Geoscience Reference
In-Depth Information
Austria (the square of the average size of
ξ
and
η
) from 30 (arc second)
2
to
5 (arc second)
2
.
So we may say that we can determine the Austrian geoid to 1-2 m without
measurements (deflections of the vertical) and without collocation, knowing
only a topographic map! This is even more surprising since Austria is not
particularly well isostatically compensated.
Of considerable interest is the effect of analytical continuation on the
isostatically (plus earth model) reduced anomalous potential
T
TI
.Itisex-
pressed by the difference
γ
−
1
T
at the earth's surface minus
γ
−
1
T
at sea
level. This difference reaches a maximum of 13 cm in the Central Alps and is
otherwise positive and negative. In the terminology of the present topic, this
is the
separation between the real geoid and the harmonic geoid
(Sect. 8.15).
Of the same interest is the difference between the height anomalies
ζ
(=
γ
−
1
T
at the earth's surface) and the geoidal heights
N
(=
γ
−
1
T
at sea
level). The maximum of 35 cm for
ζ
N
is reached at the Grossglockner
mountain (the highest peak in Austria,
H
= 3797 m). The results are in
excellent agreement with the approximate formula
−
(981 gal)
−
1
∆
g
B
H,
ζ
−
N
=
−
(11-13)
where ∆
g
B
is the Bouguer anomaly in gal and
H
is the elevation in the
same units as
ζ
and
N
. The agreement may easily be verified, since the
Bouguer anomalies in the investigated area range from 10 mgal to
−
170 mgal,
corresponding to topographic heights from 200 m to 3000 m (Sunkel 1983:
p. 140). In Sunkel et al. (1987: p. 69), the differences
ζ − N
for the whole of
Austria range between
2cm and +56cm.
All this has been computed only from the measured deflections of the
vertical. Gravity observations have been included by Kuhtreiber (2002 a,
2002 b) and Erker et al. (2003), leading to what might be a “few-centimeter
geoid”.
Important: the astrogeodetic geoid and the gravimetric geoid are com-
pared and finally combined after systematic trends have been eliminated by
Kuhtreiber (2002 b) and Erker et al. (2003).
−
11.3
Molodensky corrections
In Sect. 8.6 we have given a solutions of Molodensky's problem by means of
a series obtained on the basis of analytical continuation. It can be written
in the form of Eqs. (8-68), (8-69), (8-67),
ζ
=
ζ
0
+
ζ
1
+
ζ
2
+
ζ
3
+
···
,
(11-14)
