Priciples of Active and Passive Remote Measurements ...
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and can be formally integrated for upwarding radiation (with the convention f.1 > 0 and being
the surface coordinate) as
L1(5v,f.1, and for downwarding radiation (f.1 < 0) as
Lt(5v,f.1, " Jo
I'
( 5.56)
(5.57)
In these expressions is the total optical thickness of the atmosphere, or equivalently, the optical
depth at the surface, L1(,f.1, Lt(TOA,f.1, The physical interpretation of these solutions is as follows: the upward (downward) radiation
emerging at coordinate 5v results from the radiation produced at the lower (upper) boundary,
attenuated along its path (exponential term), plus the contributions Jvd5v/fl of each layer of
effective optical thickness d5v/ f.1 (in the direction (f.1, optical paths.
5.5.1 Shortwave: successive orders of scattering
Scattering is the dominant process in the shortwave, the source function is
(5.58)
A physically simple method consists in developing the source function according to successive
scatterings
(5.59)
where LSn-l) (5v, f.1', events. The order 0 corresponds to the direct radiation (no scattering). It can be obtained
with Beer's law,
(5.60)
where F~ is the solar flux incident at the top of the atmosphere in the direction (f.10, Physically, the radiance at 5v comes from all depths 5~ where it is scattered from direction
(f.10, Primary scattering approximation
The primary scattering approximation (Lv '::c: L~) can be made when multiple scattering is
negligible. This approximation is valid if the amount of scatterers is small that is if the optical
thickness of the total atmosphere, , is small. In this case, the upward radiation emerging at
coordinate 5v is
(5.61 )