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r dn' per, s, s') Y(r, s')
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where per, s, s') is the phase matrix giving the contributions from direction s' into the direction s. In (7.2.1), K,e is the extinction matrix for Stokes parameters due to the scatterers, ]e is the emission vector, and K ag is absorption coefficient for the background medium which is assumed to be isotropic. In general, extinction is a summation of absorption and scattering. For nonspherical particles, the extinction matrix is generally nondiagonal, and the four elements of the emission vector are all nonzero. In active remote sensing, a wave is launched by a transmitter onto the medium. In the following sections, expressions for the phase matrix, extinction matrix, and the emission vector shall be derived for the case of independent scattering. They can all be expressed in terms of the scattering function matrix.
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2.1 Phase Matrix of Independent Scattering
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The phase matrix of a collection of particles including coherent collective scattering effects will be treated in later chapters. In this section, we treat the cases of independent scattering. The Stokes matrix relates the Stokes parameters of the scattered wave to those of the incident wave, whereas the scattering function matrix relates the scattered field to the incident field. For the case of incoherent addition of scattered waves, the phase matrix will just be the averages of the Stokes matrices over orientation and size of the particles. Thus, we shall study the Stokes matrix of a single particle. Consider a plane wave
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~) = (ViEvi + hiEhi eik::;: = ei Eo eik::;:
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(7.2.2) (7.2.3) (7.2.4)
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impinging upon the particle. In spherical coordinates we have
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sin Bi cos </>d; + sin Bi sin </>i'fJ + cos BiZ = cos ()i cos </>/ + cos ()i sin </>i'fJ - sin ()i Z
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7 RADIATIVE TRANSFER THEORY
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sin <PiX + cos<Pi'fJ
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(7.2.5)
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In the direction ks , the far-field scattered wave E s will be a spherical wave and is denoted by
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E s = (Evsv s + Ehsh s )
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(7.2.6)
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(7.2.7) (7.2.8) (7.2.9)
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Vs = h s = - sin<psx + cos<psY
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sin Os cos <Psx + sin Os sin <PsY + cos Osz cos Os cos<psx + cos Os sin<psY - sinOsz
The scattered field will assume the form _ eikr = E s = - F(Os, <Ps; 0i, <Pi) . ei Eo r where F(Os, <Ps; Oi, <Pi) is the scattering function matrix. Hence ikr Evs] = e [fvv(OS,<PS:Oi,<Pi) fVh(OS'<PS~Oi'<Pi)] [EVi] [ Ehs r fhv(Os, <Ps, Oi, <Pi) fhh(Os, <Ps, Oi, <Pi) Ehi with
(7.2.10)
(7.2.11)
fab(Os, <Ps; Oi, <Pi) = as . F(Os, <Ps; Oi, <Pi) . bi (7.2.12) and a, b = v, h. To relate the scattered Stokes parameters to the incident Stokes parameters, we apply the definitions in 3:
Is = 2" L(Os, <Ps; Oi, <Pi) . Ii
(7.2.13)
where
Is and Ii are column vectors
(7.2.14)
(7.2.15)
2.1 Phase Matrix
-Im{f;hfvv) ] -Im{fhvfhh ) -Im{fvvfhh - fvhfhv ) Re{fvvfhh - fvhfhv ) (7.2.16)
Because of the incoherent addition of Stokes parameters, the phase matrix is equal to the average of the Stokes matrix over the distribution of particles in terms of size, shape, and orientation. For example, for ellipsoids with axes a, b, e and orientation Eulerian angles a, (3, ')', with respect to the principal frame, the phase matrix is
P((), </>; ()', <//) =
J JJJ J J
da db de da d(3
d')'p(a,b,e, a, (3,')')L((),</>;()',</>') (7.2.17)
where pea, b, e, a, (3, ')') is the probability density function for the quantities a, b, e, a, (3, and ')'. If the particles are all identical in shape and orientation, then
P((), </>i ()', </>') = noL((), </>; ()', </>')
(7.2.18)
The phase matrix of small spheres, small ellipsoids, and random media were treated in 3. The relation of scattering amplitudes to T-matrix was discussed in 2. In many problems of practical application, the scatterers are assumed to have an axis of symmetry. In the following, we describe how the scattering amplitudes of the body frame can be transformed to that of the principal frame for such axisymmetric objects. Consider an incident plane wave in the direction Si impinging on the axisymmetric object with axis Zb. The subscript b denotes body frame with Xb,Yb, and Zb as axes. The axis of symmetry Zb is at an angle with respect to the principal frame x, Y, and Z :