IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. … · Green’s dyadic functions, is a vector...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 1243

High-Order Generalized Extended BornApproximation for Electromagnetic Scattering

Guozhong Gao and Carlos Torres-Verdín, Member, IEEE

Abstract—Previous work on the subject of electromagnetic scat-tering has shown that the extended Born approximation (EBA) ismore accurate than the first-order Born approximation with ap-proximately the same operation count. However, the accuracy ofthe EBA degrades in cases when the source is very close to thescatterer, or when the electric field exhibits significant spatial vari-ations within the scatterer. This paper introduces a generalized ex-tended Born approximation (GEBA) and its high-order variants(Ho-GEBA) to efficiently and accurately simulate electromagneticscattering problems. We make use of a generalized series expansionof the internal electric field to construct high-order terms of thegeneralized extended Born approximation (Ho-GEBA). A salientfeature of the Ho-GEBA is its enhanced accuracy over the Bornapproximation and the EBA, even when only the first-order termof the series expansion is considered in the approximation. This be-havior is not conditioned by either the source location or the spatialdistribution of the internal electric field. A unique feature of theHo-GEBA is that it can be used to simulate electromagnetic scat-tering due to electrically anisotropic media. Such a feature is notpossible with approximations of the internal electric field that arebased on the behavior of the background electric field. Three-di-mensional (3-D) models of electromagnetic scattering are used tobenchmark the efficiency and accuracy of the Ho-GEBA, includingcomparisons against the first-order Born approximation and theEBA.

Index Terms—Anisotropy, electromagnetic (EM) scattering, ex-tended Born approximation (EBA), generalized extended Born ap-proximation (GEBA), generalized series (GS), high-order general-ized extended Born approximation (Ho-GEBA), induction logging.

I. INTRODUCTION

I NTEGRAL equations are widely used to simulate electro-magnetic (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 EMscattering by integral equations includes two sequential steps.First, the spatial distribution of electric fields within scatterersis computed through a discretization scheme. Second, the com-puted internal scattering currents are “propagated” to receiver

Manuscript received April 5, 2005; revised October 31, 2005. This workwas supported by The University of Texas at Austin’s Research Consortium onFormation Evaluation, jointly sponsored by Anadarko Petroleum Corporation,Baker Atlas, BP, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton EnergyServices, 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 78712 USA. He is now withSchlumberger Technology Corporation, EMI Technology Center, Richmond,CA 94804 USA (e-mail: [email protected]).

C. Torres-Verdín is with the Department of Petroleum and Geosystems En-gineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TAP.2006.872671

locations. The number of cells necessary to discretize the scat-terers depends on the frequency, the conductivity contrast, thesize of the scatterers, and the proximity of the source and/orthe receiver to the scatterers. This spatial discretization givesrise to a full complex linear system of equations whose solutionis the spatial distribution of internal electric fields. Computermemory requirements increase quadratically with an increase inthe number of spatial discretization cells. Furthermore, the needto solve a large, full, and complex linear system of equationsplaces significant constraints on the expedience of three-dimen-sional (3-D) integral equation methods.

Several numerical strategies are used to overcome the dif-ficulties associated with integral equation formulations of EMscattering. One strategy is to improve the efficiency of full-wavemodeling with high performance algorithms. Fang et al. [2] re-cently reported one such strategy. Fang et al. [2] algorithm ap-plies a combination of Bi-Conjugate Gradient STABilized (l)[BiCGSTAB(l)] and the fast Fourier transform (FFT) to itera-tively solve the linear system of equations. This strategy resultsin a nearly matrix-free system that reduces the computation costto compared to , where is the numberof discretization cells.

An alternative approach to expedite the solution of EM scat-tering problems is to develop approximate solutions. The latteroften represent a good compromise between computer effi-ciency and accuracy when solving large-scale inverse scatteringproblems. Several approximations of the integral equationformulation have been proposed in the past. These include theBorn 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 efficientlysimulate the EM response of electrically anisotropic mediabased on the theory of field decomposition. The Born approx-imation is restricted to low frequencies and low-conductivitycontrasts (Habashy et al., [12]). On the other hand, the EBAsignificantly improves the accuracy of the Born approximationbecause of the inclusion of multiple scattering effects (Habashyet al., [12]). It has been found, however, that the accuracy ofthe EBA degrades when the scatterer is close to the sourceregion, or else when the electric field exhibits significant spatialvariations within the scatterer (Torres-Verdín and Habashy,[11]; Gao et al., [5]). These two situations frequently arise inapplications of geophysical borehole induction logging. Gaoand Torres-Verdín [5] have made considerable progress inmaking use of the background electric fields and the spatialdistribution of conductivity to construct a preconditioning

0018-926X/$20.00 © 2006 IEEE

1244 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006

matrix that accounts for the proximity of the source to thescatterers. This method has been successfully used to solve2.5-dimensional problems in cylindrical coordinate systems.However, the method does not perform well when solving 3-DEM scattering problems (Gao and Torres-Verdín, [5]). More-over, neither the Born approximation nor the EBA effectivelyaccount for EM coupling due to electrically anisotropic media.The latter situation has been discussed in great detail in severalof our previous publications (Gao et al., [2], Gao et al., [4], andGao et al., [6]).

To properly account for the effects of source proximity, mul-tiple scattering, and EM coupling in the presence of electricallyanisotropic media, in this paper we develop a generalized ex-tended Born approximations (GEBAs) and its high-order vari-ants. We show that the EBA is a special case of GEBA. Subse-quently, we propose a high-order generalized extended Born ap-proximations (Ho-GEBA) to further improve the accuracy of theGEBA without sacrifice of computer efficiency. This is achievedby making use of a generalized series (GS) expansion of theelectric field. In the formulation of the Ho-GEBA, the GEBAacts as the residual term of the GS. Theoretical analysis and nu-merical experiments consistently confirm the high accuracy ofthe Ho-GEBA irrespective of the source position or the spatialdistribution of the internal electric field. We consider several nu-merical examples in the induction frequency range to quantifythe accuracy and efficiency of the Ho-GEBA for the cases ofvertical magnetic dipole (VMD) and transverse magnetic dipole(TMD) excitation.

This paper is organized as follows: We first introduce thetheory of integral equation modeling followed by the derivationsof the GS, the GEBA, and the Ho-GEBA. Several numerical ex-amples are included to validate the theory. We focus our atten-tion to the physical significance of the GEBA and HO-GEBAand to their numerical validity by comparing them to both thefirst-order Born approximation and the EBA.

II. THEORY OF INTEGRAL EQUATION MODELING

Assume an EM source that exhibits a time harmonic de-pendence of the form , where is angularfrequency, and is time. We consider the case of a scat-terer embedded in an unbounded homogeneous and isotropicbackground medium. The governing integral equation for theelectric field is written as (Hohmann, [10]; Habashy et al.,[12])

(1)

Likewise, the integral equation for the magnetic field iswritten as

(2)

where and are the electric and magnetic fieldvectors, respectively, associated with the background mediumand the impressed sources; is the anomalous material

complex conductivity measured with respect to the backgroundmedium and is given by

(3)

where , and is the unity dyad.The complex background conductivity in (3) is given by

(4)

where 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 func-tions, respectively, and is the spatial support of .

The dyadic electric Green’s function can be expressed inclosed form as

(5)

where the scalar Green’s function is written as

(6)

The magnetic dyadic Green’s function is related to the electricdyadic Green’s function through the expression

(7)

Equations (1) and (2) are Fredholm integral equations of thesecond kind. The solution of these equations can be approachedwith the method of moments (MoM) (Harrington, [10]). Tradi-tional implementations of the MoM yield a full matrix equationsymbolically written as [for the case of (1)]

(8)

where is a matrix involving the volume integrals of theGreen’s dyadic functions, is a vector for the total electricfield, is a vector for the background electric field, is amatrix for the anomalous electrical conductivities, and is aunity matrix.

The solution of (8) involves the following computationalissues for large-scale numerical simulation problems: a) matrixfilling time is substantial, b) large memory storage require-ments, and c) time-consuming solution of the complex linearsystem of equations. For large 3-D scatterers, often the solutionof EM scattering cannot be approached with a naïve imple-mentation of the MoM. To emphasize this point, Gao et al. [6]tabulated the most significant computer requirements associatedwith a hypothetical simulation problem. They showed that suchrequirements can easily overtax currently available computingplatforms. Some analytical techniques have been developedto efficiently evaluate the entries of the MoM matrix (Gao,Torres-Verdín, and Habashy, [8]). Still, additional numericalconsiderations are needed to circumvent the computationalissues mentioned above.

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)

where

(10)

is a linear integral operator defined by

(11)

identifies the scattered electric field, and the subscript des-ignates the spatial support of the operator.

In theory, (9) can be solved via the method of successive it-erations (Von Neumann series), namely

(12)

From the Banach theorem (Aubin, [19]), it is well known that theVon Neumann series converges if the operator is contractive,that is, if

(13)

where is the norm, , and , and areany two different solutions. In other words, to guarantee the VonNeumann series to converge, the norm of the operator mustbe less than one, i.e.

(14)

If one takes the background electric field as the initial solutionof (12), one can derive the classical Born series expansion (Born[1]) for as

(15)

where

(16)

and

(17)

Each iteration of the Born series expansion in (15) involves onlyone matrix-vector multiplication. However, usually the norm ofthe operator is greater than 1, whereupon the Born seriesexpansion of (15) does not always converge, e.g., in the case ofhighly conductive media. This situation greatly limits the rangeof applicability of the Born series expansion for simulation ofEM scattering.

Using an energy inequality, Zhdanov and Fang [13] con-structed 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 lessthan or equal to one, namely

(18)

and can be applied to any vector-valued function (see (A-9)in Appendix A).

Starting with the same energy inequality used by Zhdanovand Fang [13], in Appendix A we derive a new formulation ofthe integral equation as

(19)

where 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 contractiveintegral equation is given in Appendix A.

Based on the new integral equation (19), and following thesame procedure used in the derivation of the classical Born se-ries expansion, a new series approximation can be derived forthe 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” herehas no specific meaning. In Appendix B, we derive a series ex-pansion for the electric field as

(21)

where

(22)

and

(23)

We refer to the series expansion given by (21) as a GS expansionfor the electric field, given that any alternative series expansioncan be derived from it. For example, the classical Born seriesexpansion, the modified Born series expansion of Zhdanov andFang [13], and the quasilinear series expansion of Zhdanov andFang [13] are all special variants of (23). Table I summarizesthe relationship between the GS and other existing series ex-pansions of the electric field. A salient feature of the GS is thatit converges in the presence of arbitrary lossy media. This latterproperty is addressed in detail in Appendexes A and B.

Cui et al. [17], [18] advanced an approximation to EM scat-tering similar to the extended Born series expansion; however,their approximation does not guarantee the convergence of the


TABLE IRELATIONSHIP 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 notenforce a contractive operator.

Fig. 15 (Appendix B) compares the convergence of the EBAseries for the rock formation model shown in Fig. 14 both withand without contraction. The left-hand panel of Fig. 15 showsthe convergence behavior of the EBA series without contrac-tion (N.C.), while the right-hand panel shows the convergencebehavior of the same series with contraction (W.C.). This graph-ical comparison indicates that, without contraction, high-orderterms of the EBA series tend to diverge. We remark that thelow-order terms (i.e., the 2nd order) may accidentally producebetter 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 startingpoint does not guarantee a faster convergence of the series [seethe curve denoted by EBA series (W.C.)]. Cui et al. [17], [18]also introduced the use of a backconditioner to improve the ac-curacy of the approximation. A similar backconditioner strategywas advanced by Gao and Torres-Verdín [5] for the inversion ofborehole array induction data.

IV. THE EXTENDED BORN APPROXIMATION (EBA)

Based on (1), an extended Born approximation for EMscattering was developed that captures some of the multiplescattering effects, and that is more accurate than the first-orderBorn 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 veryclose to the scatterer or if the electric field varies significantlywithin the scatterer, such as commonly encountered in boreholeinduction logging, the accuracy of the EBA deteriorates (Gaoet al., [3]; Gao et al., [4]).

To derive the EBA, one first rewrites (1) as

(24)

Habashy et al. [12], and Torres-Verdín and Habashy [11],omitted the third term on the right-hand side of (24) by arguingthat the contribution from this term is marginal compared tothat of the second term because of the singular behavior of the

dyadic Green’s function. Thus, by omitting the third term in(24) one obtains

(25)

It immediately follows that

(26)

where is a scattering tensor, given by

(27)

The physical significance of the scattering tensor has beendetailed by Torres-Verdín and Habashy [11].

V. A GENERALIZED EXTENDED BORN APPROXIMATION

(GEBA)

In the derivation of the EBA it is not clear whether the omis-sion of the second term on the right-hand side of (24) affects thefinal solution. We proceed to derive a generalized extended Bornapproximation (GEBA) based on a mathematically and physi-cally consistent analysis.

Let M be the total number of spatial discretization cells andrewrite (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

(30)

We introduce the following Remark to define the properties ofthe above operator:

Remark 1: If there exists a spatial subdomain that satisfiesthe following two conditions.


1) Condition 1: Within , the electric field can betreated as spatially invariant.

2) Condition 2: Outside the amplitude of the Green’sdyadic function decreases sufficiently fast to have anegligible effect,

then the second term on the right-hand side of (30) can be ne-glected without affecting the accuracy of the calculation of theinternal electric field.

According to Remark 1, for such a subdomain , (30) can berewritten as

(31)

or, equivalently, as

(32)

where is a scattering tensor for the th cell, and is given by

(33)

Equation (32) is the fundamental equation of the GEBA. Themore the subdomain satisfies Remark 1, the more accurate thesolution from (32) becomes. The choice of depends primarilyon the source location(s), the frequency, and the conductivitycontrast. Notice that the center of is not necessarily the thcell. How to optimally determine goes beyond the scope ofthis paper. However, one can envision that such a subdomainwill reduce a dense matrix problem to a banded one.

We consider two special cases for the GEBA.Special Case 1: When , where is the singular

domain which only encloses the th cell. This case does notmodify (32); however, it does modify the scattering tensor givenby (33). The corresponding scattering tensor can be written as

(34)

This is the simplest case of the GEBA because the computationof the scattering tensor is trivial. However, the above expressionmay not be sufficiently accurate since it violates Condition 2 ofRemark 1, i.e., the Green’s dyadic function may not decreasesufficiently fast to cause the second term on the right-hand sideof (30) to be negligible.

Special Case 2: When , the scattering tensor be-comes

(35)

The latter result is identical to that of the EBA (Habashy et al.,[12], and Torres-Verdín and Habashy, [11]). This case is themost complex one for the GEBA, since the computation of thescattering tensor given by (35) requires numerical resources pro-portional to . Also, this treatment may not provide accu-rate simulation results, as it violates Condition 1 of Remark 1,i.e., the electric field, in general, may not be spatially invariantin the whole scattering domain.

VI. A HIGH-ORDER GENERALIZED EXTENDED BORN

APPROXIMATION (Ho-GEBA)

In the previous section, we assumed a subdomain that sat-isfied Remark 1. However, we note that the two conditions inRemark 1 are not mutually complementary. Thus, the existenceof is a tradeoff between meeting Condition 1 and Condition 2.In this section, we introduce an alternative strategy that does notneed the choice of an optimal subdomain. In such a strategy, onechooses a subdomain that satisfies Condition 1 of Remark 1as closely as possible; subsequently, one approximates the elec-tric field on the right-hand side of (30) in some fashion. Wenow develop such an approximation strategy using the GS ex-pansion of the internal electric field.

Assume that the subdomain satisfies Condition 1 and onlysatisfies Condition 2 in some fashion. Equation (30) can thus berewritten as

(36)

Notice that the second term in (30) has been split into two termsto arrive at (36). Then, by substituting the GS of E (keeping thefirst N terms, for convenience) in (21) into the right-hand sideof (36), one derives the equation for the Ho-GEBA as follows:

(37)

where is given by (C-10) and (C-11). Appendix C con-tains 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 inthe derivation of the Ho-GEBA. Therefore, (38) is a good ap-proximation of EM scattering problems. We remark here thatalthough an optimal scattering tensor is not needed for the solu-tion of (38), an optimal choice of scattering tensor can signifi-cantly improve the rate of convergence of (38).

Special Case 2: One may posit that the substitution ofin (37) for , gives rise to an approximation corresponding toSpecial Case 2 of the GEBA. As a matter of fact, we empha-size 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 nowhere. However, a similarexpression can be derived from the original equation that givesrise to the EBA. Appendix D contains a detailed mathematical


derivation for this special approximation. The final equation isgiven 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 derivationmechanisms behind the Ho-GEBA and the EBA. We remark,however, that (39) may not be a good approximation if the scat-tering tensor violates the assumption made in the derivation ofthe Ho-GEBA.

VII. THE PHYSICAL SIGNIFICANCE OF THE Ho-GEBA

From the previous discussion, it follows that the Ho-GEBA isa combination of the GS and the GEBA, in which the GEBA actsas the residual term of the GS. However, numerical exercises in-dicate that the GEBA term can dramatically increase the speedof convergence of the GS, thereby rendering the Ho-GEBA ex-tremely efficient to accurately solve EM scattering problems.We remark that the GEBA with an optimal subdomain canyield accurate solutions of EM scattering. However, as has beenpointed out by Gao et al. [3], Gao et al. [4], and Gao et al. [6], be-cause of null components in the background field vector , theGEBA may not properly reproduce cross-coupling EM terms inthe presence of electrically anisotropic media. This problem canbe circumvented with the Ho-GEBA.

The physical significance of the GEBA over the EBA hasbeen made clear in the above derivation. We now explain howthe Ho-GEBA improves the solution term by term. To do so, wefirst expand (36) as

first order

(40)

second order

(41)

and

third order

(42)

From (40), one observes that the first-order GEBAtends to keep the zeroth order scattering term intact, and henceaccounts for multiple-scattering via the interaction between thescattering tensor and the first-scattering term. Since the zerothorder scattering term is closely related to the source, one wouldexpect it to reflect some of the source effects. Because of thisproperty, it is expected that the first-order GEBA would be moreaccurate than the Born approximation, the EBA, and the GEBA.Actually, from the mathematical derivation of the GEBA and theHo-GEBA, one can expect the first-order term of the GEBA to

provide accurate simulation results, including the case of elec-trically anisotropic media.

The computation cost of low-order terms of the Ho-GEBA issimilar to that of the Born approximation. However, because theFFT can be used to compute the GS terms, the final computa-tional cost is proportional to , where is the totalnumber of spatial discretization cells (Fang et al., 2003). For theEBA, the scattering tensor can also be computed using FFTs.

VIII. NUMERICAL VALIDATION

To validate the Ho-GEBA theory, we focus on its SpecialCase 1, i.e., (38). One can envision that the accuracy of the sim-ulations could improve with a better choice of scattering tensor.In this paper, we consider examples of both conductive and re-sistive scattering in the induction frequency range. Specifically,the frequencies used are 10 and 200 KHz. For all the numer-ical examples considered in this paper, we compute results upto the 3rd-order term of the Ho-GEBA. In addition to verticalmagnetic dipole (VMD) sources, we investigate applications ofthe Ho-GEBA for the case of transverse magnetic dipole (TMD)sources due to the increasing relevance of transverse sources ingeophysical borehole induction logging (Gao et al., [3]; Gaoet al., [4]; Gao et al., [6]). We adopt the following notationto describe the simulation results: refers to the scatteredmagnetic field in the -direction due to an -directed source,and refers to the scattered magnetic field in the -direc-tion 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 approxi-mation, “EBA” designates the solution obtained with the EBA,and “HOGEBA-n” designates solutions obtainedwith the th order terms of the Ho-GEBA. The labels “REAL”and “IMAG” designate the in-phase and quadrature compo-nents, respectively.

Fig. 1 describes the scattering models used in this paper. Thebackground Ohmic resistivity is 10 m, and the backgrounddielectric constant is 1. One -directed magnetic dipole sourceand one -directed magnetic dipole source with a magnetic mo-ment of 1 are assumed located at the origin, with 20 re-ceivers deployed along the -axis uniformly separated at 0.2-mintervals. No receiver is assumed at the origin. A cubic scattererwith a side length equal to 2 m is centered about the -axis,and is symmetric about the - and -axis. Depending on the re-sistivity ( ) and the distance ( ) between the scatterer and thesource (located at the origin), the following four models are con-sidered in the simulations: Model 1: ;Model 2: ; Model 3:

; Model 4: m.Fig. 2 shows the scattered component as a function of re-

ceiver location for two different frequencies: 10 and 200 KHz.The assumed scattering model is Model 1, with the left- andright-hand panels showing results for 10 and 200 KHz, respec-tively. For each panel, the top figure describes the in-phase (real)component of , and the bottom figure describes the quadra-ture (imaginary) component of . Simulations indicate thatthe accuracy of the Ho-GEBA is superior to either the EBA orthe first-order Born approximation at both frequencies.


Fig. 1. Graphical description of the scattering models considered in this paper.The background ohmic resistivity is 10 � m and the background dielectricconstant is 1. One x-directed and one z-directed magnetic dipole sources witha magnetic moment of 1A � m are assumed located at the origin, with 20receivers deployed along the z-axis with a uniform separation of 0.2 m. Noreceiver is located at the origin. A cubic scatterer with a side length of 2 m iscentered about the x-axis, and is symmetrical about the y and z axes. Dependingon the resistivityR of the scatterer and the distanceL between the source and thescatterer, four scattering models are considered in the numerical experiments:Model 1: R = 1 � m; L = 4:0 m; Model 2: R = 1 � m; L = 0:1 m;Model 3: R = 100 �m;L = 4:0 m; Model 4: R = 100 �m;L = 0:1 m.

Fig. 2. ScatteredH component for Model 1. The left- and right-hand panelsshow simulation results for 10 and 200 KHz, respectively. For each panel, thetop and bottom figures describe the in-phase (real) and quadrature (imaginary)components of H , respectively. Simulation results obtained with the Bornapproximation, the EBA, and the Ho-GEBA (up to the third order) are plottedtogether with the exact solution.

Fig. 3 shows the scattered component as a functionof receiver location for two different frequencies: 10 and 200KHz. The assumed scattering model is Model 1, with theleft- and right-hand panels showing results for 10 and 200KHz, respectively. For each panel, the top figure describes thein-phase (real) component of , and the bottom figure de-scribes the quadrature (imaginary) component of . Again,the Ho-GEBA yields more accurate results than either theEBA or the first-order Born approximation at both frequencies.The EBA entails errors in both the in-phase or quadraturecomponents for the two frequencies, while the first-order Bornapproximation entails large errors in both the in-phase andquadrature components of .



Next, we move the scatterer closer to the source until the dis-tance between the scatterer and the source is 0.1 m (such a dis-tance is a common borehole radius in geophysical logging ap-plications). This is scattering Model 2. The remaining modelparameters are kept the same as those described for scatteringModel 1. Fig. 4 shows the scattered component as a func-tion of receiver location for two different frequencies: 10 and200 KHz. The left- and right-hand panels show simulation re-sults for 10 KHz and 200 KHz, respectively. For each panel, thetop figure describes the in-phase (real) component of , and



the bottom figure describes the quadrature (imaginary) compo-nent of . Results indicate that the Ho-GEBA is more ac-curate than the EBA and the first-order Born approximation atboth frequencies. For this particular scattering model, and bycomparison of Figs. 2 and 4, it is found that the EBA yieldsinaccurate results for regardless of both the frequency ofoperation and the distance between the source and the scatterer.

Fig. 5 shows the scattered component as a function of re-ceiver location for two different frequencies: 10 and 200 KHz.The assumed scattering model is Model 2. The left- and right-hand panels show simulation results for 10 and 200 KHz, re-spectively. For each panel, the top figure describes the in-phase(real) component of , whereas the bottom figure describesthe quadrature (imaginary) component of . We observe thatthe Ho-GEBA (especially the second and third order) is moreaccurate than the EBA and the Born approximation at both fre-quencies.

By modifying the block resistivities included in Model 1 andModel 2 from 1 to 100 , we generate two resistivescattering models: Model 3 and Model 4. Fig. 6 shows the scat-tered component as a function of receiver location at twodifferent frequencies: 10 and 200 KHz. The assumed scatteringmodel is Model 3. The left-and right-hand panels show simula-tion results for 10 and 200 KHz, respectively. For each panel,the top figure describes the in-phase (real) component of ,whereas the bottom figure describes the quadrature (imaginary)component of . Results indicate that the Ho-GEBA is moreaccurate than the EBA and the Born approximation at both fre-quencies.

Fig. 7 shows the scattered component as a functionof receiver location, at two different frequencies: 10 and 200KHz. The assumed scattering model is Model 3. The left-and right-hand panels show simulation results for 10 and 200KHz, respectively. For each panel, the top figure describes

Fig. 6. Scattered H component for Model 3. The left- and right-handpanels show simulation results for 10 KHz and 200 KHz, respectively. For eachpanel, the top and bottom figures describe the in-phase (real) and quadrature(imaginary) components ofH , respectively. Simulation results obtained withthe Born approximation, the EBA, and the Ho-GEBA (up to the third order) areplotted together with the exact solution.


the in-phase (real) component of , whereas the bottomfigure describes the quadrature (imaginary) component of .In similar fashion to Fig. 3, the Ho-GEBA is more accuratethan the EBA and considerably more accurate than the Bornapproximation at both frequencies.

We proceed to displace Model 3 closer to the source, therebyconstructing Model 4. Fig. 8 shows the scattered compo-nent as a function of receiver location at two different frequen-cies: 10 and 200 KHz. The left- and right-hand panels show sim-ulation results for 10 and 200 KHz, respectively. For each panel,




the top figure describes the in-phase (real) component of ,whereas the bottom figure describes the quadrature (imaginary)component of . Results indicate that the Ho-GEBA is moreaccurate than the EBA and the Born approximation at both fre-quencies.

Fig. 9 shows the scattered component as a function ofreceiver location at 10 and 200 KHz. The assumed scatteringmodel is Model 4. The left- and right-hand panels show simu-lation results for 10 and 200 KHz, respectively. For each panel,the top figure describes the in-phase (real) component of ,whereas the bottom figure describes the quadrature (imaginary)component of . In similar fashion to Fig. 7, results indicate

Fig. 10. Comparison of the scattered magnetic field component Hsimulated with the Born approximation and the EBA over the frequency rangefrom 10 KHz to 2 MHz. The assumed scattering model is Model 2, with onefixed receiver located at �0:1 m. The left- and right-hand panels describe thein-phase (real) and quadrature (imaginary) components of H , respectively.Simulations of H performed with the Born approximation, the EBA, and theHo-GEBA (up to the third order) are plotted together with the exact full-wavesolution.

that the Ho-GEBA is more accurate than the EBA and the Bornapproximation at both frequencies.

To further assess the accuracy of the Ho-GEBA with respectto frequency, we consider a fixed receiver located atm. The assumed scattering model is Model 2. This modelrepresents a typical conductive medium and is responsible forsubstantial near-source scattering effects. The frequency rangeconsidered for the simulations is between 10 KHz and 2 MHz,which is typical of borehole geophysical induction logging ap-plications. Figs. 10 and 11 compare the scattered magnetic fieldcomponents and , respectively, simulated with theHo-GEBA up to the fifth-order together against the full-wavesolution, the EBA, and the Born approximation. This graphicalcomparison confirms that the Ho-GEBA yields consistent andaccurate results that are superior to the EBA and the Bornapproximation over the entire frequency range.

IX. DISCUSSION AND CONCLUSIONS

The following conclusions stem from the simulation exer-cises described earlier.

1) In general, the Ho-GEBA is more accurate than theEBA regardless of both the distance between thesource and scatterer and the operating frequency. Forsome cases where the source is far from the scatterer,the EBA also provides accurate simulation results.

2) The Ho-GEBA can dramatically improve the conver-gence rate of the GS. This is best explained with asimulation exercise. Fig. 12 compares the convergenceof the GS and the Ho-GEBA for scattering Model 1

. The left- and right-hand panels in that figuredescribe simulation results for 10 and 200 KHz, re-spectively. For this simulation exercise the rate of con-vergence of the Ho-GEBA is superior to that of theGS. Also, we observe that for some cases (as shown


Fig. 11. Comparison of the scattered magnetic field componentH simulatedwith the EBA, the Born approximation, and the EBA over the frequency rangefrom 10 KHz to 2 MHz. The assumed scattering model is Model 2, with onefixed receiver located at �0:1 m. The left- and right-hand panels describe thein-phase (real) and quadrature (imaginary) components of H . Simulations ofH performed with the Born approximation, the EBA, and the Ho-GEBA (upto the third order) are plotted together with the exact full-wave solution.

Fig. 12. Comparison of the convergence behavior of the Ho-GEBA and theGS. Model 1 is the assumed scatterer. Numerical simulations correspond to thesecondary H component. The left- and right-hand panels show convergenceresults for 10 and 200 KHz, respectively.

in Fig. 12) the first-order solution is superior to thesecond-order solution. However, this behavior does notaffect the rate of convergence of the Ho-GEBA.

3) Another technical issue that needs some considera-tion is Special Case 2 of the Ho-GEBA. At the outset,we emphasized that this special case may not be ap-plicable for some cases of EM scattering. To clarifythis point, we make use of another simulation exercise.Fig. 13 describes simulation results (in-phase compo-nents of ) obtained for Model 2. In that figure, thecurves labeled HoGEBAS2-n, , describesimulation results obtained for the Special Case 2 ofthe Ho-GEBA. These results clearly indicate that Spe-cial 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 theassumed scattering model. Numerical simulations correspond to the secondaryH component. The nomenclature Ho-GEBAS2-n (n = 1; 2; 3) designatessimulation results associated with the Special Case 2 of the Ho-GEBA. The left-and right-hand panels describe the in-phase (real) and quadrature (imaginary)components of H , respectively.

problem at hand. One may then conclude that SpecialCase 2 of Ho-GEBA only applies to simulation caseswhere the EBA remains accurate.

As a general conclusion, we emphasize that the GS is a gen-eralized series expansion of the internal electric field, whereasthe GEBA is a generalized extended Born approximation. TheHo-GEBA is a combination of the GEBA and the GS. In gen-eral, the GEBA will converge substantially faster than the GS.We validated the Ho-GEBA using simple 3-D scatterers. Nu-merical experiments in the induction frequency range show thatthe Ho-GEBA is in general more accurate than both the Bornapproximation and the EBA. The total computational cost ofthe Ho-GEBA is proportional to , where isthe number of spatial discretization cells. A unique feature ofthe Ho-GEBA is that it can be used to simulate EM scatteringdue to electrically anisotropic media. This feature is not possiblewith either the Born approximation or the EBA.

APPENDIX ADERIVATION OF THE NEW INTEGRAL EQUATION

Singer [16], Pankratov [15], and Zhdanov and Fang [13] de-rived an energy inequality for the anomalous EM field. Suchan energy inequality can be generalized to the case in which anelectrical conductivity anomaly is embedded in an infinite uni-form conductive background (Singer, [16]).

Let us assume an electrical conductivity anomaly with aclosed boundary embedded in an infinite uniform conductivebackground of conductivity equal to . Following Zhdanovand Fang [13], the per-period average of energy flow ofanomalous EM field through can be expressed as

(A-1)

where is the spatial support of the conductivity anomaly,is the Poynting vector, is the outgoing unit


vector normal to the surface and are the anoma-lous electric and magnetic fields, respectively, * denotes com-plex conjugate, and designates the real part of the cor-responding quantity. According to the Poynting theorem andMaxwell’s equations (Harrington, [20]), can be rewritten as

(A-2)

where 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 followingequation holds:

(A-3)

The integrand in (A-3) can be rewritten as

(A-4)

Substitution of (A-4) into (A-3) yields the energy inequality

(A-5)

Equation (A-5) is a consequence of the physics of the interactionbetween the EM fields and the medium. This operating condi-tion represents a physical constraint for our derivations below.

Because is always positive, (A-5) is equivalent to

(A-6)

Next, we note that

(A-7)

and make use of (13) to obtain

(A-8)

where is an operator that can be applied to any vector-valuedfunction and is given by

(A-9)

From the physical constraint given by (A-6), one can derive thefollowing inequality for the operator :

(A-10)

where denotes the -norm in a Hilbert space, and is definedas

(A-11)

By making use of (14), Zhdanov and Fang [13] transformed(A-8) into

(A-12)

where

(A-13)

Equation (A-12) can be treated as an integral equation with re-spect to the product , i.e.

(A-14)

where is a new operator that remains contractive for any typeof lossy background medium (Zhdanov and Fang, [13]).

By making use of (A-12) and (13), and after some manipula-tions, one obtains

(A-15)

where

(A-16)

Following Zhdanov and Fang [13], it can be shown that for anylossy background medium , the following relationholds:

(A-17)

According to the Cauchy–Schwartz inequality, (A-10) and(A-17) guarantee that the operator be contractive, namely

(A-18)

Equation (A-15) leads to the new integral equation

(A-19)

where

(A-20)

and

(A-21)

Notice that the contraction of the new integral equation (A-19)is ensured by (A-18). Finally, is given by (A-16).


APPENDIX BDERIVATION 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

(B-2)

We note that (A-21) has also been used to derive (B-2).Now define

(B-3)

Equation (B-2) can then be rewritten as

(B-4)

Substitution of (B-4) into (23) together with (A-16) gives

(B-5)

where

(B-6)

By repeating the same procedure, one derives the following se-ries expansion:

(B-7)

where

(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 con-tractive integral equation. This result indicated that the seriesexpansion given by (B-7) was always convergent. To confirmsuch an important property, in this Appendix we consider a nu-merical example for which the classical Born series diverges.Fig. 14 describes the formation model, consisting of a conduc-tive cube with a side length of 2 m and conductivity equal to 10S/m, embedded in a background medium of conductivity equalto 1 S/m. The transmitter and the receiver are vertical magneticdipoles operating at 20 KHz. The distance between the trans-mitter and the cube is 0.1 m, and the spacing between the trans-mitter and receiver is 0.5 m. Measurements consist of the scat-tered magnetic field at the receiver. Fig. 15 compares the con-vergence of the GS (right panel) against the convergence of the

Fig. 14. Rock formation model used to numerically test the convergenceproperties of the GS. A conductive cube with a side length of 2 m and aconductivity of 10 S/m is embedded in a background medium of conductivityequal to 1 S/m. Transmitter and receiver are vertical magnetic dipoles operatingat 20 KHz. The distance between the transmitter and the cube is 0.1 m, and thespacing between transmitter and receiver is 0.5 m.

Fig. 15. Comparison of the convergence behavior of the classical Bornseries expansion, the GS (starting from the background field), the EBAseries expansion [no contraction (N.C.)], and the EBA series expansion[with contraction (W.C.)] for the rock formation model given in Fig. 14. Theleft-hand panel describes the convergence behavior of both the classical Bornseries expansion and the EBA series expansion (N.C.), while the right-handpanel describes the convergence behavior of the GS and EBA series expansion(W.C.). The exact solution (Solution Line) was calculated using a full-wave3-D integral-equation code (Fang et al., [2]).

classical Born series (left panel). On these figures, the horizontalaxis describes the iteration number, while the vertical axis de-scribes the amplitude of the scattered magnetic field. This ex-ercise clearly indicates that the classical Born series expansiondoes not converge, while the GS converges to the exact solutionin a few iterations.

APPENDIX CDERIVATION OF THE FUNDAMENTAL EQUATION OF THE

Ho-GEBA

Substitution of (25) into the right-hand side of (40) gives

(C-1)


Since is assumed spatially invariant within subdomain , onecan rewrite (C-1) as

(C-2)

From (B-3) and (B-8) one obtains

(C-3)

where

(C-4)

(C-5)

and

(C-6)

Substitution of (C-3) into (C-2) yields

(C-7)

Finally

(C-8)

where is given by (33).

APPENDIX DDERIVATION OF SPECIAL CASE 2 OF THE Ho-GEBA

First, the generalized series expansion of can be writtenas

(D-1)

Subtraction of (25) from (D-1) yields

(D-2)

For convenience, we keep the first terms in (D-2), andsubstitute the ensuing expression into (28), to obtain

(D-3)

By expanding with a Taylor series about one obtains

(D-4)

Further, by retaining only the first term on the right-hand sideof (D-4) one can write

(D-5)

Using this last expression and rearranging the terms in (D-3),one obtains

(D-6)

Substitution of (C-3) into (D-6), together with some simple ma-nipulations yields

(D-7)

where is given by (C-4) and (C-5).

ACKNOWLEDGMENT

A note of gratitude goes to Dr. D. Pardo and two anonymousreviewers for their constructive technical and editorial feedback.

REFERENCES

[1] M. Born, Optics. New York: Springer-Verlag, 1933.[2] S. Fang, G. Gao, and C. Torres-Verdín, “Efficient 3-D electromagnetic

modeling in the presence of anisotropic conductive media using integralequations,” in Proc. Third Int. 3D Electromagn. (3DEM-3) Symp., 2003.

[3] G. Gao, S. Fang, and C. Torres-Verdín, “A new approximation for 3Delectromagnetic scattering in the presence of anisotropic conductivemedia,” in Proc. Third Int. 3D Electromagn. (3DEM-3) Symposium,2003.

[4] G. Gao, C. Torres-Verdín, and S. Fang, “Fast 3D modeling of boreholeinduction data in dipping and anisotropic formations using a novel ap-proximation technique,” in Paper VV, Trans. 44th SPWLA Annu. LoggingSymp., 2003.

[5] G. Gao and C. Torres-Verdín, “Fast inversion of borehole induction datausing an inner-outer loop optimization technique,” in Paper TT, Trans.44th SPWLA Annu. Logging Symp., 2003.

[6] G. Gao, C. Torres-Verdín, and S. Fang, “Fast 3D modeling of boreholeinduction data in dipping and anisotropic formations using a novel ap-proximation technique,” Petrophys., vol. 45, no. 3, pp. 335–349, 2004.


[7] G. Gao and C. Torres-Verdín, “A high-order generalized extended Bornapproximation to simulate electromagnetic geophysical measurementsin inhomogeneous and anisotropic media,” SEG Expanded Abstracts,pp. 628–631, 2004.

[8] G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical techniques toevaluate the integrals of 3D and 2D spatial dyadic Green’s functions,”Progr. Electromagn. Res., vol. 52, pp. 47–80, 2005.

[9] R. F. Harrington, Field Computation by Moment Methods. New York:Macmillan, 1968.

[10] G. W. Hohmann, “Three-dimensional induced polarization and electro-magnetic modeling,” Geophys., vol. 40, no. 2, pp. 309–324, 1975.

[11] C. Torres-Verdín and T. M. Habashy, “Rapid 2.5-dimensional forwardmodeling and inversion via a new nonlinear scattering approximation,”Radio Sci., vol. 29, no. 4, pp. 1051–1079, 1994.

[12] T. M. Habashy, R. W. Groom, and B. Spies, “Beyond the Born and Rytovapproximations: A nonlinear approach to electromagnetic scattering,” J.Geophys. Res., vol. 98, no. B2, pp. 1759–1775, 1993.

[13] M. S. Zhdanov and S. Fang, “Quasilinear series in three-dimensionalelectromagnetic modeling,” Radio Sci., vol. 32, no. 6, pp. 2167–2188,1997.

[14] , “Quasilinear approximation in 3-D electromagnetic modeling,”Geophys., vol. 61, no. 3, pp. 646–665, 1996.

[15] O. V. Pankratov, D. B. Avdeev, and A. V. Kuvshinov, “Scattering of elec-tromagnetic field in inhomogeneous earth: Forward problem solution,”Izv. Akad. Nauk. SSSR Fiz. Zemli, vol. 3, pp. 17–25, 1995.

[16] B. S. Singer, “Method for solution of Maxwell’s equation in nonuniformmedia,” Geophys. J. Int., vol. 120, pp. 590–598, 1995.

[17] T. J. Cui, Y. Qin, G.-L. Wang, and W. C. Chew, “Low-frequency detec-tion of two-dimensional buried objects using high-order extended Bornapproximations,” Inv. Prob., vol. 20, pp. 41–62, 2004a.

[18] T. J. Cui, W. C. Chew, and W. Hong, “New approximate formulationsfor EM scattering by dielectric objects,” IEEE Trans. Antennas Propag.,vol. 52, no. 3, pp. 684–692, 2004b.

[19] J. P. Aubin, Applied Functional Analysis. New York: Wiley, 1979.[20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York:

McGraw-Hill, 1961.

Guozhong Gao received the Bachelor’s andMaster’s degrees in applied geophysics fromSouthwest Petroleum Institute (China) and BeijingPetroleum University in 1996 and 2000, respectively.He received the Ph.D. degree from the Departmentof Petroleum and Geosystems Engineering of TheUniversity of Texas at Austin in 2005.

Currently, he is a Geophysicist with SchlumbergerTechnology Corporation, Richmond, CA. His cur-rent research interests include borehole, marine,cross-well, and surface-to-borehole electromag-

netics, including modeling, inversion, and system design.

Carlos Torres-Verdín (M’82) received the Ph.D. de-gree in engineering geoscience from the University ofCalifornia, Berkeley, in 1991.

During 1991–1997, he held the position of Re-search Scientist with Schlumberger-Doll Research.From 1997 to 1999, he was Reservoir Specialistand Technology Champion with YPF, Buenos Aires,Argentina. Since 1999, he has been affiliated withthe Department of Petroleum and GeosystemsEngineering of The University of Texas at Austin,where he currently holds the position of Associate

Professor and conducts research in formation evaluation, well logging, andintegrated reservoir characterization. He has served as Guest Editor for RadioScience, and is currently a member of the Editorial Board of the Journalof Electromagnetic Waves and Applications, and an Associate Editor forPetrophysics (SPWLA) and the SPE Journal.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. … · Green’s dyadic functions, is a vector...

Documents

Transcript of IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. … · Green’s dyadic functions, is a vector...