Smooth Piecewise Algebraic Approximation as Applied to Large-Scale 2D Scattered Geodetic Data Fitting

We have developed an efficient method, Smooth Piecewise Algebraic Approximation (hereafter SPAA), to automatically compute a smooth approximation of large-scale functional scattered 2D observation points and tilt between them. The area of study is divided into patches and piecewise algebraic surfaces are fitted to the data. When the surfaces are approximated, a set of constraints is imposed in such a way that the resulting function is continuous only in the zero and first derivatives everywhere in the region, which results in a very short computation time. In other word, the surfaces are fitted simultaneously, using the constraints as set-conditions which the parameters of the surfaces must also satisfy. This method does not require a triangulation or quadrangulation of the data points and as such, it is very well suited for extremely large datasets. This method has been successfully applied to the monthly mean sea level and re-levelling data in Canada to thereby compile a map of Vertical Crustal Movements (VCM) in the region. The VCM model obtained using this method accommodates different kinds of scattered geodetic data, while yielding the optimum approximation to them. Enforcing the continuity and smoothness throughout the surfaces, the VCM model of Canada highlights the long wavelength temporal variations of the crust in the region, mainly due to Post Glacial Rebound (PGR). As a result, using the method of SPAA, a more physically meaningful VCM is modelled.