Some Point Process Models, Their Estimation and Application to New Zealand Earthquake Data

Abstract

This thesis is concerned with some point process models, their estimation and application to New Zealand earthquake data. Two point processes of first order time-homogeneous Markov-dependent intervals and their maximum likelihood estimation of parameters are studied for earthquake origin times. Some second order functions of stationary marked point processes in higher dimensional spaces and their estimation are discussed and applied to New Zealand earthquake data. This thesis is divided into two parts. Part I deals with the theoretical background, including maximum likelihood estimation of parameters for point processes specified by complete intensity functions, and the second order analysis of higher dimensional point and marked point processes. Included in this part also are a study of the asymptotic behaviour of point processes with second order tine-homogeneous Markov-dependent intervals, and a preliminary study of two distance methods for testing isotropy. Application of point processes to New Zealand earthquake data is given in part II. Two point process models of first order time-homogeneous Markov-dependent intervals are fitted to two sets of New Zealand earthquake origin times, one consisting of shallow (depthâ ¤ 60 km) shocks and the other deep (depth > 60 km) shocks Then jointly treating earthquake origin times, epicentres and magnitudes as a marked point process with magnitude as mark, a second order technique is applied to upper crustal earthquakes occurring in the main seismic region of New Zealand. The results show strong clustering of smaller shocks in the vicinity of a large earthquake within a short time before and after its occurrence, seismic gap in the occurrence of smaller earthquakes some time before a large earthquake, increase in b-value after a large earthquake, azimuthal preference of smaller shocks and azimuthal variation of magnitude.