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of downward continuation, cf. Sect. 8.6). Such a possible divergence may
concern mathematicians, but it should not concern geodesists, for several
reasons:
1. Our spherical-harmonic expansions are not infinite series but finite
polynomials, by their very determination and computations. So diver-
gence problems do not exist; the question is only good approximation.
2. Such approximating polynomials of spherical harmonics always exist
for arbitrary accuracy requirements (Frank and Mises 1930: p. 760).
In geodesy we usually speak of Runge's theorem. The whole subject is
thorougly discussed in Moritz (1980 a: Sects. 6 to 8).
3. If you use spatial collocation, the behavior (harmonic or not, conver-
gent or divergent, ...) of the solution is completely determined by the
covariance function used. One always uses “good” covariance functions,
which are harmonic and analytic down to a sphere completely inside
the earth.
So forget all about the convergence problem. It is practically solved. Further
discussions beyond the results obtained so far would have to be made at a
very high mathematical level. The question can be made as complicated as
desired; if looked at it from the right angle, it is simple.
Geoid and downward continuation
Therefore, and by the reasoning at the end of Sect. 10.1, the geoid computed
by (harmonic!) spherical-harmonic expressions and by collocation is not a
level surface of the actual geopotential W but a level surface of a harmonic
downward continuation of W , for the simple reason that the base functions
both of spherical harmonics and of collocation satisfy Laplace's equation
(8-2). We may speak of a “harmonic geoid”. This again emphasizes the
importance of analytical continuation (Sect. 8.6). We have deliberately used
the indefinite article “a” in the italicized expression above, because harmonic
downward continuation is an inverse problem and thus has no unique solution
(see below).
The application of collocation to ξ, η, g without gravity reduction gives
height anomalies ζ and undulations of the harmonic geoid, N harmonic ,by
simply varying the elevation parameter ( h and zero, respectively) in the col-
location program. A completely analogous fact was remarked at the end of
the last section for the case of height anomalies ζ c and cogeoidal heights
N c . In the case of Molodensky's problem (without or with gravity reduc-
tion), we have seen a completely similar behavior with the application of the
generalized Stokes and Vening Meinesz formulas, (8-75) and (8-76).
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