An electrothermodynamic model with distributed properties for ...

Radio Science, Volume 31, Number 2, Pages297-311, March-April 1996

An electrothermodynamicmodel with distributed properties for effective permittivities of sea ice S. V. Nghiem and R. Kwok Jet PropulsionLaboratory, CaliforniaInstitute of Technology,Pasadena

J. A. Kong and R. T. Shin Departmentof ElectricalEngineeringand ComputerScienceand ResearchLaboratoryof Electronics MassachusettsInstitute of Technology, Cambridge S. A. Arcone

and A. J. Gow

U.S. Army Cold RegionsResearchand EngineeringLaboratory, Hanover, New Hampshire

Abstract. This paper presentsa model to calculatethe temperaturedependenceof effectivepermittivitiesfor sea ice, a heterogeneousmedium containingmultiphase scatterers.With the strongpermittivity fluctuationapproachthe model accountsfor the electrodynamicscatteringeffect togetherwith the quasi-staticcharacteristicsof multiple speciesand subspeciesof inhomogeneities with distributedorientations,sizes, and shapes.Becauseof a preferentialdirectionin the orientationdistribution,the mediumis effectivelyanisotropic.The size distributionis describedwith a probability densityfunction in terms of normalizedvolumetric sizes. Scatterershapesare nonuniformand have a generalellipsoidalform characterizedby arbitrary axial ratios of correlationlengthswhich are related to physicalgeometriesof the scatterers.In this formulation, sea ice consistingof solid ice, liquid brine, and gaseousinclusionsis modeledto derive effectivepermittivitieswith thermodynamicphaseredistributionand structuralmetamorphism.Theoreticalresultsare in good agreementwith experimental data at the C band frequencyof 4.8 GHz for salineice undergoingwarming and cooling cycles. A competitiveeffect betweenthe increaseof liquid brine and the shape roundingof ellipsoidalscatterersat increasingtemperaturesexplainsthe trend observedin measureddata. Sensitivitiesof effective permittivitiesto structural and physicalparameterscharacterizingsea ice are also studied. 1.

Introduction

Most natural media are heterogeneousmixtures of materials with different phases, orientations, sizes, and shapes. Physical and structural characteristics of medium constituentsare usually interdependentand influencedby environmentalconditions such as ambient temperature, solar radiation, moisture, and precipitation. The compositemedia can be characterized by effective permittivities, which determine electromagneticwave propagation, attenuation, scattering, and emission. These processesare essentialto the interpretationof remote sensingdata of geophysicalmedia. Effective permittivitiesthemselves,however, alsooften dem-

onstrate complex environmental dependencies, such as thermal variations. This paper presents a permittivity model to explain these effects and illustrate the resultswith an applicationto sea ice. Permittivities of mixtures have long been a subject of extensive study. Many dielectric mixing formulashave been derived for multiphaseinhomogeneitieswith spherical,spheroidal,and ellipsoidal shapes.Together with a summaryof these formulas, Tinga et al. [1973] reported a formula for confocal ellipsoidal shell inclusionsand compared

Copyright1996by the American GeophysicalUnion. Paper number 95RS03429. 0048-6604/96/95RS-03429508.00 297

calculated

results with measured

data for wet wood.

In modeling of heterogeneousearth, Wait [1983] indicatedthat the particle shapeis importantto the effective electrical properties. A more complete account of electrical properties of the earth has been documented[Wait, 1989],includinga modelof coated particlesfor disseminatedsulphidemineralization. With applicationsto snow and sea ice,

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PERMITTIVITIES

Sihvola and Kong [1988]obtaineda generalmixing formula under the quasi-static condition for multiphasemixtures of scatteringellipsoids. In addition to the quasi-staticterm, a scattering term appears in the effective permittivity expressionderivedfrom the strongpermittivityfluctuation theory [Tsang and Kong, 1981a]. In this approach the quasi-staticabsorptionloss and the dispersive scatteringloss are both considered.The theory was applied to vegetation [Tsang and Kong, 1981b], snow [Tsang et al., 1982], and sea ice [Stogryn, 1987]. This approachhas been extendedto model a random medium containinguniform ellipsoidswith orientation distributions[Nghiem et al., 1993] and an isotropicheterogeneousmediumcontainingmultiple speciesof uniform size scatterers[Nghiem et al., 1995b]. In this paper a model is developedto accountfor the complexity of multiphasemixtures with multiple speciesand subspeciesof scattererscharacterized by orientation, size, and shapesdistributions. Earlier work only considers a single species of scatterer [Nghiem et al., 1993, 1995a] or multiple species only for effectively isotropic media [Nghiem et al., 1995b]without size and shapedistributions, and thermal processes in sea ice are not accounted for. With the strong fluctuation approach, derived effectivepermittivitiesare frequency-dependent,and the inherentdispersiveproperty of random media caused by scattering effects is included. The orientation distributionhas a preferential alignment direction rendering the medium effectively anisotropic.The size distributionis described in terms of number density or fractional volume

as a function

of normalized

volumetric

size.

The shape distribution affects electromagnetic propertiesof the mixture in a nonlinearmanner. To depict various shapes of scatterers in a species, scattererswith a similar shape are classifiedinto a group treated as a subspecies. The theory is used to obtain effectivepermittivities of multiphasecongelationsea ice with a columnar structure. In this formulation, physical and structural

characteristics

of the

medium

can

be

modeled more realistically. Moreover, the model allows physical and morphological variations in medium properties that are interrelated and subordinated to physical processes,such as constituent phase redistribution and structural metamorphism under

thermal

effects.

Calculated

results

are com-

pared with experimentaldata at 4.8 GHz for multi-

OF SEA

ICE

phase saline ice undergoingwarming and cooling cycles. 2.

Model

for Effective

Permittivities

In this sectionthe strongpermittivity fluctuation approachis used to formulate and derive effective permittivitiesof a heterogeneousmedium containing scattererswith different phases,permittivities, and distributed structural properties. The effective permittivities are obtained by combining the method for an inhomogeneousanisotropicmedium [Nghiem et al., 1993]with the method for a multispeciesmixture [Nghiem et al., 1995b]. 2.1.

Formulation

for a Distributed

HeterogeneousMedium

The effective permittivity tensor of the medium consistsof a quasi-staticpart and a scatteringpart correspondingto the first and secondterms, respectively, in the following expression [Tsang and Kong, 1981a]:

a•a= as+ •0[()- •a). (g)]-•-•a

(1)

where • istheunitd_yad. In (1),gaistheauxiliary permittivitytensor,S is the dyadiccoefficientof the

Diracdeltapartinthedya__dic Green's function ofan anisotropic medium,and•e• is theeffectivedyadic scatterer. Before deriving these quantitieswe need to describe the medium and define characterizing parameters.

Figure 1 illustratesthe modelingof a multiphase medium with multiple species and subspeciesof scattererssuchas sea ice. First, definea speciesas a set of scattererswith the samepermittivity. In (1),

gaand• aresolved iteratively fromasetofnonlinear coupled equations;these quantitiesdepend on ratios of correlation lengths, that is, scatterer shapes[Nghiem et al., 1995b].Thus it is necessary to classify scatterers in each species into groups consideredas subspecies;each subspeciescontains scatterersof similar shape. In this context a subspeciesis a set of scattererswith the same permittivity and the same shape. The shapes have a general ellipsoidalform with arbitrary axial ratios including spheroidsand spheresas special cases.

Whilegadepends onlyoncorrelation lengthratios, the scatteringpart in (1) does dependon the scatterer size. Size variations of scatterersin a species are accounted for with an integration over the scatterer

size distribution

scatteringpart.

in the calculation

of the

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Let % be the permittivity of a backgroundmedium hostinga total numberof N scatteringsubspecies. Permittivity e/sis for subspeciesi occupyinga fractional volume f/s in the mixture. The total

fractional volumeof allscatterers is y/N=• f/s ----1 -fo where fo is the fractional volume of the background. Note that a single subscript i is used to denotesubspeciesin all species(this simplifiesthe arrangementof subspeciesand speciesinto a single array) and that e/shas the samevalue for scatterers in the same species.

Theeffective dyadic scatterer, •eff,oftheheterogeneousanisotropic medium is given by the following expression [Tsang and Kong, 1981a; Nghiem et al., 1993, 1995b]:

Figure 2. Eulerian rotation anglesa,/3, and 3'between localcoordinates (x', y', z') andglobalcoordinates(x,y, z).

[•eff]jrn N

which isvalidunder thecondition I[Lff()]jml