Least-squares collocation - Physical Geodesy - page 369

not only the harmonic character of the anomalous potential T outside the
earth's surface is preserved, but in addition, the computational removal of
the topographic masses above sea level makes the function T harmonic down
to sea level. Hence, the collocation formula (10-24) can be applied also at
sea level, giving cogeoid heights N c . By applying the inverse reduction (the
indirect effect) to the computed height anomalies ζ c and cogeoid heights N c ,
we get actual ζ and N . It can be expected that errors in the isostatic model
used (e.g., an Airy-Heiskanen model) will largely cancel in this combined
procedure of reduction and “anti-reduction” (remove-restore technique; see
Sect. 11.1).
The procedure is theoretically optimal and practically well suited for
computer use. The integrability conditions, which in Helmert integration
are represented by the closures of the individual triangles (see Sect. 5.14),
are automatically taken into account. The fact that the deflections of the
vertical are given only in a certain region has the effect that the geoid can
only be computed in that region. Since, even by collocation, differences in
geoidal heights between two neighboring stations A and B depend essentially
only on the deflections in those two stations, the lack of data outside the
region under consideration will hardly cause a noticeable distortion. Note,
however, that the addition of a constant to all geoidal heights N will not
affect the deflections of the vertical; hence, astrogeodetic data determine
the geoidal heights only up to an additive constant. This constant may be
chosen such that the average value of the computed N is zero, and the result
of collocation comes near to this case.
To get immediately almost geocentric geoidal heights, it is appropriate to
take into consideration a global trend which mainly affects ζ and N itself, by
subtracting the effect of a suitable global gravity field, e.g., the gravity earth
model given as a spherical-harmonic expansion up to degree 180 ◦ × 180 ◦ of

Rapp (1981), say, following Sunkel (1983). This will be described in the next
section; in the present section we limit ourselves to the isostatic reduction.

Computational procedure
The computational procedure consists of the following steps:

1. Transformation of the astrogeodetic surface deflections ξ, η from the
local datum used for the geocentric Geodetic Reference System 1980
by the well-known differential formulas of Vening Meinesz (see Heiska-
nen and Moritz 1967: Eq. (5-59)). This is necessary since collocation
requires a reference system which is as realistic as possible.

2. Application of the normal plumb line curvature (8-137) to the “geo-
metric” surface deflections ξ, η gives the “dynamic” surface deflections