Comparison of two methods of astrogeodetic geoid determination based on least squares prediction and collocation

Two least squares methods used for astrogeodetic geoid determinations are described and discussed with regard to the possibilities of their practical application. The first one is based on a least squares prediction model for the geoid, namely, a linear autocorrelation of the geoid heights. The deflections of the vertical derived from this model yield observation equations for the determination of free coefficients of the linear autocorrelation by an adjustment by the method of least squares. In the second case, collocation is used as introduced in physical geodesy by T. Krarup. By applying this method, the geoid heights are directly represented by their linear correlation with the deflections of the vertical in formal accordance with the least squares prediction. The required covariance functions of the geoid heights, the components of the deflections, and between these quantities can be derived from the covariance function or the reproducing kernel of the anomalous potential by means of linear functionals. The two methods have been used for determination of a part of the geoid in the Federal Republic of Germany. A comparison of the results shows good agreement.