GENEVIÈVE LEJEUNE

PRÉDICTION DU DÉFILEMENT ET DE LA BRANCHAISON DE L’ÉPINETTE NOIRE

Mémoire présenté à la Faculté des études supérieures de l'Université Laval

dans le cadre du programme de maîtrise en Sciences Forestières pour l’obtention du grade de maître ès sciences (M.Sc.)

FACULTÉ DE FORESTERIE ET DE GÉOMATIQUE

UNIVERSITÉ LAVAL QUÉBEC

OCTOBRE 2004 © Geneviève Lejeune, 2004

Résumé

Les modèles de défilement (variation en diamètre de la tige) et de branchaison (variation du

diamètre des branches) sont utiles pour évaluer le rendement en produits finis en

optimisation du sciage. Dans cette étude, chaque équation a été calibrée avec des données

provenant d’études antérieures sur la biomasse et la morphologie de la cime de l’épinette

noire. Les paramètres de chaque équation ont été estimés par régression linéaire mixte avec

une structure d’erreur autorégressive continue de premier ordre. L’équation de défilement

résultante convient à un large gradient de peuplements et de conditions environnementales.

Les équations de branchaison obtenues décrivent correctement le patron paraboloïde du

diamètre maximum et moyen des branches à partir de la longueur de cime et du diamètre de

l'arbre. Une équation supplémentaire basée sur le diamètre de l’arbre et l’indice de qualité

de station a été développée pour prédire la hauteur de l’arbre, variable essentielle mais pas

toujours mesurée dans les inventaires réguliers.

Abstract

Stem taper (variation of stem profile) and branchiness (variation of branch basal diameter)

equations are useful in evaluating the value of lumber recovery through sawing

optimization. In this study, taper and branchiness equations were calibrated for black

spruce (Picea mariana (Mill.) BSP) using data derived from past biomass and crown

morphology studies. Equation parameters were estimated by a linear mixed effect method

employing a first-order continuous autoregressive error process. The resultant stem taper

equation captured variation across a wide range of stand and environmental conditions.

Similarly, using tree diameter at breast height (Dbh) and crown length as explanatory

variables, the resultant branchiness equations enabled the prediction of the (1) largest

branch diameter per tree, and (2) mean branch diameter per tree. Tree height measurements

are not always available in common forest inventory data, but are essential to taper and

branchiness prediction. Consequently, an additional equation to predict tree height was also

developed, based on Dbh and site index.

Avant-propos

Je tiens tout d’abord à remercier Jean-Claude Ruel, mon directeur de recherche, de m’avoir

fait confiance en m’acceptant dans son équipe. Je le remercie pour son support et la qualité

de ses conseils. Merci également à Chhun Huor Ung, co-directeur de recherche, qui m’a

permis de me dépasser et qui m’a guidée tout le long de mon apprentissage. Merci à tous

deux de m’avoir accordé la liberté d’effectuer une partie de ma maîtrise à Vancouver, à

l’Université de Colombie-Britannique, où j’ai eu la chance de travailler avec Tony Kozak,

chercheur scientifique reconnu dans l’étude du défilement. Cette expérience fut des plus

enrichissantes et a contribué à rendre inoubliable mon expérience à la maîtrise.

Je voudrais remercier le Centre de Foresterie des Laurentides de m’avoir permis d’utiliser

les indispensables données sur le défilement et la branchaison qui ont rendu possible cette

étude. Merci à Tadek Rycabel, étudiant au doctorat à l’Université Laval, de m’avoir fourni

des données de défilement additionnelles récoltées avec beaucoup de rigueur et de précision

Merci au CRSNG et à leur soutien financier qui a permis la réalisation de ce projet.

Le corps du présent mémoire est constitué d’une ébauche d’article. Les personnes

suivantes ont participé à la rédaction : Chhun Huor Ung, chercheur scientifique au Centre

de foresterie des Laurentides, Marie-Claude Lambert, statisticienne au Centre de foresterie

des Laurentides et Jean-Claude Ruel, professeur titulaire au département des sciences du

bois et de la forêt de l’Université Laval. L’article n’a pas encore été soumis pour

publication.

Le rassemblement des données, la compilation, les analyses, le document principal, ainsi

que la majeure partie de la rédaction de l’article ont été réalisés par moi-même. M. Ung a

apporté son aide au niveau de la mise en forme et de la correction de la langue pour la

partie écrite en anglais et Mme Lambert a fourni une aide au niveau statistique. M. Ruel a

apporté son expertise au niveau du perfectionnement pour répondre aux exigences de la

publication.

iv

Table des matières

Résumé.....................................................................................................................................i Abstract.................................................................................................................................. ii Avant-propos ........................................................................................................................ iii Table des matières .................................................................................................................iv Liste des tableaux....................................................................................................................v Liste des figures .....................................................................................................................vi 1.0 Introduction générale ..................................................................................................1 2.0 Predicting taper and branchiness of black spruce using inventory data .....................7

2.1 Introduction.............................................................................................................7 2.2 Material ...................................................................................................................8 2.3 Methods ................................................................................................................14

2.3.1 Mixed model .................................................................................................14 2.3.2 Taper .............................................................................................................16 2.3.3 Branch diameter ............................................................................................16 2.3.4 Total height ...................................................................................................17 2.3.5 Equation evaluation ......................................................................................18

2.4 Results and discussion ..........................................................................................20 2.4.1 Taper .............................................................................................................20 2.4.2 Branchiness...................................................................................................23 2.4.3 Height............................................................................................................28

2.5 Conclusion ............................................................................................................31 3.0 Conclusion générale..................................................................................................32 Références.............................................................................................................................34

Liste des tableaux

Table 1: Description of the sampling sites used for calibrating and validating the taper model ............................................................................................................................11

Table 2: Range of Dbh and height of black spruce in the six sites used for calibration and validation of the taper equations...................................................................................12

Table 3: Tree and branch attributes for the branchiness data set..........................................12 Table 4: Range of Dbh and total height for the tree height model .......................................14 Table 5: Fit statistics for inside and outside bark taper equations ........................................21 Table 6: Estimated absolute and relative mean biases and SEEs of diameter outside and

inside bark predictions for black spruce by height class. .............................................22 Table 7: Fit statistics for the maximum and mean branch diameter equations.....................24 Table 8: Parameters estimates and fit statistics for the black spruce height model..............29 Table 9: Estimated absolute and relative mean bias and SEEs estimates of the height

prediction by Dbh classes .............................................................................................29

Liste des figures

Figure 1: Location of the calibration and the validation data sets in relation with Canada ecozones..........................................................................................................................9

Figure 2: Detailed location of the data sets used to calibrate the taper and branchiness equations .......................................................................................................................10

Figure 3: Location of the sample plots used to derive the height equation ..........................14 Figure 4: Observed maximum branch diameter of whorl branches over depth into crown .24 Figure 5: Observed average branch diameter of whorl branches over depth into crown .....25 Figure 6: Predicted over observed maximum branch diameter ............................................26 Figure 7: Predicted over observed average branch diameter ................................................26

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1.0 Introduction générale

À cause des pressions environnementales et de la diminution des superficies forestières, la

réserve en matière ligneuse est en constante décroissance depuis le début des années 90 et,

par conséquent, le coût de cette fibre s’en trouve considérablement augmenté. Pour

maintenir l’approvisionnement en matière ligneuse pour l’industrie tout en protégeant

l’environnement, il devient critique de poursuivre un aménagement intensif des forêts pour

palier à notre faible productivité forestière. La gestion de la densité des peuplements permet

d’atteindre des objectifs d’aménagement spécifiques par la manipulation et le contrôle de la

densité du peuplement en croissance. La manipulation de la densité consiste à réguler le

nombre et l’arrangement de tiges par unité de surface par l’espacement initial ou encore,

par une séquence temporelle d’éclaircies (Newton 1997). Pour aider à la prise de décision

de gestion de peuplement, un outil appelé diagramme de gestion de la densité de

peuplement (DGDP) a été développé.

Le DGDP est un modèle d’aide à la décision développé au cours des quatre dernières

décennies pour la gestion de la densité de peuplements. Il s’agit d’un modèle moyen à

l’échelle du peuplement décrivant les relations dynamiques entre la densité du peuplement,

la taille des arbres et le volume de bois à plusieurs étapes de développement du peuplement

(Newton 1997). Le DGDP permet d’estimer le volume de bois sur pied en relation avec la

densité du peuplement selon différents stades de développement. Il est devenu un important

outil d’aide à la décision pour l’aménagement de peuplement basé sur l’espacement initial,

l’éclaircie, l’âge de récolte et la régénération.

Tous les DGDP développés ne permettent que l’estimation du volume moyen par arbre ou

du volume / ha. Autrement dit, en utilisant cet outil, on estime que l’objectif

d’aménagement est nécessairement de maximiser le volume du peuplement. D’un autre

côté, il est reconnu dans le milieu forestier que chaque unité de volume de bois n’a pas la

même valeur en termes de production de produits (ex. : rendement, qualité, valeur,

rétention de carbone). Ainsi, il devient justifiable d’étendre le concept de DGDP au

développement d’un modèle basé sur des paramètres reliés aux produits forestiers (ex. :

valeur) pour la gestion de la densité de peuplement.

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Un DGDP intégrant une notion de produits forestiers permettrait aux gestionnaires

forestiers de prendre des décisions optimales dans la gestion des peuplements forestiers

basées sur les caractéristiques des produits plutôt que sur le volume des arbres uniquement.

Les décisions relatives à la gestion des peuplements pourraient alors être prises en

considérant l’économie et le marketing des produits forestiers. Il permettrait de définir les

stratégies d’aménagement de la forêt requises pour atteindre des objectifs spécifiques (ex. :

valeur maximum, dimensions spécifiques et haute qualité).

L’étape clé du développement d’un tel modèle est l’établissement de relations dynamiques

entre les caractéristiques des arbres et du peuplement et les produits obtenus à différents

stades de développement du peuplement. Le récent développement de logiciels de

simulation de sciage rend possible la simulation du rendement en sciage et en copeaux à

partir d’arbres de toutes dimensions. Ces logiciels nécessitent toutefois la connaissance de

caractéristiques pouvant affecter le rendement et la qualité des produits et celles-ci ne

peuvent être dérivées des DGDP actuels.

Le défilement, ou la forme de la tige, est un paramètre de qualité déterminant en terme de

rendement en sciage. Il s’agit de la description de la variation en diamètre tout le long de la

tige, de la base au sommet (Larson 1963). La prédiction du volume est souvent la principale

raison pour laquelle on étudie le défilement (Garber et Maguire 2003, Sharma et Oderwald

2001, Newnham 1988, 1992, Kozak 1988), une autre preuve de l’importance que la

prédiction du volume occupe encore dans le milieu forestier. Par contre, il est reconnu par

les chercheurs forestiers que chaque mètre cube de bois n’est pas équivalent puisque le

rendement en valeur du produit fini par mètre cube de bois varie grandement

dépendamment des caractéristiques de la ressource (Zhang et Chauret 2001). En effet, deux

tiges ayant le même volume ne vont pas nécessairement produire le même rendement en

sciage car leur défilement peut être différent. Une faible variation en diamètre indique une

forme plus cylindrique de la tige, ce qui permet de produire des découpes plus longues et

donc, de plus grande valeur. De plus, un faible défilement diminue la présence de

« flache », la cause la plus importante de déclassement du bois lors d’évaluation visuelle

(Zhang et Chauret 2001). Ainsi, la prédiction du défilement en fonction des conditions de

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croissance de l'arbre permettra de déterminer l'influence des traitements sylvicoles et donc,

de la gestion de la densité sur le rendement en sciage d'un peuplement.

Larson (1963) a fait une revue exhaustive des facteurs influençant le défilement. Ces

facteurs peuvent être résumés comme suit: un défilement faible est associé à une faible

longueur de cime, à un âge avancé, à une forte densité, à une bonne qualité de site et à un

faible rapport h/d (rapport de la hauteur total / diamètre à hauteur de poitrine (dhp =

diamètre mesuré à une hauteur de 1,3 m à partir de la base de l'arbre)). Ce dernier facteur

est directement corrélé au défilement et permet, simplement, de comprendre l'influence des

conditions de croissance. Ainsi, pour une hauteur donnée, un plus gros diamètre à hauteur

de poitrine (dhp) signifie un défilement plus prononcé. Zhang et Chauret (2001) ont étudié

la variation du défilement en fonction de l’espacement initial chez l’épinette noire. Ils ont

déterminé que les arbres provenant d’un peuplement de faible densité possèdent un

défilement plus fort que ceux de même classe de diamètre situés dans un peuplement de

forte densité. D’un autre côté, le défilement augmente avec une augmentation du diamètre.

Par conséquent, à l’échelle du peuplement, le défilement moyen augmentera avec la

diminution de la densité du peuplement. Pour des arbres de même diamètre, le rendement

en valeur de produits finis par arbre augmentera avec une densité plus élevée (Zhang et

Chauret 2001).

Les nœuds sont un des paramètres les plus importants à être considérés lors d’évaluation

visuelle de classement. Ils sont la deuxième cause la plus importante en termes de

déclassement du bois (Zhang et Chauret 2001). La grosseur des nœuds est un facteur qui

affecte grandement la résistance mécanique du bois et par conséquent, la qualité et la valeur

des produits de sciage (Zhang et Chauret 2001). En effet, la présence de noeuds crée une

zone de faiblesse dans une pièce de bois et la déclasse lors d'évaluation visuelle ou de test

de résistance mécanique. Lemieux et al (2001) affirment que le diamètre de la branche est

le facteur externe le plus fortement relié à la taille des nœuds. D’ailleurs, plusieurs études

utilisent la branchaison comme élément principal pour évaluer l’effet des traitements

sylvicoles sur la qualité du bois (Colin et Houllier 1991, 1992; Maguire et al. 1991;

Maguire et al. 1994).

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Il existe une étroite dépendance entre le développement du tronc et celui du houppier, siège

de la photosynthèse (Courbet et Albouy 1994). En effet, le diamètre des branches est

fortement corrélé au dhp et à la hauteur totale de l'arbre (Colin et al 1993). Ceci signifie que

le contrôle de la compétition et de la densité affecte le développement de la tige, de la cime

et donc, les caractéristiques des branches (Carter et al 1986). La densité a un effet

considérable sur la taille de la cime et la taille des branches (Zhang et Chauret 2001). À

l’échelle du peuplement, une densité faible se traduit par un diamètre moyen plus grand,

une cime plus large et donc, des branches plus grosses. Pour une même classe de diamètre,

les arbres provenant d’un peuplement avec une densité forte auront également de plus

petites branches (Zhang et Chauret 2001). Pour ce qui est de l'effet de la provenance sur les

différences en diamètre des branches, bien que significatives, elles sont très faibles lorsque

l’effet de la vigueur individuelle est éliminé (Colin et al 1993).

La quantité et la qualité du sciage pouvant être tirés d’une bille dépendent donc largement

du patron de variation du diamètre de la tige (défilement) et du diamètre des branches

(branchaison) de la base au sommet de l’arbre. Ces deux caractéristiques sont inhérentes

aux conditions de croissance de la cime et de la tige dans son milieu, milieu qui est modelé

par l’environnement et par l’aménagement que l'on en fait.

D'un point de vue des processus physiologiques, plusieurs études se sont concentrées sur

les mécanismes de translocation de carbone à l'intérieur de l'arbre dû à des préoccupations

économiques des effets des changements de l'environnement sur l'évolution de la tige et des

propriétés du bois. Schématiquement, trois approches ont été proposées pour établir une

relation entre l'allocation de carbone et les propriétés du bois qui en découlent. La première

approche est basée sur le modèle tubulaire (Shinozaki et al 1965), et a largement été utilisée

dans les modèles de croissance plus appliquées, e.g. Mäkelä (1997, 2002). Les deux autres

approches représentent des alternatives plus flexibles de la réaction de l’arbre en termes de

partition de carbone en réponse aux changements environnementaux. Elles sont inspirées,

respectivement, de la théorie de Münch et de la théorie de l'échelle allométrique. La théorie

de Münch considère l'allocation de carbone comme le transport de soluté d’un gradient de

concentration positif à négatif (Thornley 1995, Deleuze et Houllier 1997, Mencuccini et

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Grace 1995). L'approche basée sur la théorie de l'échelle allométrique combine la loi

universelle de biologie, appelée échelle allométrique, avec les besoins de stabilité et

d'hydrologie (West et al 1999). Cela permet de prédire comment les vaisseaux (tubes

conducteurs) rétrécissent en diamètre afin de compenser pour la variation de la longueur

totale de transport. Les deux premières approches ont été partiellement testées tandis que la

troisième n'a pas été testée du tout. La validation de ces théories nécessite des équations

réalistes de branchaison et de défilement.

Actuellement, il n'existe pas de modèle pour prédire la branchaison de l'épinette noire et

une seule équation de défilement a été développée pour cette espèce pour la région du

Yukon (Bonnor et Boudewyn 1999), et un site isolé en Ontario (Newnham 1988). Les

données de défilement et de branchaison présentent une structure complexe d'auto

corrélation qui peut invalider les tests statistiques lors du choix des variables de prédiction.

À l'exception de Garber et Maguire (2003), les tentatives pour tenir compte correctement de

la structure de corrélation ont plutôt été négligées dans la plupart des études en foresterie.

Cette étude a été réalisée dans l’optique de développer un modèle d’aide à la décision, basé

sur les produits forestiers, pour la gestion de la densité des peuplements d’épinette noire. La

connaissance préalable de la dimension des arbres et de la qualité des sciages qui pourront

en être tirés, permettra aux gestionnaires de la forêt d’optimiser leurs scénarios sylvicoles

(Courbet et Albouy 1994). OPTITEK, un logiciel de simulation du sciage (Grondin et

Drouin 1998), permettra de faire le lien entre le rendement en produits finis et les

caractéristiques de l’arbre et du peuplement.

Le but de cette étude est de développer, et de valider des équations statistiques de

défilement et de branchaison pour l'épinette noire (Picea mariana (Mill.)BSP.). L’épinette

noire est l’espèce retenue pour ce projet parce qu’elle est l’espèce commerciale la plus

importante dans l’Est du Canada (Ministère des ressources naturelles 1996) et qu’elle

possède une valeur hautement reconnue dans les secteurs du sciage et des pâtes et papiers.

Le développement de chacune de ces équations se base sur deux principes: (1) une

application facile aux données d'inventaires réguliers, avec un large gradient de types de

peuplements et de sites, en rassemblant les données existantes du Québec et du Yukon, (2)

6

une estimation adéquate de la capacité de prédiction des modèles par l'utilisation de la

structure complète de la covariance des paramètres.

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2.0 Predicting taper and branchiness of black spruce using inventory data

2.1 Introduction

Black spruce (Picea mariana (Mill.) B.S.P.) is one of the major commercial species in the

Canadian boreal forest. The species is highly valued for lumber and pulp production in

eastern Canada. The total lumber value of a tree depends largely on the pattern of change in

stem diameter (taper) and basal area of branches (branchiness) at different heights along the

stem. Both stem properties are inherent to the concurrent growth of crown and stem as

controlled by the environment and management. Since the comprehensive review on stem

form theories by Larson (1963), several studies have focused on the mechanism of carbon

allocation within the tree to address the economic concern of environmental changes on

stem and wood properties. Schematically, three approaches have been proposed to establish

the relationships between carbon allocation and the derived wood properties. The first

approach is based on the pipe model theory (Shinozaki et al 1965), and has been largely

used in practical forest growth models, e.g. Mäkelä (1997, 2002). The two other approaches

represent alternatives for increasing the flexibility of carbon allocation in response to

environmental changes. They are based on Münch theory and on allometric scaling theory

respectively. The Münch theory based approach considers the carbon allocation as the

transport of solutes along a concentration gradient between sources and sinks (Thornley

1995, Deleuze and Houllier 1997, Mencuccini and Grace 1995). The allometric scaling

theory based approach combines the universal law in biology called allometric scaling with

the requirement of stability and hydrology (West et al 1999). It predicts how vessel

diameter (conducting tubes) must taper to compensate for variation in total transport path

length. The first two approaches have been partially tested while the third one has not yet

been tested. Any testing effort will require realistic taper and branchiness equations.

This requirement is strengthened by the recognition by wood scientists of the importance of

geometric and internal wood characteristics on improving lumber value recovery through

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sawing optimization during the primary log breakdown (Zhang and Morgenstern 1995).

Recently, several software programs have been proposed for sawing optimization,

especially for Scots pine (Mäkelä 1997, Ikonen et al 2002), and Norway spruce (Houllier et

al 1995). The linkage has been possible because realistic information on taper and

branchiness is available for model development and testing for these species. For boreal

black spruce however, no branchiness equation has been produced, and taper equations

were only developed for Yukon (Bonnor and Boudewyn 1999), and an isolated study site in

Ontario (Newnham 1988). Taper and branchiness data present a complex autocorrelation

structure that can invalidate tests of significance when choosing predictor variables. With

the exception of Garber and Maguire (2003), attempts to account for the correct correlation

structure has been neglected in most forestry studies. The aim of this study was to develop,

and to assess statistical equations for both black spruce taper and branchiness. The

development of both equations was govern by two requirements: (1) applicability to

inventory data covering a large range of stand and site types via the use of existing data

from Quebec and the Yukon Territory, and (2) accurate estimation of the uncertainty of

model predictions via the specification of the parameter covariance structure. Furthermore,

tree height is an essential linkage between taper and branchiness models. However, because

measurement of tree heights is both time-consuming and expensive, the standard sampling

procedure for forest inventory is often to measure heights of only a few trees in the plot

(Hann and Scrivani 1987, Dolph and Leroy 1989). Consequently, an additional equation

was developed in order to estimate total tree height based on tree diameter and stand

characteristics.

2.2 Material

Modeling the branchiness and taper of black spruce across a wide geographic area requires

the integration of diverse data sets. In this study, taper data derived from past biomass

studies conducted in Quebec and the Yukon Territory, were used. Specifically, taper data

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were derived from the ENFOR (ENergy from the FORest) biomass studies (Ouellet 1983;

Manning 1984). The data from Quebec has been utilized in a few previous studies (Ung

1990; Beaumont et al 1999; Ruel et al 2004; Rycabel 2002). The data from the Yukon

Territory was used as an independent validation data set for the taper function. Figure 1

illustrates the broad geographic extent of the datasets used to develop and validate the taper

equations. .

In contrast, the few studies in which the branchiness of black spruce has been measured

employing destructive sampling techniques, tend to be concentrated in forests characterized

by mesic drainage conditions (Ung and Ouellet 1993; Bonnet and Pastor 1997; Ruel et al

2004). Nevertheless, the data utilized to develop the branchiness model consisted of a

subset of the Quebec taper data set (Figure 2).

Figure 1: Location of the calibration and the validation data sets in relation with Canada

ecozones

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Figure 2: Detailed location of the data sets used to calibrate the taper and branchiness

equations

Black spruce is a species which occupies a wind range of sites and hence the reassembled

data used in this study is representative of the species. Particularly, given that the data were

obtained from pure and mixed stand types with regular and irregular structures. Table 1

presents the range of environmental conditions associated with each of the sample data

sites.

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Table 1: Description of the sampling sites used for calibrating and validating the taper model

Site Lebel sur

Quévillon Chibougamau Saint Camille Alma Rouyn

Noranda Baie-

Comeau Yukon

References Ouellet (1983); Ung (1990)

Ouellet (1983); Beaumont et al (1999);

Ruel et al (2004)

Bonnet and Pastor (1997)

Ung and Ouellet (1993)

Rycabel (2002)

Ouellet (1983)

Manning et al (1984)

Ecozone1 Boreal shield Boreal shield Boreal shield Boreal shield Boreal shield

Boreal shield

Boreal cordillera

Ecological domain2

Black spruce moss forest

Black spruce moss forest

Balsam fir and yellow birch

Balsam fir and

yellow birch

Balsam fir and

white birch

Black spruce

moss forest

-

Ecological Sub-domain2

West West East West West East -

Ecological Region3

12b

12b

5a 6a 8c 11a2 -

Cover Type Coniferous Coniferous Coniferous Coniferous Coniferous Coniferous Coniferous Mean annual

temperature (oC) -2.5 - 0.0 -2.5 - 0.0 1.0 – 2.5 1.0 – 2.5 0.0 – 1.0 -2.5 – 0.0 1.0 – 5.5

Degree-day (>5o)

1000 – 1250 1000 – 1250 1250 – 1500 1250 – 1500 1100 –1500 750 – 1000 -

Growing season (days)

120 - 150 120 - 150 160-170 160-170 150 - 160 100 - 140 -

Annual precipitation

(mm)

600 – 1000 600 – 1000 900 – 1100 900 – 1100 800 – 1000 1000 - 1400 300-1500

1 Ecological stratification working group (1995) 2 Ministère des ressources naturelles (2003) 3 Robitaille, A., and Saucier, J.-P. (1998)

Predicting inside bark taper based on outside bark Dbh is valuable both from a practical and

a sawing optimisation point of view. However, in order to extend the model to the largest

range of tree sizes and sites distribution, we also fitted the equation for outside bark

prediction. The ENFOR biomass data (Ouellet 1983) represents a large geographic area and

hence offers the potential to produce a taper equation with wide applicability. However, it

does not contain inside bark diameter. Consequently, two separate equations, inside and

outside bark, were developed. Table 2 presents the range of diameter at breast-height

(Dbh), and total height measurements by sample site. The 947 black spruce trees obtained

from the five Quebec sites, ranging in Dbh from 6.3 to 32.9 cm and in total height from

5.70 to 25.25 m, were used as a calibration data set for outside bark taper equation. The

inside bark taper data set contains 291 black spruce trees, ranging in Dbh from 6.3 to 32.2

cm and in total height from 7.20 to 25.25 m. The data derived from the Yukon Territory

biomass study, was used for both outside and inside bark taper validation and included 223

trees with similar but smaller ranges in Dbh and total height.

12

Table 2: Range of Dbh and height of black spruce in the six sites used for calibration and validation of the taper equations Taper / outside bark Taper / inside bark Site Number

of trees Dbh

(cm)

Height

(m)

Number of trees

Dbh

(cm)

Height

(m) Chibougamau, Lebel-sur-Quévillon

649

6.5 – 32.9

5.70 – 23.08

62

6.5 – 12.7

7.20 – 13.65

Saint Camille 19 8.3 – 24.5 9.69 – 19.35 19 8.3 – 24.5 9.69 – 19.35 Alma 32 6.3 – 24.7 8.91 – 19.17 32 6.3 – 24.7 8.91 – 19.17 Rouyn-Noranda 178 8.8 – 32.2 8.65 – 25.25 178 8.8 – 32.2 8.65 – 25.25 Baie-Comeau 69 9.7 – 26.5 8.77 – 22.31 - - - Yukon 223 6.9 – 27.5 7.20 – 20.50 223 6.9 – 27.5 7.20 – 20.50

Table 3 present tree and branch attributes for the branchiness data set. Mean and maximum

branch diameters in each whorl were calculated from 72 trees. Tree Dbh varied between 6.5

and 25.0 cm and between 6.75 and 19.65 m for total height. Contrary to the taper data set,

trees larger than 25 cm were not available for branchiness modeling. The limited size of the

branchiness data set negated the creation of both a calibration and a validation data set.

Thus all the available trees were then used for the calibration for the branchiness model.

Table 3: Tree and branch attributes for the branchiness data set Tree attribute Branch attribute Site Number

of trees Dbh

(cm)

Height

(m)

Crown Length

(m)

Number of branches

Mean diameter for whorl

(mm)

Max diameter for whorl

(mm) Chibougamau 34 6.5 – 24.2 6.75 – 19.65 1.41 – 17.45 432 1.0 - 42.3 1.0 – 46.6 Saint Camille 18 8.3 – 24.5 9.69 – 19.35 1.13 – 8.80 259 3.0 – 26.3 3.0 – 33.0 Alma 20 14.4 – 25.0 15.23 – 19.17 2.19 – 7.68 400 4.0 – 23.0 4.0 – 31.0

Reassembling data from various studies implies accepting the heterogeneous sampling

procedures with their diverse measurement protocols. Taper modeling requires stem

analysis data consisting of inside and outside bark diameter (cm) measurements obtained at

varying heights (m) along the main stem. In Quebec, the ENFOR taper data (Ouellet 1983)

consisted of outside bark diameters measured at 0.15 m and 0.8 m above ground, at 5 cm

below the first live whorl, at 1.3 m, and thereafter at 1/3, 2/3 of merchantable height (i.e.

usable portion of the stem from the stump height to an upper diameter limit of 9 cm), and at

13

the merchantable height. In ENFOR taper data from Yukon (Manning et al 1984), inside

and outside bark diameters were measured at 0.30 m and then, at intervals of 2.0 m starting

at Dbh (1.30 m). In the taper data from Ung (1990), Beaumont et al (1999), and Ruel et al

(2004), inside and outside bark diameters were measured on sections taken 0.15 m above

ground, at 1.3 m, and every meter until the first live whorl, and thereafter at every whorl

until the tip of the crown. In Rykabel’s taper data (2002), inside and outside bark diameters

were measured on sections taken 0.15 m above ground, at 0.65 m, 1.15 m, 1.3m, 2.15 m

and every meter thereafter. Thus Rykabel’s taper data has a relatively high concentration of

measurements stem diameter in butt swell region.

The branchiness data consisted of crown length (m), branch basal diameters within each

whorl (mm), and the distance of the each whorl to the apex (m). Total height and outside

bark Dbh were basic measurements that were available for each tree regardless of study

site.

For the tree height equation, the Ministry of Natural Resources of Quebec provided Dbh

and height data on 12 383 trees measured in 4108 Permanent sample plots (Direction des

inventaires forestiers 2000, 2001). These plots covered the majority of bioclimatic domains

of Quebec (Figure 3). Seventy percent of trees were used for calibration and the other 30 %

was kept to validate the height model. The range in Dbh and total height is presented in

Table 4. The tree height data covers the entire size range of the data sets used to calibrate

the taper and branchiness models.

14

Figure 3: Location of the sample plots used to derive the height equation

Table 4: Range of Dbh and total height for the tree height model Dataset subdivision Number of trees Dbh

(cm) Total height

(m) Calibration 8708 2.80-32.90 2.40-24.50 Validation 3675 3.60-32.80 3.00-24.50

2.3 Methods

2.3.1 Mixed model

The taper and branchiness sampling procedure resulted in a correlated data structure both at

the tree and the plot level. This correlated data structure has to be considered in order to

accurately estimate the uncertainty of the model predictions, i.e. to appropriately assess its

statistical significance (Gregoire et al 1995). Historically, however, only a few studies in

forestry have recognize the necessity in obtaining unbiased and minimum variance

15

estimators (e.g., Meredieu et al 1998, Garber and Maguire 2003) by correctly specifying the

covariance structure when the goal is to identify statistically significant predictor variables.

To address this concern in this study, a mixed effects model with continuous-time

autocorrelation (SP(POW)) error structure, as discussed by Gregoire et al (1995), was

applied to our irregularly spaced and unbalanced data. This covariance structure assumes

that within-subject correlation decreases proportionally with increasing (linear) distance

between measurements. Accordingly, the parameters of both the taper and branchiness

equations were estimated using the SAS/STAT PROC MIXED procedure (SAS Institute

Inc. 1999). Correlation induced by locations, plots, trees and branches were introduced in

the different models as random effects and then, were removed where justified (i.e., using

tests on the variance components employing a 30% type I error probability level (α,

Miliken et Johnson, 1984, p.262). Independent variables were dropped successively from

each equation based on parameter estimate t-values until only significant parameter

estimates remained (α = 0.05). The SAS/STAT PROC MIXED (SAS Institute Inc., 1997)

procedure was used. At each step, variance homogeneity and normality assumptions were

verified employing residual analysis. Specifically, studentized residuals were used to attest

uncorrelated and uniform variances (Gregoire et al 1995). Equations are presented in their

final form after successively dropping insignificant independent variables.

16

2.3.2 Taper

In this paper, the stem is defined as the main tree axis including both the portions within

and below the crown. Schematically, from the apex downward, the stem taper of boreal

species can successively be described as a solid of revolution resembling a cylinder, cone,

paraboloid and neiloid. To account for this geometric variation in a continuous manner,

several empirical taper equations have been proposed which can chematically classified in

three groups: trigonometric equations (Thomas and Parresol 1991, Bi and Long 2001),

polynomial equations (Bruce et al 1968, Max and Burkhart 1976), and exponent power

equations (Kozak 1988). Kozak’s exponent power equation (Eq. 1), was selected in this

study for two basic reasons: (1) it captures variation in the stem form across a wide range of

stand conditions (Garber and Maguire 2003); and (2) it can be easily linearized, facilitating

the mixed modelling approach. Thus, following a logarithmic transformation, Kozak’s

taper equation for outside or inside bark becomes:

)/)(ln()ln()ln(

)001.0ln()ln()ln()ln()ln()ln()ln(

876

52

4321

HDbhXaeXazXa

zXazXaDbhaDbhaadz +++

+++++= (1)

with d is the diameter outside or inside bark at height h (cm), h is the height above ground

(m), Dbh is the diameter outside bark at breast height (cm), H is the total tree height (m), z

is the relative height (z = h / H), ),1/()1( pzX −−= where p is the location of the

inflection point, and ai ; i=1…8 parameters to be estimated using the mixed model

procedure

2.3.3 Branch diameter

Two equations were also developed for branch diameter, the first for predicting the

maximum whorl branch diameter per tree; and the second for predicting the average whorl

17

branch diameter. Maximum branch diameter per whorl constitutes a good index of wood

quality since it represents the maximum default size at a given height (Maguire et al 1990).

On the other hand, average branch diameter can give a more valid estimate of the average

diameter profile while avoiding an over-estimate (Roeh and Maguire 1997). Branchiness

equations were specified according to the results given by Roeh and Maguire (1997):

)*ln()*ln()ln()ln( 2321 DINCCLbDINCDbhbbBDmax ++= (2)

)*ln()*ln()ln()ln( 2321 DINCCLcDINCDbhccBDmean ++= (3)

where BDmax is the maximum whorl branch diameter (mm) per tree, and BDmean mean

whorl branch diameter (mm) per tree, DINC (m) is the crown depth calculated as the

difference in height between the tree tip and the top of the subject branch (m), Dbh is the

diameter at breast height (cm), CL is the crown length (m), and bi and ci (i =1…3) are

parameters to be estimated using the mixed model procedure.

2.3.4 Total height

Measurement of all tree heights in a sample plot is not a standard forest inventory

procedure within the Province of Quebec. Due to logistical constraints, Dbh, total height

and age measurements from were obtained from only 3 site trees per plot from which site

index was calculated. On the other hand, Dbh was measured on each tree. Therefore, tree

height was estimated using Dbh and site index employing the model developed by Larsen

and Hann (1987):

)ln()ln()ln()ln( 321 SIdDbhddHT ++=

18

where HT is the total height (m), Dbh is the diameter at breast height (cm), SI is site index

(m) calculated as the potential mean dominant height at a stand age 50 yr, and di (i = 1…3)

are parameters estimated using the mixed model procedure. Note, for each plot, the mean

site index was calculated using both the black spruce site index equation developed by

Pothier and Savard (1998), and the height and age measurements obtained from the

dominant trees.

2.3.5 Equation evaluation

Fit statistics indicate how well a model fits the data set used in its construction, whereas

prediction statistics indicate how well a model may predict the dependent variable on an

independent data set (Muhairwe et al 1994). In this study, the assessment of the taper and

height functions was based on both overall fit and prediction statistics whereas the

assessment of the branch diameter equations were based on overall fit statistics only. The

fit statistics used included the standard error of the estimate (SEE) and the coefficient of

multiple determination (R2) for the logarithm of d. SEE indicates the spread of the actual

observations ( iY ) around the predicted values ( iY ):

kn

YYSEE

n

iii

−

−=∑=1

2)ˆ( (4)

where n is the number of observations and k is the number of estimated parameters used in

the estimation. The coefficient of multiple determination defines the proportion of the

variation of the dependent variable explained by the independent variables:

∑

∑

=

=

−

−−= n

ii

n

iii

YY

YYR

1

2

1

2

2

)(

)ˆ(1 (5)

19

whereY is the arithmetic mean of Y.

Expressing the SEE in units of cm for taper diameter equation and in mm for branch

diameter equations, can be very useful. Because natural logarithm transformation was

required in the process of finding the regression coefficients of the equation, reversing that

transformation will result in a systematic underestimate of the dependent variable in the

untransformed scale. According to Gregoire at al (1995), methods to correct for logarithmic

transformation bias for models with the complex error structures, such as those developed

in this study, are nonexistent. However, in practice, taper and branchiness equations will

normally be used to predict stem and branch diameters in their original (untransformed)

units. Consequently, operationally, it is very important to be able to evaluate the prediction

ability of these equations on this particular scale. Others have shown that using prediction

statistics based on the untransformed dependent variable is an acceptable solution (e.g.,

Kozak and Smith 1993, Muhairwe et al 1994).

In this study, the prediction statistics were only calculated for taper and height equations

since no independent data set was available for branchiness model. The statistics included

mean bias (bias), index of fit squared (I2, an estimate of R2), estimated standard error of

estimate (estimated SEE) (Eqs (6), (7) and (8), respectively).

nYY

biasn

i ii∑ =−

= 1)ˆ(

(6)

∑∑

=

=

−

−−= n

i i

n

i ii

YY

YYI

12

12

2

)(

)ˆ(1 (7)

21

ˆ( )

ni ii

Y Yestimated SEE

n k=

−=

−∑ (8)

20

where, this time, iY is the ith antilogarithm of predicted value of the dependant variable (ln

d). These statistics were calculated at both the overall tree scale, and along the stem at 10 %

height intervals for the taper equation. Statistics calculated by height intervals are very

important in identifying the bias of predictions along the main tree axis (Muhairwe et al

1994). The same procedure was used to validate total height equation.

2.4 Results and discussion

The multiple measurements along the bole imposed by the sampling procedure

incorporated a source of autocorrelation which can invalidate tests of significance on the

dependent variables (Garber and Maguire, 2003). The mixed effect modelling used in this

study allows analysis of data with several sources of variation. It allows taking into account

the inter-tree and intra-tree variability (Meredieu et al 1998; Brown and Prescott 1999).

Moreover, the mixed model method provides an improvement of the fit compared with

models that do not account for the error structure (Gregoire et al 1995; Meredieu et al 1998;

Brown and Prescott 1999). To the extent that a particular mixed-effects model better

portrays the pattern of explainable variation in observed phenomena, the more it can be

trusted to mimic that variation in its predictions (Gregoire et al 1995). In this study, a

continuous-time autocorrelation error structure, as discussed by Gregoire et al (1995), was

applied to an irregularly spaced and unbalanced data set. The inclusion of random subject

effects and the modelling of the correlation structure provided valid tests of significance on

model parameter estimates (Garber and Maguire 2003).

2.4.1 Taper

Table 5 lists the resultant parameter estimates, overall standard error of estimates and

coefficient of multiple determination for the taper models. All coefficients, or fixed effects,

21

were significant in both equations with the exception of the fourth term in the outside bark

equation. Both models performed very well in fitting the calibration data: overall SEEs of

0.07 (ln(cm)) for both equations, with R2s of 0.99 and 0.98 for the outside and inside bark

taper equations, respectively.

Table 5: Fit statistics for inside and outside bark taper equations

Outside bark Inside bark Parameter

estimate SE t Pr > |t|

Parameter estimate

SE t Pr > |t|

ln a1 -0.1125 0.0534 -2.11 0.04 -0.1941 0.1066 -1.82 0.07a2 1.0613 0.0296 35.88 0.00 1.0620 0.0590 18.01 0.00ln a3 -0.0085 0.0018 -4.75 0.00 -0.0073 0.0035 -2.05 0.04a4 - - - - 0.7234 0.0677 10.69 0.00a5 -0.0061 0.0023 -2.61 0.01 -0.1516 0.0127 -11.90 0.00a6 -0.4628 0.0356 -12.99 0.00 1.1009 0.1429 7.70 0.00a7 0.3650 0.0127 28.64 0.00 -0.5182 0.0789 -6.57 0.00a8 0.1276 0.0073 17.53 0.00 0.1962 0.0186 10.58 0.00SEE 0.07 0.07 R2 0.99 0.98 Note: SEE and R2 values are in term of ln(d)

Kozak’s equation is partially based on the observation that the stem form changes from

neiloid to paraboloid at a given percentage (p) of the total height of the tree. Demaerschalk

and Kozak (1977) observed that this inflection point was relatively constant within a

species regardless of tree size and ranged between 0.20 and 0.25 for British Columbia

species. Despite this, accurate determination of the p-value does not seem to be critical.

Perez et al (1990), working on Pinus oocarpa in Central Honduras, found that changing

location of the inflection point between 0.15 and 0.35 had little effect on the predictive

ability of the model. In this study, the parameter estimates of the model were insensitive to

p values varying between 0.15 and 0.35. Furthermore, no significant effects were detected

in terms of variation in R2 or SEE. This absence of a specific location might be due to the

fact that black spruce, is a species with little taper, andless butt flare than coastal BC

species. Newnham (1992) also observed little taper on other boreal species (jack pine

(Pinus banksiana Lamb.), lodgepole pine (Pinus contorta (Dougl.)), white spruce (Picea

22

glauca (Moench) Voss) and trembling aspen (Populus tremuloides Michx.). Thus in accord

with Newnhams (1992) results, a value of 0.20 was chosen for p for black spruce

Contrarily to the inflection point, butt swell strongly influences the estimated values of

taper equation parameters. When eliminating Rycabel’s taper data (2002), which is

characterized by a relatively high resolution measurement in the butt swell, the estimated

parameters become erratic producing negative diameter estimates within the butt swell

region. Consequently, the goodness of fit of the taper equation would be appreciably

improved by increasing the resolution of stem diameter measurements between the stump

and breast height, as this is region were most of the stem diameter variation occurs.

Prediction statistics of both taper equations were tested on a completely independent black

spruce data set from the Yukon Territory. Thereby, enabling the assessment of the

equations ability to capture taper variation across a wide range of stand and site conditions.

Table 6 summarizes the estimated mean bias and estimated SEEs of the equations

predictions in terms of diameter (cm) at varying heights.

Table 6: Estimated absolute and relative mean biases and SEEs of diameter outside and inside bark predictions for black spruce by height class.

Ht. from ground

(%)

n Outside bark Inside bark

Bias

(cm) Bias (%)

SEE (cm)

SEE (%)

Bias (cm)

Bias (%)

SEE (cm)

SEE (%)

0.0-0.1 290 0.43 2.39 1.16 6.53 0.36 2.20 1.07 6.470.1-0.2 177 0.14 1.05 0.37 2.68 0.05 0.37 0.37 2.910.2-0.3 95 0.08 0.56 0.76 5.03 -0.09 -0.60 0.72 5.110.3-0.4 151 0.02 0.15 0.77 6.10 -0.12 -1.02 0.74 6.340.4-0.5 115 -0.25 -2.25 0.76 6.84 -0.33 -3.23 0.77 7.480.5-0.6 131 -0.30 -2.91 0.86 8.38 -0.32 -3.38 0.86 9.050.6-0.7 145 -0.29 -3.50 0.89 10.92 -0.34 -4.50 0.86 11.570.7-0.8 124 0.05 0.77 0.89 13.01 -0.05 -0.80 0.82 13.270.8-0.9 147 0.40 8.59 0.82 17.71 0.09 2.32 0.66 16.500.9-1.0 266 0.74 27.01 0.92 33.30 0.28 12.49 0.55 24.69

Total 1641

0.19 1.84 0.86 8.29 0.02 0.26 0.76 7.98

23

The outside and inside bark taper equations slightly underestimated the taper near ground

level and at the top of the tree, while they overestimated diameter in between. Still, the

estimated biases were all below one centimetre for both equations. It should be noted that

percent bias and percent standard error can be misleading for the top stem section since

diameter increases as tree size decreases within this section (Kozak and Smith 1993).

Despite this, the overall estimated SEEs and mean biases for the outside and inside

predictions were respectively, 0.86 and 0.76 cm, and 0.19 and 0.02 cm. These results

suggest both equations exhibit good predictive ability. In addition, the equations explain 98

% of the variation in validation data set. These results suggests that the equations are

applicable to black spruce trees from the Yukon Territory.

This high accuracy of the taper equation is due essentially to the explanatory power of both

tree Dbh and tree height (HT) through the ratio Dbh/HT. This ratio accounts for stem taper

differences due to spacing (Garber and Maguire 2003). It is also a good indicator of the live

crown ratio which also affects stem taper (Newnham 1988); specifically, for a given height,

free-growing trees with large crowns will have a grater ratio Dbh/HT than forest-grown

trees with relatively small crowns. Finally, Morris and Forslund (1992) mentioned that for

jack pine, stand and site attributes (competition, climate, microsite) can explain up to 62%

of the variation in stem taper. This appreciable impact of environmental factors on tree

taper can be indirectly addressed by using site index along with Dbh for predicting missing

tree heights in inventory plots.

2.4.2 Branchiness

Table 7 shows that the two independent variables in equations 2 and 3, i.e. interactions of

Dbh with depth into crown and crown length, can explain 79 and 80% of the variation in

the maximum diameter per tree, and the mean branch diameter per tree, respectively.

Figures 4 and 5 illustrate common paraboloid pattern of increasing maximum and mean

branch diameters with crown depth.

24

Table 7: Fit statistics for the maximum and mean branch diameter equations

BDmax BDmean Parameter

estimate SE t Pr > |t| Parameter

estimate SE t Pr > |t|

ln b1 0.7148 0.1136 6.29 0.00 ln c1 0.6863 0.1172 5.86 0.00b2 0.6043 0.0604 10.01 0.00 c2 0.5765 0.0623 9.26 0.00b3 -0.0959 0.0302 -3.17 0.00 c3 -0.0897 0.0312 -2.88 0.01SEE 0.22 SEE 0.22 R2 0.79 R2 0.80 Note: SEE and R2 values are in terms of ln(BD)

0

2

4

6

8

10

12

140 10 20 30 40 50

Maximum branch diameter (mm)

Dep

th in

to c

row

n (m

)

BDmax observed

Figure 4: Observed maximum branch diameter of whorl branches over depth into crown

25

0

2

4

6

8

10

12

140 5 10 15 20 25 30 35 40 45

Average branch diameter (mm)

Dep

th in

to c

row

n (m

)

BDmean observed

Figure 5: Observed average branch diameter of whorl branches over depth into crown

Predicted values (Y-axis) are shown against the observed values (X-axis) in figures 6 and 7.

Used as a reference, the line of equivalence (diagonal) shows that predictive ability of the

equations gets less precise as depth into crown increase. This is due to the fact that

variability into branch diameter increase significantly as depth into crown increase (figure 4

and 5).

26

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40 45 50

BDmax (observed)

BD

max

(pre

dict

ed)

Figure 6: Predicted over observed maximum branch diameter

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40 45

BDmean (observed)

BD

mea

n (p

redi

cted

)

Figure 7: Predicted over observed average branch diameter

27

No direct comparison of the obtained pattern can be done with previous studies as no

information on the black spruce branchiness is available from the literature. However, an

indirect comparison can be made if one assumes that the trend of crown radius or branch

length approximately describes the trend in branch diameter along the stem. This

assumption has been proved by Deleuze et al (1996) and Madgwick et al (1986) on Norway

spruce (Picea abies (L.) Karst.). When accepting this assumption, the obtained paraboloid

pattern concords with the statistical relationship between the crown radius and the crown

depth as observed by Honer (1971) on black spruce and balsam fir. It also agrees with the

statistical relationship between leader extension and branch development observed by

Mitchell (1975) on Douglas fir (Pseudostuga menziesii (Mirb.) Franco).

When comparing the branch diameter distribution along the stem of Douglas fir with those

of Japanese cedar (Cryptomeria japonica (Thunberg ex Linnaeus f.) D.) and Scots Pine

(Pinus sylvestris L.) Maguire et al (1994) observed the presence of a peak in branch

diameter just above the lowest live whorl. This peak in branch diameter occurs somewhere

in the shaded part of the lower crown. However, no apparent separation between light and

shade crown regions exists on black spruce, resulting in the frequent occurrence of this

peak at the crown base. This is due to its narrow crown shape which induces less

competitive pressure from neighbouring trees, thus allowing deeper penetration of light into

the crown.

Apart from tree attributes, we also tested the effect of stand and site attributes on variation

in branchiness. Site index, stand density, stand basal area and stand dominant height were

not significant even when excluding tree attributes from the models. This could be due to

the fact that the sample trees were concentrated on only three intensive study sites covering

a relatively narrow environmental spectrum. In addition, there is no consensus on the

effects of stand and site attributes on branchiness in the literature. For stand density, after

mentioning that maximum branch diameter decreases with increasing stand density,

Maguire et al (1994) concluded that there is relatively little effect of stand density on live

branch diameter, at least beyond that accounted for by tree size. For site index, even

28

Maguire et al (1991) found that site index was negatively correlated with maximum branch

diameter at a given depth into crown. Maguire et al (1994) concluded that it is difficult to

interpret the direct effect of fertilization, and possibly site index, on branch diameter.

However, this conclusion has been based on the narrow gradient of stand and site

conditions as either the considered stands were young, or the sites are limited in terms of

their productivity range.

2.4.3 Height

Like most height-dbh functions found in the literature (e.g., Larsen and Hann 1987, Hann

and Scrivani 1987, Huang et al 1992), a nonlinear model was first applied to black spruce.

However, it did not accurately describe height development. Unlike other species, height

development of black spruce is not asymptotic and is best described by a linear model

(Smith and Watts 1987). Consequently, a linear model was selected.

Because height growth of dominant trees is relatively independent of stand density, it is

often used as a measure of site productivity. Site index, defined as the average height of the

dominant trees in an even-aged stand at a selected base age, is the common expression for

such a productivity measure (Hann and Scrivani 1987). This reflects the fact that trees

growing on high quality sites are generally taller for a given diameter than trees growing on

lower quality sites. Thus site index appears to be a logical variable to be incorporated into

the model in order to explain differences between stands or plots (Dolph and Leroy 1989).

On the other hand, for stands of the same age and site quality, trees growing closer together

should have smaller diameters for a given height than trees growing more widely spaced,

justifying basal area to be introduced into the model. Larsen and Hann (1987) did find that

a tree’s height is strongly correlated with its diameter and that basal area and site index are

often correlated with tree height. However, for black spruce, only diameter and site index

were significant and strongly correlated with tree height as shown in Table 8.

29

Table 8: Parameters estimates and fit statistics for the black spruce height model

Parameter estimate

SE t Pr > |t|

ln d1 -0.5722 0.0280 -20.48 0.00d2 0.5490 0.0062 87.54 0.00d3 0.6291 0.0114 55.28 0.00SEE 0.01 R2 0.78 Note: SEE and R2 values are in terms of ln(HT)

Coefficient of multiple determination (R2) shows that the equation explains 78 % of the

variation of the dependent variable. Tree height could also be predicted based on age (Hann

and Scrivani 1987, Wensel et al 1987) but this explanatory variable was not used, given

that, like height, age was not measured for all trees within a plot and hence could not be

used to predict height. The small overall estimated SEE of 0.01 suggests good predictive

accuracy. Table 9 summarizes the estimated mean biases and SEEs of the height

predictions by Dbh classes, when applying the height equation to the validation data set.

Table 9: Estimated absolute and relative mean bias and SEEs estimates of the height prediction by Dbh classes

Dbh Class (cm)

n

Bias (m)

Bias (%)

SEE (m)

SEE (%)

0-10 228 -0.14 -1.64 1.13 13.2410-15 1980 0.20 1.76 1.32 11.6815-20 1179 0.35 2.43 1.56 10.7820-25 259 0.29 1.72 1.81 10.5625 + 29 -0.77 -4.20 1.98 10.79

Total

3675

-0.23 1.79

1.43 11.36

The highest bias occurs in the Dbh class of 25 cm and greater with amean underestimated

height of –0.77 m. However, the mean bias of –0.23 m is lower than the generally accepted

error of 0.30 m for estimating the total height of coniferous species [include reference]. The

relative estimated SEEs varies little by Dbh class with an overall mean of 11.36%. The fit

30

index computed on validation data was 77 %. Therefore, the resultant prediction equation

was considered acceptable in predicting total height within the inventory sample plots.

31

2.5 Conclusion

Using available data from past studies on black spruce biomass estimation and on crown

morphology and structure, taper and branchiness equations were developed and evaluated.

Two statistical problems that are generally ignored in the literature were considered: (1)

correlation between trees within a plot; and (2) correlation between diameter and branch

measurements obtained on the same tree. Of course, the equation development was limited

by the availability of data reassembled from the past studies whose objectives do not

entirely fit with the purposes of this study. Nevertheless, based on the satisfactory

verification of the equations based on the data from the Yukon Territory, two basic results

were obtained. Firstly, taper equation fitness is more influenced by the resolution of stem

diameter measurements within the butt region of a tree than by the location of the point

where the stem form changes from neiloid to paraboloid shape. Secondly, when applying

the taper equation calibrated with eastern Canadian black spruce data to western Canadian

black spruce (Yukon Territory), the high fitting index of 98% illustrates the robustness of

the obtained equations. Consequently, the equations could be used for sawing optimization

or for validating carbon allocation theories within the context of applying standard

inventory plot data as input data. The obtained branchiness equations describe a logical

paraboloid pattern of the largest and of the mean branch diameter in a whorl using Dbh and

crown length as explanatory variables. The sparse data on branchiness limited the accuracy

of branch diameter models and negated the delineation of the influence of environmental

factors on branchiness variation. Nevertheless, the models provide an initial method for

predicting the branch diameter of mature black spruce trees. Additional branchiness data

sampled across a large spectrum of stand and environmental conditions will be required

before incorporating site index into predictive framework. Total height which is not

measured on all trees in Quebec’s forest inventory plots, but which is essential to model

taper and branchiness profile, was correctly described by a linear model using two

explanatory variables: tree Dbh and site index.

32

3.0 Conclusion générale

Avec l'utilisation de données existantes ayant été récoltées lors d'études antérieures sur la

biomasse de l'épinette noire et sur la morphologie et la structure de la cime, des équations

de défilement et de branchaison ont été développées afin de pouvoir utiliser l'étendue des

données d'inventaire disponibles en tant qu'intrant pratique pour le logiciel d'optimisation

du sciage. Deux problèmes statistiques généralement ignorés dans la littérature ont été

considérés ici: la corrélation entre les arbres d'une même placette et la corrélation entre les

branches d'un même arbre. Évidemment, le développement des équations a été limité par

les données disponibles, rassemblées à partir d'études antérieures dont les objectifs n'étaient

pas toujours entièrement compatibles avec ceux de cette étude. Néanmoins, en se basant sur

la validation satisfaisante qu'on obtient avec les données du Yukon, deux résultats peuvent

être tirés de cette étude. Premièrement, la résolution de l'équation de défilement est

grandement améliorée lorsqu'on utilise une quantité suffisante de mesures de diamètres à la

base de l'arbre, là où est situé le renflement de la tige et où la variation du défilement est

plus prononcée. Deuxièmement, lorsqu'on applique l'équation de défilement calibrée avec

des données d'épinette noire provenant de l'Est du Canada, à l'épinette noire de l'Ouest du

Canada (Yukon), l'indice de précision élevé de 98 % indique la robustesse de l'équation

obtenue. Elle présente ainsi un grand potentiel pour l’optimisation du sciage et la validation

des théories d'allocation de carbone dans un contexte d'utilisation des données d'inventaire

régulières comme intrants. L'équation de branchaison obtenue décrit correctement le patron

paraboloïde logique du plus grand diamètre et du diamètre moyen des branches par

verticille en utilisant la longueur de cime et le diamètre à hauteur de poitrine de l'arbre

comme variables explicatives. Un nombre insuffisant de données de branchaison a limité la

précision de ce modèle et a empêché de vérifier l'impact des variables environnementales

sur la variation du diamètre des branches. En dépit de la rareté des données disponibles de

branchaison chez l'épinette noire, ces modèles fournissent néanmoins une méthode initiale

pour prédire le diamètre des branches d'épinettes noires adultes. Des données de

branchaison additionnelles, récoltées dans un large spectre de conditions

33

environnementales, sont nécessaires pour incorporer la fertilité de la station dans les

prédictions. La hauteur totale, qui n'est pas mesurée sur tous les arbres lors d'inventaires

forestiers réguliers, mais qui est nécessaire pour modéliser le profil de défilement et de

branchaison, est correctement prédite par un modèle linéaire qui utilise deux variables

explicatives: le diamètre de l'arbre et l'indice de qualité de station qui traduit l'effet des

conditions environnementales sur le potentiel de croissance.

34

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