Estimating wood volume Estimating wood volume of the stem and commercial of the stem and commercial branches of hard maple branches of hard maple using a taper model using a taper model with latent variableswith latent variables

Western Mensurationists’ MeetingWestern Mensurationists’ Meeting

June 21-23, 2009 June 21-23, 2009

Vancouver, WashingtonVancouver, Washington

W.T. ZAKRZEWSKI, OMNR, Ontario Forest Research W.T. ZAKRZEWSKI, OMNR, Ontario Forest Research InstituteInstitute

Modelling wood biomass/carbon Modelling wood biomass/carbon in Ontarioin Ontario

Carbon likely will be an important component on the global Carbon likely will be an important component on the global commodity market commodity market

Future trade disputes will likely be related to carbon and its Future trade disputes will likely be related to carbon and its markets markets

Future sale prices of wood in Ontario may be based on Future sale prices of wood in Ontario may be based on harvested forest carbonharvested forest carbon

At the Ontario Forest Research Institute, Sault Ste. Marie, At the Ontario Forest Research Institute, Sault Ste. Marie, individual tree/stand-level biomass/carbon projects are individual tree/stand-level biomass/carbon projects are underway (Great Lakes Forest Carbon Program – in partnership underway (Great Lakes Forest Carbon Program – in partnership with the Michigan State University)with the Michigan State University)

Good Good taper modelstaper models and good and good specific gravity modelsspecific gravity models are are needed to support the Programneeded to support the Program

Good stem profile model crucial Good stem profile model crucial

for a tree-level quantification of carbonfor a tree-level quantification of carbon

Wood volume of individual treeWood volume of individual tree

Taper modelsTaper models are commonly used to estimate are commonly used to estimate individual tree wood volume, particularly to individual tree wood volume, particularly to quantify timber roundwood productsquantify timber roundwood products

Following models were found to perform well: Following models were found to perform well: Kozak (1969, 1988, 1994, 2002), Max and Kozak (1969, 1988, 1994, 2002), Max and Burkhart (1976), Flewelling and Raynes (1993), Burkhart (1976), Flewelling and Raynes (1993), Valentine and Gregoire (2001), Zakrzewski (1999)Valentine and Gregoire (2001), Zakrzewski (1999)

______________________________________________________________________________________________* * LeMay, V., Rathbun, L. and A. Kozak. 2008. Models of tree taper for LeMay, V., Rathbun, L. and A. Kozak. 2008. Models of tree taper for

selected species and BEC zones of British Columbia. Report for selected species and BEC zones of British Columbia. Report for Min. For. and Range, Vancouver, BC.Min. For. and Range, Vancouver, BC.

“... 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.

Range of stem profile curves as a function of Range of stem profile curves as a function of ββ in a in a flexible one-coefficient variant of a variable-form taper flexible one-coefficient variant of a variable-form taper

modelmodelused in Michigan forest inventories (Zakrzewski 2004)used in Michigan forest inventories (Zakrzewski 2004)**

ca

[m2]

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 Zakrzewski, W.T. 2004. Non-regression approach to defining a stem profile model. Proceedings, NEMO and Southern Mensurationists profile model. Proceedings, NEMO and Southern Mensurationists Conference, Roanoke, VA 2003.Conference, Roanoke, VA 2003.

Extension of the concept (HDR): Extension of the concept (HDR): Maximum Slimness LineMaximum Slimness Line

Taper of palm “tree”Taper of palm “tree”

Can a single taper model provide yield Can a single taper model provide yield information for multi-stem, branchy information for multi-stem, branchy

trees?trees?

Two approaches to model trees with commercial Two approaches to model trees with commercial branches:branches:

1) “Multi-taper approach” 1) “Multi-taper approach” 2) “A single-stem taper approach” used to define tree 2) “A single-stem taper approach” used to define tree

cumulative volume including volume of branches cumulative volume including volume of branches

Flewelling (2004)Flewelling (2004)** pioneer study generated an pioneer study generated an algorithm for California hardwoods that predicts algorithm for California hardwoods that predicts taper for individual pieces of stems.taper for individual pieces of stems.

________________________________________________________________________________________

**Flewelling, J. 2004. Compatible taper algorithms for California Flewelling, J. 2004. Compatible taper algorithms for California hardwoods. Report prepared under USDA Forest Service contract.hardwoods. Report prepared under USDA Forest Service contract.

Tree cumulative volume curveTree cumulative volume curve**

192 H4 ( )ln H U 66 H3 U2 180 H4 U 20 H U3 192 H5 ( )ln H U 96 H5 ( )ln H U 180 H3 U 36 H4 U 6 H2 U2 36 H3 U 48 H5 ( )ln H 3 U4 6 H3 U2 48 H4 ( )ln H U 48 H4 ( )ln H 48 H5 ( )ln H U ) (

H4 2 H3 H2 H4 3 H3 3 H2 H H4 4 H3 6 H2 4 H )

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________

* * Zakrzewski, W.T. and D.W. MacFarlane. 2006. Regional stem profile Zakrzewski, W.T. and D.W. MacFarlane. 2006. Regional stem profile model for cross‑border comparisons of harvested red pine (model for cross‑border comparisons of harvested red pine (Pinus Pinus resinosaresinosa Ait.) in Ontario and Michigan. Forest Science 52(4): 468-475. Ait.) in Ontario and Michigan. Forest Science 52(4): 468-475.

s-z z + z + z K = ca

432

z

Is total tree height Is total tree height HH of any help in the above of any help in the above model? model?

Can tree height Can tree height HH be treated as a latent variable? be treated as a latent variable?

Volume cumul

112

GDBH 84 H3 U 24 H2 U2 4 H U3 66 H2 U2 ( :=

96 H5 ( )ln H 192 H5 ( )ln H 84 H4 U 24 H3 U2 4 H2 U3 96 H4 ( )ln H 192 H4 ( )ln H 3 H U4 20 H2 U3 96 H4 ( )ln H U

Latent asymptotes:Latent asymptotes:multiasymptotic, polymorphic, base-age-invariant multiasymptotic, polymorphic, base-age-invariant

height growth modelheight growth model (Western Mensurationists Meeting, Rosario, WA 1991) (Western Mensurationists Meeting, Rosario, WA 1991)

Reformulated taper model that requires only DBH Reformulated taper model that requires only DBH as input variable as tree height is a latent variable as input variable as tree height is a latent variable

(Zakrzewski 2009b)(Zakrzewski 2009b)Tree total height was treated as a latent variable Tree total height was treated as a latent variable HHpp defined by the sub-defined by the sub-model using two cross-sectional areas: one at 1.3 m (model using two cross-sectional areas: one at 1.3 m (G_DBHG_DBH ) and another ) and another at the tree base (at the tree base (G_oG_o). ). G_oG_o was expressed in terms of was expressed in terms of G_DBHG_DBH ::

G_o = G_o = αα + + σσ G_DBH G_DBH

thereforetherefore

HpHp = f( = f(G_DBHG_DBH, , αα, , σσ, , ββ, , γγ))

andand

cacahh in the above is cross-sectional area at height location 0 <= in the above is cross-sectional area at height location 0 <= hh <= <= HpHp

The suggested solution is a framework The suggested solution is a framework for studying tree wood allocation for studying tree wood allocation

strategystrategy

Regression model was defined using Regression model was defined using OLS NLIN fitOLS NLIN fit

Model β γ α σ

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)

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Relative height

Diam

eter [cm

]

A

B

C

A – without commercial branches, B – with commercial branches, C – curve only for main stem of a tree with commercial branches

Predicting tree wood volume Predicting tree wood volume ((NN = 109 per category) = 109 per category)

Input variables in Zakrzewski taper model

Maple tree category

Mean error/Std. err. STD Minimum/Maximum

(%)

DBH and total height measured

With commercial branches

-2.37 / 0.98 10.25 -32.64 – 26.57

Withoutcommercial branches

-0.86 / 1.00 10.47 -20.97 – 33.29

Only DBH measured

With commercial branches

-0.003 / 1.11 11.68 -23.00 – 34.47

Withoutcommercial branches

1.15 / 1.16 12.16 -21.05 – 48.41

Evaluation of latent heights using Evaluation of latent heights using HHpp sub-sub-model for maples without commercial model for maples without commercial

branchesbranches

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60

DBH [cm]

He

igh

t [m

]

A

B

C

A – height/dbh curve implied by taper model, B – measured heights, C – height/DBH regression curve

Observed Observed G_oG_o vs. evaluated vs. evaluated G_oG_o

Regional frequencies as a base for Regional frequencies as a base for amendments in timber cruising proceduresamendments in timber cruising procedures

Branch Distribution by DBH classes

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

3 branch

2 branch

1 branch

0 branch

3 branch 0 0 0 1

2 branch 0 0 3 0

1 branch 1 50 54 0

0 branch 281 621 104 0

0-14.9(cm)15-

39.9(cm)40-

66.9(cm)67+(cm)

Based on Morawski and Basham (1958) data for Ontario (Algonquin Morawski and Basham (1958) data for Ontario (Algonquin Region)Region)

ConclusionsConclusions

In hard maple, trees of given total height and In hard maple, trees of given total height and DBHDBH, with commercial , with commercial branches, contain more merchantable wood than those with the branches, contain more merchantable wood than those with the same respective sizes but of a single stem. same respective sizes but of a single stem.

A new approach, which involves tree height as a latent variable, was A new approach, which involves tree height as a latent variable, was used to estimate wood volume for individual Ontario used to estimate wood volume for individual Ontario sugar maple sugar maple stems and branches. stems and branches. The new approach can be potentially used as a The new approach can be potentially used as a modelling framework for research on influence of stand density on modelling framework for research on influence of stand density on tree height and tree competitive status in a standtree height and tree competitive status in a stand

Total volume estimates obtained using models with and Total volume estimates obtained using models with and without without height measurementsheight measurements were reasonable were reasonable

Results indicate that the traditional formulae based on Results indicate that the traditional formulae based on HH and and DBHDBH may not be appropriate for maple if wood in branches is not may not be appropriate for maple if wood in branches is not accounted foraccounted for

Results indicate that Results indicate that carbon in maple standscarbon in maple stands could be reliably could be reliably inventoried using inventoried using DBHDBH-based biomass equations (with latent heights) -based biomass equations (with latent heights) if if DBHDBH-based taper model is integrated with the respective specific -based taper model is integrated with the respective specific gravity modelgravity model