关键词: 震级-频度关系, b值, 最小二乘法, 极大似然法, 震级的归档效应, 震级跨度

Abstract: b-value in the magnitude-frequency(G-R)relationship plays a vital role in seismicity research and seismic hazard analysis, and the most commonly used techniques to simulate it are least square approach and maximum likelihood method. Least square method is simple and easy to apply, therefore widely used in China. However, many researches show that there exist some limits in least square estimation of b-value. Earthquakes with different magnitudes are not equally weighted in this method, and larger events have higher weights, so b-value is vulnerable to the fluctuation of several big earthquakes; meanwhile, least square method needs to divide magnitude intervals artificially. With a small sample size, data points could be not enough if the magnitude interval is too wide, and events in a magnitude interval may be lacking if it is divided to be too narrow. Especially for incremental G-R relationship, it is possible that N(M_i)equals 0 in an interval with large magnitude, so log(N(M_i))loses meaning and has to be ignored, resulting in a low b-value. Therefore, under certain conditions, maximum likelihood method is recommended as an effective substitution or supplementary for least square estimation of b-value. Among numerous previous researches on maximum likelihood estimation of b-value, lots of equations have been provided, based on varied implicit assumptions and different ways of solution. A brief overview is first presented for these equations, and classification and summary are provided based on whether taking account of the effect of binned magnitude, with finite maximum magnitude, using unequal observation periods for different magnitude intervals, and with analytic solution or not. Following this, a total of 6 influential factors are analyzed, such as binning magnitude, measurement errors of magnitude, sample size, magnitude span, minimum completeness magnitude and fore- and aftershocks. At last, reasonable suggestions are provided for using those equations properly. The equations of Aki(1965), Utsu(1965), Page(1968)and Kijko and Smit(2012)are based on assumption that magnitudes are continuous random variables, and have no corrections for this, so these equations are not recommended here. For simplicity, the equations of Utsu(1966)or Tinti and Mulargia(1987)can be used, but magnitude span should be greater than 2.5 due to without finite maximum magnitude in the formulas. For researchers having capability to write code and calculate numerically, Weichert(1980)or Bender(1983)'s algorithm could be utilized. Especially when it is required to apply data with different observation periods for varied magnitudes, the formula of Weichert(1980)is recommended. This study contributes to more accurately understand and use different formulas of estimating b-value by maximum likelihood technique, which can be used as reference for peers.

Key words: magnitude-frequency relation, b-value, maximum likelihood method, least squares method, binned magnitude, magnitude span