Least-Squares Collocation With Integer Parameters

The prediction of spatially and/or temporal varying variates based on observations of these variates at some locations in space and/or instances in time, is an important topic in the various spatial and Earth sciences disciplines. This topic has been extensively studied, albeit under different names. In Geodesy it is generally referred to as least-squares collocation. The underlying model used is often of the trend-signal-noise type. This model is quite general and it encompasses many of the conceivable measurements. However, the methods of prediction based on these models have only been developed for the case the trend parameters are real-valued. In the present contribution we generalize the theory of least-squares collocation by permitting some or all of the trend parameters to be integer valued. We derive the solution of integer-based least-squares collocation and show how it compares to the solution of standard least-squares collocation.