Geoscience Reference
In-Depth Information
are much smoother and easier to interpolate. It is, however, extremely la-
borious from a computational point of view since the integration must be
performed along the earth's surface (or, what is practically the same, along
the telluroid).
We remark that the computational drawback of the present method,
the Molodensky integration along the earth's surface, can be completely
avoided if we perform our computations in space: instead of integrating
along a surface, we perform collocation in space. This modern procedure,
to be described in the next chapter, permits a simple and computationally
convenient use of surface deflections and also their combination with gravi-
metric and other data. Still, the present developments are necessary for a
full understanding of the collocation approach.
Final remarks
In these last sections we tried to apply the same principle for topographic-
isostatic reduction (the “remove-restore method”)
at point level
to all terres-
trial data related to the gravity vector: gravity anomalies and disturbances
(Sect. 8.9) and deflections of the vertical (Sect. 8.14). This unified view of
isostatically reduced data thus makes them directly suitable for combined
solutions by least-squares collocation to be treated in Chap. 10.
8.15
The meaning of the geoid
We now review the geoid and some surfaces that might be able to replace it.
We will again confirm the unique role of the geoid as a standard surface of
physical geodesy.
The meaning of the geoid is very simple. It is defined in Sect. 2.2 as
one of the equipotential surfaces (level surfaces, surfaces of constant gravity
potential)
W
(
x, y, z
) = constant. (8-162)
The constant is chosen so that, on the oceans, the geoid coincides with mean
sea level:
W
(
x, y, z
)=
W
0
.
(8-163)
This is the usual classical equation of the geoid. So what is the problem?
Well, theory and practice are different, in geodesy as well as in daily life.
First, we must disregard small tidal effects (on the order of 50 cm). This is
done by applying a suitable tidal model and is not too problematic. In fact,
we have numerous geoids determined from satellite observations. Second,
they are usually expressed in terms of a series of spherical harmonics. If
taken at sea level, such a series may diverge (this is related to the diculties
