Seismicity Models of Moderate Earthquakes in Kanto, Japan Utilizing Multiple Predictive Parameters

Abstract

We construct a single hazard function from multiple predictive parameters independently developed for moderate earthquakes in Kanto, Japan, during a learning period from 1990 to 1999, and applied to a testing period from 2000 to 2005. Here, we consider as predictive parameters the a and b values of the Gutenberg–Richter relation, the ν value (change in b value), and the Every Earthquake a Precursor According to Scale (EEPAS) model rate. To study the correlations among the parameters, we prepare two groups of space–time coordinate sets for assessment, namely the background and conditional groups selected from the learning period. The background group contains ten thousand sets of coordinates randomly selected from the space–time volume of our study. The conditional group contains 33 sets of space–time coordinates corresponding to the epicenters of the target earthquakes (M ≥ 5.0) just before their times of occurrence. Each parameter for the background group is transformed so that its distribution conforms to the standard Normal function. The mean and variance of the conditional distribution is then estimated after applying the same transformation to the conditional group. Using the means and variances of b values, ν values and EEPAS rates and the correlation matrices in the background and conditional distributions, we construct a combined hazard function following the procedure developed for normally distributed parameters. The information gain per event (IGpe) of the new hazard function is 0.26 and 0.3 units larger than that of the EEPAS rate for the learning and testing period, respectively. The R-test confirms the statistical significance of the difference in the IGpe value for the testing period.

Appendix

The proposed model could be tested with N-, L-, and R-statistics (Kagan and Jackson, 1995). In these tests, statistics obtained from the observed data are compared with a distribution of statistics, which could be empirically obtained with a large number of randomly generated sequences. Where the observed statistics are not consistent with statistics computed assuming a given model to be correct, the null hypothesis that the real earthquake sequences conform to the model may be rejected. In the present study, we will perform these three tests analytically, not with random catalogs.

If an observed score is outside a certain range (for example, ±2 standard deviations from the expected value), the model should be rejected at the respective level of significance (5%). However, a higher likelihood score, even if outside the +2 standard deviations, is regarded as a better model in the L-test (Schorlemmer et al., 2007).

The N-test checks the consistency of the observed number with the earthquake productivity of the proposed model. Assuming the study space–time domain is divided into K cells, an earthquake in each cell follows the Poisson process. The Poisson rate in the j-th cell is noted as λ _j , the expected number of event, and its variance Var[n _j ] = λ _j . The total expected number, E[n], is given by

$$ {\text{E}}[n] = \sum\limits_{j = 1}^{K} {\lambda_{j} } , $$

(A1)

where an earthquake occurrence in one cell is assumed to be independent from that in another cell. The variance of the total number is given by

$$ {\text{Var}}[n] = \sum\limits_{j = 1}^{K} {{\text{Var}}[n_{j} ]} . $$

(A2)

The L-test determines the consistency of the observed likelihood with that expected from sequences conforming to the model. The log likelihood for the j-th cell is given by

$$ l_{j} = Y_{j} \,{\text{Ln}}\,\lambda_{j} + \left( {1 - Y_{j} } \right){\text{Ln}}\left( {1 - \lambda_{j} } \right), $$

(A3)

where Y _j is 1 when an earthquake occurs and 0 otherwise, and Ln refers to the natural logarithm. It is assumed that λ _j is far less than unity and that at most one earthquake occurs in one cell. The expected value of the log-likelihood, E[l _j ] is expressed by

$$ {\text{E}}\left[ {l_{j} } \right] = \lambda_{j} \,{\text{Ln}}\,\lambda_{j} + \left( {1 - \lambda_{j} } \right){\text{Ln}}\left( {1 - \lambda_{j} } \right). $$

(A4)

The variance is given by

$$ \begin{aligned} {\text{Var}}\left[ {l_{j} } \right] & = \sum\limits_{i} {\left\{ {\left( {l_{j} - {\text{E}}\left[ {l_{j} } \right]} \right)^{2} {\text{f}}_{j} \left( {Y_{i} } \right)} \right\}} \\ & = \left\{ {{\text{Ln}}\,\lambda_{j} - {\text{Ln}}\left( {1 - \lambda_{j} } \right)} \right\}^{2} \sum\limits_{i} {\left( {Yi - \lambda_{j} } \right)}^{2} {\text{f}}_{j} \left( {Y_{i} } \right), \\ & = \left\{ {{\text{Ln}}\,\lambda_{j} - {\text{Ln}}\left( {1 - \lambda_{j} } \right)} \right\}^{2} {\text{Var}}\left[ {n_{j} } \right]. \\ \end{aligned} $$

(A5)

When an earthquake in one cell is independent from that in another cell, the mean E[l] and variance Var[l] of the log-likelihood for the whole domain are given by

$$ {\text{E}}\left[ l \right] = \sum\limits_{j = 1}^{K} {\left[ {\lambda_{j} \,{\text{Ln}}\,\lambda_{j} + \left( {1 - \lambda_{j} } \right){\text{Ln}}\left( {1 - \lambda_{j} } \right)} \right]} , $$

(A6)

and

$$ {\text{Var}}\left[ l \right] = \sum\limits_{j = 1}^{K} {\left[ {\left\{ {{\text{Ln}}\,\lambda_{j} - {\text{Ln}}\left( {1 - \lambda_{j} } \right)} \right\}^{2} \lambda_{j} } \right]} . $$

(A7)

The R-test examines whether a difference between the likelihoods of two models is significant. The expected value of the likelihood ratio, E[l ¹² _j ] is expressed by

$$ {\text{E}}\left[ {l_{j}^{12} } \right] = \lambda_{j}^{1} {\text{Ln}}\left( {{{\lambda_{j}^{1} } \mathord{\left/ {\vphantom {{\lambda_{j}^{1} } {\lambda_{j}^{2} }}} \right. \kern-\nulldelimiterspace} {\lambda_{j}^{2} }}} \right) + \left( {1 - \lambda_{j}^{1} } \right){\text{Ln}}\left( {{{\left( {1 - \lambda_{j}^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \lambda_{j}^{1} } \right)} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right), $$

(A8)

where superscripts 1 and 2 refer to the first and second models. It is assumed that earthquake sequences conform to the first model. The variance is given by

$$ \begin{aligned} {\text{Var}}\left[ {l_{j}^{12} } \right] & = \sum\limits_{i} {\left\{ {\left( {l_{j}^{12} - E\left[ {l_{j}^{12} } \right]} \right)^{2} f_{j} \left( {Y_{i}^{1} } \right)} \right\}} , \\ \, & = \left\{ {{\text{Ln}}\left( {{{\lambda_{j}^{1} } \mathord{\left/ {\vphantom {{\lambda_{j}^{1} } {\lambda_{j}^{2} }}} \right. \kern-\nulldelimiterspace} {\lambda_{j}^{2} }}} \right) - {\text{Ln}}\left( {{{\left( {1 - \lambda_{j}^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \lambda_{j}^{1} } \right)} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right)} \right\}^{2} \sum\limits_{i} {\left( {Y_{i}^{1} - \lambda_{j}^{1} } \right)^{2} {\text{f}}\left( {Y_{i}^{1} } \right)} , \\ & = \left\{ {{\text{Ln}}\left( {{{\lambda_{j}^{1} } \mathord{\left/ {\vphantom {{\lambda_{j}^{1} } {\lambda_{j}^{2} }}} \right. \kern-\nulldelimiterspace} {\lambda_{j}^{2} }}} \right) - {\text{Ln}}\left( {{{\left( {1 - \lambda_{j}^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \lambda_{j}^{1} } \right)} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \lambda_{j}^{2} } \right)}}} \right)} \right\}^{2} {\text{Var}}\left[ {n_{j}^{1} } \right]. \\ \end{aligned} $$

(A9)

The mean E[l ¹²] and variance Var[l ¹²] of the log-likelihood ratio for the whole domain are given by

$$ {\text{E}}\left[ {l^{12} } \right] = \sum\limits_{j = 1}^{\text{K}} {E\left[ {l_{j}^{12} } \right]} , $$

(A10)

and

$$ {\text{Var}}[l] = \sum\limits_{j = 1}^{K} {{\text{Var}}\left[ {l_{j}^{12} } \right]} . $$

(A11)