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

A vector radiative transfer model using the perturbation theory approach 45
Here, the inner product is defined by the phase space integration
∫ ∞ ∫ ztop
∫
〈I 1 , I 2 〉 = dλ dz dΩ I T 1 (z,Ω,λ)I 2 (z,Ω,λ) (3.14)
0
0
4π
for two arbitrary vector functions I 1 and I 2 .
For retrieval purposes of atmospheric constituents from space borne measurements one is typically
interested in a radiative effect E given by one of the four Stokes parameters (or a linear combination
of those) at a wavelength λ v , at the top of the model atmosphere pointed toward the viewing direction
of an instrument. In this particular case the response function for the i-th Stokes parameter is given
by
R i (z,Ω,λ) = δ(z−z top )δ(Ω−Ω v )δ(λ−λ v )e i , (3.15)
where e i is the unity vector in the direction of the i-th component of the intensity vector, and Ω v =
(µ v ,ϕ v ) denotes the viewing direction of the instrument.
3.2.2 Decomposition of the radiative transfer problem
The spectral coupling of the intensity vector field due to inelastic Raman scattering renders any solution
approach of the integro-differential equation (3.2) difficult. However, in the case of rotational
Raman scattering the coupling is weak. One solution technique is to solve first the radiative transfer
equation, which takes only elastic scattering processes into account and to treat the wavelength coupling
due to inelastic Raman scattering as a perturbation effect. In other words, we decompose the
transport operator ˆL in two parts,
ˆL = ˆL o + ∆ˆL. (3.16)
Here ˆL o describes a monochromatic radiative transfer problem
ˆL o I o = S, (3.17)
which can be solved with standard techniques (see e.g. Lenoble [1993]), and ∆ˆL represents a corresponding
perturbation in transport given by inelastic scattering.
For example, we can choose a radiative transfer operator ˆL o , which describes monochromatic
radiative transfer including pure Rayleigh scattering. So ˆL o is defined by Eq. (3.3) but with a scattering
source
J o (z, ˜Ω, ˜λ|z,Ω,λ) = δ(λ−˜λ) βray scat(z, ˜λ) Z ray (˜λ,
4π
˜Ω,Ω). (3.18)
Here, β ray
scat is the total Rayleigh scattering coefficient for an ensemble of N 2 , O 2 , and Ar molecules,
which is defined analogous to Eq. (3.9), and Z ray represents the corresponding effective scattering
phase matrix.