Estimating seismic-source rate parameters associated with incomplete catalogues and superimposed Poisson-renewal generating processes

Abstract

A probabilistic model is presented to quantify parameters that define the exceedance rates of earthquake magnitudes. Incompleteness of seismic catalogues and superposition of Poisson-renewal earthquake generation processes are both taken into account within a Bayesian framework. The formulation can be transformed into the parameter estimation of single Poisson or renewal process. The incomplete exceedance rate parameters are estimated from incomplete data, so that the estimated values are equal to those of the complete rate. Two cases are studied: the first one corresponds to a seismic source in the Gulf of Mexico and the other to a seismic source in the southern Pacific coast of Mexico.

Appendix: Occurrence probability of seismic events

The probability of occurrence of seismic events associated with renewal processes, whose probability density function of times between events is Gamma, can be obtained as follows.

The Laplace transform f ^∗(s) of the Gamma probability density function can be expressed as (𝜃/s+𝜃)^η, with parameters 𝜃=λ/V ² and η=1/V ², linked to the rate λ and coefficient of variation V of the times between seismic events. Similarly, if the random variable W _n =T ₁+T ₂+…+T _n describes the time to the n-th seismic event, where the times between events T=T ₁=T ₂=…=T _n , are characterized by the Gamma probability density function with parameters 𝜃 and η, then the Laplace transform of the probability density function of W _n is expressed as (𝜃/s+𝜃)^nη. This implies that the probability density function of W _n is Gamma and can be expressed as follows:

$$ f_{N}(t;\theta,\eta)=\frac{\theta_{G}^{n\eta}}{\Gamma(n\eta)}t^{n\eta-1}\exp(-\theta t) $$

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Thus, the cumulative distribution function W _n is obtained as follows:

$$\begin{array}{@{}rcl@{}} F_{N}(t) & = & {{\int}_{0}^{t}} f_{N}(x)dx \\ & = & \frac{\Gamma(n\eta,\theta t)}{\Gamma(n\eta)} \end{array} $$

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Γ(⋅,⋅) is the incomplete Gamma function and Γ(⋅) is the Gamma function. According to renewal theory (Cox 1962), the occurrence probability of the N=n seismic events at time t is estimated by the following expression:

$$\begin{array}{@{}rcl@{}} p_{N}(t) & = & F_{N}(t)-F_{N+1}(t) \\ & = & \frac{\Gamma(n\eta,\theta t)}{\Gamma(n\eta)} - \frac{\Gamma((n+1)\eta,\theta t)}{\Gamma((n+1)\eta)} \end{array} $$

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This probability can be expressed in terms of λ and V:

$$\begin{array}{@{}rcl@{}} p_{N}(t)&=&p_{N}(t,n,\lambda,V)=\frac{\Gamma(n/V^{2},\lambda t/V^{2})}{\Gamma(n/V^{2})}\\&&- \frac{\Gamma((n+1)/V^{2},\lambda t/V^{2})}{\Gamma((n+1)/V^{2})}\\ \end{array} $$

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