Bayesian analysisof a probability distribution for local intensity attenuation

Intensity attenuation and its variation as a function of the distance and earthquake size is still a critical issue in evaluating seismic hazard. We present a method that allows us to incorporate additional information from the historical earthquake felt reports in the probability estimation of local intensity attenuation. The approach is based on two ideas: a) standard intensity versus epicentral distance relationships constitute an unnecessary filter between observations and estimates; and b) the intensity decay process is affected by many, scarcely known elements (the physical parameters of the source, propagation path effects, building vulnerability, the semi-qualitative character of macroseismic scales, etc.). Hence intensity decay should be treated as a random variable as is the macroseismic intensity. We assume here that decay, defined on the set {0,1, ..., I0}, follows a binomial distribution with parameters (I0, p); p depends on the distance from the epicenter and is related to the probability of null decay at that distance. According to the Bayesian approach this p parameter is, in turn, a Beta random variable. The observations related to earthquakes with their epicenter outside the area concerned, but belonging to homogeneous zones, are used as prior knowledge of the phenomenon, while the data points of events inside the area are used to update the estimates through the posterior means of the quantities involved. Our methodology is described in detail in an application to the Umbria-Marche area in Central Italy. The data sets examined are the macroseismic intensity database DOM4.1 and the zonation ZS.4, both compiled by the Italian Group for Defence against Earthquakes (GNDT). The method is validated by comparing the observed and the estimated intensity data points of the Camerino (28/07/1799) and of the Colfiorito (26/09/1997) earthquakes