Keywords: seismology, macroseismic map, signal's random character, probabilistic approach, stochastic approach.

The probabilistic methodologies are the most used to approach the problem of the seismic waves' propagation over a territory during an earthquake. The earthquake energy distribution over the territory is a problem affected by a high degree of uncertainty, mainly as a result of the randomness in the epicentral position, in the excitation characters and in the properties of the sub-soil layers through which vibration waves travel. The question can thus be solved by means of probabilistic tools. The different seismic parameters which characterise an earthquake from a macroscopic point of view, such as the epicenter intensity, the epicenter localisation, the magnitude, the local intensity, the occurrence periods, etc., can be looked at as independent random variables.

On the other hand, some authors have proposed that uncertainty be treated from a deterministic point of view, as an optimisation, possibly convex, problem. Such an approach is assumed, and a mechanical model is introduced which aims at an analogue for the energy distribution over the territory after a seismic event. The territory is modelled as a structured solid, two-dimensional and possibly non-homogeneous. The spread of seismic intensity is assimilated to the strain-energy distribution consequent to a distortion impressed in the neighborhood of the epicenter; such distortion is assumed to be representative of the quake fracture. In other words, the procedure aims at the identification of a "heuristic Green's function" that can be looked at as a mechanical interpretation of the seismic attenuation law.

Calibration of both parameters of the macroseismic model and of the soil surface is done by forcing the solution based on recorded historical data. The optimisation problem is based on an objective function that, by expressing the "distance" of the results obtained by the model instead of the recorded data, is a function of seismological parameters and mechanical properties of the soil. The constrained optimisation problem has as objective that is set to construct a model of the propagation of energy over the territory as close as possible to that of the function of seismic intensity recorded during an earthquake.

A statistical method is explained for the characterisation of macroseismic parameters of an earthquake, seismic characterisation of the site and seismic macrozoning. This is done after building a statistical model of epicentral probability on the basis of historical earthquakes occurred in an area. The study of seismic zoning is pursued in detail through the development of a statistical method based only on knowledge of the major macro-seismic parameters (e.g. seismic magnitude, intensity and epicentre co-ordinates, etc.) characterising the historical earthquakes that occurred in the area.

The probability of locating seismic epicenter is sought after set the magnitude of earthquake expected and knowing the shape of the probability density function location that is assumed to be generated by the proper combination of n-Gaussian functions. The probability function is determined by solving some unknown parameters about single Gaussian function in combination with n-convex coefficients. A numerical application has been developed for some historic earthquakes occurred in a predefined time interval and with a medium-high intensity, whose epicenter is located in a seismic area in Italy. In this way it shows a good agreement between field experience and history, demonstrating that the proposed approach provides a good approximation of the probability curve.

By means of another statistical processing in the same area, seismic signal propagation is simulated over the territory during some large earthquakes, highlighting a good match of the simulations with real seismic propagation.

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