The geometry of the earth - Physical Geodesy - page 212

Part III: Local geodetic datums

5.11 Formulation of the problem

As we have remarked several times, the weak point of the Bruns-Hotine
method is the insu cient accuracy of the zenith angle measurement preclud-
ing the practical use of this method for larger triangulations. The trigono-
metric heights obtained in this way are significantly less accurate than the
horizontal positions.
A practical solution of this problem was to separate positions and heights.
The horizontal position was calculated on the reference ellipsoid in the way
we shall see later. Accurate heights were obtained by leveling referred to the
“actual” level surfaces, in particular to the geoid.
Thus, this theoretically and practically unsatisfactory procedure used
two different reference surfaces: the ellipsoid for horizontal position and the
geoid for heights. The mutual position of these two surfaces was not even
known because of lack of knowledge of the geoidal height N . It has been
rightfully ridiculed as “2+1-dimensional geodesy”.
There is a way out of this dilemma even for local (or rather regional)
geodetic systems. The trigonometric height h is not determined by zenith-
angle measurements but by using the simple formula

h = H + N (5-91)

from leveled orthometric heights H by adding the geoid height N !
But how do we get the geoid? Even before the satellite era, there existed
two methods:

1. the astrogeodetic method , determining N from deflections of the vertical
ξ and η ;

2. the gravimetric method , using for this purpose gravity anomalies ∆ g .

The theories of both methods were known as early as 1850, but what was
lacking were data, especially gravimetric ones. Serious practical applications
started not much before 1950, a hundred years later, just before the advent
of satellites. This will be discussed in detail later in this topic.
A reasonable measuring accuracy was achievable, but another diculty
appeared. Both methods require the evaluation of integrals of the data ( ξ and
η ,or∆ g ) as continuous functions. The data, however, are always measured
at discrete points only. Interpolation is necessary and introduces additional
errors. If the data are distributed uniformly and densely, resulting errors
may be kept small. The fundamental problem exists, however.