Occam’s Inversion of 3-D Electrical Resistivity Tomography
Douglas J. LaBrecque, Gianfranco Morelli, William Daily, Abelardo Ramirez, Paul Lundegard, 1999. "Occam’s Inversion of 3-D Electrical Resistivity Tomography", Three-Dimensional Electromagnetics, Michael Oristaglio, Brian Spies
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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.
Figures & Tables
In 1975 Jerry Hohmann published a paper1 that described his numerical implementation of an integral-equation method for three-dimensional electromagnetic (3-D EM2) modeling. The matrix equation for the simple model that he studied—a half-space containing a rectangular body discretized into 100 cubic cells—barely fit into the computer (a UNIVAC 1108 at the University of Utah). Coaxing interesting and correct results from the model and method clearly comprised much of the art and fun of the paper. And winding through the paper’s 50 or so equations and nearly 20 figures was a clear message: 3-D EM is different!
Three-dimensional electromagnetics is qualitatively different with new phenomena3 and new challenges to our understanding of how electromagnetic fields interact with Earth and other conductive bodies (including our own). In subsequent years, Jerry with his students and colleagues pursued these challenges across many fields—mining geophysics, geothermal exploration, magnetotelluric crustal studies, environmental geophysics, oil and gas exploration—in both the time and frequency domains. Of his 51 articles4 in journals and monographs, more than half dealt with three-dimensional electromagnetics.
In 1995, 20 years after Jerry’s classic paper (and three years after his death from cancer in May, 1992), nearly 200 scientists from around the world gathered at Schlumberger–Doll Research in Ridgefield, Connecticut, for a symposium in his memory, the (first) International Symposium on Three-Dimensional Electromagnetics. More than 70 papers were presented in oral and poster sessions during three days organized around the themes: Modeling, Inversion, and Practice. The quality of the work presented, the liveliness of the discussions, and the demand for the symposium proceedings were the impetus for this new volume. We invited the authors to submit longer, more tutorial versions of their articles for a book to be published by the Society of Exploration Geophysicists (SEG) in the series Geophysical Developments.
As is evident from the size of this volume, we were overwhelmed by the response. We hope that readers will find the contents equally weighty. The 44 articles collected here are the work of 97 authors, representing 55 different institutions (universities, government or industrial research labs) from 13 countries around the world. All have been reviewed and edited according to the strict standards of SEG’s lead journal, Geophysics. They represent the state of the art in 3-D EM at the time final revisions were received (from the fall of 1997 through the spring of 1998).
The lead article addresses one of Jerry’s favorite subjects—the need for independent checks on any numerical calculation; it shows how far we have come since 1975 and how far we still are from routine, confident use of 3-D EM models. We have grouped the remaining articles into nine sections:
3-D EM and parallel computers
Magnetotellurics and global induction
Mining and exploration geophysics
Borehole geophysics and logging
This division into techniques and applications is naturally very rough; many articles could easily appear in two or three different sections. The subjects covered in this volume touch, we believe, on every major technique being used today to compute, analyze, visualize, and understand 3-DEM fields in every major application of electrical geophysics (and in two applications outside geophysics: the interaction of 3-DEM fields with the human body and the non-destructive testing of aircraft). The late 1980’s saw the rapid development of 3-D seismics, which has revolutionized exploration for oil and gas in the 1990’s. The early years of the new millenium may see another revolution brought about by the rapid advances now occurring in 3-D EM.
Ridgefield, Connecticut USA
23 May 1998