A machine learning approach for characterizing soil contamination in the presence of physical site discontinuities and aggregated samples

Abstract

Rehabilitation of contaminated soils in urban areas is in high demand because of the appreciation of land value associated with the increased urbanization. Moreover, there are financial incentives to minimize soil characterization uncertainties. Minimizing uncertainty is achieved by providing models that are better representation of the true site characteristics. In this paper, we propose two new probabilistic formulations compatible with Gaussian Process Regression (GPR) and enabling (1) to model the experimental conditions where contaminant concentration is quantified from aggregated soil samples and (2) to model the effect of physical site discontinuities. The performance of approaches proposed in this paper are compared using a Leave One Out Cross-Validation procedure (LOO-CV). Results indicate that the two new probabilistic formulations proposed outperform the standard Gaussian Process Regression.

Introduction

Rehabilitation of contaminated soils in urban areas is in high demand because of the appreciation of land value associated with the increased urbanization. A common technique to rehabilitate a contaminated site is to remove contaminated soil and either treat or burry it in designated sites. Because there are important costs associated with this activity, it is essential to characterize spatial contaminant concentration in order to classify soil as either contaminated or non-contaminated based on the applicable legislation. Any cubic meter unnecessarily removed (i.e. false $+$) or any cubic meter wrongly left in place (i.e. false $-$) will increase the overall rehabilitation costs. Thus, there are financial incentives to minimize soil characterization uncertainties.

In the field of geostatistics, several researchers such as Boudreault et al. [1] and Goovaerts [2], [3] have employed the Kriging theory to characterize the spatial distribution of contaminant concentration. Historically, Kriging was proposed by Krige and later formalized by Matheron [4]. More recently, the research community has turned toward Machine Learning methods [5]. Most researchers in this field have employed Artificial Neural Networks (ANN) [6], [7], [8], [9]. ANN is a powerful tool, however it requires lots of data (up to millions of data points) to perform well [10]. This condition is seldom met in practice. In the field of Machine learning, other techniques analogous to Kriging have recently been the object of numerous publications under the name of Gaussian Process Regression, (GPR) [11]. Authors such as MacKay [12] and Rasmussen & Williams [13] have presented modern techniques to calibrate parameters efficiently, process small and large datasets, and provide enhanced formulations that increase the robustness toward numerical instabilities. These latest developments are implemented in several open-source packages such as GPML (Gaussian Process Machine Learning) [14] and GPStuff [15], both running on the Matlab/Octave language. The motivation for this paper is that current ANN and GPR formulations cannot handle two particular situations that are common during site characterization: (1) experimental conditions where contaminant concentration is quantified from aggregated soil samples and (2) the effect of physical site discontinuities. Note that even if geostatistics methods can handle aggregated soil samples using Block Kriging [16], it cannot handle the effect of physical site discontinuities.

This paper proposes a new unified formulation based on the GPR method to address the two limitations identified above. The paper is organized as follows: Section 2 introduces the standard mathematical formulation of Gaussian Process Regression along with specificities associated with soil characterization applications. Section 3 presents the two extensions to the standard GPR formulation that are proposed in this paper. The first extension account aggregated soil samples by creating virtual points that are employed to model the average contaminant concentration. The second extension proposes a new covariance function that can employ discrete attributes corresponding to physical site discontinuities. The justification for these two new probabilistic formulations comes from a case study where both features are present. In Section 4, an empirical analysis compares the performance of these new extensions with the baseline GPR model.