GEOPHYSI CS, VOL 65, NO.6 (NOVEMBER-DECE MBER 20(0 ); p. 1746-1757,1 6 FIGS., 1TABLE. Quasi-analytical approximations and series in electromagnetic modeling Michael S. Zhdanov*, Vladimir I. Dmitrlevl , Sheng Fang**, and Gabor Hursan " ABSlRACT The quasi-linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous do- main is linearly proportional to the background (nor- mal) field through an electrical reflectivity tensor t In the original formulation of the quasi-linear approxima- tion, 1 was determined by solving a minimization prob - lem based on an integral eq uation for the scattering cur- rents. This approach is much less time-consuming than the full integral eq uation method ; however, it still re- quires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation us- ing 1 through construction of quasi-analytical expres- sions for the anomalo us electromagnetic field for 3-D and 2-D models. Quasi-analytical solutions red uce dra- matically the computational effort related to forward electromagnetic modeling of inhomo geneous geoelec- trical structures. In the last sections of this paper, we extend the quasi-analytical method using iterations and develop higher order approximations resulting in quasi- analytical series which provide improved accuracy.Com- putation of these series is based on repetitive applica- tion of the given integral contraction operator, which insures rapid convergence to the correct result. Numer - ical studies demonstrate that quasi-analytical series can be treated as a new powerful method of fast but rigorous forward modeling solution . INlRODUCTION 1975; Dmitriev and Pozdnyakova , 1992). This method is based on expressing the electromagnetic fields in terms of an inte- gral equation with respect to the excess current within an in- homogeneity. The integral equation is written as a system of linear algebraic equati ons by approximating the excess cur- rent distribution ja by the piecewise constant functions. The resulting algebraic system is solved numerically (Xiong, 1992). The main difficulty of this technique is the size of the linear system of equations matrix, which demands excessive com- puter memory and calculation time to invert. This limita- tion of the integral equation technique becomes critical in in- verse problem solution which requir es multiple forward mod- eling calculations for different (updated) geoelectrical model parameters. A novel approach to 3-D electromagnetic (EM) modeling based on linearization of the integral equations for scattered E M fields has been developed recently by Zhdanov and Fang (1996a, b, 1997). Within this method , called quasi-linear (QL) approximation, the excess currents are assumed to be propor- tional to the background (normal ) field E b through an electri- cal reflectivity tensor t In the original paper s on QL approx - imations, the electrical reflectivity tensor was determined by solving a minimization problem based on an integral equ ation for the scattering currents (Zhdanov and Fang, 1996a, b). This problem is much less time-consuming than the full IE method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to estimating)." whichleads to constructin g quasi-analytical (QA) expressions for the anomalous electromagnetic field for 3-D and 2-D models. We demon strate also the connection between the QL and QA approximati ons and the localized nonlinear (LN) approximations introduced by Habashy et al. (1993) and Torres-Verdin and Habashy (1994) and conduct a comparative study of the accuracy of different approximations. The integral equation (IE) method is a powerful tool for elec- In the last sections of the paper , we extend the quasi- tromagnetic numerical modeling (Hohmann, 1975; Weidelt, analytical method using iter ative techniq ues and develop Manuscript received by the Editor July 28,1998; revised manuscript received March 9, 2000. ' University of Utah, Department of Geology and G eophysics, Salt Lake City, Utah 84112-0111. E-mail : rnzhdanov@mines,utah.edu . [Moscow State University, Faculty of Computational Math and Cybernetics, Moscow 1169899, Russia. "Formerly Universityof Utah, Department of Geology and Geophysics, Salt Lake City, Utah; presently Baker Atl as, 10205 Westh eimer, Houston, Texas, 77042. E -mail: shengJ ang@Waii.com © 2000 Society of Exploration Geophysicists. All rights reserved. 1746