Adjoint formulation of the radiative transfer equation

Problems in radiative transfer theory can be solved by various solution methods. In the previous chapter we have discussed a number of these which we will classify as the forward or the regular methods. In addition to applying the forward solutions, it is also possible to use the so-called adjoint solution techniques which offer the decisive advantage that for certain types of transfer problems the numerical effort can be drastically reduced.

In this section we will formulate the adjoint technique. It will be necessary to introduce a new terminology involving expressions such as the radiative effect and the atmospheric response due to the presence of energy sources. By means of an important but simple example we will demonstrate the numerical advantage that the adjoint technique offers in comparison to the forward formulation. In Section 5.2 we will introduce the perturbation technique and show how to apply it to the forward as well as to the adjoint formulation.

The adjoint method originated as a purely mathematical tool for the solution of linear operator equations. Discussions on this subject can be found in textbooks on principles of applied mathematics such as Courant and Hilbert (1953), Friedman (1956) and Keener (1988).