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

32 Chapter 2
scattering event. In both cases the nth order term of the Dyson series represents the contribution of
n order of Raman scattering to the Green’s operator G. For numerical applications, we truncate the
Dyson series in first order, hence
G ≈ G o − G o ∆LG o . (2.23)
Now the differences between the two applications of the perturbation theory become clear, i.e. resulting
from the the different choice of both G 0 and ∆L. In the case of the unperturbed problem
including purely Cabannes scattering, Raman scattering processes of second order and higher are ignored,
whereas in the second case these scattering processes are effectively described by Rayleigh
scattering. In that sense both applications correspond to approaches A and B mentioned above. The
corresponding numerical applications of the perturbation theory will be referred to as the PTA and the
PTB model.
First of all, we compared PTA model simulations with DA1 simulations. Here it is important
to notice that, although both models describe multiple Cabannes scattering and one order of Raman
scattering, they do not solve exactly the same problem. The DA1 model takes into account that a
Raman scattering event influences the succeeding propagation of light. This is not the case for the
PTA model. The propagation before and after the perturbation is kept fixed, as can be seen in the
second term in Eq. (2.23), where G o describes the propagation of scattered light in a pure Cabannes
scattering atmosphere. However, the model simulations show that this is only a minor effect that
causes differences less than 0.005% between the simulated radiance spectra.
Figure 2.5: Absolute difference in filling-in between the perturbation theory model result (PTA and PTB)
and the doubling-adding model result (DA). Lines are drawn for approach A (PTA, dotted line), where a
Cabannes scattering atmosphere is taken as the base problem, and for approach B (PTB, solid line), where
a Rayleigh scattering atmosphere is taken as the base problem. The same model parameters were used as
in Fig. 2.4.
Fig. 2.5 presents the comparison of both PTA and PTB model simulations with DA simulations,
which include multiple Raman scattering. Shown is the absolute difference in the filling-in spectra