) as follows:The vertical disribution of absorption of monochromatic radiation in the atmosphere was
first derived by an English geophysicist called Sydney Chapman. He found that absorption
per unit thickness of a layer may be expressed as a function of optical depth () as follows:
where
and is about 8km typically (pressure decreases by a factor e for each increase in H), z =
height, and 
is the radiation above the level z to the top of the atmosphere. Setting the
derivative to zero gives the level where absorption is greatest, which is found to be at 
, so at the
level of unit optical depth, the absorption is strongest.
Figure 17 below shows the rate of absorption as a function of height. Also shown is the density of the absorber, and the intensity I of the remaining radiation. Optical depth is shown on the right hand ordinate scale.
Figure 17: The rate of absorption / z , incident radiation I , and density as a function of height z in an isothermal atmosphere. Adapted from Wallace and Hobbs (1977), p301.
At great altitude (large z) the optical depth is very small and the density of absorbing molecules is so low that very little absorption of the incoming radiation takes place. Near the surface where density is high, the optical depth in this case is high, but there is very little radiation left which has not already been absorbed. As a result of these competing effects has a maximum when . Note that if the absorption coefficient k is large then the level of unit optical depth will be high.
Conversely if 
is small most of the downwelling radiation may reach the earth's
surface well before reaching the level of unit optical depth.
The above treatment relies on a number of simplifying assumptions (see Iribane and Cho,
1980, p 56) including which is a constant independent of height, monochromatic radiation,
one absorbing species, zenith angle for incoming radiation, Beer's law, isothermal
atmosphere, etc. However, the result remains qualitatively valid in comparison to a more
rigorous treatment.
Dr D C Griersmith
