GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 2.1 323 Considerations inspired by estimation of time completeness and seismicity rates R. Rotondi, E. Varini Istituto di Matematica Applicata e Tecnologie Informatiche, CNR, Milano, Italy One of the first steps in the computational procedure for seismic hazard evaluation is estimation of the seismicity rate of each seismogenic zone, which implies the determination of the completeness time of the corresponding data sets. If one assumes that the seismic process is stationary, or that, at least, it can be modelled in this way for long-term hazard evaluations, the answer to the double question is quite natural. The main shocks follow a homogeneous Poisson process with constant λ rate, but this rate can change because of different reasons among which the most significant one is the failure of some data recording. In fact even the simple visual inspection of the empirical cumulative count curve shows that the curve may be approximated by a piece-wise linear curve, whose final slope is related to the rate of the complete part. In other words, if y = ( y 1 , y 2 , ... y n ) are the dates of n earthquakes that occurred in the period ( t 0 , t f ) within a seismogenic zone ( t f usually denotes the end of the catalogue), then we assume that y j are a realization of a compound Poisson process with intensity function: λ ( t ) = h 1 δ ( t 0 , s ) ( t ) + h 2 δ ( s,tf ) , ( t ), (1) where h 1 and h 2 are the rate of the incomplete and complete part respectively, and s is the time from which the data set can be considered as complete, that is, s is the change-point of the rate. The unknown parameters h 1 , h 2 and s have to be estimated; to exploit prior information drawn from other databases, we estimate them in the framework of Bayesian analysis. Conditioning on h 1 , h 2 , s, the y j are independent; hence the likelihood function has the form: According to the Bayesian approach, both the heights h j ,i = 1,2, of the step function λ ( t ) and the position of the change-point s are random variables; we assume that h 1 and h 2 follow the prior distribution Gamma ( a 0 , b ), with density function b – a 0 e – h / b h a 0 –1/ Γ ( a 0 ) (mean a 0 b , variance a 0 b 2 ), while s is uniformly distributed on ( t 0 , t f ). The variability of the annual rate among various zones can be taken into account by adding another level to the model, that is, by considering also b as an InvGamma ( c 0 , d 0 ) random variable. Parameter a 0 and hyperparameters c 0 , d 0 , are fixed; their value can be derived from rough information on the average yearly number of occurrences per zone learnt from previous zonations. The model parameters x = ( h 1 , h 2 , s , b ) will be estimated through their posterior mean or mode, according to Bayesian approach. To obtain these summaries we must get the marginal posterior distribution of each parameter from the joint posterior distribution p ( x | y ) by solving multiple integrals; this analytic problem has not efficient numeric solution but can be solved by resorting to the class of methods called Markov chain Monte Carlo (MCMC). These methods are based on the simulation of a sample of dependent values which constitute a realization of a stationary Markov chain asymptotically convergent in distribution to the quantity of interest, in our case, x . If the data set consists of only a few events, the analysis may be affected by the value t 0 of the left extreme of the span in exam; consequently we have chosen t 0 so that the time interval separating it from the first event is equal to the average inter-event time, calculated taking into account the censored observation related to the time elapsed between the latest event and t f . Thus we have the relationship (2)