Pitfalls of Estimating Background Seismicity Rates from Interevent-time Statistics

The background seismicity rate (i.e. rate of mainshocks) is poorly constrained from seismicity counts because current declustering methods used to remove aftershocks rely on subjectively adjusted space-time parameters. Recently, Hainzl et al. (2006) proposed an objective method for estimating the fraction of background earthquakes in a catalog, γ, using the distribution of times between sequential events. This method is based on the assumption that the interevent times follow a gamma distribution, which is only approximate. I test the Hainzl method, and an alternative method using the theoretical interevent-time distribution of the ETAS model (Ogata, 1988). I find that both methods are plagued by a trade-off between γ and the direct Omori decay parameter pD. Hainzl et al. (2006) propose that γ for any dataset can be found from the mean over the variance of the normalized interevent times, plus a small empirical correction. I test this algorithm on 2000 ETAS simulations with varying γ, total number, duration, b-value, c-value and pD. The results of this suite of tests generally validate the method's ability to recover the correct γ. However, the tests also reveal a systematic error in the estimated γ as a function of pD. No other parameters appear to systematically affect the results. The systematic error in γ can be empirically corrected if pD is known. An alternative would be to find γ and pD that fit the theoretical equations for the interevent-time distribution derived from the ETAS model (e.g. Saichev and Sornette, 2007). However, there is a severe trade-off between γ and pD, such that different pairs of parameter values can produce nearly-identical theoretical distributions. Typically, increasing γ trades off with decreasing pD, the same sign as the correction for the Hainzl method. Either method could be used to constrain γ if pD were already accurately estimated. However, pD is not easily measured, as it is not the same as the cumulative Omori p-value measured for an aftershock sequence. Felzer et al. (2003) use forward modeling to identify a preferred pD=1.37 for California, but it is unclear how universal this value is. Therefore, caution must be used when inferring background rates from interevent-time statistics.