Abstract The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models. To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model. Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.

Summary Electrical resistivity tomography (ERT) images the electrical properties of the subsurface from dc resistivity measurements between surface and borehole electrodes. We experiment with 3-D inversion of ERT using finite-element forward solution and a conjugate-gradient inverse routine. The algorithm finds the smoothest model (Occam’s inversion) that fits the data to a given prior error level. The algorithm takes 10 to 20 iterations to converge but requires only a single forward solution per iteration and does not require direct solution of a large system of equations. Inversion of data from two sites is shown. The first site tests the ability of ERT to monitor leaks around large metal tanks at the Hanford Reservation in Washington State. Data were collected and inverted from 16 wells placed around a circular tank. The tank is of heavy-gauge steel covered with concrete, is 15 m in diameter, and extends 2 m below the ground surface. The 3-D algorithm was modified to allow the smoothness operator to be decreased at the tank boundary. The 3-D inversion was necessary to produce an accurate picture of the leak. At a second site, ERT was used to monitor the injection of air from a vertical well at a shallow petroleum remediation site. Using a cone penetrometer, three electrode strings were placed in the ground on the corners of a right triangle. The background of the site was assumed to be layered. Results of 3-D and 2-D inversion agreed well when the regions of interest were approximately 2-D. Air injection caused large changes in resistivity. At early times, these were confined to an area near the injection point. Later, the changes were along a dipping, tabular region. At the latest times, there is evidence of mixing of brackish water at the depth of the injection point with freshwater in a shallower aquifer on the site. This mixing would have decreased the resistivity and thus the apparent size and magnitude of the zone of influence of sparging.