Aftershocks and Earthquake Statistics (4) : Analyses of the Distribution of Earthquakes in Magnitude, Time and Space with Special Consideration to Clustering Characteristics of Earthquake Occurrence(2)

The first step in the analysis of the time distribution of earthquakes is to test the hypothesis that a given series of earthquake data are samples from a Poisson process. There are many independent methods for this test, e.g., the methods based on i) the time interval between events, ii) the number of events in a unit time interval, iii) the ratio of variance to mean of the number of events, iv) the autocorrelation, v) the spectrum, and many others. These tests have been applied to series of shallow and deep earthquakes in and near Japan and shallow earthquakes in the world. The results show that for series of shallow earth­ quakes from which aftershocks had been removed and for a series of deep earthquakes, the Poisson hypothesis can not be rejected by most of the methods. For series of shallow earthquakes including aftershocks, the Poisson hypothesis is rejected at very small significance levels. Instead of the Poisson process, the branching Poisson process (first used in seismology by Vere-Jones and Davies in 1966) has been adopted. Comparisons of the data with the theoretical curves for the distribution of time intervals, the variance/mean ratio, and the spectrum indicate that this model is a suitable approximation. An important parameter for this model L~ can be evaluated from the variance/mean ratio and the spectrum. The values from the both methods agree well. L ~ is related to the mean A and the variance V of the total number of aftershocks triggered by a main shock by the equation

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