The geometry of the earth - Physical Geodesy - page 220

by a numerical or graphical integration. The deflection component ε must
be given at enough stations along the profile such that the interpolation
between these stations can be done reliably. Sometimes a map of ξ and η
is available for a certain area. Such a map is constructed by interpolation
between well-distributed stations at which ξ and η have been determined
(Grafarend and Offermanns 1975). Then the profiles of integration may be
suitably selected; loops may be formed to obtain redundancies which must
be adjusted.
If the deflection components ξ and η are obtained directly from the equa-
tions

ξ =Φ − ϕ, η =(Λ − λ )cos ϕ,

(5-116)

that is, by comparing the astronomical and ellipsoidal (or geodetic) coordi-
nates of the same point, then this method is called the astrogeodetic deter-
mination of the geoid .
The astronomical coordinates are directly observed; the ellipsoidal coor-
dinates are obtained in the following way.

Determination of a local astrogeodetic datum
This is of historic interest only, but indispensible for an understanding of
the present classical triangulation system. In agreement with Part I, but in
contrast to Part II, “local” again means “regional”, referring to a country
(e.g., France) or even a continent (e.g., European Datum or North-American
Datum). In a larger triangulation system, a certain “initial point” P 1 is
chosen for which the undulation N 1 and the components ξ 1 and η 1 of the
deflection of the vertical are prescribed. Here ξ 1 ,η 1 ,and N 1 may be assumed
arbitrarily in principle; the position of the reference ellipsoid with respect
to the earth is thereby fixed. For the sake of definiteness let us consider
the case that has been of greatest practical importance, that is, the case
in which ξ 1 = η 1 = N 1 = 0. In this case, because ξ 1 = η 1 = 0, the geoid
and the ellipsoid have the same surface normal so that, because N 1 =0,the
ellipsoid is tangent to the geoid below P 1 (Fig. 5.16). The condition that
the axis of the reference ellipsoid be parallel to the earth's axis of rotation
finally determines the orientation of the triangulation net because Laplace's
equation (5-99) then gives ∆ α 1 = η 1 tan ϕ 1 =0,sothat α 1 = A 1 ;thatis,at
the initial point the ellipsoidal azimuth is equal to the astronomical azimuth.
Now we can reduce the measured distances and angles to the ellipsoid
and compute on it the position of the points of the triangulation net (their
ellipsoidal coordinates ϕ and λ ) in the usual way. After measuring the coor-
dinates Φ and Λ astronomically at the same points, we can then compute the
deflection components ξ and η by (5-116). Starting from the assumed value