Towards rapid uncertainty estimation in linear finite fault inversion with positivity constraints

Abstract:

Rapid estimation of the slip distribution for large earthquakes can assist greatly during the early phases of emergency response. These estimates can be used for rapid impact assessment and tsunami early warning. While model parameter uncertainties can be crucial for meaningful interpretation of such slip models, they are often ignored. Since the finite fault problem can be posed as a linear inverse problem (via the multiple time window method), an analytic expression for the posterior covariance matrix can be obtained, in principle. However, positivity constraints are often employed in practice, which breaks the assumption of a Gaussian posterior probability density function (PDF). To our knowledge, two solutions to this issue exist in the literature: 1) Not using positivity constraints (may lead to exotic slip patterns) or 2) to use positivity constraints but apply Bayesian sampling for the posterior. The latter is computationally expensive and currently unsuitable for rapid inversion. In this work, we explore an alternative approach in which we realize positivity by imposing a prior such that the log of each subfault scalar moment are smoothly distributed on the fault surface. This results in each scalar moment to be intrinsically non-negative while the posterior PDF can still be approximated as Gaussian. While the inversion is not linear anymore, we show that the most probable solution can be found by iterative methods which are less computationally expensive than numerical sampling of the posterior. In addition, the posterior covariance matrix (which provides uncertainties) can be estimated from the most probable solution, using an analytic expression for the Hessian of the cost function. We study this approach for both synthetic and observed W-phase data and the results suggest that a first order estimation of the uncertainty in the slip model can be obtained, therefore aiding in the interpretation of the slip distribution estimate.