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Incorporating stand density effects in modeling tree taper. Mahadev Sharma Ontario Forest Research Institute Sault Ste Marie, Canada. Ontario Forest Research Institute. Background. Taper equations are used to estimate diameters along the bole of a tree at any given height - PowerPoint PPT Presentation
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Incorporating stand density effects in modeling tree taper
Mahadev SharmaOntario Forest Research Institute
Sault Ste Marie, Canada
BackgroundBackgroundBackgroundBackground
– Taper equations are used to estimate diameters along the bole of a tree at any given height
– Individual tree volume is calculated based on these diameters and corresponding heights
– Product recoveries from different trees with the same DBH and total height could be different depending on tree shape (conic vs cylindrical)
– The shape depends on tree species
– Even within a species, the shape is influenced by stand density
– Model accuracy could be improved by incorporating stand density/characteristics
– Taper equations are used to estimate diameters along the bole of a tree at any given height
– Individual tree volume is calculated based on these diameters and corresponding heights
– Product recoveries from different trees with the same DBH and total height could be different depending on tree shape (conic vs cylindrical)
– The shape depends on tree species
– Even within a species, the shape is influenced by stand density
– Model accuracy could be improved by incorporating stand density/characteristics
ObjectiveObjectiveObjectiveObjective
– Examine the effect of stand density on taper of plantation grown jack pine and black spruce trees
– Develop taper equations that incorporate stand density information using mixed effects modeling technique
– Examine the effect of stand density on taper of plantation grown jack pine and black spruce trees
– Develop taper equations that incorporate stand density information using mixed effects modeling technique
DataDataDataData
– 1135 of jack pine and 1189 of black spruce trees sampled from 25 sites across Northern Ontario
– Disks were cut at 0.15, 0.5, 0.9, and 1.3 m up to the breast height and at 5% and 10% intervals thereafter
– 18,002 discs for jack pine and 18,852 discs for black spruce trees
– Half of the trees were used for parameter estimation and the other half for model evaluation
– 1135 of jack pine and 1189 of black spruce trees sampled from 25 sites across Northern Ontario
– Disks were cut at 0.15, 0.5, 0.9, and 1.3 m up to the breast height and at 5% and 10% intervals thereafter
– 18,002 discs for jack pine and 18,852 discs for black spruce trees
– Half of the trees were used for parameter estimation and the other half for model evaluation
DataDataDataData
Summary statistics for stand characteristics used in this study
Stand
characteristicsFrequency Mean Std. dev Minimum Maximum
Jack pine
BA/ha (m2) 75 27.46 5.78 15.28 42.25
Trees/ha 75 1773 647 884 3302
QMD (cm) 75 14.46 2.01 10.62 19.14
Black spruce
BA/ha (m2) 75 29.84 8.79 12.00 48.87
Trees/ha 75 2919 896 1471 5579
QMD (cm) 75 11.67 2.41 6.37 16.00
DataDataDataData
Summary statistics for tree characteristics used in this study
Tree
characteristicsFrequency Mean Std. dev Minimum Maximum
Jack pine
DBH (cm) 1135 17.34 4.46 6.10 34.30
Height (m) 1135 15.47 2.54 7.93 23.17
Crown ratio 1135 0.43 0.11 0.10 0.85
Black spruce
DBH (cm) 1189 13.35 3.70 2.50 24.80
Height (m) 1189 10.85 2.47 2.98 17.85
Crown ratio 1189 0.60 0.16 0.22 0.98
Taper EquationsTaper EquationsTaper EquationsTaper Equations
Sharma and Oderwald (2001)
Sharma and Zhang (2004)
where,
d = diameter inside bark at any given height h,
D = Diameter at breast height (DBH) outside bark,
H = total height, x = h/H, and
βs with and without a subscript are parameters
Sharma and Oderwald (2001)
Sharma and Zhang (2004)
where,
d = diameter inside bark at any given height h,
D = Diameter at breast height (DBH) outside bark,
H = total height, x = h/H, and
βs with and without a subscript are parameters
12
22
DD hH
hH
h
hDd
2)(2
0
22
321
D
xx
D hH
hH
h
h
D
d
Taper EquationsTaper EquationsTaper EquationsTaper Equations
Newton and Sharma (2008) evaluated Eq. (2) for the sensitivity of different disk selection protocols and found it invariant for estimating
• Inside bark diameters
• Total volume
However, Eq. (2) over-predicted diameters above 70% of total heights
The taper of these plantation grown trees were compared with those from natural stands
Trees in plantation stands tapered more than those in natural stands
Tree form was less parabolic in plantations than in natural stands
Newton and Sharma (2008) evaluated Eq. (2) for the sensitivity of different disk selection protocols and found it invariant for estimating
• Inside bark diameters
• Total volume
However, Eq. (2) over-predicted diameters above 70% of total heights
The taper of these plantation grown trees were compared with those from natural stands
Trees in plantation stands tapered more than those in natural stands
Tree form was less parabolic in plantations than in natural stands
Taper EquationsTaper EquationsTaper EquationsTaper Equations
Mathematical form assumed for Eq. (1) and (2) was
To make tree shape less parabolic the following mathematical form was assumed
Mathematical form assumed for Eq. (1) and (2) was
To make tree shape less parabolic the following mathematical form was assumed
312 hH
hDd
41
H
h
H
h
D
d
Taper EquationsTaper EquationsTaper EquationsTaper Equations
Eq. (4) results in a variable exponent taper equation as
Tree profiles generated based on the same DBH (17.0 cm) and total height (15.0 m) for jack pine
Eq. (4) results in a variable exponent taper equation as
Tree profiles generated based on the same DBH (17.0 cm) and total height (15.0 m) for jack pine
)5(
2321
0
xx
DD h
h
hH
hH
D
d
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Relative height
Dia
mete
r in
sid
e b
ark
(cm
)
Eq. 2 Eq. 5
Taper EquationsTaper EquationsTaper EquationsTaper Equations
The exponent is the only term that determines the change in taper from one point to another along the bole
Density effect on taper can be determined by incorporating the stand density information into the exponent as:
A preliminary analysis indicated that the following model with the stand basal area described the taper of plantation jack pine and black spruce
The exponent is the only term that determines the change in taper from one point to another along the bole
Density effect on taper can be determined by incorporating the stand density information into the exponent as:
A preliminary analysis indicated that the following model with the stand basal area described the taper of plantation jack pine and black spruce
)6()(
0
42
321 sdfxx
DD h
h
hH
hH
D
d
)7(4
2321
0
DBA
xx
DD h
h
hH
hH
D
d
Mixed-Effects ModelsMixed-Effects ModelsMixed-Effects ModelsMixed-Effects Models
– Data used for developing taper equations are not independent
– Discs are nested within trees and trees are nested within stands
– Variances of the parameters estimated using OLS regression methods are biased
– Mixed-effects models are used where a parameter could be a combination of fixed and random effects
– Random effects are associated with trees only
– Data used for developing taper equations are not independent
– Discs are nested within trees and trees are nested within stands
– Variances of the parameters estimated using OLS regression methods are biased
– Mixed-effects models are used where a parameter could be a combination of fixed and random effects
– Random effects are associated with trees only
Mixed-Effects ModelsMixed-Effects ModelsMixed-Effects ModelsMixed-Effects Models
Nonlinear mixed-effects variable exponent taper equation can then be written as
Eq. (8) with 5 random effects (RE) parameters could not be fitted in SAS
The best model with 4 RE parameters was
Nonlinear mixed-effects variable exponent taper equation can then be written as
Eq. (8) with 5 random effects (RE) parameters could not be fitted in SAS
The best model with 4 RE parameters was
84
2321
0 ij
DBA
xx
D
ij
Di
ijii
ij
eh
h
hH
hH
D
d iiijiijii
9)(4
2332211 )()()(
00 ij
DBA
xuxuu
D
ij
Di
ijii
ij
eh
h
hH
hHu
D
d iijiijii
Height-Diameter EquationsHeight-Diameter EquationsHeight-Diameter EquationsHeight-Diameter Equations
Fit statistics for Eq. (9) for different combinations of random-effects
parameters for jack pine and black spruce plantations
Parameters in the model
# of parms
Jack Pine Black spruce
σ2 -2Ln(L) AIC σ2 -2Ln(L) AIC
β0, β1, β2, β3 5 0.001847 - 31023 - 31013 0.001723 - 33655 - 33645
β0, β1, β2, β3, β4 6 0.001709 - 31721 - 31709 0.001552 - 34658 - 34646
β0i, β1, β2, β3 6 0.001315 - 32945 - 32933 0.001081 - 36695 - 36683
β0i, β1i, β2, β3 8 0.000866 - 35288 - 35272 0.000562 - 40942 - 40926
β0i, β1i, β2i, β3 11 0.000559 - 37746 - 37724 0.000343 - 43880 - 43858
β0i, β1i, β2i, β3i 15 0.000390 - 39503 - 39473 0.000255 - 45184 - 45154
β0i, β1i, β2i, β3i, β4 16 0.000390 - 39614 - 39582 0.000255 - 45334 - 45302
Parameter EstimatesParameter EstimatesParameter EstimatesParameter Estimates
Parameter estimates for Eq. (3) fitted using NLMIXED procedures in SAS
Parameters Jack Pine Black Spruce
Estimates SE Estimates SE
β0 0.92230 0.00108 0.90880 0.00127
β1 -0.05997 0.00251 -0.06670 0.00266
β2 0.51560 0.00746 0.54100 0.00741
β3 -0.22650 0.01026 -0.36360 0.00996
β4 0.08383 0.00756 0.07549 0.00578
σ2 0.000390 0.000006 0.000255 0.000004
EvaluationEvaluationEvaluationEvaluation
Jack Pine Black spruce
Diameter prediction bias (observed-predicted) using Eq. (9)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30 35
Predicted diameter (cm)
Bia
s (c
m)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 5 10 15 20 25 30
Predicted diameter (cm)
Bia
s (c
m)
EvaluationEvaluationEvaluationEvaluation
Jack pine Black spruce
Taper profiles for 3 randomly selected trees one from each of three classes: dominant, intermediate, and suppressed generated using Eq. (9)
0 2 4 6 8 10 12 14 16 18 20 22 240
2
4
6
8
10
12
14
16
18
20
Heig
ht
( m
)
Diameter ( cm )
Observed Mixed Fixed
0 2 4 6 8 10 12 14 16 18 20 220
2
4
6
8
10
12
14
16
Heig
ht
( m
)
Diameter ( cm )
Observed Mixed Fixed
EvaluationEvaluationEvaluationEvaluation
Jack pine Black spruce
Tree profiles (mean responses) generated from Eq. (9) using DBH = 17 cm and total height = 15 m at different stand densities (BA =10, 30, and 50 m2/ha)
0
2
4
6
8
10
12
14
16
18
20
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Relative height
Dia
met
er (c
m)
BA10 BA30 BA50
0
2
4
6
8
10
12
14
16
18
20
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Relative height
Dia
met
er (c
m)
BA10 BA30 BA50
PredictionPredictionPredictionPrediction
Jack pine Black spruce
Calibrated responses obtained using one, two, and three diameters to predict RE parameters for the trees that were closest to the average DBH and total HT
0
4
8
12
16
20
0 2 4 6 8 10 12 14 16
Height (m)
Insid
e bar
k diam
eter
(cm
)
Observed No random 35% Stump&65% Stump,35&65%
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8 9 10 11
Height (m)
Diam
eter
s ins
ide b
ark (
cm)
Observed No random 35% Stump&65% Stump,35&65%
ConclusionsConclusionsConclusionsConclusions
– Tree taper depends on stand density
– Stand basal area (BA/ha) can be included in the taper equations to account for stand density effect
– Predictive accuracy can be improved by including RE parameters
– If one diameter is used to predict RE parameters, the best choice would be at ~ 35% of total height
– If two diameters are used to predict RE parameters, the best choice would be one near the stump and the other at ~ 65% of total height
– If three diameters are used to predict RE parameters, the best choice would be one near the stump and other two at ~ 35% and ~ 65% of total height
– Tree taper depends on stand density
– Stand basal area (BA/ha) can be included in the taper equations to account for stand density effect
– Predictive accuracy can be improved by including RE parameters
– If one diameter is used to predict RE parameters, the best choice would be at ~ 35% of total height
– If two diameters are used to predict RE parameters, the best choice would be one near the stump and the other at ~ 65% of total height
– If three diameters are used to predict RE parameters, the best choice would be one near the stump and other two at ~ 35% and ~ 65% of total height
Thanks for your attention Questions?
Thanks for your attention Questions?
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