Earthquake probability based on multidisciplinary observations with correlations

Abstract

A number of researchers have formulated earthquake probabilities based on precursory anomalies of multidisciplinary observations in which the underlying assumption is that the occurrence of one precursory anomaly is independent from those of other kinds of anomalies. Observations were classified into two groups, those events followed by an earthquake and those that were not, and the ratio of observed precursors in both groups was taken into consideration. In the present report, recent advances in statistical seismology are considered within the framework of these earthquake probabilities, and the formulations are extended to cases in which precursory anomalies are observed as continuous measurements. The effects originating from mutual correlations between two precursory anomalies are also considered. It is assumed that observed values of each discipline follow a normal distribution, either as a group of observations followed by an earthquake (conditional density distribution) or as a group of observations not followed by an earthquake (background density distribution). Special attention is given to the case in which two kinds of observations are correlated in the conditional density distribution but not in the background density distribution. The results obtained are compared with cases in which the observations are independent of each other in both distributions. The geometrical mean of the probability gain is greater in the correlated case than in the independent case and becomes infinitely large when the absolute value of the correlation coefficient approaches one. This finding enables a wider application of earthquake probability than has been previously possible based on multidisciplinary observations.

Regional Earthquake Likelihood Models II: Information Gains of Multiplicative Hybrids

“…Hybrid models with multiplicative probability gains were suggested by the early work of Aki (1981) and Utsu (1982) on conditionally independent earthquake precursors. Later, Imoto (2006Imoto ( , 2007 generalized the notion of multiple earthquake precursors to earthquake probabilities or rates estimated from multidisciplinary observations and showed that multiplicative probability gains could theoretically still be obtained without the independence assumption. See also Faenza and Marzocchi (2010) and Shebalin et al (2014) for applications of a statistical model with multiplicative structure to earthquake forecasting.…”

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

“…IMOTO (2007) clarified the problem by expressing the probability gain as a quantity of information. For normally distributed observations (or predictive parameters), he derived a formula for the information gain in the general case where multidimensional observations are correlated in both the conditional and background distributions, extending earlier results for multidimensional observations correlated only in the conditional distribution (IMOTO, 2006a). Applying this procedure to three parameters (the Gutenberg-Richter a and b values and the m value, which measures changes in b value) observed in Kanto, central Japan, IMOTO (2008) constructed a seismicity model (referred to as the ABT model) in Kanto, Japan.…”

Information gain of a model based on multidisciplinary observations with correlations

“…Here, by selecting an appropriate unit and origin for variable θ 1 , equation (10) could be satisfied in general. In this case, the IGpe value for observation A 1 , IGpe( A 1 ), can be represented as follows [ Sakamoto et al , 1983; Imoto , 2006]. …”

“…If variables θ 1 , θ 2 , … θ n are mutually independent from one another in both distributions, that is, then we assume that f i ( θ i ) and g i ( θ i ) are represented by the following normal distributions: where θ denotes a column vector with components θ 1 ,θ 2 ,.. θ n , and the superscript t refers to the transpose of a matrix. In similar way to the two‐dimensional case [ Imoto , 2006], if it is assumed that for any combination of i and j , θ i and θ j are independent in both the conditional and background joint density distributions, the following equation can be obtained: …”

“…Taking the case where the observed values are normally distributed, we will then derive a formula for the general case where multidimensional observations are correlated with one another in both the conditional and background distributions. Imoto [2006] estimated the effect of the correlation between two observations in the conditional distribution. Generalizing this, we will estimate the effect for higher‐dimension cases with correlations in the conditional distribution.…”