On the uncertainties of seismic parameters: a Bayesian framework for their estimation using Brune's model

The estimation of seismic parameters from ground-motion records is subject to many uncertainties, such as: (i) parameterization, modeling procedures and underlying hypotheses, (ii) approximated input parameters, (iii) instrumental errors on records and their impact in data post-processing, (iv) procedures to estimate model's parameters. However these uncertainties are rarely treated and propagated to the final results. For example, on one side, density of rocks, velocity model, geometrical spreading, radiation pattern are just some of the common parameters needed to estimate the main seismic parameters of an earthquake and are generally used as average values. On the other side, uncertainties derived from the acquisition system and processing of the data are often neglected. Nevertheless, in many cases these uncertainties may be particularly important, as for example in the analysis of historical earthquakes, where both instrumental response and treatment of analog records intrinsically imply non negligible sources of uncertainty. Here, we present a new Bayesian procedure to estimate seismic parameters that allows: (i) to obtain a robust estimation of the Brune's model parameters (Brune 1970, 1971, J Geophys Res) and relatives uncertainties, (ii) to account for the uncertainty related to the Earth model, and (iii) to propagate such uncertainties on the estimation of seismological parameters (seismic moment, moment magnitude, radius of the circular source zone and static stress drop). It is important to highlight that this study does not want to discuss the validity or the physical significance of the Brune's model, but it is focused on the details of how to fit it on a dataset in order to evaluate the seismological parameters, accounting and properly propagating a rather large range of uncertainties. These capabilities of the proposed procedure are finally demonstrated through an illustrative application analyzing seismic records from historical events.