1 Radar Equation

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[or Conglomeration

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Since No = nodV, we have (3.1.18) The addition of scattering cross section must be applied with care. To illustrate a simple example, consider microwave scattering of >. = 1 em by an object that contains a conglomeration of particles of radius equal to a. Let V be the volume of object and let i be the fractional volume occupied by the particles. The particles are "randomly positioned." Then number of particles is (3.1.19) Then

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O"~No) = O"d(object) = NOO"d(particle) = 4~ i 3 k4a61

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11 +2

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~(Vf k)(ka)3! r + 11 47r 2

(3.1.20)

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The above equation indicates that O"~No) ~ 0 as ka ~ 0, which can be erroneous for many cases. The wrong argument is that although the positions of particles are random, the randomness of particle position can be much smaller than the wavelength so that condition (3.1.13) is not satisfied. Thus there are coherent correlated scattering from scatterers in the neighborhood of each other. Also when we apply addition of cross section to No particles in "differential volume" dV, the differential volume dV must be larger than ,\3,

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dV 2::

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(3.1.21)

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Only if (3.1.13) and (3.1.21) are obeyed do we have a valid definition of phase function p(k s , ki ) as the differential cross section per unit volume, giving us (3.1.18) when the particles are identical. Similarly, we define

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K,s K,a

= scattering cross section per unit

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volume

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(3.1.22) (3.1.23) (3.1.24)

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= absorption cross section per unit volume

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= extinction cross section per unit volume

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Then (3.1.25)

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3 FUNDAMENTALS OF RANDOM SCATTERING

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where integration is over 47r scattered directions. Under the independent scattering assumption, we have

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"'s = no

"'e =

dOsO"d(ks, k i )

= noO"s

(3.1.26) (3.1.27) (3.1.28)

"'a = noO"a

"'s + "'a = no(O"s + O"a)

The parameters "'s, "'a, and "'e are also known respectively as scattering coefficient, absorption coefficient, and extinction coefficient. Consider intensity I incident on a slab of thickness ~z and crosssectional area A. Then the power "extinguished" is

tlP = -(Intensity) (extinction cross section per unit volume) (volume) = -I K;e Atlz = tlI A (3.1.29)

Hence (3.1.30) giving the solution

I = Ioe- lieZ

(3.1.31)

Thus K;e is known as the extinction coefficient. It represents attenuation per unit distance due to absorption and scattering. If attenuation is inhomogeneous, then instead of having the product of K;e and distance, we have attenuation

'Y =

(3.1.32)

where ds is the differential distance the wave travels. Thus if we include attenuation in the radar equation, we have, instead of (3.1.5),

P: = t

where

)..2Gt (k i )Gr (k s ) ~ ~ (4 )2R R2 p(ks,kdexp(-'Yi -'Yr) dV 7r ~ r

ds ds

(3.1.33)

'Yi = 'Yr

J =J

= attenuation from

transmitter to dV

(3.1.34) (3.1.35)

= attenuation from dV to receiver

1 Radar Equation for Conglomeration

Figure 3.1.2 Derivation of narrow beam equation.

Narrow Beam Equation

The volume integration in (3.1.33) can be performed analytically if we assume narrow beam equation. If we assume half-power beamwidths lit and c/>l, respectively, in the vertical and horizontal direction for the transmitting antenna (Fig. 3.1.2), the gain pattern can be written as

G,(ki )

~ G,(k,Jexp (-ln2 [G~)' + (~;)']) ~ G,Ck,.lexp (-ln2

(3.1.36)

where C(kd is the directivity (peak gain). Similarly, for the receiver we obtain

G,(k,)

[G:)' G;)'])

(3.1.37)

In (3.1.36) and (3.1.37), lit and c/>l are the half-power beamwidths of the transmitter antenna, and lh and c/>2 are the half-power beamwidths of the receiver antenna. Substituting (3.1.36) and (3.1.37) in (3.1.33), the integration can be done assuming a narrow beam around ki and a narrow beam around kso ' This gives

A Ct(kiJCr(ksJ (A A) Pt = (471-)2 Rr R; p kso,kio exp(-')'i-')'r)~

3.1.38)

where Vc is the common volume intercepted by the narrow beams of the transmitter and the receiver, and it is given by ~shimaru, 1978]

Vc = 1.206 R;R;Ol02c/>1c/>2 _._1_ [Rrc/>r + R;c/>~] "2 sm e

(3.1.39)

where e is the angle between kio and k so ' Equation (3.1.38) is the result for bistatic radar. It is not applicable for monostatic radar when k so = -kio and

3 FUNDAMENTALS OF RANDOM SCATTERING

1800 In the monostatic Case [Ishimaru, 1978]

Pr 4 2 2 ~ P = 2.855 x 10- ,\ G (ki J()I >1

47re- 2')'

p( -kio' kiJ

(3.1.40)

is the attenuation from transmitter-receiver to R.

Particle Size Distribution

In many cases, the particles obey a size distribution n(a) so that the number of particles per unit volume with size between a and a + da is n(a) da. Thus

no =

n(a) da

(3.1.41)

Hence n(a) has dimension of (length)-4. Within the approximation of independent scattering,

10 10

dan(a) as(a)

(3.1.42)

where as (a) shows that the scattering cross section is a function of a. Also

K;a = p(ks,ki ) =

dan(a)aa(a)

(3.1.43) (3.1.44)

dan(a)ad(ks,ki;a)

For Rayleigh scattering by spheres (Section 2.2 of 1), we have

~ ~ _1121~ p(ks,ki)=k 41Er +2 ksx (~ E ksxei

)1 Jo dan(a)a roo

(3.1.45)

(3.1.46) (3.1.47)

87r = 3

r 2 k41 - - E - 11

Er -