Geoscience Reference
In-Depth Information
Thus we recognize the basic role of the covariance function in accuracy stud-
ies. The error function, on the other hand, is fundamental for problems of
error propagation.
9.6
Least-squares prediction
The values of
α
Pi
for the most accurate prediction method are obtained by
minimizing the standard prediction error expressed by (9-53) as a function
of the
α
. The familiar necessary conditions for a minimum are
∂m
2
P
∂α
Pi
≡−
2
C
Pi
+2
α
Pk
C
ik
=0
(
i
=1
,
2
,...,n
)
(9-63)
or
C
ik
α
Pk
=
C
Pi
.
(9-64)
This is a system of
n
linear equations in the
n
unknowns
α
Pk
;thesolution
is
α
Pk
=
C
(
−
1)
C
Pi
,
(9-65)
ik
where
C
(
−
1)
denote the elements of the inverse of the symmetric matrix
ik
[
C
ik
].
Substituting (9-65) into (9-41) gives
=
α
Pk
∆
g
k
=
C
(
−
1)
∆
g
P
C
Pi
∆
g
k
.
(9-66)
ik
In matrix notation this is written
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
C
11
C
12
...
C
1
n
∆
g
1
∆
g
2
.
∆
g
n
C
21
C
22
...
C
2
n
=
C
P
1
,C
P
2
, ..., C
Pn
∆
g
P
.
(9-67)
.
.
.
C
n
1
C
n
2
... C
nn
We see that
for optimal prediction we must know the statistical behavior of
the gravity anomalies
through the covariance function
C
(
s
).
There is a close connection between this optimal prediction method
and the method of least-squares adjustment. Although they refer to some-
what different problems, both are designed to give most accurate results.
The linear equations (9-64) correspond to the “normal equations” of ad-
justment computations. Prediction by means of formula (9-67) is therefore
called
“least-squares prediction”
. A generalization to heterogeneous data is
“least-squares collocation” to be treated in Chap. 10. In its most general
