Geophys. J. Int. (2003) 152, 94–112
Maximum earthquake magnitudes in the Aegean area constrained
by tectonic moment release rates
G. Ch. Koravos,1 I. G. Main,2 T. M. Tsapanos1 and R. M. W. Musson3
1 Aristotle
University of Thessaloniki, School of Geology, Geophysical Laboratory, 54006 Thessaloniki, Greece
of Edinburgh, Department of Geology and Geophysics, West Mains Road, Edinburgh EH9 3JW, UK. E-mail: [email protected]
3 British Geological Survey, Murchison House, West Mains Road, Edinburgh EH9 3JZ, UK
2 University
Accepted 2002 July 18. Received 2002 July 4; in original form 2002 March 15
SUMMARY
Seismic moment release is usually dominated by the largest but rarest events, making the estimation of seismic hazard inherently uncertain. This uncertainty can be reduced by combining
long-term tectonic deformation rates with short-term recurrence rates. Here we adopt this strategy to estimate recurrence rates and maximum magnitudes for tectonic zones in the Aegean
area. We first form a merged catalogue for historical and instrumentally recorded earthquakes
in the Aegean, based on a recently published catalogue for Greece and surrounding areas covering the time period 550 BC–2000 AD, at varying degrees of completeness. The historical
data are recalibrated to allow for changes in damping in seismic instruments around 1911.
We divide the area up into zones that correspond to recent determinations of deformation rate
from satellite data. In all zones we find that the Gutenberg–Richter (GR) law holds at low
magnitudes. We use Akaike’s information criterion to determine the best-fitting distribution
at high magnitudes, and classify the resulting frequency–magnitude distributions of the zones
as critical (GR law), subcritical (gamma density distribution) or supercritical (‘characteristic’
earthquake model) where appropriate. We determine the ratio η of seismic to tectonic moment
release rate. Low values of η (<0.5) corresponding to relatively aseismic deformation, are
associated with higher b values (>1.0). The seismic and tectonic moment release rates are then
combined to constrain recurrence rates and maximum credible magnitudes (in the range 6.7–
7.6 m W where the results are well constrained) based on extrapolating the short-term seismic
data. With current earthquake data, many of the tectonic zones show a characteristic distribution that leads to an elevated probability of magnitudes around 7, but a reduced probability of
larger magnitudes above this value when compared with the GR trend. A modification of the
generalized gamma distribution is suggested to account for this, based on a finite statistical
second moment for the seismic moment distribution.
Key words: Aegean, earthquake criticality, gamma distribution, maximum magnitude,
tectonic moment release rate.
1 INTRODUCTION
The broad area of the Aegean has been the subject of many independent studies of seismicity and tectonics because the seismicity
is significant and deformation rates are sufficiently large to measure using palaeomagnetic or geodetic techniques. There have been
several proposed models for deformation in the area (Papazachos
& Comnonakis 1971; Makris 1973; McKenzie 1978; Le Pichon &
Angelier 1979; Taymaz et al. 1991; Giunchi et al. 1996). Quantitative attempts have been made to compare these predictions with
seismic deformation rates and orientations by Jackson & McKenzie
(1988), Sonder & England (1989), Jackson et al. (1992), Kiratzi
& Papazachos (1995) and Papazachos & Kiratzi (1996). In recent
94
times, the advent of satellite data has increased the available resolution significantly (Jackson et al. 1994; Clarke et al. 1997; Davies
et al. 1997; Reilinger 1997; Kahle et al. 1999, 2000; Holt et al. 2000;
McClusky et al. 2000). It may be argued that such short-term data
(over decades) could be unrepresentative of longer-term geological
data (Myr), but a recent comparison of plate velocities from contemporary satellite data and the Nuvel-3 model showed that the two
were indistinguishable within the uncertainties involved (de Mets
1995). This implies that the process of global forcing for global
tectonics is remarkably stationary in space and time.
The results of geological, satellite and earthquake data in the
Aegean and surrounding regions show that the African plate is converging due north with respect to the Eurasian Plate, leading to
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2003 RAS
Earthquake magnitudes in the Aegean
subduction of a remnant ocean under the Aegean lithosphere. The
subduction rate is fast enough to produce a degree of roll-back at
the Hellenic trench (Le Pichon & Angelier 1979), leading to stretching of the overriding plate. Present-day satellite data show that the
trench is retreating to the southwest at around 4 cm per year (see
fig. 3 of Kahle et al. 2000). This rate of stretching is the same as
that estimated by Le Pichon & Angelier (1979) from longer-term
geological data. The strain tensor based on the moment tensor sum
(using the method of Kostrov 1974) for the Aegean backarc shows
that overall it is stretching in approximately a north–south direction,
and thinning both vertically and in the east–west direction (Sonder &
England 1989). The backarc stretching direction is therefore oblique
to the trench roll-back direction.
A detailed independent knowledge of tectonic deformation rates
can also be used to constrain seismic hazard estimates based on
extrapolations of the frequency–magnitude distribution beyond the
instrumental/historical recording period (Schwartz & Coppersmith
1984; Wesnousky et al. 1984; Main & Burton 1984, 1986). Continental deformation is usually distributed in a volume. In this case the
sum of moment tensors for individual earthquakes is proportional
to the strain contained within the boundaries of the deforming region (Kostrov 1974; Jackson & McKenzie 1988). Thus strain at the
largest scales can be considered to be plastic in this sense, even
though it occurs by summing individual brittle failure events on an
individual localized rupture area.
A long-term constraint from macroscopic strain rates is important because the seismic hazard from the largest but rarest events is
inherently uncertain. Main & Burton (1989) applied this concept to
the estimation seismic hazard in the Aegean. They split the broad
area into two zones of subduction and backarc stretching in the
Aegean area, and combined the Makropoulos (1978) catalogue of
earthquakes with the tectonic model of Le Pichon & Angelier (1979)
to estimate the frequency–magnitude relation and its error bounds
using a maximum-entropy technique. Based on the available data,
they showed that the backarc deformation is predominantly seismic,
whereas the seismic moment rates for subduction were only a small
fraction of that expected from the tectonic model. The latter implies weak coupling between the overriding and subducting plates,
a necessary condition for trench rollback, and providing a possible
mechanism for the decoupling of the directions of trench rollback
and backarc stretching noted above. More recently the combination
of seismic and tectonic data has been extended to include the estimation of maximum credible magnitudes for seismic hazard analysis
(Main 1995; Main et al. 1999).
All estimates of probabilistic seismic hazard are based on the
frequency–magnitude relationship, most commonly the Gutenberg–
Richter (GR) law:
log10 F = a − bm,
(1)
where a and b are model parameters, and F is the frequency of occurrence of earthquakes having magnitudes in the range m = ±δm/2.
Sometimes this is referred to as an ‘incremental’ or ‘discrete’ frequency, or simply ‘frequency’. Typically, the uncertainty in instrumental magnitude determination is of the order ±0.3, so magnitudes
are rarely quoted to more than one decimal point. The smallest ‘bin’
size used to plot the frequency histogram, δm, is typically 0.1. Eq. (1)
is consistent with a scale-invariant distribution of earthquake source
rupture area (Rundle 1989), commonly found precisely at the critical
point in second-order phase transitions (Main 1995, 1996). The parameters of eq. (1) are best determined by the maximum-likelihood
method (Aki 1965). This method was corrected for a finite maximum
magnitude by Page (1968) and used by Consentino et al. (1977),
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2003 RAS, GJI, 152, 94–112
95
Weichert (1980) and Kijko & Sellevoll (1989). These methods
assume a priori a sharp upper truncation of eq. (1). However,
frequency–magnitude data are often fitted to eq. (1) using cumulative frequency data, leading to a potential source of bias, specifically overestimating the b value (Page 1968). The use of cumulative
data also leads to overestimation of the statistical significance (r 2 ),
because of the smoothing (low-pass filtering) involved in taking cumulative data (even a random frequency distribution would have
a finite r 2 when cumulative data are tested in regression). Here we
use incremental frequency data to avoid such potential sources of
bias.
The Gutenberg–Richter law is usually found to apply to the
smaller-magnitude earthquakes, but both the seismic hazard and
the seismic deformation rates are dominated by the largest events,
as long as b < 1.5 (Main 1995; Amelung & King 1997a,b). Usually,
b ≈ 1, so the form of the distribution for large-magnitude events is
very important, although typically there are only a few data points
there. One possibility for deviations from the GR trend at high
magnitude is the ‘characteristic’ earthquake model of Schwartz &
Coppersmith (1984), where the probability of occurrence of large
events is greater than would be expected by linear extrapolation of
the trend at small magnitudes. In this paper we shall refer to ‘characteristic’ only in this sense, i.e. we do not imply any regular repeat
times for the characteristic events as in Schwartz & Coppersmith
(1984). Alternatively, the gamma probability density distribution,
suggested by Shen & Mansinha (1983) and Main & Burton (1984),
has a lower probability of occurrence for the largest event than the
GR trend.
One way to approach the statistics of rare, large events is to focus on these events using the statistics of extreme values (Gumbel
1958). In contrast to eq. (1), Gumbel’s third asymptotic distribution of extreme values has both a characteristic scale and an estimated maximum event size. Accordingly, Makropoulos (1978) and
Makropoulos & Burton (1985) applied this distribution to extreme
value data in the area of Greece, and determined the maximum magnitude (denoted by ω in their nomenclature) for mainland Greece
and selected cities. The use of extreme values implies that much
of the earthquake recurrence data is discarded, possibly leading to
another source of bias in the determination of the parameters of
the frequency–magnitude distribution (Knopoff & Kagan 1977).
In practice, the error bounds on the maximum magnitude using
Gumbel’s third distribution are very large, owing to a combination of the paucity of data and the large extrapolations required
(Makropoulos & Burton 1985).
An alternative for obtaining the maximum magnitude is to use the
maximum-likelihood method. For example Papadopoulos & Kijko
(1991) evaluated the maximum regional magnitude m max , assuming
a sharp truncation of eq. (1) for the seismogenic regions proposed
by Papazachos (1980). Similarly Manakou & Tsapanos (2000) and
Tsapanos (2001) estimated m max for Crete and the adjacent area
using the maximum-likelihood method of Kijko & Sellevoll (1989),
by combining historical and instrumental data on a cellular grid of
0.4 ×0.4 deg2 elements.
Here we apply recent satellite-based geodetic deformation rates to
the problem of predicting maximum magnitudes in the Aegean area,
and compare the resulting seismic and tectonic moment release rates.
Our approach is similar to that of Field et al. (1999), who examined
the problem of estimating m max in California from seismicity and
tectonic deformation rates. However, we do not fix the form of the
frequency distribution a priori, and allow trends above or below
the Gutenberg–Richter trend to occur where appropriate. We also
allow the b value to be a variable to be determined by a curve fit
96
G. C. Koravos et al.
(Field et al. 1999, assumed b = 1.0 a priori). We use a new catalogue
based on historical and instrumental data to produce a catalogue that
combines the complete reporting of small events in the instrumental
period with the more likely inclusion of larger events in the historical
catalogue. The data are then used to constrain extrapolations of the
frequency–magnitude distribution for large events and to determine
maximum magnitudes. The theoretical framework used is described
in the next section.
2 THEORY
The frequency–magnitude distribution eq. (1) implies a power-law
distribution of source rupture area, implying a system with no characteristic length-scale, and an infinite correlation length. In such
systems the energy available equates directly to the critical point
energy for the system. However, if the system is not precisely critical, then different forms of the distribution can occur depending on
whether the available energy is less than or greater than the critical value (Main 1995, 1996). A distribution that preserves the lowmagnitude form of eq. (1), but allows deviations at high magnitude is
the generalized ‘gamma’ distribution. This distribution has a powerlaw distribution of moment M at low magnitude, and an exponential
tail at high magnitude. It was first derived using maximum-entropy
techniques bounded by the mean magnitude and the mean moment
release per event (Shen & Mansinha 1983; Main & Burton 1984).
Main & Burton (1984) first applied it to the eastern Mediterranean
and southern California, and since then it has been used to describe
many other data sets (Main & Burton 1989; Kagan 1991, 1993,
1997; Leonard et al. 2001). The generalized gamma distribution of
seismic moment M has the form
Mmax −B−1
M
exp(−M /Mθ ) d M ,
(2)
N (X ≥ M) = N T MMmax
M −B−1 exp(−M /Mθ ) d M Mmin
where N is the cumulative frequency of occurrence, N T is the annual
number of events above a lower cut-off moment Mmin and Mmax is
the maximum moment. The moment–magnitude relation is
log M = cm + d,
(3)
where m denotes magnitude. For the moment–magnitude scale
c = 1.5 and d = 9.05 (Kanamori 1978). The exponent B in eq. (2)
is related to b and c through B = b/c, so typically B = 2b/3 ≈ 23 .
Mθ is a characteristic seismic moment where the probability of occurrence has dropped by a factor of e−1 compared with the GR
trend. The dominator on the right-hand side of eq. (2) is a normalizing constant, ensuring that the total probability of occurrence of
an earthquake larger than Mmin is unity. For infinite Mmax the characteristic moment Mθ can be determined from
B
Ṁ = N T Mθ1−B Mmin
B(1 − B)
(4)
(Kagan 1993), where Ṁ is the moment release rate and denotes
the gamma distribution
∞
e−t t r dt, r > −1,
(5)
(r + 1) =
0
where for integer values r ! = (r + 1). Thus, for 0 < B < 1 and
Mθ > 0, an infinite upper truncation Mmax leads to a finite moment
release. In this case the exponential tail in the density distribution in
the integral in eq. (2) implies that a finite ‘credible’ maximum moment can be defined, higher than Mθ but lower than infinity (Main
1995). For Mθ−1 > 0 (subcritical condition) the largest events occur
more rarely than the extrapolation of the GR trend. In this case a
maximum credible magnitude can be determined by the increment
m max ± δm that contributes a negligible amount (<0.1 per cent) to
the total moment release (Main 1995). However, for Mθ−1 < 0 (supercritical condition) the largest events occur more often than the
Gutenberg–Richter trend, showing a characteristic peak at the highest magnitude. In this case the constraint of finite moment release
requires an explicit finite upper bound Mmax , which dominates the
total moment release (Main 1995). It is easy to show that the GR
law, eq. (1) is retrieved for the case Mθ−1 = 0 (critical condition).
The gamma distribution is therefore more general than the truncated Gutenberg–Richter law, but requires an appropriate objective
penalty for the extra parameter Mθ before being preferred to the
truncated Gutenberg–Richter law eq. (1) as a statistical model.
The advantage of this approach is that we can place objective
constraints on the frequency–magnitude relation at high magnitudes,
including the determination of maximum magnitudes and its error
bounds, in a straightforward way where geodetic or geological data
on tectonic moment release rates are available (Kagan 1991; Main
1996). Before we undertake this here, we describe the catalogue data
that was available for this study.
3 THE EARTHQUAKE CATALOGUE
A comprehensive catalogue of instrumental and historical earthquakes in Greece and the surrounding area for the time span 550 BC–
1999 AD has recently been compiled by Papazachos et al. (2000).
The estimated completeness of the catalogue for the corresponding magnitudes and time intervals according to Papazachos et al.
(2000) is given in Table 1. Their estimated uncertainties in the
magnitudes are ±0.25 for the instrumental period (1911–1999),
±0.35 for the historical period when the number of the macroseismic observations is 10 or more and ±0.50 otherwise. Their
estimated error in the epicentre coordinates ranges from ± a few
hundred m to ±20 km for the time period 1965–1999 to ±30 km
for the period 1901–1964. For the historical earthquakes the location error can reach up to ±50 km. The shallow earthquakes in
the backarc of the Aegean have focal depths of less than 20 km,
except for those associated with subduction along the Hellenic arc
(Papazachos et al. 2000). The published catalogue is available at
(http://geohazards.cr.usgs.gov/iaspei/europe/greece/the/). For this
study, we extended the time period to the year 2000 using the preliminary annual bulletins of the seismological station of the Aristotle
University of Thessaloniki. We restricted our study to shallow earthquakes (h ≤ 50 km) in the Aegean sea and the surrounding area, and
selected a minimum threshold magnitude for the most recent events
equal to or greater than 4.5, to ensure complete reporting.
The area was first divided into the 12 zones defined by Jackson
et al. (1994) on the basis of tectonic deformation data (the location
Table 1. Completeness threshold
of the catalogue used according to
Papazachos et al. (2000).
550 BC–1500
1501–1844
1845–1901
1911–1949
1950–1963
1964–1980
1981–1999
mw
mw
mw
mw
mw
mw
mw
C
≥ 8.0
≥ 7.3
≥ 6.0
≥ 5.0
≥ 4.5
≥ 4.3
≥ 4.0
2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
97
Figure 1. Map of the area of study, divided into 16 numbered regions after Holt et al. (2000), showing the spatial distribution of the epicentres of shallow
seismicity (h ≤ 50 km) in the Aegean and the surroundings for 550 BC–2000 AD. The data are from Papazachos et al. (2000).
of the SLR measurement stations). This has the major advantage
of allowing a direct comparison of seismic and tectonic moment
release rates, although the zoning scheme for some of the regions
is different from what might have been proposed on the basis of
an independent tectonic zoning scheme. Since all zoning schemes
are to some extent arbitrary, and the major emphasis here is on
incorporating geodetic data into estimates of maximum magnitude,
we preferred to retain the scheme of Jackson et al. (1994). This
was supplemented to include the four additional zones of Holt et al.
(2000, their Fig. 7) based on more recent data. These zones represent
an optimum for adequate sampling of earthquakes and deformation
rates. A finer grid is probably not justified at this stage, although
could be adopted as more data become available. Fig. 1 depicts
the 16 examined zones and shows the spatial distribution of the
epicentres in the study area for the time period 550 BC–2000 AD.
The numbers of the tectonic zones correspond to those in Holt et al.
(2000).
In the past, there have been systematic differences in magnitude
determination for historical and instrumental data in Greece and the
surrounding area. We therefore compared the two main catalogues
that have been compiled for the area. Events that are common to
both catalogues (date, origin time and location) are plotted against
each other in Fig. 2 for the period since 1893. This shows the moment magnitude m w of Papazachos et al. (2000), plotted against the
surface wave magnitude m s of the events listed for the region in
Ambraseys (1988), Ambraseys & Jackson (1990) and Jackson et al.
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2003 RAS, GJI, 152, 94–112
(1992) for the period 1890–1986. Table 2 lists data used to plot
Fig. 2: origin time, epicentre coordinates (φ, θ), m w and m s . The
parameter δm s is the standard deviation of surface wave magnitude.
Papazachos et al. (1997) concluded that the moment magnitude m w
is equal to the surface magnitude m s in this area for the magnitude
range m s = 6.0–8.0, so we would expect a linear correlation with a
slope of unity, passing through the origin if the two catalogues were
compatible. In fact, the data of Fig. 2 show
m s = (0.90 ± 0.04)m w + (0.57 ± 0.29).
(6)
Although there is a systematic difference between the two, the maximum mean difference between the two catalogues for the range
examined here is 0.29 magnitude units. This systematic difference
is comparable to the random errors quoted above for the Papazachos et al. (2000) catalogue, and δm s ≈ 0.26 from Table 2. These
errors equate to a factor of 2–3 in seismic moment, consistent with
the scatter of the data in Fig. 2. We conclude that for the majority of events, there is a good correlation between the two catalogues
within the errors involved, but for some large events, the catalogue of
Papazachos et al. (2000) has systematically higher values. For events
near the turn of the century at 1900 up to 1912 or so, this can be
traced to the effect of using undamped narrow-band seismic instruments (Abe & Noguchi 1983). In particular, we corrected the events
of (1904 0404 1026.0 41.80N 23.00E) in zone 14 from m s = 7.7
to m s = 7.1, and (1905 1108 2206.0 40.26N 24.33E) in zone
10 from m s = 7.5 to 6.8, based on the instrumental determination of
98
G. C. Koravos et al.
8.0
ms = mw
Shallow events (h=50Km)
7.5
ms
7.0
6.5
6.0
Least Square Fit
ms=0.90( 0.04)mw+0.57( 0.29)
5.5
5.5
6.0
6.5
7.0
7.5
8.0
mw
Figure 2. Plot of m s (Ambraseys 1988; Ambraseys & Jackson 1990; Jackson et al. 1992) against m w (Papazachos et al. 2000) for shallow earthquakes in the
Aegean area and the surrounding area for the period 1899 to 1986. The dashed line illustrates the curve m s = m w , and the solid line represents the best fit to
the data using least squares regression. The dotted lines above and below the least squares best fit illustrate the 95 per cent confidence intervals on this fit. A
typical error bar for the determination of m s and m w is shown in the top left-hand corner.
Abe & Noguchi (1983) and Ambraseys (2001). Since these events
are amongst the largest recorded events in the catalogue, they also
have a disproportionate effect on the calibration of the historical
catalogue for the largest large events. We estimated this systematic
effect to be on average, the order of −0.4 magnitude units, and
applied this correction to the historical data.
4 ANALYSIS OF THE CATALOGUE
FOR EACH REGION
4.1 Frequency–magnitude data
The catalogue completeness threshold varies both in space and time
(Papazachos et al. 2000). We therefore examined the reliability of
the data as a function of the date of the recording, for the 16 regions. There are simply inadequate data at present for a realistic
analysis for regions 15 and 16. Fig. 3 shows the mean recorded
magnitude for the 14 remaining areas defined by Fig. 1 as a function
of time since 1000 AD. Prior to this date most zones contain no
historically recorded events. The time plotted in Fig. 3 is an average
within a specified window that varies with the frequency of recorded
events, and decreases up to the present day as the recorded frequency
increases. Sudden, significant decreases in the mean magnitude in
Fig. 3 can be attributed to more complete recording of smaller events
(Lomnitz 1966). This enables an estimate of catalogue completeness, based on sudden changes in mean magnitude, to be made.
Major or minor temporal subdivisions in each subcatalogue are also
highlighted in this figure.
The final subcatalogues were compiled by merging the data for
the different time periods and magnitude ranges. The time periods
were based first on the step changes identified in Fig. 3 for each
region, most associated with the improvements in station coverage
at times listed in Table 1. Fig. 4 shows a plot of the incremental frequency–magnitude distribution for different time intervals of
each subcatalogue, normalized to an annual occurrence rate by its
duration. The standard√error bars shown on the diagrams, are estimated from σ F = F/ n, where n is the number of observations
used to determine the incremental frequency F (Campbell 1983).
(This gives only a first-order estimate on the error, even assuming a
Poisson distribution, for small data samples—see Field et al. (1999)
for a discussion of more sophisticated methods of estimating such
fluctuations.)
For each region, data for the different epochs were then combined
using the vertical dashed lines shown in Fig. 4. As in Main et al.
(1999), the choice of threshold magnitudes for the different epochs
was determined by trial and error, until the optimum best-fitting regression line to GR equation was found. This criterion is consistent
with a minimum offset at the magnitude corresponding to the joins.
The final completeness thresholds determined by this procedure are
presented in Table 3. They are similar to the results shown in Table 1
(e.g. m min = 4.5 since 1950, 5.0 since 1911). For the historical
earthquakes, the epicentres have been estimated using macroseismic data for all known strong earthquakes with a magnitude greater
than 6.0. The sources of the information are from descriptions given
by historians, geographers, travellers, monks, etc. (Papazachos
& Papazachou 1997). Although Greece and the surrounding area are
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
Table 2. Seismicity data of shallow earthquakes in Aegean and its surroundings, for the period 1899–1986. m w is the moment magnitude given
by Papazachos et al. (2000), m s is the surface wave magnitude adopted by
various sources. The standard deviation, dm s , of surface wave magnitude
estimate is given in Ambraseys (1988) and Ambraseys & Jackson (1990).
Year
Date
Time
1893
1894
1894
1899
1899
1904
1909
1909
1911
1911
1912
1912
1912
1914
1914
1917
1919
1922
1923
1926
1927
1928
1928
1928
1928
1928
1930
1930
1930
1931
1932
1932
1933
1933
1935
1938
1939
1941
1941
1942
1944
1944
1947
1948
1949
1953
1954
1955
1955
1956
1956
1956
1957
1957
1957
1957
1957
1959
1963
1964
0523
0420
0427
0920
0122
0811
0119
0530
1022
0218
0809
0810
0913
1003
1017
1224
1118
0813
1205
0318
0701
0331
0414
0418
0422
0502
0223
0331
0417
0308
0926
0929
0423
0511
0104
0720
0922
0301
0523
1115
1006
0625
1006
0209
0723
0318
0430
0419
0716
0709
0709
0730
0308
0308
0308
0425
0526
0425
0918
1006
220 200
165 200
192 100
22 200
95 600
60 830
45 730
61 430
223 145
213 512
12 900
92 353
233 124
220 700
62 232
91 355
215 450
954
205 635
140 615
81 854
2 947
93 001
191 748
201 346
215 432
181 912
123 348
200 639
15 028
192 042
35 726
55 737
190 950
144 130
2 335
3 632
35 247
195 152
170 115
23 441
41 619
195 534
12 581
150 330
190 616
130 236
164 719
70 710
31 140
32 403
91 457
121 414
122 113
233 509
22 542
63 330
2 639
165 808
143 123
C
φ
λ
mw
ms
dm s
Ref.
38.31
38.60
38.66
37.82
37.20
37.66
38.20
38.44
39.60
40.90
40.62
40.60
40.10
37.70
38.31
38.40
39.20
35.00
40.00
36.10
36.78
38.20
42.15
42.10
37.94
39.40
39.60
39.47
37.78
41.28
40.45
40.97
36.80
40.40
40.40
38.29
39.00
39.67
37.10
39.40
39.51
39.05
36.96
35.50
38.58
40.02
39.28
39.37
37.55
36.64
36.60
35.90
39.30
39.38
39.20
36.50
40.60
36.90
40.67
40.10
23.25
23.00
23.04
28.25
21.60
26.93
26.50
22.14
23.30
20.80
26.88
27.20
26.20
30.20
23.34
21.70
27.40
26.80
23.40
29.60
22.36
27.50
25.28
25.00
22.98
29.50
23.10
23.03
22.99
22.49
23.76
23.23
27.30
23.70
27.50
23.79
27.00
22.54
28.20
28.10
26.57
29.26
21.68
27.20
26.23
27.53
22.29
23.00
27.05
25.96
25.70
26.00
22.70
22.63
22.80
28.60
31.20
28.70
29.00
27.93
6.3
6.7
7.2
7.0
6.5
6.8
6.0
6.2
6.0
6.7
7.6
6.2
6.7
7.1
6.0
6.0
7.0
6.8
6.4
6.9
7.1
6.5
6.8
7.0
6.3
6.2
6.0
6.1
6.0
6.7
7.0
6.2
6.6
6.3
6.4
6.0
6.6
6.3
6.0
6.2
6.9
6.1
7.0
7.1
6.7
7.4
7.0
6.2
6.9
7.5
6.9
6.0
6.5
6.8
6.0
7.2
7.1
6.2
6.3
6.9
6.0
6.4
6.9
6.9
6.1
6.2
6.0
6.3
5.9
6.7
7.4
6.3
6.9
7.0
6.2
5.8
6.9
6.7
6.4
6.8
6.4
6.5
6.8
6.9
6.3
6.2
5.9
6.0
5.9
6.7
6.9
6.2
6.5
6.3
6.4
6.1
6.5
6.1
5.9
6.2
6.8
6.0
6.9
7.2
6.6
7.2
6.7
6.2
6.8
7.5
6.6
5.8
6.5
6.6
5.9
6.7
7.0
5.7
6.4
6.9
–
0.3
0.3
0.2
0.2
0.1
–
0.1
0.3
–
0.2
–
–
0.3
0.2
0.2
0.3
0.4
0.3
0.3
0.3
0.3
–
–
0.3
0.4
0.2
0.2
0.2
–
–
–
0.2
–
0.4
0.1
0.4
0.2
0.2
0.3
0.4
0.4
0.2
0.3
0.3
0.3
0.3
0.2
0.4
0.3
0.3
0.3
0.1
0.2
0.1
0.2
0.4
0.4
0.1
0.3
1
1
1
2
1
2
3
1
1
3
2
3
3
2
1
1
2
2
1
2
1
2
3
3
1
2
1
1
1
3
3
3
2
3
2
1
2
1
2
2
2
2
1
2
2
2
1
1
2
1
1
1
1
1
1
2
2
2
2
2
2003 RAS, GJI, 152, 94–112
99
Table 2. (Continued.)
Year
Date
Time
1965
1965
1965
1966
1966
1967
1967
1967
1968
1969
1969
1969
1970
1970
1971
1975
1978
1980
1980
1981
1981
1981
1981
1981
1982
1983
1986
0309
0405
0706
1029
0205
1130
0304
0501
0219
0114
0325
0328
0328
0408
0512
0327
0620
0709
0709
1219
1227
0224
0225
0304
0118
0806
0913
175 754
31 255
31 842
23 925
20 145
72 350
175 809
70 902
224 542
231 206
132 134
14 829
210 223
135 028
62 515
51 508
200 321
21 157
23 552
141 051
173 915
205 337
23 552
215 806
192 725
154 352
172 434
φ
λ
mw
ms
dm s
Ref.
39.16
37.40
38.27
38.78
39.05
41.39
39.20
39.47
39.50
36.10
39.20
38.29
39.16
38.36
37.60
40.40
40.71
39.27
39.16
39.00
38.81
38.22
38.14
38.20
39.78
40.00
37.05
23.89
22.10
22.30
21.11
21.75
20.46
24.60
21.25
25.00
29.20
28.40
28.57
29.42
22.53
30.00
26.10
23.27
22.83
22.68
25.26
24.94
22.92
23.09
23.25
24.50
24.70
22.11
6.1
6.1
6.3
6.0
6.2
6.3
6.6
6.4
7.1
6.2
6.0
6.6
7.1
6.2
6.2
6.6
6.5
6.5
6.1
7.2
6.5
6.7
6.4
6.3
7.0
6.8
6.0
6.5
5.9
6.4
5.9
6.2
6.6
6.8
6.2
7.3
6.3
6.1
6.5
7.1
6.2
6.2
6.6
6.4
6.4
6.0
7.2
6.4
6.7
6.4
6.2
6.9
6.9
5.8
0.3
0.3
0.3
0.2
0.2
–
0.3
0.2
–
0.2
0.2
0.2
0.3
0.3
0.3
0.3
–
0.2
0.3
–
0.4
0.2
0.2
0.4
–
–
0.2
1
1
1
1
1
3
1
1
3
2
2
2
2
1
2
2
3
1
1
3
1
1
1
1
3
3
1
(1) Ambraseys (1988); (2) Ambraseys & Jackson (1990); (3) Jackson et al.
(1992).
highly populated, the final catalogue for some zones contained no
historical data. For example, zone 8 had no reported historical earthquakes, and zone 5 had insufficient data (see Fig. 4). On the other
hand, zones such as 7 or 9 have a relative large number of historical earthquakes. The overall pattern is for better coverage of wellpopulated areas on the Greek mainland and Asia minor subjected to
the highest strain rates, i.e. zones 2 and 5–12 inclusive (Holt et al.
2000).
The merged catalogues have some empty bins. In order to minimize their effect, we filtered the raw frequency data using a threepoint running mean for each data point: Fis = 14 (Fi−1 + 2Fi + Fi+1 )
as in Main et al. (1999). The filtered incremental frequency–
magnitude distribution for the 14 regions is plotted in Fig. 5 together
with error bars at one standard deviation. These plots represent our
best estimate of the overall frequency–magnitude curves for the
different regions, based on our estimate of completeness for the different magnitude bands and time intervals determined from the data
in Figs 3 and 4.
In order to determine whether eq. (2) provides a statistically significant improvement on the simpler eq. (1) we used Akaike’s (1978)
information criterion:
n AIC = − ln SR2 − p,
(7)
2
where p is the number of unknown parameters in the model, n is the
number of observations, and the residual sum of squares
SR2 =
n
[yi − γ̂ (xi )]2 ,
(8)
i=1
where yi are the data and the hat symbol denotes the bestfitting model estimate of γ (xi ). This procedure penalizes the extra
100
G. C. Koravos et al.
Figure 3. Mean magnitude m w of earthquakes of the zones of Greece and the surrounding area as a function of time AD (since 1000). Vertical dashed lines
highlight major or minor subdivisions in the quality of the earthquakes.
parameter Mθ in eq. (2) in a formal way, allowing an objective determination of which model is the simplest consistent with the data.
This method has previously been used by Utsu (1999) for the same
problem of distinguishing between eqs (1) and (2).
We fitted the incremental frequency data to the density distribution of eqs (1) and (2) using a non-linear curve-fitting based on the
Levenberg–Marquardt algorithm described in Numerical Recipes
(Press et al. 1994). In future work we will examine the effect of
using the maximum-likelihood technique (e.g. Leonard et al. 2001)
to assess the possibility that this choice of fit leads to systematic
biases in the model parameters. Formally we fitted the frequency–
magnitude data to equations of the form
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
101
Figure 3. (Continued.)
ln F(m) = α − βm,
(9)
where β is the slope and α the intercept from eq. (1) and
ln F(m) = α − βm − θ exp(c m + d ),
(10)
where θ = 1/Mθ from eqs (2) and (3), where c = c ln 10 and
d = d ln 10.
These equations correspond, respectively, to the two cases
1/Mθ = 0 (eq. 9, p = 2) and 1/Mθ = 0 (eq. 10, p = 3). The
values of α and β for the line fit, and of α, β and 1/Mθ for the
curve fit, along with the corresponding values of AIC, and equivalent b values (b = β/ ln 10), are listed for the 14 examined zones in
Table 4. For each area the best-fitting model was determined from
the minimum value of AIC. The best-fitting model for each zone is
shown in comparison with the data in Fig. 5, along with its error,
calculated at 95 per cent confidence. A majority of the zones show
supercritical or ‘characteristic’ behaviour, although exceptions such
as zone 9 (subcritical) and zones 2, 10 and 14 (critical), also exist.
Papazachos (1999) found that b ≈ 1.05 for the shallow earthquakes of Greece and the surrounding area for the range of mag-
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2003 RAS, GJI, 152, 94–112
nitude 4.5–7.0, i.e. within the typical range of 0.8–1.2 (Evernden
1970; Rhoades 1996; Abercrombie 1996; Wesnousky 1999). Our
estimated b values are also in the range 0.8–1.2 for most of the
zones, although some show higher values. In particular, zone 2 has
b = 1.7, implying that most of the seismic energy release is, in fact,
in the small-magnitude range.
4.2 Seismic moment release rate
The annual seismic moment release rate for each one of the 14 zones
can be determined
N by summing the contribution of the individual
events: Ṁ = i=1
Mi /T , where T is the catalogue duration. We
used the moment–magnitude relation derived by Papazachos et al.
(1997) for Greece and its surroundings:
log M(in N m) = 1.5 m w + 8.99,
(11)
to determine the mean seismic moment release rate for the 14 zones,
using the frequency–magnitude data of Fig. 5 (Table 5).
102
G. C. Koravos et al.
Figure 4. Frequency–magnitude plot of the epochs shown in Fig. 3, normalized to annual occurrence. The data points also show logarithmically distributed
error bars for each magnitude bin determined at one standard deviation. The vertical dashed lines delineate areas where the individual data were used or
combined in the present work.
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
103
Figure 4. (Continued.)
5 TE C T O N I C M O M E N T R E L E A S E R A T E
The seismic moment tensor (Mi j ), in a deformed volume V , is related
to the strain tensor (εi j ) by
Mi j = 2µV εi j ,
(12)
where µ is the shear modulus of the brittle crust (Kostrov 1974).
Here V = Ah s , is the volume of the seismic deformation, A is
the area of the zone and h s is its seismogenic thickness. We used
h s = 10 km here (Kiratzi et al. 1991; Taymaz et al. 1991). A similar
equation also holds for moment tensor rates and strain rates, by
dividing both sides by the catalogue duration, T.
The available geodetic data reveal only the two horizontal components of the deformation field (Holt et al. 2000). If the axes x and
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2003 RAS, GJI, 152, 94–112
y are rotated so that the maximum deformation rate is parallel to
the direction x, then the strain rate tensor has the form of a diagonal matrix. The principal average horizontal strain rates (ε̇x x , ε̇ yy )
and their orientations, used in the present study are from Holt et al.
(2000). The vertical deformation rate ε̇zz can be estimated by assuming that the volumetric component of the deformation is negligible,
and hence the trace of the moment or strain tensors are zero:
ε̇x x + ε̇ yy + ε̇zz = 0.
(13)
The total scalar moment release rate for each zone is then
Mtectonic = 2µV max(|εx x |, |ε yy |, |εzz |)
(14)
(Bowers & Hudson 1999). The value of the shear modulus (µ)
is taken to be 3 × 1010 N m—a commonly used average for the
104
G. C. Koravos et al.
Table 3. Completeness threshold for the
14 zones. The minimum cut-off magnitude
and the corresponding year is listed.
Zone 1
Zone 2
Zone 3
Zone 4
Zone 5
Zone 6
Zone 7
Zone 8
Zone 9
Zone 10
Zone 11
Zone 12
Zone 13
Zone 14
1911 m w
1950 m w
1911 m w
1950 m w
1965 m w
1609 m w
1911 m w
1950 m w
1651 m w
1900 m w
1950 m w
1900 m w
1950 m w
1546 m w
1911 m w
1950 m w
1966 m w
1595 m w
1900 m w
1950 m w
1965 m w
1901 m w
1956 m w
1540 m w
1900 m w
1911 m w
1950 m w
1471 m w
1911 m w
1950 m w
1550 m w
1911 m w
1960 m w
1967 m w
1500 m w
1911 m w
1955 m w
1890 m w
1965 m w
1677 m w
1904 m w
1948 m w
≥ 5.3
≥ 4.6
≥ 5.0
≥ 4.9
≥ 4.5
≥ 6.0
≥ 5.1
≥ 4.5
≥ 5.6
≥ 5.0
≥ 4.5
≥ 5.0
≥ 4.5
≥ 6.1
≥ 4.9
≥ 4.8
≥ 4.5
≥ 6.0
≥ 5.1
≥ 4.9
≥ 4.5
≥ 5.3
≥ 4.5
≥ 5.6
≥ 5.1
≥ 4.9
≥ 4.5
≥ 6.6
≥ 5.0
≥ 4.5
≥ 6.6
≥ 5.0
≥ 4.6
≥ 4.5
≥ 6.6
≥ 5.5
≥ 4.5
≥ 4.9
≥ 4.5
≥ 6.3
≥ 5.1
≥ 4.5
seismogenic upper crust. The resulting annual tectonic moment release rate for the 14 zones is also listed in Table 5.
6 COMPARISON OF SEISMIC AND
TECTONIC MOMENT RELEASE
The percentage of seismic strain, η = (Mseismic /Mtectonic ) × 100 is
listed for every zone in Table 5. Values of 100 per cent reflect predominantly seismic deformation. Values higher than 100 per cent
may reflect unusually high recent activity, but are more likely to be
associated with the uncertainties involved. Some of the zones with
high η correspond to the slowest-deforming parts of the Aegean, and
coincidentally to zones that border either the sea, or neighbouring
countries where the data may not have been recorded to a uniform
standard see Holt et al. (2000, Fig. 7 therein). Holt et al. (2000) came
to similar conclusions based on a similar analysis of tectonic and
seismic data using the catalogue of Ambraseys & Jackson (1990),
so this conclusion is not catalogue-dependent. In fact, our results
Table 4. The coefficients α, β, θ = 1/Mθ for the line and curve fitting
of the frequency–magnitude distribution for the 14 examined zones. The
minimum value of Akaike’s information criterion (AIC) depicts the bestfitting line or curve in Fig. 5. The final column gives the equivalent b values
of the distribution for line or curve fitting.
Zone
α
β
θ (N m)−1
AIC
b
1
11.204
15.065
18.350
18.439
12.329
14.731
12.007
16.444
8.263
9.003
11.724
15.745
12.472
14.955
11.641
12.920
15.144
15.146
11.774
11.819
8.465
9.637
8.840
10.704
9.983
13.338
9.033
9.719
2.174
3.491
3.902
3.965
2.787
3.237
2.792
3.631
1.825
2.233
2.327
3.490
2.716
3.184
2.589
2.848
3.127
3.130
2.588
2.610
2.005
2.251
2.128
2.552
2.305
2.948
2.226
2.395
–
−6.792 97E−10
–
−4.536 87E−10
–
−3.221 09E−11
–
−7.728 31E−11
–
−3.0884E−11
–
−2.5117E−11
–
−5.2677E−11
–
−3.701 67E−11
–
1.517 12E−12
–
−8.901 34E−12
–
−6.052 41E−12
–
−1.100 24E−11
–
−1.881 29E−10
–
−4.2784E−11
−14.0958
−2.461 44
−13.3657
−14.9477
−36.3361
−31.9844
−40.728 52
−36.916 92
−53.8211
−22.700 30
−70.980 33
−29.341 31
−23.648 08
−22.491 34
−28.4318
−24.9733
−19.5605
−21.791 07
−33.097 93
−34.989 62
−48.5844
−44.0549
−42.5252
−28.4276
−27.0842
−22.833 74
−36.0080
−33.2516
0.944
1.517
1.701
1.722
1.211
1.407
1.214
1.579
0.793
0.971
1.011
1.517
1.180
1.384
1.125
1.237
1.359
1.360
1.125
1.134
0.871
0.978
0.925
1.109
1.002
1.282
0.968
1.041
2
3
4
5
6
7
8
9
10
11
12
13
14
(Table 5) compare reasonably well, within the uncertainties of moment estimation, with the results of Holt et al. (2000, Table 3) for the
fastest-deforming zones of the Aegean area, suggesting that the two
catalogues are broadly compatible where the data quality is good.
7 FREQUENCY–MAGNITUDE
EXTRAPOLATION AND THE
DETERMINATION OF MAXIMUM
MAGNITUDES
In this section we use the frequency–magnitude distribution and
the seismic or tectonic moment release rates to determine maximum earthquake magnitudes as defined in Reiter (1990). For the
Gutenberg–Richter law and the characteristic distribution, the maximum possible earthquake size can be determined as the maximum
magnitude required to account for the finite measured seismic moment release rate. The equivalent relation to eq. (2) for the GR law
is, for B < 1 (b < 1.5) and Mmin Mmax ,
1−B
B
Mmin
Ṁ = N T Mmax
B
(1 − B)
(15)
(Kagan 1993). This specifies a hard maximum associated with a
sharp truncation at the maximum moment Mmax . A similar hard
maximum is required from eq. (2) for the case of negative Mθ . For
positive Mθ the maximum possible magnitude defined by a finite
moment release rate is infinity, although infinitely improbable. In
this case a maximum ‘credible’ magnitude can be defined, where
the probability of occurrence is negligible (<0.1 per cent: Main
1995).
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
105
Figure 5. Incremental frequency–magnitude distribution for the subcatalogues. Data points also show error bars for each magnitude bin. The solid line
represents the best-fitting line. The dotted lines above and below the best-fitting line illustrate the 95 per cent confidence intervals on this fit.
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2003 RAS, GJI, 152, 94–112
106
G. C. Koravos et al.
Figure 5. (Continued.)
We have already determined which distribution fits best according
to the information criterion, eq. (7). The best way of illustrating the
finite moment constraint, given the different types of distribution,
is to plot a histogram of the frequency, weighted by the moment for
a given magnitude (using eq. 11), as illustrated in Fig. 6. Such plots
show the contribution of each magnitude bin to the total moment
release rate (e.g. Main 1995; Amelung & King 1997a,b; Main et al.
1999). Such plots are analogous to plotting, say, a particle size distribution by weight fraction rather than frequency. The total moment
release rate (seismic or tectonic) in these plots is then the total area
under the curve. We can also see at a glance whether or not the
large or small events dominate the moment release, with the for-
mer being the norm in the vast majority of cases. The vertical lines
shown in Fig. 6 represent the maximum magnitudes required so that
the areas under each curve equal the seismic or tectonic moment
release rate. Given the errors involved, and applying a conservative
historic
approach, we estimated m max = max(m tectonic
, m seismic
max
max , m max ), i.e.
the maximum moments constrained, respectively, by tectonic moment release, seismic moment release or the maximum recorded
event. These estimates are listed in Table 6 as maximum magnitudes and uncertainties given the 95 per cent confidence levels introduced by the line fits, or from the error estimation in primary
magnitude determination in the case of the maximum historical
earthquake.
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
Table 5. Seismic and tectonic moment release rate (1017
N m yr−1 ) for the 14 examined zones and the ratio η =
Mseismic /Mtectonic (in per cent).
Zone
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Seismic moment
Tectonic moment
η
0.871
1.522
4.673
2.728
5.166
4.986
5.673
5.590
6.985
8.142
15.049
5.740
4.991
3.796
0.570
6.385
2.640
2.485
3.845
9.103
12.234
12.259
25.838
23.477
18.335
12.605
2.543
4.048
153.0
24.0
177.0
110.0
134.0
54.0
46.0
46.0
27.0
35.0
82.0
46.0
196.0
94.0
The largest events broadly dominate the moment release in all
but one of the zones in Fig. 6. For these zones the b value is around
1 (Table 4). For zone 2 (Fig. 5), b = 1.7, so the contribution to
the moment of small events is actually bigger than that of the big
events recorded so far. Similar observations have been reported by
Amelung & King (1997a), who found that the contributions from
each magnitude bin in the moment release are equal for the creeping
segment of the San Andreas fault, where b = 1.5. In both areas of
high b value the seismic to tectonic moment release ratio is also very
low. Zone 2 also has a very low proportion of seismic to tectonic
moment release (Table 5).
Zone 9 has a very high magnitude since it has both a relatively
low seismic efficiency and a subcritical distribution. This magnitude
is consistent with the data, but is the most uncertain of all entries in
Table 5. The absence of continuous fault breaks of several hundred
kilometres in the area almost certainly means that this value is over
estimated.
8 DISCUSSION
8.1 Comparison of seismic and tectonic moment release
We have determined recurrence rates and the maximum earthquake
magnitudes for the different tectonic zones in the Aegean area
(Fig. 7) using a combination of instrumental, historical and geodetic
data. On pragmatic grounds we have picked maximum magnitudes
that are determined either by the seismic or tectonic moment release
rates, or the maximum historical magnitude where appropriate. In
some areas the seismic moment release rate for the subcatalogue appears to exceed the tectonic moment release rate, but no more than a
factor of 2 or so, i.e. within the errors involved in the determination
of magnitude and seismic moment.
8.2 The ‘characteristic’ earthquake distribution
Most of the areas show a characteristic earthquake distribution as the
best-fitting model within the resolution of the non-linear regression
method used. Kagan (1993) has argued that there are no statistically
significant examples of the characteristic distribution for instrumentally recorded seismicity, and hence that the gamma distribution with
positive Mθ should be preferred. This also avoids the rather unphysical sharp truncations needed by the GR and characteristic distributions used here. However, characteristic distributions with a more
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2003 RAS, GJI, 152, 94–112
107
gradual tail have been seen for volcanic seismicity (Main 1987),
and have also previously been suggested for the Aegean backarc
as a whole (Main & Burton 1989). They are also a common feature of numerical models for earthquakes (Ben-Zion & Rice 1993,
Fig. 10(a) therein; Shaw et al. 1992), although sharp truncations can
also be seen in numerical models with fixed boundary conditions
(Carlson et al. 1993). A more gradual tail for the characteristic distribution, although plausible on physical grounds, would require an
extra parameter, but is unlikely to be justified by AIC for the current
data set.
What form might be suitable for such a gradual tail? The gamma
distribution (2) is the maximum entropy solution for the frequency–
magnitude distribution given knowledge of the mean magnitude m
and the mean moment M = Ṁ/N T per event (Main & Burton
1984). These are both first moments. In order to introduce a more
gradual truncation for the case of negative Mθ , we would need an
additional constraint to higher order. A potential for such a gradual truncation could come about through fluctuations in the mean
seismic moment that depend on the second moment M 2 , which
is equivalent to a finite variance on our estimate of M. We have
already seen in our basic examination of the catalogues that these
fluctuations can be very large for data sets of the duration typically
used here. Using the same method of Lagrangian undetermined multipliers, it is easy to extend the theory of Main & Burton (1984) to
include this criterion, whence
N (X ≥ M)
Mmax
2 m −B−1 exp −m /m θ1 − m /m θ2
dm ,
= NT 2 Mmax
m −B−1 exp −m /m θ1 − m /m θ2
dm Mmin
M
(16)
with one extra parameter Mθ2 in the general case representing the
finite variance. Since the second moment introduces a squared term
within the exponential in the integral, the larger magnitudes always
have a continuously reducing probability, irrespective of the sign
of Mθ2 . This second-order truncation is sharper than the first-order
truncation for positive Mθ2 , but is still continuous and differentiable.
Although eq. (17) is attractive on theoretical grounds, the data available here are unable to resolve whether or not this would be a better
fit to the data in the case of the characteristic distribution. The resolution of the problem of sharply truncated characteristic earthquakes
therefore awaits better data on the recurrence of the largest events,
from instrumental, historical or palaeoseismic data.
8.3 Regression technique
In future work the non-linear regression methodology used here
could also be improved upon. For example Leonard et al. (2001)
developed a Bayesian technique for uncertainty estimation based
on a more appropriate Poisson distribution of errors for frequency
data. (Here we have assumed a Gaussian distribution of errors, where
the variance is independent of the mean, but this can introduce a
potential source of bias in the results). The method of Leonard et al.
(2001) combines a maximum-likelihood regression technique with
a more general information criterion than AIC to determine the bestfitting distribution, and estimates the associated errors in the form
of Bayesian intervals determined from the assumption of a uniform
prior distribution. This was beyond the scope of the present work,
but in the future it will be interesting to see whether their regression
method produces significantly different results from those obtained
here.
108
G. C. Koravos et al.
Figure 6. Plots of the contribution of the magnitude interval to the total seismic moment release. The solid line represents the best-fitting line. The dotted
lines above and below the best-fitting line illustrate the 95 per cent confidence intervals on this fit. Data points also show error bars for each magnitude bin.
Thick solid vertical lines represent maximum magnitude (and their errors) estimated by seismic moment release rate while vertical thick dashed lines represent
maximum magnitudes (and their errors) estimated by tectonic moment release.
9 CONCLUSION
Maximum magnitudes for the Aegean and its surroundings have
been estimated by the application of seismic and tectonic constraints
for the most general form of the frequency–magnitude distribu-
tion. The choice of which form to use was determined objectively
by Akaike’s information criterion, applied to a recently compiled
historical and instrumental catalogue for 14 seismic zones determined by geodetic determinations of the horizontal deformation
field. We observed distributions consistent with critical (GR law)
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
109
Figure 6. (Continued.)
or subcritical-point (gamma distribution) behaviour. However, most
of the zones reveal supercritical behaviour associated with the occurrence of ‘characteristic’ earthquakes. In these zones the largest
events strongly dominate the total seismic moment release. Higher
b values tend to be associated with zones of predominantly aseismic
deformation. A new form for the frequency–magnitude distribution
based on the constraint of finite second moment for the seismic moment release is suggested to overcome the problem of an implied
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2003 RAS, GJI, 152, 94–112
sharp truncation at the maximum possible magnitude for the case
of the characteristic distribution.
ACKNOWLEDGMENTS
We thank James Jackson and John Haines for providing the coordinates of the tectonic zones, and for access to the original principal
average strain rate data used to calculate the tectonic moment release
110
G. C. Koravos et al.
Figure 6. (Continued.)
Figure 7. Map of the maximum earthquake magnitude (m max ) for the different tectonic zones in the Aegean area using a combination of instrumental,
historical, and geodetic data. The ratio of seismic to tectonic moment release rate (η) in per cent is also given.
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2003 RAS, GJI, 152, 94–112
Earthquake magnitudes in the Aegean
overall and
Table 6. Maximum magnitudes, m historical
, m tectonic
, m seismic
max
max
max , m max
the corresponding errors estimated by the use of seismic and tectonic moment
release constraints for the examined zones.
Zone
1
2
3
4
5
6
7
8
9
10
11
12
13
14
m historical
max
m tectonic
max
m seismic
max
m overall
max
6.90 (±0.40)
7.10 (±0.40)
7.20 (±0.25)
7.60 (±0.40)
7.90 (±0.40)
7.50 (±0.25)
7.00 (±0.25)
7.10 (±0.25)
7.00 (±0.25)
7.20 (±0.25)
7.60 (±0.25)
7.40 (±0.25)
6.70 (±0.25)
7.10
6.10 (±0.33)
–
7.0 0(±0.30)
7.10 (±0.10)
6.90 (±0.2)
7.60 (±0.20)
7.20 (±0.10)
7.30 (±0.20)
10.00 (±1.30)
8.50 (±0.60)
7.60 (±0.20)
7.60 (±0.20)
6.50 (±0.20)
7.10 (±0.20)
6.35 (±0.25)
6.20 (±0.40)
7.20 (±0.10)
7.10 (±0.10)
7.00 (±0.20)
7.50 (±0.10)
7.00 (±0.10)
7.10 (±0.20)
7.20 (±0.10)
7.40 (±0.40)
7.50 (±0.20)
7.40 (±0.20)
6.70 (±0.20)
7.00 (±0.20)
6.90 (±0.40)
7.10 (±0.40)
7.20 (±0.10)
7.60 (±0.40)
7.90 (±0.40)
7.60 (±0.20)
7.20 (±0.10)
7.30 (±0.20)
10.00 (±1.30)
8.50 (±0.60)
7.60 (±0.20)
7.60 (±0.20)
6.70 (±0.20)
7.10 (±0.20)
rate. This paper is published with the permission of the Executive
Director of BGS (NERC).
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