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

A doubling-adding method to include multiple orders of rotational Raman scattering 25
with n > n max are not necessarily present in the optimized grid. This small amount, which ends
up wrongly, is distributed to neighboring wavenumber bins which are present in the optimized grid.
Fig. 2.1 shows the relevance of the Raman lines versus wavenumber shift for n max =1,2,3.
Keeping only the wavenumber bins that satisfy the criterion in Eq. (2.17), we end up with a
wavenumber grid which involves considerably less wavenumber bins than the original 1 cm −1 grid.
This compressed grid is kept fixed throughout the model atmosphere. In the case of multiple layers,
the wavenumber grid of the layer with the most significant wavenumber bins is chosen and is applied
to the other layers. With the use of this optimized grid, the numerical implementation of the matrix
products of the type in Eq. (2.15) becomes feasible.
The optimization of the wavenumber grid has a dramatic impact on the calculation method of the
reflection and transmission matrices of the whole model atmosphere. Because the optimization of
the wavenumber grid is only valid for one particular wavenumber ν in, ′ the radiative transfer problem
needs to be chopped up in pieces. Instead of calculating one big matrix containing all the reflection
or transmission properties for all (ν,µ;ν ′ ,µ ′ )-combinations, a separate doubling-adding calculation
needs to be performed for each incoming wavenumber ν in.
′
In general, the choice of any finite wavenumber grid automatically leads to loss of radiation at the
borders of the spectral range of consideration. We propose to put the light that is scattered outside
this spectral range into the corresponding elastic scattering component, such that no radiation is lost.
This results in a single scattering albedo matrix ω ij = ω(ν i ;ν j ) which has both Cabannes and Raman
scattering contributions on its diagonal elements in cases where the wavenumber shift of Raman
scattering exceeds the borders of the spectral range. In this way, for each incoming wavenumber
ν j , all probabilities ω(ν i ;ν j ) add up to the total scattering probability ω ray (ν j ). The direct result is
that energy is not only conserved for single scattering, but also for higher orders of scattering. The
doubling-adding model which takes both multiple scattering and energy conservation into account as
described above will be referred to as the DA (Doubling-Adding) model in this paper.
The single scattering albedo matrix can be modified in such a way that only one order of Raman
scattering is allowed, which is very useful in the comparison with other available algorithms. This
single scattering albedo matrix,
ω DA1,ij =
{
ω(ν i ;ν j ) for ν j =ν in ′ ,
ω cab (ν j )δ ij otherwise ,
(2.19)
is constructed by setting all off-diagonal elements to zero except the ones in the column corresponding
to ν ′ in. The diagonal solely contains the scattering probability for elastic Cabannes scattering, ω cab (see
Appendix). The version of the doubling-adding model which is initialized with ω DA1,ij will be referred
to as DA1. This model properly accounts for multiple elastic Cabannes scattering and includes one
order of inelastic Raman scattering; multiple inelastic scattering processes are ignored.
2.3.2 Trade-off between accuracy and computational effort
Before modeling of a realistic Earth atmosphere can be initiated, appropriate values for n max and
ǫ need to be determined. These parameters govern the balance between accuracy and computational