Poisson tests of declustered catalogues

Abstract

The commonly enunciated reason to decluster catalogues is so that the remaining ‘main’ events will be consistent with a spatially inhomogeneous, temporally homogeneous Poisson process (SITHP) model. But are they? Conclusions depend on the declustering method, the catalogue, the magnitude range, and the statistical test. Gardner and Knopoff's (1974) conclusion that 1932–1971 southern California events with M≥ 3.8 are Poisson after declustering apparently results from their use of a test with low power. That test ignores space, is insensitive to long-term rate variations, is relatively insensitive to seismicity rate fluctuations on the scale of weeks, and uses an inaccurate approximation to the null distribution of the test statistic. Better temporal tests and a novel spatio-temporal test show that SITHP does not fit M≥ 3.8 1932–1971 or 1932–2010 Southern California Earthquake Center (SCEC) catalogues declustered using Gardner and Knopoff's windows in a linked-window or a mainshock-window algorithm. For M≥ 4.0, SCEC catalogues declustered using the Gardner-Knopoff windows in a linked-window method are far closer to SITHP, while catalogues declustered using those windows in a mainshock-window method are inconsistent with SITHP. Reasenberg's (1985) declustering method applied to southern California seismicity produces catalogues inconsistent with SITHP, even for events with M≥ 4.0. If enough events are deleted from a catalogue, the remainder always will be consistent with SITHP. This suggests posing declustering as an optimization problem: Delete the fewest events such that those left pass a particular test or suite of tests for SITHP. While that optimization problem is combinatorially complex, inexpensive suboptimal methods are surprisingly effective: Declustered catalogues can be consistent with temporal tests of SITHP at significance level 0.05 and have 50–80 per cent more events than window-declustered catalogues that are inconsistent with SITHP. However tests that incorporate spatial information reject the SITHP hypothesis for those declustered catalogues, illustrating the importance of using spatial information.