Geoscience Reference
In-Depth Information
The simplest possible example
Let the boundary surface
S
be a sphere of radius
R
. The earth is represented
by this sphere which is considered homogeneous and nonrotating. The po-
tential
W
is identical to the gravitational potential
V
, so that on the surface
S
we have constant values
W
=
GM
R
,
(8-14)
=
GM
R
2
g
.
Knowing
W
and
g
,wehave
R
=
W
g
,
(8-15)
the radius of the sphere
S
. Thus, we have solved Molodensky's problem in
this trivial but instructive example. We have indeed got geometry (i.e.,
R
)
from physics (i.e.,
g
and
W
)!
8.4
Molodensky's approach and linearization
We have just seen that the reduction of gravity to sea level necessarily in-
volves assumptions concerning the density of the masses above the geoid.
This is equally true of other geodetic computations when performed in the
conventional way.
To see this, consider the problem of computing the ellipsoidal coordi-
nates
ϕ, λ, h
from the natural coordinates Φ
,
Λ
,H
, as described in Chap. 5.
The geometric ellipsoidal height
h
above the ellipsoid is obtained from the
orthometric height
H
above the geoid and the geoidal undulation
N
by
h
=
H
+
N.
(8-16)
The determination of
N
was considered in Chap. 2 and elsewhere in this
topic. To compute
H
from the results of leveling, we need the mean gravity
g
along the plumb line between the geoid and the ground (Sect. 4.3). Since
gravity
g
cannot be measured inside the earth, we compute it by Prey's
reduction, for which we must know the density of the masses above the
geoid.
The ellipsoidal coordinates
ϕ
and
λ
are obtained from the astronomical
coordinates Φ and Λ and the deflection components
ξ
and
η
by
ϕ
=Φ
−
ξ,
λ
=Λ
−
η
sec
ϕ.
(8-17)
The coordinates Φ and Λ are measured on the ground;
ξ
and
η
can be
computed for the geoid by Vening Meinesz' formula, the indirect effect being
