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FUNDAMENTALS OF RANDOM SCATTERING

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Radar Equation for Conglomeration of Scatterers Stokes Parameters and Phase Matrices

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Elliptical Polarization, Stokes Parameters, Partial Polarization Stokes Matrix Scattering per Unit Volume and Phase Matrix Rayleigh Phase Matrix Phase Matrix of Random Media

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Fluctuating Fields

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Coherent and Incoherent Fields Probability Distribution of Scattered Fields and Polarimetric Description

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Specific Intensity Passive Remote Sensing

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Planck's Radiation Law and Brightness Temperature Kirchhoff's Law Fluctuation Dissipation Theorem Emissivity of Four Stokes Parameters

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Correlation Function of Fields References and Additional Readings

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3 FUNDAMENTALS OF RANDOM SCATTERING

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Radar Equation for Conglomeration of Scatterers

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We first derive the radar equation for scattering by a conglomeration of scatterers. Consider a volume V containing a random distribution of particles (Fig. 3.1.1). The volume is illuminated by a transmitter in the direction of ki . The scattered wave in the direction ks is received by the receiver. Consider a "differential volume" dV containing No = nodV number of particles, where no is the number of particles per unit volume. We have put differential volume in quotes " " because the physical size of dV is yet to be discussed. The Poynting's vector Si incident on dV is (3.1.1) where Pt is the power transmitted by the transmitter, Gt(ki ) is the gain of the transmitter in the direction k , and R i is the distance between transmitter i and dV. Let a~No) (k s , ki ) denote the differential scattering cross section of the No particles in dV. The physical size of dV is chosen so that

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(3.1.2)

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which means that a~No) is proportional to dV and that ferential cross section per unit volume.

p(ks , ki )

is the dif-

transmitter

receiver

Figure 3.1.1 Active remote sensing of a random collection of particles.

1 Radar Equation for Conglomeration

The measured power at receiver due to dV is

(No)'

dP = A (k )(]'d

r r s

(k s , ki ) sR2 I

(3.1.3)

where R r is the distance between dV and the receiver, and Ar(ks ) is the effective receiver area given as [Ishimaru, 1978] (3.1.4) where Gr(k s) is the gain of the receiver in direction k s. Putting together (3.1.1)-(3.1.4) and integrating over volume dV gives the receiver power Pr as

Pr =

dV ~ Gt(i~;i)Gr(i~;s) (k k ) (471-)2 R;R; P s, I

(3.1.5)

Equation (3.1.5) is the radar equation for bistatic scattering of a volume of scatterers. For monostatic radar, ks = -ki corresponding to the case of backscattering, we have Rr = Ri = R, so that (3.1.6) Equations (3.1.5) and (3.1.6) are the radar equations for a conglomeration of scatterers. They differ from radar equations of a single target in that there is an integration over volume dV in both (3.1.5) and (3.1.6).

Independent Scattering and Addition of Scattering Cross Sections

Consider the No particles in volume dV and we label the particles 1,2, ..., No. Let be the scattered field of from particle j. Since the Maxwell equations are linear, the total scattered field E is

(3.1.7) The scattered intensity is

No No

LEj LEt

j=1 l=1

(3.1.8)

In the double summation of (3.1.8), we will separate terms into l

j and

3 FUNDAMENTALS OF RANDOM SCATTERING

1 =f:. j. Thus

No No No

2 IEj l + LLEjEI

IEI 2 =

(3.1.9)

j=l 1=1

l=li

Next we take ensemble average of (3.1.9), with ensemble average denoted by angular bracket ( ),

No No

(IEI 2 ) = L(IEj I2 ) + L

Consider EjEI. Let Ej

L(EjE;)

(3.1.10)

j=l 1=1 I=h

= IEj iei8i

and EI

= lEd ei81 . Then

(3.1.11)

(EjE;) = (IEj IlEde i (8i -8d )

Here OJ - 01 = O(kdjt) where djz is the separation distance between particle j and l. If the randomness of particle separation djz is not much smaller than wavelength (Le., S.D.(djz) ~ where S.D. stands for standard deviation), then OJ - Oz has random phase between 0 and 27r radians, and

(EjE;) = 0

The condition of S.D.(djl) 2

(3.1.12) (3.1.13)

can be applied to the case when particles j and 1 are neighbors. This is because S.D.(djz) 2 S.D.(djl(neighbor)) Putting (3.1.12) in (3.1.10) gives

(3.1.14)

(IEI

L (IEj 1

(3.1.15)

Thus (3.1.15) means the addition of scattered intensity. It implies also addition of scattering cross section of particles and the addition of differential cross section of particles. Thus, if O"d(k s , k i ) is the differential cross section of one particle, then O"d

(No)

(k s , kd

= NOO"d(k s , ki )

(3.1.16)

assuming that, for simplicity, the No particles are identical. From (3.1.2) and (3.1.16), we have NOO"d(k s , ki ) = p(k s , ki)dV (3.1.17)