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International audience; This study lays the groundwork for a new generation of earthquake source models based on a general formalism that rigorously quantifies and incorporates the impact of uncertainties in fault slip inverse problems. We distinguish two sources of uncertainty when considering the discrepancy between data and forward model predictions. The first class of error is induced by imperfect measurements and is often referred to as observational error. The second source of uncertainty is generally neglected and corresponds to the prediction error, that is the uncertainty due to imperfect forward modelling. Yet the prediction error can be shown to scale approximately with the size of earthquakes and thus can dwarf the observational error, particularly for large events. Both sources of uncertainty can be formulated using the misfit covariance matrix, Cχ, which combines a covariance matrix for observation errors, Cd and a covariance matrix for prediction errors, Cp, associated with inaccurate model predictions. We develop a physically based stochastic forward model to treat the model prediction uncertainty and show how Cp can be constructed to explicitly account for some of the inaccuracies in the earth model. Based on a first-order perturbation approach, our formalism relates Cp to uncertainties on the elastic parameters of different regions (e.g. crust, mantle, etc.). We demonstrate the importance of including Cp using a simple example of an infinite strike-slip fault in the quasi-static approximation. In this toy model, we treat only uncertainties in the 1-D depth distribution of the shear modulus. We discuss how this can be extended to general 3-D cases and applied to other parameters (e.g. fault geometry) using our formalism for Cp. The improved modelling of Cp is expected to lead to more reliable images of the earthquake rupture, that are more resistant to overfitting of data and include more realistic estimates of uncertainty on inferred model parameters.

The estimation of finite fault earthquake source models is an inherently underdetermined problem: there is no unique solution to the inverse problem of determining the rupture history at depth as a function of time and space when our data are limited to observations at the Earth's surface. Bayesian methods allow us to determine the set of all plausible source model parameters that are consistent with the observations, our a priori assumptions about the physics of the earthquake source and wave propagation, and models for the observation errors and the errors due to the limitations in our forward model. Because our inversion approach does not require inverting any matrices other than covariance matrices, we can restrict our ensemble of solutions to only those models that are physically defensible while avoiding the need to restrict our class of models based on considerations of numerical invertibility. We only use prior information that is consistent with the physics of the problem rather than some artefice (such as smoothing) needed to produce a unique optimal model estimate. Bayesian inference can also be used to estimate model-dependent and internally consistent effective errors due to shortcomings in the forward model or data interpretation, such as poor Green's functions or extraneous signals recorded by our instruments. Until recently, Bayesian techniques have been of limited utility for earthquake source inversions because they are computationally intractable for problems with as many free parameters as typically used in kinematic finite fault models. Our algorithm, called cascading adaptive transitional metropolis in parallel (CATMIP), allows sampling of high-dimensional problems in a parallel computing framework. CATMIP combines the Metropolis algorithm with elements of simulated annealing and genetic algorithms to dynamically optimize the algorithm's efficiency as it runs. The algorithm is a generic Bayesian Markov Chain Monte Carlo sampler; it works independently of the model design, a priori constraints and data under consideration, and so can be used for a wide variety of scientific problems. We compare CATMIP's efficiency relative to several existing sampling algorithms and then present synthetic performance tests of finite fault earthquake rupture models computed using CATMIP.