Bulletin of the Seismological Society of America, Vol. 100, No. 6, pp. 2914–2926, December 2010, doi: 10.1785/0120100111
Ⓔ
Scaling Relations of Earthquake Source Parameter Estimates
with Special Focus on Subduction Environment
by Lilian Blaser, Frank Krüger, Matthias Ohrnberger, and Frank Scherbaum
Abstract
Earthquake rupture length and width estimates are in demand in many
seismological applications. Earthquake magnitude estimates are often available,
whereas the geometrical extensions of the rupture fault mostly are lacking. Therefore,
scaling relations are needed to derive length and width from magnitude. Most frequently used are the relationships of Wells and Coppersmith (1994) derived on
the basis of a large dataset including all slip types with the exception of thrust faulting
events in subduction environments. However, there are many applications dealing
with earthquakes in subduction zones because of their high seismic and tsunamigenic
potential. There are no well-established scaling relations for moment magnitude and
length/width for subduction events.
Within this study, we compiled a large database of source parameter estimates of
283 earthquakes. All focal mechanisms are represented, but special focus is set on
(large) subduction zone events, in particular. Scaling relations were fitted with linear
least-square as well as orthogonal regression and analyzed regarding the difference
between continental and subduction zone/oceanic relationships. Additionally, the
effect of technical progress in earthquake parameter estimation on scaling relations
was tested as well as the influence of different fault mechanisms.
For a given moment magnitude we found shorter but wider rupture areas of
thrust events compared to Wells and Coppersmith (1994). The thrust event relationships for pure continental and pure subduction zone rupture areas were found to be
almost identical. The scaling relations differ significantly for slip types. The
exclusion of events prior to 1964 when the worldwide standard seismic network
was established resulted in a remarkable effect on strike-slip scaling relations:
the data do not show any saturation of rupture width of strike-slip earthquakes.
Generally, rupture area seems to scale with mean slip independent of magnitude.
The aspect ratio L=W, however, depends on moment and differs for each slip
type.
Online Material: Table of source parameter estimates.
Introduction
In several domains of seismology it is required to estimate the geometrical dimensions of an earthquake rupture
area on the basis of the earthquake moment magnitude estimate. For the assessment of the earthquake potential of a specific region it is often necessary to estimate the size of the
largest earthquake that might be generated by a particular
fault. In history there are several examples when earthquake
magnitudes exceeded former expectations of possible maximal magnitudes along individual faults as no such event
has been recorded before; for example, the 2004 Sumatra–
Andaman earthquake Mw > 9 (e.g., Okal and Synolakis
2008) and the A.D. 365 Crete earthquake (Shaw et al. 2008).
Thus, the earthquake potential of a fault commonly is evaluated from estimates of possible fault rupture parameters that
are, in turn, related to earthquake magnitude. Similar extrapolations have to be made to compile large and complete
databases to estimate seismic or tsunami hazard.
In real-time applications, like the generation of shake
maps or the estimation of tsunami threat, slip distributions
have to be estimated (e.g., Kanamori, 2008). A first order
approximation of the ruptured fault area is generally a
representative rectangular geometry from which in the
simplest case a homogeneous average slip distribution
can be derived. Scaling relations provide then characteristic
2914
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
rupture length and width for given earthquake
magnitude.
Many applications of interest are related to subduction
zone environments. In contrast to the widely used scaling
relations of Wells and Coppersmith (1994) representing
crustal earthquakes, there are no established scaling relations
for rupture length and width of earthquakes in subduction
zones. Therefore, we compiled a database containing 283
earthquakes of all slip types and analyzed different regression approaches, focusing on the questions:
• Are there any differences between the newly compiled
scaling relations including data from subduction environment and the ones derived from the database of Wells and
Coppersmith (1994)?
• Is it possible to distinguish between continental and
subduction/oceanic earthquake scaling relations?
We distinguish between reverse faulting subduction
zone events, neighboring outer-rise events of normal faulting
slip type and oceanic strike-slip faulting earthquakes. We
compare the new oceanic/subduction zone scaling relations
with our new relationships of continental earthquakes showing equivalent focal mechanisms and with scaling relations
of other authors like Mai and Beroza (2000) and Hanks and
Bakun (2002). The product of rupture length and width for
subduction zone earthquakes and continental thrust events
will be compared to the rupture area estimates according
to the scaling relations of Murotani et al. (2008) and Somerville et al. (1999), respectively.
Furthermore, we test the impact of technical improvements on scaling relations derived from a reduced dataset
on the example of the installation of the worldwide standard
seismic network (WWSSN) during the 1960s. The installation
of the WWSSN provided an increase of available data and
established a basis for higher accuracy in parameter estimation. Therefore, we ask:
• Do the regression results differ when accounting for data
derived after 1964 only?
• Is it worth distinguishing between the scaling relations for
the different types of focal mechanism?
• And finally, do our new regression parameters follow selfsimilar scaling?
Next to a generally improved quality of the scaling
relations caused by decreased standard errors and enlarged
data ranges, the main findings can be summarized in five
points:
• Scaling relations for reverse faulting earthquakes result in
shorter but wider rupture areas compared to Wells and
Coppersmith (1994). This holds for both continental and
subduction events.
• No differing scaling relations could be found for the
rupture areas of pure continental thrust events and pure
subduction zone earthquakes.
2915
• The use of post 1964 WWSSN data has a significant effect
on scaling relationships for strike-slip earthquakes; there is
no evidence of rupture width saturation.
• The scaling relations differ significantly for different
slip types.
• Assuming constant rigidity mean slip seems to scale invariant on moment magnitude with rupture area. The aspect
ratio L=W, however, changes with moment magnitude
differently for the different slip types.
We will first describe the compilation of the new database followed by the summary of the regression methods and
the statistical analysis tool. After presenting the scaling relations, we will discuss them and conclude with a summary.
Data
The database is composed of 196 source estimates
published by Wells and Coppersmith (1994), 40 by Geller
(1976), 25 by Scholz (1982), 31 by Mai and Beroza (2000),
36 by Konstantinou et al. (2005), and 31 by several other
authors analyzing single large events (see Data and Resources section for references; Ⓔthe complete list of earthquakes is provided in Table S1 of the electronic supplement
to this paper). The dataset contains estimates of moment
magnitude, focal mechanism, rupture length, and (in most
cases) rupture width of shallow earthquakes located all over
the world including subduction zones in particular. Moment
magnitude, if not available as such, has been derived
from the seismic moment by Mw 2=3 log10 M0 10:7 dyne-cm (Hanks and Kanamori, 1979). To avoid
any inconsistencies caused by magnitude conversion formulas we restricted the dataset to events with available moment
magnitude only. Wells and Coppersmith (1994) distinguished between surface and subsurface rupture length. Surface lengths are determined from ground-surface outcrops,
whereas subsurface estimates are generally estimated from
aftershock analyses. Wells and Coppersmith (1994) as well
as Wang and Ou (1998) (they analyzed a subset of the dataset
of Wells and Coppersmith [1994] constrained to a high reliability level) showed that the surface rupture length is generally less than the subsurface rupture length. We choose to
work with the subsurface lengths and will compare our scaling relations for rupture length with the subsurface scaling
relations of Wells and Coppersmith (1994) in the discussion.
In most publications the focal mechanism was categorized into reverse, normal, or strike-slip faults according to the
dominant slip type. For the events published by Geller
(1976), the focal mechanisms were not listed and were therefore added from additional references within this study.
Hence, the database is composed of 33% reverse, 14% normal, and 53% strike-slip faulting mechanisms. We note that
this distribution does not follow the relative composition of
observed earthquakes worldwide. For example, in the
Global Centroid Moment Tensor (CMT) project catalog
(1976–2009; see the Data and Resources section), there
2916
L. Blaser, F. Krüger, M. Ohrnberger, and F. Scherbaum
are 47% reverse, 24% normal, and 39% strike-slip earthquakes larger than M > 6. This might be of importance in
the case of using the scaling relation independent of focal
mechanism when sampling, for example, a synthetic earthquake catalog with no focal mechanism information for a
hazard study.
Generally, the geometrical rupture dimensions were
obtained by seismological investigations of the aftershock
distribution. The 31 parameter estimates published for 24
specific large events in recent time were derived from 31 distinct sources in which detailed source rupture studies were
conducted in combination with geodetic data from radar
or GPS observations.
In total there are 359 entries in the database describing
283 earthquakes. For the regression analysis, multiple
parameter estimates from different authors were replaced
by their mean value.
In Figure 1 the rupture lengths are plotted versus the
earthquake moment magnitudes for different slip types indicating the original publication. Compared with Wells and
Coppersmith (1994), the supplemental data of strike-slip
events show the same magnitude range, whereas the domain
of the normal and reverse faulting earthquakes could be
expanded considerably (reverse: 4:8 ≤ Mw ≤ 9:5, normal:
5:1 ≤ Mw ≤ 8:4).
The variability of different earthquake parameters published for the same events is illustrated in Figure 2. We can
use the multiple parameter estimates to derive approximate
uncertainty bounds of those values. The differences in moment magnitude reach Δmax Mw ⪅0:3 and are independent of
the magnitude size and slip mechanism. Differences between
the logarithm of rupture lengths are found mainly to be smaller as Δmax log10 L⪅0:25. Again, no dependence on rupture
length or focal mechanism is found.
In order to detect temporal changes of data uncertainty
that may be caused by technical improvements of the global
(a)
5
6
7
8
9
10
(1)
normal
3
2
1
5
4
5
6
7
8
9
10
all slip types
Wells and Coppersmith 1994
Geller 1976
Scholz 1982
Mai and Beroza 2000
Konstantinou 2005
additional events
2
0
4
0
Mw
3 strike slip
log10L
n
X
1
ϵ^2 ;
np1 i i
where n is the P
number of samples, p 1 for linear regression, and S ni ϵ^2i sum of square residuals.
To compare the different resulting scaling relations we
calculate the coefficient of determination R2 , a measure of
how well future outcomes are likely to be predicted by
the model. There is a clear definition of R2 , only for simple
linear regression
10
4
Mw
Figure 1.
s2xy 1
1
(c)
The strong correlation between the logarithm of rupture
length and moment magnitude (between 0.9 for normal and
0.95 for reverse and strike-slip faulting earthquakes) and rupture width and moment magnitude (between 0.82 for normal
and 0.94 for reverse slip type, see Tables 1 and 2 for details)
confirms the appropriateness of scaling relations between
moment magnitude and rupture length/width. The regression
analyses were performed for the relationships of rupture
length or width (as dependent variable Y) and the moment
magnitude Mw according to log10 Y a b × Mw . Each
of these combinations was analyzed based on the focal
mechanism specific data (reverse, normal, and strike-slip)
as well as on all data independent of slip type.
Next to the regression coefficients a and b, the standard
errors of the coefficients sa, sb as well as the standard deviation of the error term sxy were compiled. The standard
deviation of the error term sxy is defined as
log L
log10L
2
0
Regression Analysis
(b)
reverse
3
observation network, the variability of parameter estimates
are plotted against the year of the earthquake origin in
Figure 2d. We are unable to detect time-dependent variability
of the epistemic uncertainties in our data set. Uncertainties
of rupture width estimates show similar properties with
Δmax log10 W⪅0:3.
6
7
Mw
8
9
10
Rupture lengths and magnitudes for all data separated by their slip type and classified according to the original publication.
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
(a)
∆max(log10L)
10
log L
(b)
all slip types
3
2
1
0
4
5
6
7
8
9
reverse
normal
strike slip
0.3
0.2
0.1
0
10
0
0.5
1
Mw
3
3.5
log L
10
0.3
0.2
∆
Mw
2.5
Mw
0.2
∆
max
2
log L
0.4
(d)
reverse
normal
strike slip
0.3
1.5
10
0.4
(c)
2917
0.1
0.1
0
4
5
6
7
Mw
8
9
0
1920
10
1940
1960
year
1980
2000
Figure 2. (a) Events with double or multiple entries are linked with a black line. Underlying gray dots are events with single entries.
(b) Maximal observation differences for the logarithm of rupture length from multiple estimates classified according to the slip type. (c) Maximal differences of multiple magnitude estimates classified according to the slip type. (d) Both magnitude and logarithmic length estimate
differences plotted at the year of the origin date.
R2
Pn
y y^ i 2
1 Pin i
;
2
i yi y
sands. The coefficient of determination takes on values between 0 and 1, where values closer to 1 imply a better fit.
We analyzed the dataset using two different regression
methods. First, we build an ordinary least-square regression
model to make our new empirical relationships comparable
to those of Wells and Coppersmith (1994). Using standard
(2)
where n is the number of data samples yi , y^ i are the predicted
values by the regression, and y is the mean of the regres-
Table 1
Scaling Relations for Rupture Length L and Moment Magnitude Mw Derived from (Subsets of)
the New Combined Catalog with Different Regression Methods*
Equation
Slip Type
Number of
Events
a
sa
b
sb
corr
sxy
R2
Mw Range
L Range (km)
log10 L a b × Mw
Orthogonal†
≥ 1964
Oceanic
Continental
Wells and Coppersmith (1994)‡
reverse
96
96
82
26
70
50
2:28
2:37
2:27
2:81
2:17
2:42
0.13
0.13
0.14
0.29
0.18
0.21
0.55
0.57
0.55
0.62
0.54
0.58
0.02
0.02
0.02
0.04
0.03
0.03
0.95
0.95
0.95
0.96
0.92
0.93
0.18
0.18
0.18
0.16
0.19
0.16
0.91
0.91
0.90
0.93
0.85
4.8–9.5
4.8–9.5
4.8–9.2
6.1–9.5
4.8–8.4
4.8–7.6
1.1–1400
1.1–1400
1.1–1400
13–1400
1.1–300
1.1–80
log10 L a b × Mw
≥ 1964
≥ 1964 and Orthogonal†
Wells and Coppersmith (1994)‡
normal
43
33
33
25
1:61
1:75
1:91
1:88
0.23
0.28
0.29
0.37
0.46
0.49
0.52
0.5
0.04
0.04
0.04
0.06
0.90
0.89
0.89
0.88
0.17
0.17
0.18
0.17
0.80
0.80
0.79
5.1–8.4
5.1–8.2
5.1–8.2
5.2–7.3
4.5–210
4.5–210
4.5–210
3.8–63
log10 L a b × Mw
Orthogonal†
≥ 1964
Oceanic
Continental
Wells and Coppersmith (1994)‡
strike-slip
144
144
116
16
128
93
2:56
2:69
2:50
2:56
2:55
2:57
0.11
0.11
0.12
0.47
0.12
0.12
0.62
0.64
0.61
0.62
0.62
0.62
0.02
0.02
0.02
0.07
0.02
0.02
0.95
0.95
0.95
0.92
0.95
0.96
0.18
0.18
0.17
0.19
0.18
0.15
0.89
0.89
0.90
0.85
0.90
4.6–8.1
4.6–8.1
4.6–8.1
5.3–8.1
4.6–8.1
4.8–8.1
1.5–450
1.5–450
1.5–427
7.0–350
1.5–450
1.5–350
log10 L a b × Mw
Orthogonal†
≥ 1964
Oceanic
Continental
Wells and Coppersmith (1994)‡
all
283
283
231
47
236
167
2:19
2:31
2:20
2:07
2:26
2:44
0.08
0.08
0.09
0.20
0.10
0.11
0.55
0.57
0.55
0.54
0.57
0.59
0.01
0.01
0.01
0.03
0.02
0.02
0.94
0.94
0.94
0.95
0.92
0.94
0.19
0.20
0.19
0.18
0.20
0.16
0.88
0.88
0.88
0.90
0.85
4.6–9.5
4.6–9.5
4.6–9.2
5.3–9.5
4.6–8.4
4.8–8.1
1.1–1400
1.1–1400
1.1–1400
7.0–1400
1.1–450
1.1–350
*Orthogonal where noted, ordinary least-square regression elsewhere.
relation.
‡
Equivalent scaling relations.
†Preferred
2918
L. Blaser, F. Krüger, M. Ohrnberger, and F. Scherbaum
Table 2
Scaling Relations for Rupture Width and Moment Magnitude Derived from (Subsets of)
the New Combined Catalog with Different Regression Methods*
Equation
Slip Type
Number of
Events
a
sa
b
sb
corr
sxy
R2
Mw Range
W Range (km)
log10 W a b × Mw
Orthogonal†
≥ 1964
Oceanic
Continental
Wells and Coppersmith (1994)‡
reverse
83
83
71
23
60
43
1:80
1:86
1:82
1:79
1:83
1:61
0.12
0.12
0.14
0.26
0.18
0.2
0.45
0.46
0.46
0.45
0.45
0.41
0.02
0.02
0.02
0.03
0.03
0.03
0.94
0.94
0.93
0.95
0.90
0.9
0.17
0.17
0.17
0.14
0.18
0.15
0.89
0.89
0.87
0.90
0.82
4.8–9.5
4.8–9.5
4.8–9.2
6.1–9.5
4.8–8.4
4.8–7.6
2–240
2–240
2–240
12–240
2–140
1.1–80
log10 W a b × Mw
Orthogonal†
≥ 1964
Wells and Coppersmith (1994)‡
normal
39
39
29
23
1:08
1:20
0:85
1:14
0.25
0.25
0.38
0.28
0.34
0.36
0.30
0.35
0.04
0.04
0.06
0.05
0.82
0.82
0.68
0.86
0.16
0.16
0.17
0.12
0.67
0.67
0.47
5.1–8.4
5.1–8.4
5.1–7.2
5.2–7.3
3–100
3–100
3–23
3.8–63
log10 W a b × Mw
≥ 1964
≥ 1964 and Orthogonal†
Oceanic
Continental
Wells and Coppersmith (1994)‡
strike-slip
129
103
103
14
115
87
0:76
1:05
1:12
0:66
0:75
0:76
0.11
0.12
0.12
0.64
0.10
0.12
0.27
0.32
0.33
0.27
0.27
0.27
0.02
0.02
0.02
0.10
0.02
0.02
0.82
0.86
0.86
0.62
0.84
0.84
0.16
0.15
0.15
0.21
0.15
0.14
0.68
0.75
0.74
0.38
0.70
4.6–8.1
4.6–7.7
4.6–7.7
5.3–7.8
4.6–8.1
4.8–8.1
2–30
2–30
2–30
4–30
2–30
1.5–350
log10 W a b × Mw
≥ 1964
≥ 1964 and Orthogonal†
Oceanic
Continental
Wells and Coppersmith (1994)‡
all
251
203
203
40
211
153
1:36
1:47
1:56
1:76
1:14
1:01
0.08
0.09
0.09
0.19
0.10
0.1
0.38
0.40
0.41
0.44
0.34
0.32
0.01
0.01
0.01
0.03
0.02
0.02
0.89
0.90
0.90
0.94
0.84
0.84
0.19
0.17
0.17
0.17
0.18
0.15
0.79
0.80
0.80
0.88
0.71
4.6–9.5
4.6–9.2
4.6–9.2
5.3–9.5
4.6–8.4
4.8–8.1
2–240
2–240
2–240
4–240
2–140
1.1–350
*Orthogonal where noted, ordinary least-square regression elsewhere.
relation.
‡
Equivalent scaling relations.
†Preferred
least-square regression relies on many assumptions.
Minimizing the vertical distances implies that errors in
the model Y a b × X are confined to the observed ycoordinates. Thus, for log10 L a b × Mw it is assumed
that the moment magnitude is measured without any error,
which does not hold as one could observe in the previous
section. Alternatively, we make use of the orthogonal (or total least-square) regression in a second approach. Orthogonal
regression minimizes the Euclidian distance to the regression
line instead of the vertical distance. In contrast to the ordinary least-square regression, the inverse problem Mw a b × log10 X can be derived by the reciprocal of the orthogonal regression parameters, as X ab b1 × Y and does not
need to be recalculated. Taking into account that the predictor variable X is also error-prone, thus, orthogonal regression
with equal weights reflects better the properties of the data
because all parameters are prone to uncertainties of about the
same range.
All scaling relation analyses, described in the following,
were performed with both regression methods. We found that
the differences between the orthogonal and ordinary relationships are minimal and for the range of available data
negligible. Standard errors of the regression coefficients
could not be reduced, and the coefficients of determination
R2 do not differ either. The errors of the data are thus symmetrically aligned around the regression line. We will discuss
in the following section the ordinary regression relations to
compare best with Wells and Coppersmith (1994) and other
published relationships, but we recommend the use of the
scaling relations derived by orthogonal regression (for clarity
highlighted in Tables 1 and 2). The results of the regression
analysis are listed in Tables 1 and 2.
Results
Are There Any Differences between the New
Scaling Relations and Those of Wells and
Coppersmith (1994)?
In the following, we compare the new scaling relations
dependent on the slip mechanism with those of Wells and
Coppersmith (1994) (Fig. 3). Rupture areas of large
(Mw > 7) thrust earthquakes are considerably shorter and
wider (where length is defined in the along-strike and width
in down-dip directions) compared with the scaling relations
of Wells and Coppersmith (1994). The slopes bW&C and bnew
of the two length-relationships differ by more than the standard deviation of the regression coefficient sbnew for the new
regression relation (see Tables 1 and 2 for details). The differences are larger for rupture width, in which the Wells and
Coppersmith (1994) slope deviates even by two standard
deviations of the new scaling relation (bW&C < bnew ). As
our dataset now contains earthquakes from subduction zone
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
(a)
3
reverse
1)
2)
(b)
2.5
reverse
log10W
log10L
2
2
1
0
log10L
3
5
6
7
Mw
8
1
0
9
(d)
normal
2)
1.5
0.5
2
log10W
(c)
new
W&C
y>1963
2919
2
5
6
7
Mw
8
9
6
7
Mw
8
9
6
7
Mw
8
9
normal
1.5
1
1
0.5
0
(e)
3
5
6
7
Mw
8
0
9
(f)
strike slip
5
strike slip
2
log10W
log10L
1.5
1
0
1
0.5
5
6
7
Mw
8
9
0
5
Figure 3. Regression solutions for the scaling relation log10 L a b × Mw and log10 W a b × Mw based on the database of
Wells and Coppersmith (1994) (dots) extended with additional events (asterisk). The circles mark the events prior to 1964. Squares denote
rupture width estimates from the Wells and Coppersmith (1994) database not judged as reliable. The solid line illustrates the regression
solution based on the entire database, the dash-dotted line illustrates the scaling relations based on the younger events only, and the dashed
line corresponds to the original Wells and Coppersmith (1994) relation. Legend in (a) is valid for all subplots. (a, b) Number 1 denotes the
1957 Aleutian earthquake (no width estimates available); number 2 denotes the Sumatra–Andaman 2004 earthquake.
thrust events, this result is not unexpected when compared to
the mere continental data accumulated by Wells and Coppersmith (1994). Generally, the brittle seismogenic layer of the
continental crust is thin compared to the brittle zone of subduction regimes extending much deeper depending on the
dip of the subducting slab. Large continental thrust earthquakes expand along the fault strike direction only after
breaking the entire crust. In contrast subduction events of
same magnitude can be wider, but they may not extend to
the same length as continental earthquakes. Only very large
subduction events are able to rupture the full extent of the
brittle zone such that further energy release leads to extent
in length only. The December 2004 earthquake in Sumatra
and the 1957 Aleutian earthquake may be of this kind as they
seem to be rather long (see Fig. 3, upper left plot; the two
events are marked).
For normal and strike-slip faulting earthquakes our newly derived scaling relations differ only marginally from Wells
and Coppersmith (1994). The differences of the regression
coefficients are smaller than one standard deviation of the
new scaling relations and are therefore considered negligible.
However, it is noteworthy that the standard deviations of all
regression coefficients were reduced considerably; for example, rupture length of normal earthquakes: sanew 0:23,
saW&C 0:37; sbnew 0:04, sbW&C 0:06.
Comparisons of our scaling relations for strike-slip
events with those of Mai and Beroza (2000), which were derived on a set of eight events spanning over the magnitude
range from Mw 5:9 to Mw 7:3, do not show considerable differences. To compare best with the results in Tables 1 and 2,
we rewrite their solution in the Mw space: log L 2:67 0:6 × Mw and log W 0:63 0:26 × Mw .
Sampling distributions from scaling relations. The decrease of the standard deviations of the regression coefficients quantifies the higher accuracy of the determination
of the regression coefficients, which has a positive impact
in many applications and is illustrated by the following
example. Assuming a synthetic database shall be compiled
using the scaling relations, in a first step a number of events
is characterized by some location and magnitude information. Second, rupture length and width are derived by
2920
L. Blaser, F. Krüger, M. Ohrnberger, and F. Scherbaum
evaluating the scaling relations given the magnitude. To
avoid having the same sizes for all rupture areas of earthquake with equal magnitudes (which would not be expected
in nature either), the rupture fault parameter would be
sampled from normal distributions according to
log10 L ∼ Na1 b1 × M; s2xy ;
(3)
a1 ∼ Na; s2a ;
(4)
b1 ∼ Nb; s2b :
(5)
The notation x ∼ Nμ; σ2 is read like variable x is sampled
from a normal distribution with mean value μ and variance
σ2 . The parameters a, b, sa, sb, and sxy are all provided in
Tables 1 and 2. The total variability of sampled length or
width estimates shrinks considerably using the regression
parameters based on our new relationships when compared
with Wells and Coppersmith (1994).
The enlargement of the dataset leads to a slight increase
of the standard deviation of the error term sxy caused by the
partially increased scatter of the data. However, the improvements on sa and sb seem to efficiently counteract the
increased scatter on sxy . Figure 4 summarizes the probability
distribution of sampled rupture lengths for magnitudes from
Mw 4 to Mw 9 derived from the scaling relations of Wells
and Coppersmith (1994) (lines) and the new relationship
(dotted lines).
When comparing the probability density functions of
distinct magnitudes in Figure 4, we note that the total variance of this model is dependent on the predictor variable
moment magnitude. This effect is called heteroscedasticity
and results from the variability of the slope; the rupture
length is drawn from a normal distribution with a mean value
depending on the regression coefficients (and their standard
deviation) and the magnitude. The spread of the sampled data
is enlarged with growing magnitude caused by the multiplication of magnitude and sampled regression coefficient b.
2.5
reverse
M= 4
5
6
7
8
As a result, the probability density functions tighten
even stronger (solid lines in Fig. 4). Further, the problem
of the dependence of the total variance on earthquake magnitude becomes negligible when accounting for the correlation of a and b.
As a first preliminary summary, we conclude that the
scaling relations based on the newly compiled dataset differ
considerably from those of Wells and Coppersmith (1994)
for reverse faulting earthquakes. The almost doubled amount
of data reduces the standard errors of the regression coefficients of all scaling relations independent on focal mechanism. This increased accuracy has a clear impact in
applications in which not only the mean value is of interest
but also its probability distribution.
In the next section we investigate more specifically differences between continental and oceanic/subduction zone
earthquakes out of the dataset.
9
2
W&C
multivariate sampling
1.5
no correlation of
regression coefficients
1
0.5
0
−2
This characteristic, however, contradicts the data in which
no dependence of the variance of rupture length/width on
the magnitude is observed.
Therefore, we analyzed the correlation of the regression
coefficients a and b using the leave-one-out cross-validation.
This resampling technique allows an estimation of the standard error of the determination of the regression parameters
by recomputing a and b N times (N equals the number of
entries in the database) excluding one sample each time. For
all relationships there was a strong correlation found
corra; b < 0:99. A stronger inclining slope results, thus,
in a lower intercept as the center of mass of the data cloud
remains close to constant.
Sampling a and b independently, therefore, leads to an
overestimation of the error. Although the range of variation
is quite small (e.g., reverse rupture length 2:34 <
ai < 2:21, 0:54 < bi < 0:57), it is worth accounting for
the correlation of the regression coefficients and sample from
a two-dimensional multivariate normal distribution (see
Table 3 for the covariance matrices) replacing equations 4
and 5 with
a1
a
(6)
∼N
;Σ :
b1
b
−1
0
1
2
3
4
5
log10(L)
Figure 4. Probability density functions of sampled (logarithmic) rupture lengths for different magnitudes using the scaling
relations of Wells and Coppersmith (1994) and our newly derived
relationships without and with accounting for the correlation of the
regression coefficients.
Is It Possible to Distinguish between Continental
and Oceanic/Subduction Scaling Relations?
We classified the 283 events in our database to be oceanic or continental. Oceanic denotes any offshore events as
well as subduction zone earthquakes, that is, hypocenter
can be attributed to subduction zone environment even if
the epicenter appears to be onshore. We found for thrust
faulting earthquakes, 70 continental events, and 25 earthquakes in subduction zones; for strike-slip events, 128 continental and 16 oceanic. The amount of data for normal
faulting oceanic earthquakes is too little (five rupture lengths,
three widths) to calculate meaningful scaling relations.
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
2921
Table 3
Covariance Matrices for the Preferred Scaling Relations in Tables 1 and 2
Slip Type
σxx × 105
σxy × 105
σyy × 105
Reverse
Orthogonal Regression
Normal*
Normal Orthogonal Regression*
Strike-Slip
Strike-Slip Orthogonal Regression
All Fault Types
All Fault Types Orthogonal Regression
Reverse
Reverse Orthogonal Regression
Normal
Normal Orthogonal Regression
Strike-Slip*
Strike-Slip Orthogonal Regression*
All Fault Types*
All Fault Types Orthogonal Regression*
24.22
26.14
197.54
222.24
11.21
12.37
2.72
3.13
28.31
27.47
299.80
264.18
12.72
13.48
4.83
4.67
3:39
3:67
29:02
32:34
1:76
1:94
0:41
0:48
3:88
3:77
47:71
42:02
2:04
2:18
0:76
0:73
0.48
0.52
4.31
4.75
0.28
0.31
0.06
0.07
0.54
0.52
7.64
6.73
0.33
0.36
0.12
0.12
Fault Parameter
Length
Length
Length
Length
Length
Length
Length
Length
Width
Width
Width
Width
Width
Width
Width
Width
*Subset of data excluding events prior to 1964.
The magnitude ranges of the available reverse faulting
continental and subduction events differ strongly: 4:8 ≤
Mw ≤ 8:4 and 6:1 ≤ Mw ≤ 9:5 for continental thrust and subduction earthquakes, respectively. The overlapping data
range spans 2.3 magnitudes only.
Our regression analysis on the available data shows
that rupture lengths of reverse faulting earthquakes on continents and in subduction environments differ considerably:
boceanic 0:62 0:04, bcontinental 0:54 0:03 (Fig. 5),
whereas surprisingly there is no obvious difference in rupture
widths. This result still holds for scaling relations derived on
the overlapping data range only. The standard deviations of
the regression coefficients increases little for continental
scaling relation but substantially for the oceanic regression
solution and the coefficients of determination decrease. A
larger dataset would definitely be desirable.
The rupture areas of reverse faulting earthquakes for
continental and subduction environment do not show
obvious differences. Figure 6 illustrates rupture areas against
magnitudes of our dataset. Because we have not derived
relationships for rupture area directly from the data, we simply multiply rupture length and width derived by our scaling
(a)
3
relations for the data range and plot the resulting line in
Figure 6. We do not see any systematic difference between
rupture areas of continental and subduction events of equal
moment magnitude despite the different material properties.
Regarding M0 μ × A × d (Kanamori and Anderson, 1975),
we conclude that in general mean slip d is reciprocally proportional to the rigidity μ for given seismic moment M0 and
rupture area A. This is consistent with the assumptions of
constant stress drop made by Bilek and Lay (1999) and supports the hypothesis that tsunami earthquakes (characterized
by extensive tsunami generation for given seismic moment;
Kanamori, 1972) result because of small rigidity of sediments in shallow subduction environments and the consequent large displacements.
However, the conclusion that rupture areas of thrust faulting continental and subduction zone events are of the same
size is in contradiction to the conclusions of Murotani et al.
(2008) stating that rupture areas of plate-boundary earthquakes are larger than those of crustal earthquakes. Murotani
et al. (2008) analyzed fault areas from source rupture models
with heterogeneous slip of 26 estimates of 11 plate-boundary
earthquakes in the Japan region between 1923 and 2003. They
(b)
reverse
reverse
2
log10W
2.5
log10L
2.5
2
1.5
1
continenal
all reverse
subdubtion
0.5
0
1.5
1
0.5
0
5
6
7
Mw
8
9
5
6
7
8
9
Mw
Figure 5. Scaling relations of rupture length and width depending on magnitude based on mere continental thrust (stars) or subduction
(open circles) earthquakes.
2922
L. Blaser, F. Krüger, M. Ohrnberger, and F. Scherbaum
6
5
log10A
Somerville et al. (1999)
3
Murotani et al. (2008)
Somerville et al. (1999)
reverse oceanic
all continental
all continental
2
1
0
reverse oceanic
Murotani et al. (2008)
4
4
5
6
7
Mw
8
9
10
Figure 6. Rupture areas of continental and oceanic/subduction
zone events versus moment magnitudes. Somerville et al. (1999):
rupture areas derived by slip model analyses of crustal earthquakes
of all slip; all continental: rupture areas (L × W) of continental
earthquakes in our new compiled dataset, all slip types; Murotani
et al. (2008): rupture areas of Japanese plate-boundary earthquakes
(partly multiple estimates) derived from slip model analysis; reverse
oceanic: rupture areas (L × W) of subduction zone earthquakes in
our new compiled dataset.
compared their results with scaling relations based on crustal
events derived by Somerville et al. (1999) (data and scaling
relations of Murotani et al. [2008] and Somerville et al. [1999]
are shown in Figure 6). Whereas the data of the crustal events
of Somerville et al. (1999) match well with our database, there
is a clear offset of about factor 2 in rupture area at given
moment magnitude between the scaling relations of Murotani
et al. (2008) and our subduction earthquake scaling relationship. The offset may result from regional differences as
Murotani et al. (2008) focused on Japanese events and
because of the different methods estimating the rupture areas.
All techniques of estimating rupture area are based on many
assumptions and simplifications. Beresnev (2003) summarized several sources of uncertainties in the process of rupture
area estimates based on slip distributions. On the other hand
the evaluation of aftershock distribution are also based on subjective judgment and other sources of uncertainty. This illustrates that the total uncertainties in rupture area estimation are
large in general. Therefore, it is questionable whether the differentiation between subduction zone and continental rupture
areas is reasonable at all.
For completeness we note that there is no difference
between the scaling relations of continental and oceanic
strike-slip events found in the data. This is remarkable
because the rigidity of the uppermost oceanic lithosphere
is higher than rigidity of the upper continental crust; therefore, for a constant seismic moment and equal rupture area
the mean slip d of oceanic
according to M0 μ × A × d,
events consequently would be smaller than that of continental earthquakes.
If continental and oceanic rupture areas can extend to the
same size, although the oceanic crust is generally thinner
than the continental one, this would support the hypothesis
that earthquakes can not only rupture the crust but are able to
propagate dynamically into the uppermost mantle. McKen-
zie et al. (2005) and Geli and Sclater (2008) found transform
faulting earthquakes rupturing within the upper oceanic mantle, whereas earthquakes in old continental lithosphere occur
almost entirely within the crust as McKenzie et al. (2005)
pointed out. They conclude that the mechanical behavior
of oceanic and continental upper mantle appears to depend
on temperature alone.
However, it should be noted that the relationships for
oceanic strike-slip events are prone to much larger uncertainties than the equivalent continental solution. The uncertainties may be simply too large to allow for such detailed
conclusions.
Do the Regression Results Differ When Accounting
for Data Derived after 1964 Only?
The quality of earthquake source parameter estimates
depends certainly on the quality of the recorded waveform
data. From the beginning of the establishment of global seismological observation, technology has continuously evolved.
In this realm we investigate the changes in geographic station
density, timing accuracy, and standardization of recording
equipment. The installation of the worldwide standard seismic
network (WWSSN) in the sixties led to a major improvement in
the amount and quality of recorded data. The recent use of
geodetic networks for improved earthquake size (rupture geometry and slip distribution) estimation will probably have a
similar impact in the advance of observation accuracy.
Whereas there are not yet enough data to calculate scaling
relations for earthquakes for which new array-based geodetic
data are available, we will perform a test on the effect on scaling relations when accounting for data since 1964 only as we
consider these data to be more reliable. In particular, the
reliability of geometrical fault parameters estimated from
aftershock distributions before 1964 is questionable.
We recalculated all regression relations on the dataset
excluding the events before 1964 (dotted and dashed lines
in Fig. 3). Differences of more than one standard deviation
between regression parameters from the two distinct datasets
are observed for normal faulting earthquake parameters as
well as for the rupture width of strike-slip events. Thrust
faulting scaling relations are not affected by the reduction
of the dataset. We will first discuss the new normal faulting
scaling relations and then those of strike-slip events.
The dataset for normal slip type is reduced substantially.
The overall range of rupture width estimates shrinks considerably as there is only a single estimate of a large earthquake
(Saitama 1933, Mw 8:4). The differences of the scaling
relations for the remaining data range (5:1 ≤ Mw ≤ 7:2)
are negligible. Rupture length estimates however are still
available for a wide data range due to the two outer-rise
events of Sumbawa 1977 (Mw 8:2) and Kuril 2007
(Mw 8:1). The exclusion of events prior to 1964 results in
longer rupture length estimates for normal faulting earthquakes. In general, we note that the differences between
the relationships are smaller than the standard deviations
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
of the regression coefficients. Nevertheless, we prefer the
scaling relations for normal faulting earthquakes based on
events after 1964.
For strike-slip events the rupture widths are much wider
for earthquakes larger than Mw ⪆6 considering recent events
only. In particular, one cluster of 12 width estimates between
10 and 20 km corresponding to earthquakes of magnitudes
larger than Mw > 7:5 is excluded from the full dataset. Only
three larger events (Mw ≥ 7:6) that occurred after 1964 with a
rupture width estimate around 10 km remain in the catalog
(gray squares in Fig. 3f). Investigations of the original literature showed that the width of these was set ad hoc without
any seismological evidence or direct observations. In conclusion, we excluded those from our database (Ⓔin Table S1,
available as an electronic supplement to this paper, the corresponding widths are set in parentheses). Rupture widths of
strike-slip events with Mw > 7 thus appear to be considerably
larger compared to the initial dataset. No obvious saturation
of the rupture width can be observed.
Hence, rupture areas of large strike-slip events
become larger according to our new scaling relations than
those derived by the relationships of Wells and Coppersmith
(1994) for a given moment magnitude. This is contradictory
to the findings of Hanks and Bakun (2002, 2008). They state
that Wells and Coppersmith (1994) are overestimating rupture area for a given moment magnitude. Hanks and Bakun
(2002) propose a bilinear source scaling model for rupture
area of large (Mw > 7) continental strike-slip earthquakes.
Our reanalyzed dataset does not give any indication for such
a bilinear model as there is no obvious saturation of rupture
width at the thickness of the crust. For rupture width of
strike-slip events, we propose to use the scaling relations
based on the data recorded after 1964. Furthermore, in the
case of slip type independent scaling, we suggest using scaling relations derived on the subset ≥ 1964, too.
Is It Worth Distinguishing between the Scaling
Relations for the Different Types of Focal
Mechanism?
Wells and Coppersmith (1994) stated that most of their
relationships as a function of slip type are not statistically
3.5
(a)
different at a 95% significance level. Thus, they consider
the regression for all types to be appropriate for most
applications.
We found clear statistical difference at a 95% level of significance (Student’s t-test) between all combinations of slip
types for the intercepts of the regression relations (Fig. 7).
The slopes also differ significantly for most combinations except for: reverse versus normal (length and width), reverse and
all (length and width), and strike-slip versus all (width only).
In order to not misinterpret apparent differences due to
distinct magnitude ranges, the t-tests were repeated for
regression relations derived on the basis of a dataset limited
to the maximal magnitude of 8.1. We still find the same analysis results and therefore exclude biased values due to
sampling.
In conclusion, we do not recommend the use of one scaling relation for all different kinds of slip types. It is better
to distinguish between the scaling relations for different slip
types.
Self-Similar Scaling
Finally, we will analyze the scaling relations in the context of self-similar earthquake scaling. There is an ongoing
debate whether the earthquake source parameters rupture
length, rupture width, and mean slip are scale invariant with
magnitude (Romanowicz, 1992,1994; Scholz, 1994,1997;
Wang and Ou, 1998; Mai and Beroza, 2000; and many more).
The discussion has been started with the definition of L and W
models by Scholz (1982) in which stress drop and mean slip
are determined by fault length and width, respectively.
The slopes of the regression relations of moment magnitude Mw and the logarithm of either rupture length, width,
or mean slip should be close to 0.5 in the case of self-similar
earthquake scaling (scaling with seismic moment the fault
parameters should scale like M0 ∝ L1=3 , M0 ∝ W 1=3 ,
M0 ∝ d1=3 , which follows from M0 μLW d assuming constant rigidity μ). Considering our (favorite) scaling relations
(marked with a dagger symbol in Tables 1 and 2), it becomes
obvious that the slopes b differ considerably from 0.5 at least
for rupture length and width of strike-slip events (b 0:64 0:02 length, b 0:33 0:02 width), reverse rupture
3
(b)
3
2.5
1.5
1
0.5
5
6
7
Mw
8
9
10
log (W)
2
reverse
normal
strike slip
reverse
normal
strike slip
all
2
10
log (L)
2.5
0
2923
1.5
1
0.5
0
5
6
7
Mw
8
9
Figure 7. Different regression solutions for the scaling relation log10 L a b × Mw and log10 W a b × Mw dependent on slip
type. The scaling relations of rupture length of normal faulting earthquakes and of rupture width of strike-slip as well as slip-independent
events are based on the reduced dataset, excluding all events older than 1964.
2924
L. Blaser, F. Krüger, M. Ohrnberger, and F. Scherbaum
length (b 0:57 0:02), and normal faulting width (b 0:36 0:04). Interestingly, the slopes of rupture length and
width sum up close to one independent on focal mechanism.
The size of the rupture area therefore is linearly proportional
to the moment magnitude. The aspect ratio L=W, however, is
dependent on earthquake magnitude. Considering our data,
we do not observe the L model nor the W model but rather an
A model in which mean slip scales with the rupture area
(under the assumption that μ is constant) and the magnitude
dependent aspect ratio varies for the different slip types.
•
Summary
For many seismological applications, the size and shape
of rupture areas has to be estimated because of the lack of
observations. Scaling relations are often in demand for subduction environments, in particular, because of the high seismic and tsunamigenic potential. Wells and Coppersmith
(1994) provide a set of well-established scaling relations
based on a large dataset of all slip types but with the explicit
exclusion of subduction zone events. To our knowledge there
are currently no published equivalent relationships feasible
for subduction zones. With the analysis of a large database
containing source parameter estimates of 283 continental and
oceanic/subduction zone earthquakes, we present and discuss scaling relations filling this gap. We calculated ordinary
least-square and orthogonal regression relations for moment magnitude and rupture length or width, respectively.
Orthogonal regression analysis accounts for uncertainties
of the predictor variable and therefore better reflects the uncertainty characteristics of the data. In order to compare with
previous works (Wells and Coppersmith, 1994; Somerville
et al., 1999; Mai and Beroza, 2000; Hanks and Bakun,
2002; Murotani et al., 2008), we always additionally used
the standard regression approach. The differences between
the scaling relations derived by the two methods are negligible within the range of available data.
The new combined catalog contains next 196 source
estimates of the Wells and Coppersmith (1994) dataset, other
carefully determined rupture geometry parameters published
by Geller (1976), Scholz (1982), Mai and Beroza (2000), and
Konstantinou et al. (2005), as well as particular publications
from 24 large and more recent earthquakes (Data and Resources section). Compared with Wells and Coppersmith
(1994), the amount of data could almost be doubled, not only
for reverse faulting but also for normal and strike-slip faulting earthquakes. The range of the earthquake magnitude,
however, was increased for reverse and normal faulting
earthquakes only.
The analysis of the combined catalog can be summarized as follows:
• The newly derived linear regression relations for moment
magnitude and the logarithm of rupture length and width,
respectively, differ considerably for large (Mw > 7) thrust
earthquakes. As expected for a model representing also
•
•
•
earthquakes from subduction zones, rupture length is
shorter compared to the estimates of Wells and Coppersmith (1994), whereas rupture width is larger.
The analysis of scaling relations of pure continental thrust
events and pure subduction zone earthquakes showed no
systematic difference between their size of rupture areas.
This is in contradiction to the findings of Murotani et al.
(2008). They compared rupture areas from plate-boundary
earthquakes with rupture areas of continental events from
Somerville et al. (1999), both derived by slip map inversions. The clear offset of about a factor of 2 between the
data from Murotani et al. (2008) and our data points to the
large uncertainties in rupture area estimation and raises the
question if the distinction between continental and subduction zone rupture areas can be resolved at all.
In 1964, the WWSSN was established, providing an increase of available data and the basis of higher accuracy
of earthquake source parameter estimation. We took this
technical improvement as an example to test the reliability
of the data and its corresponding influence on the derived
scaling relations. Restricting the dataset to the time interval
of the last four decades has a large influence on the scaling
relations of strike-slip rupture widths. According to the
data, there is no evidence of a saturation of rupture width.
This supports the hypothesis that (oceanic) earthquakes are
able to rupture not only the brittle crust but even further
down into the uppermost mantle as found at transform
faults by McKenzie et al. (2005) and Geli and Sclater
(2008). This interpretation is supported by the fact that we
could not observe differences between oceanic and continental rupture area sizes of strike-slip events. However, we
have to point out the large uncertainties in the more recent
data. As soon as sufficient rupture area estimates derived
by geodetic or other advanced techniques are available, an
effort should be made to analyze the effect of reduced
uncertainties and biases.
Wells and Coppersmith (1994) stated that there is no statistically significant difference between the scaling relations depending on the different slip types. The analysis
of the scaling relations based on the enlarged database
showed clear statistical differences at a 95% level of
significance. Therefore, we recommend using different
scaling relations depending on the focal mechanism.
The analysis of the slope values results in a new type of
model regarding the subject of self-similar earthquake scaling: mean slip seems to scale invariant to moment magnitude with the rupture area, whereas the aspect ratio L=W is
dependent on the moment magnitude and varies for the different slip types. The combined database does not support
the L model (Scholz, 1982) nor the W model (Romanowicz, 1992) but rather an A model.
Note Added in Proof
It has come to our attention that another set of scaling
relationships for subduction earthquakes has been produced
Scaling Relations of Earthquake Source Parameter Estimates with Focus on Subduction Environment
by a team at the Imperial College in London (Strasser et al.,
2010), and we would like to alert the reader to that model as
well since this alternative provides a means of addressing
epistemic uncertainty in such empirical relationships between magnitude and rupture dimensions.
Data and Resources
ⒺThe entire list of source parameter estimates is provided in Table S1 as an electronic supplement to this paper.
Global Centroid Moment Tensor (CMT) project catalog
(1976–2009) is available at www.globalcmt.org (last accessed March 2010). The source parameters of the 24 earthquakes added individually to the dataset were published by
Johnson et al. (1994); Johnson et al. (2001); Antonioli et al.
(2002); Delouis et al. (2002); Ammon et al. (2008); Aoi et al.
(2008); Briggs et al. (2006); Burchfiel et al. (2008); Carlo
et al. (1999); Catherine et al. (2005); Huang et al. (2008);
Klinger et al. (2005); Lay et al. (2005, 2009); Li et al.
(2005); Lin et al. (2002); Lynnes and Lay (1988); McGinty
and Robinson (2007); Ozacar and Beck (2004); Petersen
et al. (2009); Robinson et al. (2003); Satake and Tanioka
(1999); Shabestari et al. (2004); Stramondo et al. (2005);
Tatar et al. (2005); Umutlu et al. (2004); Yamanaka and
Kikuchi (2003); Yu et al. (2001).
Acknowledgments
We are grateful to Martin Mai, who suggested checking the regression
parameters with respect to self-similar scaling; John Ristau, who provided
important additional data; and editor Arthur McGarr for his careful and constructive reviews that helped to improve the manuscript. This work is funded
by the German Federal Ministry of Education and Research, Geotechnologien Frühwarnsysteme, Number 03G0648D.
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Institute of Earth and Environmental Sciences
University of Potsdam
Karl-Liebknecht-Str. 24/25
14476 Potsdam, Germany
lilian@geo.uni‑potsdam.de
kruegerf@geo.uni‑potsdam.de
mao@geo.uni‑potsdam.de
fs@geo.uni‑potsdam.de
Manuscript received 22 April 2010