Emissivities in Java

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Emissivities
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The emissivity of a rough surface can be calculated from the bistatic reflection coefficient. We have
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1.2 Dielectric Rough Surfaces
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The coherent and incoherent bistatic reflection coefficients derived from the Kirchhoff-approximated diffraction integrals in (2.1.93) can be used to calculate the emissivity of the rough surface. In terms of the coherent and incoherent reflectivity functions, the emissivity is given by (2.1.158) where, for a obtain
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c or i, denoting coherent and incoherent, respectively, we
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(2.1.159) In the above equation, the dependence on the azimuthal angle of incidence cPi is dropped because the rough surface is isotropic and the results are independent of cPi. After substituting in the explicit expressions for the bistatic scattering coefficients from (2.1.93), carrying out the angular integration and assuming that C(p) = exp( _p2 /l2), we obtain (2.1.160)
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where a = v, h, and (2.1.162) (2.1.163) and 10 and It are the zeroth- and first-order modified Bessel functions. We note that if the scattered field is used instead of the total field in the diffraction integral, then for a = v or h we have
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2 KIRCHHOFF AND RELATED METHODS
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ria (B i ) =
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cos i
2 [2 B
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2 dBssinBsexp [-h2k2(cosBi
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+ cos Bs)2]
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B )2h(xm)
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x L.J
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~ (kh(cosBs + cosBi))2m
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{(1 +
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(Io(x m)- h~:m))}
(2.1.164)
The difference between the incoherent reflectivities obtained using the total and the scattered field in the diffraction integral is due to the approximation made on the integrand F(a, 13). The emissivity may also be calculated in terms of the bistatic transmission coefficients from medium 1 to medium 0, 1 ea (Bi,<Pi)=-4
I: l
1r 2 /
dB1sinB1
b=v,
1d<pn~b'(Bi,<Pi;Bl,<Pl)
n cos B ~ B~ no cos 1
(2.1.165)
1.3 Second-Order Slope Corrections
We next extend the results of the Kirchhoff approximation up to the second order in slope. The scattered field is
Es(r) =ls,dS' {iWftoG(r, 1") . nx H(r')+ \l x G(r, 1") . nX E(r')}
It can readily be shown that
(2.1.166)
dS' {iwftoG(r, 1") . n X Hi(r')
+ \l X G(r,r') . n X Ei(r') } =
that is, the surface integral of the incident field is equal to zero. Then
Es(r)
ls, dS' {iWftoG(r, 1") . n X Hs(r')
+ \l x G(r,r'). it x Es(r')}
(2.1.167) In this section we will use the scattered field formulation. In the far field we have
Es(r)
= ike
47fr
(1 -ksks )' r dS' {k s x (n
x Es(r')) + T/ (n x Hs(r'))} e- ik .r'
(2.1.168)
The local angle of incidence is Bl i such that
COSB li
-n ki
}1 +
(2.1.169)
--;=====;;==:====;;:: 2 2 00
-ax - f3fJ + z
+ 13
(2.1.170)
1.3 Second-Order Slope Corrections
where a and {3 are defined in (2.1.53), and cos eli
= -n ki = a sin ei cos <Pi + {3 sin ei sin <Pi + cos ei VI + a 2 + (32
-----;::=~=::=,:c----
(2.1.171)
The local perpendicular and parallel polarization reflection coefficients are
k cos eli - ) k 2 sin2 eli Rh = - - - - ' - ; = = = = = = k cos eli + ) kr - k 2 sin2 eli
kr -
(2.1.172)
qk cos eli - Eo)kr - k 2 sin2 eli
R v = ------'--;====== Elk cos eli + Eo) k 2 sin2 eli
kr -
(2.1.173)
Thus (2.1.174) where
F( a,(3)
VI + a 2 + {32 {k s x (n x qi)Rh(ei . qi) +Rh(ei' qi)(n. ki)qi
(ei . Pi)(n . ki)(k s x qi)Rv (2.1.175)
+ (ei' Pi)(n x qdRv}
Next we expand F( a, (3) about zero slope to the second order in slope [Leader, 1971]. Thus
- a, (3) = F( F(O,O)
8FI + a-;:) a,(3=O + {3 8FI a,(3=O ""{3 va v 82F + a{3 oaO{3la=o,(3=o
Es(r) =
{32 82F oa2 Ia=o,(3=o + 2 0{32Ia=(3=o
(2.1.176)
(1 - ksk s) Eo Jr dx'dy' {F(O,O) + aFa(O, 0) + (3F f3(O, 0) A
~2 Faa(O,O) + ~2 F(3(3(O, 0) + a{3Faf3 (O,O)} ikd r'
(2.1.177)
Integrating by parts and ignoring edge effects give
Ao kdz dz
r dx'dy,O~eikd3XI+ik'J)lyl+ik,jzf(xl,y/) = _ kdx Jr dx'dy'eik,j.rl J 8x r dx'dy'{3/kd.r' ~dY lAo dx'dy'eikd.r' r lA,
(2.1.178) (2.1.179)
2 KIRCHHOFF AND RELATED METHODS
(2.1.183)
EsCi')
i:~:r (1 - ieslcs)Eo' Lo dx'dy' {F(kl. s,kl. i ) + 2:dZ fxx(x', y')Faa(O, 0)
+ 2Lz fyy(x ' , y')F(3{3(O, 0) + k:
zfxy(x', y')Fa{3(O, O)} i
kd r'
(2.1.184)
First, we calculate the coherent field. Note that (f(x, y)f(x', y')} = h 2C(x - x', Y - y') = h 2 C(lx - x'l,
Iy -
y'l)
(2.1.185)
Since f(x, y) is a Gaussian random process, then f(x, y), ax(x, y), f3y(x, y), and fxy(x, y) are joint Gaussian random processes with correlation functions. 282C (x-x',y-y') (2.1.186) (ax (x, y )f( x' , Y'))=h 8x 2 2 ((3y(x, y)f(x', y')) = h 28 C(x ~:~' y - y') (2.1.187)
1 - x', (fxy (x, y )f( x, Y')) = h282C(x 8x8y Y - y')
(2.1.188)
The variances are
(aAx, y)ax(x, y)) = h2Cxxxx(0, 0) (f3y(x, y)f3y(x, y)) = h2Cyyyy(0, 0) (fxy(x, y)fxy(x, y)) = h2CXYXY(0, 0)
(2.1.189) (2.1.190) (2.1.191)
1.3 Second-Order Slope Corrections
Using the correlation functions and variances, the joint probability density functions of the joint Gaussian processes can be calculated readily. These joint probability density functions can then be used to calculate statistical averages. For coherent field, we have
(eikdz!(x,y)) = e-~h2k~z
(2.1.192)
For two zero-mean jointly Gaussian random variables Xl and X2, we have (2.1.193) where 0"1 and 0"2 are the standard deviations of Xl and X2, respectively, and p is the correlation coefficient. Then
(X2 exp(ivxl)) = iV(XlX2) exp ( --2-1 2 2 V (X ))
(2.1.194)