Bayesian inference of kinematic earthquake rupture parameters through fitting of strong motion data

Due to uncertainties in data and in forward modelling, the inherent limitations in data coverage and the non-linearity of the governing equation, earthquake source imaging is a problem with multiple solutions. The multiplicity of solutions can be conveniently expressed using a Bayesian approach, which allow to state inferences on model parameters in terms of probability density functions. The estimation of the posterior state of information, expressing the combination of the a priori knowledge on model parameters with the information contained in the data, is achieved in two steps. First, we explore the model space using an evolutionary algorithm to identify good data fitting regions. Secondly, using a neighbourhood algorithm and considering the entire ensemble of models found during the search stage, we compute a geometric approximation of the true posterior that is used to generate a second ensemble of models from which Bayesian inference can be performed. We apply this methodology to infer kinematic parameters of a synthetic fault rupture through fitting of strong motion data. We show how multiple rupture models are able to reproduce the observed waveforms within the same level of fit, suggesting therefore that the solution of the inversion cannot be expressed in terms of a single model but rather as a set of models which show certain statistical properties. For all model parameters we compute the posterior marginal distribution. We show how for some parameters the posterior do not follow a Gaussian distribution rendering the usual characterization in terms of mean value and standard deviation not correct. We compare the posterior marginal distributions with the 'raw' marginal distributions computed from the ensemble of models generated by the evolutionary algorithm. We show how they are systematically different proving therefore that the search algorithm we adopt cannot be directly used to estimate uncertainties. We also analyse the stability of our inferences comparing the posterior marginals computed by different independent ensembles. The solutions provided by independent explorations are similar but not identical because each ensemble searches the model space differently resulting in different reconstructed posteriors. Our study illustrates how uncertainty estimates derive from the topology of the objective function, and how accurate and reliable resolution analysis is limited by the intrinsic difficulty of mapping the 'true' structure of the objective function.