| New least-squares estimators (nLSE) were developed for seismicity parameters: Richter-b
and annual rate of earthquakes. The estimators are derived by minimizing the sum of squared
errors,
SSE = -I (n - Ï )2,
i=1 i i
where I is the number of magnitude intervals in [mmin,mmax],ni is the number of
earthquakes in the i-th magnitude interval, mi, and Ïi = νtipi is the expectation of ni where
ν, ti, and pi are the mean annual rate of earthquakes with magnitude larger than
mmin, the period of completeness, and the probability of mi, respectively. For the
doubly-truncated exponential distribution of magnitude, the probability is given
by
-I
pj = exp(- βmj )- i=1 exp(- βmi), β = bln(10).
Unlike the conventional least-squares estimators (cLSE), the nLSE does not violate the
assumption of independency of data because the nLSE uses the interval frequency. Moreover,
the nLSE can accommodate the period of completeness, the maximum magnitude, and the
effect of magnitude grouping with a finite interval width. The accuracy of the nLSE, the
cLSE, and the maximum likelihood estimators (MLE) were compared by using the ensemble
of simulated earthquake catalogs. The comparison revealed that the sample means of the
three estimators are comparable, while the sample standard deviations show large
differences; the sample standard deviation of MLE is smallest and that of cLSE
largest.
The reason why we don’t discard the LSE is that there are some applications where the
MLE is not efficient. An example is the analysis of minimum magnitude above which an
earthquake catalog is complete. |