DC Electrical Resistance Tomography Inversion

Document Type : Research

Authors

1 Assistant professor of Institute of Geophysics, University of Tehran

2 Institute of Geophysics

Abstract

Geo-electrical approaches are intrinsically sensitive to discontinuities within the electrical characteristics of subsurface structures. Electrical resistivity tomography (ERT) is well-established and widely used to solve a variety of subsurface detection problems, e.g., engineering studies, environmental and hydro-geophysical investigations, and archaeological exploration. The ERT measurements are also applied to map geologic features such as lithology, structure, fractures, and stratigraphy; hydrologic features such as depth to the water table, depth to aquitard (impermeable layer), and groundwater salinity; and to delineate groundwater contaminants. As in any geophysical procedures, surface or borehole resistivity measurements do not provide a direct image of the Earth’s subsurface but simply the integrated effect of the subsurface properties which could be far removed from the ground truth, in particular, in cases of complex subsurface property distribution. To infer an image of the subsurface resistivity distribution from a limited number of uncertain observations, a non-linear inverse problem needs to be formulated. The inverse direct current resistivity problem is generally ill-posed with respect to data uncertainties and incomplete data sets. Hence, regularization schemes must be incorporated in the inverse problem to find a unique and stable solution. Even though regularization plays a significant role in inverse problem theory, there is a large ambiguity in choosing it (Scales and Snieder, 1997). The most commonly used techniques for regularization of inverse problems are 1) the projection methods, such as truncated singular value decomposition and 2) penalty methods, such as Tikhonov-Phillips regularization, and total variation methods. In the Tikhonov-Phillips method, the regularization term consists of the squared norm of the sought solutions. The quadratic regularization methods smear out the edges of the desired model but non-quadratic regularizations address the issue of stability without penalizing the required sharp boundaries. They usually require the solution to be sparse in a specific domain. Although non-quadratic regularization methods lead to further complexity of the problem compared to the quadratic approaches. For both of these approaches, a suitable regularization parameter should yield a fair balance between the perturbation error and regularized solution. Whereas in most cases the true subsurface geology exhibits a gradual variation in the electrical properties of layer boundaries, using the smoothness-constrained method is more suitable to visualize the Earth’s structures. However, smoothing constraints are inconsistent with realistic circumstances when sharp bulk conductivity contrast exists in the subsurface. The development of direct resistivity inversions has progressed successfully. The first attempt for inversion of 2D resistivity data made by Pelton et al (1978). Although their algorithm is not well suited to complex cases. Smith and Vozoff (1984) and Tripp et al (1984) presented a 2D resistivity inversion using a finite-difference method. The schemes proposed by them are suitable for complicated 2D models but do not incorporate the effects of topography on resistivity data in the inversion algorithm. Toy and Yang (1990) developed an algorithm for 2D resistivity inversion where the topography is considered in the model. The paper of Loke and Barker (1996) proposed a Gauss-Newton-based algorithm for ERT inversion in the framework of finite-difference. In recent years, the introduction of multi-channel instrument resulted in a renaissance of the geo-electrical data acquisition, and consequently, significant progress in 2D and 3D inverse modeling algorithms (e.g., Dahlin, 2001; Dahlin and Zhou, 2003; Günther et al, 2006; Chambers et al, 2006; Oldenborger et al, 2007; Wilkinson et al, 2010; Zhou et al, 2014; Pang et al, 2020). In addition, an integral part of every electrical resistivity inversion is an accurate and efficient forward modeling resulting in the numerical simulation of responses for a given conductivity model. A numerical technique for the 2.5 dimensional DC resistivity forward calculation based on the finite-difference method is provided. In this study, we developed and applied a model-space Occam’s method to the electrical resistivity tomography inversion. Mathematically, Occam's inversion is a generalized least-squares inversion method under some specified model property constraint (Constable et al., 1987; De Groot-Hedlin and Constable, 1990). Thus make the inversion method more stable and robust. This algorithm converges even in complex subsurface property distribution where other inversion algorithms may fail. The efficiency and applicability of our numerical strategy for 2D resistivity inverse modeling is tested using two synthetic case studies as well as a real dataset. Furthermore, the reliability of the recovered models is dealt with through a model sensitivity analysis based on the resolution density distribution. The rest of the paper is structured as follows: Section 2 gives a brief review of the forward and inverse modeling formulation. Next, section 3 verifies the functionality of the inversion algorithm using synthetic and real resistivity data sets. Finally, section 4 provides a short conclusion and summary.

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