A Statistical Study of Some Aftershock Problems

Abstract

According to T. Utsu, the largest shock in an aftershock sequence is by about 1.4 smaller than its main shock on magnitude scale. On the other hand, M. Båth has independently given 1.2 for the value. It is one of the main aims of this paper to deduce the above mentioned experiential results by the theory of probability.
Now, the number of aftershocks is written by the equation,
n(t)dt=A/t+Bdt, (1)
where n(t) is the number of aftershocks occurring in a time interval between t and (t+dt). The relation between M (Magnitude) of main shock and A has been found to be expressed by the equation,
logA=0.79M-3.8, (2)
whereas B is independent of the magnitude of main shock.
On the other hand, the frequency distribution of M of aftershocks is written in the form,
n(M)dM=const.10^-dMbM. (3)
The relations (2) and (3) are assumed in deriving a POPULATION for a sequence of aftershocks. Then, using the theory of distribution of extreme values, we can find the most probable value of M of the largest aftershock. The result shows that the most probable value of the magnitude of the largest aftershock is about 1 to 1.4 smaller than that of the main shock.
Another thing treated in this paper is to explain the statistical fact that the percentage of earthquakes accompanied by aftershocks decreases as their magnitueds become smaller. The percentages are calculated according to the extreme value theory. Good agreement is found between the calculated and the actual percentages.