Geoscience Reference
In-Depth Information
This means that the difference between the geoidal undulation
N
and the
height anomaly
ζ
is equal to the difference between the normal height
H
∗
and
the orthometric height
H
.Since
ζ
is also the undulation of the quasigeoid,
this difference is also the distance between geoid and quasigeoid.
According to Sect. 4.5, the two heights are defined by
H
=
C
g
H
∗
=
C
γ
,
,
(8-109)
where
C
is the geopotential number,
g
is the mean gravity along the plumb
line between geoid and ground, and
γ
is the mean normal gravity along the
normal plumb line between ellipsoid and telluroid. By eliminating
C
between
these two equations, we readily find
g − γ
γ
H
∗
−
H
=
H,
(8-110)
which is also the distance between the geoid and the quasigeoid, see (8-108);
hence
N
=
ζ
+
g
−
γ
H.
(8-111)
γ
The height anomaly
ζ
may be expressed, for instance, by Molodensky's
formula (8-57). Then we obtain
∆
gS
(
ψ
)
dσ
+
g
1
S
(
ψ
)
dσ
+
g − γ
γ
R
4
πγ
0
R
4
πγ
0
N
=
H,
(8-112)
σ
σ
where
g
1
is the term (8-62). Thus
N
is given by Stokes' integral,
applied to
free-air anomalies at ground level
, and two small corrections, where
1. the term containing
g
1
represents the effect of topography;
2. the term containing
g
−
γ
represents the distance between the geoid
and the quasigeoid.
If we neglect these two corrections, then the geoidal undulations
N
are
given by Stokes' integral using free-air anomalies. This was first noted by
Stokes in 1849. A new approach by Jeffreys (1931) by means of Green's iden-
tities started several developments which culminated in the work of Molo-
densky and others.
The advantage of this method for the determination of
N
is that the
density of the masses above sea level enters only indirectly, as an effect
on the orthometric height
H
through the mean gravity
g
,whichmustbe
computed by a Prey reduction (Sect. 3.5). Hence, as far as errors in the
