TBf3(Oo, cPo) = If3(Oo, cPo) K

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(3.5.13)

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5.2 Kirchhoff's Law

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From (3.5.12) and (3.5.13), we get for a half-space medium

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TB,B((}o, <Po) = T(1 - TlO,B((}l))

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It is convenient to define emissivity e,B((}o, <Po) as

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(3.5.14)

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(3.5.15)

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so that

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(3.5.16)

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The reflection rlO,B obeys a symmetry relation that is a result of reciprocity and energy conservation. From reciprocity, the transmission from either direction is the same. By energy conservation, reflectivity is one minus transmissivity. Thus combining reciprocity and energy conservation gives the following symmetry relation:

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(3.5.17)

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where T01,B((}o) denotes reflection for wave when incident from region 0 to region 1. Thus, from (3.5.16) and (3.5.17), the emissivity becomes

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(3.5.18)

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The result that the emissivity is equal to one minus reflectivity is a consequence of energy conservation and reciprocity.

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5.2 Kirchhoff's Law

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Kirchhoff's law generalizes the concept of emissivity equal to one minus reflectivity to the case where there is bistatic scattering due to rough surface and volume inhomogeneities. Consider a plane wave with polarization a incident onto a scattering medium with area A. Then the power intercepted by the surface area A is IE~12AcoS(}i/(21J). The power scattered into the upper hemisphere is equal to

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j3=v,h

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upper hemisphere

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(3.5.19)

Thus the fractional power that is absorbed by the surface is given by one minus the fractional power that is scattered back into the upper hemisphere and is known as the absorptivity of the surface.

3 FUNDAMENTALS OF RANDOM SCATTERING

incoming blackbody radiation

dPe dPi

+ dPr

/(O,r/J)

(Oi,(/1;)

Figure 3.5.3 Geometry for derivation of Kirchhoff's law.

P~h /.'/' dB. sin B,

aa(()i,<Pi) = 1= 1-

d<i>. r'IEp.I'

IE~12AcoS()i

4~ I:

!3=v,h

r/ d()s Jo

sin()s

r Jo

d<ps 'Y(3a(()s,<Ps; ()i,<Pi)

(3.5.20)

Kirchhoff's law describes the relation between the emissivity and the absorptivity of the body. Consider the body to be in temperature equilibrium with blackbody radiation in the half-space above it (Fig. 3.5.3). Under this equilibrium condition, it will be assumed that just as much energy of a given polarization leaves the surface in a given direction as falls upon it from the same direction with the same polarization. The power incident on the surface with polarization f3 from direction (Oi, <pd is, from (3.4.1) and (3.5.10)

KT >.2 dO i A cos Oi dv

(3.5.21)

The power leaving the surface in the same direction is the sum of two parts: (1) the thermal emission from the surface and (2) the external black-body radiation scattered by the surface in direction (()i, <Pi)' The first part is given by

dPe = e(3(()i, <Pi)

>.2 cos ()i A dO i dv

(3.5.22)

The second part arises from incoming radiation from all directions (0, <p) that are scattered into the direction (Oil <Pi) with polarization f3 and are described

5.2 Kirchhoff's Law

by the bistatic scattering coefficients. Thus, the power scattered into solid angle dOi with polarization (3 is

dPr =

J ~;

upper

dO cos OA{"Y/1v((ti, <Pi; 0, <p)

+ "Y/1h(Oi, <Pi; 0, <p) } 4~ dO i dv

hemisphere

(3.5.23) Since the body is in thermodynamic equilibrium with the half-space above it, we have

dPi = dPe

+ dPr

(3.5.24)

Equating (3.5.21) to the sum of (3.5.22) and (3.5.23) gives 1 dO cos ~. ["Y(3v(Oi' <Pi, 0, <p) + "Y(3h(Oi, <Pi, 0, <p)] 1 = e(3(Oi, <Pi) + 4 1f J 7r+ cos 172 2 We next make use of reciprocity relation of (2.5.37). Thus,

(3.5.25) d<P"Yex(3(O, <P; Oi, <Pi) ex Equation (3.5.25) is a formula that calculates the emissivity from the bistatic scattering coefficient "y. It also relates active and passive remote-sensing measurements. To illustrate the use of (3.5.25), let us consider the case of reflection by a plane dielectric interface. We note that the power reflected will be in the specular direction and has the same polarization as the incident wave and the fractional reflected power is IR(3(Oi)1 2 = T(3(Oi) with R(3(Oi) the Fresnel reflection coefficient for polarization (3 and incident direction 0i' Therefore,

"Y(3o:(fJs , <Ps;fJi, <pd = Ro:(Od 2471" J( cos Os - cos Oi) J( <Ps - <Pi)Jex (3

(3.5.26)

1 e(3(Oi, <Pi) = 1 - 471"

r/ I: Jo

dO sin 0 Jo

and substituting (3.5.26) into (3.5.25) gives

e(3(Oi,<pd = l-IR(3(Oi)!2

(3.5.27)

Relation (3.5.25) is an exact relation. However, it must be applied with caution in actual practice. The problem arises when scattering effects of rough surfaces and volume inhomogeneities are to be taken into account. (i) In the absence of rough surface and volume inhomogeneities, there is no backscattering. Thus monostatic measurements in active remote sensing directly measures the effects of the scattering by random media. For passive remote sensing, one measures the emissivity which has a finite result even when there is no rough surface or no volume scattering. Thus rough surface and volume scattering gives only a differential change (which can be small) from the flat surface homogeneous medium emissivity.