Earthquake prediction analysis based on empirical seismic rate: the M8 algorithm

The quality of space-time earthquake prediction is usually characterized by a 2-D error diagram (n, tau), where n is the fraction of failures-to-predict and tau is the local rate of alarm averaged in space. The most reasonable averaging measure for analysis of a prediction strategy is the normalized rate of target events lambda(dg) in a subarea dg. In that case the quantity H = 1 - (n + tau) determines the prediction capability of the strategy. The uncertainty of lambda(dg) causes difficulties in estimating H and the statistical significance, alpha, of prediction results. We investigate this problem theoretically and show how the uncertainty of the measure can be taken into account in two situations, viz., the estimation of alpha and the construction of a confidence zone for the (n, tau)-parameters of the random strategies. We use our approach to analyse the results from prediction of M >= 8.0 events by the M8 method for the period 1985-2009 (the M8.0+ test). The model of lambda(dg) based on the events M-w >= 5.5, 1977-2004, and the magnitude range of target events 8.0 <= M < 8.5 are considered as basic to this M8 analysis. We find the point and upper estimates of alpha and show that they are still unstable because the number of target events in the experiment is small. However, our results argue in favour of non-triviality of the M8 prediction algorithm.