Correlation between the parameters of the rate equation for simple aftershock sequences: implications for the forecasting of rates and probabilities

Abstract

We analyzed the correlations among the parameters of the Reasenberg and Jones (1989) formula describing the aftershock rate after a mainshock as a function of time and magnitude, on the basis of parameter estimates made in previous works for New Zealand, Italy and California. For all of three datasets we found that the magnitude-independent productivity a is significantly correlated with the b-value of the Gutenberg-Richter law and, in some cases, with parameters p and c of the modified Omori's law. We argued that the correlation between a and b can be ascribed to an inappropriate definition of the coefficient of mainshock magnitude as the correlation becomes insignificant if the latter is assumed to be $α\simeq$ 2/3b rather than b. This interpretation well agrees with the results of direct a estimates we made, by an epidemic type model (ETAS), from the data of some large Italian sequences. We also verified that assuming $α$ about 2/3 of the average b value estimated from Italian sequences occurred in the time interval 1981-1996 improves the ability to predict the behavior of most recent sequences (from 1997 to 2003). Our results indicate a partial inadequacy of the original Reasenberg and Jones (1989) formulation when used to forecast the productivity of future sequences. In particular, the aftershock rates and probabilities tend to be overestimated for stronger mainshocks and conversely underestimated for weaker ones.