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Mapping by sampling and prediction of local and regional values of two-dimensional surfaces is a frequent, complex task in geographical information systems. This article describes a method for the approximation of two-dimensional surfaces by optimizing sample size, arrangement and prediction accuracy simultaneously. First, a grid of an ancillary data set is approximated by a quadtree to determine a predefined number of homogeneous mapping units. This approximation is optimal in the sense of minimizing Kullback-divergence between the quadtree and the grid of ancillary data. Then, samples are taken from each mapping unit. The performance of this sampling has been tested against other sampling strategies (regular and random) and found to be superior in reconstructing the grid using three interpolation techniques (inverse squared Euclidean distance, kriging, and Thiessen-polygonization). Finally, the …