Correlation between the parameters of the aftershock rate equation: Implications for the forecasting of future sequences

We analyzed the correlations among the parameters of the Reasenberg and Jones [Reasenberg, P.A., Jones, L.M., 1989. Earthquake hazard after a mainshock in California, Science 243, 1173–1176] 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 Omoriʹs law. We also found significant correlations between p and c but, different from some previous works, not between p and b. We verified that assuming a coefficient for mainshock magnitude α ≈ 2/3b (instead of b) removes the correlation between a and b and improves the ability to forecast the behavior of Italian sequences occurred from 1997 to 2003 on the basis of average parameters estimated from sequences occurred from 1981 to 1996. This assumption well agrees with direct α estimates made in the framework of an epidemic type model (ETAS) from the data of some large Italian sequences. Our results suggest a modification of the original Reasenberg and Jones (1989) formulation leading to predict lower rates (and probabilities) for stronger mainshocks and conversely higher rates for weaker ones. We also inferred that the correlation of a with p and c might be the consequence of the trade-off between the two parameters of the modified Omoriʹs law. In this case the correlation can be partially removed by renormalizing the time-dependent part of the rate equation. Finally, the absence of correlation between p and b, observed for all the examined datasets, indicates that such correlation, previously inferred from theoretical considerations and empirical results in some regions, does not represent a common property of aftershock sequences in different part of the world.