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estimate was at a minimum, that is, Dx i
x i is the estimate of the x0i i
D
, where
~
~
parameter obtained by any other method.
And
finally, third, let the procedure of estimation of the xi i parameters be as
simple as possible and permit to judge about the best discrete set of radiometric
channels needed to realize the two conditions above.
Let the totality of the xi i
-solution of the
system of linear equations with the disturbed right-hand part. Multiply step by step
the ith solution of the system ( 2.11 )byc 1i ,
estimates meeting these conditions be
˃
, c mi , and let
X
n
c ji A il ¼ d jl ;
ð 2
:
12 Þ
i¼1
where
1
for j ¼ l
d jl ¼
0
for j
6 ¼ l
As a result, we have
x 1 ¼ X
x 1 ¼ X
c 1i T i þ X
¼ X
n
n
n
n
c 1i r
2
i
c 1i T i ;
~
c 1i n i ;
D
~
i¼1
i¼1
i¼1
i¼1
~
For
x i (i
2) we write similar relationships. Derive an auxiliary expression:
!
Þ ¼ X
i þ l 1 X
þ X
j¼2 l j X
n
n
m
n
c 1i r
2
u
ð
c 11 ; ...;
c 1n
c 1i A i1 1
c 1i A ij ;
i¼1
i¼1
i¼1
where
μ j are the non-estimated Lagrange multipliers. Putting
first derivative func-
tions
ˆ
to zero, we obtain:
k þ X
m
j¼1 l j A kj ¼ 0
2
2c 1k r
ð k ¼ 1
; ...;
n Þ
These equalities together with conditions ( 2.12 ) form the system (m + n)of
equations whose solution enables one to
find desired optimal values of c ij .Asa
result, D[x 1 ]=
−μ 1 /2, and the remaining
μ j (j
2) values meet a system m
×
n of
equations:
X
j¼1 l j X
X
j¼1 l j X
m
n
m
n
2
i
2
i
A ij A i1 r
¼ 2
;
A ij A i1¼ r
¼ 0
;
ð l ¼ 2
; ...;
m Þ
i¼1
i¼1
For m = 2 we have:
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