Geoscience Reference
In-Depth Information
Least-squares interpolation
Let us consider a function
K
=
K
(
P, Q
)
,
(10-16)
in which two points
P
and
Q
are the independent variables. Let this function
K
be
•
symmetric with respect to
P
and
Q
,
•
harmonic with respect to both points, everywhere outside a certain
sphere, and
positive-definite (the positive definitiveness of a function is defined
similarly as in the case of a matrix).
Then the function
K
(
P, Q
) is called a (harmonic) kernel function (Moritz
1980 a: p. 205). A kernel function
K
(
P, Q
) may serve as “building material”
from which we can construct base functions. Taking for the base functions
the form
•
ϕ
k
(
P
)=
K
(
P, P
k
)
,
(10-17)
where
P
denotes the variable point and
P
k
is a fixed point in space, we obtain
least-squares interpolation
already treated by a quite different approach in
Chap. 9.
This name originates from the statistical interpretation of the kernel
function as a
covariance function
(Sect. 9.2); then least-squares interpola-
tion has some minimum properties (least-error variance, similarly as in least-
squares adjustment). This interpretation is not essential, however; one may
also work with arbitrary analytical kernel functions, considering the proce-
dure as a purely analytical mathematical approximation technique. Normally
one tries to combine both aspects in a reasonable way.
Substituting (10-17) into (10-6), we get
A
ik
=
K
(
P
i
,P
k
)=
C
ik
;
(10-18)
this square matrix now is symmetric (in the general case,
A
ik
is not sym-
metric!) and positive definite because of the corresponding properties of the
function
K
(
P, Q
). Then the coecients
b
k
follow from (10-8) and may be
substituted into (10-2). With the notation
ϕ
k
(
P
)=
K
(
P, P
k
)=
C
Pk
,
(10-19)
the result may be written in the form
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
C
11
C
12
... C
1
q
f
1
f
2
.
f
q
C
21
C
22
... C
2
q
f
(
P
)=
C
P
1
... C
Pq
C
P
2
,
(10-20)
.
.
.
C
q
1
C
q
2
... C
qq
