SUPPLEMENTARY MATERIAL
INDIVIDUAL VARIABILITY IN TREE ALLOMETRY DETERMINES LIGHT RESOURCE ALLOCATION IN FOREST ECOSYSTEMS – A HIERARCHICAL BAYESIAN APPROACH.
in Oecologia
Ghislain Vieilledent?,1,2,3
Benoît Courbaud1,6
Jean-François Dhôte4,5
and
Georges Kunstler1
James S. Clark6
[?] Corresponding author: \E-mail:
[email protected] \Phone: 00.33.4.67.59.37.48 \Fax: 00.33.4.67.59.37.33 [1] Cemagref –Mountain Ecosystems Research Unit, 2 rue de la Papeterie, BP 76, F–38402 Saint-Martin-d’Hères cedex, France [2] AgroParisTech–UMR1092, Laboratoire d’Etude des Ressources Forêt Bois, 14 rue Girardet, F–54000 Nancy, France [3] Cirad–UR105 Forest Ecosystem Goods and Services, TA C-105/D, Campus International de Baillarguet, F–34398 Montpellier Cedex 5, France [4] INRA–UMR1092, Laboratoire d’Etude des Ressources Forêt Bois, 14 rue Girardet, F–54000 Nancy, France [5] ONF–Département Recherche, Boulevard de Constance, F–77300 Fontainebleau, France [6] Duke University–Nicholas School of the Environment and Earth Sciences, box 90328 Durham NC, 27708, USA
1
Site Site name number 1
Luan
Country
Alps region
Elevation (m)
Latitude
Longitude
Switzerland
Canton of Vaud
1442
46° 21' 45” N
6° 58' 16” E
Species (% of stems at Number of first census) Surface First Second trees (first (ha) census census census) Abies Picea Others alba abies 1.00
339
24
22
54
2004
–
2
Miroir1
France
Tarentaise
1357
45° 36' 18" N
6° 53' 07" E
0.25
375
95
5
0
1994
2006
3
Miroir3
France
Tarentaise
1377
45° 36' 19" N
6° 53' 09" E
0.25
319
91
9
0
1994
2006
4
Premol
France
Belledone
1434
45° 06' 41" N
5° 51' 26" E
0.80
503
34
45
20
2005
–
5
Queige
France
Beaufortain
1358
45° 41' 57'' N
6° 27' 30'' E
0.50
285
51
49
0
2002
–
6
Sixt
France
Haut Giffre
1520
46° 01' 16" N
6° 48' 51" E
0.25
608
1
95
5
1994
2006
7
SteFoy
France
Tarentaise
1642
45° 33' 08'' N
6° 54' 23'' E
0.25
219
0
99
1
1994
2006
8
StRhemy
Italy
Aosta Valley
1874
45° 50’ 16” N
7° 11’ 18” E
0.30
96
0
91
9
2003
–
9
Teppas
Italy
Aosta Valley
1720
45° 02' 36'' N
6° 40' 30'' E
2.00
939
73
21
6
1998
–
Appendix S1: Plot characteristics. Trees were measured on nine different plots ranging in size from 0.25 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.
2
25
50
20
40
Mean Linear Power Quotient Michaelis−Menten Gompertz Mean by DBH class
Mean Linear Power Mean by H class
● ● ●
●
CH (m)
●
H (m)
●
●
15
30
● ●
●
●
10
20
●
●
● ●
5
10
●
● ●
0
0
●
0
10
20
30
40
50
60
70
80
90
100
110
0
10
20
30
40
H (m)
DBH (cm)
(b)
10
(a)
8
Mean Linear Power Mean by DBH class
CR (m)
6
●
●
● ●
4
● ●
● ● ●
2
●
0
●
0
10
20
30
40
50
60
70
80
90
100
110
DBH (cm)
(c)
Appendix S2: 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
(a) H-DBH Model description
Model number
Effects (Y: yes, n: no) Mathematical function
Posterior mean of deviance
pD
DIC
Covariate DBH
Parameters
Mean model
H1
n
4012.59
1.99
4014.58
Linear model
H2
Y
276.47
1.94
278.41
Power model
H3
Y
271.90
3.25
275.15
Monod model
H4
Y
176.10
2.97
179.07
Michaelis-Menten model
H5
Y
168.09
3.18
171.26
Gompertz model
H6
Y
-198.82
3.05
-195.78
Effects (Y: yes, n: no)
pD
DIC
Covariate H
Posterior mean of deviance
(b) CH-H Model description
Model number
Mathematical function
Parameters
Mean model
CH1
n
2868.67
1.98
2870.65
Power model
CH2
Y
1521.62
2.90
1524.52
Linear model
CH3
Y
1529.80
2.01
1531.81
Effects (Y: yes, n: no)
Posterior mean of deviance
pD
DIC
Covariate DBH
(c) CR-DBH Model description
Model number
Mathematical function
Parameters
Mean model
CR1
n
1685.17
1.99
1687.15
Linear model
CR2
Y
1862.47
2.01
1864.48
Power model
CR3
Y
301.95
2.96
304.91
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
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). zit : Measurement t of variable for tree i. z can be DBH, height, crown height or crown radius. Z: Vector of observed values zit . zi,0 : Latent variable (“true value”) z for tree i. Z0 : Vector of “true values” zi,0 . σz2 : 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: Q Q p(Z|Z0 , σz2 ) = Tt=1 Ii=1 LN (zit |log(zi,o ), σz2 ) Priors p(log(zi,0 )) = N (log(zi,0 )|ui , vi ), with ui = 0 and vi = 1.0 × 106 p(σz2 ) = IG(σz2 |s1 , s2 ), with s1 = 1.0 × 10−3 and s2 = 1.0 × 10−3 Joint posterior Q Q p(Z0 , σz2 |Z, priors) ∝ Tt=1 Ii=1 p(zit |zi,o , σz2 )p(zi,0 )p(σz2 ) Q Q p(Z0 , σz2 |Z, priors) ∝ Tt=1 Ii=1 LN (zit |log(zi,o ), σz2 )N (log(zi,0 )|ui , vi )IG(σz2 |s1 , s2 ) Conditional posterior for parameter σz2 Q Q p(σz2 |Z, Z0 , priors) ∝ Tt=1 Ii=1 LN (zit |log(zi,o ), σz2 )IG(σz2 |s1 , s2 )
5
Measurement error results MCMC provided 1000 estimates for σz2 . The mean and standard variation for σz2 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: zit = zi,0 exp(it ). For a 95% confidence interval: −2¯ σz ≤ it ≤ +2¯ σz exp(−2¯ σz ) ≤ exp(it ) ≤ exp(+2¯ σz ) 100(exp(−2¯ σz ) − 1)(%) ≤ measurement error(%) ≤ 100(exp(+2¯ σz ) − 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 (σ²z)
Sd (σ²z)
Measurement error (%) confidence interval at 95% lower bound
upper bound
DBH
2.21E-05
3.42E-06
-0.93
0.94
H
3.97E-03
6.13E-04
-11.84
13.43
CH
7.78E-02
1.23E-02
-42.76
74.69
CR
2.42E-02
3.75E-03
-26.74
36.50
Appendix S4: Means and standard deviations for variance associated to measurement errors. Means and variances were calculated from the thousand simulations of σz2 obtained with MCMC. Credible interval at 95% for the measurement errors were computed. As errors were multiplicative they were expressed in percentage.
6
Abies alba µj densities
r density
15
3.2
3.3
3.4
3.5
3.6
50 0.082
0.084
0.086
0.088
0.090
3.1
3.2
3.3
3.4
3.5
3.6
3.7
0.072
0.074
0.076
0.078
r τ2 density
0.082
50
100
150
0.080
0.03
0.04
0.05
0.06
0.020
0.025
0.030
0.035
0.040
0.000
0.005
0.010
0.015
0.020
0.030
0.035
0.040
τ2 σ2y density
2e−05
3e−05
4e−05
0.002
0.004
0.006
σy2
0.008
0.006
0.008
500
0.050
0 100
0 0.000
σx2
0.045
300
Density
80000
Density
40000
500 0 100
300
Density
80000 40000
1e−05
700
Vδ σ2x density
120000
τ2 σ2y density
700
Vδ σ2x density
0 0e+00
0
0
20 0
10
50
100
Density
Density
150
100 80 60
Density
40
50 40 30
200
µj V δ density
200
r τ2 density
20
Density
0
0 0.080
µj V δ density
0
0.02
120000
0.01
Density
200
Density
5
50 0
0
3.1
60
3.0
100
10
Density
150
Density
100
10 5
Density
200
15
250
r density
300
µj densities
Picea abies
0e+00
1e−05
2e−05
3e−05
4e−05
0.000
0.002
0.004
σy2
σx2
(a) µj densities
µj densities 8
6
6 4
Density
4 3
Density
2
2 0.0
0.5
1.0
0.10
0.15
0.20
0.25
0.30
0.35
−0.5
0.0
0.5
1.0
1.5
0.35
0.40
Vδ
τ2 density
σx2
τ2 density
σx2
0.03
0.04
0.001
0.003
0.005
0.007
600 0.07
0.001
40 20
30
σ2y
0.10
0.12
0.14
0.02
(b)
7
0.04
0.06
0.08
σ2y
0.003
0.005
σ2x
10 0.08
0.55
400
Density 0.05
σ2y
0
10
20
Density
30
40
0.03
τ2
0
0.06
0.50
0 0.01
σ2x
σ2y
0.45
200
30 10 0
0 0.02
20
Density
400
Density
200
60 80
600
40
µj
0.01
0.04
0.30
Vδ
τ2
0.02
0.25
µj
20 40 0.00
0
0
1
2 0
0
−0.5
0
Density
−1.0
Density
V δ density
5
8 10 6
Density
4
4 2
Density
6
8
V δ density
0.10
0.12
0.14
0.007
35
Density
25
15 10
Density
5
15
Density
25
35
20 15 10
b densities
−0.6
−0.5
−0.4
−0.3
−0.2
0.40
0.42
0.44
0.46
0.48
0.50
−0.4
0.48
0.50
0.52
V δ densities
τ2 densities
0.010
0.015
0.020
0.025
3.5e−05
σ2y
0.010
0.015
densities
0.030
0.040
5.0e−06
σ2y
Density
densities
0
Density 0.020
0.005
τ2
0 0.010
0.025
60
Density 0.040
100000
densities
100
Density 2.5e−05
σ2x
0.030
Vδ
0
40000 0
1.5e−05
0.020
σ2x
50
100000
150
densities
0.020
0 20 0.010
τ2 σ2y
150
0.005
100
0.025
0.56
50
0.020
60 80
Density
0
0 0.015
20 40
80
Density
40
100
0.010
0.54
100
τ2 densities 120
V δ densities
50
Density
−0.6
b
Vδ
Density
−0.8
µj
σ2x
5.0e−06
−1.0
b
0 0.005
0 5 −1.2
µj
40000
−0.7
150
−0.8
0
0
0 5
5
Density
µj densities
b densities
15
µj densities
1.5e−05
2.5e−05
σ2x
3.5e−05
0.010
0.020
σ2y
0.030
0.040
(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 (σy2 ) and on covariate (σx2 ). All other priors were taken non-informative.
8
(a) H-DBH Parameter
Abies alba
Picea abies
Mean
Sd
Mean
Sd
3.34E+00
4.74E-02
3.42E+00
3.94E-02
µ [Miroir1]
3.24E+00
4.03E-02
−
−
µ [Miroir3]
3.35E+00
4.20E-02
−
−
µ [Premol]
3.19E+00
4.02E-02
3.39E+00
4.17E-02
µ [Queige]
Signification
3.28E+00
3.59E-02
3.53E+00
2.84E-02
µ [Sixt]
site fixed effects
−
−
3.52E+00
2.93E-02
µ [SteFoy]
−
−
3.44E+00
2.89E-02
µ [StRhemy]
−
−
3.30E+00
3.26E-02
3.34E+00
3.15E-02
3.23E+00
3.96E-02
8.46E-02
1.55E-03
7.67E-02
1.45E-03
3.12E-02
7.42E-03
1.08E-02
2.33E-03
3.97E-03
6.14E-04
3.97E-03
6.14E-04
2.21E-05
3.40E-06
2.21E-05
3.40E-06
2.94E-02
3.61E-03
4.03E-02
2.45E-03
µ [Teppas] r V
σ²y σ²x τ²
fixed f parameter variance of individual random effects variance of log-measurement errors on response H variance of log-measurement errors on covariate DBH variance of log-errors
(b) CH-H Parameter
Abies alba Mean
Sd
Mean
Sd
3.67E-01
1.26E-01
7.72E-01
1.39E-01
µ [Miroir1]
-6.04E-01
1.12E-01
−
−
µ [Miroir3]
-2.76E-01
1.11E-01
−
−
µ [Premol]
-5.19E-01
1.38E-01
1.37E-02
1.46E-01
µ [Queige]
Signification
Picea abies
site fixed effects
7.14E-01
6.16E-02
4.88E-01
8.45E-02
µ [Sixt]
−
−
-6.95E-02
1.33E-01
µ [SteFoy]
−
−
4.01E-01
6.99E-02
µ [Teppas]
2.09E-01
5.00E-02
3.03E-01
1.12E-01
2.13E-01
3.44E-02
3.80E-01
5.10E-02
7.79E-02
1.19E-02
7.79E-02
1.19E-02
3.97E-03
6.14E-04
3.97E-03
6.14E-04
1.78E-02
5.36E-03
4.48E-02
1.16E-02
V
σ²y σ²x τ²
variance of individual random effects variance of log-measurement errors on response CH variance of log-measurement errors on covariate H variance of log-errors
(c) CR-DBH Parameter
Abies alba Mean
Sd
Mean
Sd
-3.27E-01
6.45E-02
-6.41E-01
6.46E-02
µ [Miroir1]
-6.38E-01
4.57E-02
−
−
µ [Miroir3]
-5.88E-01
5.26E-02
−
−
-6.07E-01
5.51E-02
-9.18E-01
5.55E-02
-3.54E-01
4.30E-02
-7.74E-01
4.48E-02
−
−
-1.03E+00
4.56E-02 3.99E-02
µ [Premol] µ [Queige]
Signification
Picea abies
site fixed effects
µ [Sixt] µ [SteFoy]
−
−
-9.78E-01
µ [Teppas]
-4.77E-01
3.26E-02
-7.74E-01
4.19E-02
4.54E-01
1.33E-02
5.25E-01
1.20E-02
1.48E-02
2.78E-03
2.49E-02
4.23E-03
2.42E-02
3.73E-03
2.42E-02
3.73E-03
2.21E-05
3.40E-06
2.21E-05
3.40E-06
1.34E-02
2.70E-03
1.48E-02
3.35E-03
b V
σ²y σ²x τ²
fixed f parameter variance of individual random effects variance of log-measurement errors on response CR variance of log-measurement errors on covariate DBH variance of log-errors
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
