Geoscience Reference
In-Depth Information
In the same way we find the error covariance in the points
P
and
Q
:
C
(
−
1)
ik
σ
PQ
=
C
PQ
−
C
Pi
C
Qk
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
1
C
11
C
12
...
C
1
n
C
Q
1
C
Q
2
.
C
Qn
C
21
C
22
...
C
2
n
=
C
PQ
−
C
P
1
,C
P
2
, ..., C
Pn
.
.
.
.
C
n
1
C
n
2
... C
nn
(9-74)
These two formulas give the error covariance function for least-squares pre-
diction. Both formulas have a form similar to that of (9-67) and are equally
well suited for computations so that ∆
g
and its accuracy can be calculated
at the same time.
It is clear that, after appropriate slight changes, this theory applies au-
tomatically to
gravity disturbances δg
.
Practical considerations
Geometric interpolation (Sect. 9.4) is suited for the interpolation of point
anomalies in a dense gravity net, with station distances of 10 km or less. If
mean anomalies for blocks of 5
×
5
or larger are needed rather than point
anomalies, then some kind of representation, such as that considered in the
previous section, may be simpler and hardly less accurate.
Least-squares prediction is, by its very definition, more accurate than
either geometric interpolation or representation, but the improvement in ac-
curacy is not striking. The main advantage of least-squares prediction is
that it permits a systematic, purely numerical, digital processing of gravity
data; gravity anomalies are stored in data bases, and gravity anomaly maps,
if necessary, are generated automatically. The same formula applies to both
interpolation and extrapolation so that gaps in the gravity data make no dif-
ference in the method of computation, which becomes completely schematic
(Moritz 1963). For practical and computational details see Rapp (1964) and
many other papers published since.
For larger station distances, of 50 km or more, prediction of individual
point values becomes meaningless. In this case we must work with mean
anomalies of, say, 1
◦
×
1
◦
blocks.
9.7
Correlation with height
So far we have taken into account only the mutual correlation of the gravity
anomalies, their autocorrelation, disregarding the correlation with height,
