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

14 Chapter 1
1.4.2 Inclusion of inelastic scattering in radiative transfer modeling
The radiation field that leaves the Earth’s atmosphere can be simulated by solving the vector radiative
transfer equation in the plane-parallel approximation given the appropriate boundary conditions
[Chandrasekhar, 1960]. This equation, which can be derived from the Maxwell equations
[Mishchenko et al., 2006], is a balance equation that links the intensity vector I to sinks and sources
of radiation in the medium, i.e.
µ d dz I(z,Ω,λ) = −n(z)σ ext(z,λ)I(z,Ω,λ) + J(z,Ω,λ) (1.8)
Here, the Stokes parameters depend on the altitude z in the model atmosphere, the wavelength λ and
the propagation direction of the radiation Ω = (µ,ϕ). Here, µ is the cosine of the solar zenith angle
and ϕ is the relative azimuthal angle. The left-hand side in Eq. (1.8) is called the streaming term.
It involves the change of the intensity vector in the direction Ω along a path-length dz/µ through
the medium. The first term on the right hand side of Eq. (1.8) describes the extinction of radiation,
where σ ext is the extinction cross section (in cm 2 ) of the ensemble of particles and n(z) is the particle
number density (in cm −3 ) at altitude z. The extinction term describes the removal of radiation from
the beam by (1) elastic scattering, (2) inelastic scattering, and (3) absorption.
The second term on the right hand side of Eq. (1.8) is a source term. Thermal emission by the
Earth’s atmosphere and surface can be neglected in the ultraviolet, visible and near infrared wavelength
range and therefore the source function is given by
J(z,Ω,λ) = n(z)σ ext(z,λ)
4π
∮
4π
∫ ∞
dΩ ′ dλ ′ Φ(z,Ω,λ|z,Ω ′ ,λ ′ )I(z,Ω ′ ,λ ′ ). (1.9)
0
Here, Φ is the 4 × 4 transformation matrix. The source function J describes how incident polarized
radiation from all possible directions (Ω ′ ) and with all possible wavelengths λ ′ is added to the beam of
interest with direction Ω and wavelength λ. Compared to the standard elastic vector radiative transfer
equation the source function has two additional dimensions, namely the wavelength of the incident
light λ ′ and the wavelength of the scattered light λ. The coupling of light with different wavelengths,
from different directions, and the coupling of the Stokes parameters is implemented by using the
extended source function in Eq. (1.9).
The solution of the vector radiative transfer equation (Eq. (1.8)) involves thus multiple elastic
and inelastic scattering processes, which together build up the radiation field that emerges from the
terrestrial atmosphere into the satellite instrument’s line-of-sight.
1.4.3 Current status
To interpret the reflectivity spectra that are measured by instruments such as GOME, SCIAMACHY,
OMI and GOME-2 accurate radiative transfer simulations are required. Many well-verified methods
exist to solve the vector radiative transfer equation including multiple elastic (Rayleigh) scattering
(see e.g. [Lenoble, 1985] for an overview). Several approximations to include rotational Raman