Geoscience Reference
In-Depth Information
divergences and the a priori restraints to the albedo:
F
i
+1
+
F
i
−
F
i
−
F
i
+1
≥
=
0,
i
1,...,
N
i
−1,
F
↑
(
P
i
F
↓
(
P
i
=
|
=
≥
1000mbar)
1000mbar)
A
(−)
,
(3.8)
F
↑
(
P
i
=
|
F
↓
(
P
i
=
1000mbar)
1000mbar)
≤
A
(+)
,
F
i
|
F
i
=
≤
A
(max)
,
i
1...,
N
i
.
The second and third lines in the set of restraints (3.8) account for the known
range of the spectral albedo of the surface:
A
(−)
is a minimal possible mag-
nitude,
A
(+)
is a maximal possible magnitude. These magnitudes
A
(−)
and
A
(+)
have been chosen from the spectral reflectivity data of similar surfaces
(Chapurskiy 1986; Vasilyev A et al. 1997a, 1997b, 1997c) (spectral brightness
coefficients to nadir with the approximation of the orthotropic surface equal
to the albedo of sand, snow and pure lake water). These data will be considered
inSect.3.4.Themaximalalbedoofthesystem“atmosphereplussurface”is
assumed as
A
(max)
=
0.95.
The solution of equation system (3.7) together with restraints (3.8) was
accomplished with the iteration technique. Firstly, (3.7) was solved with the
LST without accounting for restraints (3.8), and the fulfilling of restraints
(3.8) was tested for the obtained solution. The iterations were broken when all
these conditions had been fulfilled. Otherwise, the solution of system (3.7) was
searchedwith restraints (3.8). Restraints (3.8) were transformed to the rigorous
equality and the variables were excluded from system (3.7) by substitution of
these equalities. The corresponding formulas expressing this solution will be
presented in Chap. 4. The iteration scheme was constructed as follows. Firstly,
values
F
i
+1
were excluded from the restraints for the irradiances and values
F
i
- from the restraints for the albedo. The solution of system (3.7) was inferred
for every excluded variable separately (2
N
i
solutions as a whole) with the LST,
and the one with the smallest error was chosen. For this solution, restraints
(3.8)weretestedagain.Iftheyfailedtheiterationswerecontinued,andthe
couple of restraints were excluded, then three restraints, and so on. As the
worse variant it was to examine 3
2
2
N
i
−2
solutionsanditwastheappropriate
number for modern computers as in our experiments
N
i
·
=
6.
The final result of the secondary processing of the sounding data are the
desired values of irradiances
F
i
and
F
i
=
1,...,
N
i
together with the
covariance matrix of the errors. It should be emphasized that further interpre-
tation of the results is to obtain the matrix as a whole and not only its diagonal
(the variance of the irradiances values). If the solution has been obtained using
restraints (3.8), the part of the irradiances is linearly dependent and hence
non-informative. The indicator of the linear dependence has also been written
totheoutputfileofthesecondaryprocessing.Wewouldliketopointoutthat
owing to the individual solution of system (3.7) while accounting for (3.8) for
every wavelength the number of the independent (informative) irradiance val-
ues are essentially different for different wavelength. Coefficients
D
,
a
1
,...,
a
5
,
b
1
,...,
b
5
and their standard deviations are additional result of the secondary
processing.
for
i
Search WWH ::
Custom Search