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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
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Background

• 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

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Objective

• 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

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Data

• 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

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Data
Summary statistics for stand characteristics used
in this study
Stand characteristics Frequency Mean Std. dev Minimum Maximum
Jack pine Jack pine Jack pine Jack pine Jack pine 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 Black spruce Black spruce Black spruce Black spruce 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
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Data
Summary statistics for tree characteristics used
in this study
Tree characteristics Frequency Mean Std. dev Minimum Maximum
Jack pine Jack pine Jack pine Jack pine Jack pine 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 Black spruce Black spruce Black spruce Black spruce 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
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Taper 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
• ßs with and without a subscript are parameters

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Taper 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

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Taper Equations

• Mathematical form assumed for Eq. (1) and (2)
  was
• To make tree shape less parabolic the following
  mathematical form was assumed

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Taper 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

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Taper 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

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Mixed-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

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Mixed-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

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Height-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 Jack Pine Jack Pine Black spruce Black spruce Black spruce
Parameters in the model of parms s2 -2Ln(L) AIC s2 -2Ln(L) AIC
ß0, ß1, ß2, ß3 5 0.001847 - 31023 - 31013 0.001723 - 33655 - 33645
ß0, ß1, ß2, ß3, ß4 6 0.001709 - 31721 - 31709 0.001552 - 34658 - 34646
ß0i, ß1, ß2, ß3 6 0.001315 - 32945 - 32933 0.001081 - 36695 - 36683
ß0i, ß1i, ß2, ß3 8 0.000866 - 35288 - 35272 0.000562 - 40942 - 40926
ß0i, ß1i, ß2i, ß3 11 0.000559 - 37746 - 37724 0.000343 - 43880 - 43858
ß0i, ß1i, ß2i, ß3i 15 0.000390 - 39503 - 39473 0.000255 - 45184 - 45154
ß0i, ß1i, ß2i, ß3i, ß4 16 0.000390 - 39614 - 39582 0.000255 - 45334 - 45302
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Parameter Estimates
Parameter estimates for Eq. (3) fitted using
NLMIXED procedures in SAS
Parameters Jack Pine Jack Pine Black Spruce Black Spruce
Parameters Estimates SE Estimates SE
ß0 0.92230 0.00108 0.90880 0.00127
ß1 -0.05997 0.00251 -0.06670 0.00266
ß2 0.51560 0.00746 0.54100 0.00741
ß3 -0.22650 0.01026 -0.36360 0.00996
ß4 0.08383 0.00756 0.07549 0.00578
s2 0.000390 0.000006 0.000255 0.000004
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Evaluation
Diameter prediction bias (observed-predicted)
using Eq. (9)
Jack Pine
Black spruce
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Evaluation
Taper profiles for 3 randomly selected trees one
from each of three classes dominant,
intermediate, and suppressed generated using Eq.
(9)
Jack pine
Black spruce
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Evaluation
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)
Black spruce
Jack pine
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Prediction
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
Black spruce
Jack pine
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Conclusions

• 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

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• Thanks for your attention
• Questions?
  

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