Multidimensional Residual Analysis of Point Process Models for Earthquake Occurrences

Residual analysis methods for examining the fit of multidimensional point process models are applied to point process models for the space–time–magnitude distribution of earthquake occurrences, using, in particular, the multidimensional version of Ogata's epidemic-type aftershock sequence (ETAS) model and a 30-year catalog of 580 earthquakes occurring in Bear Valley, California. One method involves rescaled residuals, obtained by transforming points along one coordinate to form a homogeneous Poisson process inside a random, irregular boundary. Another method involves thinning the point process according to the conditional intensity to form a homogeneous Poisson process on the original, untransformed space. The thinned residuals suggest that the fit of the model may be significantly improved by using an anisotropic spatial distance function in the estimation of the spatially varying background rate. Using rescaled residuals, it is shown that the temporal–magnitude distribution of aftershock activity is not separable, and that, in particular, in contrast to the ETAS model, the triggering density of earthquakes appears to depend on the magnitude of the secondary events in question. The residual analysis highlights that the fit of the space–time ETAS model may be improved by allowing the parameters governing the triggering density to vary for earthquakes of different magnitudes. Such modifications may be important because the ETAS model is widely used in seismology for hazard analysis.

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