Modified GR

1. Bulletin ofthe SeismologicalSocietyof America,Vol.75, No. 1, pp. 319-322, February 1985
A MODIFIED FORM OF THE GUTENBERG-RICHTER MAGNITUDE-
FREQUENCY RELATION: MAXIMUM LIKELIHOOD ESTIMATION
OF ITS PARAMETERS
BY ANDRZEJ KIJKO
In the two recent papers, Lomnitz-Adler and Lomnitz (1978, 1979) have proposed
a double exponential magnitude-frequency relation
In G(x) = A - B exp(ax), (1)
where a, A and B are constants, and G(x) is the cumulative exceedance of the
magnitude x*. The same relation was used by Lomnitz-Adler (1983) who presented
a statistical model of spatial and stress drop frequency distribution. The same
magnitude-frequency relation results from the solution of statistical dynamic equa-
tion for the threshold and asperity model of fault rupture (Lomnitz-Adler, 1984).
Introduction of formula (1) has initiated a number of studies considering physical
implications and the range of application of this relation (e.g., Kijko and Sellevoll,
1981; Jones et al., 1982; Kijko, 1982; Dessokey, 1983; Gan and Tung, 1983).
In this paper, taking into account the restraints of different fitting techniques,
the maximum likelihood procedure for the estimation of parameters a and B is
described. The maximum likelihood method was introduced previously by Utsu
(1965) and Aki (1965) to obtain b value in the Gutenberg-Richter magnitude-
frequency relation. Utsu and Aki formulas, derived originally for exact continuous
magnitudes and an infinite maximum magnitude, were modified by Page (1968) for
continuous magnitudes and a finite maximum. Karnik (1971) has derived similar
formulas for interval magnitude data and Bender (1983) for magnitude grouped
data. The maximum likelihood estimation can be found by setting the partial
derivatives of the logarithm of the likelihood function
N
L(XI O) = YI g(X~ ]O) (2)
i=1
equal to zero, where g(Xi JO) is the density function of the following form
g(x [O) = G(x). a.B exp(ax), (3)
in which O -- (a, B) are the sought parameters and X = (X1, ..., XN) are the
earthquake magnitudes, complete for magnitude ->Mmin in a given interval of time.
Setting 3 In L/aa = 0 gives
1/a + (X) - B((XY) - X,Y~) = O, (4)
* For the normalized version of equation (1), the A value is equal B exp(aMmi.), where Mminis the
threshold magnitude in a catalog. An alternative, formal normalization, was proposed by Lomnitz-Adler
and Lomnitz (1982).
319

2. 320
where
LETTERS TO THE EDITOR
Zl = Mmin,
}'1 = exp(aX~),
N
<x> = Y X~/N,
i=1
N
(Y) = Z YjN
i=1
Yi = exp(~Xi),
and setting ~ In L/~ B = 0 gives
1
B <Y) + Y~= 0. (5)
Substituting B in relation (4) by B calculated from relation (5), we obtain
1 (XY> - X~Y~
- + <X> - (6)
<Y) - y~
Since relation (6) is independent of the parameter B, it provides an equation useful
for the maximum likelihood estimation of a. Equation (6) can be solved by an
iterative procedure which can be performed even by a small desk computer. The
parameter B can be estimated from relation (4) or (5) after replacing the parameter
by ~ obtained from equation (6).
The variance of ~ and/~ can be found from the following relations
~,2 = -c/d.N,
gB2 = -aid.N, (7)
where
a = _[~-2 + j~((X2y> _ X12Y1)].
b = _(~-1 _]_ {X))//},
c -- 1//} 2,
and
d = ac - b2.
The described procedure for the estimation of a, B, and A was applied to the
same data as those used by Lomnitz-Adler and Lomnitz (1979), i.e., Catalogue of
Chinese earthquakes, 780 B.C. to 1973 A.D. (Academia Sinica, 1974). This catalog
contains events 629 with the threshold magnitude Mmin = 6.0. The following values

3. LETTERS TO THE EDITOR 321
were obtained: 0.10 + 0.03, 10.34 ___4.90, and 19.46 for h + ~,/} + ~ and A,
respectively. In order to fit the same empirical data, Lomnitz-Adler and Lomnitz
(1979) have set a = 0.17.
Some inconsistency following from the estimated value of a = 0.10 should be
pointed out. The constant is none other than the well-known parameter ~ from the
following relation
In Mo = aM + ~, (8)
which except for very large earthquakes has a value of about 1.5 In 10 = 3.45 (e.g.,
Kanamori and Anderson, 1975; Jones et al., 1982). This inconsistency was pointed
out by P. Mechler (Lomnitz-Adler and Lomnitz, 1978).
We hope that application of the relations presented here to different seismic
regions would be helpful for better understanding of the new magnitude-frequency
relation's applicability and its restraints.
ACKNOWLEDGMENTS
I wish to thank Dr. I. F. Jones from University of British Columbia, Vancouver, Canada for helpful
suggestions and to Dr. J. Lomnitz-Adler from the National Universityof Mexico for sendinga preprint
of his paper "The Time-DependentStatistics of Threshold and Asperity Models."
REFERENCES
Academia Sinica {1974). Catalogue of great earthquakes in China, 780 B.C. to 1973 A.D. (in Chinese),
Institute of Geophysics, Peking, 31 pp.
Aki, K. (1965). Maximum likelihood estimate of b in the formula log N = a - bM and its confidence
limits, Bull. Earthquake Res. Inst., Tokyo Univ. 43, 237-239.
Bender, B. (1983). Maximum likelihood estimation of b values for magnitude grouped data, Bull. Seism.
Soc. Am. 73,831-851.
Dessokey, M. M. (1983). Statistical models of the seismic risk analysis for miningtremors and natural
earthquakes, Ph.D. Thesis, Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland,
120 pp.
Gan, Z. J. and C. C. Tung (1983). Extreme value distribution of earthquake magnitude, Phys. Earth
Planet. Interiors 32, 325-330.
Jones, I. F., L. Mansinha, and P.-Y. Shen (1982). On the double exponential frequency-magnitude
relation of earthquakes, Bull. Seism. Soc. Am. 72, 2373-2375.
Kanamori, H. and D. L. Anderson (1975). Theoretical basis of some empirical relations in seismology,
Bull. Seism. Soc. Am. 65, 1073-1096.
Karnik, V. K. (1971). Seismicity of the European Area, Part 2, Academia, Publishing House of the
Czechoslovak Academy of Sciences, Praha, Czechoslovakia, 123-169.
Kijko, A. (1982). A comment on "a modified form of the Gutenberg-Richter magnitude-frequency
relation," Bull. Seism. Soc. Am. 72, 1759-1762.
Kijko, A. and M. A. Sellevoll (1981).Triple exponentialdistribution,a modified model for the occurrence
of large earthquakes, Bull. Seism. Soc. Am. 71, 2097-2101.
Lomnitz-Adler, J. (1983). A statistical model of the earthquake process, Bull. Seism. Soc. Am. 73, 853-
862.
Lomnitz-Adler, J. (1984). The time dependent statistics of threshold and asperity models, preprint, 39
PP.
Lomnitz-Adler, J. and C. Lomnitz (1978).A new magnitude-frequency relation, Tectonophysics 49, 237-
245.
Lomnitz-Adler, J. and C. Lomnitz (1979). A modified form of the Gutenberg-Richter magnitude-
frequency relation, Bull. Seism. Soc. Am. 69, 1209-1214.
Lomnitz-Adler, J. and C. Lomnitz (1982). Reply to A. Kijko's "a comment on a modified form of the
Gutenberg-Richtermagnitude-frequency relation," Bull. Seism. Soc. Am. 72, 1763.

4. 322 LETTERS TO THE EDITOR
Page, R. (1968). Aftershocks and microaftershocks of the great Alaska earthquake of 1964, Bull. Seism.
Soc. Am. 58, 1131-1168.
Utsu, T. (1965). A method for determining the value of b in a formula log n = a - bM showing the
magnitude-frequency relation for earthquakes (with English summary), Geophys. Bull. Hokkaido
Univ. 13, 99-103.
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Manuscript received 13 April 1984