Seismic risk analyses aim at establishing a relation that links the earthquake activity rate to the magnitude, using earthquake catalogs. The most widely used relation is the log-linear relation proposed by Gutenberg and Richter (Science 83:183–185, 1936) and Gutenberg and Richter (Bull Seismol Soc Am 46(3):105–145, 1945): logENm=a-bm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log {\mathbb {E}}\left[ N_m \right] = a-bm$$\end{document}, where ENm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}\left[ N_m \right] $$\end{document} is the mean number of earthquakes which magnitude is greater than m, with modification at larger magnitudes by Cosentino et al. (Bull Seismol Soc Am 67:1615–1623, 1977), Kijko and Sellevoll (Bull Seismol Soc Am 79(3):644–654, 1989), Page (Bull Seismol Soc Am 58:1131–1168, 1968), Pisarenko and Sornette (Pure Appl Geophys 160:2343–2364, 2003) and Weichert (Bull Seismol Soc Am 70(4):1337–1346, 1980). That relation leads to an Exponential distribution for the magnitudes, that we assume to be truncated to a maximum magnitude mmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\rm {max}}$$\end{document}, a priori fixed under geophysical considerations. The objective of this paper is to simultaneously estimate the parameters (a, b) from data which main features are: (a) data are observed on unequal observation periods; (b) magnitudes are imprecisely known. Within the assumption that earthquakes are modeled by a continuous Poisson point process, we propose an estimator of (a, b) that maximizes the likelihood of that Poisson point process which has been discretized on classes of magnitudes. The asymptotic distribution of the estimator is a Normal distribution of dimension 2. That distribution is used to propagate the estimation uncertainties on (a, b) until the recurrence model. Uncertainty analyses enable to draw the 5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\%$$\end{document} and 95%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$95\%$$\end{document} quantile curves around the estimated recurrence model.

中文翻译：

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不等观测周期和不精确震级下古腾堡-里希特地震复发参数的估计

地震风险分析旨在使用地震目录建立将地震活动率与震级联系起来的关系。最广泛使用的关系是 Gutenberg 和 Richter (Science 83:183–185, 1936) 和 Gutenberg and Richter (Bull Seismol Soc Am 46(3):105–145, 1945) 提出的对数线性关系：logENm=a -bm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin }{-69pt} \begin{document}$$\log {\mathbb {E}}\left[ N_m \right] = a-bm$$\end{document}, 其中 ENm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin }{-69pt} \begin{document}$${\mathbb {E}}\left[ N_m \right] $$\end{document} 是震级大于 m 的平均地震次数，在更大的修改Cosentino 等人的幅度。(Bull Seismol Soc Am 67:1615–1623, 1977), Kijko and Sellevoll (Bull Seismol Soc Am 79(3):644–654, 1989), Page (Bull Seismol Soc Am 58:1131–19688), 1 Pisa和 Sornette (Pure Appl Geophys 160:2343–2364, 2003) 和 Weichert (Bull Seismol Soc Am 70(4):1337–1346, 1980)。这种关系导致幅度的指数分布，我们假设被截断到最大幅度 mmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \ usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\rm {max}}$$\end{document}，在地球物理考虑下先验固定。本文的目的是从数据中同时估计参数（a，b），其主要特征是：（a）数据是在不等的观察周期上观察到的；(b) 幅度未知。在地震由连续泊松点过程建模的假设下，我们提出了 (a, b) 的估计器，该估计器最大化已在震级类别上离散化的泊松点过程的可能性。估计量的渐近分布是维数为 2 的正态分布。该分布用于传播 (a, b) 上的估计不确定性，直到递归模型。不确定性分析能够绘制 5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\%$$\end{document} 和 95%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage {amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$95\%$$\end{document} 分位数围绕估计的递归模型曲线。该分布用于传播（a，b）上的估计不确定性，直到递归模型。不确定性分析能够绘制 5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\%$$\end{document} 和 95%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage {amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$95\%$$\end{document} 分位数围绕估计的递归模型的曲线。该分布用于传播（a，b）上的估计不确定性，直到递归模型。不确定性分析能够绘制 5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\%$$\end{document} 和 95%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage {amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$95\%$$\end{document} 分位数围绕估计的递归模型的曲线。