Modern views on the determination of the figure of the earth - Physical Geodesy - page 303

Also in three dimensions, functions satisfying Laplace's equation are
called harmonic, as we know well. Analytical continuation is again best de-
fined by Taylor series, and analytical continuation is frequently called har-
monic continuation (Kellogg 1929: Chap. X).
Above we have been misusing the all-round word “reduced” in the sense
of “analytical” or “harmonic” continuation and will continue to do so for
brevity. As we have seen in Sect. 8.2 and will see in Sect. 8.9, it is not a
gravity reduction in the standard sense of explicit mass removal. The Taylor
series whose first term is (8-54) is an analytical operation performed on the
external potential directly at ground level, preserving the Laplace equation
∆ W =0.(Infact,∆ W =2 ω 2 , but let us, as we did in (8-2), for a while for-
get earth rotation, which implies ω =0and W = V .) Thus, it is a harmonic
function and our “reduction” is really analytical continuation as a harmonic
function or briefly harmonic continuation . Harmonic continuation is the
keynotioninmodernphysicalgeodesy,fromMolodensky'sprob-
lem to least-squares collocation. Its full meaning will gradually emerge
in what follows, as a notion which is surprisingly simple and general. Sym-
bols like ∆ g harmonic will relate to harmonic continuation. In what follows, we
shall sometimes continue to use “reduce downward” or “continue downward”
instead of “harmonically continue downward” and use “reduce upward” in
a similar sense. We also use “continue upward”. Only in doubt, the clumsy
expression “harmonically continue upward” should be employed. Also “ana-
lytical continuation” is used. It all means the same. In the present context,
confusion is hardly possible.
Hence we see why gravity anomalies ∆ g at ground level may be used
for f in (8-58), whereas the equivalent expression (2-394) was originally
derived for gravity anomalies at sea level .Since∆ g harmonic and ∆ g differ only
by terms of the order of h , the difference between using ∆ g harmonic or ∆ g
in (8-62) causes only an error of the order of h 2 , which is negligible in the
linear approximation.
Analytical continuation: historical remarks
The use of analytical continuation has an interesting history. It was first
considered as a possibility by Molodensky himself, already before 1945, but
he soon rejected this method! Molodensky was a profound mathematician,
with a high regard for mathematical rigor. He would not be satisfied with
intuitive heuristic approaches so common in mathematical physics, also in
the present topic.
In fact, the analytical continuation of the external gravitational potential
into the interior of the earth's masses is very likely to become singular at
some points. As a serious mathematician, Molodensky rejected the use of