volume | PIER Journals

Abstract

The limited-memory quasi-Newton optimization method with simple bounds has been applied to develop a novel fully threedimensional (3-D) magnetotelluric (MT) inversion technique. This nonlinear inversion is based on iterative minimization of a classical Tikhonov-type regularized penalty functional. But instead of the usual model space of log resistivities, the approach iterates in a model space with simple bounds imposed on the conductivities of the 3-D target. The method requires storage that is proportional to n_cpÃ—N, where the N is the number of conductivities to be recovered and ncp is the number of the correction pairs (practically, only a few). This is much less than requirements imposed by other Newton type methods (that usually require storage proportional to NÃ—M, or NÃ—N, where M is the number of data to be inverted). Using an adjoint method to calculate the gradients of the misfit drastically accelerates the inversion. The inversion also involves all four entries of the MTimp edance matrix. The integral equation forward modelling code x3d by Avdeev et al. [1, 2] is employed as an engine for this inversion. Convergence, performance and accuracy of the inversion are demonstrated on a 3D MTsyn thetic, but realistic, example.

1. Avdeev, D. B., A. V. Kuvshinov, O. V. Pankratov, and G. A. Newman, "High-performance three-dimensional electromagnetic modeling using modified Neumann series. Wide-band numerical solution and examples," J. Geomagn. Geoelectr., Vol. 49, 1519-1539, 1997.

2. Avdeev, D. B., A. V. Kuvshinov, O. V. Pankratov, and G. A. Newman, "Three-dimensional induction logging problems, Part I, An integral equation solution and model comparisons," Geophysics, Vol. 67, 413-426, 2002.
doi:10.1190/1.1468601

3. Avdeeva, A. D. and D. B. Avdeev, "QN inversion of large-scale MTdata," Progress in Electromagnetic Research Symposium, 200-213, 2006.

4. Avdeev, D. B., "Three-dimensional electromagnetic modelling and inversion from theory to application," Surveys in Geophysics, Vol. 26, 767-799, 2005.
doi:10.1007/s10712-005-1836-x

5. Chen, J., D. W. Oldenburg, and E. Haber, "Reciprocity in electromagnetics: application to modeling marine magnetometric resistivity data," Phys. Earth Planet. Inter., Vol. 150, 45-61, 2005.
doi:10.1016/j.pepi.2004.08.015

6. Dorn, O., H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, "A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields," Inverse Problems, Vol. 15, 1523-1558, 1999.
doi:10.1088/0266-5611/15/6/309

7. Haber, E., "Quasi-Newton methods for large-scale electromagnetic inverse problems," Inverse Problems, Vol. 21, 305-323, 2005.
doi:10.1088/0266-5611/21/1/019

8. Mackie, R. L., W. Rodi, and M. D. Watts, "3-D magnetotelluric inversion for resource exploration," 71st Annual Internat. Mtg. Extended Abstracts, 1501-1504, 2001.

9. McGillivray, P. R. and D. W. Oldenburg, "Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problems," Geophysics, Vol. 60, 899-911, 1990.

10. Newman, G. A., S. Recher, B. Tezkan, and F. M. Neubauer, "3D inversion of a scalar radio magnetotelluric field data set," Geophysics, Vol. 68, 791-802, 2003.
doi:10.1190/1.1581032

11. Newman, G. A. and P. T. Boggs, "Solution accelerators for largescale three-dimensional electromagnetic inverse problem," Inverse Problems, Vol. 20, 151, 2004.
doi:10.1088/0266-5611/20/6/S10

12. Nocedal, J. and S. J. Wright, Numerical Optimization, Springer, 1999.

13. Rodi, W. and R. L. Mackie, "Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion," Geophysics, Vol. 66, 174-187, 2001.
doi:10.1190/1.1444893

14. Siripunvaraporn, W., M. Uyeshima, and G. Egbert, "Threedimensional inversion for network-magnetotelluric data," Earth Planets Space, Vol. 56, 893-902, 2004.

15. Tikhonov, A. N. and V. Y. Arsenin, Solutions of Ill-posed Problems, Wiley, 1977.

16. Weidelt, P., "Inversion of two-dimensional conductivity structures," Phys. Earth Planet. Inter., Vol. 10, 281-291, 1975.
doi:10.1016/0031-9201(75)90054-0

17. Zhdanov, M. S. and E. Tolstaya, "Minimum support nonlinear parametrization in the solution of a 3D magnetotelluric inverse problem," Inverse Problems, Vol. 20, 937-952, 2004.
doi:10.1088/0266-5611/20/3/017

Dr. Dmitry Avdeev

Dr. Anna Avdeeva