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

A vector radiative transfer model using the perturbation theory approach 47
G
G
G
o
= − + − ...
G
o
o
∆ L
G
o
∆ L
G
G
o
o
∆ L
Rayleigh
Rayleigh +
1. order Raman correction
Rayleigh +
2. order Raman correction
Figure 3.1: Illustration of the perturbation series in Eq. (3.28). The Green’s operator Ĝ may be approximated
by the unperturbed Green’s operator Ĝo plus higher order corrections. Here, each correction term
consists of a chain of elements built up from the unperturbed Green’s operator Ĝo, which describes the
propagation of light to the point of perturbation, and the perturbation of transport itself, represented by
the operator ∆ˆL.
for any radiation source S and any response function R. The operator Ĝ can be given in the form of
an integral, viz.
∫ ∫ ztop
∫ ∞
Ĝ = dΩ ′ dz ′ dλ ′ G(z,Ω,λ|z ′ ,Ω ′ ,λ ′ ) ◦ (3.24)
4π
0
0
where the kernel G is a function with a 4 × 4 matrix structure. Because of Eq. (3.21) the kernel G
has to obey the relation
ˆLG(z,Ω,λ|z ′ ,Ω ′ ,λ ′ ) = δ(z−z ′ )δ(Ω−Ω ′ )δ(λ−λ ′ )1. (3.25)
This equation can be written also as
ˆLg i (z,Ω,λ|z ′ ,Ω ′ ,λ ′ ) = δ(z−z ′ )δ(Ω−Ω ′ )δ(λ−λ ′ )e i , (3.26)
where g i is the i-th column vector of G. Here, g i represents the intensity vector field at the target
point (z, Ω,λ) for a given unity light source in the i-th Stokes parameter at the source point (z ′ , Ω ′ ,λ ′ ).
The matrix G is commonly called the Green’s function of the corresponding differential equation and
accordingly we call the inverse operator Ĝ= ˆL −1 the Green’s operator.
If the calculation of the Green’s function intends to solve the actual problem for a particular source
S, this formulation does not provide any simplification. However, the concept is needed for the perturbation
theory approach. Suppose that a simplified problem, which in our case is the monochromatic
radiative transfer problem in Eq. (3.17), is already solved with a corresponding Green’s operator Ĝo.
Then the operator Ĝ and ˆL, which belong to the radiative transfer problem including inelastic Raman
scattering, satisfy the relation
Ĝ = Ĝo − Ĝ ∆ˆLĜo. (3.27)