1. Estimating wood volume of the stem and commercial branches of hard maple using a taper model with latent variables Western Mensurationists’ Meeting June 21-23, 2009 Vancouver, Washington W.T. ZAKRZEWSKI, OMNR, Ontario Forest Research Institute

2. Modelling wood biomass/carbon in Ontario Carbon likely will be an important component on the global commodity market Carbon likely will be an important component on the global commodity market Future trade disputes will likely be related to carbon and its markets Future trade disputes will likely be related to carbon and its markets Future sale prices of wood in Ontario may be based on harvested forest carbon Future sale prices of wood in Ontario may be based on harvested forest carbon At the Ontario Forest Research Institute, Sault Ste. Marie, individual tree/stand-level biomass/carbon projects are underway (Great Lakes Forest Carbon Program – in partnership with the Michigan State University) At the Ontario Forest Research Institute, Sault Ste. Marie, individual tree/stand-level biomass/carbon projects are underway (Great Lakes Forest Carbon Program – in partnership with the Michigan State University) Good taper models and good specific gravity models are needed to support the Program Good taper models and good specific gravity models are needed to support the Program

3. Good stem profile model crucial for a tree-level quantification of carbon

4. Wood volume of individual tree Taper models are commonly used to estimate individual tree wood volume, particularly to quantify timber roundwood products Taper models are commonly used to estimate individual tree wood volume, particularly to quantify timber roundwood products Following models were found to perform well: Kozak (1969, 1988, 1994, 2002), Max and Burkhart (1976), Flewelling and Raynes (1993), Valentine and Gregoire (2001), Zakrzewski (1999) Following models were found to perform well: Kozak (1969, 1988, 1994, 2002), Max and Burkhart (1976), Flewelling and Raynes (1993), Valentine and Gregoire (2001), Zakrzewski (1999)_______________________________________________ * LeMay, V., Rathbun, L. and A. Kozak. 2008. Models of tree taper for selected species and BEC zones of British Columbia. Report for Min. For. and Range, Vancouver, BC.

5. “... A surprising result was the accuracy of relatively simple model, the Zakrzewski (1999) model, in terms of tree volume estimates…” Executive Summary, Report for Ministry of Forest and Range BC 2008 “ The (Zakrzewski) equation… provided results equivalent to our Max and Burkhart model. However, computation… …was deemed too complex for most practitioners in this region.” Jiang L. and J.R. Brooks 2008. Taper, Volume and Weight Equations for Red Pine in West Virginia. North. J. Appl. For. 25(3): 151-153.

6. Range of stem profile curves as a function of β in a flexible one- coefficient variant of a variable-form taper model used in Michigan forest inventories (Zakrzewski 2004) * ca [m 2 ] h [m] In the above model ca is cross-sectional area, K is a DBH-based scaling constant, H is tree total height, and z = 1 – h/H is a relative tree height. Model input variables: DBH and H. *Zakrzewski, W.T. 2004. Non-regression approach to defining a stem profile model. Proceedings, NEMO and Southern Mensurationists Conference, Roanoke, VA 2003.

7. Extension of the concept (HDR): Maximum Slimness Line

8. Taper of palm “tree”

9. Can a single taper model provide yield information for multi-stem, branchy trees?

10. Two approaches to model trees with commercial branches: 1) “Multi-taper approach” 2) “A single-stem taper approach” used to define tree cumulative volume including volume of branches Flewelling (2004) * pioneer study generated an algorithm for California hardwoods that predicts taper for individual pieces of stems. Flewelling (2004) * pioneer study generated an algorithm for California hardwoods that predicts taper for individual pieces of stems. ____________________________________________ * Flewelling, J. 2004. Compatible taper algorithms for California hardwoods. Report prepared under USDA Forest Service contract. * Flewelling, J. 2004. Compatible taper algorithms for California hardwoods. Report prepared under USDA Forest Service contract.

11. Tree cumulative volume curve* _________________________________________________________________________________________________________________ * Zakrzewski, W.T. and D.W. MacFarlane. 2006. Regional stem profile model for cross ‑ border comparisons of harvested red pine (Pinus resinosa Ait.) in Ontario and Michigan. Forest Science 52(4): 468-475. Is total tree height H of any help in the above model? Can tree height H be treated as a latent variable?

12. Latent asymptotes: multiasymptotic, polymorphic, base-age-invariant height growth model (Western Mensurationists Meeting, Rosario, WA 1991)

13. Reformulated taper model that requires only DBH as input variable as tree height is a latent variable (Zakrzewski 2009b) Tree total height was treated as a latent variable H p defined by the sub-model using two cross-sectional areas: one at 1.3 m (G_DBH ) and another at the tree base (G_o). G_o was expressed in terms of G_DBH : G_o = α + σ G_DBH therefore Hp = f(G_DBH, α, σ, β, γ) and and ca h in the above is cross-sectional area at height location 0 <= h <= Hp

14. The suggested solution is a framework for studying tree wood allocation strategy

15. Regression model was defined using OLS NLIN fit Modelβγασ Without commercial branches (N = 366) -1.754 (-1.980 - -1.528) 0.898 (0.713 – 1.082) 0.0012 (0.0002 – 0.0022) 1.137 (1.082 – 1.192) With commercial branches (N = 367) -1.767 (-1.951 – -1.582) 0.906 (0.756 – 1.055) 0.0015 (0.0005 – 0.0025) 1.126 (1.083 – 1.168) A – without commercial branches, B – with commercial branches, C – curve only for main stem of a tree with commercial branches

16. Predicting tree wood volume (N = 109 per category) Input variables in Zakrzewski taper model Maple tree category Mean error/ Std. err.STDMinimum/Maximum (%) DBH and total height measured With commercial branches -2.37 / 0.9810.25-32.64 – 26.57 Without commercial branches -0.86 / 1.0010.47-20.97 – 33.29 Only DBH measured With commercial branches -0.003 / 1.1111.68-23.00 – 34.47 Without commercial branches 1.15 / 1.1612.16-21.05 – 48.41

17. Evaluation of latent heights using H p sub-model for maples without commercial branches A – height/dbh curve implied by taper model, B – measured heights, C – height/DBH regression curve

18. Observed G_o vs. evaluated G_o

19. Regional frequencies as a base for amendments in timber cruising procedures Morawski and Basham (1958) data for Ontario (Algonquin Region) Based on Morawski and Basham (1958) data for Ontario (Algonquin Region)

20. Conclusions In hard maple, trees of given total height and DBH, with commercial branches, contain more merchantable wood than those with the same respective sizes but of a single stem. In hard maple, trees of given total height and DBH, with commercial branches, contain more merchantable wood than those with the same respective sizes but of a single stem. A new approach, which involves tree height as a latent variable, was used to estimate wood volume for individual Ontario sugar maple stems and branches. The new approach can be potentially used as a modelling framework for research on influence of stand density on tree height and tree competitive status in a stand A new approach, which involves tree height as a latent variable, was used to estimate wood volume for individual Ontario sugar maple stems and branches. The new approach can be potentially used as a modelling framework for research on influence of stand density on tree height and tree competitive status in a stand Total volume estimates obtained using models with and without height measurements were reasonable Total volume estimates obtained using models with and without height measurements were reasonable Results indicate that the traditional formulae based on H and DBH may not be appropriate for maple if wood in branches is not accounted for Results indicate that the traditional formulae based on H and DBH may not be appropriate for maple if wood in branches is not accounted for Results indicate that carbon in maple stands could be reliably inventoried using DBH-based biomass equations (with latent heights) if DBH-based taper model is integrated with the respective specific gravity model Results indicate that carbon in maple stands could be reliably inventoried using DBH-based biomass equations (with latent heights) if DBH-based taper model is integrated with the respective specific gravity model