Approaches to Solving the Maximum Possible Earthquake Magnitude (M_max) Problem

Abstract

The problem of evaluation of the maximum possible regional earthquake magnitude (M_max) is reviewed and analyzed. Two aspects of this topic are specified: statistical, and historical and paleoseismic. The frequentist and the fiducial approaches used in the problem are analyzed and compared. General features of the Bayesian approach are discussed within the framework of the M_max problem. A useful connection between quantiles of a single event and maximum event in a future time interval T is derived. Various estimators of M_max used in seismological practice are considered and classified. Different methods of estimation are compared: the statistical moment method, the Bayesian method, the estimators based on the extreme value theory (EVT), the estimators using order statistics. A comparison of several well-known estimators of M_max in the framework of the truncated Gutenberg–Richer law is made. As a more adequate and stable alternative to M_max the quantiles Q_q(T) of maximum earthquake considered in future time horizon T are proposed and analyzed. These quantiles permit us to select a time horizon T and quantile level q for a reliable estimation of maximum possible magnitudes. The instability of M_max-estimates compared to Q_q(T)-estimates is demonstrated. The main steps of the Q_q(T)-quantile estimation procedure are highlighted. The historical and paleoseismic data are used, and an additional evidence of low robustness of M_max-parameter is found. The evidence of possibility of earthquake magnitudes well exceeding the M_max-value obtained for the truncated Gutenberg–Richter law is found also. The present situation in the domain of the M_max-evaluation is discussed.

Appendix

Let us consider function

$$G(M_{\max } ) = M_{\max } - \mathop \int \limits_{{m_{0} }}^{{M_{\max } }} \left[ {\frac{{1 - \exp \left( { - \beta \left( {x - m_{0} } \right)} \right)}}{{1 - \exp \left( { - \beta (M_{\max } - m_{0} } \right))}}} \right]^{n} dx$$

The integral in the right part can be transformed as follows. We take a new variable

$$y = \frac{{1 - \exp \left( { - \beta \left( {x - m_{0 } } \right)} \right)}}{{1 - \exp \left( { - \beta \left( {M_{\max } - m_{0 } } \right)} \right)}}$$

and denote

$$u = 1 \, {-} \, \exp \left( { - \beta \left( {M_{\max } {-} \, m_{0} } \right)} \right)$$

We get

$$M_{\max } = \, m_{0} - \frac{{\log \left( {1 - u} \right)}}{\beta }$$

$$\begin{aligned} G\left( {M_{max} } \right) \, & = M_{max } - \mathop \int \limits_{0}^{1} y^{n} \frac{u}{{\beta \left( {1 - uy} \right)}}dy \, = \, m_{0} - \frac{{{\text{log}}\left( {1 - u} \right)}}{\beta } - \frac{1}{{\beta u^{n} }} \cdot \mathop \int \limits_{0}^{u} \frac{{z^{n} dz}}{1 - z} \\ & = m_{0} - \frac{{{\text{log}}\left( {1 - u} \right)}}{\beta } - \frac{1}{{\beta u^{n} }}\mathop \int \limits_{0}^{u(} \frac{{z^{n} - 1 + 1)dz}}{1 - z} = \, m_{0} - \frac{{{\text{log}}\left( {1 - u} \right)}}{\beta } - \frac{1}{{\beta u^{n} }}\mathop \int \limits_{0}^{u(} \frac{{z^{n} - 1)dz}}{1 - z} - \frac{1}{{\beta u^{n} }}\mathop \int \limits_{0}^{u} \frac{dz}{{1 - z}} \\ \end{aligned}$$

$$\begin{gathered} m_{0} - + \frac{{\log \left( {1 - u} \right)}}{\beta } + \frac{1}{{\beta u^{n} }}\mathop \int \limits_{0}^{u} (1 + z + z^{2} + \ldots + z^{n - 1} )dz + \frac{{\log \left( {1 - u} \right)}}{{\beta u^{n} }} \hfill \\ = m_{0} + \frac{1}{\beta }\left\{ {\left( {\frac{1}{{u^{n} }} - \, 1} \right)\log \left( {1 - u} \right) + \frac{{S_{u} }}{{u^{n} }}} \right\} \hfill \\ \end{gathered}$$

where $S_{u} = \, u \, + \frac{{u^{2} }}{2} + \ldots + \frac{{u^{n} }}{n}$, $u \, = \, 1 \, {-} \, \exp \left( { - \beta \left( {M_{\max } {-} \, m_{0} } \right)} \right)$.