Geoscience Reference
In-Depth Information
In Stokes' formula, however, the integration should be extended over
the whole earth. The gravity anomaly ∆
g
must be known all over the earth;
however, accurate gravity measurements at sea are possible. The gravimetric
method yields, for the whole earth, absolute geoidal undulations: the center
of the reference ellipsoid coincides with the center of the earth. Nowadays,
this is only a theoretical possibility because the required complete cover-
age of the whole earth is not available; again, GPS helps. Nevertheless, the
gravimetric method is still basic: it furnishes, not the geocenter, but details
of the geoid, together with the astrogeodetic method!
The astrogeodetic method has often been applied to the determination
of geoidal sections. We mention, because of its pioneering character and its
romantic title, “Das Geoid im Harz” by Galle (1914). In the years following
1970 it is becoming rare to use Helmert's integral formula in its original
form, and deflections of the vertical are more and more combined with other
data (gravity, GPS, and other satellite data) for a uniform determination of
geoid and gravity field (see Chaps. 10 and 11).
Adjustment of nets of astrogeodetic geoidal heights
With a suciently dense net of astrogeodetic stations (preferably Laplace
points) with an average station distance of 10-20 km, the Helmert integral
(5-119) can be approximated by
B
B
ε
A
+
ε
B
2
∆
N
AB
≡
N
B
−
N
A
=
−
εds
=
−
ds
(5-121)
A
A
or
ε
A
+
ε
B
2
∆
N
AB
=
−
s
AB
.
(5-122)
Thus, the undulation difference can be computed for the line
AB
,andsimi-
larly for other lines
BC
and
CA
in the triangle
ABC
(Fig. 5.17). The closure
condition
∆
N
AB
+∆
N
BC
+∆
N
CA
= 0
(5-123)
must be satisfied and imposed as a condition in the least-squares adjustment
C
A
s
AB
B
Fig. 5.17. Triangular net for an astrogeodetic geoid
