The Climate Machine: The Greenhouse Effect

Evidence for the greenhouse effect can be seen in:
  1. the longwave spectral opacity of the atmosphere and
  2. the difference between the observed surface temperature of the Earth and the effective radiating temperature as produced by a simple climate model.
Due to the greenhouse effect, the surface temperature (Ts) is greater than the effective radiating temperature (Te) by an amount given by,
Ts = Te (2 / (2-e) )1/4
where the emissivity of the atmosphere is given by e. The calculation is explained here.

For optically thick atmospheres (several optical depths) the calculation can be extended further.