A probabilistic model of seismicity: Kamchatka earthquakes

Abstract

The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω, \( \tilde F \)P}. Each earthquake is treated as an outcome ωi in the space of elementary events Ω whose cardinality for the period under consideration is given by the number of events. In turn, ωi is characterized by a system of random variables, viz., energy class ki, latitude φi, longitude λi, and depth hi. The time of an outcome has been eliminated from this system in this study. The random variables make up subsets in the set \( \tilde F \) and are defined by multivariate distributions, either by the distribution function \( \tilde F \) (φ, λ, h, k) or by the probability density f(φ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequency interpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law for probability density f(k), where the initial value of the distribution function f(k0) is the basic data [Bogdanov, 2006] rather than the seismic activity A0, we proceed to show that for different intervals of coordinates and time the distribution felim(k) of an earthquake catalog with the aftershocks eliminated is identical to the distribution ffull(k), which corresponds to the full catalog. It follows from our calculations that f0(k) takes on nearly identical numeral values for different initial values of energy class k0 (8 ≤ k0 ≤ 12) f(k0). The difference decreases with an increasing number of events. We put forward the hypothesis that the values of f(k0) tend to cluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesis that statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a distribution function with the initial value f(k0) = 2/3. We discuss statistical distributions of earthquake hypocenters over depth and the epicenters over various areas for several periods