Least squares 2D bi-cubic spline approximation: Theory and applications

Smooth surface approximation plays an important role in many applications. As an extension of the 2D bi-cubic spline interpolation, we propose the least squares 2D bi-cubic spline approximation (LS-BICSA). LS-BICSA, generally applicable to a set of irregular data, is formulated by the pure bi-cubic spline functions. The method is flexible because the user can impose the appropriate continuity and differentiability conditions at the boundary points using the available constrained least squares theory. Two scenarios having two and four knots at the boundary points are described in this paper. Two examples are presented to illustrate the theory. The first example approximates a known mathematical function on an irregular grid. The second example uses a real data set to approximate the geoid height. LS-BICSA can thus provide reliable results for many geoscience applications where it is required to approximate the function values using a smooth spline surface in the least squares sense.