Three-Dimensional Inversion of MT Fields Using Bayesian Statistics
Vjacheslav Spichak, Michel Menvielle, Michel Roussignol, 1999. "Three-Dimensional Inversion of MT Fields Using Bayesian Statistics", Three-Dimensional Electromagnetics, Michael Oristaglio, Brian Spies
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Bayesian statistics provide a formalism for inversion of magnetotelluric (MT) data in 3-D structures composed of elementary homogeneous domains. Available information (including assumptions about the model) is put in a probability density function (PDF) for prior values of the conductivities in the region of the search; the parameters to be found are the posterior values of the conductivity.
A stochastic algorithm called a Gibbs sampler estimates the posterior PDF. The outer cycle of the iterative inversion consists of scanning the homogeneous domains in the region of the search; the inner cycle involves solution of the forward problem for a set of models. This process represents a Markov chain, whose transition law converges to the marginal PDF of the parameters.
The inner cycle uses the finite-difference program FDM3D-MT, which computes electromagnetic responses in the frequency domain for 1-D, 2-D, or 3-D models. Iterative solution of the finite-difference equations is very fast and reduces greately the total CPU time since the results of the previous cycle are used as starting points for the next forward model. In most of our tests, the outer iteration converges in 15–20 iterations, which allows an attack on 3-D problems even on microcomputers. Examples show how the quantity and quality of data and the prior information affect the results of inversion.
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