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.