Harmonization of design-based mapping for spatial populations

The mapping of a survey variable throughout a continuum or for finite populations of units is usually performed from a model-dependent perspective. Nevertheless, when a sample of locations/units is selected by a probabilistic sampling scheme, the complex task of modelling can be avoided by using the inverse distance weighting interpolator and deriving the properties of maps in a design-based perspective. Conditions ensuring consistency of maps can be derived mainly based on some obvious assumptions about the pattern of the survey variable throughout the study region as well from the feature of the sampling scheme adopted to select locations/units. Nevertheless, in a design-based setting the totals of the survey variable for a set of domains partitioning the study region are commonly estimated by traditional estimators such as the Horvitz–Thompson estimator in the case of finite populations or the Monte-Carlo estimator in the case of continuous populations or by related estimators exploiting the information of auxiliary variables. That necessarily gives rise to different total estimates with respect to those achieved from the resulting maps as the sum of the interpolated values within domains. To obtain non-discrepant results, a harmonization of maps is here suggested, in such a way that the resulting totals arising from maps coincide with those achieved by traditional estimation. The capacity of the harmonization procedure to maintain consistency is argued theoretically and checked by a simulation study performed on some real populations.