Nat Hazards
DOI 10.1007/s11069-010-9531-8
ORIGINAL PAPER
Magnitude conversion problem for the Turkish
earthquake data
Aykut Deniz • M. Semih Yucemen
Received: 22 July 2008 / Accepted: 24 March 2010
Ó Springer Science+Business Media B.V. 2010
Abstract Earthquake catalogues which form the main input in seismic hazard analysis
generally report earthquake magnitudes in different scales. Magnitudes reported in different scales have to be converted to a common scale while compiling a seismic data base
to be utilized in seismic hazard analysis. This study aims at developing empirical relationships to convert earthquake magnitudes reported in different scales, namely, surface
wave magnitude, MS, local magnitude, ML, body wave magnitude, mb and duration
magnitude, Md, to the moment magnitude (Mw). For this purpose, an earthquake data
catalogue is compiled from domestic and international data bases for the earthquakes
occurred in Turkey. The earthquake reporting differences of various data sources are
assessed. Conversion relationships are established between the same earthquake magnitude
scale of different data sources and different earthquake magnitude scales. Appropriate
statistical methods are employed iteratively, considering the random errors both in the
independent and dependent variables. The results are found to be sensitive to the choice of
the analysis methods.
Keywords Earthquake magnitude conversion Moment magnitude Orthogonal regression Standard least squares regression
1 Introduction
An important primary step in carrying out a seismic hazard analysis is the compilation of a
comprehensive earthquake catalogue. Earthquake data can be obtained from various
sources, each of which may have different ways of reporting an earthquake, different
history in the development of recording instruments or different instrument spread, besides
the uncertainty in the nature of earthquakes. Additionally, there exist a number of widely
A. Deniz
Technological Engineering Services Co. Ltd, PO Box 45, Aksu, Antalya 07110, Turkey
M. S. Yucemen (&)
Department of Earthquake Studies, Middle East Technical University, Ankara 06531, Turkey
e-mail: [email protected]
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used different magnitude scales, such as body wave magnitude (mb), duration magnitude
(Md), local magnitude (ML), surface wave magnitude (Ms) and moment magnitude (Mw).
There are also various differences among these magnitude scales. For these reasons, it is
necessary to perform a comprehensive statistical analysis for the comparison of the same
earthquake magnitude scale records of different institutions and develop relationships for
the conversion and calibration of different earthquake magnitude scales. Especially in
seismic hazard analyses, it is quite important to use the appropriate conversion equations,
since biases introduced through conversion equations cause biases in the b-value of the
Gutenberg–Richter frequency–magnitude distribution and in the seismic hazard estimates
(Castellaro et al. 2006). Also, biases introduced through conversion equations cause biases
in the empirical strong ground motion prediction relationships, which are usually specified
in terms of an earthquake magnitude scale.
A number of relationships have been developed empirically between moment magnitude and other magnitude scales. Ulusay et al. (2004) performed a regression analysis on
221 records from 122 earthquakes that occurred in Turkey and proposed a set of equations
to convert the different magnitude scales to the moment magnitude scale. The relationships
developed in that study were obtained by using the standard least squares regression
analysis in which it is assumed that the independent variable (each magnitude scale to be
converted) is free from measurement error and considers the random error on the dependent variable (Mw) only. However, a certain degree of error is present also on the independent variables and thus the assumptions of the classical least squares regression fail.
Instead, a statistical method, considering the error in both the independent and dependent
variables should be utilized. One of the methods is the orthogonal regression, which is the
most reliable conversion method provided that the ratio between the variances of errors in
the magnitudes to be related is known (Castellaro and Bormann, 2007). Castellaro et al.
(2006) investigated the bias introduced if the analysis is based on standard regression when
compared to the analysis based on orthogonal regression and found out that large errors (up
to 0.4 m.u. in some magnitude scales) are possible for the Unified Italian Catalogue that
they have developed.
In addition to the studies mentioned above, Grünthal and Wahlström (2003) and
Stromeyer et al. (2004) have employed the chi-square maximum likelihood method. This
method also takes into account the errors both in the independent and in the dependent
variables. Utsu (2002) performed a comprehensive study for the earthquake magnitude
conversion problem and proposed non-linear conversion equations, but did not consider the
error in the dependent variable.
In this study, various data sets (which are listed in the next section) are utilized to form
the most complete earthquake catalogue of Turkish events. In view of the findings of
Castellaro and Bormann (2007), the orthogonal regression is selected as the basic statistical
model for the development of the empirical conversion equations among the earthquake
records in five different magnitude scales (mb, Md, ML, Ms and Mw). Here, an additional
deviation from the previous studies is the screening of the earthquake catalogues with
respect to a lower bound magnitude (in Mw) in an iterative way. This is because in seismic
hazard analysis, generally a lower bound is specified for the magnitudes. The iterative
analysis is carried out such that, due to the differences in magnitude scales in which
earthquakes are reported, initial minimum guesses for each magnitude scale should be
assigned to satisfy a minimum lower bound in the moment magnitude scale. The corresponding lower bounds in other scales are updated as the conversion equations are
developed and the sensitivity of the resulting conversion equations to these lower bounds is
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checked. The magnitude conversion problem as well as the associated statistical models is
evaluated and discussed in full detail throughout the study in the relevant sections.
2 Initial data and the pooled catalogue
Five large data sets were available for the study. These were provided by the Earthquake
Research Department (ERD) of the General Directorate of Disaster Affairs of Turkey
(GDDA-ERD 2004), Kandilli Observatory and Earthquake Research Institute of the
Bogazici University (KOERI 2004), International Seismological Centre (ISC 2004a, b) and
United States Geological Survey (USGS 2004a). Also, a small data set reported by ERD
was accessible in which earthquakes are given in more than one magnitude scale (Inan
et al. 1996). Another small data set of large magnitude earthquakes was made available by
USGS, in which large magnitude earthquakes prior to 1973 (the starting year of the main
USGS catalogue) were present (USGS 2004b). Inan et al. (1996), GDDA-ERD (2004) and
KOERI (2004) are based on the studies of domestic researchers and agencies.
From the available data sets, instrumental earthquake records between January 01, 1900,
and July 30, 2004, having epicentres in the area bounded by 34°–44° north latitudes and
24°–47° east longitudes (2° north–south and east–west extension of coordinates of Turkey)
are selected and included into the processed data.
Due to the differences among different magnitude scales, the minimum magnitude
corresponding to Mw = 4.5 has to be guessed for each one of the magnitude scales considered in the study. For this purpose, the conversion equation set of Ulusay et al. (2004) is
utilized. In the literature, there are a number of conversion relationships among the different magnitude scales. However, all are empirical and depend on the specific seismic data
base used, which more or less reflect the earthquake characteristics of the corresponding
locations as well as instrument calibration.
A Mw of 4.5 corresponds approximately to Ms = 3.6, mb = 4.4, Md = 4.3 and
ML = 3.7 in the other magnitude scales according to the equation set of Ulusay et al.
(2004). Earthquakes with magnitude Mw B 4.5 are assumed to cause no damage to
engineering structures (Yucemen 1992). Iterative application of statistical procedures to
the combination of the previously mentioned earthquake catalogues at various stages may
require smaller magnitude lower bounds than the above set (i.e. Ms = 3.6, mb = 4.4,
Md = 4.3 and ML = 3.7). To avoid possible missing data at the beginning, the initial
minimum values are set to slightly smaller values as: Ms = 3.5, mb = 4.2, Md = 4.2 and
ML = 3.6.
After compiling the preliminary pooled catalogue, time and space windows are specified to identify the same earthquake given by different data sources. For screening especially the old records, a time window of 3 min and a space window of 0.2° latitudes and
longitudes are selected. In other words, any two events reported in two different catalogues, which fall within the bounds of these time and space windows, are assumed to
belong to the same event. For relatively recent data, required precautions are taken
manually to avoid any mistake due to this assumption. Initially, 28,530 events are accumulated from the component catalogues. Total number of different events is determined to
be 16,015; approximately 44% of the records belong to other events with the specified
same-event-neighbourhood.
The small differences in geographic location, time and depth of an event as reported by
different catalogues are eliminated by assigning the average values of the corresponding
parameters to that event. Magnitude records of all institutions in all scales are tabulated and
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it is observed that even records of different institutions in the same scale differ for some of
the earthquakes. This is attributed to the way of processing the available instrumental data
or data collection capabilities of these institutions (Ambraseys and Melville 1982).
3 The magnitude conversion procedure
For the calibration of magnitudes in a given magnitude scale, two approaches are developed. In the first approach, the values reported by all data bases for an event in each
magnitude scale are averaged and this average value is taken as the magnitude of that event
in the selected scale. This is called as intra-scale averaging. In this approach, standard
error decreases as more data become available in the same scale. The moment magnitude
corresponding to this average value is denoted by Mw0 .
The second approach is based on running a series of statistical regression analyses to
investigate whether there is a systematic deviation between the reported magnitudes of
different catalogues in a single scale and making the necessary modifications between the
same-magnitude-scale records of these catalogues. This is called as intra-scale conversion.
The resulting moment magnitude corresponding to the intra-scale converted magnitudes is
denoted by M*w. In this approach, one of the data sources has to be chosen as the reference
catalogue, and magnitudes in each magnitude scale should be converted to the magnitude
scale of the reference catalogue. Then, these converted intra-scale magnitudes are averaged
so that a representative magnitude is obtained for the earthquake under consideration.
Intra-scale averaging and intra-scale conversion approaches are components of the intrascale adjustments.
The procedure of expressing a magnitude record in terms of an equivalent value in
another scale (namely, Mw) is named inter-scale conversion. Accounting the intra-scale
average as both the representative property of an event and performing intra-scale conversions introduce uncertainties prior to the inter-scale conversion. The relative significance of such uncertainties is investigated later in Sect. 7.
4 An overview of the regression analysis applied in the study
Regression analysis aims at the computation of a probabilistic relationship between several
variables described in terms of the mean and variance of a random variable as a function of
the value of other variables (Ang and Tang 1975). This is a probabilistic approach, because
there is no unique relationship but a range of possible values for the regression output. As
stated by Ang and Tang (1975), this range is represented by the mean and variance of the
relationship.
While the regression equation, in terms of its form and constants (i.e. the mean of the
regression equation), is given a great importance, its variance around the mean is often
ignored in engineering applications. This might result in the loss of information regarding
the relative uncertainty of the available regression outputs as indicated by their variances.
In the case of the earthquake magnitude conversion problem and in many similar
problems in earthquake engineering, the data set of concern is usually bounded by a lower
limit, since earthquakes with magnitudes less than this lower limit are expected to cause no
damage to engineering structures. For this reason, in such circumstances, the regression
analysis should be only capable to represent the relationship of the random variables under
consideration above the specified lower limit. Elements of the data set below this limit
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should be left out of the analysis. If the lower limit of the response variable is set to a
certain value and regression analysis is carried out on an arbitrarily selected data set, it
cannot be known, in advance, whether the resultant equation will satisfy the pre-specified
lower limit or not. Consequently, couple of iterative analyses may be required. At the end
of each iteration, by backward substituting the lower limit of the response variable into the
regression equation, the required lower bound of the predictor variable should be revised
and after removing the data below the new lower bound, the regression analysis should be
repeated. (Due to various reasons which will be discussed later, backward substitution of
the response variable into the regression equation to obtain the predictor variable in the
standard least squares regression is not correct. Standard least squares regression is not
reversible. But revising the lower bound in this way is applicable to orthogonal regression,
which is reversible.)
The iterative application of the regression analysis is shown in Fig. 1. In this figure, a
set of magnitude pairs is plotted. A single point, especially in the lower tail of the distribution, may stand for several observations. A magnitude m0 corresponding to the lower
bound with respect to the scale on the y-axis is to be specified. Initially, all points are
included into the analysis and line A is obtained. However, line A necessitates the consideration of magnitudes greater than the m1 limit on the x-axis. Ignoring smaller observations and repeating the regression analysis, line B is obtained. Lower limit on the x-axis,
corresponding to m0 on the y-axis now shifts to m2. Redoing the regression analysis, line C
is obtained, which almost coincides with line B, and does not require any further iteration.
Regression analysis has converged and using the equation of line C together with the data
set, for which the lower bound magnitude of the predictor variable is greater than m2, the
lower bound of the response variable is now obtained to be exactly m0. If such a
Fig. 1 Iterative regression analysis (m0: lower bound magnitude with respect to the Mresponse scale; m1, m2:
lower bound magnitudes with respect to the Mpredictor scale corresponding to m0 before and after first
iteration, respectively)
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convergence is not achieved, and for example the analysis is stopped after the initial run,
values below the lower limit in the predictor variable will create a bias in the results. Since
the lower tail is much populated than the upper portion, significant errors may occur. The
lower bound in the predictor variable might be selected arbitrarily or subjectively. But as
stated previously, in our study, instead of utilizing arbitrary data sets in the initial stage,
equations of Ulusay et al. (2004) were employed in order to establish the initial guesses of
the required lower bound magnitudes in each predictor scale.
The usual procedure while employing regression analysis is the standard least squares
regression because of its simplicity (Kendall and Stuart 1979). This method assumes that
the independent (predictor) variable (each magnitude scale to be converted) is free from
measurement error and that the random error on the dependent (response) variable (Mw) is
normally distributed and have a constant standard deviation in the whole regression
domain. However, error is present also on the predictor variables, which are the earthquake
magnitudes in this case, and thus the assumptions of the classical least squares regression
fail.
5 Orthogonal regression
Kendall and Stuart (1979) have proposed the orthogonal regression to account also for the
effects of measurement errors in the predictor variables. In the very general sense,
orthogonal regression has two variables denoted by y and x (y, standing for the dependent
variable and x, standing for the independent variable), linearly related and that their
measurement errors are independent normal variates with variances denoted by ry2 and
rx2, respectively. Assumption of being independent normal variates for these measurement
errors of the regression data is valid for earthquake records, for which systematic errors are
eliminated and uncertainty in the records is only due to the earthquake excitation arrival to
the station.
Although the steps seem straightforward and do not to introduce much extra effort
compared to the standard least squares regression, an additional knowledge is required in
orthogonal regression. This challenging unknown is the orthogonal regression estimator, g,
which is defined as:
.
ð1Þ
g ¼ r2y r2x
In this case, the slope, b, of the linear conversion relationship becomes a function of g.
The orthogonal regression is schematically illustrated in Fig. 2a. As observed in this figure,
the orthogonal distances between the observed data points and the regression line are
minimized instead of minimizing the vertical or horizontal distances as in the ordinary
standard least squares and inverted least squares regression (Fig. 2b, c, respectively).
In quantifying the orthogonal regression estimator, a number of researchers, which were
referred in Castellaro et al. (2006), have accepted g = 1.0 during their inter-scale conversion analyses. However, the reasoning behind the selection of the moment magnitude as
the representative scale of an earthquake is the accuracy in its determination. As the
response parameter in regression is Mw and variance of the error in Mw(ry2) is in the
numerator, it is not very likely to have g to be equal to or greater than 1.0. For example, if
the accuracy in the determination of Mw is assumed to be twice that of other scales in terms
of the variance of measurement errors, g may be taken as 0.5. Or alternatively, one may
perform a sensitivity analysis to investigate the variation of b as a function of g (since other
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Fig. 2 Schematic illustration of a orthogonal regression; b standard least squares regression; c inverted
standard least squares regression
parameters in the formulation of the orthogonal estimator of slope are fixed for a certain
data set) and select reasonable values.
In the following sections, first the intra-scale averaging procedure is applied; then, the
orthogonal regression is used for intra-scale conversion. Afterwards, both standard least
squares regression and inverted standard least squares regression are applied together with
the orthogonal regression for inter-scale conversion to the available data sets in a comparative manner and the results are discussed.
6 Intra-scale adjustments
The composition of the combined catalogue from a couple of component catalogues
reporting different estimators of earthquake magnitude is investigated by generating single
magnitude scale tables of the combined catalogue. In other words, all available mb, Md,
ML, Ms and Mw values of each earthquake are collected in the corresponding magnitude
scale table with respect to the reporting institutions. Averaging of the available records of
each earthquake in a selected magnitude scale is straightforward and the inter-scale conversion procedure is repeated alternatively based on the intra-scale averaged earthquake
magnitudes, which is discussed in the next section.
For intra-scale conversion, ISC (2004b) is selected as the reference catalogue and
variations of records in the same-magnitude-scales are sketched in Fig. 3. The reason for
the selection of ISC (2004b) as the reference catalogue is that: firstly, it provides records in
all magnitude scales including the Mw scale, allowing an inter-scale comparison; and
secondly, it is the most comprehensive component catalogue in terms of the number of
reported events after the KOERI (2004) catalogue, which does not contain any records in
the Mw scale. A series of orthogonal regression analyses are carried out among the same
magnitude scale records of different institutions. For the corresponding orthogonal
regression analyses, g is assumed to be equal to 1.0, because the variables being compared
are the same-magnitude-scale-records.
In Fig. 3, the correspondence between ISC (2004b) and GDDA-ERD (2004) is almost
perfect in all scales, except Md, since GDDA-ERD (2004) is mostly based on ISC (2004b).
Because there is not enough number of records, it is not possible to conduct an intra-scale
comparison in the Mw scale. In the body wave magnitude (mb) scale, there is no significant
deviation of records in different catalogues. On the other hand, in the case of duration
magnitude (Md) scale analysis, significant deviations for the same event from that reported
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Fig. 3 Variations of all catalogue records in the same magnitude scale (i.e. mb, Md, ML and Ms) with respect
to the reference catalogue (ISC 2004b)
in the reference catalogue are observed. In the Md scale, records due to GDDA-ERD (2004)
indicate a very small resultant slope of 0.32, whereas a relatively large intercept of 2.86 in
the correlation. This does not seem to explain reasonably the relationship between the
records of two different institutions in the same scale, which can be attributed to the higher
degree of uncertainty in the determination of duration magnitude as pointed out by Del
Pezzo et al. (2003).
In the local magnitude (ML) scale, KOERI (2004) slightly underestimates the magnitudes with the difference increasing as the earthquake magnitude increases. Finally, the
surface wave magnitude (Ms) scale is examined. The records in the surface wave magnitude (Ms) scale due to USGS (2004a, b) are systematically underestimated, based on a
considerable number of records. But the difference this time is only minor. Ms records of
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Fig. 4 Analysis of the reporting differences between the reference catalogue (ISC 2004b) and the KOERI
(2004) catalogue in the surface wave magnitude (Ms) scale
KOERI (2004) also show various differences; small magnitude events are significantly
overestimated in this catalogue, whereas large magnitude events show no deviation from
the reference catalogue. Since the number of small magnitude events is large in nature, any
linear regression will be highly affected from the crowded lower tail of the analysis
domain, as mentioned previously. Such a deviation will result in a diversion of the estimation of large magnitude events from the actual state. For this reason, and for only Ms
records of KOERI (2004), a bilinear intra-scale conversion relationship is proposed. The
slope change point between the two linear segments is determined after an analysis of the
reporting differences between the Ms records of the reference catalogue and the KOERI
(2004) catalogue as shown in Fig. 4. Here, the average of the two Ms records is given on
the x-axis, whereas the average difference of the records between the reference catalogue
and the KOERI (2004) catalogue is displayed on the y-axis. As it can be seen, at the
magnitude level of 5.5, the tendency changes. Below this level, KOERI (2004) records are
greater than the reference catalogue records, whereas above this level, the opposite is valid
except for magnitude level of 7.0. The set of conversion equations obtained based on the
above-mentioned intra-scale conversion considerations are also presented in Fig. 3. Note
that all results are expressed in terms of the magnitude scales reported by ISC (2004b) after
utilizing these relationships.
As mentioned before, the uncertainty and error introduced by the intra-scale adjustments deserve further attention especially during the inter-scale conversions. In the
orthogonal regression, the estimated covariance matrix of the regression equation constants
a and b can also be obtained (Fuller 1987). But this requires the detailed knowledge of the
variances of errors in the regression elements quantitatively. For this reason, only an
average estimate of converted magnitudes can be reached and quantification of the
underlying uncertainty is not possible with the data sets used in our study.
Parameters of the intra-scale conversion analyses are summarized in Table 1 (number of
observations, variances of the dependent and the independent variables, their covariance, b
value in the analysis of each correlation). In this table, s2x and s2y are the sample variances of
x and y, respectively, and sxy is the sample covariance between x and y.
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Table 1 Intra-scale orthogonal
regression analysis results for
each magnitude scale in the
standard case where g = 1.0.
The target magnitudes are
expressed in terms of the scales
reported by ISC (2004b)
Scale Predictor catalogue
No. of
s2x
observations
mb
GDDA-ERD (2004)
3,514
0.106 0.107 0.106 1.00
USGS (2004a)
1,571
0.088 0.089 0.078 1.01
Md
ML
Ms
s2y
sxy
b
Inan et al. (1996)
38
0.232 0.207 0.214 0.94
GDDA-ERD (2004)
15
0.043 0.024 0.007 0.32
KOERI (2004)
32
0.063 0.053 0.052 0.91
GDDA-ERD (2004)
17
0.039 0.039 0.039 1.00
USGS (2004a)
48
0.123 0.135 0.113 1.05
KOERI (2004)
50
0.111 0.098 0.081 0.92
GDDA-ERD (2004)
939
0.812 0.834 0.816 1.01
USGS (2004b)
42
1.025 1.119 1.033 1.05
USGS (2004a)
50
0.778 0.813 0.744 1.02
Inan et al. (1996)
26
0.610 0.638 0.604 1.02
KOERI (2004)
(linear intra-scale
conversion)
920
0.427 0.872 0.517 1.52
KOERI (2004)
(bilinear intrascale conversion,
for \5.5)
780
0.136 0.319 0.122 2.00
KOERI (2004)
(bilinear intrascale conversion,
for C5.5)
140
0.483 0.372 0.353 0.86
In the intra-scale conversions, assuming the orthogonal regression estimator, g, equal to
1.0 requires further elaboration. For this, a sensitivity analysis is performed with the data
provided in Table 1 and variation of b, with respect to g, is recorded while the other
variables used in the computation of b (i.e. s2x , s2y , sxy) are kept constant. It is observed that
the g = 1.0 assumption seems quite reasonable and provides approximately 15–30%
higher values than the standard least squares option. Only for the duration magnitude scale
(Md) of GDDA-ERD (2004) and the surface wave magnitude scale (Ms) of KOERI (2004),
unstable and quite variable trends are obtained. For this reason, selection of the appropriate
value of g becomes an important matter for these two magnitude scales.
7 Inter-scale conversions
Carrying out the regression analyses among different scales is not different from the intrascale conversion analyses, except for the selection of g. This time we set g = 0.5 assuming
that Mw estimates are twice more reliable than the estimates of the magnitudes in other
scales, the comparison being made in terms of the respective variances. This selection is
consistent in reflecting the reliability of the earthquake recordings in Turkey with respect to
different scales. However, it is further elaborated by conducting a sensitivity study, where
the variation of the b value (slope of the regression equation) with respect to g is examined
and the result of this sensitivity analysis is summarized and discussed later in this section.
After setting the value of g equal to 0.5, Eqs. (2.a–5.b) are obtained as the conversion
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relationships between Mw and the rest of the other magnitude scales. Note that the resultant
Mw obtained by performing averaging in the intra-scale considerations is denoted by Mw0 ,
whereas resultant Mw obtained by performing intra-scale conversions is denoted by M*w. In
these equations, subscript ‘‘ave’’ corresponds to the intra-scale averaged case, whereas
subscript ‘‘ref’’ corresponds to the intra-scale converted case.
Mw0 ¼ 2:34mbave 6:61
ð2:aÞ
Mw
¼ 2:34mbref 6:61
ð2:bÞ
Mw0 ¼ 1:27Mdave 1:12
ð3:aÞ
Mw
¼ 1:85Mdref 3:36
ð3:bÞ
Mw0
¼ 1:57MLave 2:66
ð4:aÞ
Mw ¼ 1:63MLref 2:88
ð4:bÞ
Mw0
¼ 0:54Msave þ 2:81
ð5:aÞ
Mw ¼ 0:54Msref þ 2:81
ð5:bÞ
As it can clearly be seen from these equations, results of the ‘‘ref’’ and ‘‘ave’’ methods
show no difference in the mb and Ms scales. However, in the duration magnitude (Md) and
in the local magnitude (ML) scales, both the slopes and the intercepts of the regression
equations show large differences. The difference in the duration magnitude scale is
attributed to the fact that most of the duration magnitude records were provided by GDDAERD (2004) and these records were previously intra-scale converted to duration magnitudes as reported by the reference catalogue (ISC 2004b), in which the conversion
uncertainty was large. On the other hand, the difference in the local magnitude scale is
attributed to the inconsistent local magnitude data set of KOERI (2004) when compared to
the reference catalogue, ISC (2004b).
In order to quantify the differences in the overall catalogue, based on intra-scale
averaging and intra-scale conversions, the distribution of (M*w - Mw0 ) is examined and the
resulting values are given in Table 2. In Table 2, the maximum M*w values are too large to
represent the real physical conditions, in both cases (a) and (b). These extreme magnitude
values are due to the intra-scale conversion relationship of GDDA-ERD (2004) catalogue
Table 2 Properties of (M*w – Mw0 ) distribution
(a)
(b)
max(M*w)
9.59
max(M*w)
9.59
min(M*w)
1.94
min(M*w)
1.94
max(Mw0 )
8.60
max(Mw0 )
8.60
min(Mw0 )
1.94
min(Mw0 )
1.94
Maximum difference
Mean difference
Variance
1.82
-0.161
0.024
Maximum difference
Mean difference
Variance
(a): Intra-scale conversion is applied in all scales in a linear form in obtaining
all scales in obtaining Mw0
1.82
-0.272
0.065
M*w.
Averaging is applied in
(b): Intra-scale conversion is applied in all scales in a linear form, apart from the bilinear Ms conversion of
KOERI (2004) records in obtaining M*w. Averaging is applied in all scales in obtaining Mw0
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in the Md scale. It is to be noted that, among the total 795 Md records, only 287 were
reported by GDDA-ERD (2004). The mean differences in these two cases result in the fact
that converted moment magnitudes based on intra-scale averaged earthquake catalogue
data (Mw0 ) are greater than the converted moment magnitudes based on intra-scale converted earthquake catalogue data (M*w). From seismic hazard analysis point of view,
although this remains on the safe side, it might yield an overestimated seismicity.
Here, alternatively only Ms records of KOERI (2004) (contributing to 83% of the total
number of earthquake records in the Ms scale) are subjected to intra-scale conversion (both
linear and bilinear). Then, the results are averaged together with the Ms records of other
catalogues. The average magnitudes in all scales are converted to the moment magnitude.
Then, the final converted moment magnitudes due to each of mb, Md, ML, Ms are averaged.
This aims at determining the significance of the overestimation of only Ms records by
0y
KOERI (2004). The moment magnitude obtained as such is denoted to be Mw since it
mostly displays the characteristics of converted moment magnitudes based on intra-scale
0y
averaged earthquake catalogue. Statistical parameters of the (Mw - Mw0 ) distribution is
illustrated in Table 3.
0y
The maximum Mw values given for cases (a) and (b) seem reasonably reflect the real
physical conditions this time. Still the mean differences in these two cases result in the fact
that converted moment magnitudes based on intra-scale averaged earthquake catalogue
data (Mw0 ) are greater. A comparison of Tables 2 and 3 reveals that there is no significant
0y
difference in the distribution of and M*w and Mw . This is certainly attributed to the following facts: the dominant magnitude scale in the combined catalogue is the Ms scale,
almost all the Ms records were provided by the KOERI (2004) catalogue as well as the Ms
records of KOERI (2004) are converted in both cases in the same way. Considering these
facts and recalling the need to employ a bilinear intra-scale conversion relationship for
KOERI (2004) records (Fig. 4), it is concluded that for all inter-scale conversions from mb,
Md and ML to Mw, there is no need to perform intra-scale conversion analyses among the
same-scale-records of different catalogues. For the Ms conversion of KOERI (2004)
records on the other hand, it is more favourable to employ a bilinear intra-scale conversion
relationship prior to inter-scale conversion. Finally, case (b) of Table 3 is selected to be
appropriate for our analyses.
In the case of orthogonal regression, sensitivity analyses for b are also performed, and
the parameters used are given in Table 4. As a function of g, the best estimate b varies
0y
Table 3 Properties of (Mw – Mw0 ) distribution
(a)
0y
max(Mw )
0y
min(Mw )
(b)
1.94
0y
max(Mw )
0y
min(Mw )
max(Mw0 )
8.60
max(Mw0 )
8.60
min(Mw0 )
1.94
min(Mw0 )
1.94
Maximum difference
Mean difference
Variance
8.60
0.34
-0.166
0.019
Maximum difference
Mean difference
Variance
8.60
1.94
0.61
-0.276
0.060
(a): Averaging is performed in all scales, only Ms records of KOERI (2004) are converted to the reference
0y
catalogue with a linear equation in obtaining Mw . Averaging is applied in all scales in obtaining Mw0
(b): Averaging is performed in all scales, only Ms records of KOERI (2004) are converted to the reference
0y
catalogue with a bilinear equation in obtaining Mw . Averaging is applied in all scales in obtaining Mw0
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Table 4 Inter-scale orthogonal regression sensitivity analysis parameters for the standard case where
g = 0.5
Parameter
mb ? Mw
conversion
mb-ave
No. of observations 92
Md ? Mw
conversion
ML ? Mw
conversion
Ms ? Mw
conversion
mb-ref
Md-ave
Md-ref
ML-ave
ML-ref
Mas-ave
Ms-ref
92
35
35
45
45
62
62
s2x
0.209
0.209
0.501
0.282
0.320
0.303
0.345
0.345
s2y
0.888
0.888
0.730
0.730
0.670
0.670
0.108
0.108
sxy
0.369
0.369
0.545
0.374
0.407
0.392
0.162
0.162
b
2.34
2.34
1.27
1.85
1.57
1.63
0.54
0.54
a
-6.61
-6.61
-1.12
-3.36
-2.66
-2.88
2.81
2.81
a
In this calculation, Ms reports of the KOERI (2004) catalogue are converted to the reference catalogue by
a bilinear intra-scale conversion equation
from 2.40 to 2.00 for both Mw0 and M*w analyses in converting mb to Mw; it varies from 1.35
to 1.15 for Mw0 and from 1.95 to 1.50 for M*w analyses in converting Md to Mw; it varies
from 1.65 to 1.35 for Mw0 and from 1.70 to 1.40 for M*w analyses in converting ML to Mw.
On the other hand, for Ms vs. Mw conversions, the best estimate b varies from 0.65 to 0.48
0y
for both Mw and M*w analyses. Furthermore, based on this sensitivity study, it is observed
that the b value corresponding to g = 0.5 is 2–4% larger than the b value corresponding to
g = 1.0 in all of the conversion equations, indicating a small difference in the results if the
more common assumption g = 1.0 has been used. However, even this small difference
may cause underestimation of seismic hazard due to the underestimation in the conversion
of large magnitudes if g = 1.0 has been used instead of g = 0.5.
The value of b asymptotically approaches to the b value obtained from the standard least
squares regression as g gets larger. This is due to the fact that since g represents the ratio of
variance of error in the response variable to the variance of error in the predictor variable, it
will diverge to infinity if the error in the predictor variable is very small when compared to
that in the response variable, reducing the orthogonal regression to the standard least
squares regression. Whereas, for values of g close to zero, the opposite is valid. g being
zero is unlikely, but it is very likely that g is less than 1.0 in the inter-scale conversion case.
Our analyses show that for the data set under consideration, the difference between the b
values of the standard least squares regression and the most critical case of orthogonal
regression (g close to zero) is in the order of 20%. g = 0.5 selection yields very close
results to the most critical case with approximately 15% larger slopes than the conventional
least squares method. Twenty percentage of difference between the two extremes is relatively low, and selection of g remains a minor problem in these analyses.
The standard least squares method gives the set of Eqs. 6–9. When compared to Eqs.
2.a–5.b, slopes of all four equations are smaller and intercepts are larger.
Mw ¼ 1:77mbave 3:80
ð6Þ
Mw ¼ 1:09Mdave 0:24
ð7Þ
Mw ¼ 1:27MLave 1:41
ð8Þ
Mw ¼ 0:47Msave þ 3:16
ð9Þ
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Table 5 Inter-scale standard least squares regression analysis parameters
Parameter
mb ? Mw
conversion
mb-ave
Md ? Mw
conversion
Md-ave
ML ? Mw
conversion
ML-ave
Ms ? Mw
conversion
Mas-ave
No. of
observations
92
35
45
62
Mean mx
4.9232
4.9057
4.1833
5.0161
Mean Mw
P
(xi yi)
P 2
(xi )
4.9092
5.1114
3.9044
5.5153
2,257.10
896.16
752.93
1,725.14
2,248.89
859.36
801.59
1,581.03
b
1.77
1.09
1.27
0.47
a
-3.80
-0.24
-1.41
3.16
a
In this calculation, Ms reports of the KOERI (2004) catalogue are converted to the reference catalogue by
a bilinear intra-scale conversion equation
Parameters of the standard least squares regression are summarized in Table 5 for the
inter-scale conversion, where mx represents either mb, Md, ML or Ms scales.
For the sake of comparison and for checking the validity of g = 0.5 assumption,
equations corresponding to the inverted (reverse) standard least squares regression are also
obtained. In this case, Mw is taken as the predictor variable and all of the other magnitude
scales are treated as the response variables. Accordingly, in the inverted standard least
squares regression, the horizontal distances between the observed data points and the
regression line are minimized, instead of minimizing the vertical distances as in the case of
standard least squares regression (see Fig. 2c). The conversion equations derived in this
way are as follows:
mbave ¼ 0:42Mw þ 2:89
ð10Þ
Mdave ¼ 0:75Mw þ 1:09
ð11Þ
MLave ¼ 0:61Mw þ 1:81
ð12Þ
Ms¼ave ¼ 1:49Mw 3:22
ð13Þ
In order to compare the slopes of the conversion equations obtained from the inverted
least squares regression (Eqs. 10–13) with those obtained from the orthogonal (Eqs. 2.a–
5.b) and the standard least squares (Eqs. 6–9) regressions, the inverse of the slopes should
be taken in Eqs. 10–13, which yields to: 2.38, 1.33, 1.64 and 0.67 for mb, Md, ML and Ms
conversions, respectively. In all cases, it is observed that the slope obtained from the
orthogonal regression is greater than that of the standard least squares regression but
slightly less than the inverse of the slope of the inverted (reverse) least squares regression.
This is expected and consistent with the theoretical basis of the regression procedures
implemented as well as our assumption for the value of g.
The above analyses were based on a catalogue, lower bounds of which (in all scales)
were guessed, based on the set of equations provided by Ulusay et al. (2004). At the
moment, with the presence of the pooled catalogue and our inter-scale conversion equations, those bounds can be updated to produce again a minimum Mw of 4.5 by a backward
substitution to Eqs. 2.a–5.b. Lower bound magnitude values obtained in this way are shown
in Table 6.
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Table 6 Updated lower bound
magnitudes to produce a minimum Mw of 4.5
Scale
Initial lower
bound
Lower bound due to
derived relations
mb
4.2
4.7
Md
4.2
4.4
ML
3.6
4.5
Ms
3.5
3.1
As can be seen, these are different from the selected initial values. mb lower bound has
increased from 4.2 to 4.7, Md lower bound has increased from 4.2 to 4.4, ML lower bound
has increased from 3.6 to 4.5 and Ms lower bound has decreased from 3.5 to 3.1. Actually,
a decrease in the Ms lower bound, which is in contrast to what is expected in view of the
other magnitude scales, draws significant attention. A detailed look to this phenomenon
reveals the following. Because most of the earthquake records in the surface wave magnitude scale are due to the KOERI (2004) catalogue and a bilinear intra-scale conversion
equation is employed to convert the Ms records of this catalogue to the reference catalogue,
an equivalent lower bound for the KOERI (2004) records should be obtained individually
for comparison with the initial lower bound in the corresponding scale. A lower bound of
3.1 with respect to the Ms scale of the reference catalogue corresponds to a lower bound of
4.1 in the same scale as reported by the KOERI (2004) catalogue (using the equation for
magnitudes less than 5.5 displayed in Fig. 3). Since all Ms records are not due to the
KOERI (2004) catalogue and they are not converted using the same intra-scale bilinear
correlation, a revised lower bound of 4.1 for KOERI (2004) catalogue is probably overestimated. However, it is adequate to observe the increase in Ms lower bound also, as in the
other cases.
Elimination of smaller records will have an influence on the regression constants.
Hence, all analyses are performed for the second time, with these new updated lower
bounds, to check whether there will be a significant change in the relationships. This is
named as the first iteration. Redoing the necessary calculations (assuming again g = 0.5),
the new conversion equations obtained are given in Eqs. 14.a–17.b for the orthogonal
regression. Iteration with updated lower bounds is applied only for the intra-scale averaged
magnitude data base for mb, Md and ML scales (new lower bounds to be 4.7, 4.4 and 4.5,
respectively) and intra-scale averaged magnitude data base for Ms scale (new lower bound
to be 4.1) including the bilinear intra-scale converted data base records of KOERI (2004).
Also, standard least squares regression analyses are performed for the sake of comparison
and the resulting conversion equations are presented in Eqs. 14.a–17.b. (Sub-index (a)
refers to the results of the orthogonal regression, whereas sub-index (b) refers to the results
of the standard least squares regression.). This time equations corresponding to the inverted
standard least squares regression are not derived.
Mw ¼ 2:25mbave 6:14
ð14:aÞ
Mw ¼ 0:71mbave þ 1:79
ð14:bÞ
Mw ¼ 0:83Mdave þ 1:20
ð15:aÞ
Mw ¼ 0:64Mdave þ 2:19
ð15:bÞ
Mw ¼ 6:38MLave 26:84
ð16:aÞ
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Mw ¼ 0:26MLave þ 3:81
ð16:bÞ
Mw ¼ 0:54Msave þ 2:81
ð17:aÞ
Mw ¼ 0:47Msave þ 3:16
ð17:bÞ
Sensitivity analyses of b, using the combined catalogue based on the updated lower
bound magnitudes, show that the variation with respect to the selection of g is more critical
than what was observed in the previous run for all magnitudes. The orthogonal estimator of
the slope (for g = 0.5) is 3 and 25 times greater than the conventional standard least
squares estimation for the mb and ML scales, respectively. For the remaining scales (Md and
Ms), a smaller change has occurred and approximately 30% larger values are obtained. For
instance, at the magnitude level of 6.0 for mb, Md, ML and Ms scales, the orthogonal
regression estimation of the corresponding Mw value is 1.22, 1.02, 2.13 and 1.01 times
greater than the standard least squares estimation of the corresponding Mw value.
In Table 7, the values of the parameters employed during the first iteration with updated
lower bounds are given, where s2x and s2y are the sample variances of x and y, respectively,
and sxy is the sample covariance between x and y. In Table 8, standard least squares
regression parameters are summarized in tabular form. In this table, mx again represents
either of the mb, Md, ML or Ms scales.
First, iteration results do not require any further modification of lower bounds of mb and
Ms scales. On the other hand, the required Md lower bound decreased from 4.4 to 4.0 and
the required ML lower bound increased from 4.5 to 4.9. A decrease in the lower bound after
an increase in the previous iteration for the Md scale indicates the lack of robustness of the
Table 7 Inter-scale orthogonal
regression sensitivity analysis
parameters for the standard case
where g = 0.5 (first iteration)
Parameter
mb ? Mw
conversion
mb-ave
No. of
66
observations
Table 8 Inter-scale standard
least squares regression analysis
parameters (first iteration)
27
ML ? Mw
conversion
ML-ave
12
Ms ? Mw
conversion
Ms-ave
62
s2x
0.088
0.264
0.137
0.345
s2y
0.170
0.170
0.291
0.108
sxy
0.062
0.168
0.035
0.162
b
2.25
0.83
6.38
0.54
a
-6.14
1.20
-26.84
2.81
Parameter
mb ? Mw
conversion
mb-ave
No. of
66
observations
123
Md ? Mw
conversion
Md-ave
Md ? Mw
conversion
Md-ave
ML ? Mw
conversion
ML-ave
Ms ? Mw
conversion
Ms-ave
27
12
62
5.0161
Mean mx
5.1535
5.1907
5.0083
Mean Mw
P
(xi yi)
P 2
(xi )
5.4508
5.5111
5.1167
5.5153
1,853.00
776.755
307.90
1,725.14
1,758.57
734.343
302.51
1,581.03
b
0.71
0.64
0.26
0.47
a
1.79
2.19
3.81
3.16
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statistical procedure with respect to this magnitude scale. It should be noted that during the
first iteration analysis between Md and Mw, the number of observations has decreased from
35 to 27. A similar observation can be made in the case of ML scale. Although the lower
magnitude limit continues to increase after the first iteration, Equation (16.a) might be
considered to be far from representing an earthquake magnitude conversion relationship.
Using this equation, magnitudes at high levels in the ML scale correspond to very large
magnitudes in the Mw scale and magnitudes at low levels in the ML scale correspond to
very low magnitudes in the Mw scale, both of which are physically not possible. During the
first iteration analysis between ML and Mw, the number of observations has decreased from
45 to 12, with which it is not possible to conduct a reliable regression analysis.
Finally, since the analyses of mb and Ms scales have converged at the end of the first
iteration, these results can be used with confidence. In order to overcome the difficulty in
concluding the magnitude conversion problem in the remaining two scales, the conversion
equations obtained at the stage of the initial analyses for Md and ML scales are adopted, as
these analyses are based on a larger number of observations. It is to be noted that the nonconverging results of Md and ML scales create a minor overall uncertainty, because the
number of Md and ML records are relatively small (4.53 and 12.64%, respectively) in the
whole pooled catalogue.
Comparison of the first iteration results with the initial results reveals that the discussion
of the need to employ an iterative regression analysis as presented in Sect. 4 with the
preparation of Fig. 1 seems to have minor influence on our results. Certainly, this issue
depends very much on the data base under consideration. In our data base, further revision
of the lower bounds in the Md and ML scales makes it impossible to carry out a statistical
treatment of the problem. Hence, it is concluded that the conversion relationships in their
final form should be as follows:
Mw ¼ 2:25mbave 6:14
ð18Þ
Mw ¼ 1:27Mdave 1:12
ð19Þ
Mw ¼ 1:57MLave 2:66
ð20Þ
Mw ¼ 0:54Msave þ 2:81
ð21Þ
Magnitude lower bounds corresponding to Mw = 4.5 are, on the other hand, determined
to be 4.7, 4.2, 3.6 and 4.1 for mb, Md, ML and Ms scales, respectively. Final earthquake
catalogue, after applying Eqs. 18–21, consists of 4,752 records with Mw C 4.5, forming the
Unified Turkish Earthquake Catalogue that is given in Deniz (2006) and can be accessed
at: http://etd.lib.metu.edu.tr/upload/12607063/index.pdf.
8 Summary and conclusions
A set of empirical equations are developed to convert earthquake magnitudes in the mb,
Md, ML and Ms scales to the Mw scale. For this purpose, a Turkish earthquake data
catalogue (1900–2004) is compiled from domestic and international sources. The earthquake reporting differences of these data sources are assessed. Conversion relationships are
also established between the same earthquake magnitude scale of different data sources
and different earthquake magnitude scales. It is to be noted that all of the earlier regression
equations proposed in Turkey to convert magnitudes from one scale to another were based
on the standard least squares method. In this study, the conversion equations are developed
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by using the orthogonal regression procedure as recommended in the recent literature on
the magnitude conversion problem (e.g. Castellaro et al. 2006, Castellaro and Bormann
2007). Standard and inverted (reverse) least squares regression analyses are also calculated
for the sake of comparison. The results are found to be sensitive to the choice of the
analysis methods. Consistent with the underlying theory, methods considering the random
error both in the independent and in the dependent variables are observed to yield higher
converted earthquake magnitudes at large magnitude levels and vice versa at small magnitude levels when compared to the conventional statistical approach.
The past earthquake data for Turkey, reported by the main data bases, are pooled and
the earthquakes reported in different scales are all converted to the moment magnitude
scale by using the conversion equations derived in the study. In order to achieve the direct
use of this catalogue in seismic hazard studies, the lower bound is set to Mw = 4.5, which
is generally assumed as the lower bound for civil engineering purposes. The earthquake
catalogue obtained in this way is called the Unified Turkish Earthquake Catalogue and is
made available at the following address: http://etd.lib.metu.edu.tr/upload/12607063/
index.pdf. The catalogue covers a period lying between January 1, 1900, and July 30,
2004, and contains 4,752 records with Mw C 4.5.
The main conclusions of our study are itemized below:
(i) In order to compile the most complete earthquake catalogue, all the available seismic
databases providing past earthquake data for Turkey are utilized. Various differences
of reporting in the data bases of different data sources within a selected earthquake
magnitude scale are observed. However, no systematic deviation of reports, except
the Ms reports of the Kandilli Observatory and Earthquake Research Institute, is
seen.
(ii) In processing the catalogue data for inter-scale conversions from mb, Md and ML to
Mw, it is observed that there is no need to perform intra-scale conversion analyses
among the same-scale-records of different catalogues, and the average of the reports
provided by different data sources can be considered as the representative property
of an earthquake in a selected scale. However, prior to the inter-scale conversion of
Ms records of KOERI (2004) to Mw, it is recommended to employ a bilinear intrascale conversion relationship and then again the average of the Ms records due to
different data sources and the intra-scale converted Ms records of KOERI (2004) can
be used.
(iii) The presence of five different magnitude scales (mb, Md, ML, Ms and Mw) in the
available earthquake catalogues makes it necessary to perform regression analyses to
develop empirical conversion equations with respect to a selected magnitude scale.
In the Unified Turkish Earthquake Catalogue, which is compiled as a part of this
study, the moment magnitude (Mw) is selected as the target scale. While determining
the lower bound magnitudes in each scale corresponding to a certain converted lower
bound moment magnitude (Mw = 4.5 in our study), an iterative approach is
employed since the lower bound magnitudes in the initial scales and the target lower
bound magnitude depend on each other.
(iv) Comparison of the first iteration results with the initial results reveals that iterative
regression analysis seems to have a minor influence on the results. Certainly, this
issue depends very much on the data base under consideration. In our data base,
further revision of the lower bounds in the Md and ML scales reduces the sample size
significantly, which hinders the conduct of reliable regression analyses.
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(v)
During the development of the conversion equations among different magnitude
scales, statistical methods considering the random errors both in the independent
and in the dependent variables should be employed. In our study, for this purpose,
the orthogonal regression is utilized. When compared to the standard least squares
method, orthogonal regression yielded larger slopes but smaller intercepts in all
cases considered in our study as expected and consistent with the underlying theory
of the orthogonal regression.
(vi) Based on our experience within the context of this study, we also recommend that
the orthogonal regression be utilized in driving magnitude conversion equations.
However, orthogonal regression requires an additional input parameter compared to
the standard least squares regression. This parameter is the orthogonal regression
estimator, g, which is defined as the ratio of the variance of the error in the moment
magnitude (or the selected magnitude scale) to the variance of the error in the
magnitude to be converted. Estimating the g values is a current research subject, and
the guidelines given by Castellaro and Bormann (2007) provide a useful reference in
this respect. The assumed values in our study are consistent with the values
recommended by other researchers (such as, Castellaro et al. 2006, Castellaro and
Bormann 2007).
(vii) The final equations recommended for use in converting the earthquake magnitudes
reported in body wave magnitude (mb), duration magnitude (Md), local magnitude
(ML) and surface wave magnitude (Ms) scales to the moment magnitude (Mw) scale
are given by Eqs. 18–21, respectively.
Acknowledgments The authors thank the journal reviewers for their critical reading of the manuscript and
providing valuable comments on the initial manuscript.
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