Active Remote Sensing of a Layer of Nonspherical Particles in .NET

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1.4 Active Remote Sensing of a Layer of Nonspherical Particles
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and at z ==-d
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1(0, <jJ, z
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= -d) = R(O) 1(1f -
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0, <jJ, z = -d)
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(8.1.68)
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for 0 ~ 0 ~ 1f/2. In (8.1.68), R(O) is the reflectivity matrix for the interface separating region 1 and region 2 and is given by
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-1m (Rv(O)Rh(O)) o Re (Rv(O)Rh(O)) (8.1.69) where R v and Rh are, respectively, the Fresnel reflection coefficients for vertically and horizontally polarized waves. To derive integral equations, we can regard S(O, <jJ, z) and W(O, <jJ, z) in (8.1.63) and (8.1.64) as source terms. The homogeneous solutions are then given by the eigensolutions of coherent propagation in 7. The particular solution can be calculated by the method of variation of parameters. The arbitrary constants of the homogeneous solution are solved by imposing the boundary conditions of (8.1.67) and (8.1.68). The two coupled integral equations for the upward- and downward-going specific intensities are 1(1f - 0, <jJ, z) = E(1f - 0, <jJ)D ((3(1f - 0, <jJ)z sec 0) 0 E- 1 (1f - 0, <jJ)106(cosO - cosOo)6( > - cPo)
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R(O) - [
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Re (Rv(O)Rh(O)) 1m (Rv(O)Rh(O))
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1 dz' { E(1f -
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0, <jJ)D ((3( 1f - 0, <jJ )(z - z') sec 0)
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E- 1 (1f - 0, W(O, <jJ,
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(8.1.70)
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1(0, >, z) = E(0, > )D (-(3(0, <jJ) sec O(z + d)) E- 1 ( 0, <jJ )R(0) E( 1f - 0, <jJ )D (-(3( 1f - 0, <jJ)d sec 0)
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E- 1 (1f - O,<jJ)106(cosO - cosOo)6(<jJ - <jJo)
+ E(0, <jJ )D (- (3( 0,
sec 0
sec O(z
+ d))
0, <jJ) 0, >)W (0, >, z') }
fO dz' {E- 1 <jJ)R(O)E(1f (O,
D (- (3 (1f - 0, <jJ) sec 0(z'
+ sec 0
+ d)) E- 1 (1f -
dz'E(O,<jJ)D ((3(0, <jJ) secO(z' - z))
(8.1.71)
E- 1 (0, <jJ)S(O, <jJ, z')
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
where D(f3(e, cP)z sec e) is a 4 x 4 diagonal matrix with the iith element equal to exp((3i(e,cP)zsece). The (3i are the eigenvalues of coherent propagation and E is the associated eigenmatrix ( 7). In applying the iteration method, we treat the first term on the righthand side of (8.1.70) and (8.1.71) as the zeroth-order solution for the downward-going and upward-going specific intensities, respectively. The zeroth-order solution corresponds to coherent wave propagation of Stokes parameters. The first-order upward-going specific intensity at z = 0 is listed below. The first-order solution is also known as first-order multiple scattering.
IP)(O, cP, z = 0)
L. sec e{ E(e, cP)D (-(3( e, cP)d sec e) E- (e, cP)R(e)E(
7r -
e, cP) }
{E- 1 (7r - e,cP)p(7r - e,cP;eo,cPo)E(eo,<po)} ki
1e-{3dn:-8,cP)dsec8-{3i(80,cPo)dsec80
(3k(7r - e, cP) sece + (3i(eo,cPo) sec eo
{= (eo, cPo)R(e ) = E1
. E( 7r
eo, cPo)D (-(3( 7r - eo, cPo)d sec eo) E- 1 ( 7r - eo, cPo)!0 }
sec e{ E(e, cP)D (-(3( e, cP)d sec e) E- 1 (e, <p)R(e)E( 7r
e, cP) } lk
x {E- 1 (7r - e, <p)P(7r - e, cP; 7r - eo, cPo)E(7r - eo, <Po)} ki
e-{3dn:-8,cP)dsec(} - e-{3,(n:-8 0 ,cPo )dsec80
(3i (7r -
eo, <Po) sec eo - (3k( 7r - e, <p) sec e
{=-1 (7r-e ,<po)I _} E
+ sec e L
E1k(e, <p) { E- 1 (e, cP)P(e, <p; eo, cPo)E(eo,<Po) } ki
e-{3i(80,cPo)dsec(}0
e-{3k(8,cP)dsec8 -
(3i(eo,cPo) sec eo - (3k(e,cP)sece
{= (0 E1
<Po)R(Oo)
. E( 7r - eo, <po)D (-(3( 7r - eo, <Po)d sec 00 ) E- 1 (7r
eo, cPo)!0
(8.1.72)
denotes
+ sec e L
Elk(e, <p) { E- 1 (e, cP)P(e, cP; 7r - eo, <po)E( 7r - eo, cPo) } ki
e-{3k (8,cP)dsec 8-{3i (n:-8 0 ,cPo)d sec
(3 k(e, <p) sec e + (3i (7r - eo, <Po) sec eo where the summation over indices k and i are from 1 to 4 and {
8 {=-1 (7r-e ,cPo)I } E
1.4 Active Remote Sensing of a Layer of Nonspherical Particles
the ijth element of the 4 x 4 matrix. Given the vector radiative transfer equation (8.1.63) and the boundary conditions in (8.1.67) and (8.1.68), the Stokes vector can be calculated either numerically or iteratively. Once that is solved, the overall scattered Stokes vector in direction (Os, <Ps) that is observed by the receiver is Is(Os, <Ps) = I(Os, <Ps, z = 0) = [Ivs , hs, Us, 11,,] and can be calculated. It is proportional to 10 with the proportionality represented by the 4 x 4 averaged Mueller matrix M(Os, <Ps; 7r - 00 , <Po) as follows (8.1.73) The Mueller matrix represents the overall polarimetric characteristics of the layer of random discrete scatterers including all multiple scattering effects and boundary reflections that are included in the vector radiative transfer theory. The results based on the Mueller matrix will be illustrated for the backscattering direction with Os = 00 and <Ps = 7r + <Po. Based on the Mueller matrix, we shall illustrate four polarimetric signatures: the phase difference, the copolarized return, the depolarized return, and the degree of polarization. The phase difference between vv and hh waves, <Pvh, (the phase at which the probability density function of phase difference is maximum [Sarabandi, 1992]) is