Modern views on the determination of the figure of the earth - Physical Geodesy - page 305

you are not satisfied with 1 mgal, prescribe 10 − 3 mgal or 10 − 1000 mgal!
Bjerhammar has pointed out that the assumption of a complete contin-
uous gravity coverage at every point of the earth's surface, from which the
above negative answer follows, is unrealistic because we can measure gravity
only at discrete points. If the purpose of physical geodesy is understood as
the determination of a gravity field that is compatible with the given discrete
observations , then it is always possible to find a potential that can be an-
alytically continued down to the ellipsoid. This is the theoretical basis for
least-squares collocation .
Here we need only one result: Do not worry about analytical contin-
uation! It is always possible with an arbitrarily small error being
not equal to 0 (though not for one being 0).
So, in the same year 1969, Marych and Moritz independently found an el-
ementary solution by analytical continuation in the form of an infinite series
denoted as “Molodensky series”. Details can be found in Moritz (1980 a):
The original form of Molodensky's series obtained by solving an integral
equation is found in Sect. 45. Pellinen's equivalence proof that the simple
“analytical continuation solution” and Molodensky's integral equation solu-
tion are equivalent (that means, the series are termwise equal!) is found in
Sect. 46.
We remark that analytical continuation is a purely mathematical concept
independent of the density of the topographic masses. Thus, it is not an
“introduction of gravity reduction through the backdoor”, which would be
contrary to the spirit of Molodensky's theory.

8.6.5 Another perspective

Consider Fig. 8.6. Let us assume that the analytical downward continuation
of ∆ g to the sea level surface has been performed, obtaining ∆ g harmonic .The
sea-level anomalies ∆ g harmonic then generate, on the physical surface of the
earth, a field of gravity anomalies ∆ g that is identical with the actual gravity
anomalies on the earth's surface as obtained from observation. Therefore, the
gravity anomalies that they generate outside the earth must also be identical

P

g

h P ground

h

sea level

g harmonic

Fig. 8.6. Free-air anomalies at ground level, ∆ g , and at sea level, ∆ g harmonic