Uncertainty estimations for seismic source inversions

Source inversion is a widely used practice in seismology. Magnitudes, moment tensors, slip distributions are now routinely calculated and disseminated whenever an earthquake occurs. The accuracy of such models depends on many aspects like the event magnitude, the data coverage and the data quality (instrument response, isolation, timing, etc.). Here, like in any observational problem, the error estimation should be part of the solution. It is however very rare to find a source inversion algorithm which includes realistic error analyses, and the solutions are often given without any estimates of uncertainties. Our goal here is to stress the importance of such estimation and to explore different techniques aimed at achieving such analyses. In this perspective, we use the W phase source inversion algorithm recently developed to provide fast CMT estimations for large earthquakes. We focus in particular on the linear-inverse problem of estimating the moment tensor components at a given source location. We assume that the initial probability densities can be modelled by Gaussian distributions. Formally, we can separate two sources of error which generally contribute to the model parameter uncertainties. The first source of uncertainty is the error introduced by the more or less imperfect data. This is carried by the covariance matrix for the data (C_d). The second source of uncertainty, often overlooked, is associated with modelling error or mismodelling. This is represented by the covariance matrix on the theory, C_T. Among the different sources of mismodelling, we focus here on the modelling error associated with the mislocation of the centroid position. Both C_d and C_T describe probability densities in the data space and it is well known that it is in fact C_D = C_d + C_T that should be included into the error propagation process. In source inversion problems, like in many other fields of geophysics, the data covariance (C_D) is often considered as diagonal or even proportional to the identity matrix. In this work, we demonstrate the importance of using a more realistic form for C_D. If we incorporate accurate covariance components during the inversion process, it refines the posterior error estimates but also improves the solution itself. We discuss these issues using several synthetic tests and by applying the W phase source inversion algorithm to several large earthquakes such as the recent 2011 Tohoku-oki earthquake.