EECS 730, Winter 2009 c© K. Sarabandi

Radiative Transfer Theory

1 Introduction

Previously the behavior of electromagnetic waves in random media was studied rigoruslyusing Maxwell’s equations. However, as discussed, these analytical solutions are onlyvalid for tenuous media and therfore have limited use for practical problems. For prob-lems where the medium is made up of discrete scatterers and the number density ofscatterers is low, the single scattering theory may be applied and the scattered fieldincluding multiple scattering from all scatterers may be added coherently. However,keeping track of all scatterers and scattering components is a very difficult task. Ra-diative transfer (RT) theory provides a heuristic approach that does this task in anefficient way. RT theory is based on the law of conservation of energy and makes useof the single scattering properties of scatterers in the medium. The radiative transfertheory was develped by Schuster in the study of light propagation in foggy atmosphere[3]. Later on this theory was used extensively by astrophysicists [1]. In recent years thistheory has received significant attention for its applications in number of microwave andmillimeter-wave active and passive remote sensing [4, 5, 6, 7].

2 Specific Intensity

One of the fundamental quantities in the radiative transfer (RT) theory is the specificintensity denoted by I(r, s) when r denotes the position at which the specific intensity isdefined and s denotes the direction of energy flow of the specific intensity at r. Since thephysical basis of RT theory is the law of the conservation of energy, the specific intensitymust be related to the flow of energy in a given direction at a point in the randommedium. However, to arrive at a simple form of the transport equations, the specificintensity is defined in terms of power density per unit solid angle per unit bandwidth[1]. With this defninition the propagation decay of spherical waves ( 1

R2 ) is embedded inthe definition of the specifice intensity, as will be shown later. For monochromatic wavesthe specific intensity is simply defined as the power density per unit solid angle. Thisis the definition we will adapt throughout the rest of this chapter. With this definition,the power emitted from a hypothetical unit area dA and flowing in direction s within asolid angle dΩ can be obtained from [2]

dP = I(r, s) cos θdAdΩ (1)

1

where θ is the angle between unit normal to dA and s is shown in Figure 1.

Figure 1: The power emitted from dA and flowing along s within solid angle dΩ is usedto define the specific intensity.

The unit of the specific intensity is Wm−2sr−1. We may also consider a hypotheticalarea dA in the random medium to intercept the specific intensity flowing in the medium.Consider a receiver with beamwidth dΩ and aperture dA, the power received is given by

dP = I(r, s) cos θdAdΩ (2)

as shown in Figure 2.

It is obvious from Figures 1 and 2 that the intensity entering a point and the intensityleaving that point is the same (in the absence of any scatterer in that point).

As mentioned earlier, the definition of the specific intensity is made in such a waythat it is spatially invariant in a homogeneous medium. To demonstrate this let usconsider the specific intensity in a homogeneous loss less medium at two points r1 andr2. Representing the specific intensity of r1 by I1(r1,s) and at r2 by I(r2, s) and twodifferential areas at each point by dA1 and dA2 perpendicular to s, the power receivedby dA2 according to (2) is given by

dP2 = I2dΩ2dA2. (3)

According to Figure 3 this power in terms of I1 is

dP2 = I1dΩ1dA1 (4)

Referring to Figure 3, dΩ1 = dA2

R2 and dΩ2 = dA1

R2 where R = |r1− r2|. Substituting thesein (3) and (4), it is obvious that I1(r1, s) = I2(r2, s).

In the definition of the specific intensity no reference to the polarization of electromag-netic waves has been made so far. For completely unpolarized waves (purely random

2

Figure 2: The power intercepted by dA4 from a solid angle dΩ along s.

Figure 3: The geometry solid angles and differential areas employed to show the spatialinvariance of specific intensity in homogenous and loss less medium.

3

polarization) the specific intensity is considered scalar and contains the total power. Inradar remote sensing where the source of the electromagnetic wave in a random mediumis completely polarized, the specific intensity can no longer be assumed to be scalar. Infact, as the wave propagats and scatters by the particles in the medium, energy transferbetween different polarization channels take place. Study of the content of each polariza-tion channel of the scattered intensity reveals the polarimetric behavior of the randommedium.

To keep track of the polarization state of the wave in a random medium a vector specificintensity is defined and denoted by I [6]. For an elliptically polarized monochromaticplane wave I is defined using the modified Stokes parameters Iυ, Ih, U, and V as follows:

I =

Iυ

Ih

UV

=1

Z0

|Eυ|2|Eh|2

2Re(EυE∗

h)2Im(EυE

∗

h)

δ(Ω − Ω0) (5)

where δ(.) represents Dirac delta function and Ω0 denotes the direction of propagation(φ0, θ0). In (5) Z0 is the characteristic impedance of the average or effective medium.

As the incident plane wave enters a random medium and encounters scatterers, part ofits power scatters into other directions and produces specific intensities propagating inall directions. The scattered fields from the inhomogeneities in the medium are sphericalwaves. For a spherical wave, the vector specific intensity also is defined in terms of Stokesparameters. Hence, the scattered intensity emanating from a random medium is definedas:

Is =

Isυ

Ish

Us

V s

=1

Z0∆Ωs

〈|Esυ|2〉

〈|Esh|2〉

2Re(〈EsυE

s∗h 〉)

2Im(〈EsυE

s∗h 〉)

(6)

where 〈 〉 denotes the ensemble average. For a distributed target with an illimunationarea A and for an observation point at (φs, θs) the solid angle ∆Ωs = A cos θs/r

2, wherer is the distance between the distributed target and the observation point. Here θs isthe angle between the outward normal to A and the vector defining the direction fromthe target to the obeservation point.

The bistatic scattering coefficient corresponding to a q-polarized incident plane wavewhich gives rise to a p-polarized spherical wave is defined as:

σ0pq =

4πr2

A· 〈|E

sp|2〉

|Eiq|2

, (7)

4

where p, q = υ or h polarization. Using the first two components of the incident andscattered specific intensities in (7), the bistatic scattering coefficient may be written as:

σ0pq(π − θ0, φ0; θs, φs) =

4π cos θsIsp(θs, φs)

I iq(π − θ0, φ0)

(8)

where (π − θ0, φ0) denotes the direction of the downward-going incident intensity and(θs, φs) denotes the direction of the upward-going scattered intensity. We note thatθs = θ0 and φs = π + φ0 corresponds to scattering in the backward direction.

3 Boundary Conditions for the Specific Intensities

At the interface between two dielectric media the boundary conditions for the electric andmagnetic fields are used to relate the field quantities in each region. Similar boundaryconditions are needed to relate the specific intensities reflected and transmitted at aplanar interface between two dielectric media. This can be obtained easily from theFresnel reflection and transmission coefficients. It should be noted that for random mediaproblems it is usually one of the two or both dielectric media that are inhomogenous.In this case an effective homogeneous dielectric for the random medium is substituted.As discussed previously the effective homogeneous medium supports a wave equal to themean-field in the random medium.

Consider two homogeneous dielectric media with a planar interface as shown in Fig. 4.Suppose a plane wave in the medium 1 with index of refraction n1 is impingent uponthe interface at an angle θ1. A portion of the incident wave is reflected and the restis transmitted to the lower medium with an index of refraction n2. The reflected andtransmitted fields are predicted by the Fresnel reflection and transmission coefficientsgive by

rυ12 =n2 cos θ1 − n1 cos θ2

n2 cos θ1 + n1 cos θ2, (9)

rh12 =n1 cos θ1 − n2 cos θ2

n1 cos θ1 + n2 cos θ2,

tυ12 =2n2 cos θ1

n2 cos θ1 + n1 cos θ2,

th12 =2n1 cos θ1

n1 cos θ1 + n2 cos θ2,

where θ2 is the refraction angle obtained from the Snell’s law (θ2 = sin−1(n1

n2sin θ1)). For

a differential area ∆A at the interface the power received, reflected, and transmitted interms of the specific intensities are given by

5

Figure 4: Specific intensities at the interface between two homogeneous dielectric media.

Pi = I ip∆A cos θidΩi (10)

Pr = Irp∆A cos θrdΩr (11)

Pt = I tp∆A cos θtdΩ2. (12)

Noting that dΩr = dΩi, θr = θi = θ1, and the fact that

Pr = Rp12Pi

where Rp12 = |rp12|2 is the power reflectivitiy, from (10) and (11) it can easily be shownthat

Irp = Rp12I

ip = |rp12|2I i

p (13)

where p denotes the polarization of the incident wave. Applying the law of conservationof energy (Pi = Pr + Pt) it can also be shown that

I ip = Ir

p + I tp

cos θ2dΩ2

cos θ1dΩ1.

But dΩ1 = sin θ1dθ1dφ1, and dΩ2 = sin θ2dθ2dφ2. From Snell’s law (n1 sin θ1 = n2 sin θ2)the relation between dθ1 and dθ2 can be obtained and is given by

dθ2 =n1 cos θ1

n2 cos θ2dθ1.

6

Hence

I ip = Ir

p +n2

1

n22

I tp (14)

which renders

I tp =

n22

n21

(1 − Rp12)Iip

Recognizing (1 − Rp12) = Tp12 as the power trasmissivity given in terms of the Fresneltransmission coefficient by

Tp12 =n2 cos θ2

n1 cos θ1|tp12|2,

the transmitted intensity may finally be written as [2]

I tp =

n32 cos θ2

n31 cos θ1

|tp12|2I ip. (15)

Equations (13) and (15) only relate the first two components of the reflected and trans-mitted vector intensities to the incident vector intensity. Expressing all components ofthe reflected and transmitted Stokes parameters of the incident wave and (15), it canbe shown that [4]

Ir = ℜ12(θ1)Ii

where ℜ12(θ1) is the reflectivity matrix given by

ℜ12(θ1) =

|rυ12|2 0 0 00 |rh12|2 0 00 0 ℜ(rυ12r

∗

h12) −ℑ(rυ12r∗

h12)0 0 ℑ(rυ12r

∗

h12) ℜ(rυ12r∗

h12)

. (16)

In a similar manner, the relation between the incident and transmitted Stokes vector inintensities is found to be

It = T12(θ1)Ii (17)

where T12(θ1) is the transmissivity matrix, given by

7

Figure 5: The geometry of a differential volume of a random medium containing anaverage scatterer and its interaction with the specific intensities flowing the medium.

T12(θ1) =n3

2 cos θ2

n31 cos θ1

|tυ12|2 0 0 00 |th12|2 0 00 0 ℜ(tυ12t

∗

h12) −ℑ(tυ12t∗

h12)0 0 ℑ(tυ12t

∗

h12) ℜ(tυ12t∗

h12)

. (18)

Equation (18) applies when θ1 is less than the critical angle. For θ1 greater than thecritical angle, T12(θ1) = 0.

4 Radiative Transfer Equation

In this section the basic formulation of the radiative transfer equation is derived. Asmentioned earlier the physical basis of this theory is the law of conservation of energy.The specific intensity is used as the quantity that contains the incident and scatteredwave power in the random medium. Since the scattered waves in the medium propa-gate in all directions, specific intensities flow in all directions in the medium as well.In radiative transfer formulation, instead of evaluating the interaction of waves withindividual scatterers, the interaction of the specific intensity, with the ensemble aver-age of scattering and absorption properties of the scatterers or a cluster of scatterers isconsidered.

To demonstrate this let us consider a differential volume of the random medium con-taining typical scatterers of the constituent particles in the medium. As shown in 5the intensity entering this differential volume and propagating analog s experiences twophenomena: 1) attenuation or power loss caused by the absorption and scattering crosssection of the average scatterer within the differential volume, and 2) amplification or apower boost caused by the bistatic scattering cross section of the average particle whichredirects the intensities entering the unit volume from other directions to s direction.

According to the law of conservation of energy, the difference between the intensity thatleaves the differential volume and the intensity that enters it (∆I(r, s)) must be equal

8

to the algebraic sum of the power loss due to absorption and scattering (extinction) andthe total bistatic contribution. Let us first consider the extinction part. Assume thatthe average scatterer has an extinction cross section σext = σa +σs where σa and σs are,respectively, the absorption and scattering cross section of the average scatterer. Thenthe power loss is

∆P = −I(r, s)dΩσext

where the negative sign denotes the power loss. The reduction in intensity therefore isgiven by

∆Iext =∆P

∆AdΩ= I(r, s)

σext

∆A. (19)

Denoting extinction per unit volume by κe = δext

∆A∆S, (19) is simply give by

∆Iext = −I(r, s)κe∆S.

The second component which appears as a source within the unit volume is proportionalto the bistatic scattering cross section of the average scatterer. Denoting the bistaticscattering cross section by σb(s, s′) where the s′ and s are, respectively, denoting thedirection of incidence and scattering. The power density on the scatterer generated byan intensity flowing along s′ isI(r, s′)dΩ′ and the power density scattered along s at adistance R from the differential volume is

∆Pb =1

4πR2σb(s, s′)I(r, s′)dΩ′. (20)

Denoting the bistatic scattering cross section per unit volume by σ0b(s, s

′) = σb(s, s′)/(∆A∆s),

the net change in the intensity for this bistatic direction is

δIb =∆Pb

dΩ=

1

4πσ0

b(s, s′)I(r, s′)dΩ′∆s

The quantity 14π

σ0b(s, s′) is known as the phase function which is denoted by

P(s, s′) = N〈|S(s, s′)|2〉 (21)

where S(s, s′) is the far-field amplitude of a typical scatterer and N is the number ofscatterers per unit volume. The total increase in the intensity is the sum of δIb over allincidence directions. That is,

∆Ib =∫

4πδIbdΩ′ = ∆s

∫

4πP(s, s′)I(r, s′)dΩ′.

9

Therefore in limit as ∆s → 0, we have

dI(r, s)

ds= −κeI(r, s) +

∫

4πP(s, s′)I(r, s′)dΩ′ (22)

which is the statement of the scalar radiative transfer equation.

The vector form of the radiative transfer can be obtained in a similar manner. In thevector case 4 × 1 vector specific intensity (Stokes vector) is used and therefore 4 × 4matrices, instead of scalars, must be used for the extinction and phase functions. Thevector form of the radiative transfer equation has the following form:

dI(r, s)

ds= −κeI(r, s) +

∫

4πP(s, s′)I(r, s′)dΩ′ (23)

κe and P(s, s′) are the fundamental quantities of the vector radiative transfer equation.Once these quantities are obtained, the solution to the intero-differential equation (23)can be obrained rather easily. However, it should be mentioned that relating theseparameters to the physical parameters of the medium accurately is not straightforward.For sparse random media these quantities can be obtained from the single scatteringbehavior of individual particles as will be shown next. However, for dense randommedia calculation of κe and P(s, s′) is very difficult.

5 Phase and Extinction Matrices

In this section the phase and extinction matrices used in vector radiative transfer arerelated to the scattering matrix elements of individual scatterers in a random medium.In this process it is assumed that the statistical properties of the constituent particlessuch as orientation angles, and size are known.

5.1 Phase Matrix

For a single particle in the global coordinate system (X, Y, Z,) with some orientationangles (θj , φj) suppose the scattering matrix S(θs, φs; θi, φi; θj , φj) is known where (θi, φi)and (θs, φs) denote, respectively, the directions of incidence and scattering. As discussedbefore, the counterpart of the scattering matrix for the modified Stokes vector is theStokes matrix L(θs, φs; θi, φi; θj , φj) which relates the scattered modified Stokes vectorIs

to the incident modified Stokes vector Ii by [6]

Is =1

r2L(θs, φs; θi, φi; θj, φj) · Ii (24)

where L(θs, φs; θi, φi; θj, φj) is the Stokes matrix and is given by

10

L =

|Sυυ|2 |Sυh|2 Re(S∗

υhSυυ) −Im(S∗

υhSυυ)|Shυ|2 |Shh|2 Re(S∗

hhShυ) −Im(S∗

hhShυ)2Re(SυυS

∗

hυ) 2Re(SυhS∗

hh) Re(SυυS∗

hh + SυhS∗

hυ) −Im(SυυS∗

hh − SυhS∗

hυ)2Im(SυυS

∗

hυ) 2Im(SυhS∗

hh) Im(SυυS∗

hh + SυhS∗

hυ) Re(SυυS∗

hh − SυhS∗

hυ)

(25)

If the particles in the medium are positioned and oriented randomly, we may add thescattered Stokes vectors of the particles incoherently. This can be justified at highfrequencies by noting that the phase of the scattered wave from a particle depends onits position and if the distribution of the scatterers is sufficiently random, the pahsedistribution of the scattered field becomes uniform. If particles are electrically closeto each other, the relative position of the particles with respect to each other must beaccounted for. Suppose the number of particles per unit volume is denoted by N , thenreferring to (21) the phase matrix of this medium can simply be computed from

P(θs, φs; θi, φi) = N〈L〉 (26)

where 〈 〉 is the ensemble average over the size and orientation angles of the particles.The phase matrix P(θs, φs; θi, φi) relates the average intensity scattered by a unit volumeof the medium into the direction (θs, φs) to the intensity incident upon the unit volumefrom the direction (θi, φi).

5.2 Extinction Matrix

The extinction matrix characterizes the attenuation of the Stokes parameters due toabsorption and scattering. For a medium with a low concentration of particles, theattenuation rate can be obtained from the extinction cross section of the individualscatterers. By applying the optical theorem, the extinction cross section, σp

ext (p =υ orh), is given by

σpext =

4π

k0

Im[Spp(θi, φi; θi, φi; θj, φj)],

from which the extinction coefficient can be obtained through

κp = N〈σpext〉 (27)

To find a better estimate of the coherent field along the propagation direction (θi, φi),Foldy’s approximation can be employed. The coupled equations for the vertical andhorizontal mean field in this approximation is of the following form [8]

11

dEυ

ds= (ik0 + Mυυ)Eυ + MυhEh (28)

dEh

ds= MhυEυ + (ik0 + Mhh)Eh (29)

where s is the distance along the direction of propagation and

Mmn =i2πN

k0

〈Smn(θi, φi; θi, φi; θj, φj)〉 m, n = υ, h. (30)

Using the definition of the modified Stokes parameters and (28) and (29) the followingcoupled differential equation is found

d

dsI = −κeI (31)

where κ is the extinction matrix and is given by

κe =

−2Re(Mυυ) 0 −Re(Mυh) −Im(Mυh)0 −2Re(Mhh) −Re(Mhυ) −Im(Mhυ)

−2Re(Mhυ) −2Re(Mυh) −[Re(Mυυ) + Re(Mhh)] [Im(Mυυ) − Im(Mhh)]2Im(Mhυ) 2Im(Mυh) −[Im(Mυυ) − Im(Mhh)] −[Re(Mυυ) + Re(Mhh)]

(32)

To solve a system of coupled linear differential equations, eigen-analysis is usually used.The solutions of such a system are exponential functions whose exponents are the eigen-values of the coefficient matrix. The eigen-values in this case are the possible propagationconstants. These effective propagation constants (for the mean field)can be obtainedfrom the eigen-values of the matrix formed by the righd hand side coefficients of equations(28) and (29). The two distinct eigen-values are found

K1 = k0 − i2[Mυυ + Mhh + r] (33)

K2 = k0 − i2[Mυυ + Mhh − r] (34)

wherer = [(Mυυ − Mhh)

2 + 4MhυMυh]1

2 (35)

The eigen-vectors corresponding to K1 and K2 are denoted by [1, b1] and [b2, 1] respec-tively, and b1 and b2 are given by

b1 =2Mhυ

Mυυ − Mhh + r

12

b2 =2Mυh

−Mυυ + Mhh − r

The corresponding effective propagation constant of the coherent Stokes parameters canbe obtained from the eigen-value solution of equation (31). The eigen-values of extinctionmatrix are given by [8]

Λ(θ, φ) =

λ1

λ2

λ3

λ4

=

2Im[K1]iK∗

2 − iK1

iK∗

1 − iK2

2Im[K2]

(36)

The eigen-matrix Q(θ, φ) is also defined in such a way that the columns of Q(θ, φ) arethe eigen-vectors of the extinction matrix and is given by

Q(θ, φ) =

1 b∗2 b2 |b2|2|b1|2 b1 b∗1 1

2Re[b1] 1 + b1b∗

2 1 + b2b∗

1 2Re[b2]−2Im[b1] −i(1 − b1b

∗

2) i(1 − b2b∗

1) 2Im[b2]

(37)

It is worth noting that at high frequencies or generally for particles with no depolarizationwhere Mυh = Mhυ = 0, the eigen-matrix Q is independent of the particles shape andorientation. In this case the eigen-matrix and its inverse are given by

Q =

1 0 0 00 0 0 10 1 1 00 −i i 0

, Q−1 =

1 0 0 00 0 1

2i2

0 0 12

− i2

0 1 0 0

.

6 Rayleigh Model for Phase and Extinction Matri-

ces

At low frequencies where the typical dimensions of scatterers are small compared tothe wavelength, simple expressions for the phase and extinction matrices can be ob-tained. This simplification is a consequence of the fact that the scattering matrix at lowfrequency regimes can be expressed explicitly in terms of the angles of incidence andscattering. As it was shown before the scattered field of a small particle resembles thatof a dipole and is given by

Es

= −ωk0Z0

4π

eikr

rks × (ks × p) (38)

13

where ks is the unit vector along the direction of scattering and p is the dipole moment.However, under the Rayleigh approximation the dipole moment, independent of particlegeometry and dielectric constant, is linearly related to the inciddent field [9]

p = ǫ0

=

A ·Ei(39)

where=

A is the normalized polarizability tensor (dimensionless) of a particle in themedium. Expressing E

sand Ei in (υ, h, k) coordinate systems

Es =eikor

r

=

S Ei (40)

where the pq element of=

S in terms of=

A is given by

Spq = − k20

4πps ·

[

ks × (ks×=

A ·qi)]

(41)

Noting that

ks ×[

ks × (=

A ·qi)]

=[

ks · (=

A ·qi)]

ks − (=

A ·qi)

and the fact that ps · ks = 0, elements of the scattering matrix in terms of the elements

of=

A can be obtained from

Spq(θs, φs, ; θi, φi, ; θj, φj) =k2

0

4πps · (

=

A ·qi) (42)

where (θj, φj) denote the orientation angles of the particle in the global coordinatesystems. Equation (42) clearly shows the explicit dependence of Spq on the incidenceand scattering directions which will be exploited for the efficient calculation of the phaseand extinction matrices. This can be done since all ensemble averaging operations overthe size and orientation angles, required for the evaluation of the phase and extinctionmatrices, can be carried only once independent of the incidence and observation angles.

6.1 Extinction Matrix Calculation

The extinction matrix is given by (32) and requires computation of 〈Spq〉 in forwarddirection which is given by

〈Spq〉 =k2

0

4π〈pi · (

=

A ·qi)〉

14

However, pi and qi are deterministic quantities independent of the orientation angle ofthe particles, then

〈Spq〉 =k2

0

4πpi · 〈

=

A〉 · qi

Assuming the polarizability tensor in a local coordinate system is denoted by=

Aℓ throughEulerian rotation angle α, β and γ, in the global coordinate system we have

=

A==

T t=

Aℓ

=

T

where the ijth element of=

A in terms of the entries of=

Aℓ and=

T can easily be evaluatedfrom [10]

aij =3

∑

m=1

3∑

n=1

(tmitnj)aℓmn. (43)

Equation (43) clearly shows that the symmetry property of the polarizability tensor is

preserved across a coordinate transformation. Elements of=

T are simple trigonometericfunctions of α, β, and γ, e.g. [8]

t11 = cos γ cos β cos α − sin γ sin α

t12 = cos γ cos β sin α + sin γ cos α...

and once the joint pdf of α, β, γ, fαβγ(α, β, γ) is known, 〈aij〉 can easily be computedfrom integrals like

I ijmn =

∫ 2π

0

∫ π

0

∫ 2π

0tmitnjfαβγ(α, β, γ)dαdβdγ.

The local coordinate system can usually be chosen so that=

Aℓ is diagonal; in this case

〈aij〉 =3

∑

m=1

〈tmitmj〉aℓmm =

3∑

m=1

I ijmmaℓ

mm. (44)

In the special case of uniform distribution where

15

fαβγ(α, β, γ) =1

8π2sin β

I ijmm can be computed easily, and it can be shown that

I ijmm =

13

i = j0 i 6= j

∀m .

Uniform angular distribution renders a diagonal extinction matrix with elements

κe =Nk0

3ℑ

[

3∑

m=1

aℓmm

]

. (45)

As mentioned previously in Rayleigh region results based on the optical theorem canonly account for the absorption loss and no scattering loss is included in this formulation.

For spherical particles, no ensemble averaging over orientation angles is necessary dueto their symmetry. Substituting the polarizability tensor element of spherical particlesin (45), the absorption coefficient is found to be

κa = 3kfℑ

ǫs − ǫ

ǫs + 2ǫ

where f = Nυ is the volume fraction of particles and ǫs and ǫ are, respectively, therelative permittivity of sphere and the background. Previously we also found the atten-tuation rate due to scattering which is given by

κs = 2fa3k4| ǫs − ǫ

ǫs + 2ǫ|2.

where a is the sphere radius. The extinction coefficient for the spherical medium is thus

κe = κa + κs.

6.2 Phase Matrix Elements

Determination of the phase matrix requires calculation of ensemble average of terms like〈SpqS

∗

mn〉. Starting from (42) we have

16

〈SpqS∗

mn〉 =k4

0

(4π)2〈(ps·

=

A ·qi)(ms·=

A∗

·ni)〉. (46)

To simplify (46) the notion of Kronecker product can be used [11]. Recalling that

kronecker product of an i × j matrix=

B with j × k matrix=

C is given by a (ij) × (jk)matrix

=

B ⊗=

C=

b11

=

C b12

=

C b1j

=

C

b21

=

C...

bi1

=

C bij

=

C

(46) may be written as

〈SpqS∗

mn〉 =k4

0

(4π)2ps · 〈

=

G〉ni. (47)

In (47)

〈=

G〉 = 〈=

A=

U=

A∗〉

where

=

U= qi ⊗ ms

Using the Kronecker product again

〈G〉 = 〈=

A ⊗=

A∗〉U (48)

where 〈G〉 and U are 9 × 1 vectors and 〈=

A ⊗=

A∗〉 is a 9 × 9 matrix. Equation (48)

clearly shows that the quantity 〈=

A ⊗=

A∗〉 is independent of angles of incidence andscattering. This quantity is only a function of orientation angles of the particles whichare embedded in the transformation matrix T through the Eulerian angles α, β, andγ.

Spherical particles are orientation independent and hence using (46) a closed form ex-pression for their phase matrix at low frequencies can be obtained. The polarizabilitytensor of spherical particles was found to be

=

A= 3υǫǫs − ǫ

ǫs + 2ǫ

=

I (49)

17

where υ is the volume of the spherical particle. Substituting (49) into (46) we have

N〈SpqS∗

mn〉 =3k4a3

4πf

∣

∣

∣

∣

ǫs − ǫ

ǫs + 2ǫ

∣

∣

∣

∣

2

(ps · qi)(ms · ni) (50)

where f is the volume fraction (Nυ), and k = k0

√ǫ. Expanding the dot products in (50)

for different values of p, q, m, n, (υ or h) the phase matrix is found to have the followingform

P (θs, φs; θi, φi) =3k4a3f

4π

∣

∣

∣

∣

ǫs − ǫ

ǫs + 2ǫ

∣

∣

∣

∣

2

p11 p12 p13 0p21 p22 p23 0p31 p32 p33 00 0 0 p44

with

p11 = sin2 θs sin2 θi + cos2 θs cos2 θi cos2(φs − φi)

+2 sin θs sin θi cos θs cos θi cos(φs − φi)

p12 = cos2 θs sin2(φs − φi)

p13 = cos θs sin θs sin θi sin(φs − φi)

+ cos2 θs cos θi sin(φs − φi) cos(φs − φi)

p21 = cos2 θi sin2(φs − φi)

p22 = cos2(φs − φi)

p23 = − cos θi sin(φs − φi) cos(φs − φi)

p31 = −2 cos θs sin θi cos θi sin(φs − φi)

− cos θs cos2 θi cos(φs − φi) sin(φs − φi)

p32 = 2 cos θ−s sin(φs − φi) cos(φs − φi)

p33 = sin θs sin θi cos(φs − φi)

+ cos θs cos θi cos(2(φs − φi))

p44 = sin θs sin θi cos(φs − φi) + cos θscosθi

7 Solution of the Radiative Transfer Equation

The solution to the vector radiative transfer equation can be obtained either iterativelyor numberically using the discrete ordinate eigen-analysis. Both of these approacheshave been studied extensively in the literature [1, 2, 8, 6]. The iterative approach isapplicable for random media with weak scattering where the scattering source function

18

(the contribution of the integral in the RT equation) is much smaller than the extinctionterm. In such cases the integro-differential equation can be expressed in terms of anintegral equation which is then solved iteratively. The discrete ordinate approach doesnot put any restrictions on the medium parameters. However, it assumes that thespecific intensity flows only along certain discrete directions in the medium. Using thisapproximation together with the application of the truncated Fourier series expansion ofthe phase matrix and the specific intensity, the RT equation is cast in terms of a matrixequation which is then solved using the eigen-analysis. The solutions provided here arevalid for stratified random media which implies that the random layers of the mediumare statisticlly homogeneous. In other words, the specific intensities are independent ofazimuthal position vector ρ = xx + yy.

7.1 Iterative Approach

To demonstrate the solution RT equation iteratively, let us consider the problem ofscattering from a forest canopy. A forest canopy may be modeled as a two-layer randommedium above a homogeneous dielectric half-space as shown in Figure 6 [12]. The toprandom layer is assumed to have thickness d including leaves and branches. This layer,henceforth will be referred to as the crown layer, and the lower layer with thicknessHt which includes the tree trunks will be referred to as the trunk layer. The trunklayer is assumed to consist of vertical, homogeneous, dielectric cylinders. The air-crowninterface and the crown-trunk interface will be treated as diffused boundaries, and theinterface between the trunk layer and the ground layer is considered a specular surface.For this problem the specific intensity in each layer is only a function z. The layeredrandom medium model is shown in Figure 7.

When formulating the radiative transfer problem for bounded layer media, the standardpractice is to split the intensity vector into upward-going (I+(θ, φ, z)) and downward-going (I−(θ, φ, z)) components, by restricting variations of θ between 0 and π/2 [6]. Inthe crown layer, the intensity (I+

c(θ, φ, z)) travelling in the upward direction (θ, φ) and

the intensity (I−c(θ, φ, z)) traveling in the downward direction (π− θ, φ) must satisfy the

coupled radiative transfer equations

d

dzI+c(µ, φ, z) = −κ

+c

µI+c(µ, φ, z) + F+

c(µ, φ, z), −d ≤ z ≤ 0

− d

dzI−c(−µ, φ, z) = −κ

−

c

µI−c(−µ, φ, z) + F−

c(−µ, φ, z), −d ≤ z ≤ 0 (51)

where κ±

c is the extinction matrix of the crown layer, µ = cos θ, and −µ = cos(π−θ). Thesource functions F+

c(µ, φ, z) and F−

c(−µ, φ, z) account for directing the energy incident

upon an elemental volume from all directions into the direction (θ, φ) and (π − θ, φ),respectively, and are given by

19

Figure 6: A simplified forest canopy composed of a crown and an trunk layer.

F+c(µ, φ, z) =

1

µ[∫ 2π

0

∫ 1

0Pc(µ, φ; µ′, φ′)I+

c(µ′, φ′, z)dΩ′

+∫ 2π

0

∫ 1

0Pc(µ, φ;−µ′, φ′)I−

c(−µ′, φ′, z)dΩ′]

F−

c(−µ, φ, z) =

1

µ[∫ 2π

0

∫ 1

0Pc(−µ, φ; µ′, φ′)I+

c(µ′, φ′, z)dΩ′

+∫ 2π

0

∫ 1

0Pc(−µ, φ;−µ′, φ′)I−

c(−µ′, φ′, z)dΩ′] (52)

where dΩ′ = dµ′dφ′ = sin θ′dθ′dφ′, and Pc(µ, φ; µ′, φ′) is the phase matrix of the crownlayer. Equations identical to (51) and (52) may be written for the trunk layer (−d′ ≤z ≤ −d) upon replacing the subscript c with the subscript t. It can be shown thatthe farfield amplitude of a long cylinder is proportional to sin V/V and thus the phasematrix of the trunk layer must have a similar dependence, i.e. [12]

Pt(θs.φs; θi, φi) ∼

sin[k0Ht(cos θi − cos θs)/2]

k0Ht(cos θi − cos θs)/2

2

For forest canopies the height of the trunk layer is assumed to be much larger than thewavelength (Ht ≫ λ). Thus the following approximation can be employed

sin[k0Ht(cos θi − cos θs)/2]

k0Ht(cos θi − cos θs)/2≈ δk(µs − µi)

20

Figure 7: Layered medium representation of the forest medium shown in 6

where δk is the Kronecker delta function and is defined by

δk(µs − µi) =

1; µs = µi

0; otherwise

As a result of this approximation the cylinders in the trunk layer can only generateupward-going (downward-going) intensity when they are illuminated by an upward-going (downward-going) intensity. Therefore the source functions in the trunk layer aregiven by

F+t(µ, φ, z) =

1

µ

∫ 2π

0Pt(µ, φ; µ, φ′)[

∫

1

0

I+t(µ′, φ′, z)δk(µ − µ′)dµ′]dφ′

F−

t(µ, φ, z) =

1

µ

∫ 2π

0Pt(−µ, φ;−µ, φ′)[

∫

1

0

I−t(−µ′, φ′, z)δk(µ − µ′)dµ′]dφ′ (53)

where the quantity in the bracket is the representation of the specific intensity for a twodimensional problem. The solution to differential equations (51) and (52) can formallybe expressed as

I+c (µ, φ, z) = e−κ

+c (z+d)/µI+

c (µ, φ,−d) +∫ z

−deκ

+c (z−z′)/µF+

c (µ, φ, z′)dz′ (54)

I−c (−µ, φ, z) = eκ−

c z/µI−c (−µ, φ, 0) +∫ 0

zeκ

−

c (z−z′)/µF−

c (−µ, φ, z′)dz′ (55)

21

where

κ+c = κc(θ, φ),

κ−

c = κc(π − θ, φ), (56)

and the following notation has be adopted

e−κcz/µ = Qc(µ, φ)Dc(µ, φ′;−z/µ)Q−1c (µ, φ) (57)

where Qc(µ, φ) is a matrix whose columns are eigen vectors of the extinction matrix κ,and Dc(µ, φ;−z/µ) is a diagonal matrix whose diagonal elements are of the followingform

[Dc(µ, φ;−z/µ)]ii = eλi(µ,φ)z/µ

with λi(µ, φ) being the ith eigen value of κc(µ, φ).

Similarly the vector specific intensities I+t and I−t in the trunk layer are given by

I+t (µ, φ, z) = e−κ

+t

(z+d′)/µI+t (µ, φ,−d′) +

∫ z

−d′e−κ

+t

(z−z′)/µF+t (µ, φ, z′)dz′ (58)

I−t (−µ, φ, z) = eκ−

t(z+d)/µI−t (−µ, φ,−d) +

∫

−d

zeκ

−

t(z−z′)/µF−

t (−µ, φ, z′)dz′ (59)

where the subscript t denotes that the propagation and scattering processes are takingplace in the trunk layer. If we limit our solution to first-order scattering, the first-ordercontribution of the trunk layer is observable only on the surface of a cone with generatingangel θ0 which includes the backscattering direction.

8 First-Order Solution for Bistatic Scattering

Because there is no reflection at the (diffuse) air-crown boundary (z = 0) and the crown-trunk boundary (z = −d), the following boundary conditions must be satisfied:

I−c (−µ, φ, 0) = I0δ(µ − µ0)δ(φ − φ0), (60)

I+c (µ, φ,−d) = I+

t (µ, φ,−d), (61)

I−t (−µ, φ,−d) = I−c (−µ, φ,−d). (62)

At the bottom boundary (z = −d′), the boundary condition is

22

I+t (µ, φ, d′) = R(µ)I−t (−µ, φ,−d′), (63)

where R(µ), the reflectivity matrix of the specular surface, is given by (16).

In order to obtain the scattering behavior of the layered media, we need to solve forI+c (µ, φ, z) and then evaluate it at z = 0. Upon setting z = −d in (55), inserting the

result in (59), and then evaluating the resultant expression at z = −d′, the downward-going intensity at the bottom surface can be obtained. Now the upward-going intensityat the bottom surface (I+

t (µ, φ,−d′)) can be found by using the boundary condition(63). The expression for I+

t (µ, φ,−d′) can be inserted into equation (58) and then thelatter can be evaluated at z = −d to obtain an expression for I+

t (µ, φ,−d). Finally byinserting the resultant expression for I+

t (µ, φ,−d) into equation (54), we end up withthe expression

I+c (µ, φ, z) = e−κ

+c (z+d)/µR′(µ, φ) · [e−κ

−

c d/µI−c (−µ, φ, 0)

+∫ 0

−de−κ

−

c (d+z′)/µF−

c (−µ, φ, z′)dz′]

+∫ z

−de−κ

+c (z−z′)/µF+

c (µ, φ, z′)dz′

e−κ+c (z+d)/µ[e−κ

+

tHt/µR

∫

−d

−d′e−κ

−

t(z′+d′)/µF−

t (−µ, φ, z′)dz′

∫

−d

−d′e−κ

+

t(z′+d)/µF+

t (µ, φ, z′)dz′] (64)

where

R′(µ, φ) = e−κ+t

ht/µ · R(µ) · eκ−

tHt/µ

The matrix R′(µ, φ) accounts for extinction in the trunk layer and reflection at thespecular surface. The above expression is given in terms of the source functions F±

c

and F±

t which in turn are given by (52) and (53) in terms of I±c

and I±t . Thus we needto solve the coupled integral equations (53), (58), (59), and (64) to obtain I±c (µ, φ, 0).If the scattering albedo of the medium is small, we can solve the integral equationsusing an iterative approach. We shall start with the zeroth-order solutions, which areobtained by setting Pc(µ, φ; µ′, φ′) = Pt(µ, φ; µ′, φ′) = 0 in (52) and (53) which rendersF±

c (µ, φ, z) = F±

t (−µ, φ, z) = 0. Using the boundary conditions given by (60)-(64) thezeroth-order specific intensities are given by

I−c0(−µ, φ, z) = eκ−

c z/µI0δ(µ − µ0)δ(φ − φ0)

I−t0(−µ, φ, z) = eκ−

t(z+d)/µe−κ

−

c d/µI0δ(µ − µ0)δ(φ − φ′

0)

23

I+t0(−µ, φ, z) = e−κ

+

t(z+d)/µRe−κ

−

tHt/µe−κ

−

c d/µI0δ(µ − µ0)δ(φ − φ0) (65)

I+c0(−µ, φ, z) = e−κ

+c (z+d)/µe−κ

+

tHt/µRe−κ

−

tHt/µe−κ

−

c d/µI0δ(µ − µ0)δ(φ − φ0)

where the symbols I±c0 and I±t0 are used to denote the zeroth-order solutions of I±c andI±t , respectively.

The zeroth-order solution corresponds to propagation of the coherent wave through themedium with the scattering ignored, except for its contribution to extinction. To obtainthe first-order solution, we first need to use (66) in (52) and (53) to compute the first-order source functions F±

c1 and F±

t1 and then insert the result in (64). This process leadsto

I+c (µ, φ, z) = e−κ

+c (z+d)/µR′(µ, φ)e−κ

−

c d/µδ(µ − µ0)δ(φ − φ0)I0

+1

µe−κ

+c (z+d)/µR′(µ, φ)

·

∫ 0

−d[e−κ

−

c (z′+d)/µPc(−µ, φ; µ0, φ0)e−κ+

c (z′+d)/µ0R′(µ0, φ0)

·eκ−

c d/µ0 + e−κ−

c (z′+d)/µPc(−µ, φ;−µ0, φ0)eκ

−

c z′/µ0 ]dz′

I0

+1

µ∫ z

−d[e−κ

+c (z−z′)/µPc(µ, φ; µ0, φ0)e

−κ+c (z′+d)/µ0R′(µ0, φ0)e

−κ−

c d/µ0

+e−κ+c (z−z′)/µPc(µ, φ;−µ0, φ0)e

κ−

c z′/µ0 ]dz′I0+

1

µe−κ

+c (z+d)/µ0e−κ

+

tHt/µR[

∫

−d

−d′e−κ

−

t(z′+d′)/µPt(−µ, φ;−µ, φ0)

eκ−

t(z′+d)/µ0dz′] · e−κ

−

c d/µ0δk(µ − µ0)I0

+1

µe−κ

+c (z+d)/µ

∫

−d

−d′eκ

+

t(z′+d′)/µPt(µ, φ; µ, φ0)e

−κ+

t(z′+d′)/µ0dz′

·Re−κ−

tHt/µ0e−κ

−

c d/µ0δk(µ − µ0)I0 (66)

where it is understood that κ±

c,t/µ = κ±

c,t(µ, φ)/µ and κ±

c,t/µ0 = κ±

c,t(µ0, φ0)/µ0. To findan expression for the intensity emerging from the crown layer at z = 0, we shall firstdefine the integrals in (66) in terms of equivalent matrices. Using the definition givenin (57)

A1(µ, φ; µ0, φ0) =∫ 0

−dDc(−µ, φ;−(z′ + d)/µ)Q−1

c (−µ, φ)Pc(−µ, φ; µ0, φ0)

Qc(µ0, φ0)Dc(µ0, φ0;−(z′ + d)/µ0)dz′, (67)

whose (ij)th element is given by

24

[A1(µ, φ; µ0, φ0)]ij =1 − exp[−(λi(−µ, φ)/µ + λj(µ0, φ0)/µ0)d]

λi(−µ, φ)/µ + λj(µ0, φ0)/µ0

·[Q−1c (−µ, φ)Pc(−µ, φ; µ0, φ0)Qc(µ0, φ0)]ij .

In a similar manner,

A2(µ, φ; µ0, φ0) =∫ 0

−dDc(−µ, φ;−(z′ + d)/µ)Q−1

c (−µ, φ)Pc(−µ, φ;−µ0, φ0)

Qc(−µ0, φ0)Dc(−µ0, φ0; z′/µ0)dz′ (68)

[A2(µ, φ; µ0, φ0)]ij =exp[−λi(−µ, φ)d/µ] − exp[−λj(−µ0, φ0)d/µ0]

−λi(−µ, φ)/µ + λj(µ0, φ0)/µ0

·[Q−1c (−µ, φ)Pc(−µ, φ; µ0, φ0)Qc(−µ0, φ0)]ij ,

A3(µ, φ; µ0φ0) =∫ 0

−dDc(µ, φ; z′/µ)Q−1

c(µ, φ)Pc(µ, φ; µ0, φ0)

Qc(µ0, φ0)Dc(µ0, φ0;−(z′ + d)/µ0)dz′ (69)

[A3(µ, φ; µ0, φ0)]ij =exp[−λj(µ0, φ0)d/µ0] − exp[−λi(µ, φ)d/µ]

λi(µ, φ)/µ − λj(µ0, φ0)/µ0

·[Q−1c (µ, φ)Pc(µ, φ; µ0, φ0)Qc(µ0, φ0)]ij ,

and

A4(µ, φ; µ0, φ0) =∫ 0

−dDc(µ, φ; z′/µ)Q−1

c(µ, φ)Pc(µ, φ;−µ0, φ0)

Qc(−µ0, φ0)Dc(−µ0, φ0; z′/µ0)dz′ (70)

[A4(µ, φ; µ0, φ0)]ij =1 − exp[−(λi(µ, φ)/µ + λj(−µ0, φ0)/µ0)d]

λi(µ, φ)/µ + λj(−µ0, φ0)/µ0

·[Q−1c (µ, φ)Pc(µ, φ;−µ0, φ0)Qc(−µ0, φ0)]ij ,

25

A5(µ, φ; µ0, φ0) =∫

−d

−dDt(−µ, φ;−(z′ + d)/µ)Q−1

t (−µ, φ)Pt(−µ, φ;−µ0, φ0)

Qt(−µ0, φ0)Dt(−µ0, φ0; (z′ + d)/µ0)dz′ (71)

[A5(µ, φ; µ0, φ0)]ij =exp[−λi(−µ, φ)Ht/µ] − exp[−λj(−µ0, φ0)Ht/µ0]

−λi(−µ, φ)/µ + λj(−µ0, φ0)/µ0

·[Q−1t (−µ, φ)Pt(−µ, φ; µ0, φ0)Qt(−µ0, φ0)]ij ,

A6(µ, φ; µ0, φ0) =∫

−d

−dDt(µ, φ; (z′ + d)/µ)Q−1

t (µ, φ)Pt(µ, φ;−µ0, φ0)

Qt(µ0, φ0)Dt(µ0, φ0;−(z′ + d′)/µ0)dz′ (72)

[A6(µ, φ; µ0, φ0)]ij =exp[−λj(µ0, φ0)Ht/µ0] − exp[−λi(−µ, φ)Ht/µ]

λi(µ, φ)/µ − λj(µ0, φ0)/µ0

·[Q−1t (µ, φ)Pt(µ, φ; µ, φ0)Qt(µ0, φ0)]ij .

In view of these matrices, (66) can be evaluated at z = 0 and written in the form

Is(µ, φ) = I+1 (µ, φ, 0)

= e−κ+c d/µR′(µ0, φ0)e

−κ−

c d/µ0δ(µ − µ0)δ(φ − φ0)I0

+1

µe−κ

+c d/µR′(µ, φ)Qc(−µ, φ)A1Q

−1c (µ0, φ0)R

′(µ0, φ0)e−κ

−

c d/µ0I0

+1

µe−κ

+c d/µR′(µ, φ)Qc(−µ, φ)A2Q

−1

c(−µ0, φ0)I0

+1

µQc(µ, φ)A3Q

−1c (µ0, φ0)R

′(µ0, φ0)e−κ

−

c d/µ0I0

+1

µQc(µ, φ)A4Q

−1c (−µ0, φ0)I0

+1

µe−κ

+c d/µe−κ

+

tHt/µRQt(−µ, φ)A5Q

−1t (−µ0, φ0)e

−κ−

c d/µ0δk(µ − µ0)I0

+1

µe−κ

+c d/µQt(µ, φ)A6Q

−1t (µ0, φ0)Re−κ

−

tHt/µ0e−κ

−

c d/µ0δk(µ − µ0)I0

(73)

26

The seven terms contained in (73), which are diagrammed in Fig. 8, represent:

Term ds This is a coherent reflection term resulting from direct propagation of theincident intensity through the layers down to the bottom boundary, followed by specular

reflection by the specular boundary, and then followed with direct propagation throughthe layers to the upper boundary. This term exists only in the specular direction (θ, φ) =(θ0, φ0), and in that case, its magnitude is equal to I0, reduced by the product of thetwo-way attenuation and the reflection coefficient.

Term 1 This term represents propagation of the incident intensity through the two layersto the bottom boundary, followed with specular reflection at θ0, then bistatic scatteringby the vegetation material in the crown layer downward in a direction (π−θ, φ) such thatafter specular reflection by the lower surface a second time at θ, the reflected intensitypropagates upward through the two layers along the direction (θ, φ).

Term 2a This term represents propagation of the incident intensity I0 into the crownlayer along the direction (θ0, φ0), followed with bistatic scattering by the vegetation ma-terial downward along the direction (π−θ, φ), and then followed with specular reflectionby the lower boundary upward through the layers along the direction (θ, φ).

Term 2b This is the complement of term 2a. It represents propagation of the incidentintensity I0 through the layers down to the bottom boundary, specular reflection at θ0,upward propagation at θ0, then bistatic scattering by the vegetation in the direction(θ, φ).

Term 3 This term does not involve reflection by the bottom boundary. It representsincidence at (π−θ0, φ0) followed by bistatic scattering upward along the direction (θ, φ).For a semi-infinite layer, the other four terms vanish, and only this term remains.

Term 4a This term represents the propagation of the incident intensity in the crownlayer (i.e. attenuation only but no change in direction), followed by bistatic scatteringby the trunks down to the ground surface, followed by bistatic reflection by the surfaceboundary, and then direct propagation through the trunk and crown layers.

Term 4b This term is the complement of term 4a (same path, but in reverse direction).This term and term 4a will be referred to as the ground-trunk term. Contribution ofthese terms is observable only on the surface of a cone with generating angle θ0.

With Is(µ, φ) given by (73), the expression for the bistatic scattering coefficient may now

27

be readily found by inserting (73) into (8). For the backscattering case, we set θ = θ0

and φ− φ0 + π in (67)-(73) and the first-order scattering mechanism is shown in 9. Foran azimuthally symmetric medium, this condition causes the diagonal components ofA2, A3, A5, and A6 to become indeterminate. Application of L’Hopital’s rule, however,leads to

limµ→µ0

[A2(µ, φ; µ0, φ0)]ii = de−λi(µ0)d/µ0 · [Q−1c (−µ, φ)Pc(−µ, φ;−µ0, φ0)Qc(−µ0, φ0)]ii .

The same result applies to A3,A5 and A6. The backscattered intensity can be relatedto the incident intensity through a matrix T(θ0, φ0),

Is = T(θ0, φ0)I0

References

[1] Chandrasekhar, S., Radiative Transfer, Dover, New york, 1960.

[2] Ishimaru, A., Wave Propagation and Scattering in Random Media, IEEE Press, pp.148-155, New York, 1997.

[3] Schuster, A., “ Radiation Through a Foggy Atmosphere,” Astrophys. J., vol. 21,pp. 1-22, 1905.

[4] Tsang, L., and J. A. Kong, “Radiative Transfer Theory for Active Remote Sensingof Half-Space Random Media,” Radio Sci., 13, pp. 763-773, 1978.

[5] Tsang, L., J. A. Kong, and R. T. Shin, “Radiative Transfer Theory for ActiveRemote Sensing of a Layer of Nonspherical Particles,” Radio Sci., 19, pp. 629-642,1984.

[6] Ulaby, F. T. , R. K. Moore, and A. K. Fung,Microwave Remote Sensing: Active and Passive, Vol. III – Volume Scatteringand Emission Theory, Advanced Systems and Applications, Artech House, Inc.Dedham, MA: 1986.

[7] Karam, M. A., and A. K. Fung, “Electromagnetic Scattering from a Layer of FiniteLength, Randomly Oriented, Dielectric, Circular Cylinders Over a Rough Interfacewith Application to Vegetation,” International Journal of Remote Sensing, 9, pp.1109-1134, 1988.

[8] Tsang, L., J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing, JohnWiley and Sons, New York, 1985.

28

[9] Kleinman, R. E. and T. B. A. Senior, Low and High frequency Asymptotics, Chapter1, V. K. Vardan and V. V. Vardan eds., North-Holland, Amsterdam: No Year Given.

[10] Kendra, J. R., and K. sarabandi, “ A Hybrid Experimental/Theoretical ScatteringModel for Dense Random Media,” IEEE Trans. Geosci. and Remote Sensing, Vol.37, no. 1, pp. 21-35, 1999.

[11] Whitt, M. W., “Microwave Scattering from Periodic Row-structured Vegetation,”Ph. D. thesis, the University of Michigan, 1991.

[12] Sarabandi, K., “Electromagnetic Scattering from Vegetation Canopies,” Ph. D.thesis, the University of Michigan, 1989.

Figure 8: First-order scattering terms in a forest medium for the bistatic case.

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Figure 9: First-order scattering terms in a forest medium for the bistatic case.

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