Ontario Forest Research Institute

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


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


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


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


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)


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



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


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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Ontario Forest Research Institute