Likelihood analysis of earthquake focal mechanism distributions

TL;DR

This paper applies a likelihood-based method to evaluate earthquake focal mechanism forecasts, addressing previous empirical limitations and exploring the complexities of calculating likelihood scores for source orientation distributions.

Contribution

It introduces a likelihood approach for assessing earthquake focal mechanism forecasts, incorporating complex orientation distributions and proposing preliminary solutions for statistical analysis challenges.

Findings

Likelihood scores can effectively evaluate forecast skill.

Complex orientation distributions like Cauchy and von Mises-Fisher are used.

Preliminary methods address issues of data resolution and variability.

Abstract

In our paper published earlier we discussed forecasts of earthquake focal mechanism and ways to test the forecast efficiency. Several verification methods were proposed, but they were based on ad-hoc, empirical assumptions, thus their performance is questionable. In this work we apply a conventional likelihood method to measure a skill of forecast. The advantage of such an approach is that earthquake rate prediction can in principle be adequately combined with focal mechanism forecast, if both are based on the likelihood scores, resulting in a general forecast optimization. To calculate the likelihood score we need to compare actual forecasts or occurrences of predicted events with the null hypothesis that the mechanism's 3-D orientation is random. For double-couple source orientation the random probability distribution function is not uniform, which complicates the calculation of the likelihood value. To better understand the resulting complexities we calculate the information (likelihood) score for two rotational distributions (Cauchy and von Mises-Fisher), which are used to approximate earthquake source orientation pattern. We then calculate the likelihood score for earthquake source forecasts and for their validation by future seismicity data. Several issues need to be explored when analyzing observational results: their dependence on forecast and data resolution, internal dependence of scores on forecasted angle, and random variability of likelihood scores. In this work we propose a preliminary solution to these complex problems, as these issues need to be explored by a more extensive statistical analysis.