Estimation of Mc and parameters for Recurrence models in Earthquake Catalog
Speaker: Hung-Jung Yeh
Abstract
wiemer and Wyss used Minimum Mc [magnitude of completeness] and used maximum likelihood estimate to defined a&b-value, and Utsu used 4 models [weibull, lognormal, gamma, exponential probability] to discussed recurrence of earthquake in some seismic zones.
Estimation of Mc is based on a simple power law can approximate the FMD [frequency-magnitude distribution], the FMD describes the relationship between the frequency of occurrence and magnitude of earthquakes: logN=a-bM, and the data quality, as reflected by the Mc level. we use the method to calculated the goodness of fit between the data and the predicted FMD as a function of lower cutoff the magnitude data. Mc is defined as the magnitude at which 90% of the data can be modeled by a power law fit, that is absolute difference R.
Author use MLE to obtain the a&b value with Mc, than they got GR law [logN=abM], after that they want to obtain mean and variance of the catalog to calculate recurrence probability, they used CV value to calculate, and CV is equal to standard deviation of catalog divided to length of catalog divided to N, But catalog does not have data in big magnitude, cannot be used to estimate recurrence probability, must use GRlaw and mean of CV to estimate standard deviation of catalog.
Recurrence of earthquake in some seismic zones has been discussed by weibull, gamma, lognormal, exponential probability model, the parameter estimate and the corresponding ln L for the four models are given for three data sets, the parameter values are expressed using the time unit of years, estimate parameter use MOM & MLE, but there are no actually differences between the values estimated by the two methods.
We have compared four renewal models using different distributions of time intervals, it’s found that different models are best suited for different sets of data, anyway the differences among the four models are usually small, and it’s hard to say which is the best model.
Reference
Stefan Wiemer, and Max Wyss, 2000, Minimum Magnitude of Completeness in Earthquake Catalogs : Example from Alaska, the Western United states, and Japan, Bulletin of Seismological Society of America, 90, 4, pp. 859-869, August 2000
Tokuji Utsu, 1984, Estimation of Parameters for Recurrence Models of Earthquakes, Bulletin of the Earthquake Research Institute University od Tokyo, Vol. 59(1984) pp. 53-66