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Annals of Forest Science

, Volume 68, Issue 2, pp 325–335 | Cite as

Modelling the diameter distribution of eucalyptus plantations with Johnson’s S B probability density function: parameters recovery from a compatible system of equations to predict stand variables

  • Ayana MateusEmail author
  • Margarida Tomé
Article

Abstract

  • Introduction  The simulation of diameter distributions is the basis for predicting volume in the so-called diameter distribution models. Combined with volume, volume ratio and taper equations, these models allow the prediction of volume assortments according to user needs. The simulation of diameter distributions is also essential in initialising individual tree models. It is also a useful aid for planning harvesting operations.

  • Methods  In this paper, Johnson’s distribution was used to model the diameter distribution of Eucalyptus globulus in Portugal. When a predefined probability density function is used as part of a growth and yield model, the parameters of the function must be estimated for each year during the simulation period.

  • Results  The development of a system of equations that relates stand characteristics to mathematical functions of the distribution, such as the moments of the distribution, allows for the estimation of parameters (i.e. parameter recovery). This method assures compatibility between the characteristics of the observed population used in parameter recovery and those obtained through simulation.

  • Conclusions  The system of equations was built in such a way that the observed well-established biological processes between stand variables is maintained, and the equations were simultaneously fitted to minimise the determinant of the covariance matrix of errors. Based on validation with an independent data set, the model provides precise estimates of total stand volume.

Keywords

Probability density function Johnson’s SB distribution Diameter distribution Forest planning 

Notes

Acknowledgements

This work was supported by the FP6 EFORWOOD IP project (contract 518128) and the FCT project CarbWoodCork (POCI/AGR/57279/2004). The authors gratefully acknowledge Bernard Parresol (USDA Forest Service Southern Research Station) for facilitating the use of a SAS program for the Johnson’s S B distribution parameter recovery that was used in an initial stage of this research. The authors also thank the pulp companies Celbi and Silvicaima for providing a large portion of the data used in this study.

References

  1. Aranda UD (2004) Modelo dinámico de crescimento para masas de Pinus sylvestris L. procedentes de repablación en Galicia. Tesis Doctoral, Universidade de Santiago de Compostela -Escuela Politécnica Superior-Depart. de Ingenieria Agroflorestal, 310 pGoogle Scholar
  2. Bailey RL, Dell TR (1973) Quantifying diameter distributions with the Weibull function. For Sci 19(2):97–104Google Scholar
  3. Cao QV (2004) Predicting parameters of a Weibull function for modeling diameter distribution. For Sci 50(5):682–685Google Scholar
  4. Fonseca TF (2004) Modelação do crescimento, mortalidade e distribuição diamétrica, do pinhal bravo no vale do Tâmega. Tese de Doutoramento, Universidade de Trás-os-Montes e Alto Douro. 247 p.Google Scholar
  5. Fonseca TF, Marques CP, Parresol BR (2009) Describing maritime pine diameter distributions with Johnson’s S B distribution using a new all-parameter recovery approach. For Sci 55(4):367–373Google Scholar
  6. Furtado AX (2006) Modelação da estrutura dinâmica de povoamentos de Eucalyptus globulus em primeira rotação. Tese de Doutoramento, Faculdade de Ciências e tecnologia da Universidade Nova de Lisboa, 192 pGoogle Scholar
  7. Gallant A (1987) Nonlinear statistical models. Wiley, New York, 624 pCrossRefGoogle Scholar
  8. Hafley WL, Buford MA (1985) A bivariate model for growth and yield prediction. For Sci 31:237–247Google Scholar
  9. Hafley WL, Schreuder HT (1977) Statistical distributions for fitting diameter and height data in even-ages stands. Can J For Res 7:481–487CrossRefGoogle Scholar
  10. Hahn GJ, Shapiro SS (1967) Statistics for engineers. Wiley, New York, p 200Google Scholar
  11. Hyink DM, Moser Jr JW (1983) A generalised framework for projecting forest yield and stand structure using diameter distributions. For Sci 29(1):85–95Google Scholar
  12. Johnson NL (1949) Systems of frequency curves generated by methods of translation. Biometrika 36:147–176Google Scholar
  13. Johnson N, Kotz S (1970) Continuous univariate distribution, vol 1. Wiley, New York, 761 pGoogle Scholar
  14. Kamziah AK, Ahmad MI, Lapongan J (1999) Nonlinear regression approach to estimating Johnson SB parameters for diameter data. Can J For Res 29:310–314CrossRefGoogle Scholar
  15. Kiviste A, Nilson A, Hordo M, Merenakk M (2003) Diameter distribution models and height-diameter equations for Estonian forest. In: Amaro A, Reed D, Soares P (eds) Modelling forest systems. CABI, Wallingford, pp 169–179Google Scholar
  16. Law AM, Kelton WD (1982) Simulation modelling and analysis. McGraw-Hill, New York, 400 pGoogle Scholar
  17. Li F, Zhang L, Davis CJ (2002) Modelling the joint distribution of tree diameters and heights by bivariate generalised beta distribution. For Sci 48(1):47–58Google Scholar
  18. Lilliefors HW (1967) On the Kolmogorov–Smirnov test for normality with mean and variance unknown. Am Stat Assoc J 62:399–402CrossRefGoogle Scholar
  19. Maltamo M, Puumalainen J, Paivinen R (1995) Comparison of Beta and Weibull functions for modelling basal areas diameter distributions in stands of Pinus sylvestris and Picea abies. Scand J For Res 10:284–295CrossRefGoogle Scholar
  20. Marto M, Palma J, Mateus A, Tomé M (2009) Computer program for estimation of Johnson’s S B parameters using a parameter recovery method. Publicações Científicas Forchange PC-X/2009. Centro de Estudos Florestais, Instituto Superior de Agronomia, Universidade Técnica de Lisboa, LisboaGoogle Scholar
  21. Massey FJ (1951) The Kolmogorov–Smirnov test for goodness of fit. Am Stat Assoc J 46:68–78CrossRefGoogle Scholar
  22. Ek AR, Monserud RA (1979) Performance and comparisons of stand growth models based on individual tree and diameter class growth. Can J For Res 9:231–244CrossRefGoogle Scholar
  23. Myers RH (1990) Classical and modern regression with applications, 2nd edn. Duxbury, Belmont, 488 pGoogle Scholar
  24. Palahí M, Pukkala T, Blasco E, Trasobares A (2007) A comparison of beta, Johnsons SB, Weibull and truncated Weibull functions for modelling the diameter distribution of forest stands in Catalonia (north-east of Spain). Eur J For Res 126(4):563–571Google Scholar
  25. Parresol B (2003) Recovering parameters of johnson’s S B distribution. Res. pap. SRS-31. USDA For. Ser., Southern Research Station, Asheville, p 9Google Scholar
  26. Páscoa F (1987) Estrutura, Crescimento e Produção em Povoamentos de Pinheiro Bravo. Um Modelo de Simulação. Tese de Doutoramento, Universidade Técnica de Lisboa—Instituto Superior de AgronomiaGoogle Scholar
  27. Perry DA (1985) The competition process in forest stands. In: Cannel MGR, Jackson JE (eds) Attributes of trees as crop plants. Institute of Terrestrial Ecology, Abbots Ripton, pp 481–506Google Scholar
  28. Rennolls K, Wang M (2005) A new parameterisation of Johnson’s S B distribution with application to fitting forest tree diameter data. Can J For Res 35:575–579CrossRefGoogle Scholar
  29. Reynolds MR, Burk TE, Huang W (1988) Goodness-of-fit tests and model selection procedures for diameter distributions models For Sci 34:377–399Google Scholar
  30. SAS Institute (2005) The SAS System for windows, 9.1 edn. SAS Institute, CaryGoogle Scholar
  31. Soares P, Tomé M (1996) Changes in eucalypt plantations structure, variability and relative growth pattern under different intraspecific competition gradients. In: Skovsgaard JP, Johannsen VK (eds) Modelling regeneration success and early growth of forest stands. Proceedings from the IUFRO conference, Copenhagen. Danish Forest and Landscape Research Institute, Hörsholm, pp 270–284Google Scholar
  32. Tomé M, Ribeiro F, Soares P (2001) O modelo GLOBULUS 2.1-Relatórios técnico científicos do GIMREF no 1/2001. Universidade Técnica de Lisboa, Instituto Superior de Agronomia, Centro de Estudos Florestais, LisboaGoogle Scholar
  33. Tomé M, Barreiro S, Cortiçada A, Meyer A, Ramos T, Malico P (2007) Inventário florestal 2005-2006. Áreas, volumes e biomassas dos povoamentos florestais. Resultados Nacionais e por Nut’s II e III. Publicações GIMREF RT 5/2007. Universidade Técnica de Lisboa, Instituto Superior de Agronomia, Centro de Estudos Florestais, LisboaGoogle Scholar
  34. Zhang L, Packard KC, Liu C (2003) A comparison of estimation methods for fitting Weibull and Johnson’s SB distribution to mixed spruce-fir stands in northeastern North America. Can J For Res 33(7):1340–1347CrossRefGoogle Scholar
  35. Zhou B, McTague JP (1996) Comparison and evaluation of five methods of estimation of the Johnson system parameters. Can J For Res 26:928–935CrossRefGoogle Scholar

Copyright information

© INRA and Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.CMA, Departamento de Matemática, Faculdade de Ciências e Tecnologia, FCTUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Centro de Estudos Florestais (CEF), Instituto Superior de Agronomia (ISA)Universidade Técnica de Lisboa (UTL)LisbonPortugal

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