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 (Mi) equals 0 in an interval with large magnitude, so log (N (Mi)) 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.%b值在地震活动性研究和地震危险性分析中起着十分重要的作用,通常拟合b值的方法有最小二乘法(Least Squares Method)和极大似然法(Maximum Likelihood Estimation).最小二乘法简单易行,得到了广泛的应用.然而很多研究表明该方法存在一定的局限性,极大似然法在特定条件下可以作为最小二乘法的一种可行的替代或补充方法.前人对极大似然法的研究非常繁杂,提出了各种各样的方程式,每个方程式的隐含假设和求解方式各不相同.文中对主要方程式进行了简要的回顾,并按照是否考虑震级的归档效应、是否设定有限最大震级、是否对不同震级档数据取不同的观察时段和是否具有解析解这4个方面,对这些方程式进行了分类和总结.进而对震级的归档效应、震级的测量误差、样本量、震级跨度、最小完整震级和前余震共6个可能影响极大似然法估计b值的因素进行了分析和总结.最后对正确使用这些方程式提出了合理的建议.文中的分析和总结有助于更准确地理解和使用不同的极大似然法估计b值的方程式,以供相关研究者参考.
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