Statistics on the Earthquake Occurrences (I)

Time Intervals of Successive Earthquakes

Abstract

A new statistical model is presented for describing the frequency distribution of the time intervals between the successive earthquakes. In this modelling, we consider a general differential equation for the probability distribution function f (x) as
df(x)/dx=-g(x, f), (1)
where x is the time interval of the successive events. Assuming that the function g depends only on f and expanding g(f) with respects to f and neglecting higher terms than the second order, we obtain the following equation.
df(x)/dx=-β(1±αf)f, (2)
where α and β are constants. In the limit of α→0, we get the familiar exponential distribution. The general solution of this equation is given as
f(x)dx=1/exp[β(x+ξ)]±1 dx/α. (3)
This function form is equivalent to the familiar distributions known as the Bose-Einstein (-) and Fermi-Dirac (+) distribution in statistical physics. By using this model, the frequency distributions of the intervals of actual seismic series which deviate from the exponential distribution are successfully described. The result shows that the second term of the right side of ed. (2) which characterizes the clustering effect of the events is essentially important in the earthquake occurrences.