t. This paper surveys some of the work our group has done in electrical impedance tomography.

Electrical Impedance Tomography

We review theoretical and numerical studies of the inverse problem of electrical impedance tomography which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.

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Electrical impedance tomography

An improved electrical impedance tomographic reconstruction algorithm is presented that is generally guaranteed to converge. The algorithm is attractive for several reasons. A modified Newton?Raphson method varies a finite-element model of resistivities to fit a set of voltage measurements in a least-squared sense. Two procedures for calculating the Jacobian matrix are derived. One is standard, while the other is based on the compensation theorem. This second procedure is more efficient for computations, and just as accurate as the standard one. The inherent ill-conditioning in the approximate Hessian matrix of the linearized system is eliminated using the Marquardt method. Results from two-dimensional computer simulations are compared to four other reconstruction algorithms, which are based on methods proposed by other authors. The modified Newton?Raphson method provided significantly better reconstructions than any of the other methods. The algorithms compared are the perturbation, equipotential, iterative-equipotential, and the double-constraint methods. The modified Newton?Raphson method was found to be sensitive to measurement error, but future work in designing electrode-probing configurations is expected to reduce this sensitivity.

Comparing Reconstruction Algorithms for Electrical Impedance Tomography

The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problem. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newton's method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newton's One-Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.

NOSER: An algorithm for solving the inverse conductivity problem

Cross borehole electrical resistivity tomography (ERT) was used to image the resistivity distribution before and during two infiltration experiments. In both cases water was introduced into the vadose zone, and the change in resistivity associated with the plume of wetted soil was imaged as a function of time. The primary purpose of this work was to study the capabilities and limitations of ERT to image underground structure and ground water movement in the vadose zone. A secondary goal was to learn specifics of unsaturated flow in a complex geologic setting. Tomographs of electrical resistivity taken before infiltration image coarser, well-drained soils (sands and gravels) as more resistive zones, whereas finer grained soils (silts and clays), which hold more water by capillarity, are imaged as more conductive. Images of changes in resistivity during infiltration show growth of the water infiltration plume with time that is consistent with known geology. In the ERT images we see the effects of capillary barriers and infer differences between capillary-driven flow through fine sediments and gravity-driven flow through very permeable sediments. Images are consistent with numerical flow simulations using hydrological parameter values consistent with soil types inferred from well logs. ERT can be a useful tool to monitor movement of circuitous moisture fronts in a heterogeneous field setting that would go undetected by borehole measurements.

Electrical resistivity tomography of vadose water movement

An electrical impedance-based method and apparatus for imaging the interior of a structure, such as a mineral or human body, having regions of differing impedance, are described. The technique involves impressing a number of excitation patterns on the body and measuring the resulting voltages at a number of sites. Because voltages need not be measured at current-injection sites, contact resistance problems are ameliorated. This impedance-computed tomography reconstruction process employs the solution of the Poisson equation and makes no assumptions of raylike behavior of the current flow paths. Results, based on computer-simulated data and real measurements, are presented.

A Wexler

Papers

Impedance computed tomography algorithm and system.