Abstract
The old-growth forests are characterized by multi-aged structures for which the De Liocourt model with the Meyer negative exponential function is the most common descriptor of tree diameter distributions. In this study, we put forward a mixture of Gaussian functions model for describing the diameter at breast height distribution in uneven-aged mixed forest stands. For this purpose, a permanent plot of one hectare was installed in the natural silver fir-beech mixed old-growth forest Şinca Veche (Făgăraş Mountains, Romania). The sample with 396 trees was split into homogeneous age groups using k-means clustering analysis and by limiting the coefficient of variation to 30%, both for the diameter and for the maximum cambial age at breast height. The proposed cumulative distribution function is generated by summing the Gaussian probability density functions obtained for each age group. For verifying the efficacy of the parameter estimation procedure and to assess the goodness-of-fit, suitable statistical criteria were used: Shapiro–Wilk normality test and the root mean square error (RMSE). The developed mixture distribution and the negative exponential distribution were both compared to the empirical diameter distribution. The comparison of the observed RMSE values (3.12 < 5.93) shows the suitability of the proposed method, with a better degree of accuracy than that achieved by applying the negative exponential model. The results of this study confirm the applicability of Gaussian multi-component models for describing tree diameter distributions in natural silver fir-beech mixed stands.
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References
Alessandrini A, Biondi F, Di Filippo A, Ziaco E, Piovesan G (2011) Tree size distribution at increasing spatial scales converges to the rotated sigmoid curve in two old-growth beech stands of the Italian Apennines. For Ecol Manage 262:1950–1962. https://doi.org/10.1016/j.foreco.2011.08.025
Assmann E (1970) The principles of forest yield study: studies in the organic production, structure, increment, and yield of forest stands. Pergamon Press, Oxford
Barati R (2013) Application of Excel solver for parameter estimation of the nonlinear Muskingum models. KSCE J Civ Eng 17(5):1139–1148. https://doi.org/10.1007/s12205-013-0037-2
Brown CE (1998) Chapter 13: coefficient of variation. In: Applied multivariate statistics in geohydrology and related sciences. Springer, Berlin, pp 155–157. https://doi.org/10.1007/978-3-642-80328-4_13
Buksnowitz C, Teischinger A, Grabner M, Müller U, Mahn L (2010) Tracheid length in Norway spruce (Picea abies (L.) Karst.). Analysis of three databases regarding tree age, cambial age, tree height, inter-annual variation, radial distance to pith and log qualities. Wood Res 55(4):1–14
Calama R, Barbeito I, Pardos M, del Río M, Montero G (2008) Adapting a model for even-aged Pinus pinea L. stands to complex multi-aged structures. For Ecol Manage 256:1390–1399. https://doi.org/10.1016/j.foreco.2008.06.050
Cancino J, Gadow KV (2002) Stem number guide curves for uneven-aged forests, development and limitations. In: Gadow KV, Nagel J, Saborowski J (eds) Continuous cover forestry. Kluwer Academic Publishers, Dordrecht, pp 163–174. https://doi.org/10.1007/978-94-015-9886-6_13
Cook ER, Kairiukstis LA (eds) (1990) Methods of dendrochronology: applications in the environmental sciences. Kluwer Academic Publishers and International Institute for Applied Systems Analysis, Dordrecht
de Lima RAF, Batista JLF, Prado PI (2015) Modeling tree diameter distributions in natural forests: an evaluation of 10 statistical models. For Sci 61(2):320–327. https://doi.org/10.5849/forsci.14-070
De Liocourt F (1898) De l’amenagement des sapinières. Bull Trimestriel Soc For Franche-Comté Belfort Besançon 4:396–409
Duduman G (2011) A forest management planning tool to create highly diverse uneven-aged stands. Forestry 84(3):301–314. https://doi.org/10.1093/forestry/cpr014
Emborg J, Christensen M, Heilmann-Clausen J (1996) The structure of Suserup skov, a near-natural temperate deciduous forest in Denmark. For Landsc Res 1:311–333
Fries C, Johansson O, Pettersson B, Simonsson P (1997) Silvicultural models to maintain and restore natural stand structures in Swedish boreal forests. For Ecol Manage 94:89–103. https://doi.org/10.1016/S0378-1127(97)00003-0
Gadow KV, Zhang CY, Wehenkel C, Pommerening A, Corral-Rivas J, Korol M, Myklush S, Hui GY, Kiviste A, Zhao XH (2012) Chapter 2: forest structure and diversity. In: Pukkala T, Gadow KV (eds) Continuous cover forestry. Managing forest ecosystems 23. Springer, Berlin, pp 29–83. https://doi.org/10.1007/978-94-007-2202-6_2
Gamborg C, Larsen JB (2003) ‘Back to nature’—a sustainable future for forestry? For Ecol Manage 179:559–571. https://doi.org/10.1016/S0378-1127(02)00553-4
Gauss CF (1823) Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Gottingen, Germany
Goff FG, West D (1975) Canopy-understory interaction effects on forest population structure. For Sci 21:98–108
Gove JH, Ducey MJ, Leak WB, Zhang L (2008) Rotated sigmoid structures in managed uneven-aged northern hardwood stands: a look at the Burr Type III distribution. Forestry 81(2):161–176. https://doi.org/10.1093/forestry/cpm025
Grubbs F (1969) Procedures for detecting outlying observations in samples. Technometrics 11(1):1–21. https://doi.org/10.2307/1266761
Gül AU, Misir M, Misir N, Yavuz H (2005) Calculation of uneven-aged stand structures with the negative exponential diameter distribution and Sterba’s modified competition density rule. For Ecol Manage 214:212–220. https://doi.org/10.1016/j.foreco.2005.04.012
Guldin JM (1996) The role of uneven-aged silviculture in the context of ecosystem management. West J Appl For 11:4–12
Haight RG (1987) Evaluating the efficiency of even-aged and uneven-aged stand management. For Sci 33:116–134
Henderson AR (2006) Testing experimental data for univariate normality. Clin Chim Acta 366:112–129. https://doi.org/10.1016/j.cca.2005.11.007
Horodnic SA (2012) Jugglers in statistics of normality. In: Horodnic SA, Duduman ML, Palaghianu C (eds) Proceedings of the international conference integrated management of environmental resources—Suceava, November 4–6th, 2011. Editura Universităţii “Ştefan cel Mare” Suceava, Romania, pp 102–109. https://doi.org/10.13140/2.1.3931.2162
IUCN (2017) IUCN evaluations of nominations of natural and mixed properties to the world heritage list—WHC/17/41.COM/INF.8B2. IUCN report for the world heritage list. 41st session, Kraków, Poland, 2–12 July
Janowiak M, Nagel LM, Webster C (2008) Spatial scale and stand structure in northern hardwood forests: implications for quantifying diameter distributions. For Sci 54:497–506
Jaworski A, Podlaski R (2012) Modelling irregular and multimodal tree diameter distributions by finite mixture models: an approach to stand structure characterization. J For Res 17:79–88. https://doi.org/10.1007/s10310-011-0254-9
Kimmins JP (1987) Forest ecology. Macmillan, New York
Korpel S (1995) Die Urwälder der Westkarpaten. Fischer, Stuttgart (in German)
Laplace PS (1812) Théorie analytique des probabilités. Courcier, Paris
Lasdon LS, Smith S (1992) Solving sparse nonlinear programs using GRG. ORSA J Comput 4(1):2–15. https://doi.org/10.1287/ijoc.4.1.2
Leak WB (1964) An expression of diameter distribution for unbalanced, uneven-aged stands and forests. For Sci 10:39–50
Leak WB (1996) Long-term structural change in uneven-aged northern hardwoods. For Sci 42:160–165
Liu C, Zhang L, Davis CJ, Solomon DS, Grove JH (2002) A finite mixture model for characterizing the diameter distribution of mixed-species forest stands. For Sci 48:653–661
Loewenstein EF, Johnson PS, Garrett HE (2000) Age and diameter structure of a managed uneven-aged oak forest. Can J For Res 30:1060–1070. https://doi.org/10.1139/x00-036
Lorimer CG (1980) Age structure and disturbance history of a Southern Appalachian virgin forest. Ecology 61:1169–1184. https://doi.org/10.2307/1936836
Lorimer CG, Krug AG (1983) Diameter distributions in even-aged stands of shade-tolerant and midtolerant tree species. Am Midl Nat 109(2):331–345. https://doi.org/10.2307/2425414
Macdonald PDM, Pitcher TJ (1979) Age-groups from size-frequency data: a versatile and efficient method of analyzing distribution mixtures. J Fish Res Board Can 36(8):987–1001. https://doi.org/10.1139/f79-137
Marín-Pageo F, Rapp-Arrarás Í (2013) The application of the Liocourt model to uneven-aged cork oak stands. L’Italia Forestale e Montana 68(2):85–93. https://doi.org/10.4129/ifm.2013.2.03
McLachlan G, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, Hoboken
Meyer HA (1933) Eine mathematisch-statistische Untersuchung über den Aufbau des Plenterwaldes. Schweizerische Zeitschrift für Forstwesen 84:33–46, 88–103, 124–131
Meyer HA (1952) Structure, growth, and drain in uneven-aged forests. J For 50:85–92
Meyer HA, Stevenson DD (1943) The structure and growth of virgin beech-birch-maple-hemlock forests in northern Pennsylvania. J Agric Res 67:465–484
O’Hara KL (1998) Silviculture for structure diversity: a new look at multiaged systems. J Forest 96:4–10
Oheimb GV, Westphal C, Tempel H, Härdtle W (2005) Structural pattern of a near-natural beech forest (Fagus sylvatica) (Serrahn, north-east Germany). For Ecol Manage 212:253–263. https://doi.org/10.1016/j.foreco.2005.03.033
Oliver CD, Larson BC (1996) Forest stand dynamics, update edn. Wiley, New York
Pach M, Podlaski R (2015) Tree diameter structural diversity in Central European forests with Abies alba and Fagus sylvatica: managed versus unmanaged forest stands. Ecol Res 30(2):367–384. https://doi.org/10.1007/s11284-014-1232-4
Peng C (2000) Growth and yield models for uneven-aged stands: past, present and future. For Ecol Manage 132:259–279. https://doi.org/10.1016/S0378-1127(99)00229-7
Peterken GF (1996) Natural woodland: ecology and conservation in northern temperate regions. Cambridge University Press, Cambridge
Podlaski R (2017) Forest modelling: the gamma shape mixture model and simulation of tree diameter distributions. Ann For Sci 74(2):29. https://doi.org/10.1007/s13595-017-0629-y
Podlaski R, Roesch FA (2014) Modelling diameter distributions of two-cohort forest stands with various proportions of dominant species: a two-component mixture model approach. Math Biosci 249:60–74. https://doi.org/10.1016/j.mbs.2014.01.007
Pontailler JY, Faille A, Lemée G (1997) Storms drive successional dynamics in natural forests: a case study in Fontainebleau forest (France). For Ecol Manage 98:1–15. https://doi.org/10.1016/S0378-1127(97)00073-X
Prodan M (1965) Holzmesslehre. J.D. Sauerländer’s Verlag, Frankfurt am Main (in German)
Pryseley A, Mintiens K, Knapen K, der Stede YV, Molenberghs G (2010) Estimating precision, repeatability, and reproducibility from Gaussian and non-Gaussian data: a mixed models approach. J Appl Stat 37(9–10):1729–1747. https://doi.org/10.1080/02664760903150706
Razali NM, Wah YB (2011) Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests. J Stat Model Anal 2(1):21–33
Reynolds MR, Burk T, Huang WH (1988) Goodness-of-fit tests and model selection procedures for diameter distribution models. For Sci 34:373–399
Rinn F (2003) TSAP-win user reference manual. Rinntech, Heidelberg. http://www.rinntech.com. Accessed January 2017
Rondeux J (1993) Le mesure des arbres et des peuplements forestieres. Les Presses Agronomiques de Gembloux, Gembloux
Royston P (1992) Approximating the Shapiro–Wilk W test for non-normality. Stat Comput 2:117–119. https://doi.org/10.1007/BF01891203
Rubin BD, Manion PD, Faber-Langendoen D (2006) Diameter distributions and structural sustainability in forests. For Ecol Manage 222:427–438. https://doi.org/10.1016/j.foreco.2005.10.049
Schütz JP (1999) Close-to-nature silviculture: is this concept compatible with species diversity? Forestry 72:359–366. https://doi.org/10.1093/forestry/72.4.359
Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52(3/4):591–611. https://doi.org/10.2307/2333709
Smejkal GM, Bîndiu C, Vişoiu-Smejkal D (1995) Banater Urwälder. Mirton Verlag, Temeswar (in German)
Smith DM, Larson BC, Kelty MJ, Ashton PMS (1997) The practice of silviculture: applied forest ecology, 9th edn. Wiley, New York
Vrška T, Hort L, Adam D, Odehnalová P, Horal D (2002) Developmental dynamics of virgin forest reserves in the Czech Republic. Academia, Prague
Wang X, Hao Z, Zhang J, Lian J, Li B, Ye J, Yao X (2009) Tree size distributions in an old-growth temperate forest. Oikos 118:25–36. https://doi.org/10.1111/j.0030-1299.2008.16598.x
Westphal C, Tremer N, von Oheimb G, Hansen J, von Gadow K, Härdtle W (2006) Is the reverse J-shaped diameter distribution universally applicable in European virgin beech forests? For Ecol Manage 223:75–83. https://doi.org/10.1016/j.foreco.2005.10.057
WSL (2017) Swiss Federal Institute for Forest, Snow and Landscape Research. http://www.wsl.ch/dendro/products/dendro_glossary/Details_EN?id=35&language=English. Accessed January 2017
Zaiontz C (2015) Real statistics using Excel. http://www.real-statistics.com. Accessed April 2017
Zenner EK (2005) Development of tree size distributions in douglas-fir forests under differing disturbance regimes. Ecol Appl 15:701–714. https://doi.org/10.1890/04-0150
Zhang L, Liu C (2006) Fitting irregular diameter distributions of forest stands by Weibull, modified Weibull, and mixture Weibull models. J For Res 11:369–372. https://doi.org/10.1007/s10310-006-0218-7
Zhang L, Gove JH, Liu C, Leak WB (2001) A finite mixture of two Weibull distributions for modeling the diameter distributions of rotated-sigmoid, uneven-aged stands. Can J For Res 31:1654–1659. https://doi.org/10.1139/cjfr-31-9-1654
Acknowledgements
The research leading to these results has received funding from the EEA Financial Mechanism 2009–2014 under the Project Contract No. 18SEE. Also, we are grateful to WWF Romania for the financial support for the field activities.
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Communicated by Arne Nothdurft.
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Horodnic, S.A., Roibu, C.C. A Gaussian multi-component model for the tree diameter distribution in old-growth forests. Eur J Forest Res 137, 185–196 (2018). https://doi.org/10.1007/s10342-017-1097-5
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DOI: https://doi.org/10.1007/s10342-017-1097-5