Geoscience Reference
In-Depth Information
form, least-squares collocation also includes parameter estimation by least-
squares adjustment. This is an advanced subject treated in great detail in
Moritz (1980 a).
It is easy to determine the accuracy of least-squares prediction. Insert
the
α
of Eq. (9-65) into (9-53), after appropriate changes in the indices of
summation. This gives
m
2
=
C
0
−
2
α
Pk
C
Pk
+
α
Pk
α
Pl
C
kl
P
(9-68)
=
C
0
−
2
C
(
−
1)
C
Pi
C
Pk
+
C
(
−
1)
C
Pi
C
(
−
1)
C
Pj
C
kl
.
ik
ik
jl
For the reader to appreciate the Einstein summation convention, we give
this equation in its original form:
m
2
P
=
C
0
−
2
k
α
Pk
C
Pk
+
k
α
Pk
α
Pl
C
kl
l
2
i
C
Pi
C
Pk
+
i
C
(
−
1)
C
(
−
1)
C
Pi
C
(
−
1)
=
C
0
−
C
Pj
C
kl
.
ik
ik
jl
k
j
k
l
(9-69)
But now back to normal! We have
C
(
−
1)
C
kl
=
δ
jk
=
1if
j
=
k
0if
j
(9-70)
jl
=
k
.
The matrix [
δ
kl
] is the unit matrix. This formula states that the product of
a matrix and its inverse is the unit matrix. Thus, we further have
C
(
−
1)
C
(
−
1)
C
kl
=
C
(
−
1)
δ
jk
=
C
(
−
1)
(9-71)
because a matrix remains unchanged on multiplication by the unit matrix.
Hence, we get
ik
jl
ik
ij
2
C
(
−
1)
C
Pi
C
Pk
+
C
(
−
1)
m
2
=
C
0
−
C
Pi
C
Pj
P
ik
ij
2
C
(
−
1)
C
Pi
C
Pk
+
C
(
−
1)
(9-72)
=
C
0
−
C
Pi
C
Pk
ik
ik
C
(
−
1)
=
C
0
−
C
Pi
C
Pk
.
ik
Thus, the standard error of least-squares prediction is given by
m
2
C
(
−
1)
=
C
0
−
C
Pi
C
Pk
P
ik
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
C
11
C
12
...
C
1
n
C
P
1
C
P
2
.
C
Pn
C
21
C
22
...
C
2
n
=
C
0
−
C
P
1
,C
P
2
, ..., C
Pn
.
.
.
.
C
n
1
C
n
2
... C
nn
(9-73)
