An iterative optimization scheme to accommodate inequality constraints in air quality geostatistical estimation of multivariate PM

The kriging-based estimation of the different types of atmospheric particulate matter (PM) pollutions defined in the air quality regulation raises some operational problems because the (co)kriging equations are obtained by minimizing a linear combination of the estimation variances subject to unbiasedness constraints. As a consequence, the estimation process can result in total PM(10) concentrations that are less than the PM(2.5) concentrations which would be physically impossible. In a previous publication, it was shown that a convenient external drift modeling can reduce the number of spatial locations where the inequality constraint is not satisfied, without completely solving the problem. In this work, the formulation of the cokriging system is modified, inspired by previous works focusing on positive kriging. The introduction of additional constraints on the cokriging weights are presented, leading to a unique and optimal solution to the problem of cokriging under inequality constraints between two variables. Some computational and algorithmic details are introduced. An evaluation of the penalized cokriging is provided by using the European PM monitoring sites dataset: some maps and performance scores are given to assess the relevance of our iterative optimization scheme.