Earthquake probability based on multidisciplinary observations with correlations

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.