Research Program: Statistical Evaluation of Earthquake Occurrance Data using Point Process Techniques.

Principal Investigator: Rick Paik Schoenberg

In seismology, multi-dimensional point process data are routinely collected in the form of catalogs of earthquake origin times and locations, along with associated magnitudes, orientations, and other information on each earthquake. The statistical analysis of earthquake catalogs has been significantly enhanced by the recent advancement of conditional intensity models for point processes. Such models have contributed greatly to the accuracy of seismic hazard estimates obtained from catalogs of earthquake data.

However, the problem of discriminating between competing models and the question of how to assess the validity of hazard estimates based on these models have not been adequately addressed. In most cases, multi-dimensional models for earthquake data are evaluated by inspecting marginal distributions, one at a time. Such methods are generally very weak; hence only gross inconsistencies between the model and data may be observed. It is imperative that more sophisticated and powerful means of evaluating the fit of multi-dimensional models for earthquake catalogs be employed.

The goal of this project is to improve the state of knowledge about procedures for analyzing multi-dimensional point processes and to apply these methods in clear, intelligible ways to the solution of real scientific problems. Two goals in particular are to obtain accurate estimates of seismic hazard, and to establish more powerful tools for evaluating multi-dimensional models for earthquake occurrences. This project has been given some funding support via UCLA's FCDA (Faculty Career Development Award) for the period July 1, 2001 to June 30, 2002.


Prior results.

Recent findings (e.g. Ogata 1988, Schoenberg 1997 , Ogata 1998, Schoenberg 1999 ) have suggested that point process residual analysis is a very powerful tool for evaluating multi-dimensional earthquake models. Such residual analysis in Schoenberg (1997) highlighted the distinction between the fit of various multivariate models to micro-earthquake data from Parkfield, California, for example, and also suggested ways in which existing models could be improved.

Further study on the application of point process residual analysis to models for earthquake catalogs is necessary. The research in Schoenberg (1997) only involved a single test case: a seven-year catalog of micro-earthquakes from Parkfield, California. Investigation of appropriate models from other sorts of catalogs, e.g. catalogs of different lengths, over different time scales, and most importantly with different magnitude ranges, is needed. We plan to explore the use of multi-dimensional residual analysis to assess models for other catalogs, such as the Harvard catalog and Council of the National Seismic System catalogs, which contain decades of observations of larger, more significant events.

In Schoenberg 1999 , we explored multi-dimensional point process residual analysis, a powerful method for evaluating point process models. The main result was the derivation of a novel method of conditioning, which resulted in a more general method of residual analysis. This method, which was an extension of methods explored by previous authors, enabled residual analysis to be used for a wider variety of models, including those used by seismologists to describe earthquake occurrences.

In Schoenberg and Bolt (2000) , we explored a novel class of models for earthquake occurrences -- models which, incidentally, were suggested in part by the residual analysis of Schoenberg (1997). Earthquakes have been shown to exhibit different types of behavior on different time scales. The methods used in Schoenberg and Bolt (2000) enable one to discriminate between the time scales of these behaviors and to see which of these observed patterns are significant and which may be spurious. The models in Schoenberg and Bolt (2000) were shown to fit well to two different catalogs, including small and moderate earthquakes in California, and provided meaningful, interpretable seismological results. In Kagan and Schoenberg (2001) , we explored the frequency-magnitude distributions of earthquakes, and improved ways of estimating the relevant parameters involved.