| A new method based on the Chi-square test was developed for the assessment of the
completeness of earthquake catalogues, with the null hypothesis,
H0 : pk = p0kfor all k
where pk and p0k are probabilities of the k-th magnitude interval calculated from an
earthquake catalogue and a model distribution assumed, respectively. Doubly-truncated
exponential function was assumed for the model distribution. Given an earthquake catalogue,
the Pearson’s test statistic is calculated for each magnitude interval and compared with the
Chi-square of the significance level of α, X1-α2. All the magnitude intervals at which the
Pearson’s test statistics are less than the Chi-square values are identified, then the minimum
magnitude interval of them is chosen as the magnitude of catalogue completeness,
mc.
To determine a proper range of the significance level, the performance test was carried out
with synthesized incomplete earthquake catalogues where the Richter-b value, the maximum
magnitude, and the magnitude range of complete portion are given. The goal of the
performance test was not to see how accurately mc is located but to see how accurately the
Richter-b value and the maximum magnitude are estimated, which is in line with the reason
why we assess the completeness of earthquake catalogues. The performance test showed that
the significance level of 5~10 % yields reasonable estimates of the Richter-b value and the
maximum magnitude. At larger significance levels, while the estimates of the Richter-b
value and the maximum magnitude are little improved, mc increases and more
catalogues fail to have mc estimate, which means we discard more earthquake data. |