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geoid. The avoidance of such assumptions has been the guiding idea of Molo-
densky's research. However, orthometric heights are but little affected by er-
rors in density. The error in H due to the imperfect knowledge of the density
hardly ever exceeds 1-2 decimeters even in extreme cases (Sect. 4.3). It is
presumably smaller than the inaccuracy of the corresponding ζ even with
very good gravity coverage, because of inevitable errors of interpolation, etc.
If, therefore, the method of Sect. 8.10 is used, the geoid can be determined
with virtually the same accuracy as the quasigeoid. Note that it is theoret-
ically even possible to eliminate completely the errors arising from the use
of the geoid (Moritz 1962, 1964). Thus, we may well retain the geoid with
its physical significance and its other advantages.
How much do Molodensky's formulas differ from the corresponding equa-
tions of Stokes and Vening Meinesz? The deviation of ζ from the result of
the original Stokes formula is given by the equivalent expressions
g 1 S ( ψ )
g
∂h
R
4 πγ 0
R
4 πγ 0
ζ 1 =
ζ 1 =
( h
h P ) S ( ψ )
or
σ
σ
(8-120)
according to Eqs. (8-62) and (8-66). This correction may even be smaller
than the difference ζ
N (see Sect. 11.3).
It is appropriate again to point out that the deflection of the vertical
is relatively more affected by the Molodensky correction than is the height
anomaly. In extreme cases, this correction may attain values of a few seconds,
as studies of models by Molodensky (Molodenski et al. 1962: pp. 217-225)
indicate. This is considerable, since 1 in the deflection corresponds to 30 m
in position. Numerical estimates will be found in Chap. 11.
We may summarize the result of applying Stokes' and Vening Meinesz'
formulas to free-air anomalies directly, without any corrections. Stokes' for-
mula yields height anomalies ζ with high accuracy; for many practical pur-
poses, we may, in addition, identify these height anomalies with the corre-
sponding geoidal undulations N . Vening Meinesz' formula gives deflections
of the vertical at ground level that are relatively less accurate but often
acceptable.
An advantage of the modern theory is its direct relation to the external
gravity field of the earth, which is particularly important nowadays for the
computation of the effect of gravitational disturbances on spacecraft trajec-
tories and satellite orbits. It is immediately clear that ground-level quan-
tities, such as free-air gravity anomalies, are better suited for this purpose
than the corresponding quantities referred to the geoid, which is separated
from the external field by the outer masses. For the computation of the ex-
ternal field and of spherical harmonics, the method described in Sect. 8.6.5
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