degree of fit of ob-
servations to a power law (through m), and the degree of
agreement of properties of the examined region to preshock
properties expressed by relations (5, 7, 8, and 9) through P.
We should also point out that the critical region of an
oncoming mainshock, where a preshock accelerating defor-
mation is observed, coincides with the preparing region for
this earthquake, where other precursory phenomena are ob-
served. This is of importance from a practical point of view,
because these regions, which are easily defined by the
method presented in the present article, can be searched to
identify other precursors of oncoming mainshocks.
Acknowledgments
We would like to thank two anonymous reviewers for their construc-
tive comments and suggestions, which helped to improve our work. Thanks
are also due to Wessel and Smith (1995) for their generous distribution of
the GMT software, which was used to generate some of the figures of this
study. This work has been partly financed by the Greek Earthquake Plan-
ning & Protection Organization (OASP) under project 20242 and the Py-
thagoras EPEAEK project 21945 Aristotle Univ. Thessaloniki Research
Committee, and it is Geophysical Lab. Univ. Thessaloniki contribution
#631/2003.
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Geophysical Laboratory
University of Thessaloniki
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Thessaloniki, Greece
Manuscript received 1 September 2004.
