Geoscience Reference
In-Depth Information
A
), where
A
=(
A
+
B
)
/
2,
between the principal moments of inertia
G
(
C
−
and the angular velocity
ω
.
It is also possible to define the mean earth ellipsoid geometrically as
that ellipsoid which approximates the geoid most closely. This definition is
perhaps more appealing to the geodesist; it may, for instance, be formulated
by the condition that the sum of the squares of the deviations
N
of the geoid
from the ellipsoid be a minimum:
N
2
dσ
= minimum
(5-148)
σ
(this integral is to be considered the limit of a sum). The condition of clos-
est approximation may also be expressed in terms of the deflections of the
vertical:
(
ξ
2
+
η
2
)
dσ
= minimum
,
(5-149)
σ
minimizing the sum of the squares of the total deflection of the vertical
ϑ
=
ξ
2
+
η
2
.
(5-150)
Many other similar definitions of closest approximation are possible.
The first definition, based on the condition (5-148), is the most plausible
and the most appropriate intuitively, as has been already noted by Helmert;
in principle, however, all definitions are more or less conventional and are
equivalent theoretically as we shall see below.
The second definition, based on the condition (5-149), uses deflections
of the vertical and is, thus, particularly well adapted to the astrogeodetic
method. However, since this method can be applied only over limited areas,
at most spanning the continents, the integral (5-149) must be replaced by a
sum covering the astronomical stations of a restricted region:
(
ξ
2
+
η
2
)
dσ
= minimum
.
(5-151)
In this way, we can get only the best-fitting ellipsoid for the region consid-
ered, rather than a general earth ellipsoid. As Fig. 5.20 indicates, a
locally
best-fitting ellipsoid
may be quite different from the mean earth ellipsoid,
which can be considered a best-fitting ellipsoid for the whole earth.
If a reasonably good approximation of the earth ellipsoid by a local best-
fitting ellipsoid is desired, it is advisable to subtract the effect of the topog-
raphy and of its isostatic compensation from the astrogeodetic deflections of
the vertical before the minimum condition (5-151) is applied. The purpose
of this procedure is to smooth the irregularities of the geoid. In this way,
