A Kriging Approach based on Aitchison Geometry for the Characterization of Particle-Size Curves in Heterogeneous Aquifers

We consider the problem of predicting the spatial field of particle-size curves (PSCs) from a sample observed at a finite set of locations within an alluvial aquifer near the city of T\"{u}bingen, Germany. We interpret particle-size curves as cumulative distribution functions and their derivatives as probability density functions. We thus (a) embed the available data into an infinite-dimensional Hilbert Space of compositional functions endowed with the Aitchison geometry and (b) develop new geostatistical methods for the analysis of spatially dependent functional compositional data. This approach enables one to provide predictions at unsampled locations for these types of data, which are commonly available in hydrogeological applications, together with a quantification of the associated uncertainty. The proposed functional compositional kriging (FCK) predictor is tested on a one-dimensional application relying on a set of 60 particle-size curves collected along a 5-m deep borehole at the test site. The quality of FCK predictions of PSCs is evaluated through leave-one-out cross-validation on the available data, smoothed by means of Bernstein Polynomials. A comparison of estimates of hydraulic conductivity obtained via our FCK approach against those rendered by classical kriging of effective particle diameters (i.e., quantiles of the PSCs) is provided. Unlike traditional approaches, our method fully exploits the functional form of particle-size curves and enables one to project the complete information content embedded in the PSC to unsampled locations in the system.