Geoscience Reference
In-Depth Information
Instead of the average square of ∆
g
, consider the average product of
the gravity anomalies ∆
g
∆
g
at each pair of points
P
and
P
that are at a
constant distance
s
apart. This average product is called the
covariance
of
the gravity anomalies for the distance
s
and is defined by
cov
s
{
∆
g}≡M{
∆
g
∆
g
} .
(9-5)
The average is to be extended over all pairs of points
P
and
P
for which
PP
=
s
= constant.
The covariances characterize the
statistical correlation
of the gravity
anomalies ∆
g
and ∆
g
, which is their tendency to have about the same
size and sign. If the covariance is zero, then the anomalies ∆
g
and ∆
g
are
uncorrelated or independent of one another (note that in the precise lan-
guage of mathematical statistics, zero correlation and independence are not
quite the same, but we may neglect the difference here!); in other words,
thesizeorsignof∆
g
has no influence on the size or sign of ∆
g
.Gravity
anomalies at points that are far apart may be considered uncorrelated or
independent because the local disturbances that cause ∆
g
have almost no
influence on ∆
g
and vice versa.
If we consider the covariance as a function of distance
s
=
PP
,thenwe
get the
covariance function C
(
s
) mentioned at the beginning:
∆
g
∆
g
}
(
PP
=
s
)
.
C
(
s
)
≡
cov
s
{
∆
g
}
=
M
{
(9-6)
For
s
=0,wehave
∆
g
2
C
(0) =
M
{
}
=var
{
∆
g
}
(9-7)
according to (9-2). The covariance for
s
= 0 is the variance.
A typical form of the function
C
(
s
) is shown in Fig. 9.1. For small dis-
tances
s
(1 km, say), ∆
g
is almost equal to ∆
g
, so that the covariance is
almost equal to the variance; in other words, there is a very strong corre-
lation. The covariance
C
(
s
) decreases with increasing
s
because then the
Cs)
(
s
Fig. 9.1. The covariance function
