Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 35

approximation. Considering different orientations of the micro-roughness of
the natural surfaces it is possible to assert that as reflection is closer to the
orthotropic, then reflected radiation is less polarized. The homogeneous dis-
tribution of reflected radiation over directions is corresponded to the fully
chaotic orientation of the micro-reflectors that causes the chaotic distribution
of the polarization ellipses, i. e. the unpolarized light. Thus, the orthotropic
reflection means also the absence of the dependence upon the polarization.
Otherwise, when the anisotropy is stronger the dependence is clearer. The
water surface is the most anisotropic surface, therefore, in this case the ques-
tion about the exactness of the approximation of unpolarized radiation needs
special study.
The function R ( ϕ ), defined from the relation between the radiances

incoming on the surface I (

µ ϕ µ , , ,

τ 0 , µ ζ ϕ ), ( µ , , > 0) and reflected from the surface

τ 0 , µ ζ ϕ µ I ( , , ), ( < 0), characterizes the radiation reflection from the surface:

π

2 1

1
π

τ 0 , µ µ 0 , ϕ = ϕ µ ϕ µ , ϕ ) I ( τ 0 , µ , µ 0 , ϕ ) µ d µ I ( , ) d R ( , , . (1.73)

0 0

It is easy to test that for the orthotropic surface (1.71) and (1.73) yield the
equality R ( µ |π

for normalizing in (1.73). Equation (1.73) in the operator form is written as:

µ ϕ µ , ϕ ) = , , A and just it defines the existence of the factor

I ↑ = RI ↓ , (1.74)

where: I ↑ = τ 0 , µ µ 0 , ϕ ) is the reflected radiance, I ↓ = τ 0 , µ µ 0 , ϕ I ( , I ( , )isthe

= µ

µ ϕ µ , ϕ ) is the operator of the reflection incoming radiance, and r R ( , ,

π

from the surface.
The necessity of accounting the reflection from the surface in the radiative
transfer theory is based on the evident assumption that the reflection is equal
to the illumination of the atmosphere from the bottom (i. e. from the bottom
boundary of the atmosphere τ = τ 0 ). Thus, it is enough to solve the radiative

transfer problems for diffused radiation in the atmosphere first with the illu-
minationfromthetopandthenwiththeilluminationfromthebottom,and
after all, it is necessary to add both results.
Introduce the following notation system:

1. the values related to the system “atmosphere plus surface” are specified
with the upper line;
2. thevaluesrelatedtotheatmosphereilluminatedfromthebottomwithout
surface are specified with the symbol ;

3.thevaluesrelatedtotheatmosphereilluminatedfromthetopwithout
surface are specified without special marks.

∼

Then the solution of the radiative transfer problem, written in the operator
form (1.57), will be the following: I = TI 0 where I 0 is the radiance incoming to