1. Incorporating stand density effects in modeling tree taper Mahadev Sharma Ontario Forest Research Institute Sault Ste Marie, Canada

2. BackgroundBackground –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

3. ObjectiveObjective –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

4. DataData –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

5. DataData Summary statistics for stand characteristics used in this study Stand characteristics FrequencyMeanStd. devMinimumMaximum Jack pine BA/ha (m 2 )7527.465.7815.2842.25 Trees/ha7517736478843302 QMD (cm)7514.462.0110.6219.14 Black spruce BA/ha (m 2 )7529.848.7912.0048.87 Trees/ha75291989614715579 QMD (cm)7511.672.416.3716.00

6. DataData Summary statistics for tree characteristics used in this study Tree characteristics FrequencyMeanStd. devMinimumMaximum Jack pine DBH (cm)113517.344.466.1034.30 Height (m)113515.472.547.9323.17 Crown ratio11350.430.110.100.85 Black spruce DBH (cm)118913.353.702.5024.80 Height (m)118910.852.472.9817.85 Crown ratio11890.600.160.220.98

7. 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 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

8. 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  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

9. Taper 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

10. 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  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

11. 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  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

12. 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 –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

13. 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  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

14. 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 PineBlack spruce σ2σ2 -2Ln(L)AICσ2σ2 -2Ln(L)AIC β 0, β 1, β 2, β 3 50.001847- 31023- 310130.001723- 33655- 33645 β 0, β 1, β 2, β 3, β 4 60.001709- 31721- 317090.001552- 34658- 34646 β 0i, β 1, β 2, β 3 60.001315- 32945- 329330.001081- 36695- 36683 β 0i, β 1i, β 2, β 3 80.000866- 35288- 352720.000562- 40942- 40926 β 0i, β 1i, β 2i, β 3 110.000559- 37746- 377240.000343- 43880- 43858 β 0i, β 1i, β 2i, β 3i 150.000390- 39503- 394730.000255- 45184- 45154 β 0i, β 1i, β 2i, β 3i, β 4 160.000390- 39614- 395820.000255- 45334- 45302

15. Parameter Estimates Parameter estimates for Eq. (3) fitted using NLMIXED procedures in SAS ParametersJack PineBlack Spruce EstimatesSEEstimatesSE β0β0 0.922300.001080.908800.00127 β1β1 -0.059970.00251-0.066700.00266 β2β2 0.515600.007460.541000.00741 β3β3 -0.226500.01026-0.363600.00996 β4β4 0.083830.007560.075490.00578 σ2σ2 0.0003900.0000060.0002550.000004

16. EvaluationEvaluation Jack Pine Black spruce Diameter prediction bias (observed-predicted) using Eq. (9)

17. EvaluationEvaluation 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)

18. EvaluationEvaluation 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 m 2 /ha)

19. PredictionPrediction 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

20. ConclusionsConclusions –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

21. Thanks for your attention Questions ? Thanks for your attention Questions ?