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

A doubling-adding method to include multiple orders of rotational Raman scattering 21
where ∆τ ′ ≡ ∆τ(ν ′ ) is the optical thickness of the model atmosphere layer at wavenumber ν ′ , and
δ represents Dirac’s delta function. Analogous to Eq. (2.3), the transmission operator for the diffuse
light, T dif , is defined with the integral kernel T dif .
Let us assume that the reflection and transmission properties are known for two separate layers a
and b. Then, with the definitions of the reflection and transmission operator in Eqs. (2.2) and (2.3),
the reflection and transmission properties of the combined layer ab are given by the adding equations.
These can be adopted in a straightforward manner from the corresponding equations including only
elastic scattering (e.g. Lacis and Hansen [1974]), viz.
R ab = R a + T 0,a U + T ∗ dif,a U ,
R ∗ ab = R ∗ b + T 0,b U∗ + T dif,b U ∗ ,
T dif,ab = T dif,b T 0,a + T 0,b D + T dif,b D ,
T ∗ dif,ab = T ∗ dif,a T 0,b + T 0,aD ∗ + T ∗ dif,a D∗ .
The auxiliary operators
D = T dif,a + ST 0,a + ST dif,a ,
D ∗ = T ∗ dif,b + S∗ T 0,b + S ∗ T ∗ dif,b ,
U = R b T 0,a + R b D ,
U ∗ = R ∗ aT 0,b + R ∗ aD ∗ ,
describe the transmission and reflection properties at the layer interface between a and b. Here, the
operators
S =
∑ ∞ Q n ,
S ∗ =
n=1
∑ ∞ (2.8)
Q ∗ n ,
n=1
describe multiple reflections of light between the layers with
and
Q n = Q 1 Q n−1 ,
Q ∗ n = Q ∗ 1 Q ∗ n−1 ,
Q 1 = R ∗ aR b ,
Q ∗ 1 = R b R ∗ a .
(2.6)
(2.7)
(2.9)
(2.10)
At first glance, the operator products in Eqs.(2.6), (2.7), (2.9) and (2.10) might look simple, but
each product involves an integration over all possible angles and wavenumbers. For example, the
integral kernel for R ∗ aR b is calculated as
R a R ∗ b (ν,µ,ϕ;ν′ ,µ ′ ,ϕ ′ ) =
1
π
∫ ∞
0
∫ 2π ∫ 1
dν ′′ dϕ ′′ dµ ′′ µ ′′ R a (ν,µ,ϕ;ν ′′ ,µ ′′ ,ϕ ′′ )Rb(ν ∗ ′′ ,µ ′′ ,ϕ ′′ ;ν ′ ,µ ′ ,ϕ ′ ).
0
0
(2.11)