An analytical solution for the Bayesian estimation of ground motion from macroseismic intensity data

In this paper we present an analytical solution for the Bayesian update of ground motion estimates using macroseismic intensity (MI) observations. This solution builds on an earlier work of Ebel and Wald (Earthq Spectra 19:511–529, 2003), who proposed a discrete version of the updating problem. We show that the updated probability distribution of the ground motion has the logarithm of the median that is the weighted average of the respective logarithms of its prior value and a value computed with a ground motion-MI relation, with weights that are proportional to the predictive power of each source of information, measured with its corresponding variance. We also show that the posterior standard deviation is always smaller than the prior. In our opinion, besides its numerical simplicity, the solution presented here allows for a much clearer view of the role that the different parameters play.