Multiple indicator cokriging with application to optimal sampling for environmental monitoring

Abstract

A probabilistic solution to the problem of spatial interpolation of a variable at an unsampled location consists of estimating the local cumulative distribution function (cdf) of the variable at that location from values measured at neighbouring locations. As this distribution is conditional to the data available at neighbouring locations it incorporates the uncertainty of the value of the variable at the unsampled location. Geostatistics provides a non-parametric solution to such problems via the various forms of indicator kriging. In a least squares sense indicator cokriging is theoretically the best estimator but in practice its use has been inhibited by problems such as an increased number of violations of order relations constraints when compared with simpler forms of indicator kriging. In this paper, we describe a methodology and an accompanying computer program for estimating a vector of indicators by simple indicator cokriging, i.e. simultaneous estimation of the cdf for K different thresholds, {F(u, zk) k = 1,..., K}, by solving a unique cokriging system for each location at which an estimate is required. This approach produces a variance-covariance matrix of the estimated vector of indicators which is used to fit a model to the estimated local cdf by logistic regression. This model is used to correct any violations of order relations and automatically ensures that all order relations are satisfied, i.e. the estimated cumulative distribution function, F̂(u, zk), is such that: F̂(u, zk) ∈ [0, 1], ∀ zk, and F̂(u, zk) ≤ ̂F (u, zℓ) for zk < zℓ. A case study, in which an optimal spatial sampling strategy is required for environmental monitoring, is used to demonstrate the methodology and the use of the program. The purpose of the sampling is to determine the spatial extent of a specified phase (e.g. areal distribution of a pollutant or contaminant). The sampling is sequential with a new sample being added to an existing network at each step of the procedure. The optimal locations for additional samples are determined from two maps (provided by the indicator cokriging program): a map of the probability of belonging to the specified phase and a map of the associated estimation variances. Optimality is achieved by minimising simultaneously the uncertainty of belonging to the phase and the variance of the error of estimating the uncertainty, thus guaranteeing that maximum information is gained from additional samples. © 2004 Elsevier Ltd. All rights reserved.