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

A doubling-adding method to include multiple orders of rotational Raman scattering 23
transmission from wavenumber ν j to wavenumber ν i ; the next two indices, p and q, describe reflection
or transmission from zenith angle µ q to µ p in the mth Fourier mode. Evaluation of an operator product
of the type R ∗ aR b in Eq. (2.11) simplifies to calculating the matrix product
[R a R ∗ b] m ij;pq = [1 + δ 0m]
k∑
max r∑
max
k=k min r=1
∆ν k w r µ r [R a ] m ik;pr [R∗ b] m kj;rq , (2.15)
where summation over all Gaussian points r = 1,..,r max and all wavenumber bins k = k min ,...,k max
takes place. The factor π [1 + δ 0m ] in Eq. (2.15) arises from the integration over azimuthal angle of
products of cos m[ϕ − ϕ ′ ]-type terms which appear in the Fourier expansion of Eq. (2.14). Here, δ 0m
is the so-called Kronecker-delta.
Since the computation time to calculate a matrix product of the type in Eq. (2.15) is proportional
to the number of wavenumbers used to the third power, and since the accessible computer memory
is limited, this number should be reduced as much as possible to make a numerical implementation
feasible. We propose to evaluate a compact wavenumber grid with only the rotational Raman lines
which are significant with respect to a certain threshold, for incoming sunlight at a given wavenumber
ν ′ in. Significant lines are determined by convolving the single scattering albedo ω(ν;ν ′ ) with itself to
get the scattering probability distribution for the nth order of scattering:
ω n (ν;ν ′ in) =
∫ ∞
0
dν ′′ ω(ν;ν ′′ )ω n−1 (ν ′′ ;ν ′ in). (2.16)
In other words, ω n (ν;ν ′ in) describes the probability that incoming light at wavenumber ν ′ in ends up
at a wavenumber ν after n times of scattering. We compare this with the corresponding scattering
probability of the nth order of elastic Rayleigh scattering, ω n ray(ν ′ in), which can be interpreted as the
probability that scattering in general takes place. Then, significant lines are defined to be lines that
satisfy
r(n max ,ν;ν ′ in) > ǫ, (2.17)
with
r(n max ,ν;ν ′ in) =
n∑
max
n=1
n max
∑
n=1
ω n (ν;ν ′ in)
ω n ray(ν ′ in ) , (2.18)
where ǫ is a given threshold and n max is the maximum order of inelastic scattering in the determination
of the optimized wavenumber grid. The threshold ǫ and n max should be chosen in such a way that
they have little influence on the end result. It is important to realize that fixing n max does not mean
that scattering of inelastic scattering of orders n > n max does not occur; it means that higher order
scattered light can end up in the incorrect wavenumber bins, because the wavenumber bins associated