Geoscience Reference
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is the modern equivalent of the classical geoidal height,
N
=
Q
0
P
0
.
(8-124)
Using the anomalous potential
T
=
W
−
U,
(8-125)
we have according to Bruns' theorem
ζ
=
T
γ
N
=
T
γ
,
,
(8-126)
Q
Q
0
where
γ
denotes the ellipsoidal normal gravity.
The points
P
0
form the geoid, and the points
Q
0
constitute the ellipsoid,
both being level surfaces (of
W
and
U
, respectively). On the other hand, the
points
P
form the earth's surface, and the set of points
Q
defines an auxiliary
surface, denoted as telluroid according to R.A. Hirvonen. As a matter of fact,
neither the earth's surface nor the telluroid are level surfaces, which makes
matters more complicated than in the classical situation, where we deal with
level surfaces.
Following a suggestion of Molodensky, one could plot the height anoma-
lies
ζ
as vertical distances from the reference ellipsoid. Thus one obtains a
geoid-like surface, the quasigeoid, and
ζ
could be considered as quasigeoidal
heights. In contrast to the geoid, however, the quasigeoid is not a level surface
and does not admit of a natural physical interpretation. Therefore, working
with height anomalies
ζ
, it is best to consistently consider them quantities
referred to the earth's surface (vertical distances between earth surface and
telluroid), rather than using the quasigeoidal concept. A summary will be
given in Sect. 8.15.
The classical gravity anomaly ∆
g
0
at sea level is defined as
∆
g
0
=
g
(
P
0
)
−
γ
(
Q
0
)
,
(8-127)
where
g
denotes gravity and
γ
normal gravity. So far,
g
(
P
0
) denotes the ac-
tual gravity on the geoid; we are not yet here considering mass-transporting
gravity reductions.
Analogously we have according to Molodensky:
∆
g
=
g
(
P
)
− γ
(
Q
)
.
(8-128)
Generally we will, as far as feasible, use the subscript “0” to designate quan-
tities referred to sea level, to distinguish them from quantities referred to
the earth's surface, which do not carry such a subscript. For instance, ∆
g
0
