地震のマグニチュード分布式のパラメータの推定

最大地震のマグニチュードcを含む場合

抄録

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≤c
n(M)=0 M>c} (2)
and log n(M)=a-bM+log(c-M) M<c
n(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-1e^-Bx(C-x) C>x≥0 (6)
where x=M-M_S, B=b ln 10, C=c-M_S, and M_S 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
expC(C-2x)/Cx-x²=C²-2Cx-x²/2x²-x² (16)
B=(2x-C)/(x²-Cx) (15)
or x²/x²=2-BC(BC+2)/(e^BC-1)/1-BC/(e^BC-1) (18)
Bx=1-BC/(e^BC-1) (19)
The maximum likelihood method for equation (5) yields only one equation (equation (19), Page (1968), Okada (1970)). If we adopt C=Max(x_i) as the second equation, the C value is considerably biased. To correct the bias, a correction ΔC 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+4BC+6)+2BC-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 ∑^S_i=1 1/C-x_i=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(x_i)+ΔC) 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.