Accounting for prediction uncertainty when inferring subsurface fault slip

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, C_χ, which combines a covariance matrix for observation errors, C_d and a covariance matrix for prediction errors, C_p, associated with inaccurate model predictions. We develop a physically based stochastic forward model to treat the model prediction uncertainty and show how C_p 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 C_p to uncertainties on the elastic parameters of different regions (e.g. crust, mantle, etc.). We demonstrate the importance of including C_p 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 C_p. The improved modelling of C_p 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.