A W F Projects   |   Prof. Dr. Christoph Kleinn - Dr. Lutz Fehrmann
Development of design unbiased estimators for the restricted k-tree sampling techniques PCM (point-centered quarter method) and T-square sampling


Field sampling is one of the major sources of information in empirical research in ecology, forestry and related disciplines. When decisions are made about sampling and plot techniques the researcher needs to take into account the tradeoff between precision/accuracy and cost. Point to object distance sampling, also called k-tree sampling or fixed-count sampling (because per sample a fixed number of k nearest trees are being taken as sample trees), belongs to a class of plot design techniques that is very practical and is easily implemented in the field. Its practicability is frequently stressed as a major advantage over, for example, fixed area plots. In ecology, k-tree sampling is frequently applied (Krebs 1999), while forest inventory statisticians tend to advise against it (Pyandeh and Ek 1986, Mandallaz 1995, Schreuder 2004) because of concern for bias. An unbiased estimator had only been developed recently – resulting from DFG project KL894-7 (Kleinn and Vilčko 2006b).

However, that unbiased estimator requires determining (mapping) the position of various neighbouring trees to be able to determine the per-sample-tree selection probability which is required for application of the Horwitz-Thompson estimator.

A number of research questions arise from project KL894-7, in particular regarding the practicability of the unbiased estimator approach, the possibility to develop easier to implement proxies and, above all, whether the approach can also be applied to a class of what we call here restricted k-tree sampling, where the k “nearest” trees around a sample point are selected subject to particular conditions; specifically, the point-centered quarter method and the T-square sampling technique.


Overall objective of this study is to support sustainable forest management and research into ecological and forest management issues by increasing the efficiency of data and information gathering by means of k-tree plot design options.

In order to achieve the above specified overall and primary technical objectives, the following technical objectives are defined:

  1. Literature review.

  2. Analysis of the geometric characteristics of the point centered quarter method (PCQM) and derivation of an approach to calculate per-tree inclusion zones for any (reasonably large) number of sectors and any (reasonably large) number of nearest trees per sector.

  3. Translation of that algorithm for the generalized PCQM into an efficient software solution.

  4. Analysis of the geometric characteristics of the T-square technique and derivation of an approach to calculate per-tree inclusion zones for any (reasonably large) values of k.

  5. Translation of that algorithm for T-square sampling into an efficient software solution.

  6. Generate artificial tree maps with at least three different spatial tree arrangements: random, systematic and clustered.

  7. Provide at least three real tree maps for simulation studies.

  8. Simulation studies:

    a) Carry out simulation studies with the artificial and the real tree maps comparing the performance among the k-tree sampling approaches for equal values of k each.

    b) Carry out simulation studies comparing the k-tree techniques to fixed area plot sampling and relascope sampling (for the same average number of k trees selected per sample point) for the per-hectare estimation of stem density and basal area.

  9. Develop an efficient approach for practical implementation of inclusion zone calculations.

  10. Develop approximation techniques, for example by regressing inclusion zones against distance measures taken from the sample point to the neighbouring trees.

  11. Apply the techniques in the field to various target objects and target variables under contrasting conditions (cooperation with partners in Chile and South Africa) and conduct time studies to test the hypothesis of the practical superiority of k-tree sampling.

  12. Report the results and present them for discussion in at least two congresses in forestry and one in quantitative ecology.


11.08 - 04.09: Update literature review
12.08 - 05.09: PCSM: inclusion zones algorithm
05.09 - 09.09: Software implementation PCSM
08.09 - 01.10: T-square: inclusion zones algorithm
11.09 - 03.10: Software implementation T-square
02.10 - 04.10: Generate artificial tree maps
03.10 - 03.10: Provide real tree maps
03.10 - 12.10: Sampling simulation on tree maps
08.10 - 01.11: Implementation approach
05.10 - 04.11: Field application and time studies
02.11 - 07.11: Modeling inclusion zones
06.11 - 10.11: Report and present results


Eberhardt, L.L. 1967. Some developments in distance sampling. Biometrics (23):207-216.

Engeman, R., R. Sugihara, L. Pank and W.E. Dusenberry. 1994. A comparison of plotless density estimators using Monte Carlo simulation. Ecology 75:1769-1779.

Essed, D. 1957. Estimation of standing timber. Doctoral Thesis. Wageningen, 60p.

Gregoire, T. and H. Valentine. 2008. Sampling Strategies for Natural Resources and the Environment (Applied Environmental Statistics). Chapman & Hall. 496p.

Hall, J.B. 1991. Multiple-nearest-tree sampling in an ecological survey of Afromontane catchment forest. Forest Ecology and Management 42:245-266.

Kleinn, C. und F. Vilčko. 2006a. A new empirical approximation for estimation in k-tree sampling. Forest Ecology and Management 237(2):522-533.

Kleinn, C. and F. Vilčko. 2006b. Design unbiased estimation for point to tree distance sampling. Canadian Journal of Forest Research 36(6):1407-1414

König, G. 1835. Die Forstmathematik mit Anweisung zur Forstvermessung, Holzschätzung und Waldwerthrechnung, nebst Hüfstafeln für Forstschätzer.Gotha, in Commission der Becker´schen Buchhandlung. X, 436, 56p.

Krebs, C.J. 1999. Ecological methodology. Second edition. Addison Wesley Longman. 620p.

Magnussen, S. and C. Kleinn. In Press. Two new density estimators for distance sampling. European Journal of Forest Research.

Mandallaz, D. 1995. “Les hazard fait bien les choses”: statistische Methoden für die Waldinventur. Schweiz. Z. Forstwes. 146(12):1015-1032.

Okabe, A., B. Boots, K. Sugihara and S.N. Chiu. 1999. Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. Second Edition. Wiley. 671p.

Payandeh, B. and A.R. Ek. 1986. Distance methods and density estimators. Can. J. For. Res. 16:918-924.

Picard, N., A.M. Kouyaté and H. Dessard. 2005. Tree Density Estimations Using a Distance Method in Mali Savanna. Forest Science 51(1):7-18.

Pielou, E.C. 1959. The use of point-to-plant distances in the study of pattern of plant population. J. Ecology 47:603-613.

Prodan, M. 1968. Punktstichprobe für die Forsteinrichtung (A point sample for forest management planning). Forst- und Holzwirt 23(11):225-226.

Roesch, F.A., E.J. Green and C. Scott. 1993. An Alternative View on Forest Sampling. Survey Methodology 19(2):199.204.

Schreuder, H. T. 2004. Sampling using a fixed number of trees per plot. Research Note RMRS-RN-17. USDA Forest Service. p.1-3.

Sheil, D., M.J. Ducey and I. Samsoedin. 2003. A new type of sample unit for the efficient assessment of diverse tree communities in complex forest landscapes. Journal of Tropical Forest Science 15(1):117-135.

Stoffels, A. 1955. Die Genauigkeit der Bestimmung der Stammzahl pro Hektar durch Messung von Stammabständen (The accuracy of estimation of number of stems per hectare with distance measurements). Forstwiss. Centralbl. 74:211-218.

Sutherland, W.J. (editor) 1998. Ecological Census Techniques. A Handbook. Cambridge University Press. 336p.

Valentine, H.T., J.H. Gove and T.G. Gregoire. 2001. Monte Carlo approaches to sampling forested tracts with lines and points. Can. J. For. Res. 31:1410-1424.


Prof. Dr. Martin Schlather
Institute of Mathematical Stochastics
University of Göttingen

Prof. van Laar and Dr. Coert Geldenhuys
Stellenbosch University
South Africa

Prof. Dr. Victor Sandoval
Universidad Austral de Chile


Prof. Dr.Christoph Kleinn
Institut für Waldinventur und Waldwachstum
Büsgenweg 5, 37077 Göttingen
Tel. +49 551 39 3473

 Participating Scientists

Dr. Lutz Fehrmann
Institut für Waldinventur und Waldwachstum
Büsgenweg 5, 37077 Göttingen
Tel. +49 551 39 3826


DFG (Deutsche Forschungsgemeinschaft)

 Time Frame

November 2008 – October 2011