Non-Linear Effects in the Geodetic Version of the Free Geodetic Boundary Value Problem Based on Higher Order Reference Fields

The first step in any evaluation of the Geodetic Boundary Value Problem (GBVP) consists of a linearization of the originally non-linear observation equations, based on a chosen reference potential. Neglecting non-linear terms produces a bias in the observation equations as well as in the derived solution functions. In the following the impact of non-linear terms of second order is considered for the case of the geodetic version of the free GBVP, assuming that the “horizontal” coordinates of the surface points — e.g. the geodetic latitudes ϕg and longitudes λ related to a specific reference ellipsoid — are given, while the “vertical” component of the position vector (e.g. ellipsoidal height H) is unknown; occasionally this type of GBVP is denoted as the scalar free GBVP. The most obvious choice of observables consists in using the gravity potential W and the modulus of gravity Г as boundary data, balancing the two types of unknowns, namely the external gravity potential and the ellipsoidal heights of the surface points. Theoretical aspects of existence and uniqueness of the geodetic version of the free GBVP have been evaluated by Sacerdote and Sanso (1986) and Otero (1987).