Likelihood analysis of earthquake focal mechanism distributions

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. We apply a conventional likelihood method to measure the skill of earthquake focal mechanism orientation forecasts. The advantage of such an approach is that earthquake rate prediction can be adequately combined with focal mechanism forecast, if both are based on the likelihood scores, resulting in a general forecast optimization. We measure the difference between two double-couple sources as the minimum rotation angle that transforms one into the other. We measure the uncertainty of a focal mechanism forecast (the variability), and the difference between observed and forecasted orientations (the prediction error), in terms of theseminimum rotation angles. 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 (or equally probable). For 3-D rotation the random rotation angle distribution is not uniform. To better understand the resulting complexities, we calculate the information (likelihood) score for two theoretical 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 exploredwhen analyzing observational results: their dependence on forecast and data resolution, internal dependence of scores on forecasted angle and random variability of likelihood scores. Here, we propose a simple tentative solution but extensive theoretical and statistical analysis is needed.