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µ
)and
X
2
(
ϕ
)arederived:
X
1
(
µ
π
ϕ
1−
2
−
µ
)
d
µ
=
ϕ
)
d
ϕ
=
X
1
(
1,
X
2
(
1 .
(3.15)
µ
ϕ
−
−
Conditions (3.15) are similar thus, it is possible to construct the functions
X
1
(
ϕ
) with the same analytical expression. These functions are
to reach maximums for
µ
)and
X
2
(
µ
=
µ
and
ϕ
=
ϕ
µ
=
ϕ
=
0andto
describe the decreasing of the reflection for other quantities. The classical
and well known Henyey-Greenstein function defined by (1.31) satisfies these
conditions. It is rather appropriate for the application, and the integrals of
types (3.15) are easy calculated for it. So the approximation on the base of the
Henyey-Greenstein function is proposed:
i. e.
0and
g
1
1
(1 +
g
1
−2
g
1
µ
)
=
X
1
(
µ
)
1
|
2
·
,
1
((1−
g
1
)
2
+2
g
1
1
(1+
g
1
+2
g
1
µ
)
3
|
2
µ
)
1
|
2
−
(3.16)
1
(1+
g
2
−
g
2
g
2
2
ϕ
|π
|
)
3
2
ϕ
)
=
X
2
(
π
·
,
1
((1−
g
2
)
2
+
g
2
1
(1+
g
2
+
g
2
2
−
|
|
ϕ
|π
)
1
ϕ
|π
)
1
2
where 0
<g
1
<
1and0
<g
2
<
1 are the approximation parameters.
Va l u e s
A
,
g
1
,and
g
2
have been obtained by the simple counting of all possible
SBCmagnitudesoverallmeasurements(overtheinitialdatabutnotclassifi-
cation results) to estimate the accuracy of proposed approximations (3.14) and
(3.16) and to evaluate the approximation parameters. The procedure has been
conducted for the corresponding data of water and moss marsh surfaces with
choosing the concrete magnitudes of the parameters providing the minimal
standard deviation of the approximation. As the viewing angle did not exceed
45
◦
(Sect. 3.1), the obtained values of albedo
A
have had no physical meaning (it
has not been the albedo but a certain coefficient). Concerning the anisotropy
parameters it has been obtained for water surface
g
1
=
=
0.7,
g
2
0.7 and for
=
=
moss marsh:
g
1
0.5.
Mention that the observational grid is not rather detailed over viewing
angle and azimuth (Vasilyev A et al. 1997a, 1997b, 1997c), so the accuracy of
the obtained coefficients is not rather high, namely they could be considered
as a rough estimation with assuming their standard deviation is equal to
0.05. According to the same reason the spectral dependence of coefficients
g
1
and
g
2
(increasing from the UV to NIR spectral regions) could be ignored
and the equal magnitudes could be attributed to all wavelengths. It should be
emphasized, that the formally calculated standard deviation of approximations
(3.14) and (3.16) turns out about 10%, which is close to the observational
uncertainty. Hence, we can conclude that (3.14) and (3.16) are describing the
anisotropy of the natural surfaces reflection exactly enough, though the small
valuesofthestandarddeviationcouldonlybetheconsequenceofasmall
amount of the grid points.
0.2,
g
2
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