- 著者
-
宇津 徳治
- 出版者
- 公益社団法人 日本地震学会
- 雑誌
- 地震 第2輯 (ISSN:00371114)
- 巻号頁・発行日
- vol.31, no.4, pp.367-382, 1978-12-25 (Released:2010-03-11)
- 参考文献数
- 7
- 被引用文献数
-
5
18
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-1e-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 areexpC(C-2x)/Cx-x2=C2-2Cx-x2/2x2-x2 (16)B=(2x-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 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 ∑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)+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.