IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL Guozhong Gao and Carlos Torres-Verdín, Member, IEEE

Transcription

1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL High-Order Generalized Extended Born Approximation for Electromagnetic Scattering Guozhong Gao and Carlos Torres-Verdín, Member, IEEE Abstract Previous work on the subject of electromagnetic scattering has shown that the extended Born approximation (EBA) is more accurate than the first-order Born approximation with approximately the same operation count. However, the accuracy of the EBA degrades in cases when the source is very close to the scatterer, or when the electric field exhibits significant spatial variations within the scatterer. This paper introduces a generalized extended Born approximation (GEBA) and its high-order variants (Ho-GEBA) to efficiently and accurately simulate electromagnetic scattering problems. We make use of a generalized series expansion of the internal electric field to construct high-order terms of the generalized extended Born approximation (Ho-GEBA). A salient feature of the Ho-GEBA is its enhanced accuracy over the Born approximation and the EBA, even when only the first-order term of the series expansion is considered in the approximation. This behavior is not conditioned by either the source location or the spatial distribution of the internal electric field. A unique feature of the Ho-GEBA is that it can be used to simulate electromagnetic scattering due to electrically anisotropic media. Such a feature is not possible with approximations of the internal electric field that are based on the behavior of the background electric field. Three-dimensional (3-D) models of electromagnetic scattering are used to benchmark the efficiency and accuracy of the Ho-GEBA, including comparisons against the first-order Born approximation and the EBA. Index Terms Anisotropy, electromagnetic (EM) scattering, extended Born approximation (EBA), generalized extended Born approximation (GEBA), generalized series (GS), high-order generalized extended Born approximation (Ho-GEBA), induction logging. I. INTRODUCTION INTEGRAL equations are widely used to simulate electromagnetic (EM) scattering problems, including applications in geophysical prospecting (Hohmann, [10], Fang et al., [2], Gao et al., [3], Gao et al., [4], among others). The solution of EM scattering by integral equations includes two sequential steps. First, the spatial distribution of electric fields within scatterers is computed through a discretization scheme. Second, the computed internal scattering currents are propagated to receiver Manuscript received April 5, 2005; revised October 31, This work was supported by The University of Texas at Austin s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko Petroleum Corporation, Baker Atlas, BP, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton Energy Services, Mexican Institute for Petroleum, Occidental Petroleum Corporation, Petrobras, Precision Energy Services, Schlumberger, Shell International E&P, Statoil, and TOTAL. G. Gao was with the Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, Texas USA. He is now with Schlumberger Technology Corporation, EMI Technology Center, Richmond, CA USA ( ggao@slb.com). C. Torres-Verdín is with the Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX USA ( cverdin@uts.cc.utexas.edu). Digital Object Identifier /TAP locations. The number of cells necessary to discretize the scatterers depends on the frequency, the conductivity contrast, the size of the scatterers, and the proximity of the source and/or the receiver to the scatterers. This spatial discretization gives rise to a full complex linear system of equations whose solution is the spatial distribution of internal electric fields. Computer memory requirements increase quadratically with an increase in the number of spatial discretization cells. Furthermore, the need to solve a large, full, and complex linear system of equations places significant constraints on the expedience of three-dimensional (3-D) integral equation methods. Several numerical strategies are used to overcome the difficulties associated with integral equation formulations of EM scattering. One strategy is to improve the efficiency of full-wave modeling with high performance algorithms. Fang et al. [2] recently reported one such strategy. Fang et al. [2] algorithm applies a combination of Bi-Conjugate Gradient STABilized (l) [BiCGSTAB(l)] and the fast Fourier transform (FFT) to iteratively solve the linear system of equations. This strategy results in a nearly matrix-free system that reduces the computation cost to compared to, is the number of discretization cells. An alternative approach to expedite the solution of EM scattering problems is to develop approximate solutions. The latter often represent a good compromise between computer efficiency and accuracy when solving large-scale inverse scattering problems. Several approximations of the integral equation formulation have been proposed in the past. These include the Born approximation (1933), the extended Born approximation (EBA) (Habashy et al., [12]; and Torres-Verdín and Habashy, [11]), and the quasi-linear approximation (Zhdanov and Fang, [14]). In addition, a smooth approximation (Gao et al., [2]; Gao et al., [4]; Gao et al., [6]) was developed to efficiently simulate the EM response of electrically anisotropic media based on the theory of field decomposition. The Born approximation is restricted to low frequencies and low-conductivity contrasts (Habashy et al., [12]). On the other hand, the EBA significantly improves the accuracy of the Born approximation because of the inclusion of multiple scattering effects (Habashy et al., [12]). It has been found, however, that the accuracy of the EBA degrades when the scatterer is close to the source region, or else when the electric field exhibits significant spatial variations within the scatterer (Torres-Verdín and Habashy, [11]; Gao et al., [5]). These two situations frequently arise in applications of geophysical borehole induction logging. Gao and Torres-Verdín [5] have made considerable progress in making use of the background electric fields and the spatial distribution of conductivity to construct a preconditioning X/$ IEEE

2 1244 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 matrix that accounts for the proximity of the source to the scatterers. This method has been successfully used to solve 2.5-dimensional problems in cylindrical coordinate systems. However, the method does not perform well when solving 3-D EM scattering problems (Gao and Torres-Verdín, [5]). Moreover, neither the Born approximation nor the EBA effectively account for EM coupling due to electrically anisotropic media. The latter situation has been discussed in great detail in several of our previous publications (Gao et al., [2], Gao et al., [4], and Gao et al., [6]). To properly account for the effects of source proximity, multiple scattering, and EM coupling in the presence of electrically anisotropic media, in this paper we develop a generalized extended Born approximations (GEBAs) and its high-order variants. We show that the EBA is a special case of GEBA. Subsequently, we propose a high-order generalized extended Born approximations (Ho-GEBA) to further improve the accuracy of the GEBA without sacrifice of computer efficiency. This is achieved by making use of a generalized series (GS) expansion of the electric field. In the formulation of the Ho-GEBA, the GEBA acts as the residual term of the GS. Theoretical analysis and numerical experiments consistently confirm the high accuracy of the Ho-GEBA irrespective of the source position or the spatial distribution of the internal electric field. We consider several numerical examples in the induction frequency range to quantify the accuracy and efficiency of the Ho-GEBA for the cases of vertical magnetic dipole (VMD) and transverse magnetic dipole (TMD) excitation. This paper is organized as follows: We first introduce the theory of integral equation modeling followed by the derivations of the GS, the GEBA, and the Ho-GEBA. Several numerical examples are included to validate the theory. We focus our attention to the physical significance of the GEBA and HO-GEBA and to their numerical validity by comparing them to both the first-order Born approximation and the EBA. II. THEORY OF INTEGRAL EQUATION MODELING Assume an EM source that exhibits a time harmonic dependence of the form, is angular frequency, and is time. We consider the case of a scatterer embedded in an unbounded homogeneous and isotropic background medium. The governing integral equation for the electric field is written as (Hohmann, [10]; Habashy et al., [12]) Likewise, the integral equation for the magnetic field written as and are the electric and magnetic field vectors, respectively, associated with the background medium and the impressed sources; is the anomalous material (1) is (2) complex conductivity measured with respect to the background medium and is given by, and is the unity dyad. The complex background conductivity in (3) is given by is the ohmic conductivity of the background medium, is the dielectric constant of the background medium, and is the electrical permittivity of free space. In (1), and are the electric and magnetic dyadic Green s functions, respectively, and is the spatial support of. The dyadic electric Green s function can be expressed in closed form as the scalar Green s function is written as The magnetic dyadic Green s function is related to the electric dyadic Green s function through the expression Equations (1) and (2) are Fredholm integral equations of the second kind. The solution of these equations can be approached with the method of moments (MoM) (Harrington, [10]). Traditional implementations of the MoM yield a full matrix equation symbolically written as [for the case of (1)] is a matrix involving the volume integrals of the Green s dyadic functions, is a vector for the total electric field, is a vector for the background electric field, is a matrix for the anomalous electrical conductivities, and is a unity matrix. The solution of (8) involves the following computational issues for large-scale numerical simulation problems: a) matrix filling time is substantial, b) large memory storage requirements, and c) time-consuming solution of the complex linear system of equations. For large 3-D scatterers, often the solution of EM scattering cannot be approached with a naïve implementation of the MoM. To emphasize this point, Gao et al. [6] tabulated the most significant computer requirements associated with a hypothetical simulation problem. They showed that such requirements can easily overtax currently available computing platforms. Some analytical techniques have been developed to efficiently evaluate the entries of the MoM matrix (Gao, Torres-Verdín, and Habashy, [8]). Still, additional numerical considerations are needed to circumvent the computational issues mentioned above. (3) (4) (5) (6) (7) (8)

3 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING 1245 III. A GENERALIZED SERIES (GS) EXPANSION OF THE ELECTRIC FIELD For convenience, we rewrite (1) using operator notation as (9) Using an energy inequality, Zhdanov and Fang [13] constructed a globally convergent modified Born series expansion. In that work, a linear map was used to transform the operator into a new operator,. The norm of is always less than or equal to one, namely (18) is a linear integral operator defined by (10) (11) and can be applied to any vector-valued function (see (A-9) in Appendix A). Starting with the same energy inequality used by Zhdanov and Fang [13], in Appendix A we derive a new formulation of the integral equation as identifies the scattered electric field, and the subscript designates the spatial support of the operator. In theory, (9) can be solved via the method of successive iterations (Von Neumann series), namely (12) From the Banach theorem (Aubin, [19]), it is well known that the Von Neumann series converges if the operator is contractive, that is, if (13) is the norm,, and, and are any two different solutions. In other words, to guarantee the Von Neumann series to converge, the norm of the operator must be less than one, i.e. (14) If one takes the background electric field as the initial solution of (12), one can derive the classical Born series expansion (Born [1]) for as and (15) (16) (17) Each iteration of the Born series expansion in (15) involves only one matrix-vector multiplication. However, usually the norm of the operator is greater than 1, upon the Born series expansion of (15) does not always converge, e.g., in the case of highly conductive media. This situation greatly limits the range of applicability of the Born series expansion for simulation of EM scattering. (19) the tensors, and are given by (A-13), (A-18) and (A-19), respectively. The electric field is computed via (A-16) after is solved from (19). A proof that (19) is a contractive integral equation is given in Appendix A. Based on the new integral equation (19), and following the same procedure used in the derivation of the classical Born series expansion, a new series approximation can be derived for the electric field. We start by assuming that the initial value of in (19) is, namely (20) Notice that is unknown and that the subscript CB here has no specific meaning. In Appendix B, we derive a series expansion for the electric field as and (21) (22) (23) We refer to the series expansion given by (21) as a GS expansion for the electric field, given that any alternative series expansion can be derived from it. For example, the classical Born series expansion, the modified Born series expansion of Zhdanov and Fang [13], and the quasilinear series expansion of Zhdanov and Fang [13] are all special variants of (23). Table I summarizes the relationship between the GS and other existing series expansions of the electric field. A salient feature of the GS is that it converges in the presence of arbitrary lossy media. This latter property is addressed in detail in Appendexes A and B. Cui et al. [17], [18] advanced an approximation to EM scattering similar to the extended Born series expansion; however, their approximation does not guarantee the convergence of the

4 1246 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 TABLE I RELATIONSHIP BETWEEN THE GS AND OTHER SERIES EXPANSIONS OF THE INTERNAL ELECTRIC FIELD REPORTED IN THE OPEN TECHNICAL LITERATURE high-order terms of the series because the formulation does not enforce a contractive operator. Fig. 15 (Appendix B) compares the convergence of the EBA series for the rock formation model shown in Fig. 14 both with and without contraction. The left-hand panel of Fig. 15 shows the convergence behavior of the EBA series without contraction (N.C.), while the right-hand panel shows the convergence behavior of the same series with contraction (W.C.). This graphical comparison indicates that, without contraction, high-order terms of the EBA series tend to diverge. We remark that the low-order terms (i.e., the 2nd order) may accidentally produce better results for some cases (see, for example, Cui et al., [17], [18]). However, the overall behavior of the series is divergent. Fig. 15 (right panel) also indicates that the use of a better starting point does not guarantee a faster convergence of the series [see the curve denoted by EBA series (W.C.)]. Cui et al. [17], [18] also introduced the use of a backconditioner to improve the accuracy of the approximation. A similar backconditioner strategy was advanced by Gao and Torres-Verdín [5] for the inversion of borehole array induction data. IV. THE EXTENDED BORN APPROXIMATION (EBA) Based on (1), an extended Born approximation for EM scattering was developed that captures some of the multiple scattering effects, and that is more accurate than the first-order Born approximation for some practical simulation problems (Habashy et al., [12], and Torres-Verdín and Habashy, [11]). However, it has also been shown that if the source is very close to the scatterer or if the electric field varies significantly within the scatterer, such as commonly encountered in borehole induction logging, the accuracy of the EBA deteriorates (Gao et al., [3]; Gao et al., [4]). To derive the EBA, one first rewrites (1) as dyadic Green s function. Thus, by omitting the third term in (24) one obtains It immediately follows that is a scattering tensor, given by The physical significance of the scattering tensor detailed by Torres-Verdín and Habashy [11]. (25) (26) (27) has been V. A GENERALIZED EXTENDED BORN APPROXIMATION (GEBA) In the derivation of the EBA it is not clear whether the omission of the second term on the right-hand side of (24) affects the final solution. We proceed to derive a generalized extended Born approximation (GEBA) based on a mathematically and physically consistent analysis. Let M be the total number of spatial discretization cells and rewrite (9) into component form as (28) We proceed to decompose the domain into two subdomains, and, in which is a subdomain which encloses the th cell. Thus, (28) can be rewritten as (29) By transferring the second term on the right-hand side of (29) to the left-hand side one obtains (24) Habashy et al. [12], and Torres-Verdín and Habashy [11], omitted the third term on the right-hand side of (24) by arguing that the contribution from this term is marginal compared to that of the second term because of the singular behavior of the (30) We introduce the following Remark to define the properties of the above operator: Remark 1: If there exists a spatial subdomain that satisfies the following two conditions.

5 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING ) Condition 1: Within, the electric field can be treated as spatially invariant. 2) Condition 2: Outside the amplitude of the Green s dyadic function decreases sufficiently fast to have a negligible effect, then the second term on the right-hand side of (30) can be neglected without affecting the accuracy of the calculation of the internal electric field. According to Remark 1, for such a subdomain, (30) can be rewritten as or, equivalently, as (31) (32) VI. A HIGH-ORDER GENERALIZED EXTENDED BORN APPROXIMATION (Ho-GEBA) In the previous section, we assumed a subdomain that satisfied Remark 1. However, we note that the two conditions in Remark 1 are not mutually complementary. Thus, the existence of is a tradeoff between meeting Condition 1 and Condition 2. In this section, we introduce an alternative strategy that does not need the choice of an optimal subdomain. In such a strategy, one chooses a subdomain that satisfies Condition 1 of Remark 1 as closely as possible; subsequently, one approximates the electric field on the right-hand side of (30) in some fashion. We now develop such an approximation strategy using the GS expansion of the internal electric field. Assume that the subdomain satisfies Condition 1 and only satisfies Condition 2 in some fashion. Equation (30) can thus be rewritten as is a scattering tensor for the th cell, and is given by (33) Equation (32) is the fundamental equation of the GEBA. The more the subdomain satisfies Remark 1, the more accurate the solution from (32) becomes. The choice of depends primarily on the source location(s), the frequency, and the conductivity contrast. Notice that the center of is not necessarily the th cell. How to optimally determine goes beyond the scope of this paper. However, one can envision that such a subdomain will reduce a dense matrix problem to a banded one. We consider two special cases for the GEBA. Special Case 1: When, is the singular domain which only encloses the th cell. This case does not modify (32); however, it does modify the scattering tensor given by (33). The corresponding scattering tensor can be written as (34) This is the simplest case of the GEBA because the computation of the scattering tensor is trivial. However, the above expression may not be sufficiently accurate since it violates Condition 2 of Remark 1, i.e., the Green s dyadic function may not decrease sufficiently fast to cause the second term on the right-hand side of (30) to be negligible. Special Case 2: When, the scattering tensor becomes (35) The latter result is identical to that of the EBA (Habashy et al., [12], and Torres-Verdín and Habashy, [11]). This case is the most complex one for the GEBA, since the computation of the scattering tensor given by (35) requires numerical resources proportional to. Also, this treatment may not provide accurate simulation results, as it violates Condition 1 of Remark 1, i.e., the electric field, in general, may not be spatially invariant in the whole scattering domain. (36) Notice that the second term in (30) has been split into two terms to arrive at (36). Then, by substituting the GS of E (keeping the first N terms, for convenience) in (21) into the right-hand side of (36), one derives the equation for the Ho-GEBA as follows: (37) is given by (C-10) and (C-11). Appendix C contains a detailed mathematical derivation of (37). We remark that (37) is the fundamental equation of the HO-GEBA. Two special cases can also be considered for the Ho-GEBA. Special Case 1: Substitution of in (37) for yields (38) This approximation closely follows the assumptions made in the derivation of the Ho-GEBA. Therefore, (38) is a good approximation of EM scattering problems. We remark here that although an optimal scattering tensor is not needed for the solution of (38), an optimal choice of scattering tensor can significantly improve the rate of convergence of (38). Special Case 2: One may posit that the substitution of in (37) for, gives rise to an approximation corresponding to Special Case 2 of the GEBA. As a matter of fact, we emphasize that one cannot directly derive such an approximation from (36) because when, the term involving in (36) automatically approaches zero, and only the term remains. In such a case the GS can be used no. However, a similar expression can be derived from the original equation that gives rise to the EBA. Appendix D contains a detailed mathematical

6 1248 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 derivation for this special approximation. The final equation is given by (39) Incidentally, by making a simple substitution from to, one obtains exactly the same form given by (39). In some sense, this exercise sheds light on the difference between the derivation mechanisms behind the Ho-GEBA and the EBA. We remark, however, that (39) may not be a good approximation if the scattering tensor violates the assumption made in the derivation of the Ho-GEBA. VII. THE PHYSICAL SIGNIFICANCE OF THE Ho-GEBA From the previous discussion, it follows that the Ho-GEBA is a combination of the GS and the GEBA, in which the GEBA acts as the residual term of the GS. However, numerical exercises indicate that the GEBA term can dramatically increase the speed of convergence of the GS, thereby rendering the Ho-GEBA extremely efficient to accurately solve EM scattering problems. We remark that the GEBA with an optimal subdomain can yield accurate solutions of EM scattering. However, as has been pointed out by Gao et al. [3], Gao et al. [4], and Gao et al. [6], because of null components in the background field vector, the GEBA may not properly reproduce cross-coupling EM terms in the presence of electrically anisotropic media. This problem can be circumvented with the Ho-GEBA. The physical significance of the GEBA over the EBA has been made clear in the above derivation. We now explain how the Ho-GEBA improves the solution term by term. To do so, we first expand (36) as first order second order and third order (40) (41) (42) From (40), one observes that the first-order GEBA tends to keep the zeroth order scattering term intact, and hence accounts for multiple-scattering via the interaction between the scattering tensor and the first-scattering term. Since the zeroth order scattering term is closely related to the source, one would expect it to reflect some of the source effects. Because of this property, it is expected that the first-order GEBA would be more accurate than the Born approximation, the EBA, and the GEBA. Actually, from the mathematical derivation of the GEBA and the Ho-GEBA, one can expect the first-order term of the GEBA to provide accurate simulation results, including the case of electrically anisotropic media. The computation cost of low-order terms of the Ho-GEBA is similar to that of the Born approximation. However, because the FFT can be used to compute the GS terms, the final computational cost is proportional to, is the total number of spatial discretization cells (Fang et al., 2003). For the EBA, the scattering tensor can also be computed using FFTs. VIII. NUMERICAL VALIDATION To validate the Ho-GEBA theory, we focus on its Special Case 1, i.e., (38). One can envision that the accuracy of the simulations could improve with a better choice of scattering tensor. In this paper, we consider examples of both conductive and resistive scattering in the induction frequency range. Specifically, the frequencies used are 10 and 200 KHz. For all the numerical examples considered in this paper, we compute results up to the 3rd-order term of the Ho-GEBA. In addition to vertical magnetic dipole (VMD) sources, we investigate applications of the Ho-GEBA for the case of transverse magnetic dipole (TMD) sources due to the increasing relevance of transverse sources in geophysical borehole induction logging (Gao et al., [3]; Gao et al., [4]; Gao et al., [6]). We adopt the following notation to describe the simulation results: refers to the scattered magnetic field in the -direction due to an -directed source, and refers to the scattered magnetic field in the -direction due to a -directed source. In the figures, the label Exact designates the solution obtained with a full-wave 3-D IE code, Born designates the solution obtained with the Born approximation, EBA designates the solution obtained with the EBA, and HOGEBA-n designates solutions obtained with the th order terms of the Ho-GEBA. The labels REAL and IMAG designate the in-phase and quadrature components, respectively. Fig. 1 describes the scattering models used in this paper. The background Ohmic resistivity is 10 m, and the background dielectric constant is 1. One -directed magnetic dipole source and one -directed magnetic dipole source with a magnetic moment of 1 are assumed located at the origin, with 20 receivers deployed along the -axis uniformly separated at 0.2-m intervals. No receiver is assumed at the origin. A cubic scatterer with a side length equal to 2 m is centered about the -axis, and is symmetric about the - and -axis. Depending on the resistivity ( ) and the distance ( ) between the scatterer and the source (located at the origin), the following four models are considered in the simulations: Model 1: ; Model 2: ; Model 3: ; Model 4: m. Fig. 2 shows the scattered component as a function of receiver location for two different frequencies: 10 and 200 KHz. The assumed scattering model is Model 1, with the left- and right-hand panels showing results for 10 and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, and the bottom figure describes the quadrature (imaginary) component of. Simulations indicate that the accuracy of the Ho-GEBA is superior to either the EBA or the first-order Born approximation at both frequencies.

7 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING 1249 Fig. 1. Graphical description of the scattering models considered in this paper. The background ohmic resistivity is 10 1 m and the background dielectric constant is 1. One x-directed and one z-directed magnetic dipole sources with a magnetic moment of 1 A 1 m are assumed located at the origin, with 20 receivers deployed along the z-axis with a uniform separation of 0.2 m. No receiver is located at the origin. A cubic scatterer with a side length of 2 m is centered about the x-axis, and is symmetrical about the y and z axes. Depending on the resistivity R of the scatterer and the distance L between the source and the scatterer, four scattering models are considered in the numerical experiments: Model 1: R =1 1 m; L =4:0m; Model 2: R =1 1 m; L =0:1m; Model 3: R = m; L =4:0m; Model 4: R = m; L =0:1m. Fig. 3. Scattered H component for Model 1. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. Fig. 2. Scattered H component for Model 1. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. Fig. 3 shows the scattered component as a function of receiver location for two different frequencies: 10 and 200 KHz. The assumed scattering model is Model 1, with the left- and right-hand panels showing results for 10 and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, and the bottom figure describes the quadrature (imaginary) component of. Again, the Ho-GEBA yields more accurate results than either the EBA or the first-order Born approximation at both frequencies. The EBA entails errors in both the in-phase or quadrature components for the two frequencies, while the first-order Born approximation entails large errors in both the in-phase and quadrature components of. Fig. 4. Scattered H component for Model 2. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. Next, we move the scatterer closer to the source until the distance between the scatterer and the source is 0.1 m (such a distance is a common borehole radius in geophysical logging applications). This is scattering Model 2. The remaining model parameters are kept the same as those described for scattering Model 1. Fig. 4 shows the scattered component as a function of receiver location for two different frequencies: 10 and 200 KHz. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, and

8 1250 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 Fig. 5. Scattered H component for Model 2. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. the bottom figure describes the quadrature (imaginary) component of. Results indicate that the Ho-GEBA is more accurate than the EBA and the first-order Born approximation at both frequencies. For this particular scattering model, and by comparison of Figs. 2 and 4, it is found that the EBA yields inaccurate results for regardless of both the frequency of operation and the distance between the source and the scatterer. Fig. 5 shows the scattered component as a function of receiver location for two different frequencies: 10 and 200 KHz. The assumed scattering model is Model 2. The left- and righthand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, as the bottom figure describes the quadrature (imaginary) component of. We observe that the Ho-GEBA (especially the second and third order) is more accurate than the EBA and the Born approximation at both frequencies. By modifying the block resistivities included in Model 1 and Model 2 from 1 to 100, we generate two resistive scattering models: Model 3 and Model 4. Fig. 6 shows the scattered component as a function of receiver location at two different frequencies: 10 and 200 KHz. The assumed scattering model is Model 3. The left-and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, as the bottom figure describes the quadrature (imaginary) component of. Results indicate that the Ho-GEBA is more accurate than the EBA and the Born approximation at both frequencies. Fig. 7 shows the scattered component as a function of receiver location, at two different frequencies: 10 and 200 KHz. The assumed scattering model is Model 3. The leftand right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top figure describes Fig. 6. Scattered H component for Model 3. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. Fig. 7. Scattered H component for Model 3. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. the in-phase (real) component of, as the bottom figure describes the quadrature (imaginary) component of. In similar fashion to Fig. 3, the Ho-GEBA is more accurate than the EBA and considerably more accurate than the Born approximation at both frequencies. We proceed to displace Model 3 closer to the source, thereby constructing Model 4. Fig. 8 shows the scattered component as a function of receiver location at two different frequencies: 10 and 200 KHz. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel,

9 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING 1251 Fig. 8. Scattered H component for Model 4. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. Fig. 10. Comparison of the scattered magnetic field component H simulated with the Born approximation and the EBA over the frequency range from 10 KHz to 2 MHz. The assumed scattering model is Model 2, with one fixed receiver located at 00:1 m. The left- and right-hand panels describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulations of H performed with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact full-wave solution. that the Ho-GEBA is more accurate than the EBA and the Born approximation at both frequencies. To further assess the accuracy of the Ho-GEBA with respect to frequency, we consider a fixed receiver located at m. The assumed scattering model is Model 2. This model represents a typical conductive medium and is responsible for substantial near-source scattering effects. The frequency range considered for the simulations is between 10 KHz and 2 MHz, which is typical of borehole geophysical induction logging applications. Figs. 10 and 11 compare the scattered magnetic field components and, respectively, simulated with the Ho-GEBA up to the fifth-order together against the full-wave solution, the EBA, and the Born approximation. This graphical comparison confirms that the Ho-GEBA yields consistent and accurate results that are superior to the EBA and the Born approximation over the entire frequency range. Fig. 9. Scattered H component for Model 4. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top and bottom figures describe the in-phase (real) and quadrature (imaginary) components of H, respectively. Simulation results obtained with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact solution. the top figure describes the in-phase (real) component of, as the bottom figure describes the quadrature (imaginary) component of. Results indicate that the Ho-GEBA is more accurate than the EBA and the Born approximation at both frequencies. Fig. 9 shows the scattered component as a function of receiver location at 10 and 200 KHz. The assumed scattering model is Model 4. The left- and right-hand panels show simulation results for 10 and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of, as the bottom figure describes the quadrature (imaginary) component of. In similar fashion to Fig. 7, results indicate IX. DISCUSSION AND CONCLUSIONS The following conclusions stem from the simulation exercises described earlier. 1) In general, the Ho-GEBA is more accurate than the EBA regardless of both the distance between the source and scatterer and the operating frequency. For some cases the source is far from the scatterer, the EBA also provides accurate simulation results. 2) The Ho-GEBA can dramatically improve the convergence rate of the GS. This is best explained with a simulation exercise. Fig. 12 compares the convergence of the GS and the Ho-GEBA for scattering Model 1. The left- and right-hand panels in that figure describe simulation results for 10 and 200 KHz, respectively. For this simulation exercise the rate of convergence of the Ho-GEBA is superior to that of the GS. Also, we observe that for some cases (as shown

10 1252 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 Fig. 11. Comparison of the scattered magnetic field component H simulated with the EBA, the Born approximation, and the EBA over the frequency range from 10 KHz to 2 MHz. The assumed scattering model is Model 2, with one fixed receiver located at 00:1 m. The left- and right-hand panels describe the in-phase (real) and quadrature (imaginary) components of H. Simulations of H performed with the Born approximation, the EBA, and the Ho-GEBA (up to the third order) are plotted together with the exact full-wave solution. Fig. 12. Comparison of the convergence behavior of the Ho-GEBA and the GS. Model 1 is the assumed scatterer. Numerical simulations correspond to the secondary H component. The left- and right-hand panels show convergence results for 10 and 200 KHz, respectively. in Fig. 12) the first-order solution is superior to the second-order solution. However, this behavior does not affect the rate of convergence of the Ho-GEBA. 3) Another technical issue that needs some consideration is Special Case 2 of the Ho-GEBA. At the outset, we emphasized that this special case may not be applicable for some cases of EM scattering. To clarify this point, we make use of another simulation exercise. Fig. 13 describes simulation results (in-phase components of ) obtained for Model 2. In that figure, the curves labeled HoGEBAS2-n,, describe simulation results obtained for the Special Case 2 of the Ho-GEBA. These results clearly indicate that Special Case 2 of the Ho-GEBA is not applicable to the Fig. 13. Simulation results for Special Case 2 of the Ho-GEBA. Model 2 is the assumed scattering model. Numerical simulations correspond to the secondary H component. The nomenclature Ho-GEBAS2-n (n =1; 2; 3) designates simulation results associated with the Special Case 2 of the Ho-GEBA. The leftand right-hand panels describe the in-phase (real) and quadrature (imaginary) components of H, respectively. problem at hand. One may then conclude that Special Case 2 of Ho-GEBA only applies to simulation cases the EBA remains accurate. As a general conclusion, we emphasize that the GS is a generalized series expansion of the internal electric field, as the GEBA is a generalized extended Born approximation. The Ho-GEBA is a combination of the GEBA and the GS. In general, the GEBA will converge substantially faster than the GS. We validated the Ho-GEBA using simple 3-D scatterers. Numerical experiments in the induction frequency range show that the Ho-GEBA is in general more accurate than both the Born approximation and the EBA. The total computational cost of the Ho-GEBA is proportional to, is the number of spatial discretization cells. A unique feature of the Ho-GEBA is that it can be used to simulate EM scattering due to electrically anisotropic media. This feature is not possible with either the Born approximation or the EBA. APPENDIX A DERIVATION OF THE NEW INTEGRAL EQUATION Singer [16], Pankratov [15], and Zhdanov and Fang [13] derived an energy inequality for the anomalous EM field. Such an energy inequality can be generalized to the case in which an electrical conductivity anomaly is embedded in an infinite uniform conductive background (Singer, [16]). Let us assume an electrical conductivity anomaly with a closed boundary embedded in an infinite uniform conductive background of conductivity equal to. Following Zhdanov and Fang [13], the per-period average of energy flow of anomalous EM field through can be expressed as (A-1) is the spatial support of the conductivity anomaly, is the Poynting vector, is the outgoing unit

11 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING 1253 vector normal to the surface and are the anomalous electric and magnetic fields, respectively, * denotes complex conjugate, and designates the real part of the corresponding quantity. According to the Poynting theorem and Maxwell s equations (Harrington, [20]), can be rewritten as (A-2) is the anomalous electric current vector. It has been shown that the energy flow of the anomalous field must be nonnegative (Pankratov, [15]). Thus, the following equation holds: The integrand in (A-3) can be rewritten as Substitution of (A-4) into (A-3) yields the energy inequality (A-3) (A-4) (A-5) Equation (A-5) is a consequence of the physics of the interaction between the EM fields and the medium. This operating condition represents a physical constraint for our derivations below. Because is always positive, (A-5) is equivalent to From the physical constraint given by (A-6), one can derive the following inequality for the operator : (A-10) denotes the -norm in a Hilbert space, and is defined as (A-11) By making use of (14), Zhdanov and Fang [13] transformed (A-8) into (A-12) (A-13) Equation (A-12) can be treated as an integral equation with respect to the product, i.e. (A-14) is a new operator that remains contractive for any type of lossy background medium (Zhdanov and Fang, [13]). By making use of (A-12) and (13), and after some manipulations, one obtains (A-15) Next, we note that and make use of (13) to obtain (A-6) (A-7) (A-16) Following Zhdanov and Fang [13], it can be shown that for any lossy background medium, the following relation holds: (A-17) According to the Cauchy Schwartz inequality, (A-10) and (A-17) guarantee that the operator be contractive, namely Equation (A-15) leads to the new integral equation (A-18) (A-19) (A-8) is an operator that can be applied to any vector-valued function and is given by (A-9) and (A-20) (A-21) Notice that the contraction of the new integral equation (A-19) is ensured by (A-18). Finally, is given by (A-16).

12 1254 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 APPENDIX B DERIVATION OF THE GENERALIZED SERIES EXPANSION FOR THE INTERNAL ELECTRIC FIELD Assume that the initial guess of in (23) is given by, namely (B-1) We remark that is unknown and that the subscript CB here has no specific meaning. Substitution of (B-1) into (23) together with (A-16) yields Fig. 14. Rock formation model used to numerically test the convergence properties of the GS. A conductive cube with a side length of 2 m and a conductivity of 10 S/m is embedded in a background medium of conductivity equal to 1 S/m. Transmitter and receiver are vertical magnetic dipoles operating at 20 KHz. The distance between the transmitter and the cube is 0.1 m, and the spacing between transmitter and receiver is 0.5 m. We note that (A-21) has also been used to derive (B-2). Now define Equation (B-2) can then be rewritten as Substitution of (B-4) into (23) together with (A-16) gives (B-2) (B-3) (B-4) (B-5) (B-6) By repeating the same procedure, one derives the following series expansion: (B-7) (B-8) and is given by (B-3). In Appendix A, we demonstrated that the integral equation from which the series expansion (B-7) was derived is a contractive integral equation. This result indicated that the series expansion given by (B-7) was always convergent. To confirm such an important property, in this Appendix we consider a numerical example for which the classical Born series diverges. Fig. 14 describes the formation model, consisting of a conductive cube with a side length of 2 m and conductivity equal to 10 S/m, embedded in a background medium of conductivity equal to 1 S/m. The transmitter and the receiver are vertical magnetic dipoles operating at 20 KHz. The distance between the transmitter and the cube is 0.1 m, and the spacing between the transmitter and receiver is 0.5 m. Measurements consist of the scattered magnetic field at the receiver. Fig. 15 compares the convergence of the GS (right panel) against the convergence of the Fig. 15. Comparison of the convergence behavior of the classical Born series expansion, the GS (starting from the background field), the EBA series expansion [no contraction (N.C.)], and the EBA series expansion [with contraction (W.C.)] for the rock formation model given in Fig. 14. The left-hand panel describes the convergence behavior of both the classical Born series expansion and the EBA series expansion (N.C.), while the right-hand panel describes the convergence behavior of the GS and EBA series expansion (W.C.). The exact solution (Solution Line) was calculated using a full-wave 3-D integral-equation code (Fang et al., [2]). classical Born series (left panel). On these figures, the horizontal axis describes the iteration number, while the vertical axis describes the amplitude of the scattered magnetic field. This exercise clearly indicates that the classical Born series expansion does not converge, while the GS converges to the exact solution in a few iterations. APPENDIX C DERIVATION OF THE FUNDAMENTAL EQUATION OF THE Ho-GEBA Substitution of (25) into the right-hand side of (40) gives (C-1)

13 GAO AND TORRES-VERDÍN: HIGH-ORDER GENERALIZED EBA FOR EM SCATTERING 1255 Since is assumed spatially invariant within subdomain, one can rewrite (C-1) as For convenience, we keep the first terms in (D-2), and substitute the ensuing expression into (28), to obtain (C-2) From (B-3) and (B-8) one obtains (D-3) By expanding with a Taylor series about one obtains (C-3) (D-4) Further, by retaining only the first term on the right-hand side of (D-4) one can write (D-5) Using this last expression and rearranging the terms in (D-3), one obtains and Substitution of (C-3) into (C-2) yields (C-4) (C-5) (C-6) (D-6) Substitution of (C-3) into (D-6), together with some simple manipulations yields (C-7) (D-7) Finally is given by (C-4) and (C-5). is given by (33). APPENDIX D DERIVATION OF SPECIAL CASE 2 OF THE Ho-GEBA First, the generalized series expansion of as Subtraction of (25) from (D-1) yields (C-8) can be written (D-1) (D-2) ACKNOWLEDGMENT A note of gratitude goes to Dr. D. Pardo and two anonymous reviewers for their constructive technical and editorial feedback. REFERENCES [1] M. Born, Optics. New York: Springer-Verlag, [2] S. Fang, G. Gao, and C. Torres-Verdín, Efficient 3-D electromagnetic modeling in the presence of anisotropic conductive media using integral equations, in Proc. Third Int. 3D Electromagn. (3DEM-3) Symp., [3] G. Gao, S. Fang, and C. Torres-Verdín, A new approximation for 3D electromagnetic scattering in the presence of anisotropic conductive media, in Proc. Third Int. 3D Electromagn. (3DEM-3) Symposium, [4] G. Gao, C. Torres-Verdín, and S. Fang, Fast 3D modeling of borehole induction data in dipping and anisotropic formations using a novel approximation technique, in Paper VV, Trans. 44th SPWLA Annu. Logging Symp., [5] G. Gao and C. Torres-Verdín, Fast inversion of borehole induction data using an inner-outer loop optimization technique, in Paper TT, Trans. 44th SPWLA Annu. Logging Symp., [6] G. Gao, C. Torres-Verdín, and S. Fang, Fast 3D modeling of borehole induction data in dipping and anisotropic formations using a novel approximation technique, Petrophys., vol. 45, no. 3, pp , 2004.

14 1256 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 [7] G. Gao and C. Torres-Verdín, A high-order generalized extended Born approximation to simulate electromagnetic geophysical measurements in inhomogeneous and anisotropic media, SEG Expanded Abstracts, pp , [8] G. Gao, C. Torres-Verdin, and T. M. Habashy, Analytical techniques to evaluate the integrals of 3D and 2D spatial dyadic Green s functions, Progr. Electromagn. Res., vol. 52, pp , [9] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, [10] G. W. Hohmann, Three-dimensional induced polarization and electromagnetic modeling, Geophys., vol. 40, no. 2, pp , [11] C. Torres-Verdín and T. M. Habashy, Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear scattering approximation, Radio Sci., vol. 29, no. 4, pp , [12] T. M. Habashy, R. W. Groom, and B. Spies, Beyond the Born and Rytov approximations: A nonlinear approach to electromagnetic scattering, J. Geophys. Res., vol. 98, no. B2, pp , [13] M. S. Zhdanov and S. Fang, Quasilinear series in three-dimensional electromagnetic modeling, Radio Sci., vol. 32, no. 6, pp , [14], Quasilinear approximation in 3-D electromagnetic modeling, Geophys., vol. 61, no. 3, pp , [15] O. V. Pankratov, D. B. Avdeev, and A. V. Kuvshinov, Scattering of electromagnetic field in inhomogeneous earth: Forward problem solution, Izv. Akad. Nauk. SSSR Fiz. Zemli, vol. 3, pp , [16] B. S. Singer, Method for solution of Maxwell s equation in nonuniform media, Geophys. J. Int., vol. 120, pp , [17] T. J. Cui, Y. Qin, G.-L. Wang, and W. C. Chew, Low-frequency detection of two-dimensional buried objects using high-order extended Born approximations, Inv. Prob., vol. 20, pp , 2004a. [18] T. J. Cui, W. C. Chew, and W. Hong, New approximate formulations for EM scattering by dielectric objects, IEEE Trans. Antennas Propag., vol. 52, no. 3, pp , 2004b. [19] J. P. Aubin, Applied Functional Analysis. New York: Wiley, [20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, Guozhong Gao received the Bachelor s and Master s degrees in applied geophysics from Southwest Petroleum Institute (China) and Beijing Petroleum University in 1996 and 2000, respectively. He received the Ph.D. degree from the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin in Currently, he is a Geophysicist with Schlumberger Technology Corporation, Richmond, CA. His current research interests include borehole, marine, cross-well, and surface-to-borehole electromagnetics, including modeling, inversion, and system design. Carlos Torres-Verdín (M 82) received the Ph.D. degree in engineering geoscience from the University of California, Berkeley, in During , he held the position of Research Scientist with Schlumberger-Doll Research. From 1997 to 1999, he was Reservoir Specialist and Technology Champion with YPF, Buenos Aires, Argentina. Since 1999, he has been affiliated with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, he currently holds the position of Associate Professor and conducts research in formation evaluation, well logging, and integrated reservoir characterization. He has served as Guest Editor for Radio Science, and is currently a member of the Editorial Board of the Journal of Electromagnetic Waves and Applications, and an Associate Editor for Petrophysics (SPWLA) and the SPE Journal.