THREE-DIMENSIONAL NONLINEAR INVERSION OF ELECTRICAL CONDUCTIVITY FROM
ELECTROMAGNETIC MEASUREMENTS IN THE EARTH
Aria Abubakar
Laboratory of Electromagnetic Research, Faculty of Information Technology and Systems
The main emphasis of this project is to develop an efficient method to image the subsurface around a borehole and between two boreholes in an arbitrary three-dimensional formation from electromagnetic measurements in the borehole.
Introduction
In this project it is proposed to image the subsurface around a borehole (single-well) and between two boreholes (cross-well) by reconstructing the electrical conductivity of the earth from electromagnetic measurements in the borehole. These measurements are either of the electrode type (static problem) or of the induction type (magnetic dipole, 27 kHz). The conductivity distribution is seen as a contrasting domain with respect to a known embedding. The response of the conductivity contrast is measured in terms of electric and/or magnetic field at the borehole axis. From these electromagnetic field quantities the conductivity contrast has to be reconstructed, which is a highly nonlinear problem. Furthermore, in the single-well case the angle through which the borehole penetrates the rock formation can be more or less arbitrary and the measurements are always affected by environmental effect such as those from the borehole, invaded zones, layering, and adjacent thin beds. This three-dimensional nonlinear problem has to be solved efficiently, such that imaging results are available readily.
This research project started in April 1997 finished in October 2000.
Results in 2000
The method, which has been successfully applied to handle the electrode logging problem in a formation with arbitrary number of layer, dip, and invasion zones has been extended to handle the problem of induction logging and full-wave electromagnetic problem. Further, it is shown that the multi-frequency data, preconditioning operators based on extended born, and total variation regularization term can significantly improved the reconstruction results of the cross-well, single-well induction/electrode logging and surface sensing problem.
Articles in international jornals
Abubakar, A., P.M. van den Berg; Non-linear three-dimensional inversion of cross-well electrical measurements, Geophysical Prospecting, Vol. 48, No. 1, 2000, pp. 109-134.
Abubakar, A., P.M. van den Berg; Nonlinear inversion in electrode logging in a highly deviated formation with invasion using an oblique coordinate system, IEEE Transactions on Geoscience and Remote Sensing,
Vol. 38, No. 1, January 2000, pp. 25-38.
Abubakar, A., P.M. van den Berg; Three-dimensional inverse scattering applied to cross-well induction sensors, IEEE Transactions on Geoscience and Remote Sensing, Vol. 38, No. 4, July 2000, pp. 1669-1681.
Abubakar, A., P.M. van den Berg; Iterative reconstructions of electrical conductivity from multiexperiment low-frequency electromagnetic data, Radio Science, Vol. 35, No. 6, 2000, pp. 1293-1306.
Abubakar, A., P.M. van den Berg, B.J. Kooij; A conjugate gradient contrast source technique for 3D profile inversion, Special Issue on Problems of Random Scattering and Electromagnetic Wave Sensing, IEICE
Transactions on Electronics, Vol. E83-C, No. 12, December 2000, pp. 1864.

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Conference paper
Abubakar, A., P.M. van den Berg; Joint inversion of electrode and induction logging data, Extended Abstracts
Book 62th European Association of Geoscientists and Engineers, Glasgow, Scotland, May 29 – June 2,
2000, D-04, 4 pp.
Abubakar, A., P.M. van den Berg; Forward and inverse CG methods with extended Born as preconditioners,
Expanded Abstracts 70th Annual Meeting of Society of Exploration Geophysicists, Calgary, Canada, 6 –
11 August, 2000, CD-ROM 4p.
Abubakar, A., P.M. van den Berg; Extended Born approximation as a preconditioning operator for conjugate gradient forward and inverse methods, Proceedings IEEE 2000 International Geoscience and Remote
Sensing Symposium IGARS 2000, Honolulu, Hawaii, U.S.A., 24-28 July 2000, Vol. VII, pp. 3126-3129.
Abubakar, A., P.M. van den Berg; Three-dimensional extended contrast source inversion, Proceedings ISAP
International Symposium on Antennas and Propagation 2000, Fukuoka, Japan, 21-25 August 2000, Vol. 3, pp. 935-938.
Abubakar, P.M. van den Berg; A total variation enhanced contrast source inversion method for threedimensional profile reconstruction, Proceedings ECCOMAS 2000, Barcelona, Spain, 11-14 September
2000, CD-ROM, 19 p. + abstract, Book of abstracts, p. 1132.
PhD Thesis
Abubakar, A.; Three-dimensional nonlinear inversion of electrical conductivity, Ph.D. thesis Delft University of Technology, promotor Prof.dr.ir. P.M. van den Berg, 2000, ISBN 90 407 2105 X, 258pp.