Geoscience Reference
In-Depth Information
net. Longitude and azimuth are often measured at the same point. Then
Laplace's condition
∆
α
=∆
λ
sin
ϕ
(5-118)
furnishes an important check on the correct orientation of the net and forces
the axis of the ellipsoid to be parallel with the earth's axis of rotation. Thus
it may be used for adjustment purposes. Astronomical stations with longi-
tude and azimuth observations are, therefore, called
Laplace stations
.For
these purposes, the measuring accuracy of astronomical field observations is
sucient, in contrast to the use for directly determining horizontal positions
by
ϕ
=Φ
− ξ
,etc.inSect.2.21.
The astrogeodetic determination of the geoid, also called
astronomical
leveling
, was known to Helmert (1880) and even before.
Comparison with the Stokes method
It is quite instructive to compare Helmert's formula
B
N
=
N
A
−
εds
(5-119)
A
for the astrogeodetic method with Stokes' formula
∆
gS
(
ψ
)
dσ
R
4
πγ
0
N
=
(5-120)
σ
for the gravimetric method. Both methods use the gravity vector
g
.Itis
compared with a normal gravity vector
. The components
ξ
=∆
ϕ
and
η
=∆
λ
cos
ϕ
of the deflection of the vertical represent the differences in
direction
, and the gravity anomaly ∆
g
represents the difference in
magnitude
of the two vectors. Helmert's formula determines the geoidal undulation
N
from
ξ
and
η
, that is, by means of the direction of
g
, and Stokes' formula
determines
N
from ∆
g
, that is, by means of the magnitude of
g
.Both
formulas are somewhat similar: they are integrals which contain
ε
,or
ξ
and
η
,and∆
g
in linear form.
Otherwise, the two formulas show marked differences which are charac-
teristic for the respective method. In Helmert's formula, the integration is
extended over part of a profile; thus, it is sucient to know the deflection
of the vertical in a limited area. The position of the reference ellipsoid with
respect to the earth's center of gravity is unknown, however, and can be
determined only by means of the gravimetric method or, more practically,
the analysis of satellite orbits (Sect. 7.2). Furthermore, the astrogeodetic
method can be used only on land, because the necessary measurements are
impossible at sea.
γ
