The Gutenberg–Richter (G–R) relationship can be derived as the Gibbs distribution. For a given earthquake set (all earthquakes in a given region, time period, magnitude range, tectonic settings) the Gibbs probability density function for magnitudes, with a given b value in its exponent, is the most uniform distribution under the constraints of the magnitude range and mean value. Therefore, it represents our limited knowledge about the system output: the only pieces of information are the mean value and the magnitude range. Honest earthquake forecasts can be based on such a distribution, since it represents all and only available information about the seismic system. The b value can change among different earthquake sets (in time, space, magnitude ranges, or tectonic settings), since it is related to earthquake rupture dynamics, or seismic source characteristics. The relationship between the b value and the exponent β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} in the rupture area vs. maximum slip scaling, A∝Dβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\propto D^{\beta }$$\end{document}, results from viewing earthquake recurrence time in connection with the slip budget. This makes a link between earthquake statistics (the G–R law) and physics (fault characteristics). Specifically, the relationship enables us to explain different ranges of b values at megathrust faults, in dependence on interplate asperity and coupling distributions, as well as on amounts of sediments and fluids in subduction channels. The approach differs from common interpretations of the G–R law in that the b value becomes a field variable, not a constant. It is always the Gibbs distribution for a given magnitude range that we use due to our ignorance about the system outcome, and it is the b value that variates, depending on our knowledge about the system physics. This is important for seismic forecasts, which are mostly based on the G–R relationship. First, because the physical processes leading to the largest earthquakes can be revealed by observing the b value variations. Second, because earthquake generation process can be thought of as sampling with constraints, where the b value and the magnitude range are the constraints.

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Gutenberg-Richter 的 b 值和地震粗糙模型

Gutenberg-Richter (G-R) 关系可以导出为 Gibbs 分布。对于给定的地震集（给定区域、时间段、震级范围、构造环境中的所有地震），震级的吉布斯概率密度函数（其指数中具有给定的 b 值）是震级约束下最均匀的分布范围和平均值。因此，它代表了我们对系统输出的有限了解：唯一的信息是平均值和幅度范围。诚实的地震预测可以基于这样的分布，因为它代表了关于地震系统的所有可用信息。b 值可以在不同的地震集（时间、空间、震级范围或构造设置）之间变化，因为它与地震破裂动力学有关，或震源特征。b 值与指数的关系 β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage {upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} 在破裂区域与最大滑移比例，A∝Dβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A\propto D^{\beta }$$\end{document}，根据滑动预算查看地震复发时间的结果。这在地震统计（G-R 定律）和物理学（断层特征）之间建立了联系。具体来说，这种关系使我们能够解释巨型逆冲断层处不同范围的 b 值，这取决于板间粗糙度和耦合分布，以及俯冲通道中沉积物和流体的数量。该方法与 G-R 定律的常见解释不同，因为 b 值变成了一个场变量，而不是一个常数。由于我们对系统结果的无知，我们始终使用给定幅度范围内的吉布斯分布，并且根据我们对系统物理的了解，b 值会发生变化。这对于主要基于 G-R 关系的地震预测很重要。首先，因为可以通过观察 b 值的变化来揭示导致最大地震的物理过程。第二，