Design-based mapping for finite populations of marked points

The estimation of marks for a finite population of points scattered onto a study region is considered when a sample of these points is selected by a probabilistic sampling scheme. At each point, the mark is estimated by means of an inverse distance weighting interpolator. The design-based asymptotic properties of the resulting maps are derived when the study area remains fixed, a sequence of nested populations with increasing size is considered and samples of increasing size are selected. Conditions ensuring design-based asymptotic unbiasedness and consistency are given. They essentially require that marks are the values of a pointwise or uniformly continuous deterministic function, the enlargement of the populations is rather regular and the sequence of sampling designs ensures an asymptotic spatial balance. A computationally simple mean squared error estimator is proposed. A simulation study is performed to assess the theoretical results on artificial populations. Finally, an application for mapping the values of the height of trees in a forest stand located in North Italy is reported.