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

50 Chapter 3
The forward and adjoint formulations of radiative transfer do not describe two independent radiative
transfer problems. The corresponding intensity fields I and I † are linked by the relation
〈S † , I〉 = 〈I † , S〉 , (3.39)
which can be derived using Eqs. (3.2), (3.32) and (3.33). For the simulation of a radiation effect E we
choose the corresponding response function R as the adjoint source S † . In this particular case, the left
hand side of Eq. (3.39) represents the definition of the radiation effect E in Eq. (3.13) and the right
hand side provides a second recipe to calculate E using the adjoint field. Here, one weighs the source
S by the adjoint intensity field I † and integrates the product over the phase space of the problem. In
other words, the adjoint field I † (z,Ω,λ) can be interpreted as the importance of the source S(z,Ω,λ)
with respect to the radiation effect E [Lewins, 1965, Gerstl, 1980]. In particular, due to the specific
response function R i in Eq. (3.15) the solution of the adjoint problem
has the form
ˆL † o I † i,o = R i (3.40)
I † i,o (z,Ω,λ) = I† i,o (z,Ω,λ v)δ(λ−λ v ). (3.41)
The use of the response function as an adjoint source allows one to rewrite the perturbation series in
Eq. (3.31) to first order as
E i = E i,o + 〈I † i,o , ∆ˆLI o 〉 + O(∆ 2 ), (3.42)
where I † o and I o are the adjoint and forward solutions of the unperturbed radiative transfer problem 1 .
Here, O(∆ 2 ) indicates higher orders of perturbation. Thus, the first order perturbation effect is solely
determined by the adjoint and forward intensity field. This represents a significant simplification for
any computational effort, because the numerically expensive calculation of the full Green’s function
is avoided.
For the numerical calculation of I † i,o another advantage of the adjoint formulation is that both the
forward and the adjoint formulation can be solved with one radiative transfer code. The definition of
the vector field
Ψ i (z,Ω,λ) = I † i,o (z, −Ω,λ) (3.43)
allows one to transform the adjoint equation (3.33) into a corresponding pseudo forward equation
[Bell and Glasstone, 1970, Box et al., 1988, Hasekamp and Landgraf , 2002], viz.
ˆL ps
o Ψ i = S ps
i (3.44)
1 To derive Eq. (3.42) one makes use of the fact that the operator Ĝ† is the adjoint operator of Ĝ. To show this, we
write Eq. (3.39) in the form 〈S † , ĜS〉 = 〈Ĝ† S † , S〉 using Eqs. (3.22) and (3.38). This is true for any sources S † and S
and thus Ĝ† is the adjoint operator of Ĝ, according to the definition in Eq. (3.32).