Geoscience Reference
In-Depth Information
These are particular values of the covariance function
C
(
s
), for
s
=
ik
,
s
=
Pi
,and
s
= 0; for instance,
ik
is the distance between the gravity
stations
i
and
k
. The abbreviated notations
C
ik
and
C
Pi
are self-explanatory.
We further set
ε
2
=
m
2
P
.
(9-51)
Thus
m
P
is the root mean square error of a predicted gravity anomaly at
P
,
or briefly, the standard
error of prediction
(interpolation or extrapolation).
Taking all these relations into account, we find the average
M
of (9-49)
to be
M
{
P
}
n
n
n
m
2
=
C
0
−
2
α
Pi
C
Pi
+
α
Pi
α
Pk
C
ik
.
(9-52)
P
i
=1
i
=1
k
=1
This is the fundamental formula for the standard error of the general predic-
tion formula (9-41). For the special cases described in the preceding section,
the particular values of
α
Pi
are to be inserted.
Einstein's summation convention
At least at this point the reader will be grateful to Albert Einstein for having
invented not only the theory of relativity - well, even the general theory of
relativity has been used in geodesy (Moritz and Hofmann-Wellenhof 1993),
but the reader of the present topic will be saved from it - but also the very
practical summation convention which has eradicated myriads of unneces-
sary summation signs from the mathematical literature. This convention
simply says that, if an index occurs twice in a product, summation is auto-
matically implied. Using this convention, the preceding equation is simply
written
m
2
P
=
C
0
−
2
α
Pi
C
Pi
+
α
Pi
α
Pk
C
ik
.
(9-53)
In the future we shall take this equation for granted unless stated otherwise.
Such formulas are also handsome for programming (a loop).
Now back to reality in the form of examples.
As an example consider the case of representation, Eq. (9-44); all
α
are
zero except one. Here (9-53) yields
m
2
=
C
0
−
2
C
P
1
+
C
0
=2
C
0
−
2
C
P
1
.
(9-54)
P
For the case of zero anomaly, there is
m
2
p
=
C
0
, as should be expected.
Often we need not only the standard error
m
P
of prediction but also the
correlation of the prediction errors
ε
P
and
ε
Q
at two different points
P
and
Q
, expressed by the
“error covariance” σ
PQ
, which is defined by
σ
PQ
=
M
{
ε
P
ε
Q
}
.
(9-55)
