Fluctuating Fields

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In random scattering problems, since the inhomogeneities are randomly distributed, the fields and intensity are fluctuating. In the case of random discrete scatterers, the particles are randomly distributed in positions, shapes, sizes, and orientations so that the scattered fields are randomly fluctuating. For the case of random rough surface, the height profile is randomly distributed. In this section, we examine some of the basic relations of fluctuating fields. The concept of random fields is the foundation of random media and random rough surface scattering problems. Unlike deterministic problem in which there is only one solution, random media problem only has an unique solution for a single realization. As realization changes, the positions of the scatterers change and the height profiles change, and the fields will fluctuate. Realization changes can be due to the motion of the random media and rough surfaces or to the motion of the transmitter and receiver as they would view different parts of the surfaces and random media. Classic works on the concept of random fields include Booker and Gordon [1950], Foldy [1945], Lax [1951, 1952], Twersky [1962, 1964], Keller [1964], Frisch [1968], Beran [1968], Tatarskii [1971], and Ishimaru [1978].

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3.1 Coherent and Incoherent Fields

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Let E(r) be the field. In general, E(r) is a random function of position. We can write E(r) as an average field (E) and the fluctuating field :

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E(r)

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(E(r))

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+ (r)

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(3.3.1) (3.3.2)

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( (r)) = 0

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where angular bracket ( ) stands for ensemble averaging. The average field is also called the coherent field, and the fluctuating field is called the incoherent

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3 FUNDAMENTALS OF RANDOM SCATTERING

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field. However, although the fluctuating field is called the "incoherent" field, it can contain partial coherent effects. From (3.3.1)

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EE* = (E)(E*)

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Thus

+ (E) * + (E*) + ( *) + ( (rl) *(r2))

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(3.3.3)

(E(rI)E*(r2)) = (E(rl))(E*(r2))

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(3.3.4)

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The quantity (E(rI)E*(r2)) is called the field correlation function at rl and r2. It is the sum of a coherent part represented by the first term of (3.3.4) and the incoherent part as represented by the second term of (3.3.4). When rl = r2, we get the intensity from (3.3.4). Let

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I = E(r)E* (r)

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be the intensity. Then

(3.3.5)

(1) = Ie + If

where

(3.3.6)

Ie = I(E(r))1

is the coherent intensity and

(3.3.7) (3.3.8)

If = (1 (r)1

is the incoherent intensity. The intensity can also be fluctuating. The variance a} of the intensity fluctuation is given by

a} = (1 2 ) -

(1)2 = (IEI

(Ie + I f )2 = (IEI 4 )

(I(E(r))1

+ (1 (r)1 2)

(3.3.9)

3.2 Probability Distribution of Scattered Fields and Polarimetric Description

Consider; a random field created by scattering from many different particles or by random rough surface

E = Aexp(i4 = X

+ iY

(3.3.10)

where A is amplitude, 4> is the phase, and X and Yare called the quadrature components. Since E is sum of fields from many different particles or from many peaks and valleys in random rough surfaces, we can write [Ishimaru, 1978; Sarabandi, 1992]

L (Xn+ iYn)= LAn exp(i4>n)

n=l n=l

(3.3.11)

3.2 Probability Distribution

where n is the particle index and N is the number of particles. It is assumed that the Xn's and Yn's are independent random variables. Then, the central limit theorem states that the probability distribution of a sum of Nindependent random variables will approach a normal distribution as N - t 00, regardless of the probability distribution of each random variable. Based on this, X and Yare normally distributed. It is further assumed that the phase has no preferred value and is uniformly distributed between 0 and 27r. The uniform distribution of p( ) is independent of A. Thus, the joint probability density function p(A, ) is

p(A, ) = p(A)p( ) 1 p( ) = 27r 0 ::; ::; 27r

(3.3.12) (3.3.13)

If we take an average using the probability density function above, we obtain

(X) = (A cos ) = (A) (cos )

Similarly,

loo dAp(A) l27r d p( ) cos = 0

(Y) = 0

(XY) = (A sin cos )

(3.3.14)

(3.3.15) (3.3.16)