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

22 Chapter 2
When layers a and b are the same, the adding is called doubling. In this case, the number of calculations
can be reduced considerably due to symmetry considerations [de Haan et al., 1987].
To calculate the reflection and transmission functions for a homogeneous model layer, the doubling
scheme is applied to a very thin sublayer. In general, the optical properties of a homogeneous
model layer are characterized by its optical thickness ∆τ, its single scattering albedo ω, and its phase
function P . For a very thin layer, the single scattering approximation suffices and yields
{ }
R(ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) = 1 4 µ ′ ∆τ+µ∆τ 1−e
−∆τ/µ−∆τ ′ /µ ′ ω(ν;ν ′ )P(−µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) , (2.12)
′
⎧
1 ∆τ ′
e −∆τ/µ ω(ν;ν ′ )P(µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) if µ ′ ∆τ =µ∆τ ′ ,
4 µ µ
⎪⎨
′
T dif (ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) = {
1 ∆τ ′
4 µ ⎪⎩
′ ∆τ−µ∆τ e
−∆τ ′ /µ ′ −e −∆τ/µ} (2.13)
ω(ν;ν ′ )P(µ,ϕ;ν ′ ,µ ′ ,ϕ ′ )
′
otherwise .
∆τ ′
Here, ∆τ ′ ≡ ∆τ(ν ′ ) and ∆τ ≡ ∆τ(ν). Furthermore, we assume that the phase function P depends
only on the wavenumber of the incident light. The scattering probability from wavenumber ν ′ to ν
is given by the single scattering albedo ω(ν;ν ′ ). Besides R and T dif in Eqs.(2.12) and (2.13), which
describe the diffuse light, T 0 in Eq. (2.5) is used to describe the transmission of the direct light.
2.3 Numerical implementation
2.3.1 Discretization
In order to implement the adding equations (2.6–2.10), a proper discretization of the operator product
in Eq. (2.11) is required. First of all, the dependence on azimuthal angle needs to be circumvented
by representing the reflectivity and transmittance as a Fourier series. In the scalar approximation this
is done by expanding the reflection and transmission functions as a cosine Fourier series of the order
m max ,
R(ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) =
m∑
max
m=0
R m (ν,µ;ν ′ ,µ ′ ) cos m[ϕ−ϕ ′ ], (2.14)
where R m is the mth Fourier component [Liou, 1992]. The adding equations remain valid for each
Fourier component separately (e.g. Hansen and Travis [1974b]).
The integral over zenith angle in the operator product in Eq. (2.11) can be approximated by a
Gaussian quadrature of order r max with quadrature points µ r and weights w r with r = 1,..,r max . In
addition, the wavenumber interval ν ∈ [0, ∞] needs to be gridded. We decided to use a finite 1 cm −1
grid for this purpose, with wavenumber bins ν k and bin-widths ∆ν k = 1 cm −1 . This wavenumber
grid seems sufficient for the simulation of measurements with a spectral resolution of approximately
0.2 nm.
After a successful discretization, the reflection and transmission functions become matrices with
elements [R] m ij;pq, [T ] m ij;pq, [T ∗ ] m ij;pq and [R ∗ ] m ij;pq. The first two indices i and j describe reflection or