Statistical methods in physical geodesy - Physical Geodesy - page 342

anomalies ∆ g and ∆ g become more and more independent. For very large
distances, the covariance will be very small but not in general exactly zero
because the gravity anomalies are affected not only by local mass distur-
bances but also by regional factors. Therefore, we may expect an oscillation
of the covariance between small positive and negative values.
Note that positive covariances mean that ∆ g and ∆ g tend to have the
same size and the same sign; negative covariances mean that ∆ g and ∆ g
tend to have the same size and opposite sign. The stronger this tendency,
the larger is C ( s ); the absolute value of C ( s ) can, however, never exceed the
variance C (0).
The practical determination of the covariance function C ( s )issomewhat
problematic. If we were to determine it exactly, we should have to know grav-
ity at every point of the earth's surface. This we obviously do not know; and
if we knew it, then the covariance function would have lost most of its signif-
icance because then we could solve our problems rigorously without needing
statistics. As a matter of fact, we can only estimate the covariance function
from samples distributed over the whole earth. But even this is not quite
possible at present because of the imperfect or completely missing gravity
data over the oceans. For a discussion of sampling and related problems see
Kaula (1963, 1966 b).
The first comprehensive estimate of the covariance function was made by
Kaula (1959). Some of his values are given in Table 9.1 for historical interest.
They refer to free-air anomalies. The argument is the spherical distance

s
R

ψ = (9-8)

corresponding to a linear distance s measured on the earth's surface; R is a
mean radius of the earth. The rms free-air anomaly is

= √ 1201 = rms { ∆ g } ± 35 mgal . (9-9)

We see that C ( s ) decreases with increasing s and that, for s/R > 30 ◦ ,very
small values oscillate between plus and minus.
For some purposes we need a local covariance function rather than a
global one; then the average M is extended over a limited area only, instead
of over the whole earth as above. Such a local covariance function is useful
for more detailed studies in a limited area - for instance, for interpolation
problems. As an example we mention that Hirvonen (1962), investigating the
local covariance function of the free-air anomalies in Ohio, found numerical
values that are well represented by an analytical expression of the form

C 0
1+( s/d ) 2 ,

C ( s )=

(9-10)