Geoscience Reference
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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