Bilinear surface smoothing for spatial interpolation with optional incorporation of an explanatory variable. Part 1:Theory

Bilinear surface smoothing is an alternative concept which provides flexible means for spatial interpolation. Interpolation is accomplished by means of fitting a bilinear surface into a regression model with known break points and adjustable smoothing terms. Additionally, as an option, the incorporation in an objective manner, of the influence of an explanatory variable available at a considerable denser dataset is possible. The parameters involved in each case (with or without an explanatory variable) are determined by a nonparametric approach based on the generalized cross-validation (GCV) methodology. A convenient search technique of the smoothing parameters was achieved by transforming them in terms of tension parameters, with values restricted in the interval [0, 1). The mathematical framework, the computational implementation and details concerning both versions of the methodology, as well as practical aspects of their application are presented and discussed. In a companion paper, examples using both synthesized and real world (hydrological) data are presented to illustrate the methodology. The proposed mathematical framework constitutes a simple alternative to existing spatial interpolation methodologies.