Some Methods of Assessing and Estimating Point Processes Models for Earthquake Occurrences

Alejandro Veen
Ph.D., 2006
Advisor: Frederic Paik Schoenberg

This dissertation presents methods of assessing and estimating point process models and applies them to Southern California earthquake occurrence data. The first part provides an alternative derivation of the asymptotic distribution of Ripley's K-function for a homogeneous Poisson process and shows how it can be combined with point process residual analysis in order to test for different classes of point process models. This is done with the mean K-function of thinned residuals (KM) or a weighted analogue called the weighted or inhomogeneous K-function (KW). This work derives the asymptotic distributions of KM and KW for an inhomogeneous Poisson process. Both statistics can be used as measures of goodness-of-fit for a variety of classes of point process models. The second part deals with the estimation of branching process models. The traditional approach by Maximum Likelihood can be a very unstable and computationally difficult. Viewing branching processes as incomplete data problems suggests using the Expectation-Maximization algorithm as a practical alternative. A particularly efficient procedure based on maximizing the partial log-likelihood function is proposed for the Epidemic-type Aftershock Sequence (ETAS) model, one of the most widely used seismological branching process models.
The third part of this work applies the weighted K-function to assess the goodness-of-fit of a class of point process models for the spatial distribution of earthquakes in Southern California. Then, the proposed EM-type algorithm is used to estimate declustered background seismicity rates of geologically distinct regions in Southern California.