A stochastic approach to reconstruction of faults in elastic half space

We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minimized for building the image. We first prove that the minimum of that functional converges to the unique solution of the related fault inverse problem. Due to inherent uncertainties in measurements, rather than seeking a deterministic solution to the fault inverse problem, we then consider a Bayesian approach. In this approach the geometry of the fault is assumed to be planar, it can thus be modeled by a three dimensional random variable whose probability density has to be determined knowing surface measurements. The randomness involved in the unknown slip is teased out by assuming independence of the priors, and we show how the regularized error functional introduced earlier can be used to recover the probability density of the geometry parameter. The advantage of the Bayesian approach is that we obtain a way of quantifying uncertainties as part of our final answer. On the downside, this approach leads to a very large computation since the slip is unknown. To contend with the size of this computation we developed an algorithm for the numerical solution to the stochastic minimization problem which can be easily implemented on a parallel multi-core platform and we discuss techniques aimed at saving on computational time. After showing how this algorithm performs on simulated data, we apply it to measured data. The data was recorded during a slow slip event in Guerrero, Mexico.