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

20 Chapter 2
small [Landgraf et al., 2004]. In addition to the scalar approach we consider a plane-parallel model
atmosphere, which is a fair approximation for the simulation of space-borne measurements with a
nadir viewing geometry for moderate solar zenith angles (e.g. [Walter et al., 2004]).
The doubling-adding approach [Stokes, 1862, Peebles and Plasset, 1951, van de Hulst, 1963,
1980, Hovenier, 1971, Hansen, 1971, de Haan et al., 1987] offers an alternative solution technique
of a radiative transfer problem compared to solving the radiative transfer equation directly. Based on
the reflection and transmission properties of optically thin model layers, the corresponding properties
of the model atmosphere can be calculated using the adding scheme. Until now, this approach has
only included elastic scattering, but it can be extended in a straightforward manner to also account for
multiple inelastic scattering.
The doubling-adding method aims to calculate the reflection and transmission properties of a
model atmosphere given by the reflection and transmission operators R, R ∗ , T and T ∗ . These operators
relate the incoming radiation defined at the top and bottom of the model atmosphere to the
reflected and transmitted radiation:
I + top = R I − top ,
I − bot
= R ∗ I + bot ,
I − bot
= T I − top ,
I + top = T ∗ I + bot . (2.1)
Here, I top + and Itop − represent the intensity at the top of the model atmosphere, where superscripts “+”
and “-” indicate upward and downward directions. Similarly, I + bot and I− bot
indicate the intensity at the
lower boundary of the model atmosphere. For a plane parallel atmosphere, the Stokes parameters are
functions of wavenumber ν and of the direction of the radiation. The direction is characterized by the
cosine of the zenith angle µ, which is defined with respect to the outward normal, and the azimuthal
angle φ, which is measured counterclockwise when looking downward.
The reflection and transmission operators can be written as integral operators of the type
R I − top(ν,µ,ϕ) = 1 π
∫ ∞
0
∫ ∞
∫ 2π ∫ 1
dν ′ dϕ ′ dµ ′ µ ′ R(ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ )Itop(ν − ′ ,µ ′ ,ϕ ′ ) , (2.2)
0
0
T Itop(ν,µ,ϕ) − = 1 ∫ 2π ∫ 1
dν ′ dϕ ′ dµ ′ µ ′ T(ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ )I
π
top(ν − ′ ,µ ′ ,ϕ ′ ) , (2.3)
0 0 0
where integral kernels R and T denote the reflection and transmission functions. Analogous expressions
hold for the operators R ∗ and T ∗ .
For the numerical implementation, it is useful to separate the transmission properties of the direct
beam from those of the scattered radiation, namely
T = T 0 + T dif . (2.4)
Here, the transmission operator for the direct beam, T 0 , is defined with the integral kernel
T 0 (ν,µ,ϕ;ν ′ ,µ ′ ,ϕ ′ ) = π µ ′ e−∆τ ′ /µ ′ δ(ν−ν ′ )δ(µ−µ ′ )δ(ϕ−ϕ ′ ), (2.5)