A stochastic approach to reconstruction of faults in elastic half space

Abstract

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.

Dislocations in a layered elastic medium with applications to fault detection

“…Reconstruction has been tested primarily through iterative algorithms [36], based on Newton's methods or constrained optimization of a suitable misfit functional, using either Boundary Integral methods or Finite Element methods, as well as Green's function methods to solve the direct problem. For stochastic and statistical approaches to inversion we mention [20,35] and references therein. We do not address here the question of reconstruction and its stability (see [11,32]).…”

Stability properties for a class of inverse problems

“…Here, R is the square [−150, 150] 2 in R 2 and V is the square [−200, 200] in the plane with equation x 3 = 0. These numbers were chosen to facilitate comparison to previous studies [18,19,21,23]. On V we choose a uniform 11 by 11 grid for the points P j , 1 ≤ j ≤ M .…”

A stochastic algorithm for fault inverse problems in elastic half space with proof of convergence

“…We consider data generated in a configuration closely related to studies involving field data for a particular region and a specific seismic event [13,15]. To ensure that we perform a simulation with realistic orders of magnitude, the scaling is such that the unit for x in R 3 is in kilometers, while ũ and G are in meters, as in [15].…”

“…However, the fault inverse problem is nonlinear in m. If one were to apply any of these methods to select a fixed C, the selection would depend on m and as a result different candidates for m would be unfairly compared, [12]. Better results are obtained if we fix the same value for C for all m in B, [13]. Even then, determining the optimal value for C is not possible.…”