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singular functions for regular purposes.
Still, analytical continuation continued to exert an irresistable fascina-
tion because its use is so easy. It was rediscovered around 1960 by A. Bjer-
hammar. At the General Assembly of the International Union of Geodesy
and Geophysics in Berkeley, California, in 1963, one of the authors (H.M.)
talked to Bjerhammer about these diculties, but Bjerhammar refused to
take them seriously. After a long discussion he convinced H.M. that analyt-
ical continuation was rigorously possible for discrete boundary data (all our
terrestrial gravity measurements are discrete) and approximately possible
for continuous boundary data.
This admittedly intuitive thinking was made rigorous by the idea of
Krarup (1969) that Runge's theorem, well known for approximation of ana-
lytical functions of a complex variable, should be applied to the problem of
analytical continuation of harmonic functions in space. Runge's theorem, in
the form of Krarup, loosely speaking says that, even if the external geopoten-
tial cannot be regularly continued from the earth surface
S
into its interior,
it can be made continuable by an arbitrarily small change of the geopotential
at
S
. Another historical remark: the Krarup-Runge theorem for harmonic
functions in space goes back at least to Szego and to Walsh (both around
1929), cf. Frank and Mises (1930: pp. 760-762). It is always dangerous to
talk about priorities! A detailed discussion will be found in Moritz (1980 a:
Sects. 6 to 8).
More on the validity of this method
Let us summarize. The presupposition of this method is that the earth's
external gravitational potential can be continued, as a regular harmonic
function, analytically down to sea level. This is the case if and only if it
is possible to shift the masses outside the ellipsoid into its interior in such
a way that the potential outside the earth remains unchanged or, in other
words, if the analytical continuation of the disturbing potential
T
is a regular
function everywhere between the earth's surface and the ellipsoid. Thus, the
question arises whether the external potential can be analytically continued
down to sea level.
Rigorously
, as we have just remarked, the answer must be in the negative,
in view of the irregularities of topography (Molodenski et al. 1962: p. 120;
Moritz 1965: Sect. 6.4). This fact is also related to the divergence at the
earth's surface of the spherical-harmonic expansion for the external potential
(Sect. 2.5).
However, by Krarup-Runge's theorem, the analytical continuation of the
external potential down to sea level is possible
with su
cient accuracy for
all practical purposes
. Actually it is possible with any accuracy you wish; if
