Design-unbiased estimation for point-to-tree distance sampling

Point-to-tree distance sampling designs, sometimes also referred to as k-tree sampling or fixed-count sampling, are practical response design options for field sampling in forest inventories and ecological surveys. While practitioners accept and use several approaches to estimate stem density and other stand attributes, a major concern from a statistical point of view is the lack of a general unbiased estimator for this class of sampling strategies. In this paper we analyse point-to-tree distance sampling in the framework of design-based probabilistic sampling and present an unbiased estimator valid for estimation of any stand attribute. This estimator draws upon the idea of defining an inclusion zone around each tree. A tree is taken as a sample tree if a selected sample point falls into its inclusion zone. The size of the inclusion zone is therefore a measure of the individual tree's inclusion probability when sampling is done with random sample points. Once the inclusion probabilities are known for all sampled trees, the Horwitz-Thompson estimator can be used as an unbiased estimator for any stand variable. In point-to-tree distance sampling, the inclusion zone of a particular tree depends exclusively on the spatial arrangement of the neighbouring trees. Such inclusion zones are determined by k-order Voronoi polygons, where k is the number of trees being sampled per sample point. The approach, however, requires the positions of the k sample trees and a number of surrounding trees to be mapped. Field application is therefore difficult, but a comparison of plot designs by simulation studies in fully mapped stands can now also be done with an unbiased estimator for k-tree sampling.