Environmental and Ecological Statistics

, Volume 19, Issue 1, pp 107–123 | Cite as

Triangulation based inclusion probabilities: a design-unbiased sampling approach

  • Lutz Fehrmann
  • Timothy G. Gregoire
  • Christoph Kleinn
Open Access


A probabilistic sampling approach for design-unbiased estimation of area-related quantitative characteristics of spatially dispersed population units is proposed. The developed field protocol includes a fixed number of 3 units per sampling location and is based on partial triangulations over their natural neighbors to derive the individual inclusion probabilities. The performance of the proposed design is tested in comparison to fixed area sample plots in a simulation with two forest stands. Evaluation is based on a general approach for areal sampling in which all characteristics of the resulting population of possible samples is derived analytically by means of a complete tessellation of the areal sampling frame. The example simulation shows promising results. Expected errors under this design are comparable to sample plots including a much greater number of trees per plot.


Design based inference Inclusion probability Delaunay Triangulation Plot design Continuous population 



This research was supported by the German Research Foundation DFG (KL 894/ 13-1). We thank Sebastian Schnell for helpful methodological and technical discussions. Further we thank two anonymous reviewers. Their extensive contributions to formulate the idea was substantial. We appreciated constructive comments, help and valuable suggestions to improve the manuscript.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Lutz Fehrmann
    • 1
  • Timothy G. Gregoire
    • 2
  • Christoph Kleinn
    • 1
  1. 1.Chair of Forest Inventory and Remote SensingGeorg-August-Universität GöttingenGöttingenGermany
  2. 2.Yale School of Forestry and Environmental StudiesNew HavenUSA

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