Geoscience Reference
In-Depth Information
Table 11.4. Comparison of direct numerical integration and fast
Fourier transform (FFT): maximum and (arithmetic) mean val-
ues of Molodensky corrections
ζ
i
,ξ
i
,η
i
for
i
=1
,
2; test area:
46
.
788
◦
≤
46
.
512
◦
,13
.
438
◦
≤
14
.
646
◦
, 600 m
ϕ
≤
λ
≤
≤
topographic
height
≤
2400 m
ξ
1
[
]
η
1
[
]
ξ
2
[
]
η
2
[
]
ζ
1
[cm]
ζ
2
[cm]
maximum
40.8
2.0
2.0
0.8
0.2
0.2
direct int.
values
47.6
1.5
1.4
0.7
0.1
0.1
FFT
mean
31.3
0.4
0.4
0.2
0.03
0.03
direct int.
values
36.7
0.3
0.4
0.5
0.02
0.02
FFT
conclusions:
1. The method of Molodensky corrections depends very much on the de-
tails of numerical integration (data density, smoothing, etc.).
2. The corrections decrease for increasing
i
=1
,
2
,
3
,...
. This is what
they have to do. Higher corrections may be expected finally to consist
of “pure noise” because of general roughening and increasing round-off
errors, so that the question of convergence becomes practically as well
as theoretically meaningless: higher terms must simply be put equal to
zero by higher force.
3. The Molodensky correction
ζ
1
may reach a few decimeters,
ζ
2
and
higher-order terms might frequently be negligible.
4. At the end of Sects. 2.21 and 8.8, we have remarked a curious phe-
nomenon. Using the same data, gravimetric methods seem to furnish
the vertical position (expressed by
ζ
or
N
) roughly by one order of
magnitude better than the horizontal position (as expressed by
ξ, η
).
If we take the old astronomer's rule that 1
=
30 m in position, then
1 m corresponds to 0
.
03
. Assume that we get 1 m in vertical position
and wish to get the same accuracy for horizontal position. This would
mean that we have to get the astronomical measurements Φ
,
Λand
the deflections of the vertical
ξ, η
with better than 0
.
03
.Thisalso
seems to apply with the order of magnitude of the Molodensky correc-
tions, where a Molodensky correction
ζ
1
=0
.
41 m comes along with a
ξ
1
=
η
1
=2
, which corresponds to 60 m.
In this sense, gravimetry is weaker by one order of magnitude in deter-
mining the horizontal than the vertical position. This is an admittedly
