Least-squares collocation - Physical Geodesy - page 368

because r ∆ g is harmonic, and similarly for r .Thefactor
R 2
rr

(10-31)

is chosen to become equal to 1 if both points P and Q lie on sea level; in
this case, Eq. (10-29) reduces to (10-28).
So, each of the terms of (10-29) is harmonic, that is, it satisfies Laplace's
equation. Thus, the whole series (10-29) is harmonic (if it converges), being
a linear combination of harmonic terms. This is a well-known consequence
of the linearity of Laplace's equation: the linear combination of solutions of
any linear equation is itself a solution of this equation.
Thus, also the spherical harmonics series of T = r ∆ g is harmonic down
to the reference sphere r = R , with respect both to r and r .Harmonic
functions, by their very definition, are regular analytic functions down to
r = R ,so T and all its linear combinations are regular and thus admit
downward continuation down to the reference sphere (cf. Sect. 8.6).

10.2 Application of collocation to geoid

determination

It is well known that the direct interpolation of free-air gravity anomalies,
which essentially are surface gravity anomalies (8-128) in high mountains,
e.g., by least-squares interpolation, leads to relatively poor results because
of the correlation of the free-air anomalies with elevation (Sect. 9.7). This
correlation with elevation constitutes a considerable trend which must be
removed before the interpolation. Bouguer anomalies take care of the de-
pendence on the local irregularities of elevation; isostatic anomalies are, in
addition, also largely independent on the regional features of topography; in
Sect. 11.1 we shall consider, in addition, also the removal of global trends
by spherical-harmonic earth gravity models (e.g., EGM 96, see www.iges
.polimi.it/index/geoid repo/global models.htm) obtainable from the inter-
net.
In exactly the same way we must remove the main trend of the vertical
deflections ξ, η and the gravity anomalies ∆ g by an isostatic reduction be-
fore applying collocation. Thus, isostatic reduction, pragmatically regarded
as trend removal, is essential for the practical application of least-squares
collocation in mountainous regions (Forsberg and Tscherning 1981).
Physically speaking, we transport the topographic masses to the interior
of the geoid in such a way that the isostatic mass deficiencies are filled. The
observation point P remains in its position on the earth's surface. In this way,