In many cases, the log-frequency vs magnitude curves for earthquakes show considerable curvature, though the well-known G-R (Gutenberg-Richter) formula predicts a straight line. To represent such data, two modifications of the G-R formula have been proposed.
log
n(
M)=
a-
bM M≤
cn(
M)=0
M>
c} (2)
and log
n(
M)=
a-
bM+log(
c-
M)
M<
cn(
M)=0
M≥
c} (3)
These are called here the truncated G-R formula and the modified G-R formula, respectively. These equations can be written in the form of probability density function:
f(
x)=
B/1-
e-BCe-Bx C≥
x≥0 (5)
f(
x)=
B2/
e-BC+
BC-1
e-Bx(
C-
x)
C>
x≥0 (6)
where
x=
M-
MS,
B=
b ln 10,
C=
c-
MS, and
MS is the lower limit of magnitude above which the data is complete.
The estimation of
B and
C in equation (5) by the method of moments was discussed by Okada (1970) and Cosentino
et al. (1977). The equations proposed here are
exp
C(
C-2
x)/
Cx-
x2=
C2-2
Cx-
x2/2
x2-
x2 (16)
B=(2
x-
C)/(
x2-
Cx) (15)
or
x2/
x2=2-
BC(
BC+2)/(
eBC-1)/1-
BC/(
eBC-1) (18)
Bx=1-
BC/(
eBC-1) (19)
The maximum likelihood method for equation (5) yields only one equation (equation (19), Page (1968), Okada (1970)). If we adopt
C=Max(
xi) as the second equation, the
C value is considerably biased. To correct the bias, a correction Δ
C which is a function of
B and
C is proposed. For this correction we must use some estimated values for
B and
C.
To estimate
B and
C in equation (6) by the method of moments the following equations are used.
x2/
x2(
e-BC+
BC-1){
e-BC(
B2C2+4
BC+6)+2
BC-6}/{
e-BC(
BC+2)+
BC-2}
2 (25)
Bx=
e-BC(
BC+2)+
BC-2/
e-BC+
BC-1 (23)
The maximum likelihood estimates of
B and
C in equation (6) can be obtained by the equation:
2-
Bx=
C/
S ∑
Si=1 1/
C-
xi=
BC(1-
e-BC)/
e-BC+
BC-1 (27), (29)
The accuracy of
B and
C values determined by the above methods is estimated by Monte Carlo technique for the cases of
S=50, 100, 200, 400, and 800 and several values of
B and
C. If we adopt the truncated G-R formula, the second method (which uses
C=Max(
xi)+Δ
C) gives more accurate
C values, whereas the accuracy of
B values is almost the same as that obtained by the method of moments. If we adopt the modified G-R formula, the maximum likelihood method gives more accurate
B and
C values than the method of moments. The η value (η=
x2/
x2) is a useful index for the deviation of the distribution of data from the G-R formula (for the G-R formula, theoretical value for η is 2). An application of the present methods shows regional variations in
b,
c, and η values of shallow earthquakes in Japan.
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