SUPPLEMENTARY MATERIAL

Transcription

1 SUPPLEMENTARY MATERIAL INDIVIDUAL VARIABILITY IN TREE ALLOMETRY DETERMINES LIGHT RESOURCE ALLOCATION IN FOREST ECOSYSTEMS A HIERARCHICAL BAYESIAN APPROACH. in Oecologia Ghislain Vieilledent,1,,3 Benoît Courbaud 1,6 Georges Kunstler 1 Jean-François Dhôte 4,5 and James S. Clark 6 [ ] Corresponding author: \ ghislain.vieilledent@cirad.fr \Phone: \Fax: [1] Cemagref Mountain Ecosystems Research Unit, rue de la Papeterie, BP 76, F 3840 Saint-Martin-d Hères cedex, France [] AgroParisTech UMR109, Laboratoire d Etude des Ressources Forêt Bois, 14 rue Girardet, F Nancy, France [3] Cirad UR105 Forest Ecosystem Goods and Services, TA C-105/D, Campus International de Baillarguet, F Montpellier Cedex 5, France [4] INRA UMR109, Laboratoire d Etude des Ressources Forêt Bois, 14 rue Girardet, F Nancy, France [5] ONF Département Recherche, Boulevard de Constance, F Fontainebleau, France [6] Duke University Nicholas School of the Environment and Earth Sciences, box 9038 Durham NC, 7708, USA 1

2 Species (% of stems at Number of Site Elevation Surface first census) First Second Site name Country Alps region Latitude Longitude trees (first number (m) (ha) census) Abies Picea census census Others alba abies 1 Luan Switzerland Canton of Vaud ' 45 N 6 58' 16 E Miroir1 France Tarentaise ' 18" N 6 53' 07" E Miroir3 France Tarentaise ' 19" N 6 53' 09" E Premol France Belledone ' 41" N 5 51' 6" E Queige France Beaufortain ' 57'' N 6 7' 30'' E Sixt France Haut Giffre ' 16" N 6 48' 51" E SteFoy France Tarentaise ' 08'' N 6 54' 3'' E StRhemy Italy Aosta Valley N E Teppas Italy Aosta Valley ' 36'' N 6 40' 30'' E Appendix S1: Plot characteristics. Trees were measured on nine different plots ranging in size from 0.5 ha to 1 ha. Six plots were located in the French Alps, two in the Italian Alps and one in the Swiss Alps. Stands are uneaven-aged. Abies alba Mill. (Silver Fir) and Picea abies (L.) Karst. (Norway spruce) are the dominant species. All sites are situated at the mountain-belt elevation from 800 to 1800 m.

3 H (m) Mean Linear Power Quotient Michaelis Menten Gompertz Mean by DBH class CH (m) Mean Linear Power Mean by H class DBH (cm) (a) H (m) (b) CR (m) Mean Linear Power Mean by DBH class DBH (cm) (c) Appendix S: Graphical superposition of calibrated mathematical functions with points representing the mean of the response by covariate class. Allometries are: (a) height as a function of DBH, (b) crown height as a function of height and (c) crown radius as a function of DBH. Some parametric functions may be too much constrained by an unbalanced data-set, where the number of smaller trees is much more important than the number of bigger trees. Here we show that the graphical superposition of the mathematical function selected and the mean by DBH class (or H class) was good and that selected models were not biased because of an unbalanced data-set. 3

4 (a) H-DBH Model description Model number Mathematical function Parameters Effects (Y: yes, n: no) Covariate DBH Posterior mean of deviance pd DIC Mean model H1 n Linear model H Y Power model H3 Y Monod model H4 Y Michaelis-Menten model H5 Y Gompertz model H6 Y (b) CH-H Model description Model number Mathematical function Parameters Effects (Y: yes, n: no) Covariate H Posterior mean of deviance pd DIC Mean model CH1 n Power model CH Y Linear model CH3 Y (c) CR-DBH Model description Model number Mathematical function Parameters Effects (Y: yes, n: no) Covariate DBH Posterior mean of deviance pd DIC Mean model CR1 n Linear model CR Y Power model CR3 Y Appendix S3: Model comparison for the three allometric relations. Allometries are (a) height as a function of DBH, (b) crown height as a function of height and (c) crown radius as a function of DBH. The lower the DIC, the best the model. A difference of more than 10 in the DIC rules out the model with the higher DIC. For equivalent DIC, we selected the model with the lower deviance. If the deviance difference was inferior to 10, we applied the parsimonious principle selecting the model with fewer parameters (with the lowest pd). 4

5 Appendix S4: Measurement errors Model for measurement errors Indexes and notations i: Index of the tree. t: Index of the measuring team. T : Number of measurements for each tree (T = 3). I: Number of trees in the measurement error protocol (I = 50). z it : Measurement t of variable for tree i. z can be DBH, height, crown height or crown radius. Z: Vector of observed values z it. z i,0 : Latent variable ( true value ) z for tree i. Z 0 : Vector of true values z i,0. σ z: Variance for measurement errors. N: Normal distribution. LN: Log-normal distribution. IG: Inverse-gamma distribution. Bayes formula p(parameter data, model) Likelihood Prior Likelihood The likelihood is defined as the probability of observing the data under the assumption that the model is true: p(z Z 0, σ z) = T t=1 Ii=1 LN(z it log(z i,o ), σ z) Priors p(log(z i,0 )) = N(log(z i,0 ) u i, v i ), with u i = 0 and v i = p(σ z) = IG(σ z s 1, s ), with s 1 = and s = Joint posterior p(z 0, σ z Z, priors) T t=1 Ii=1 p(z it z i,o, σ z)p(z i,0 )p(σ z) p(z 0, σ z Z, priors) T t=1 Ii=1 LN(z it log(z i,o ), σ z)n(log(z i,0 ) u i, v i )IG(σ z s 1, s ) Conditional posterior for parameter σ z p(σ z Z, Z 0, priors) T t=1 Ii=1 LN(z it log(z i,o ), σ z)ig(σ z s 1, s ) 5

6 Measurement error results MCMC provided 1000 estimates for σ z. The mean and standard variation for σ z were calculated for each dendrometric variable (Tab. S4). We were able to estimate the precision of our measurement as a percentage (Tab. S4) because we considered multiplicative errors: z it = z i,0 exp(ɛ it ). For a 95% confidence interval: σ z ɛ it + σ z exp( σ z ) exp(ɛ it ) exp(+ σ z ) 100(exp( σ z ) 1)(%) measurement error(%) 100(exp(+ σ z ) 1)(%) Results showed a very good estimation of the DBH with a low measurement error (0.93%). Height was also quite well estimated with an error close to 10%. The two other variables, crown height and crown radius, were quite difficult to measure in the field and had a range of precision of approximately 50% and 30%, respectively. Variable Model Mean (σ² z ) Sd (σ² z ) Measurement error (%) confidence interval at 95% lower bound upper bound DBH.1E E H 3.97E E CH 7.78E-0 1.3E CR.4E E Appendix S4: Means and standard deviations for variance associated to measurement errors. Means and variances were calculated from the thousand simulations of σz obtained with MCMC. Credible interval at 95% for the measurement errors were computed. As errors were multiplicative they were expressed in percentage. 6

7 Abies alba Picea abies densities r density densities r density r density τ density r density τ density τ τ density density density density 0e+00 1e 05 e 05 3e 05 4e e+00 1e 05 e 05 3e 05 4e (a) densities density densities density τ density τ τ density τ (b) 7

8 densities b densities densities b densities b b densities τ densities densities τ densities τ τ densities densities densities densities 5.0e e 05.5e e e e 05.5e e (c) Appendix S5: Posteriors and priors for parameters. Allometries are: (a) height as a function of DBH, (b) crown height as a function of height and (c) crown radius as a function of DBH. Priors are represented with dashed lines (- - -) and posteriors with plain lines ( ). We used informative priors for the measurement error variance on response (σ y) and on covariate (σ x). All other priors were taken non-informative. 8

9 (a) H-DBH Parameter Signification Mean Sd Mean Sd µ [Miroir1] µ [Miroir3] µ [Premol] µ [Queige] µ [Sixt] µ [SteFoy] µ [StRhemy] µ [Teppas] 3.34E E-0 3.4E E-0 3.4E E E E E E E E-0 3.8E E E+00.84E-0 3.5E+00.93E E+00.89E E E E E-0 3.3E E-0 r fixed f parameter 8.46E E E E-03 (b) CH-H 3.1E-0 7.4E E-0.33E E E E E-04.1E E-06.1E E-06.94E E E-0.45E-03 Parameter Signification Mean Sd Mean Sd (c) CR-DBH site fixed effects V variance of individual random effects σ²y errors on response H σ²x errors on covariate DBH τ² variance of log-errors µ [Miroir1] µ [Miroir3] µ [Premol] µ [Queige] µ [Sixt] µ [SteFoy] site fixed effects µ [Teppas] variance of individual random V effects σ²y errors on response CH σ²x errors on covariate H τ² variance of log-errors Abies alba Abies alba Picea abies Picea abies 3.67E E E E E E E E E E E E E E E E E E E E-0.09E E E E-01.13E E E E E E E E E E E E E E E E-0 Abies alba Picea abies Parameter Signification Mean Sd Mean Sd µ [Miroir1] µ [Miroir3] µ [Premol] µ [Queige] µ [Sixt] µ [SteFoy] µ [Teppas] site fixed effects -3.7E E E E E E E E E E E E E E E E E E E E E E E E-0 b fixed f parameter 4.54E E-0 5.5E E-0 V variance of individual random effects σ²y errors on response CR σ²x errors on covariate DBH τ² variance of log-errors 1.48E-0.78E-03.49E-0 4.3E-03.4E E-03.4E E-03.1E E-06.1E E E-0.70E E E-03 Appendix S6: Means and standard deviations of the estimated parameters for the best allometric models. Allometries are: (a) height as a function of DBH, (b) crown height as a function of height and (c) crown radius as a function of DBH. 9