Design-based consistency of the Horvitz–Thompson estimator under spatial sampling with applications to environmental surveys

Spatial populations are usually located on a continuous support. They can be surfaces representing the values of the survey variable at any location, finite collections of units with the corresponding values of the survey variable, or finite collections of areal units partitioning the support, where the survey variable is the total amount of an attribute within. We derive conditions on the design sequence ensuring consistency of the Horvitz–Thompson estimator of spatial population totals, supposing minimal requirements on the survey variable. A simulation study is performed to check theoretical results. Consistency and its implications in real surveys are discussed with focus on environmental surveys.