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从世界上M≥7.5浅源地震伴随的81个余震序列,以及日本及其周围大地震的67个余震序列,确定了古登堡—里克特定律中b值的先验分布。我们假设先验统计总体中的b值服从伽马分布Γ(φ,ζ),其中φ和ζ分别代表形状和标度参数。对每个序列所得到的b值都与统计总体中的随机采样一致,但有统计误差。在本文中我们用最大似然法估计总体分布中的φ和平均值,b_(avg)=φζ。对世界上M≥5.0的余震序列估计的φ和b_(avg)分别是55和1.13,而对D<2.55的余震序列的估计值分别为28和1.22,其中D为主震和余震的震级差。对于日本的余震序列我们得到φ=28,b_(avg)=0.97。蒙特卡罗法的数值检验表明,所估计的b_(avg)相当接近于输入值,而φ却往往比输入值大。那些较小的值,如全球序列的φ=26～31,日本序列φ=20～25,或许比由最大似然法得到的估计值更合适些。假设b值的先验分布是Γ(φ,b_(avg)/φ),对一组N个地震,由最大似然法得到的估计值b=(N+φ-1)/(N/b_U+φ/b_(avg)),其中b_U是没用宇津所提出的先验分布而用最大似然法得到的估计值。所估计的值b比b_U更稳定可靠。特别是对于只有很少地震数据时,这里提出的方法很有价值。
Based on the 81 aftershocks accompanied by M≥7.5 shallow earthquakes in the world and 67 aftershocks of Japan and the surrounding earthquakes, the prior distribution of b in Gutenberg-Rickett law is determined. We assume that the b value in the priori statistical population obeys the gamma distribution Γ (φ, ζ), where φ and ζ represent the shape and scale parameters, respectively. The b values obtained for each sequence are consistent with the random sampling in the population, but with statistical errors. In this paper, we use the maximum likelihood method to estimate the overall distribution of φ and average, b_ (avg) = φζ. The estimated φ and b_ (avg) of the aftershock sequence with M≥5.0 in the world are 55 and 1.13 respectively, while the estimated aftershock sequences with D <2.55 are 28 and 1.22, respectively, where D is the magnitude difference between the main and aftershocks . For the aftershock sequence in Japan we have φ = 28 and b_ (avg) = 0.97. Numerical tests of the Monte Carlo method show that the estimated b_ (avg) is quite close to the input value, whereas φ is often larger than the input value. Those smaller values, such as φ = 26-31 in the global sequence and φ = 20-25 in the Japanese sequence, may be better than the estimates obtained by the maximum likelihood method. Suppose the prior distribution of b is Γ (φ, b_ (avg) / φ). For a set of N earthquakes, the estimated maximum likelihood b = (N + φ-1) / (N / b_U + φ / b_ (avg)), where b_U is the estimate obtained by maximum likelihood method without the prior distribution proposed by Tsuzuchi. The estimated value b is more stable and reliable than b_U. The method presented here is valuable, especially with very little seismic data.