APPROXIMATION

ABSTRACT
The theory behind the use of least squares collocation (LSC) for
the determination of an approximation (7) to the anomalous gravity poten-
tial, T, and the steps leading to the implementation of the method are
reviewed. LSC is described as an approximation method in a reproducing
kernel Hi l bert space (RKHS) of harmonic functions.
The first step in the implementation is the selection of an appro-
priate inner product or reproducing kernel for the RKHS. It is explained
how an isotropic kernel may be selected so that it represents the main
features of an empirically determined covariance function, which here is
defined completely without reference to probabilistic concepts. The
choice of this type of kernel assures, that T will have the smallest possible
difference from T in a least-squares sense.
The use of LSC, for the determination of 7 and contingently of a
set of related parameters, necessitates that a set of linear equations is
soived with as many unknowns as the number of observations plus the num-
ber of parameters. It is explained how (and under which conditions) the
number of observations can be limited by constructing local solutions,
valid for small, but overlapping areas.
Furthermore a smoothing of the gravity field will have the effect
that fewer observations are needed in order to achieve a given resolution.
This may be achieved by subtracting out the contribution from local topo-
graphic masses and more or less well-known geological structures. Alter-
natively or in combination with this the method of mixed collocation may
be used. Here a RKHS is conctructed, which gives the (theoretical ) possi-
bility of estimating anomalous densities.
Finally, the implementation of LSC on a computer is described by
dividing it into separate steps of which several may be accomplished
using published software. Each step is illustrated by describing the ac-
tions taken when determining a quasi-geoid for the Nordic countries.