Fitting diameter distribution models to data from forest inventories with concentric plot design

Nikos Nanos, Sara Sjöstedt de Luna


Aim: Several national forest inventories use a complex plot design based on multiple concentric subplots where smaller diameter trees are inventoried when lying in the smaller-radius subplots and ignored otherwise. Data from these plots are truncated with threshold (truncation) diameters varying according to the distance from the plot centre. In this paper we designed a maximum likelihood method to fit the Weibull diameter distribution to data from concentric plots.

Material and methods: Our method (M1) was based on multiple truncated probability density functions to build the likelihood. In addition, we used an alternative method (M2) presented recently. We used methods M1 and M2 as well as two other reference methods to estimate the Weibull parameters in 40000 simulated plots. The spatial tree pattern of the simulated plots was generated using four models of spatial point patterns. Two error indices were used to assess the relative performance of M1 and M2 in estimating relevant stand-level variables. In addition, we estimated the Quadratic Mean plot Diameter (QMD) using Expansion Factors (EFs).

Main results: Methods M1 and M2 produced comparable estimation errors in random and cluster tree spatial patterns. Method M2 produced biased parameter estimates in plots with inhomogeneous Poisson patterns. Estimation of QMD using EFs produced biased results in plots within inhomogeneous intensity Poisson patterns.

Research highlights:We designed a new method to fit the Weibull distribution to forest inventory data from concentric plots that achieves high accuracy and precision in parameter estimates regardless of the within-plot spatial tree pattern. 



expansion factors; forest growth and yield; National Forest Inventory; spatial point pattern; Weibull

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Adame P, Del Rìo M, Cañellas I, 2010. Ingrowth model for pyrenean oak stands in north-western Spain using continuous forest inventory data. Eur J For Res 129 (4): 669-678.

Alberdi I, Condés S, Martínez-Millán J, Saura S, Sánchez G, Pérez F, Villanueva JA, Vallejo R, 2010. National Forest Inventory Report: Spain. In: National Forest Inventories: pathways for harmonised reporting; Tomppo E, Gschwantner T, Lawrence M, McRoberts RE (eds). pp. 527-554. Springer.

Aunós A, Riba A, Blanco R, 2010. Caracterización selvícola de las masas monoespecíficas de pino laricio en Cataluña. Forest Syst 18 (3): 338-349.

Baddeley A, 2010. Analysing spatial point patterns in R. [September 14 2016].

Baddeley A, Turner R, 2005. Spatstat: An R package for analyzing spatial point patterns. J Stat Softw 12 (6): 1-42.

Bailey RL, Dell T, 1973. Quantifying diameter distributions with the Weibull function. Forest Sci 19 (2): 97-104.

Balakrishnan N, Kateri M, 2008. On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data. Stat Probab Lett 78 (17): 2971-2975.

Barreiro S, Godinho Ferreira P, Azevedo A, 2010. National Forest Inventory Report: Portugal. In: National Forest Inventories: pathways for harmonised reporting; Tomppo E, Gschwantner T, Lawrence M, McRoberts RE (eds). pp. 437-464. Springer.

Böhl J, Brändli UB, 2007. Deadwood volume assessment in the third Swiss National Forest Inventory: methods and first results. Eur J For Res 126 (3): 449-457.

Bravo F, Álvarez-González JG, Río M, Barrio M, Bonet JA, Bravo-Oviedo A, Calama R, Castedo-Dorado F, Crecente-Campo F, Condes S, et al., 2011. Growth and yield models in Spain: historical overview, contemporary examples and perspectives. Forest Syst 20 (2): 315-328.

Breidenbach J, Antón-Fernández C, Petersson H, McRoberts RE, Astrup R, 2014. Quantifying the model-related variability of biomass stock and change estimates in the Norwegian National Forest Inventory. Forest Sci 60 (1): 25-33.

Cañadas N, Montero G, Güemes C, 2002. Modelización de la distribución diamétrica de las masas de Pinus pinea L. de Valladolid (España) mediante la función Weibull. Invest Agrar: Sist Recur For 11 (2): 263-282.

Casella G, Berger RL, 2001. Statistical inference, 2nd ed. Dusbury, Thomson Learning, 660 pp.

Chirici G, McRoberts RE, Winter S, Bertini R, Brändli UB, Asensio IA, Bastrup-Birk A, Rondeux J, Barsoum N, Marchetti M, 2012. National forest inventory contributions to forest biodiversity monitoring. Forest Sci 58 (3): 257-268.

de Lima RAF, Batista JLF, Prado PI, 2015. Modeling tree diameter distributions in natural forests: An evaluation of 10 statistical models. Forest Sci 61 (2): 320-327.

Del Río M, Montes F, Cañellas I, Montero G, 2003. Revisión: Índices de diversidad estructural en masas forestales. Invest Agrar: Sist Recur For 12 (1): 159-176.

Diamantopoulou MJ, Özçelik R, Crecente-Campo F, Eler Ü, 2015. Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods. Biosyst Eng 133: 33-45.

Diggle P, 2003. Statistical analysis of spatial point patterns, 2nd ed. Arnold Publishers. 159 pp.

Dimov LD, Chambers JL, Lockhart BR, 2005. Spatial continuity of tree attributes in bottomland hardwood forests in the southeastern United States. Forest Sci 51 (6): 532-540.

Fraver S, D'Amato AW, Bradford JB, Jonsson BG, Jönsson M, Esseen PA, 2014. Tree growth and competition in an old-growth Picea abies forest of boreal Sweden: influence of tree spatial patterning. J Veg Sci 25 (2): 374-385.

Fridman J, Holm S, Nilsson M, Nilsson P, Ringvall AH, Ståhl G, 2014. Adapting National Forest Inventories to changing requirements-The case of the Swedish National Forest Inventory at the turn of the 20th century. Silva Fenn 48 (3): 1095.

Gobakken T, Næsset E, Nelson R, Bollandsås OM, Gregoire TG, Ståhl G, Holm S, Ørka HO, Astrup R, 2012. Estimating biomass in Hedmark County, Norway using national forest inventory field plots and airborne laser scanning. Remote Sens Environ 123: 443-456.

González JR, Trasobares A, Palahí M, Pukkala T, 2007. Predicting stand damage and tree survival in burned forests in Catalonia (North-East Spain). Ann For Sci 64 (7): 733-742.

Gorgoso J, Álvarez González J, Rojo A, Grandas-Arias J, 2007. Modelling diameter distributions of Betula alba L. stands in northwest Spain with the two-parameter Weibull function. Forest Syst 16 (2): 113-123.

Henttonen HM, Kangas A, 2015. Optimal plot design in a multipurpose forest inventory. Forest Ecosyst 2 (1): 1-14.

Husch B, Beers TW, Kershaw Jr, JA, 2003. Forest mensuration, 4th ed. John Wiley & Sons. 456 pp.

Johnson NL, Kotz S, Balakrishnan N, 1995. Continuous univariate distributions, 2nd ed. Wiley. 756 pp.

Kangas A, Maltamo M (eds), 2006. Forest inventory: methodology and applications. Springer, Dordrecht, The Netherlands. 362 pp.

Kitahara F, Mizoue N, Yoshida S, 2009. Evaluation of data quality in Japanese National Forest Inventory. Environ Monit Assess 159 (1): 331-340.

Madrigal A, Álvarez JG, Rodríguez R, Rojo A, 1999. Tablas de producción para los montes españoles. Fundación Conde del Valle de Salazar. UPM, Escuela Técnica Superior de Ingenieros de Montes, Madrid, 253 pp.

Mcgarrigle E, Kershaw JA, Lavigne MB, Weiskittel AR, Ducey M, 2011. Predicting the number of trees in small diameter classes using predictions from a two-parameter Weibull distribution. Forestry 84 (4): 431-439.

McRoberts RE, Ståhl G, Vidal C, Lawrence M, Tomppo E, Schadauer K, Chirici G, Bastrup-Birk A, 2010a. National Forest Inventories: Prospects for harmonised international reporting. In: National Forest Inventories: Pathways for common reporting; Tomppo E, Gschwantner T, Lawrence M, McRoberts ER (eds). pp. 33-43. Springer Netherlands, Dordrecht.

McRoberts RE, Tomppo EO, Næsset E, 2010b. Advances and emerging issues in national forest inventories. Scand J Forest Res 25 (4): 368-381.

Mehtätalo L, Comas C, Pukkala T, Palahí M, 2011. Combining a predicted diameter distribution with an estimate based on a small sample of diameters. Can J For Res 41 (4): 750-762.

Montero G, Ruiz-Peinado R, Muñoz M, 2005. Producción de biomasa y fijación de CO2 por los bosques españoles. Monografias INIA: Serie Forestal Nº 13. INIA, Madrid. 264 pp.

Montoya D, Zavala MA, Rodríguez MA, Purves DW, 2008. Animal versus wind dispersal and the robustness of tree species to deforestation. Science 320 (5882): 1502-1504.

Nanos N, Montero G, 2002. Spatial prediction of diameter distribution models. For Ecol Manage 161 (1): 147-158.

Nanos N, Gil L, Montero G, 2002. Análisis espacial de los datos del Inventario Forestal Nacional utilizando técnicas geoestadísticas. In : El inventario forestal nacional: elemento clave para la gestión forestal sostenible; Bravo F, del-Río M, del-Peso C (eds). pp 149-158. Fundación General de la Universidad de Valladolid, Spain.

Nanos N, Calama R, Montero G, Gil L, 2004a. Geostatistical prediction of height/diameter models. For Ecol Manage 195 (1): 221-235.

Nanos N, González-Martı́nez SC, Bravo F, 2004b. Studying within-stand structure and dynamics with geostatistical and molecular marker tools. For Ecol Manage 189 (1): 223-240.

Ng H, Chan P, Balakrishnan N, 2002. Estimation of parameters from progressively censored data using EM algorithm. Comput Stat Data An 39 (4): 371-386.

Packalen P, Vauhkonen J, Kallio E, Peuhkurinen J, Pitkänen J, Pippuri I, Strunk J, Maltamo M, 2013. Predicting the spatial pattern of trees by airborne laser scanning. Int J Remote Sens 34 (14): 5154-5165.

Palahí M, Pukkala T, Trasobares A, 2006. Modelling the diameter distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis forest stands in Catalonia using the truncated Weibull function. Forestry 79 (5): 553-562.

Palahí M, Pukkala T, Blasco E, Trasobares A, 2007. Comparison of beta, Johnson's SB, Weibull and truncated Weibull functions for modeling the diameter distribution of forest stands in Catalonia (north-east of Spain). Eur J For Res 126 (4): 563-571.

Pique-Nicolau M, del-Rio M, Calama R, Montero G, 2011. Modelling silviculture alternatives for managing Pinus pinea L. forest in North-East Spain. Forest Syst 20 (1): 3-20.

Pommerening A, 2006. Evaluating structural indices by reversing forest structural analysis. For Ecol Manage 224 (3): 266-277.

Pommerening A, Särkkä A, 2013. What mark variograms tell about spatial plant interactions. Ecol Model 251: 64-72.

R Core Team, 2013. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Rennolls K, Geary D, Rollinson T, 1985. Characterizing diameter distributions by the use of the Weibull distribution. Forestry 58 (1): 57-66.

Ruiz-Peinado R, del Rio M, Montero G, 2011. New models for estimating the carbon sink capacity of Spanish softwood species. Forest Syst 20 (1): 176-188.

Ruiz-Peinado R, Bravo-Oviedo A, Montero G, del Río M, 2016. Carbon stocks in a Scots pine afforestation under different thinning intensities management. Mitigation and Adaptation Strategies for Global Change 21 (7): 1059-1072.

Schlather M, Ribeiro PJ, Diggle PJ, 2004. Detecting dependence between marks and locations of marked point processes. J R Stat Soc Ser B 66 (1): 79-93.

Sghaier T, Cañellas I, Calama R, Sánchez-González M, 2016. Modelling diameter distribution of Tetraclinis articulata in Tunisia using normal and Weibull distributions with parameters depending on stand variables. IFOREST: 705.

Sharma RP, Breidenbach J, 2015. Modeling height-diameter relationships for Norway spruce, Scots pine, and downy birch using Norwegian national forest inventory data. Forest Sci Technol 11 (1): 44-53.

Teissier du Cros R, Lopez S, 2009. Preliminary study on the assessment of dead wood volume by the French national forest inventory. Ann For Sci 66 (3): 302-302.

Thomas M, 1949. A generalization of Poisson's binomial limit for use in ecology. Biometrika 36 (1/2): 18-25.

Tomppo E, Gschwantner T, Lawrence M, 2010. National forest inventories: pathways for common reporting. Springer Verlag. 612 pp.

Valbuena-Rabadán MA, Santamaría-Peña J, Sanz-Adán F, 2016. Estimation of diameter and height of individual trees for Pinus sylvestris L. based on the individualising of crowns using airborne LiDAR and the National Forestry Inventory data. Forest Syst 25 (1): e046.

Weiskittel AR, Hann DW, Kershaw Jr JA, Vanclay JK, 2011. Forest growth and yield modeling. John Wiley & Sons. 344 pp.

Winter S, Chirici G, McRoberts RE, Hauk E, 2008. Possibilities for harmonizing national forest inventory data for use in forest biodiversity assessments. Forestry 81 (1): 33-44.

Wulff S, Hansson P, Witzell J, 2006. The applicability of national forest inventories for estimating forest damage outbreaks-Experiences from a Gremmeniella outbreak in Sweden. Can J For Res 36 (10): 2605-2613.

Zeng W, Tomppo E, Healey SP, Gadow KV, 2015, The national forest inventory in China: history-results-international context. Forest Ecosyst 2 (1): 1-16.

DOI: 10.5424/fs/2017262-10486