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and the formula of Neumann-Koch
K ( ψ ) δg dσ ,
R
4 π
T =
(8-47)
σ
where
K ( ψ )=
2 n +1
n +1 P n (cos ψ )
(8-48)
n =0
and, by summation of this series,
sin( ψ/ 2) ln 1+
1
1
sin( ψ/ 2)
K ( ψ )=
(8-49)
being the Neumann-Koch function.
So in the GPS boundary problem on the sphere, the solution (8-47)
is completely analogous to the formula of Stokes (8-43) for the classical
problem.
The fact that the GPS problem is conceptually simpler (fixed-boundary
surface) than Molodensky's problem (free-boundary surface) is expressed by
the fact that Stokes' function must start with n =2,since n = 1 gives a zero
denominator, whereas Neumann-Koch's function (8-48) is regular for all n .
In both cases, the height anomaly ζ (here the geoidal height) is given by
Bruns' formula
ζ = T
γ
T
γ 0 .
=
(8-50)
In the spherical approximation, γ may be, in formulas of Bruns' and Stokes'
type, replaced by our usual mean value γ 0 = γ 45 .
We will see that these spherical solutions form the base for an elemen-
tary solution of Molodensky's problem and the GPS problem for the earth's
surface. We only mention the well-known fact that, for the earth's surface S ,
these two problems are oblique-derivative problems , since the direction of the
plumb line does not coincide with the normal to the earth's surface, at least
on land. Thus the GPS boundary problem for S is not a spherical Neumann
problem, which always involves the normal derivative!
8.6
Solution by analytical continuation
8.6.1
The idea
The idea is very simple (Fig. 8.4). Our observations ∆ g or δg ,givenonthe
earth's surface S , are “reduced”, or rather “analytically continued” (upward
or downward, see below and Fig. 8.5), to a level surface (or normal level
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