Geoscience Reference
In-Depth Information
where the symbol
M
{
h
}
denotes the mean height of the whole area consid-
ered.
If ∆
g
and ∆
h
are not correlated, then the function
B
(
s
) is identically
zero. If this is not the case, then we should also take the height into account
in our interpolation.
It is easy to extend the prediction formula (9-41) for this purpose, but
this has turned out to be of little practical importance.
Application to Bouguer anomalies
Of great practical importance, however, is the question whether it is possible
to render the free-air anomalies independent of height by adding a term that
is proportional to the height. In other words, when is the quantity
z
=∆
g
−
b
∆
h,
(9-79)
with a certain coecient
b
, uncorrelated with height? In statistical termi-
nology, correlation with height is a
trend
, which may be capable of being
removed.
The trend
z
has the form of a Bouguer anomaly; for a real Bouguer
anomaly we have, according to Sect. 3.4,
b
=2
πG.
(9-80)
For the density
=2
.
67 g/cm
3
we get
b
=+0
.
112 mgal/m
.
(9-81)
Let us form the covariance function
Z
(
s
) between the “Bouguer anomaly”
z
of (9-79) and height difference ∆
h
z
∆
h
}
∆
g
∆
h
−
b
∆
h
∆
h
}
Z
(
s
)
≡
M
{
=
M
{
=
B
(
s
)
−
bA
(
s
)
.
(9-82)
If
z
is to be uncorrelated with
h
,then
Z
(
s
) must be identically zero. The
condition is
B
(
s
)
− bA
(
s
)
≡
0
,
(9-83)
which must be satisfied for all
s
and a certain constant
b
at least approxi-
mately.
We see that the “Bouguer anomaly”
z
is uncorrelated with height if
the functions
A
(
s
)and
B
(
s
) are proportional for the area considered; the
constant
b
is then represented by
b
=
B
(
s
)
A
(
s
)
.
(9-84)
