If N is constant, Eq. (3) can be integrated to yield

r where Tr is the transmittance over path length r and is given by

Rearranging Eq. (4) yields

where the first term on the right of Eq. (6) is the residual image-forming radiance, while the second term is the path radiance (airlight), N*, which results from scattering processes throughout the sight path. The parameter N*^ is the sky radiance:

If Tœ is approximately zero, then Nq = N*œ = Ns and

where Ns is sky radiance. Equation (8) allows for a simple approximation of N* when Ns is known.

The explicit dependence of N* on illumination and directional scattering proper-tics of the atmosphere are best examined by considering where

The second term on the right-hand side is the contribution to Nt from sky, cloud, and earth radiance and dQ is an element of solid angle. The parameter hs is sun irra-diance, and a is the volume scattering function defined in such a way that bs = a dQ.

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Therefore, a describes the amount of radiant energy (light) scattered in some direction, while the sum of radiant energy scattered in all directions is proportional to the scattering coefficient bs. The amount of energy scattered out of and into a sight path over some incremental distance, Ar, is proportional to bs. It is a fundamental optical property of the atmosphere. Its measurement and characterization have been the focus of a number of studies.

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