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If, like me, you find physics more easy to follow given the mathematics rather than the explanation, below you will find a simple derivation to calculate Irradiance at the ground!

If we look to the original equation - if I = the Irradiance (intensity of flux, per unit area) then dI = the absorption dependent intensity change.

Assuming in the atmosphere radiation is at a vertical incident, we can write:

Where I0= Irradiance at the top of the atmosphere, IG = Irradiance at ground level and Z = given height of the atmosphere. Assuming the extinction co-efficient is constant, we can then write:

With N being the number (or concentration) of molecules, we need to consider that N is not constant across the atmosphere. As so, we can re - write the integral as NT. Over a unit area of ground, the Number of molecules in the atmosphere will  = NT

Equations derived from previous study, all images created as own using Microsoft Word