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

44 Chapter 3
if we use the effective scattering phase matrix
Z cab (˜λ, ˜Ω,Ω) = ∑ N=
N 2 ,O 2 ,Ar
χ N
β cab
scat,N
with the total scattering coefficients
β cab
scat(z,
(z, ˜λ)
˜λ)
Zcab
N (˜λ, ˜Ω,Ω) (3.8)
βscat(z, cab ˜λ) = ∑ χ N βscat,N(z, cab ˜λ) (3.9)
N=
N 2 ,O 2 ,Ar
and an analogous expression for the Raman scattering contribution. The explicit form of the scattering
phase matrices as well as the scattering coefficients are given in Appendix 3.A. The right-hand side
of Eq. (3.2) provides the source of light and can either be a volume source inside the atmosphere or a
surface source chosen to reproduce the incident flux conditions at the boundaries of the atmosphere,
or some combination of the two. In the ultraviolet and visible part of the spectrum the radiation source
S is determined by the unpolarized sunlight that illuminates the top of the Earth atmosphere:
S(z,Ω,λ) = µ 0 δ(z−z top )δ(Ω−Ω 0 )F 0 (λ). (3.10)
Here, z top is the height of the model atmosphere, Ω 0 = (−µ 0 ,ϕ 0 ) describes the geometry of the
incoming solar beam (we define µ 0 > 0), and F 0 is given by
F 0 (λ) = [F 0 (λ), 0, 0, 0] T , (3.11)
where F 0 is the solar flux per unit area perpendicular to the direction of the solar beam.
Because the reflection of light at the ground surface is already included in the radiative transfer
equation (3.2) and the incoming solar beam is represented by the radiation source S in Eq. (3.10), the
intensity vector I is subject to homogeneous boundary conditions:
I(z top ,Ω,λ) = [0, 0, 0, 0] T for µ < 0
I(0,Ω,λ) = [0, 0, 0, 0] T for µ > 0. (3.12)
In combination with these boundary conditions, the radiation source S can be interpreted as located
a vanishingly small distance below the upper boundary. Similarly, the surface reflection takes place a
vanishingly small distance above the lower boundary (see e.g. Morse and Feshbach [1953]).
The solution of Eq. (3.2) gives the internal intensity vector field inside the model atmosphere.
In the context of measurement simulations, one is generally interested in a certain radiative effect E
of this field. This scalar quantity can be derived from the vector intensity field with an appropriate
response vector function R through the inner product [Marchuk, 1964, Bell and Glasstone, 1970, Box
et al., 1988]
E = 〈R, I〉 . (3.13)