Rotational Raman scattering in the Earth's atmosphere ... - SRON

74 Chapter 3
where S l is the expansion coefficient matrix and P l m is the generalized spherical function matrix
[Hovenier and van der Mee, 1983, de Haan et al., 1987]. For Rayleigh, Cabannes and rotational
Raman scattering these coefficient matrices can be determined straightforwardly [Stam et al., 2002].
In the corresponding Fourier expansion of the intensity field
I(z,Ω) =
∞∑
(2 − δ m0 ) [ B +m (ϕ 0 − ϕ)I +m (z,µ) + B −m (ϕ 0 − ϕ)I −m (z,µ) ] , (3.86)
m=0
two types of Fourier coefficient vectors are needed
I +m (z,µ) = 1
2π
I −m (z,µ) = 1
2π
∫ 2π
0
∫ 2π
0
dϕ B +m (ϕ 0 − ϕ) I(z,Ω) ,
dϕ B −m (ϕ 0 − ϕ) I(z,Ω). (3.87)
An expansion analogous to (3.86) holds for the radiation source S. The Fourier coefficients of the
solar source in Eq. (3.10) are given by
S +m (z,µ,λ) = 1
2π µ 0δ(z−z top )δ(µ+µ 0 )F 0 (λ) ,
S −m (z,µ,λ) = [0, 0, 0, 0] T . (3.88)
On the other hand, the Fourier coefficients of the source S ps in Eq. (3.44), which is defined by the
response function R i in Eq. (3.15), are
S +m
Ψ (z,µ,λ) = 1
2π δ(z−z top)δ(µ+µ v )e i ,
S −m
Ψ (z,µ,λ) = [0, 0, 0, 0]T . (3.89)
for i = 1, 2, but
S +m
Ψ (z,µ,λ) = [0, 0, 0, 0]T ,
S −m
Ψ (z,µ,λ) = 1
2π δ(z−z top)δ(µ+µ v )e i , (3.90)
for i = 3, 4. Hence, due to the Fourier coefficients of the solar source in Eq. (3.88) the Fourier
coefficients I −m of the forward intensity field are zero. For the adjoint fields the coefficients Ψ −m
are zero for the response function R i with i = 1, 2, while for the response function R i with i = 3, 4
the coefficients Ψ +m are zero. This simplification is possible due to the special form of the Fourier
expansion in Eqs. (3.81) and (3.86). For the types of radiation sources considered in this paper, this
reduces significantly the numerical efforts required for solving the radiative transfer equation.