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≤
n(M)=0 M>} (2)
and log n(M)=a-bM+log(-M) M<
n(M)=0 M≥} (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)=/1--BCe^-Bx ≥x≥0 ()
f(x)=/-+--Bx(-x) >x≥0 (6)
where x=M-M_S, = ln 10, =-M_S, and M_S is the lower limit of magnitude above which the data is complete.
The estimation of and in equation () by the method of moments was discussed by Okada (1970) and Cosentino et al. (1977). The equations proposed here are
expC(-2x)/Cx-x²=-2Cx-x²/2x²-x² (16)
=(2x-)/(x²-Cx) (15)
or x²/x²=2-(+2)/(e^BC-1)/1-/(e^BC-1) (18)
Bx=1-/(e^BC-1) (19)
The maximum likelihood method for equation () yields only one equation (equation (19), Page (1968), Okada (1970)). If we adopt =Max(x_i) as the second equation, the value is considerably biased. To correct the bias, a correction ΔC which is a function of and is proposed. For this correction we must use some estimated values for and .
To estimate and in equation (6) by the method of moments the following equations are used.
x²/x²(-+-1){-(++6)+-6}/{-(+2)+-2}² (25)
Bx=-(+2)+-2/-+-1 (23)
The maximum likelihood estimates of and in equation (6) can be obtained by the equation:
2-Bx=/S ∑^S_i=1 1/-x_i=(1--)/-+-1 (27), (29)
The accuracy of and 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 and . If we adopt the truncated G-R formula, the second method (which uses =Max(x_i)+ΔC) gives more accurate values, whereas the accuracy of 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 and 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 , , and η values of shallow earthquakes in Japan.

抄録全体を表示