A kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers

Abstract

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 Tübingen, Germany. We interpret PSCs 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 PSCs 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 PSCs and enables one to project the complete information content embedded in the PSC to unsampled locations in the system.

Improving processing by adaption to conditional geostatistical simulation of block compositions

“…Geostatistics for positive data (Dowd, 1982;Rivoirard, 1990) is well established. Textural data-sets have been studied by van den Boogaart and Schaeben (2002), while one-dimensional distributional data-sets were treated by Delicado et al (2008) and Menafoglio et al (2014). Compositional data geostatistics was studied in depth by Pawlowsky-Glahn and Olea (2004) and Tolosana-Delgado (2006), and is applied here.…”

Preprocessing of centred logratio transformed density functions using smoothing splines

“…Neither smoothing the original discretised densities [8] nor using of Bernstein polynomials, that is proposed in [21], is coherent with the Bayes space methodology.…”

“…Although the clr transformation may lead to computational problems for some of statistical methods due to constraint (3), it is still well acceptable for distance-based methods or functional principal component analysis, similarly as for the case of compositional data, and could thus extend the currently existing analytical tools [8,21]. However, density functions (as well as functional data in general) occur in the practice rarely in their continuous form.…”

Profile Monitoring of Probability Density Functions via Simplicial Functional PCA With Application to Image Data

“…This may have a detrimental effect both on process monitoring performances and on dimensionality reduction capabilities. A number of authors (e.g., , Van den Boogaart et al, 2010, Delicado (2011, Van den Boogaart et al (2014), Menafoglio et al (2014Menafoglio et al ( , 2016aMenafoglio et al ( , 2016b, Hron et al (2016)) pointed out that PDFs can be interpreted as functional compositional data, i.e., functional observations carrying only relative information, which are usually collected in the form of constrained data integrating to a constant. Traditional FDA techniques operate in the space of square-integrable real measurable functions L 2 , whereas compositional data entails the use of a different space, known as Bayes space, B 2 , Van den Boogaart et al, 2010, Egozcue et al (2013, Van den Boogaart et al (2014)), that generalizes to the functional setting the well-known Aitchison geometry for compositional data (Aitchison, 1986;Pawlowsky-Glahn and Egozcue, 2001;Egozcue, 2009;Pawlowsky-Glahn and Buccianti, 2011).…”