Scale-Invariance and Fuzzy Techniques Explain the Empirical Success of Inverse Distance Weighting and of Dual Inverse Distance Weighting in Geosciences

Authors

Laxman Bokati, The University of Texas at El PasoFollow
Aaron A. Velasco, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-2020

Comments

Technical Report: UTEP-CS-20-24

Abstract

Once we measure the values of a physical quantity at certain spatial locations, we need to interpolate these values to estimate the value of this quantity at other locations x. In geosciences, one of the most widely used interpolation techniques is inverse distance weighting, when we combine the available measurement results with the weights inverse proportional to some power of the distance from x to the measurement location. This empirical formula works well when measurement locations are uniformly distributed, but it leads to biased estimates otherwise. To decrease this bias, researchers recently proposed a more complex dual inverse distance weighting technique. In this paper, we provide a theoretical explanation both for the inverse distance weighting and for the dual inverse distance weighting. Specifically, we show that if we use the general fuzzy ideas to formally describe the desired property of the interpolation procedure, then physically natural scale-invariance requirement select only these two distance weighting techniques.