The principle of invariance in the original form was stated by Ambartsumian (1942) expressing the invariance of the diffusely reflected radiation emerging from a semi-infinite atmosphere to the addition or subtraction of an infinitely thin atmospheric layer. Chandrasekhar (1960) advanced the original from and stated four general principles of invariance which apply to finite atmospheric layers. These principles are not based on the radiative transfer equation, but they are of equal physical validity. We accept Goody's (1964a) statement that the principles of invariance may be viewed as a series of common-sense relations between the scattering and transmission functions with the radiances emerging from the upper and lower boundaries of an atmospheric layer and at some intermediate variable level.

Definitions of the scattering and transmission functions

Let us consider a plane–parallel atmospheric layer of vertical optical thickness τ1 bounded on both sides by a vacuum, see Figure 3.1. The upper boundary of this layer is illuminated by a beam of parallel downward directed radiation S0, while at τ = τ1 no radiation is incident in the upward direction. For simplicity, only short-wave radiation will be considered. However, inclusion of infrared radiation causes no particular difficulties. We call this situation the restricted or standard problem.