Geoscience Reference
In-Depth Information
matrix
G
0
with a strengthening of the selection conditions for the guarantee,
i. e. comparing the eigenvalues not with unity but with the less magnitude.
Finally, we present the concrete calculation algorithms of the nonlinear
inverse problems. The general algorithm of the penalty functions method
(4.30)isconvertedtotheform:
=
X
0
+
P
m
(
P
m
G
n
S
−
Y
G
n
P
m
+
L
−
m
+
P
m
C
n
HC
n
P
m
)
−1
P
m
X
n
+1
(4.52)
G
n
S
−1
(
Y
−
G
(
X
n
)+
G
n
(
X
n
−
X
0
)) +
C
n
H
(−
C
(
X
n
)+
C
n
(
X
n
−
X
0
))
[
]
.
Y
The algorithm with improved convergence (4.32), which has been used in the
sounding data processing, transforms to:
=
X
0
+
P
m
(
P
m
G
n
S
−
Y
G
n
P
m
+
L
−
m
+
P
m
HP
m
)
−1
P
m
×
[
X
n
+1
G
n
S
−
Y
Y
−
G
(
X
n
)
G
n
(
X
n
−
X
0
)
+
H
(
X
n
−
X
0
)
(4.53)
]
.
In both cases the posterior covariance matrix is calculated with the following
formula:
=
P
m
(
P
m
G
n
S
−
Y
G
n
P
m
+
L
−
m
)
−1
P
m
.
S
X
4.4
Selection of Retrieved Parameters in Short-Wave Spectral Ranges
Hereinbefore the mathematical aspects of the inverse problems have been
mainly considered. In addition to the availability of the formal-mathematical
algorithms, the analysis of the physical meaning of the obtained results is of
great importance. In particular, for the inverse problems of atmospheric optics
it is important to answer the question: to what extent the retrieved parame-
ters correspond to their real values in the atmosphere at the moment of the
observation.Thecomparisonoftheresultsoftheinverseproblemsolution
with the data of direct measurements of the retrieved parameters answers this
question sufficiently clearly and unambiguously. However, in the general case,
the possibility of parallel direct measurements is limited. For example, during
the airborne observations the vertical profiles of the temperature, contents of
absorbing gases and parameters of the aerosols would have been measured
simultaneously with the radiances and irradiances, if there had been an op-
portunity. The situation with the satellite observations is even worse; because
the simultaneous airborne observations of the mentioned parameters are nec-
essary, that needs developing and financing the scientific programs at the state
level. Thus, the simultaneous direct measurements to test the retrieved param-
eters are too expensive. In this connection, the way proposed by the authors of
the topic by Gorelik and Skripkin (1989) has to be mentioned, where the ex-
penditures for the technical solution of the problem (costs of the instruments,
experiments, data processing etc.) are included in the total value, which is
assumed as the minimum for the inverse problem solution. In that statement,
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