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

A doubling-adding method to include multiple orders of rotational Raman scattering 19
sented by Kattawar et al. [1981]. Since then, several radiative transfer models were presented which
account for multiple scattering, but include one order of Raman scattering. The radiative transfer
models of Joiner et al. [1995, 2004], and of Sioris and Evans [2002b] are based on a successive order
of scattering scheme which calculates the contribution of each order of scattering for a Rayleigh scattering
model atmosphere. Each scattering order is then corrected for one possible Raman scattering
event. The models provide the exact single scattering solution including inelastic Raman scattering.
For Raman scattering occurring in second and higher scattering orders, the Raman scattering source
function has to be simplified to achieve a feasible numerical implementation [Joiner et al., 1995,
2004, Sioris and Evans, 2002b]. A different approach was presented by Vountas et al. [1998] and
Landgraf et al. [2004] (see Chapter 3). Both models divide the radiative transfer equation including
inelastic Raman scattering in the corresponding equation for a Rayleigh scattering atmosphere and a
perturbation term. The effect of this perturbation on the solution of the Rayleigh scattering problem
is described by Vountas et al. [1998] with a Picard series, whereas Landgraf et al. [2004] use a Dyson
series known from perturbation theory. The truncation of both series takes place in first order, which
in both cases corresponds to one order of Raman scattering.
The first model approach accounting for two orders of Raman scattering was presented by Stam
et al. [2002] who use the analytical solution for an atmosphere layer with second order scattering.
Therefore, two inelastic scattering processes are also included. This vector model produces the differential
Ring effect structures well. The continuum height was calculated with a separate doublingadding
model that takes multiple elastic Rayleigh scattering into account.
In this paper, we present an extension of the adding method including inelastic scattering which is
not limited with respect to the order of inelastic scattering. In Section 2, the theory of the doublingadding
method is extended to include inelastic scattering. Then, in Section 3, we discuss the numerical
implementation of the method. In Section 4 we present filling-in results for a model of the Earth’s
atmosphere at 0.2 nm spectral resolution. For the first time, we quantify the contribution of multiple
inelastically scattered radiation. Then, in Section 5, we compare the extended doubling-adding model
with the model of Landgraf et al. [2004], which takes one order of Raman scattering into account, but
which compensates for the neglect of multiple inelastic scattering to a certain degree. We summarize
our findings in Section 6.
2.2 The adding method including inelastic scattering
The interaction of radiation with the Earth’s atmosphere and the Earth’s surface is described by the
radiative transfer equation. In its scalar approach, the radiative transfer equation neglects any polarization
aspects of light. Therefore, the radiation field is characterized by the intensity field I only.
This approach simplifies the radiative transfer problem extensively. Although it is known that the
scalar approximation introduces errors [Chandrasekhar, 1960, Lacis et al., 1998, Hasekamp et al.,
2002], it is well suited to study the principal effect of multiple orders of inelastic Raman scattering.
Furthermore, we know that rotational Raman scattering has a minor effect on the Stokes parameters
Q, U and V , and that the errors in the Ring structures in I introduced by the scalar approach are