GNGTS 2019 - Atti del 38° Convegno Nazionale

312 GNGTS 2019 S essione 2.1 ANOTHER STEP IN THE PATH FROM MACROSEISMIC FIELDS TO PROBABILISTIC MODELING OF ATTENUATION R. Rotondi, E. Varini Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Milano, Italy It is beyond doubt that information provided by historical seismology can provide a valuable aid to understanding the evolution of the seismic phenomenon; we cannot therefore renounce to analyze macroseismic intensity data in order to assign the seismic hazard of an area; well aware that this requires the development of specific models and techniques different from those adopted in the study of instrumental data. In this context the issue of macroseismic attenuation is also included. From the beginning of the 2000s, a path has been undertaken in the direction of probabilistic modeling of macroseismic attenuation, from the perspective of treating the whole process as random and not only of adding a Gaussian error to the empirical relationships between magnitude, distance and intensity at site. In fact, the irregularities in the shape and extent of macroseismic fields depend on various factors and their potential interactions, like topography, pattern of population density, characteristics of regional geology, that cannot be reduced to a measurement error. Among the cornerstones of this approach there is the respect, as far as possible, of the ordinal nature of macroseismic intensity. According to this assumption, the intensity at any site is considered as an integer random variable that varies between 1 and the epicentral intensity I 0 . Beta-binomial model. As for any model, we started by simplifying reality and assuming that the attenuation trend is circular. Conditioned on the epicentral intensity I 0 and on a fixed epicentral distance, the intensity I s at a given site was assumed to have a binomial distribution with parameter p . Given a point source and circular isoseismal lines bounding the points of equal intensity, one draws J circular bins around the epicenter and supposes that in all of the sites within each j -th bin, I s , so as ∆ I , has the same binomial distribution with parameter p j , i.e.: (1) In its turn, each p j has a beta distribution with hyperparameters α j and β j that, according to the Bayesian approach, are assigned by exploiting the information drawn from previous databases; then the posterior mean of each p j provides the estimate of these paramaters. To extend the value of p at any epicentral-site distance one approximates the estimates pˆ j by the smoothing inverse power function g(d) = , whose coefficients c 1 , c 2 are estimated by the method of least squares. In this way one is able to forecast in terms of macroseismic intensity I s at site the damage scenario that a future earthquake of given intensity I 0 could cause by the smoothed binomial probability distribution obtained by replacing p j with g(d) in Eq. (1) and by using the mode i smooth of this distribution as forecast value of the intensity I s at any site distant d from the epicenter [Rotondi, Zonno (2004)]. Three criteria were used to validate the results: the logarithmic scoring rule, the ratio between the probability that the fitted model assigns to an observation and the probability of the forecast value, and the absolute discrepancy between observed and estimated intensities at site. Construction of learning sets. Another crucial point is the continuous updating of the model parameters in the light of the most recent databases and on the basis of the identification of sets of macroseismic fields homogeneous from the attenuation viewpoint. As regards the first point, we refer to the results obtained from the analysis of the Italian Databases DBMI15; about the second point, each of 538 macroseimic fields of DBMI15 has been characterized through summaries of the spatial distribution of the intensity decay; in particular, location and dispersion measures (mean and median values and 3rd quantile) have been computed for each