IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 8, AUGUST 2010 3139

Electromagnetic Scattering by Bicontinuous RandomMicrostructures With Discrete Permittivities

Kung-Hau Ding, Senior Member, IEEE, Xiaolan Xu, and Leung Tsang, Fellow, IEEE

Abstract—For electromagnetic (EM) scattering by dense media,the traditional approach is to use particles of spheres or ellipsoidsthat are densely and randomly packed in a background medium.The particles have discrete permittivities that are different fromthe background medium. The dense-medium model has beenapplied to the microwave remote sensing of terrestrial snow. In thispaper, we propose a different approach of using a bicontinuousmedium with discrete permittivities and study the EM scatteringproperties using analytical and numerical methods. The bicontin-uous medium is a continuous representation of interfaces betweeninhomogeneities within the medium. Discrete permittivities arethen assigned to the inhomogeneities of the structure. The analyti-cal approach is based on the Born approximation using the derivedanalytical correlation functions. The numerical method is basedon the numerical Maxwell model of 3-D (NMM3D) approach. Inparticular, the discrete-dipole approximation and the conjugategradient-squared method accelerated by the fast Fourier trans-form technique are used in solving the volume integral equation.Scattering results of analytical and numerical approaches arecompared. Numerical results are illustrated using parameters inmicrowave remote sensing of terrestrial snow. In the NMM3Dsimulations, three kinds of convergence tests are conducted, viz.,convergence with respect to the discretization size, convergencewith respect to the sample size, and convergence with respectto the number of realization. The NMM3D results indicate thatthe scattering by the bicontinuous medium with a broader sizedistribution has a weaker frequency dependence than that by themedium with a more narrow size distribution. The frequency-dependence power law index can be lower than two, which is verymuch lower than the power of four in Rayleigh scattering. TheNMM3D results also exhibit fairly large cross-polarization returnswhich account for the local nonisotropic microstructures of bicon-tinuous media, although the medium is statistically isotropic.

Index Terms—Bicontinuous media, dense media, discrete dipoleapproximation, discrete ransom media, frequency dependence,NMM3D, volume scattering.

I. INTRODUCTION

THE MODELING of wave propagation and scattering inrandom media with discrete permittivities has applications

in microwaves and optics [1]–[4]. Discrete permittivities referto cases when the permittivities change with discrete values

Manuscript received March 31, 2009; revised September 10, 2009. Date ofpublication April 8, 2010; date of current version July 21, 2010. This work wassupported in part by the Air Force Office of Scientific Research and in part bythe U.S. National Aeronautics and Space Administration.

K.-H. Ding is with the Sensors Directorate, Air Force Research Lab-oratory, Hanscom Air Force Base, MA 01731 USA (e-mail: kung-hau.ding@hanscom.af.mil).

X. Xu and L. Tsang are with the Department of Electrical Engineer-ing, University of Washington, Seattle, WA 98195 USA (e-mail: xlxu@ee.washington.edu; tsang@ee.washington.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2010.2043953

from one part to the other part of the medium. Examples includeporous media and dense media. For example, snow consistsof mixtures of ice particles and air. Thus, the permittivitiesvary discretely assuming the values of 3.2 for ice and 1 for air[1]. Studies of such problems include examining the effectiveconductivities and effective permittivities which are very lowfrequency behavior when the medium is well characterized byeffective values and then replaced by the macroscopic Maxwellequations with effective permittivities and conductivities[1]–[3]. However, in situations when the grain sizes or thescales of inhomogeneities are not too small compared with thewavelengths of the electromagnetic (EM) waves, then there arestochastic fluctuations of the EM waves with significant contri-butions of incoherent fields that have random phases. For suchcases, the bistatic scattering phase matrices are the quantities ofinterest which can be used in a dense-medium radiative transfer(DMRT) theory [1], [3], [5]–[9]. In the past, the dense-mediummodel consisted of particles of spheres or ellipsoids densely andrandomly packed in a background medium. The particles havediscrete permittivities that are different from the backgroundmedium. The dense-medium model has been applied to themicrowave remote sensing of terrestrial snow [5]–[12]. Ana-lytical methods of dense-medium scattering include using thequasicrystalline approximation (QCA) with the Percus–Yevickpair distribution function to account for the relative positionsof particles. The particles can have adhesive force to form ag-gregates which are also exhibited in the pair distribution func-tions. In the numerical approach, we have used the Metropolisshuffling method to simulate the positions of particles whichcan have adhesive properties [10]–[12]. Using the computer-generated particle positions, we have used Foldy–Lax multiplescattering equations to calculate bistatic scattering properties.The results of both analytical and numerical methods exhibitfrequency dependence of the power law of less than four eventhough the grain sizes are much smaller than the wavelength, aresult that comes from the aggregate scattering properties. Thephase matrix also exhibits cross-polarization effects that arisefrom coherent near-field interactions [11], [12].

In this paper, we propose a different approach of using abicontinuous medium with discrete permittivities and study theEM scattering properties using analytical and numerical meth-ods. Instead of using particles, the mathematical formalismdeveloped by Cahn [13] is applied in this study to simulate themorphologies of random porous structures. The model is basedon a continuous representation of interfaces between differentphases and domains within the medium. The interfaces areconstructed from a large number of stochastic sinusoidal waveswith random phases and directions [14], [15]. Discrete permit-tivities are then assigned to different domains. We also assumethat the wavenumbers of sinusoidal waves follow the gamma

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3140 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 8, AUGUST 2010

Fig. 1. Simulated bicontinuous random structures with fv = 0.3 and α =0.371. The wavenumber distribution parameters are (a) 〈ζ〉 = 3000 m−1 andb = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10. Each side of slices is equal to64 mm.

distribution [16]. Physically, the wavenumber corresponds tothe inverse of the size of the inhomogeneity. Assigning dif-ferent values of mean and variance for the polydispersity ofthe wavenumber, a variety of bicontinuous microstructures canbe created. The 3-D bicontinuous random structures can becharacterized by analytical correlation functions [14]–[16].

The Born approximation [1], [3] is applied in the analyti-cal approach using analytical correlation functions of bicon-tinuous media [14]–[16]. The numerical approach is basedon the numerical Maxwell model of 3-D (NMM3D) method[8]–[12], which is used to solve the volume integral equation. Inparticular, the discrete-dipole approximation (DDA) approachis applied [2], [17]. The conjugate gradient-squared (CGS)method [18] with the fast Fourier transform (FFT) acceler-ation is employed to solve the matrix equation iteratively.The advantage of the CGS method is that it avoids usingthe adjoint of the impedance matrix to gain faster conver-gence [18]. For the cases studied in this paper, we found thatthe CGS-with-FFT method converges about six times fasterthan the conjugate-gradient (CG)-with-FFT method [2]. InSection II, we summarize the bicontinuous random morphologyand illustrate two examples of simulated bicontinuous randommedia with different microstructures and average domain sizes.The analytical correlation functions for bicontinuous mediaare demonstrated in Section III. Section IV gives the volumeintegral equation formulation for the bicontinuous mediumwith discrete permittivity, the Born approach [1], [3], and theDDA treatment for the volume scattering problem [2]. The1–2 polarization system [2], [12] and the computations of phasematrices are also described in this section. The phase matricesare best represented in the 1–2 polarization frame, in which thephase matrices only depend on the angle between the incidentand scattered directions. The phase matrices are obtained by

subtraction of the coherent field from the total field and consistof the incoherent field only [2], [10]–[12]. In Section V, weillustrate the numerical results for the scattering coefficients, theco- and cross-polarization phase matrices, and their ratios usingboth analytical and numerical methods. Scattering results ofanalytical and NMM3D approaches are compared. In NMM3Dsimulations, three kinds of convergence tests are conducted,viz., convergence with respect to the discretization size, con-vergence with respect to the sample size, and convergence withrespect to the number of realizations. We illustrate the resultsusing parameters in microwave remote sensing of terrestrialsnow. The frequency-dependence power law index can be lowerthan two, which is very much lower than the power of four inRayleigh scattering. The results also exhibit fairly large cross-polarization returns which account for the local nonisotropicmicrostructures of bicontinuous media, although the medium isstatistically isotropic. Finally, conclusion remarks are given inSection VI.

II. COMPUTER GENERATION OF

BICONTINUOUS STRUCTURE

The bicontinuous structure can be generated by using the fol-lowing algorithm which is a simpler version of Berk’s method[14], [15]. Let

S(r) =1√N

N∑n=1

cos(ζn · r + φn). (1)

Then, the random function S(r) can be constructed stochasti-cally by choosing a large value of N , for example, N = 104. In(1), the wave vector ζn = ζnζn and the phase φn are randomvariables and assumed to be independent. We choose theserandom numbers as follows. The phase φn is uniformly randombetween 0 and 2π, the unit vector ζn is uniformly distributed ona unit spherical surface, and the wavenumbers ζn are distributedbetween 0 and ∞ and have a probability density functionp(ζ). In this paper, we choose that p(ζ) follows the gammadistribution function [16]

p(ζ) =1

Γ(b + 1)(b + 1)b+1

〈ζ〉

(ζ

〈ζ〉

)b

e−(b+1) ζ〈ζ〉 (2)

where Γ() is the gamma function and 〈ζ〉 is the mean wavenum-ber. Physically, the wavenumber ζ corresponds to the inverse ofthe size of the inhomogeneity. The standard deviation of ζ, i.e.,Δζ, is given by

Δζ =√〈ζ2〉 − 〈ζ〉2 =

〈ζ〉√b + 1

. (3)

We can see that the distribution is narrow when b is large, andthe distribution becomes broader when b decreases. A large bcorresponds to a single grain size, while a smaller b correspondsto a broad size distribution. Thus, we can use (1) to generateS(r) for every position r.

Next, for every r and the value of α, we define a function

Θα (S(r)) ={

1, for S(r) > α0, otherwise

(4)

where α is the cutting level for specifying the system. Inthis way, the entire space is divided randomly into regions

DING et al.: EM SCATTERING BY BICONTINUOUS RANDOM MICROSTRUCTURES 3141

Fig. 2. Correlation functions Cs(r) and Cα(r) for bicontinuous random structures with fv = 0.3 and α = 0.371. The wavenumber distribution parameters are(a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

of Θα = 1 and Θα = 0. Examples are shown in Fig. 1. Forterrestrial snow as ice grains embedded in air, we can as-sign the Θα(r) = 1 regions as ice and the Θα(r) = 0 regionsas air.

The fractional volume of ice fv is related to the value ofα. Based on the central limit theorem, S(r) is a Gaussianrandom process with zero mean and 〈S2〉 = 1/2. The mean ofΘα(S(r)) is equal to volume fraction fv

fv = 〈Θα (S(r))〉 =12

[1 − erf (α)] (5)

where erf() is the error function. Thus, by choosing the frac-tional volume fv , we can find α from (5).

Two computer-generated bicontinuous random structures areshown in Fig. 1. Each side of slices is equal to 64 mm. Thesemicrostructures are formed by superposition of 104 sinusoidalwaves according to (1) for a medium with fv = 0.3 of inhomo-geneities. The cutting level is at α = 0.371. The parameters ofthe wavenumber distribution are 〈ζ〉 = 3000 m−1 and b = 100for the structure shown in Fig. 1(a) and 〈ζ〉 = 1500 m−1 andb = 10 for that shown in Fig. 1(b). As shown in the slicesof horizontal and vertical cross sections, the different phasesshown in Fig. 1(a) are more uniform, while the microstructureshown in Fig. 1(b) has a more variability than that in Fig. 1(a).Thus, the leveled Gaussian random field model given by (1)and (4) could be useful in modeling complicated morpholo-gies. For the sake of illustration, the units have been chosento correspond to scattering by terrestrial snow at microwavefrequencies. For terrestrial snow, the ice grains are of the sizeof 1 mm in diameter. By using submicrometer sizes, the theoryis also applicable to optical scattering.

III. STATISTICAL CHARACTERIZATION

OF BICONTINUOUS MEDIA

In Fig. 1, we have shown the simulated morphologies of thebicontinuous structures. Some real snow images can be found in[19] and [20]. There are good resemblances between them. Theanalytical expression for the correlation function of a leveledGaussian random field Γα(r) = 〈Θα(S(r))Θα(S(0))〉 can bederived from (1) and (4). The correlation function Γα(r) isexpressed as

Γα(r) = f2v +

∞∑m=1

Cm(α) [Cs(r)]m . (6)

Note that Γα(0) = fv and Γα(∞) = f2v . The coefficients

Cm(α) in (6) are given by

Cm(α) =e−2α2

[Hm−1(α)]2

πm!2m(7)

where Hm()’s are the Hermite polynomials and Cs(r) is theautocorrelation function of S(r)

Cs(r) =

∞∫0

dζ p(ζ)j0(ζr) (8)

here, j0(ζr) = sin ζr/ζr, and p(ζ) is the distribution func-tion of wavenumber ζ. As r = 0, Cs(0) = 1, and as r → ∞,Cs(∞) = 0. For the case of the gamma distribution (2), Cs(r)is given by

Cs(r) = cosb+1 ϕ

(ϕ

sin ϕ

)[sin(bϕ)

bϕ

](9)

in which ϕ = tan−1(〈ζ〉r/(b + 1)).

3142 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 8, AUGUST 2010

The two autocovariance functions Cs(r) and Cα(r) ≡Γα(r) − f2

v , corresponding to the structures shown in Fig. 1(a)and (b), are plotted in Fig. 2(a) and (b), respectively. Note thatCα(r)/fv(1 − fv) is also known as the Porod–Debye correla-tion function [14], [15] or the normalized correlation function[1]. We can see that the oscillatory behavior and the negativecorrelation for the bicontinuous media are quite distinct fromthe usual assumption of an exponential-type correlation func-tion used in the continuous random medium model [1], [3].The snow samples had been characterized using correlationfunctions which can be measured using an image analysistechnique [19].

For bicontinuous media, the analytical expression of surfaceper unit volume or specific surface area (SSA) can also beobtained [15]. The SSA for a medium with the wavenumberdistribution given by (2) is expressed as

SSA =2〈ζ〉π√

3

√b + 2b + 1

e−2α2. (10)

The SSA of snow has been measured using various tech-niques [21].

IV. MICROWAVE SCATTERING FROM

TERRESTRIAL SNOW

Because of the resemblance between the simulated bicontin-uous morphologies and the real snow images, the mathematicalformalism in (1) might be usefully applied to model the snowmicrostructures. Thus, in this paper, the bicontinuous struc-tures with discrete permittivities are used to model microwavescattering from terrestrial snow. For a simulated bicontinuoussample, we assign the permittivity of ice to regions whereΘα(S(r)) = 1 and the permittivity of air to regions whereΘα(S(r)) = 0. The SSA can be computed analytically forbicontinuous random media [15].

The spatial distribution of permittivity εp(r) for a two-phasebicontinuous random medium can be expressed as

εp(r) = (εp − ε)Θα (S(r)) + ε (11)

where ε and εp are the respective permittivities of the twophases. Θα(S(r)) is the indicator function given in (4). For thecase of microwave remote sensing of snow, ε is the permittivityof air, and εp is the permittivity of ice.

The EM response of a volume V with permittivity distribu-tion εp(r) is governed by the volume integral equation

E(r) = Einc

(r) +k2

ε

∫V

dr′G(r, r′) · [εp(r′) − ε] E(r′)

(12)

where k2 = ω2με, Einc

is the incident electric field, and Gand μ are the dyadic Green function and permeability, respec-tively. Substituting (11) into (12), the scattered field can beexpressed as

Es(r) = k2(εp

ε− 1

) ∫V

dr′G(r, r′) · Θα (S(r′)) E(r′).

(13)

Fig. 3. Geometry for defining the 1–2 system based on a scattering plane. The

scattering plane contains ki and ks. The angle between ki and ks is Θ.

A. 1–2 Polarization System

Consider a volume V containing many randomly positionedscatterers (Fig. 3). The phase matrix is the matrix describingthe relation between each polarization of the scattered intensitydue to the respective polarization of the incident wave. Thephase matrices are best represented in the 1–2 polarizationframe [2], [12], in which the phase matrices only depend onthe angle Θ between the incident and scattered directions. Letki be the propagation direction of the incident field and ks

be the direction of the scattered field. The unit vector 1i isperpendicular to both ki and ks, and 2i is perpendicular to bothki and 1i. Let 1s = 1i; then, 2s is perpendicular to both ks and1s. These relations can be written as

1i = 1s =ks × ki

|ks × ki|2i = ki × 1i

2s = ks × 1s. (14)

The polarization vectors 1i, 2i, 1s, and 2s can be related tothe vertical and horizontal polarizations of vi, hi, vs, and hs bysome transformations.

In the 1–2 polarization system, the phase matrices, namely,P11, P21, P12, and P22, obey the relation [2], [12][

|E1s|2|E2s|2

]=

[P11 P12

P21 P22

] ⎡⎣∣∣∣Einc

1

∣∣∣2∣∣∣Einc2

∣∣∣2⎤⎦ (15)

where the subscripts 1 and 2 denote the polarizations of theelectric fields. In the far field, |r| |r′|, the scattered field, inthe 1–2 polarization frame becomes

Es =eikr

4πrk2

(εp

ε− 1

)(1s1s + 2s2s)

·∫V

dr′e−iks·r′Θα (S(r′)) E(r′). (16)

The phase matrices are the bistatic cross sections normalizedby the volume containing the scatterers.

B. NMM3D: Calculation of Phase Matrix andScattering Coefficient

The approach of NMM3D is used for the cases of ex-act numerical Maxwell solutions of 3-D random medium

DING et al.: EM SCATTERING BY BICONTINUOUS RANDOM MICROSTRUCTURES 3143

simulations [8]–[12]. For each realization Θα(S(r)), (12) ofthe wave scattering by bicontinuous random structures is solvednumerically. The scattered fields are obtained from (13). Then,the numerical values of the scattering coefficients and phasematrices can be computed.

Due to the presence of inhomogeneities, the effects of wavescattering remove power from the coherent wave into the in-coherent field. To separate the coherent and incoherent com-ponents, the scattered field is averaged to obtain the coherentscattered wave, which is calculated by [2], [10]–[12]

〈Es〉 =1

Nr

Nr∑τ=1

Eτs (17)

where τ is the index of realization with τ = 1, 2, . . . , Nr real-izations. E

τs is the scattered field of the τ th sample from the

ensemble. The incoherent scattered field is expressed as [2],[10]–[12]

Eτs = E

τs − 〈Es〉. (18)

The bistatic scattering cross section is the averaged incoher-ent scattered wave intensity normalized by the incident waveintensity [2], [10]–[12]

σbi(ks, ki) =r2

|Einc|2

(1

Nr

Nr∑τ=1

∣∣Eτs

∣∣2) (19)

where ki = x sin θi cos φi + y sin θi sinφi + z cos θi and ks =x sin θs cos φs + y sin θs sinφs + z cos θs. The phase matrixP (ks, ki) is defined to be the averaged bistatic scattering crosssection per unit volume [2], [10]–[12]

P (ks, ki) =1V

σbi(ks, ki). (20)

Note that the phase matrix consists of the incoherent fieldonly.

To obtain the phase matrices, namely, P11, P21, P12, and P22,

we let the incident wave be Einc

= yeikz and ki = z. From(14), the polarization vectors 1i, 1s, 2i, and 2s are given by

1i = 1s = x sin φs − y cos φs

2i = x cos φs + y sin φs

2s = x cos θs cos φs + y cos θs sinφs − z sin θs. (21)

Note that θs = Θ in the 1–2 polarization system. Then, theincident wave can be expressed as

Einc

= yeikz = (−1i cos φs + 2i sinφs)eikz. (22)

When φs = 90◦, the incident wave becomes Einc

= 2ieikz ,

which is 2i polarized. Thus, P12(θs) and P22(θs) can be de-

termined. When φ = 180◦, the incident wave becomes Einc

=1ie

ikz , which is 1i polarized. Thus, P11(θs) and P21(θs) can bedetermined. P11 and P22 are copolarization phase matrices, andP21 and P12 are cross-polarization phase matrices.

The scattering coefficient κs is obtained by integrating thebistatic scattering cross section over the solid angle 4π and then

normalized by the volume [2], [10]–[12]

κs =1V

π∫0

dθs sin θs

2π∫0

dφsσbi(ks, ki). (23)

C. Analytical Model: Born Approximation

Let Einc

= eieiki·r, where ei is the polarization vector. Us-

ing the Born approximation approach [1], [3] E ≈ Einc

, thefar-field scattered wave becomes

Es =eikr

4πrk2

(εp

ε− 1

)(1s1s + 2s2s)

· ei

∫V

dr′eikd·r′Θα (S(r′)) (24)

where kd = ki − ks. From (24), the mean scattered field isproportional to the fractional volume and

〈Es〉 = fveikr

4πrk2

(εp

ε− 1

)(1s1s + 2s2s) · ei

∫V

dr′eikd·r′

(25)

which is in the specular direction ks = ki. The incoherentscattered field is expressed as

Es =eikr

4πrk2

(εp

ε− 1

)(1s1s + 2s2s)

· ei

∫V

dr′eikd·r′[Θα (S(r′)) − fv] . (26)

If the incident wave is 1i polarized, since 1s = 1i and2s · 1i = 0, the scattered wave is 1s polarized only. Then,PB21(θs) = 0 since 1i is perpendicular to 2s, and PB11(θs) isgiven by

PB11(θs) =k4

4π

∣∣∣εp

ε− 1

∣∣∣2 ∞∫0

drr2j0(kdr)[Γα(r) − f2

v

](27)

where kd = |kd|, j0(kdr) = sin kdr/kdr, and Γα(r) is givenin (6). Similarly, if the incident wave is 2i polarized, with1s · 1i = 0 and 2s · 2i = cos θs, the scattered wave becomes 2s

polarized only. Then, PB12(θs) = 0, and PB22(θs) is given by

PB22(θs)=cos2 θsk4

4π

∣∣∣εp

ε−1

∣∣∣2 ∞∫0

drr2j0(kdr)[Γα(r)−f2

v

].

(28)

Note that both PB11 and PB22 depend on the same integral∫ ∞0 drr2j0(kdr)[Γα(r) − f2

v ], but PB22 has the cos2 θs factor,which is different from PB11. Thus, under the Born approxima-tion, there are no cross-polarization returns, and the frequencydependence of scattering is the fourth power of frequency. Thescattering coefficient κsB is obtained as

κsB = π

π∫0

dθs sin θs [PB11(θs) + PB22(θs)] . (29)

3144 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 8, AUGUST 2010

Fig. 4. Phase matrices based on Born approximation as a function of scattering angles for bicontinuous structures with wavenumber distributions (a) 〈ζ〉 =3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10. The volume fraction is fv = 0.3, the permittivity is εp = 1.1ε0, and the frequency is 37 GHz.

D. DDA/CGS/FFT: DDA With CGS and FFT

In the numerical approach, the leveled wave method de-scribed in Section II is applied to generate various configura-tions of bicontinuous microstructures in an example cube. TheEM scattering for each realization is then calculated by solvingthe volume integral equation (12) using the method of mo-ments. Based on the DDA [2], [17], the region V is subdividedinto N cubic subcells of equal volume ΔV = (V/N) = d3,where d is the side length of the elemental cube. Then, weobtain a discretized version of the volume integral equation

pi = αiEinci − αi

N∑j=1,j �=i

Aij · pj (30)

in which pi = ΔV (εpi − ε)Ei is the polarization of the ith

elemental cube and εpi, Ei, Einci , and αi are the values of

the permittivity, the internal field, the incident field, and thepolarizability in the ith subcell, respectively. The polarizabilityαi is

αi =αC

i

1 − αCi

4πεd3

[(4π3

) 13 k2d2 + i2k3d3

3

] (31)

in which αCi = 3εd3((εpi − ε)/(εpi + 2ε)). The matrix ele-

ments Aij in (30) are Aij = −(k2/ε)G(ri, rj).After the system of equations (30) is solved, the scattered

field at position r is computed using the discretized form of(13). In this paper, the matrix equation (30) is solved usingthe CGS method [18] that is further accelerated by the FFTtechnique. The CGS method avoids using the adjoint of theimpedance matrix A. For the cases studied in this paper, theCGS method converges about six times faster than the CG

method. The error norm criterion for the iterative solutions hasbeen set equal to 1 × 10−4 for the numerical results presentedin the next section.

V. NUMERICAL RESULTS

Terrestrial snow consists of ice and air. The grain size of icein the snow varies from 0.2 to 1.6 mm, whereas the volumefraction of ice ranges from 20% to 40% [22]. The relativepermittivity of ice is about 3.2. Satellite passive microwaveremote sensing of snow is presently conducted at 18.7 and37 GHz. The electromagnetic scattering by the two randomstructures shown in Fig. 1(a) and (b), with respective 〈ζ〉 =3000 m−1 and b = 100 and 〈ζ〉 = 1500 m−1 and b = 10, isstudied in this section. Two permittivity contrast cases, namely,εp = 1.1ε0 and εp = 3.2ε0, will be examined in this section.The numerical results are produced by means of a 3.2-GHzworkstation with 32-GB RAM.

A. Results of Born Approximation

We first consider the cases of media with a small permittivitycontrast εp = 1.1ε0. Fig. 4 shows the phase matrices computedusing the Born approximation approach at 37 GHz. The per-mittivities of the two phases are ε = ε0 and εp = 1.1ε0, and thevolume fraction is fv = 0.3. We can see the different scatteringbehaviors between the two bicontinuous systems. Although thetwo structures have the same amount of inhomogeneities, themicrostructure with 〈ζ〉 = 1500 m−1 and b = 10 scatters muchstronger than the system with 〈ζ〉 = 3000 m−1 and b = 100.In addition, the forward scattering shown in the structure with〈ζ〉 = 3000 m−1 and b = 100 is stronger than the backwardscattering. However, for the system with 〈ζ〉 = 1500 m−1 andb = 10, the backscattering becomes stronger than the forward

DING et al.: EM SCATTERING BY BICONTINUOUS RANDOM MICROSTRUCTURES 3145

TABLE ISCATTERING COEFFICIENTS OBTAINED FROM BORN APPROXIMATION AND THE NMM3D METHOD

Fig. 5. Comparison of phase matrices between the Born approximation and NMM3D solutions. The volume fraction is fv = 0.3, the permittivity isεp = 1.1ε0, and the frequency is 37 GHz. The level parameters of the simulated microstructure are α = 0.371. The wavenumber distribution parameters are(a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

scattering. These anomalous scattering behaviors might be dueto the broader size distribution associated with the porous struc-ture shown in Fig. 1(b). The scattering coefficients calculatedby using the Born approximation are given in Table I.

B. Results of NMM3D

The 3-D bicontinuous random media are generated based on(1) with a gamma distribution of wavenumber (2). The volumefraction is fv = 0.3. The simulated sample is of cubic sizewith L = 64 mm. The discretized subcell size is d = 0.5 mm.For each case, Nr = 50 samples are created. The averagefractional volume is equal to 0.299873 with the standard de-viation 4.128 × 10−4 for the case with 〈ζ〉 = 3000 m−1 andb = 100. For the structure with 〈ζ〉 = 1500 m−1 and b = 10,the average fractional volume is equal to 0.299869 with thestandard deviation 9.925 × 10−4.

The EM scattering is solved using the NMM3Dapproach described in the previous section. In NMM3Dsimulations, three kinds of convergence tests are conducted,viz., convergence with respect to the discretization size(d), convergence with respect to the sample size (L), andconvergence with respect to the number of realization (Nr).

We also compare the NMM3D approach with the analyticalBorn approximation for low and high permittivity contrasts.The frequency dependence of scattering by the bicontinuousrandom structure is also investigated.

1) Comparison Between NMM3D and Born Approximation:The scattering results of analytical Born approximation arecompared with those of NMM3D. We first consider the casesof media with a small permittivity contrast εp = 1.1ε0. Fig. 5shows the comparison for 37 GHz, in which the results of Bornapproximation have been shown in Fig. 4. Overall, we can seethat the NMM3D method and the analytical Born approachmatch well for the low-permittivity-contrast case. However, wedo observe some fluctuations shown in the NMM3D solutions.This is because the numerical method only considers samples offinite sizes and takes the ensemble average of a finite number ofsamples. Moreover, large random phase fluctuations are usuallyobserved from sample to sample in the NMM3D approach.The scattering coefficients of the NMM3D simulations arealso given in Table I. For this case, the agreement of scat-tering coefficients between the Born and NMM3D methodsis good.

Next, we compare the scattering results of Born approx-imation with the NMM3D method for media with a higher

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Fig. 6. Comparison of phase matrices between the Born approximation and NMM3D solutions. The volume fraction is fv = 0.3, the permittivity isεp = 3.2ε0, and the frequency is 37 GHz. The level parameters of the simulated microstructure are α = 0.371. The wavenumber distribution parameters are(a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

TABLE IISCATTERING COEFFICIENTS AND FREQUENCY DEPENDENCE

OF BORN APPROXIMATION AND THE NMM3D METHOD

permittivity contrast εp = 3.2ε0. Fig. 6 shows the comparisonfor 37 GHz. As we expected, the analytical Born approxima-tion does not agree with the NMM3D method for the casewith higher permittivity contrast, particularly for the higherfrequency and the microstructure with 〈ζ〉 = 1500 m−1 andb = 10. The comparison of scattering coefficients between theBorn and NMM3D approaches is presented in Table II. Thescattering coefficients of the Born method deviate very muchfrom the NMM3D results. This higher contrast case is alreadybeyond the validation region of the Born approximation. Notethat the meaning of n in Table II will be given in Section III.

2) Convergence Test of NMM3D Parameters: In theNMM3D simulations of bicontinuous media, several numericalparameters are used: the discretization subcell size (d), the sam-ple size (L), and the number of realization (Nr). However, thephysical result of scattering coefficient κs should not dependon these numerical parameters. In the following, we use thebicontinuous system with the wavenumber distribution 〈ζ〉 =1500 m−1 and b = 10 to show the convergences of NMM3Dsimulations with regard to these parameters. The frequency is

at 18.7 GHz. Convergence tests of the other microstructure andthe other frequency are similar.

When using numerical approaches, the accuracy of thenumerical results may depend on the resolution, i.e., thenumber of grid points per wavelength or the discretization size.In Fig. 7, we compare the NMM3D scattering results of usingtwo discretization sizes d = 0.5 mm and d = 0.25 mm, whichare about 1/32 and 1/64 wavelength, respectively. The othernumerical parameters are fixed with L = 32 mm and Nr = 50.The phase matrices P11 and P21 are plotted as a function ofscattering angles. We can see that the copolarization phasematrices P11 agree very well for two discretizations. On theother hand, the finer discretization slightly increases the cross-polarization return. Convergence tests for P22 and P12 aresimilar. The convergence test of scattering coefficient κs is alsoshown in Fig. 7. The results are shown up to 50 realizations. Thescattering coefficients are κs = 6.0029 for d = 0.5 mm andκs = 6.0589 for d = 0.25 mm. We can see that the scatteringcoefficient converges similarly for the two discretization sizes.

In order to obtain a sufficient amount of incoherent wave andrandom phase, it is essential to have the distribution of randominhomogeneities over a spatial domain of several wavelengthsin the NMM3D simulations. Fig. 8 shows the convergence testfor two sample sizes L = 32 mm and L = 64 mm, which areabout two and four wavelengths, respectively. The discretiza-tion size is d = 0.5 mm, and Nr = 50 samples are used. Thephase matrices P22 and P12 of the two sample sizes are plottedas a function of scattering angles. From the results, we can seethat the phase matrices have more fluctuations for the largersample size as it contains more random phase information. Thecomparison of copolarization is good for the two sample sizes.However, the larger sample size gives a higher return in the

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Fig. 7. Convergences of phase matrices and κs with respect to the discretization size (d) for the bicontinuous system with fv = 0.3 and α = 0.371. Thepermittivity is εp = 3.2ε0. The discretization sizes are d = 0.25 mm and d = 0.5 mm. The sample size is L = 32 mm, and Nr = 50. The frequency is18.7 GHz. The wavenumber distribution parameters are 〈ζ〉 = 1500 m−1 and b = 10.

Fig. 8. Convergences of phase matrices and κs with respect to the sample size (L) for the bicontinuous system with fv = 0.3 and α = 0.371. The permittivityis εp = 3.2ε0. The sample sizes are L = 32 mm and L = 64 mm. The discretization size is d = 0.5 mm, and Nr = 50. The frequency is 18.7 GHz. Thewavenumber distribution parameters are 〈ζ〉 = 1500 m−1 and b = 10. The scattering coefficients are κs = 6.0029 for L = 32 mm and κs = 5.9539 forL = 64 mm.

cross-polarization. This is due to which a larger sample is moreopaque than a smaller sample; a wave is then multiply scatteredmore in a larger sample than in a smaller sample. Convergencetests for P11 and P21 are similar. The convergence of scatteringcoefficient κs is also shown in Fig. 8. The scattering coefficientsare κs = 6.0029 for L = 32 mm and κs = 5.9539 for L =64 mm. They are quite close to each other. We can see that

the scattering coefficient converges faster against the realizationnumber when the sample size is doubled.

The convergence test with increasing number of realization(Nr) is shown in Fig. 9. The discretization and sample sizesare d = 0.5 mm and L = 64 mm, respectively. The phasematrices P22 and P12 computed from 30, 40, and 50 samplesare plotted and compared as a function of scattering angles.

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Fig. 9. Convergences of phase matrices with respect to the number of realization (Nr) for the bicontinuous system with fv = 0.3 and α = 0.371. Thepermittivity is εp = 3.2ε0. The discretization and sample sizes are d = 0.5 mm and L = 64 mm, respectively. The frequency is 18.7 GHz. The wavenumberdistribution parameters are 〈ζ〉 = 1500 m−1 and b = 10.

The results show some variations for the phase matrices withregard to the number of realizations. This might be due tothe large amount of fluctuating fields in the dense randommedia. However, the overall convergence of the phase matrixis good. The convergence of the scattering coefficient withrespect to the number of realization is shown in Fig. 8. Similarconvergences for the other frequency and microstructure arealso obtained. The scattering coefficients computed using theNMM3D approach are summarized in Table II. It shows that,while with the same fractional volume, the medium with thewavenumber distribution 〈ζ〉 = 1500 m−1 and b = 10 gives astronger scattering than the medium with 〈ζ〉 = 3000 m−1 andb = 100. This is because the medium with a smaller z tends tohave a broad size distribution of inhomogeneities.

3) Frequency Dependence of Scattering: It is noted fromTable II that the scattering coefficients are a function of both mi-crostructure and frequency. To study the frequency dependenceof scattering by bicontinuous structures, we define a power lawsuch that (κs(at f2)/κs(at f1)) = (f2/f1)n, where f1 and f2

are the two frequencies. The frequency-dependence index n iscomputed from [2], [10]–[12]

n =log

[κs(at f2)κs(at f1)

]log

(f2f1

) . (32)

We use the scattering coefficients at 18.7 GHz as reference.The frequency dependences are n = 3.4197, for the mediumwith 〈ζ〉 = 3000 m−1 and b = 100, and n = 1.7314, for themedium with 〈ζ〉 = 1500 m−1 and b = 10. We note that theindex n can become much smaller than four (for independentscattering) for the bicontinuous structure with a broader sizedistribution. The frequency-dependence index is about 2.5 byusing the Foldy–Lax multiple scattering approach [12]. Thefrequency dependences based on the Born method are alsogiven in Table II. However, the Born approximation predicts

a much higher frequency dependence of scattering than theNMM3D method.

4) Normalized Co- and Cross-Polarization Phase Matrices:In Figs. 10 and 11, we illustrate the normalized phase matricesat two frequencies for bicontinuous systems with εp = 3.2ε0.Figs. 10(a) and 11(a) show the structure with 〈ζ〉 = 3000 m−1

and b = 100 at 18.7 and 37 GHz, respectively. For thismicrostructure, both co- and cross-polarization phase matricespredict a higher peak at the forward scattering direction.Moreover, the normalized forward scattering increases withincreasing frequency. Similar scattering predictions have beenshown in the phase matrices based on the QCA dense-mediumapproach [12]. However, the NMM3D simulations givenonzero cross-polarization returns that are approximatelyone-tenth of the copolarization intensities. On the contrary,the scattering by the microstructure with 〈ζ〉 = 1500 m−1

and b = 10, as shown in Figs. 10(b) and 11(b), predicts apeak at the backscattering direction in the copolarizationphase matrices. Moreover, the normalized backward scatteringdecreases with increasing frequency. For the higher frequencycase, the cross-polarization becomes comparable with thecopolarization intensity. These scattering behaviors are verydifferent from the phase matrices based on the QCA and Mieapproaches.

Figs. 12 and 13 show the plots of the ratio (ρ) of the cross- tothe copolarization phase matrix as a function of the scatteringangle. From these results, we observe that the ratio P12/P22 islarger than the ratio P21/P11 for most of the scattering angles.The results also show that the ratio P12/P22 may be largerthan one at some scattering directions. For the structure with〈ζ〉 = 3000 m−1 and b = 100, the ratios at the backscatteringdirection are ρ = 0.106 at 18.7 GHz and ρ = 0.157 at 37 GHz.For the system with 〈ζ〉 = 1500 m−1 and b = 10, the ratiosat the backscattering direction are ρ = 0.041 at 18.7 GHz andρ = 0.274 at 37 GHz, which are a fairly large cross-polarizationbackscattering return.

DING et al.: EM SCATTERING BY BICONTINUOUS RANDOM MICROSTRUCTURES 3149

Fig. 10. Normalized phase matrices for the bicontinuous systems with fv = 0.3 and α = 0.371. The permittivity is εp = 3.2ε0. The frequency is 18.7 GHz.The wavenumber distribution parameters are (a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

Fig. 11. Normalized phase matrices for the bicontinuous systems with fv = 0.3 and α = 0.371. The permittivity is εp = 3.2ε0. The frequency is 37 GHz. Thewavenumber distribution parameters are (a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

VI. CONCLUSION

In this paper, we have introduced the bicontinuous structureto model random media with discrete dielectric properties.However, the advantage of using sticky spherical particles inthe microwave snow model is that fewer number of modelparameters are needed. The QCA/DMRT model with singlegrain size has been applied in multichannel active and passive

snow remote sensing [7], [8]. We have demonstrated that, as-suming the polydispersity of the wavenumber, the bicontinuousstructure model can be applied to describe a variety of randommedia with complex microstructures. We have presented boththe analytical Born approximation and the NMM3D simulationresults of electromagnetic scattering by bicontinuous randommedia. The analytical Born method is based on the analyticalcorrelation functions of bicontinuous media. The numerical

3150 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 8, AUGUST 2010

Fig. 12. Ratio of cross- and copolarization phase matrices for the bicontinuous systems with fv = 0.3 and α = 0.371. The permittivity is εp = 3.2ε0. Thefrequency is 18.7 GHz. The wavenumber distribution parameters are (a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

Fig. 13. Ratio of cross- and copolarization phase matrices for the bicontinuous systems with fv = 0.3 and α = 0.371. The permittivity is εp = 3.2ε0. Thefrequency is 37 GHz. The wavenumber distribution parameters are (a) 〈ζ〉 = 3000 m−1 and b = 100 and (b) 〈ζ〉 = 1500 m−1 and b = 10.

approach is based on the NMM3D method that rigorouslysolves Maxwell’s equations. The electromagnetic scatteringresults are illustrated for the scattering coefficients, the co-and cross-polarization phase matrices, and their ratios. Thebistatic scattering behavior depends on the bicontinuous mi-crostructures. The backscattering is shown to be larger thanthe forward scattering for the bicontinuous structure with abroader size distribution. The analytical and NMM3D scat-

tering results are compared. In the case of low permittivitycontrast, the NMM3D method agrees with the analytical Bornapproach. However, they do not match for the higher con-trast case. In the NMM3D simulations, the scattering by thebicontinuous medium with a broader size distribution showsa much weaker frequency dependence than the medium witha more narrow size distribution. The frequency-dependenceindex can be lower than two. The NMM3D method also

DING et al.: EM SCATTERING BY BICONTINUOUS RANDOM MICROSTRUCTURES 3151

predicts fairly large cross-polarization returns which accountfor the asymmetric microstructures of bicontinuous media.Future work will apply the phase matrix and the scattering co-efficient in the DMRT theory for the microwave remote sensingof snow.

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[1] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave RemoteSensing. New York: Wiley-Interscience, 1985.

[2] L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electro-magnetic Waves. New York: Wiley-Interscience, 2001.

[3] L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves. NewYork: Wiley-Interscience, 2001.

[4] K. K. Tse, L. Tsang, C. H. Chan, and K.-H. Ding, “Multiple scatteringof waves by random distribution of particles for applications in lightscattering by metal nanoparticles,” in Light Scattering and NanoscaleSurface Roughness, A. Maradudin, Ed. New York: Springer-Verlag,2007, ch. 13, pp. 337–366.

[5] C.-T. Chen, B. Nijssen, J. Guo, L. Tsang, A. Wood, J.-N. Hwang, andD. P. Lettenmaier, “Passive microwave remote sensing of snow con-strained by hydrological simulations,” IEEE Trans. Geosci. Remote Sens.,vol. 39, no. 8, pp. 1744–1756, Aug. 2001.

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Kung-Hau Ding (S’85–M’85–SM’03) received theB.S. degree in physics from National Tsing HuaUniversity, Hsinchu, Taiwan, and the M.S. degrees inboth physics and electrical engineering and the Ph.D.degree in electrical engineering from the Universityof Washington, Seattle.

From 1989 to 1993, he was with the PhysicalScience Laboratory, New Mexico State University,Las Cruces. From 1993 to 1998, he was with theResearch Laboratory of Electronics, MassachusettsInstitute of Technology, Cambridge. In May 1998,

he joined the Air Force Research Laboratory, Hanscom Air Force Base, MA,where he is currently a Research Engineer with the Sensors Directorate. Hisresearch interests include wave propagation and scattering in random media,microwave remote sensing, and radar clutter characterization.

Xiaolan Xu received the B.Eng. degree in electricalengineering from Zhejiang University, Hangzhou,China. She is currently working toward the Ph.D.degree in the Department of Electrical Engineering,University of Washington, Seattle.

She is currently a Research Assistant with the Lab-oratory of Applications and Computations in Elec-tromagnetics and Optics, Department of ElectricalEngineering, University of Washington. Her researchinterests include theoretical and numerical studiesof electromagnetic wave propagation, scattering in

random media, and snow remote sensing.

Leung Tsang (S’73–M’75–SM’85–F’90) receivedthe B.S., M.S., and Ph.D. degrees from theMassachusetts Institute of Technology, Cambridge.

Between 2001 and 2004, while on leave, he wasthe Chair Professor and the Assistant Head with theDepartment of Electronic Engineering, City Univer-sity of Hong Kong, Kowloon, Hong Kong. He iscurrently a Professor and the Associate Chairmanfor Education with the Department of Electrical En-gineering, University of Washington, Seattle, wherehe has been teaching since 1983. He is the author of

four books, namely, Theory of Microwave Remote Sensing (Wiley-Interscience,1985), Scattering of Electromagnetic Waves, Vol. 1: Theory and Applica-tions (Wiley-Interscience, 2000), Scattering of Electromagnetic Waves, Vol. 2:Numerical Simulations (Wiley-Interscience, 2001), and Scattering of Elec-tromagnetic Waves, Vol. 3: Advanced Topics (Wiley-Interscience, 2001). Hiscurrent research interests include remote sensing and geoscience applications,multiple scattering of waves, signal integrity, computational electromagnetics,and wireless communications.

Dr. Tsang is a Fellow of the Optical Society of America. He has been thePresident of the IEEE Geoscience and Remote Sensing Society (GRSS) sinceJanuary 2006. He was the Editor-in-Chief of the IEEE TRANSACTIONS ON

GEOSCIENCE AND REMOTE SENSING from 1996 to 2001. He was the GeneralChairman of the 1998 IEEE International Geoscience and Remote SensingSymposium. He was the recipient of the IEEE GRSS Outstanding ServiceAward in 2000. He was also the recipient of the IEEE Third Millennium Medalin 2000.