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

A vector radiative transfer model using the perturbation theory approach 77
with coefficients
Λ m k =
N∑
i,j=−N
j≠0
Υ m k = 1
2π
Γ m k = 1
2π
N∑
i=−N
i≠0
a i a j 〈Ψ +mT
d
(−µ i ,λ v )〉 k ∆Z m k (µ j ,µ i ) 〈I +m
d
(µ j ,λ)〉 k
1 ∑ N
µ v
a i 〈Ψ +mT
d
(−µ i ,λ v )〉 k ∆Z m k (−µ 0 ,µ i ) F 0
i=−N
i≠0
a i e T i Z m k (µ i ,µ v ) 〈I +m
d
(µ i ,λ)〉 k
Θ m k = 1
4π 2 1
µ v
e T i Z m k (−µ 0 ,µ v ) F 0 . (3.99)
Here z k−1 and z k indicate the lower and upper layer boundary, respectively, β scat,k represents the
scattering coefficient, and Z m k is the Fourier component of the scattering phase matrix in layer k.
Quantities of the form 〈f〉 k denotes a layer average defined by
〈f〉 k = 1 2 [f(z k−1) + f(z k )] , (3.100)
where f is a function of altitude z. The remainder of the integrals in Eq. (3.98) can be calculated in a
straightforward manner, which gives
[
M∑
K m (λ,λ v ) ≈ β scat,k (λ,λ v ) Λ m k + Υ m µ 0
k
β ext,k (λ) [1 − e∆τ k(λ)/µ 0
] e −τ k−1(λ)/µ 0
k=1
+Γ m k
+Θ m k
1
β ext,k (λ v ) [1 − e∆τ k(λ v)/µ v
] e −τ k−1(λ v)/µ v
µ 0 e −τ k−1(λ v)/µ v
e −τ k−1(λ)/µ 0
β ext,k (λ v )µ 0 + β ext,k (λ)µ v
[1 − e ∆τ k(λ v)/µ v
e ∆τ k(λ)/µ 0
]
]
. (3.101)
An analogous approach holds for the coefficients (3.54), which allows one to evaluate the perturbation
integral in Eq. (3.46).