Fractional Helmholtz to Help Detect Fine Scale Features

Fine scale features in the subsurface are notoriously difficult to detect and characterize using electromagnetic source and sensor systems. One approach is to make use of large scale inversion in which the target inversion parameters are material properties. The technique consists of a least squares objective function that attempts to reconcile the difference between numerical predictions and experimental observations. Based on several simplifying assumptions, we concentrate on the Helmholtz equation as a constraint to our optimization formulation. Because this is a large scale, nonlinear optimization problem, we make use of adjoints and Newton-Krylov based solvers with trust-region globalization. However, a material inversion with the standard Helmholtz equation does not provide any fine scale characterization without a series of fine scale discretizations, which is especially complicated in the case of fractures. To that end, we consider the use of fractional exponents on the differential operators for the Helmholtz equation to help characterize fine scale features in the subsurface. This super-diffusion approach has been known to achieve non-local phenomena. Our solution technique involves the decomposition of Helmholtz with boundary conditions into four (two real and two imaginary) equations and a decomposition of the boundary conditions with the intent of assigning simple boundary conditions to the fractional operator-based equations and the remaining boundary conditions to the non-fractional operator-based equations. All four equations are discretized with finite elements. The fractional-operators are replaced with a summation of standard differential operators according to the Dunford-Taylor approach. We demonstrate our approach on a prototype with "truth measurements" from a synthetically generated fine scale system.