A geometric approach to joint inversion with applications to contaminant source zone characterization

Abstract

This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inverse problems, the main motivation here is to characterize subsurface contaminant source-zones by processing down gradient hydrological data and cross-gradient electrical resistance tomography (ERT) observations. Inspired by Newton's method for multi-objective optimization, we present an iterative inversion scheme in which descent steps are chosen to simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixelbased inversion over the whole domain. Instead, we employ a parametric level set (PaLS) technique that substantially reduces the number of underlying parameters, making the inversion computationally tractable. The performance of the technique is examined and discussed through the reconstruction of source zone architectures that are representative of dense non-aqueous phase liquid (DNAPL) contaminant release in a statistically homogenous sandy aquifer. In these examples, the geometric configuration of the DNAPL mass is considered along with additional information about its spatial variability within the contaminated zone, such as the identification of low and high saturation regions. Comparison of the reconstructions with the true DNAPL architectures highlights the superior performance of the model-based technique and joint inversion scheme. *

Identifying attributes of CO2 leakage zones in shallow aquifers using a parametric level set method

“…Those authors then identified all unknown parameters using the adjoint-state equation approach. In a follow-on study, Aghasi et al 52 acknowledged the potential issue of local minima. A potential issue with such an approach is that different types of parameters may have 'aliasing effect' on each other (i.e., the result of adjusting weights may be equivalent to scaling the CSRBF or shifting the CSRBF centers), which in the worst case may even prevent the solution from converging.…”

Radial-basis-function level-set-based regularized Gauss–Newton-filter reconstruction scheme for dynamic shape tomography

“…In an explicit parametrization, this boundary has been described in terms of a spline basis [9,12,14] in two dimensions or with spherical harmonics [10] in three dimensions. In implicit formulations [6,[17][18][19], typically the shape unknowns are the values of the function sr on the reconstruction grid. In [17] with the objective of retaining an implicit representation coupled with significant search-space-dimensionality reduction (as in explicit schemes), we represent sr as a RBF via a Hermite interpolation scheme to fit a few on-curve points (called centers of the RBF, and denoted by r c 1 …r c m ) and the normal unit vectors at those points (denoted by n 1 …n m , where n i ≡ cos θ c i ; sin θ c i for some θ c i ).…”

“…While the first (explicit-representation) class of schemes (as in [9][10][11][12][13][14]) has the advantage of fewer unknowns, which is useful in potential threedimensional reconstructions, the second (implicitrepresentation) class [6,15,16] is better suited to handle topological changes in the evolving shape of the boundary. Radial basis function (RBF)-based implicit-representation reconstruction schemes were first suggested in [17] followed by recent works such as [18,19], extending the capability of the approaches in [10,14] by allowing for topological changes, while retaining their advantage over conventional implicitrepresentation schemes of having few unknowns. A detailed literature survey of these various classes of schemes is given in [10,13,14,17,18].…”

“…Radial basis function (RBF)-based implicit-representation reconstruction schemes were first suggested in [17] followed by recent works such as [18,19], extending the capability of the approaches in [10,14] by allowing for topological changes, while retaining their advantage over conventional implicitrepresentation schemes of having few unknowns. A detailed literature survey of these various classes of schemes is given in [10,13,14,17,18].…”

“…In recent works [18,19], a compactly supported RBF-based parameterized level-set representation with centers in the interior of the domain (the centers in [17] are placed on the boundary) is used to solve single and multi-objective reconstruction problems in static nonlinear tomographic settings.…”