Doubly balanced samples with dynamic sample sizes

A spatial sampling design determines where sample locations are placed in a study area to achieve precise estimates of population parameters. Many environmental variables have positive spatial associations, and spatially balanced designs perform well. The recently published dynamic assignment sampling (DAS) design draws spatially balanced master or over-samples in auxiliary spaces. This article proposes a new objective function for DAS to draw doubly balanced master or over-samples, where two balancing properties are satisfied: approximately balanced on auxiliary variables and spatially balanced. All we require is a measure of the distance between population units. Numerical results show that the method generates spatially balanced, balanced, or doubly balanced master or over-samples and compares favorably with established fixed sample size designs. We provide an example application using total aboveground biomass over a large study area in Eastern Amazonia, Brazil, and design-based variance estimators.