applications of dense media radiative transfer theory for passive microwave remote sensing of foam...

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 5, MAY 2001 1019

Applications of Dense Media Radiative TransferTheory for Passive Microwave Remote Sensing of

Foam Covered OceanJianjun Guo, Leung Tsang, Fellow, IEEE, William Asher, Kung-Hau Ding, and Chi-Te Chen

Abstract—The effect of the foam covered ocean surface onthe passive microwave remote sensing measurements is studiedbased on the electromagnetic scattering theory. In fomulating anelectromagnetic scattering model, we treat the foam as denselypacked sticky air bubbles coated with thin seawater coating. Thelayer of foam covers the ocean surface that has air bubbles. Wethen use dense media radiative transfer (DMRT) theory withquasi-crystalline approximation (QCA) for densely distributedsticky moderate size particles to calculate the brightness tempera-tures of the foam-covered ocean surface. Results are illustrated for19 GHz and 37 GHz and for both vertical and horizontal polar-izations as a function of foam microstructure properties and foamlayer thickness. Comparisons are also made with experimentalmeasurements.

Index Terms—Dense media radiative transfer, electromagneticwave scattering, microwave emissivity.

I. INTRODUCTION

T O estimate the effect of the foam above the ocean surfaceon the passive microwave remote sensing measurements,

various empirical microwave emissivity models were used[1]–[6]. Williams [1] measured emissivities of foam in awaveguide and found that at X-band, the emissitivity of foamdepends strongly on the thickness of the foam layer. Wilheit’smodel [2] treats foam as having neither polarization nor viewingangle dependence. In Pandey’s empirical emissivity model[3], the effect of foam was taken into account by couplingtheoretical expressions of specular ocean surface emissivitywith empirical expressions from ocean tower observationsand from the analysis of published measurements. Smith [4]described the brightness temperature over foam-covered oceanas a function of incidence angle and frequency. He related theemissivities of foam at the three channels (vertical polarizationat 19 GHz and both polarizations at 37 GHz) to one anotherby linear regression. Stogryn [5] used a least squares fit ofa polynomial to the data of measurements and derived anexpression for the foam emissivity as a function of incidenceangle and frequency. All these models are empirical fittingprocedures using experimental data. The empirical models do

Manuscript received June 8, 1999; revised November 13, 2000. This workwas supported by the Office of Naval Research under Grant N00014-99-1-0190.

J. Guo, L. Tsang, and C.-T. Chen are with the Department of Electrical En-gineering, University of Washington, Seattle, WA 98195-2500 USA.

W. Asher is with the Applied Physics Laboratory, University of Washington,Seattle, WA 98195 USA.

K.-H. Ding is with the Air Force Research Lab, Sensors Directorate/SNHE,Hanscom AFB, MA 01731-2909 USA.

Publisher Item Identifier S 0196-2892(01)03830-X.

not take into account the physical microstructure propertiesof foam and the foam layer thickness.

It is known that foam on the ocean surface can affect thebrightness temperatures measured by microwave radiometers.However, there is relatively little known definitively concerninghow foam affects such fundamental parameters like the dragcoefficient or the impact of foam on the retrieval of theocean surface wind vector from satellite-mounted microwaveinstruments. This gap in knowledge is due in large part to thedifficulty in making measurements at high wind speeds, whensignificant foam coverage is present. Understanding how foamincreases microwave emissivity, and developing quantitativeemissivity models for a foam-covered water surface, wouldbe of great help in evaluating radiometer performance at highwind speeds.

The subject of foam dynamics has attracted attention recently.Controlled experiments are performed for foam dynamics andmicrowave emissivity measurements [6]. The microstructuresof foam are also studied [7]. In this paper, we apply the recentlydeveloped dense media radiative transfer (DMRT) theory [8]based on sticky particle model to analyze this problem, takinginto account the physical microstructure properties of foam andthe foam layer thickness.

We model the foam as densely packed sticky air bubblescoated with thin seawater coating. The model of medium withdensely distributed coated particles of moderate size can beused. In the past, both theoretical and experimental studies [9],[10] of the propagation and scattering of waves in dense mediashow that the assumption of independent scattering is not validand the correlated scattering effects and coherent near fieldinteraction effects must be taken into account.

The quasi-crystalline approximation (QCA) accounts forthe pair distribution function of the particle positions andcoherent wave interactions [11]. The Percus–Yevick equation,which compares well with the Monte Carlo simulations [12],is used to describe the pair distribution functions [11]. Thescattering results of QCA also compare well with the MonteCarlo simulations of exact solutions of Maxwell’s equations[10] of randomly distributed finite size spheres. Recently, wehave extended the dense media theory to include cases whenthe particles have sticky force that make them adhere to formaggregates [8]. A key difference between sticky particle modeland nonsticky particle model is the varied frequency depen-dence of scattering by changing the stickiness parameter.

We will focus on the foam impact. By using the distortedBorn approximation with QCA, we calculate numerical results

0196–2892/01$10.00 ©2001 IEEE

1020 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 5, MAY 2001

of the complex effective propagation constants, scattering coef-ficients and phase matrix, of densely packed air bubbles coatedwith sea water. The brightness temperature of foam-coveredocean surface can be simulated. We describe the dynamic prop-erties of foam in Section II. In Section III, the QCA theory fordensely packed coated particles of moderate sizes is presented.For particles of moderate size, the QCA equations are formu-lated in terms of the -matrix formulism and utilizing vectorspherical waves as basis functions. Also, we discuss the pairdistribution functions of sticky particles of multiple sizes basedon the Percus–Yevick pair distribution functions. In Section IV,the DMRT theory is discussed. Derived from QCA, the DMRTtheory describes the scattering, absorption and emission of adense medium and relates the physical parameters of air bub-bles coated with sea water in foam, such as the size, density andcoating thickness, and layer thickness of the foam, to the bright-ness temperatures. The theoretical results of brightness temper-atures with typical parameters of foam in passive remote sensingat 19 GHz and 37 GHz are illustrated numerically in Section V.Comparisons are also made with experimental measurementsfor vertical and horizontal polarizations [6].

II. DYNAMIC DESCRIPTION OFFOAM

In order to model the microwave radiometric properties offoam, it is critical to have an accurate model for the physicalstructure of the bubbles therein. Foam on a water surface is adynamic system, and the defining physical properties such asmean bubble size and liquid water content are functions of theage of the foam. In general, the mean bubble size increases withage as bubbles coalesce and the water content decreases as waterdrains from the interstitial spaces [7]. In addition, the shapeof the bubbles changes from nearly spherical for very youngfoams with relatively large liquid water content to polyhedronsfor aged foams with small amounts of water in the interstitialspaces. Modeling the radiometric properties of foam is greatlysimplified if it can be assumed that foam on the ocean surfaceis composed of nearly spherical bubbles. It is also necessary tohave realistic values for the foam void fraction (i.e., the ratioof the volume of air to the total volume of the foam) and meanbubble size.

Fig. 1 is a video photograph of the bubble structure of foamon the ocean surface. The image was acquired using a partiallysubmerged underwater CCD video camera equipped with amacro telecentric lens whose focal plane lay precisely at thefront window of the camera housing. The overall thickness ofthe foam layer was on the order of 5 cm and it is estimatedthat this image was acquired in the bottom half of the foamlayer. Because the foam is formed from bubbles rising to thesurface from below and the lifetime of the foam layer wason the order of 4 s, this image is therefore of foam having amean age of under 2 s. In agreement with previous data on thebubble structure of young water-based foams [7], the majorityof bubbles are nearly spherical and the average thickness of theinterstitial water layer between bubbles in the foam is seen tobe on the order of 80m. Furthermore, digital image analysisof 20 similar foam images showed that the mode of the bubblesize distribution is approximately 500m. Based on these

Fig. 1. Video microphotograph of bubble structure in foam on the surface ofthe ocean.

Fig. 2. Spherical dielectric coated particle.b anda are the inner and outerradii of spherical shells, respectively." and" are permittivities of thematerials in the core and within the shell, respectively.

numbers, the void fraction of the foam is on the order of 0.75(i.e., 75% of the total volume of the foam is air). These valueswere used in formulating the physical model for the foam usedin the electromagnetic scattering calculations.

III. QUASI-CRYSTALLINE APPROXIMATION FORMULATION FOR

MEDIA WITH MODERATE SIZE COATED PARTICLES

Consider a distribution of coated particles centered atin a volume . We assume that there are

species of coated particles in the medium. The physical andgeometric structure of a spherical dielectric coated particle isshown in Fig. 2, where denotes that the coated particle is ofthe th species. The random distributed coated particles neednot be identical in size, permittivity and the thickness of thecoating. However, interpenetration of particles is not allowed.Let the incident wave impinging in the direction of. TheFoldy-Lax multiple scattering equation can be written in matrixform as

(1)

GUO et al.: DENSE MEDIA RADIATIVE TRANSFER THEORY 1021

whereand exciting field coefficients of the th particle

and the th particle, respectively;

-matrix of the th particle;incident field coefficient.

Equation (1) can be interpreted as the field exciting theth par-ticle is the sum of incident wave and the scattered wave fromall other particles. The matrix is the matrix that de-scribes the vector translation formula transforming sphericalwaves centered at to spherical waves centered at. By takingthe conditional average by holding positionfixed, we have

(2)

whereconditional average given the position and state ofthe th particle;conditional probability of finding theth particle at

given the th particle at ;conditional average given the properties and posi-tions of particles and .

Based on the QCA

(3)

we can obtain the averaged exciting field as follows:

(4)

wherenumber of particles per unit volume of thespecies;conditional average of the exciting field of parti-cles at of species ;

matrix representation of the matrix of species;

cross pair distribution function of two speciesand ;

column matrix representing the coefficient ofthe incident wave when expanded into sphericalwaves.

The integral equation in (4) can be solved by using vectorspherical wave expansions. They lead into two sets of equations,the Lorentz–Lorenz law and the Ewald–Oseen extinction the-orem [13].

The conditional averaged exciting field coefficients obey thefollowing homogeneous system of equations (Lorentz–Lorenzlaw). Let be expanded into vector sphericalwaves with and as coefficients where

and represents the and vectorspherical waves, respectively

(5)

(6)

In (5) and (6)

(7)

where and are coefficients defined in [11,p. 496], and are terms ofWigner symbols defined in [11, pp. 449–450];is the effective propagation constant; and are sphericalBessel function and its derivative; and are spherical Hankel

function and its derivative. The scattering coefficients andfor coated spheres ofth species with outer radius ,

inner radius , core wavenumber and shell wavenumber, are those of Mie scattering-matrix coefficients expressed

by [14] in (8) and (9), shown at the bottom of the page,

(8)

(9)


where

(10)

(11)

and is the spherical Neumann function. The symbols,, , and are defined as , ,

, and , respectively.The ratios of exciting field coefficients and

are determined by setting the determinant of (5)and (6) equal to zero. The determinant equation gives thesolution for the effective propagation constant. They canbe solved numerically by using Muller’s method to searchthe root . Therefore, the value of andcan be determined within a single constant. The effectivepropagation constant is complex, , and theextinction rate is . In principle, both andcan be computed by the Lorentz–Lorenz law. However, since

, calculated in this manner may not have thedesired accuracy. Also, the imaginary part is affected by thesticky particle pair distribution functions that have sharp peaksthat may be difficult to evaluate numerically. To ensure energyconservation, we proceed as follows. We use the real partascalculated from (5) and (6) of Lorentz–Lorenz law. We also usethe relative values of ’s as determined by (5) and (6). Forthe single constant, it is determined by Ewald–Oseen extinctiontheorem. The generalized Ewald–Oseen extinction theoremis obtained by balancing the incident wave term and the termof the same phase dependence that is a result of the integralin (4). We then use the coherent exciting field to calculatethe absorption coefficient. The distorted Born approximationis applied to calculate the phase matrix elements and thescattering coefficient. The phase matrix is the bistatic crosssection per unit volume. In passive remote sensing, we furtherintegrate over the azimuthal angle

(12)

where , , and is given by [8, eqs. (37)–(40)].Then, the extinction coefficient is calculated by the addition ofabsorption and scattering coefficients.

By employing the coherent exciting field on a particleas solved by QCA, we can obtain the reflection

coefficient which is given in [8] as

(13)

whereassociated Legendre polynomial ofth order anddegree 1;Legendre polynomial;

and incident and transmitted angles, respectively.And then we can get the absorption coefficient as follows:

(14)

The scattering coefficient can be calculated by the following

(15)

The extinction coefficient is and the Albedois . The effective propagation constant is

.Physically, the air bubbles in the foam layer adhere to each

other, which is shown in Fig. 1. To simulate the foam’s effect,we use the sticky particle model in which the particles are al-lowed to adhere together to form clusters to better represent thephysical property of the foam. The sticky particle has a sticki-ness parameter. The smaller the is, the more sticky the parti-cles are. The Percus–Yevick approximation of the pair distribu-tion function for the sticky spherical particles can be solvedanalytically using the factorization method of Baxter [15]. Thecalculations of are given by equations in [16]. In the fol-lowing sections, we use particles of identical size but with dif-ferent thicknesses of coating.

IV. DENSEMEDIA RADIATIVE TRANSFERTHEORY

Consider thermal emission from a layered medium withcoated dielectric particles embedded in a background medium,as indicated in Fig. 3. The layer consisting of coated particles(in region 1) covers a half space of ocean (region 2) embeddedwith air bubbles. The radiative transfer (RT) equations forpassive remote sensing in region 1 are in the following form:

(16)

(17)


Fig. 3. Geometrical configuration for thermal emission from foam covered ocean. The foam layer is region 1 and is absoptive and scattering. Region 2 isairbubbles embedded in sea water and is absorptive.

whereequal to ;Boltzman’s constant;

, vertical and horizontal specific intensities, respec-tively.

The boundary conditions are the Fresnel type using the effec-tive propagation constant for the dense medium. For the upperboundary, the air/region 1 interface at

(18)

(19)

where and are the Fresnel reflectivities with effectivepropagation constant

(20)

(21)

In the lower boundary, the region 1 and region 2 interface,

(22)

(23)

with (24) and (25), shown at the bottom of the page, whereis the permittivity of the media in region 2. For the case of foamcovered ocean, region 2 consists of air bubbles embedded in theocean background so that we will use the effective permittivity

of medium 2 in this set of equations, instead of. Region 2is assumed to be absorptive. To calculate , a simple physicalmodel based on induced dipoles is used [13]. Letdenote thepermittivity of the seawater, the fractional volume occupiedby the air bubbles. Then the effective permittivity , is givenby the Maxwell-Garnett mixing formula

(26)

where

(27)

Note that the effective permittivity here does not includescattering attenuation which is small due to the fact that the sea-water is heavily absorptive.

After solving the eigenvalue problem and imposing on theboundary conditions, the brightness temperatures in the direc-tion , where is related to by Snells’law, are given by

(28)

(29)

V. NUMERICAL SIMULATIONS OF BRIGHTNESSTEMPERATURE

AND COMPARISON WITHEXPERIMENTAL MEASUREMENTS

In the following, we illustrate the numerical results of thebrightness temperatures based on the sticky particle model. Theeffective propagation constant, scattering rate, extinction rate,


albedo, and the phase functions are first calculated by QCA.Consequently, these parameters are put into the RT equationsto compute the brightness temperatures.

As described in the foregoing sections, the foam is modeledas densely packed sticky air bubbles with thin seawater coatingand the background medium is air. In this model, region 0 isair, region 1 is foam layer consists of coated air bubbles em-bedded in a background of air, and region 2 consists of air bub-bles and ocean as its background. The geometrical configurationis shown in Fig. 3.

We define the foam parameters as follows:foam layer thickness;outer radius of coated air bubbles of theth species;inner radius of coated air bubbles of theth species;fractional volume of coated air bubbles of theth species

(30)

total fractional volume of air bubbles

(31)

number of species, we choose two in the numerical simu-lations;fractional volume of sea water in foam

(32)

stickiness parameter for the sticky model;permittivity of seawater;the temperature of the foam;observation angle;fractional volume of air bubble in medium 2 (used in (26)to calculate ).

Note that is the thickness of coating of theth species.And we assume that the permittivity of the sea water in this layeris the same as that in foam, also they have the same temperature.

According to the microstructure of foam, is about 4%.The diameter of coated air bubbles ranges from 200 mto several millimeters. The thickness of the coating of the air

Fig. 4. Absorption coefficient as a function of the ratio of the inner radiusb tothe outer radiusa of the coated air bubble.

bubbles in foam varies. The foam extends to a layer thicknessranging from several millimeters to several centimeters. Note

that the permittivity of seawater is a function of the frequencyand other physical parameters such as the temperatureandsalinity. To calculate the permittivity of seawater at microwaveremote sensing frequencies, we set the temperature of 284 Kand salinity of 20 per thousand, and the model of Klein andSwift [17] is applied. Based on this model, the permittivity ofthe sea water at 19 GHz and 37 GHz areand , respectively.

First, we present the absorption coefficient dependence on thecoating thickness for several in Fig. 4. It is clear that, withthe same water fractional volume , the absorption coefficient

for air bubbles with thinner coating is much larger than thatof bubbles with thicker coating. Also, while keeping the coatingthickness fixed, the absorption coefficient will increase with theincrease of water fractional volume. On the other hand, we cansee that the coated air bubble is more absorptive at 37 GHz thanthat at 19 GHz.

The following results show the brightness temperatures asa function of the observation angle, the thickness of the foamlayer, and the size of the coated air bubbles in the foam layer.All the parameters for the simulation results shown in Figs. 5–8are listed in Table I. For simplicity, we have used two speciesof coated air bubbles having identical size but with different

(24)

(25)


Fig. 5. Brightness temperature as a function of observation angle. Thediameter of the coated air bubble is 500�m.

Fig. 6. Brightness temperature as a function of the thickness of the foam layerat observation angle 53. The diameter of the coated air bubble is 500�m.

coating thickness and fractional volume, namely thicker coatedone with small fractional volume and thinner coated one withlarger fractional volume, in the foam description.

Fig. 5 shows the calculated brightness temperatures as a func-tion of observation angle at 19 GHz and 37 GHz, for both hor-izontal polarization and vertical polarization, respectively. InFigs. 6–8, we plot the brightness temperatures of horizontal po-larization and vertical polarization, at 19 GHz and 37 GHz also,as a function of thickness of the foam layer with different size ofthe coated air bubbles. The diameters of air bubbles are 500m,500 m, 1000 m, and 1500 m in Figs. 4–7, respectively. Asthe size of bubbles increases the scattering effects increases andthe albedo also increases, which is reflected by the decrease ofthe corresponding saturated brightness temperatures. The ob-servation angle is 53for Figs. 5–7. Fig. 5 shows the polariza-tion dependence as a function of viewing angles. We note that19 GHz has a stronger polarization dependence than 37 GHz.The polarization difference are respectively 43 K and 32 K at



TABLE IPARAMETERS FOR THESIMULATED RESULTSSHOWN IN FIGS. 5–8

53 observation angle. Fig. 6 indicates that, once the brightnesstemperatures at 19 GHz increase to the values of that at 37 GHz,then saturations for 19 GHz take place. The frequency depen-dence is weak after saturation. The polarization dependence isalso weak for a large thickness of foam layer of small coatedair bubbles. From these three figures, we can conclude that as


TABLE IINUMERICAL RESULTS FROM THEQCA-STICKY SIMULATION FOR FIGS. 5 AND 6

TABLE IIINUMERICAL RESULTS FROM THEQCA-STICKY SIMULATION FOR FIG. 7

TABLE IVNUMERICAL RESULTS FROM THEQCA-STICKY SIMULATION FOR FIG. 8

the thickness of the foam layer increases, the brightness tem-peratures of all four channels will increase correspondingly andthen saturate at particular thickness of the foam layer. For ver-tical polarization, the saturation at 19 GHz occurs at 21.9 cm,16.4 cm and 15.9 cm, for the three cases of coated air bubble sizeof 500 m, 1000 m, and 1500 m, respectively. At 37 GHz,saturation of vertical polarization occurs at smaller layer thick-ness of 8.3 cm, 4.5 cm, and 4.0 cm, respectively. The saturationpoint of horizontal polarization is slightly different from thatof vertical polarization. For layer thickness larger than satura-tion thickness, the difference between brightness temperaturesof vertical polarization and horizontal polarization is a functionof the size of coated air bubbles. The extinction rates, effectivewavenumbers and albedo calculated from QCA for these casesare shown in Tables II–IV, respectively. It can be seen that theseawater is absorptive and absorption dominates over the extinc-tion rates for small particle sizes. However, as the particle sizeincreases, the albedo increases with the size of coated air bub-bles. In these tables, the effective permittivity of region 1 isdefined as , where is the effective propagationconstant and is the wave number in free space.

Next, in Fig. 9 we compare the simulation results with the ex-perimental measurements in [6]. In the experiment, brightnesstemperatures of vertical polarization and horizontal polarizationwere measured at 19 GHz. The incidence angle was set at 53.To reduce noise, Asheret al. [6] averaged the time series ofbrightness temperature for successive bubble plumes by usingthe tipping bucket to generate reproducible bubble plumes. Fol-lowing the computational procedure described in the foregoingsections, we simulated the corresponding time series of bright-ness temperature. The experimental observation indicates thatthe brightness temperatures increases rapidly with a time in-terval of 1 s and then gradually decreases to the ocean values

Fig. 9. Comparison between the simulation results and the measurements forthe time series of the ocean surface microwave brightness temperature at 19GHz for horizontal polarization and vertical polarization, respectively.

of brightness temperatures in about 8 s as the foam dissipates.The thickness of the foam layer increases from 0 to 2 cm, andthen down to 0 which ends the simulation. The parameters forthis figure are listed below.

increasing from 0 to 2 cm very quickly and then de-creasing to 0 exponentially;400 m;200 m–395 m;50%;4%;increasing from 0 to 50% very quickly and then down to0;0.1;28.9541 i36.8340;284 K.

From Fig. 9, we see that the simulation results compare wellwith the experimental measurements. At a single time instant,the simulated brightness temperatures of vertical polarizationand horizontal polarization agree well with the experimentalmeasurements. Moreover, from the point of view of time series,the simulated brightness temperatures agree well with those ofmeasurements. Also, the polarization dependence for the timeseries (corresponding to different thickness of foam layer) iswell predicted by the sticky particle model.

VI. CONCLUSIONS

By formulating an electromagnetic scattering model of foam,we apply electromagnetic scattering theory and dense media ra-diative transfer theory to analyze the effects of foam on the pas-sive microwave remote sensing measurements rigorously. Wemodel the foam as densely packed sticky air bubbles coated withthin seawater coating. The DMRT for moderate size sticky par-ticles based on QCA is applied to calculate the brightness tem-peratures of the foam-covered ocean surface. Numerical simu-lations show that the polarization and frequency dependencesof brightness temperatures on microstructure properties such asfoam layer thickness, foam air bubble size, coating thickness


and coated bubbles fractional volume. Simulation results are ingood agreement with experimental measurements.

REFERENCES

[1] G. F. Williams, “Microwave Emissivity Measurements of Bubbles andFoam,” IEEE Trans. Geosci. Electron., vol. GE-9, pp. 221–224, July1971.

[2] T. T. Wilheit Jr, “A Model for the Microwave Emissivity of the Ocean’sSurface as a Function of Wind speed,”IEEE Trans. Geosci. Electron.,vol. GE-17, pp. 244–249, Apr. 1979.

[3] P. C. Pandey and R. K. Kakar, “An Empirical Microwave emissivityModel for a Foam-Covered Sea,”IEEE J. Oceanic Eng., vol. OE-7, no.3, pp. 135–140, 1982.

[4] P. M. Smith, “The Emissivity of Sea Foam at 19 and 37 GHz,”IEEETrans. Geosci. Remote Sensing, vol. 26, pp. 541–547, Sept. 1988.

[5] A. Stogryn, “The Emissivity of Sea Foam at microwave frequencies,”J.Geophys. Res., vol. 77, pp. 1658–1666, 1972.

[6] W. Asher, Q. Wang, E. C. Monahan, and P. M. Smith, “Estimation ofAir-Sea Gas Transfer Velocities from Apparent Microwave BrightnessTemperature,”Mar. Technol. Soc. J., vol. 32, no. 2, pp. 32–40, 1998.

[7] R. de Vries,Absorptive Bubble Separation Techniques, R. Lemlich,Ed. New York: Academic, 1972, p. 331.

[8] L. Tsang, C.-T. Chen, A. T. C. Chang, J. Guo, and K.-H. Ding, “DenseMedia Relative Transfer Based on Quasicrystalline Approximation withApplications to Passive Microwave Remote Sensing of Snow,”RadioSci., vol. 35, no. 3, pp. 731–749, 2000.

[9] A. Ishimaru and Y. Kuga, “Attenuation constant of coherent field in adense distribution of particles,”J. Opt. Soc. Amer., vol. 72, no. 10, pp.1317–20, 1982.

[10] L. Tsang, C. E. Mandt, and K.-H. Ding, “Monte Carlo simulations ofthe extinction rate of dense media with randomly distributed dielectricspheres based on solution of Maxwell’s equations,”Opt. Lett., vol. 17,no. 5, pp. 314–316, 1992.

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

[12] K.-H. Ding, C. Mandt, L. Tsang, and J. A. Kong, “Monte Carlo simula-tions of pair distribution functions of dense discrete random media withmultiple sizes of particles,”J. Electromagn. Waves Applicat., vol. 6, no.8, pp. 1015–1030, 1992.

[13] L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from adense medium consisting of correlated Mie scatterers with size distri-butions and applications to dry snow,”J. Electromagn. Waves Applicat.,vol. 6, no. 3, pp. 265–286, 1992.

[14] K. H. Ding and L. Tsang, “Effective Propagation Constants and Atten-uation Rates in Media of Densely Distributed Dielectric Particles withSize Distributions,”J. Electromagn. Waves Applicat., vol. 5, no. 2, pp.117–142, 1991.

[15] R. J. Baxter, “Ornstein-Zernike Relation and Percus–Yevick Approxi-mation for Fluid Mixtures,”J. Chem. Phys., vol. 52, pp. 4559–4562,1970.

[16] L. Zurk, “Electromagnetic wave propagation and scattering in dense,discrete random media with application to remote sensing of snow,”Ph.D. dissertation, Univ. Washington, Seattle, 1995.

[17] L. A. Klein and C. T. Swift, “An Improved Model for the Dielectric Con-stant of Sea Water at Microwave Frequencies,”IEEE Trans. AntennasPropagat., vol. AP-25, no. 1, pp. 104–111, 1977.

Jianjun Guo was born in Tianmen, Hubei Province,China. He received the B.S. degree in electrical engi-neering from Wuhan University, China, in 1995 andthe M.S. degree in electrical engineering from theUniversity of Science and Technology of China in1998. He is currently pursuing the Ph.D. degree withthe Department of Electrical Engineering, Universityof Washington, Seattle.

Leung Tsang (S’73–M’75–SM’85–F’90) was bornin Hong Kong. He received the S.B., S.M., E.E., andPh.D. degrees from the Massachusetts Institute ofTechnology, Cambridge.

He is currently a Professor of electrical en-gineering with the University of Washington,Seattle. He is co-author of the bookTheory ofMicrowave Remote Sensing(New York: Wiley,1985). His current research interests include wavepropagation in random media and rough surfaces,remote sensing, opto-electronics, and computational

electromagnetics.Dr. Tsang has been Editor-in-Chief of IEEE TRANSACTIONS ONGEOSCIENCE

AND REMOTE SENSINGsince 1996. He was the Technical Program Chairman ofthe 1994 IEEE Antennas and Propagation International Symposium and URSIRadio Science Meeting, the Technical Program Chairman of the 1995 Progressin Electromagnetics Research Symposium, and the General Chairman of the1998 IEEE International Geoscience and Remote Sensing Symposium. He is aFellow of the Optical Society of America and the recipient of the OutstandingService Award of the IEEE Geoscience and Remote Sensing Society for 2000.He is also the Recipient of the IEEE Third Millenium Medal.

William Asher received the B.A. degree in chem-istry from Reed College, Portland, OR, in 1980 andthe Ph.D. degree in environmental science and engi-neering from Oregon Graduate Institute for Scienceand Technology, Beaverton, in 1987.

He was a Senior Research Scientist at PacificNorthwest National Laboratory, Sequim, WA, forfive years and was then at the Joint Institute for theStudy of the Atmosphere and the Ocean, Universityof Washington, Seattle. He is currently a SeniorOceanographer at the Applied Physics Laboratory,

University of Washington.

Kung-Hau Ding, photograph and biography not available at the time of publi-cation.

Chi-Te Chen, photograph and biography not available at the time of publication.

applications of dense media radiative transfer theory for passive microwave remote sensing of foam...

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