Geoscience Reference
In-Depth Information
9.2
The covariance function
It is quite remarkable that all the problems mentioned above can be solved
by means of only one function of one variable, without any other information.
This is the
covariance function
of the gravity anomalies.
First we need a measure of the average size of the gravity anomalies ∆
g
.
If we form the average of ∆
g
over the whole earth, we get the value zero:
∆
gdσ
=0
.
1
4
π
M
{
∆
g
}≡
(9-1)
σ
The symbol
M
stands for the average over the whole earth (over the unit
sphere); this average is equal to the integral over the unit sphere divided
by its area 4
π
. The integral is zero if there is no term of degree zero in the
expansion of the gravity anomalies ∆
g
into spherical harmonics, that is, if a
reference ellipsoid of the same mass as the earth and of the same potential
as the geoid is used. This will be assumed throughout this chapter.
Note that if this is not the case, that is, if
M
=0,thenwe
may form new gravity anomalies ∆
g
∗
=∆
g − m
by subtracting the average
value
m
.Then
M
{
∆
g
}
=
m
∆
g
∗
}
= 0 and all the following developments apply to the
“centered” anomalies ∆
g
∗
.
Clearly, the quantity
M{
∆
g}
, which is zero, cannot be used to charac-
terize the average size of the gravity anomalies. Consider then the average
square of ∆
g
,
{
∆
g
2
dσ .
1
4
π
var
{
∆
g}≡M{
∆
g
2
}
=
(9-2)
σ
It is called the
variance
of the gravity anomalies. Its square root is the
root
mean square (rms) anomaly
:
var
=
M
∆
g
2
rms
{
∆
g
}≡
{
∆
g
}
{
}
.
(9-3)
The rms anomaly is a very useful measure of the average size of the gravity
anomalies; it is usually given in the form
rms
{
∆
g
}
=
±
35 mgal ;
(9-4)
the sign
expresses the ambiguity of the sign of the square root and sym-
bolizes that ∆
g
may be either positive or negative. The rms anomaly is very
intuitive; but the variance of ∆
g
is more convenient to handle mathemati-
cally and admits an important generalization.
±
