Geoscience Reference
In-Depth Information
Note that in practice especially with the equality of all
h
j
, explicit expressions
(4.28) are to be used for the calculations. In the nonlinear case functions
g
i
and
c
j
are expanded into Taylor series and with considering only the linear term
the equation for the iteration is obtained:
(
G
n
WG
n
+
C
n
HC
n
)(
X
n
+1
−
X
n
)
=
G
n
W
(
Y
−
G
(
X
n
)) −
C
n
HC
(
X
n
) ,
(4.30)
∂
|∂
∂
|∂
where
G
n
and
C
n
are the matrices of partial derivatives (
g
i
x
k
)and(
c
j
x
k
)
=
=
for
i
1,...,
J
,correspondingly,and
C
(
X
n
)isthevectoroffunc-
tion
c
j
(4.26). All vectors andmatrices are calculated for argument
X
n
.Applying
to (4.30) the above-described approach of improving the iterations conver-
gence, namely adding to both parts combination (
G
n
WG
n
+
C
n
HC
n
)(
X
n
−
X
0
)
the iteration algorithm of LST is obtained with taking into account conditions
(4.26) according to the penalty functions method:
1,...,
N
and
j
X
0
+(
G
n
WG
n
+
C
n
HC
n
)
−1
=
X
n
+1
(4.31)
G
n
W
(
Y
−
G
(
X
n
)+
G
n
(
X
n
−
X
0
)) +
C
n
H
(−
C
(
X
n
)+
C
n
(
X
n
−
X
0
))
[
]
.
An important point of general expression (4.31) is that the parameter values
of the previous step of the iterations are defined in the range of the current
iteration; hence, they can be used as constants in the penalty functions. For
example, demand that the desired parameters of the current iteration don't
differtoomuchfromtheirvaluesatthepreviousiteration,i.e.weusetheabove-
discussed approach of the convergence retarding. The constraint conditions
evidently correspond to it:
=
X
n
+1
−
X
n
0 ,
(4.32)
and
X
n
here is not a variable but the constant. In this case, matrix
C
n
HC
n
coincides with matrix
H
,as
C
n
is the identity matrix. Also equality
C
(
X
n
)
=
0
is correct as per to conditions (4.32) and (4.31) converts to the algorithm with
improved convergence, proposed in the study by Polyakov (1996):
=
X
0
+(
G
n
WG
n
+
H
)
−1
×
[
X
n
+1
(4.33)
G
n
W
(
Y
−
G
(
X
n
)+
G
n
(
X
n
−
X
0
)) +
H
(
X
n
−
X
0
)
]
.
In algorithm (4.33) the greater the weight magnitudes are, the closer the values
ofthepreviousandfollowingiterations,thusthesmooth(withoutspread)
convergence of the iterations with correct selections of
h
j
could be provided.
We should mention one other particular case of applying the penalty func-
tion method (Gorelik and Skripkin 1989), which could be used for the inverse
problems of atmospheric optics. The situation frequently met in practice, is
thecase,wherethepartofthedesiredparameters(orevenall)couldbeequal
toanintegeronly.Forexample,theproblemofaccountingforacertainfactor
influencing the radiative transfer, which could be described by introducing to
vector
X
a certain parameter that can be equal to zero or unity, i. e. “to turn
on” or “turn off” this factor. These problems can be solved with the method of
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