Geoscience Reference
In-Depth Information
formally identical with Eq. (9-67) obtained in a completely different way.
Least-squares collocation
Here we again derive the base functions from a kernel function
K
(
P, Q
), but
in a way slightly different from (10-17): we put
ϕ
k
(
P
)=
L
k
K
(
P, Q
)
,
(10-21)
where
L
k
means that the functional
L
k
is applied to the variable
Q
;the
result no longer depends on
Q
(since the application of a functional results
in a definite number). Thus, in (10-14) we must put
B
ik
=
L
i
L
k
K
(
P, Q
)=
C
ik
,
(10-22)
which gives a matrix which again is symmetric. Solving (10-14) for
b
k
and
substituting into (10-2) gives with
ϕ
k
(
P
)=
L
k
K
(
P, Q
)=
C
Pk
(10-23)
the formula
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
C
11
C
12
... C
1
q
1
2
.
q
C
21
C
22
... C
2
q
f
(
P
)=
C
P
1
... C
Pq
C
P
2
.
(10-24)
.
.
.
C
q
1
C
q
2
... C
qq
This is formally the same expression as (10-20), but with
f
i
replaced by
i
and with “covariances”
C
ik
and
C
Pi
defined by
“covariance propagation”
(10-22) and (10-23). The concept of covariance propagation is a straight-
forward generalization of the formal structure of error propagation known
from adjustment computations. However, this structure as such is purely
mathematical rather than statistical. We know that a “linear functional” is
the continuous analogue (in infinite-dimensional Hilbert space) to the usual
concept of a linear function in
n
-dimensional vector space. We try not to bur-
den the reader with too much mathematical formalism, but this is treated in
great detail in Moritz (1980 a) and in Moritz and Hofmann-Wellenhof (1993:
Chap. 10). We cannot, however, resist the temptation to compare the struc-
ture
b
i
=
L
i
a
j
(10-25)
leading to
cov(
b
i
,b
j
)=
L
i
L
j
cov(
a
k
,a
l
)
(10-26)
