Variable lag variography using k-means clustering

Abstract

Experimental variography in three dimensions based on drillhole data and current modelling software requires the selection of particular directions (azimuth and plunge) and a basic lag distance. Variogram points are then calculated on distances which are multiples of that basic lag. As samples rarely follow a regular grid, directional and distance tolerances are applied in order to have sufficient pairs to calculate reliable variogram points. This process is adequate when drillholes follow a drilling pattern (even if not an exactly regular grid) but can be time consuming and hard when the drilling pattern is irregular or when drillhole orientations vary considerably. Having all variogram points being calculated on multiples of a fixed lag, and the same tolerance being applied throughout the range of distances used, can be very restrictive and a reason for considerable time wasting or even failure to calculate an interpretable experimental variogram. The method discussed in this paper is using k-means clustering of sample pairs based on pair separation distance leading to a number of clusters each representing a different variogram point. This way, lag parameters are adjusted automatically to match the spatial distribution of sample locations and the resulting variogram is improved. Case studies are provided showing the benefits of this method over current fixed-lag experimental variogram calculation techniques.