Fuzzy membership function for weighting pairs in variographical analysis

Abstract

Prior to mapping a regionalized variable by geostatistical interpolators, variographical analysis should be performed to quantify the spatial variability of sampled values. Variographical analysis is based on pairing the samples, calculating a measure of dissimilarity between the paired samples and averaging them, called semi-variogram. Since averaging is a sensitive operator to the extreme values, the semi-variogram might show instabilities if the database contains extreme values, for example when its statistical distribution is lognormal. In addition, in case the sample locations are irregularly spaced, the directional sample pairing might not find sufficient number of pairs in some directions, so average of the pair’s dissimilarities might not be representative. In order to reduce the instability of semi-variograms, fuzzy membership function is proposed in this article to weight the sample pairs according to the vector of variography. The script of fuzzy variographical analysis is developed in the Python language, and is tested on the soil samples acquired in the Fukushima region in 2011 to monitor cesium contamination in the top soil. It is illustrated that the fuzzy semi-variogram is more coherent in different directions and lag distances, hence more certain to be used for anisotropy modeling without manual masking of heterogeneous samples. So, fuzzy semi-variogram could be the right choice for automatic fitting of mathematical models. At the end, kriging interpolation based on classic and fuzzy semi-variograms is compared qualitatively and quantitatively, indicating the reduce of interpolation error by about 50%.