prÉdiction du dÉfilement et de la branchaison de … · 2018-04-12 · equations are useful in...
TRANSCRIPT
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
1
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.
2
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
3
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).
4
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
5
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.
7
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
8
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
9
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
10
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.
11
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
Références Beaumont J.-F., Ung, C.-H. et Bernier-Cardou, M. 1999. Relating site index to ecological
factors in black spruce stands: tests of hypotheses. For. Sci. 45(4): 484-491. Bi, H. and Long, Y. 2001. Flexible taper equation for site-specific management of Pinus
radiata in New South Wales, Australia. For. Ecol. Manage. 148: 79-91. Bonnet, P-A and Pastor, M. 1997. Utilisation de la dimension fractale du houppier pour la
vérification de la loi d'auto-éclaircie. Unpublished. Rapport de stage. Centre de foresterie des Laurentides. 30 p + annexes.
Bonnor, G.M. and Boudewyn, P. 1999. Taper-volume equations for major tree species of the Yukon territory. Pacific Forest Research Centre, Canadian Forestry Service. Information Report BC-X-323, 18 p.
Brown, H and Prescott, R. 1999. Applied mixed models in medicine. Statistics in practice. Chichester; New York: J. Wiley & Sons, 428 p.
Bruce, D., Curtis, R. O. and Vancoevering, C. 1968. Development of a system of taper and volume tables for red alder. For. Sci. 14: 339-350.
Carter, R.E., Miller, I.M. and Klinka, K. 1986. Relationships between growth form and stand density in immature Douglas-fir. Forestry Chronicle 62: 440-445.
Colin, F. and Houllier, F. 1991. Branchiness of Norway spruce in north-eastern France: modelling vertical trends in maximum nodal branch size. Ann. Sci. For. 48: 679-693.
Colin, F. and Houllier, F. 1992. Branchiness of Norway spruce in northeastern France: predicting the main crown characteristics from usual tree measurements. Ann. Sci. For. 49: 511-528.
Colin, F., Houllier, F., Joannes, H. and Haddaoui, A., 1993. Modélisation du profil vertical des diamètres, angles et nombres de branches pour trois provenances d’épicea commun. Silvae Genetica 42 (4-5): 206-222.
Courbet, F. et Albouy, A., 1994. Modélisation dendrométrique de l’architecture du Cèdre de l’Atlas en peuplement. Comptes-rendus du colloque « Architecture des arbres fruitiers et forestiers ». 23-25 novembre 1993. Montpellier, France, Les Colloques, no 74 : 191-207.
Deleuze C. and Houllier F. 1997. A transport model for tree ring width. Silva Fennica 31(3): 239-250.
Deleuze C., Hervé J-C., Colin F., and Ribeyrolles L. 1996. Modelling crown shape of Picea abies: spacing effects. Can. J. For. Res. 26: 1957-1996.
Demaershalk, J.P., and Kozak, A. 1977. The whole-bole system: a conditioned dual-equation system for precise prediction of tree profiles. Can. J. For. Res. 7: 488-497.
Direction des inventaires forestiers. 2000. Normes d’inventaire forestier : les placettes échantillons temporaires, peuplement de 7 m et plus de hauteur. Ministère des Ressources naturelles, 169 p.
Direction des inventaires forestiers. 2001. Normes d’inventaire forestier : placettes échantillons permanentes. Ministère des Ressources naturelles, 233 p.
Dolph, K. and Leroy,1989. Height-diameter equations for young-growth Red fir in California and southern Oregon. USDA Forest Service. Pacific Southwest Forest and Range Experiment Station. Res. Note PSW-408, 4 p.
35
Ecological stratification working group, 1995. A national framework for Canada. Agriculture and agri-food Canada, Research brand, Centre for land and biological resources research and environment Canada. Ottawa.
Garber, S.M. and Maguire, D.A. 2003. Modeling stem taper of three central Oregon species using nonlinear mixed effects models and autoregressive error structures. For. Ecol. Manage. 179: 507-522.
Gregoire, T.G., Schabenberger, O. and J.P. Barret. 1995. Linear modelling of irregularly spaced, unbalanced, longitudinal data from permanent-plot measurements. Can. J. For. Res. 25: 137-156.
Grondin, F., and N. Drouin. 1998. Optitek Sawmill Simulator-User’s Guide. Forintek Canada Corporation, Québec, Canada.
Hann, D.W. and Scrivani, J. A. 1987. Dominant-height-growth and site-index equations for Douglas fir and Ponderosa pine in southwest Oregon. Res. Paper 59. Corvallis, OR: Forest Research Laboratory, Oregon State University. 13p.
Honer, T. G. 1971. Crown shape in open- and forest-grown balsam fir and black spruce. Can. J. For. Res. 1: 203-207.
Houllier, F., Lebanc, J.-M. and Colin, F. 1995. Linking growth modelling to timber quality assessment for Norway spruce. For. Ecol. Manage. 74: 91-102.
Huang, S., Titus, S.J. and Wiens, D.P. 1992. Comparison of nonlinear height-diameter functions for major Alberta tree species. Can. J. For. Res. 22: 1297-1304.
Ikonen, V.-P., Kellomäki, S. and Peltola, H. 2002. Linking tree stem properties of Scots pine (Pinus sylvestris L.) to sawn timber properties through simulated sawing. For. Ecol. Manage. 174: 251-263.
Kozak, A. 1988. A variable-exponent taper equation. Can. J. For. Res. 18 : 1363-1368. Kozak, A. and Smith, J.H.G. 1993. Standards for evaluating taper estimating systems. For.
Chron. 69(4): 438-444. Larsen, D.R. and Hann, D.W. 1987. Height-diameter equations for seventeen tree species in
southwest Oregon. Res. Paper 49. Corvallis, OR: Forest Research Laboratory, Oregon State University. 16 p.
Larson P. R. 1963. Stem form development of forest trees. For. Sci. Monograph 5, 42 p. Lemieux, H. Beaudoin, M. and Zhang, S.Y. 2001. Characterization and modeling of knots
in black spruce (Picea mariana) logs. Wood and Fiber Science 33(3): 465-475. Madgwick, H. A. I., Tamm, C. O. and Mao-Yi, F. 1986. Crown development in young
Picea abies stands. Scand. J. For. Res. 1: 195-204. Maguire, D.M., Moeur, M. and Bennett, W.S. 1990. Simulating branch diameter and
branch distribution in young Douglas fir. Publ.FWS-2-90, Virginia Polytechnic Institute and State University, School of Forestry and Wildlife Resources, Blacksburg, VA, pp.85-94.
Maguire, D.A., Kershaw, Jr., J.A. and Hann, D.W. 1991. Predicting the effects of silvicultural regime on branch size and crown wood core in Douglas-fir. For. Sci., 37:1409-1428.
Maguire, D.A., Moeur, M., Bennett, W.S. 1994. Models describing basal diameter and vertical distribution of primary branches on young Douglas-fir. For. Ecol. Manage. 63: 23-55.
36
Manning, G.H., Massie, M.R.C. and Rudd, J. 1984. Metric single-tree weight tables for the Yukon Territory. Pacific Forest Research Centre, Canadian Forestry Service. Information Report BC-X-250. 60 p.
Max, T and Burkhart, H.E. 1976. Segmented polynomial regression applied to taper equations. For. Sci. 22: 283-289.
Mencuccini M. and Grace J. 1995. Climate influences the leaf area/sapwood area ratio in Scots pine. Tree Physiol. 15: 1-10.
Meredieu, C., Colin, F. and Hervé, J-C. 1998. Modelling branchiness of Corsican pine with mixed effects models. Ann. Sci. For. 55: 359-374.
Miliken, G.A. and Johnson, D.E. 1984. The analysis of messy data. Volume1: Designed experiments. Van Nostrand-Reinhold. New York, 473 pp.
Ministère des ressources naturelles, 1996. L’industrie québécoise des produits du bois-Situation et perspectives d’avenir. Les publications du Québec. Québec, Canada. 225 p.
Ministère des ressources naturelles, 2003. Vegetation zones and bioclimatic domains in Québec. [ http://www.mrnfp.gouv.qc.ca/english/forest/quebec/ quebec-environment-zones.jsp]
Mitchell, K.J. 1975. Dynamcis and simulated yield of Douglas-fir. For. Sci. Monograph 17, 39 p.
Morris, D.M. and Forslund, R.R. 1992. The relative importance of competition, microsite, and climate in controlling the stem taper and profile shape in jack pine. Can. J. For. Res. 22: 1999-2003.
Muhairwe, C.K., LeMay, V.M., and Kozak, A. 1994. Effects of adding tree, stand, and site variables to Kozak’s variable-exponent taper equation. Can. J. For. Res. 24: 252-259.
Newnham, R. M. 1988. A variable-form taper function. Canadian Forest Service, Petawawa National Forest Institute. Inf. Rep. OI-X-83, 33 p.
Newnham, R.M. 1992. Variable-form taper functions for four Alberta tree species. Can. J. For. Res. 22: 210-223.
Newton, P.F. 1997. Stand density management diagram : review of their development and utility in stand-level management planning. Forest Ecology and Management 98: 251-265.
Ouellet, D. 1983. Biomass equations for black spruce in Quebec. Canadian Forest Service. Information Report LAU-X-60E. 27 p.
Perez, D.N., Burkhart, H.E., and Stiff, C.T. 1990. A variable-form taper function for Pinus oocarpa Schiede in central Honduras. For. Sci. 36: 186-191.
Pothier, D. et Savard, F. 1998. Actualisation des tables de production pour les principales espèces forestières du Québec. Ministère des Ressources naturelles-Forêt Québec. Gouvernement du Québec. 183 p.
Robitaille, A., and Saucier, J.-P. 1998. Paysages régionaux du Québec méridional. Les Publications du Québec. 213 p.
Roeh, R.L. and Maguire, D.A., 1997. Crown profile models based on branch attributes in coastal Douglas-fir. For. Ecol. Manage. 96: 77-100.
Ruel, J.-C., Horvath, R. Ung, C.-H., and Munson, A. 2004. Comparing height growth and biomass production of black spruce trees in logged and burned stands. Forest Ecology and Management 193:371-384.
37
Rycabel, T. 2002. Research proposal for linking black spruce wood attributes with stand conditions. Faculté de foresterie et de géomatique. Université Laval. (Unpublished).
SAS Institute Inc., 1997. SAS/STAT Software: Changes and enhancements through release 6.12, SAS Institute Inc., Cary, NC, USA: 1167 pp.
SAS Institute Inc, 1999. SAS/STAT User’s guide, Changes and enhancements to SAS/STAT Software in V7 and V8. SAS Institute Inc, Cary, NC, USA.
Sharma, M. and Oderwald, R. G. 2001. Dimensionally compatible volume and taper equations. Can. J. For. Res. 31: 797-803.
Shinozaki, K., Yoda, K., Hozimu, K. and Kira, T. 1965. A quantitative analysis of plant form – the pipe model theory. I. Basic analysis. Jpn. J. Ecol. 14: 97-105.
Smith, V.G. and Watts, M. 1987. An assesment of the structural method of deriving a black spruce site equation. Can. J. For. Res. 17: 1181-1189.
Thomas, C.E. and Parresol, B.R. 1991. Simple, flexible, trigonometric taper equations. Can. J. For. Res. 21: 1132-1137.
Thornley, J.H.M. 1995. Shoot:root allocation with respect to C, N and P: an investigation and comparison of resistance and telenomic models. Ann. Bot. 75: 391-405.
Ung, C.-H. 1990. Tarifs de cubage paramétrés: application à l'épinette noire de Lebel-sur-Quévillon. Can. J. For. Res. 20: 1471-1478.
Ung, C.-H., and Ouellet, D. 1993. Croissance et branchaison de l'épinette noire Picea mariana (Mill.) B.S.P.. in C.H. Ung (ed.). Les modèles de croissance forestière et leur utilisation, Colloque international tenu les 18 et 19 novembre 1993, Centre municipal des Congrès, Québec (Canada), 45-57.
Wensel, L.C., Walter, J. M. and Biging, G.S. 1987. Tree height and diameter growth models for northern California conifers. Hilgardia, 55(8): 1-20.
West, G.B., Brown, J.H. and Enquist, B.J. 1999. A general model for the structure and allometry of plant vascular systems. Nature 400: 664-667.
Zhang, S.Y. and Chauret, G. 2001. Impact of initial spacing on tree and wood characteristics, product quality and value recovery in black spruce (Picea mariana).. Project Report, Forintek Canada Corp., Sainte-Foy, 47p.
Zhang, S.Y. and Morgenstern, E.K. 1995. Genetic variation and inheritance of wood density in black spruce (Picea mariana) and its relationship with growth: implications for tree breeding. Wood Sci. Technol. 30: 63-75.