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(3.5.35) where "7 is the wave impedance in region O. For & = ~ = v, Pvv corresponds to the power of thermal emission from area Ao in solid angle dO. in polarization v in the direction So and for frequency interval !:::..wj(21f). For & = ~ = h, Phh corresponds to power of emission of horizontal polarization. For & = v and ~ = h, 2 Re( Pvh) and 2 Im( P vh ) correspond to correlation of the emission in vertical and horizontal polarizations and are known as third and fourth Stokes parameters. These correlation parameters can be nonzero even though the thermal emission as given by (3.5.32) is uncorrelated. From (3.5.33) and (3.5.34) we obtain
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-* 2 . (E(ro,w)E (ro,w'))r o 1 1m "0---+ 00 2"7
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3 r2 = 2W J..L2 8(w - w')
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ME"(r)KT(r)Gol(ro,r). GOl(ro,r) (3.5.36)
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where COl is the dyadic Green's function of half-space medium with source in region 1 and field point in region O. It is the exact solution of the dyadic Green's function of the half-space medium problem including surface and volume scattering effects. Equation (3.5.35) is a result of the field approach and will include coherent effects in thermal emission problem and can be
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5.4 Emissivity of Four Stokes Parameters
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used in the case where there is a temperature distribution. There are four brightness temperatures TBv(So), TBh(So), UB(So), and VB(So); these are known, respectively, as vertical polarization, horizontal polarization, and third and fourth Stokes parameters. The relations between brightness temperature and Paf3 are defined as follows:
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Pvv ] [TBV(SO)] Phh TBh(So) K ~w 2Re(Pvh ) = UB(so) ).227fAocosOodOo [ 2Im(Pvh) VB(so)
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(3.5.37)
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5.4 Emissivity of Four Stokes Parameters
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The emsissivity of the third and fourth stokes parameters in passive remote sensing was recognized in the mid-1980s [Tsang, 1984]. It was later extended to rough surface with anisotropy [Tsang and Chen, 1990; Tsang, 1991]. Subsequently, extensive theoretical and numerical calculations were performed [Yueh et al. 1994, 1997; Li et al. 1994; Johnson et al. 1994; Gasiewski and Kunkee, 1994]. Observations Were made on soil surfaces [Veysoglu et al. 1991]
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3 FUNDAMENTALS OF RANDOM SCATTERING
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Figure 3.5.5 Active remote sensing from an inhomogeneous medium with non-uniform temperature distribution.
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and later by aircraft measurements [Yueh et al. 1995, 1997]. Satellite measurements are being planned. In this section, we shall derive emissivity in terms of bistatic scattering coefficients. This has been done for vertical and horizontal polarized emissivities using Kirchhoff's law in Section 5.2. In this section, the derivation is based on the result of fluctuation-dissipation theorem as given by (3.5.38). Furthermore, we shall derive emissivities for all four Stokes parameters. Consider active remote sensing of the same medium of Fig. 3.5.4 with an incident plane wave launched in the direction of Sob = -So that is opposite to the emission direction (Fig. 3.5.5). Let the incident electric field be of unit amplitude with the polarization O:b. Then the incident field is (3.5.39) Let E s be the scattered field in region 0 and let the electric field inside the scatterer due to the incident wave be E1('f). We express E1(f) in terms of G lO (f, fo). By definition of the index notation of dyadic Green's function G lO (f,f o ) . ei is the electric field in medium 1 due to an electric current source abo(f - fo)/(iw/-L). In the limit of To ---+ 00, this corresponds to a plane incident wave in direction Sob with incident electric field (3.5.40)
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5.4 Emissivity of Four Stokes Parameters
Thus lim G lO (r,1' o )'
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r 0--+00
is the electric field in medium 1 due to incident
plane wave of (3.5.39). Using the linear property of Maxwell's equations and comparing (3.5.40) with (3.5.39), we obtain
ro-.oo
lim G lO (r,r o )'
= -El(r)
41fr o
eikro _
(3.5.41)
Maxwell's equations in region 1 are
\7 X El = iWIlB 1 \7 x HI = -iwq(1')El
The boundary conditions on S are
(3.5.42) (3.5.43) (3.5.44) (3.5.45)