Geoscience Reference
In-Depth Information
Geoid, harmonic geoids, and quasigeoid
The
geoid in the usual sense
of Eqs. (2-18) or (8-163) is defined purely by
nature and is independent of geodetic observations (except for the tidal cor-
rections). Its disadvantage is that it depends on the “topographic masses”
above the geoid whose density is unknown, at least in principle. This draw-
back seems to be theoretical rather than practical.
The
harmonic geoids
are equipotential surfaces of an analytical down-
ward continuation. We shall be careful to denote the harmonically continued
potential by
W
harmonic
so that
W
harmonic
=
W
0
= constant
(8-164)
denote harmonic geoid(s).
To repeat, analytical downward continuation based on discrete data at
the earth's surface is an
inverse problem
(Sect. 1.13; for more informa-
tion see www.inas.tugraz.at/forschung/InverseProblems/AngerMoritz.html)
which has infinitely many possible solutions. For collocation, e.g., each solu-
tion corresponds to the choice of a different covariance function.
Thus, the “harmonic geoid” is not uniquely defined. It is a product not
only of nature but also of the computational method used. It cannot, there-
fore, replace the real geoid as a standard surface.
The “cogeoids” of the various gravity reductions (Sect. 8.2) are inter-
mediate computational concepts and should never be used in place of the
geoid. The topographic-isostatic height anomalies at point level,
ζ
c
,andthe
heights of the topographic-isostatic cogeoid,
N
c
, are related to each other by
analytical continuation
. The same collocation formula applies if the height
anomaly
f
(
P
) is computed at sea level with elevation parameter 0 to give
N
c
,or,if
f
(
P
) is computed at point level with elevation parameter
h
,to
give
ζ
c
. (The elevation parameter
h
is a height above sea level in any of the
definitions of Chap. 4.) See item 5 at the end of Sect. 10.2.
For the limiting case of Fig. 8.5 c, take the question: “How is the undu-
lation
N
harmonic
of a 'harmonic geoid' related to the height anomaly
ζ
above
it on the ground and on the same vertical?” Answer: “By analytic continu-
ation!”
Another special question to which the answer is also easy: “Which gravity
reduction leaves the geoid unchanged?” Answer: “The Rudzki reduction”
(Sect. 3.8). So why not use it? It changes the external potential, which today
is of paramount importance.
“What is the difference between the Rudzki reduction and the harmonic
downward continuation?” Answer: “The Rudzki reduction leaves the geoid
unchanged but changes the external geopotential: there is
W
c
=
W
=
W
0
only on the geoid,
but
W
c
=
W
outside the earth, which is inadmissible.
