A Special Analytical Methodology for Variogram Modeling and Interpolation of Terrain Elevation Data by Kriging Method

Abstract

This paper presents a special scientific analytical methodology to conduct geostatistical spatial analysis, Variogram modeling and interpolation by Kriging method using terrain elevation data measured over geographical spatial unit, while accounting for anisotropic behavior of terrain within this unit. The methodology which includes the design of surface interpolation that gives weights to all data points, starts by performing geostatistical analysis and building the Variogram chart. The Variogram models that best representing the data is computed by using standard mathematical regression functions. The modeling process is achieved by using iterative methods and nonlinear least squares optimization process. The coherence between Variogram models constraint and the weights used in the kriging system ensures statistically the best unbiased estimators as well as minimum variances for the interpolated values. Kriging reduces the unrealistic smoothing surfaces inherited in other interpolation methods. It is also robust with respect to very small spatial differences in data points positions, where they are included in the process. There are a large number of semi-Variogram models that could be employed, although different models may lead to different interpolations. The study focuses on the ten most popular models (some of them recently discovered). The mean value of absolute variances provides valuable information help us to select which model is the best from several candidates. If anisotropy exists in variography according to different directions, then several Variogram models needs to be determined.

Special Matlab programs were written by the author for implementing all stages of the above methodology. The study has shown that the interpolation process by Kriging fails in some cases and inaccurate in other cases Thus we need easy and fast computational tools performing many experiments at the same time giving clear representation results and final error analysis, so that the best solution is reached at last. This was the main and most important achievement of this study.