Evaluation and Calibration of the CroBas-PipeQual Model for Jack Pine (Pinus banksiana Lamb.) using

Bayesian Melding Hybridization of a process-based forest growth model with

empirical yield curves

Mémoire

Stephanie Ewen

Maîtrise en sciences forestières

Maître ès sciences (M.Sc.)

Québec, Canada © Stephanie Ewen, 2013

iii

Résumé

CroBas-PipeQual a été élaboré pour étudier les effets de croissance des arbres sur la qualité du bois. Ainsi, il

s’agit d’un modèle d’intérêt pour maximiser la valeur des produits extraits des forêts.

Nous avons évalué qualitativement une version de CroBas-PipeQual calibrée pour le pin gris (Pinus banksiana

Lamb.) de façon à vérifier l’intérêt de l’utiliser comme outil de planification forestière. Par la suite, nous avons

fait une analyse de sensibilité et une calibration bayesienne à partir d’une table de production utilisée au

Québec.

Les principales conclusions sont:

1. Les prédictions de hauteur sont les plus sensibles aux intrants et aux paramètres liés à la

photosynthèse;

2. La performance de CroBas est améliorée en tenant compte de la relation observée entre deux

paramètres utilisés pour estimer la productivité nette et l'indice de qualité de station; et

3. CroBas requiert d’autres améliorations avant de pouvoir être utilisé comme outil de planification.

v

Abstract

CroBas-PipeQual is a process-based forest growth model designed to study foliage development and how

growth processes relate to changes in wood quality. As such, CroBas-PipeQual is of interest as a component

model in a forest level decision support model for value assessment.

In this thesis, the version of CroBas-PipeQual calibrated for jack pine (Pinus banksiana Lamb.) in Québec,

Canada was qualitatively evaluated for use in forest management decision-making. Then, sensitivity analyses

and Bayesian melding were used to create and calibrate a stand-level version of CroBas-PipeQual to local

empirical height yield models in a hybrid-modelling approach.

Key findings included:

1. Height predictions were most sensitive to input values and to parameters related to net

photosynthesis;

2. Model performance was improved by varying two net-productivity parameters with site quality; and

3. Model performance needs further improvement before CroBas-PipeQual can be used as a component

of a forest-management decision tool.

vii

Table of Contents

Résumé ............................................................................................................................................................... iii

Abstract ............................................................................................................................................................... v

Table of Contents ............................................................................................................................................... vii

List of Tables ...................................................................................................................................................... ix

List of Figures ..................................................................................................................................................... xi

Foreword ........................................................................................................................................................... xiii

General Introduction ............................................................................................................................................ 1

Article 1. Qualitative assessment of a process-based forest growth model for use in a value-optimization

decision support tool ........................................................................................................................................... 5

1.01 Abstract.............................................................................................................................................. 5

1.02 Résumé ............................................................................................................................................. 7

1.03 Introduction ........................................................................................................................................ 9

1.04 Methods ........................................................................................................................................... 13

1.04.01 PSP Data ................................................................................................................................ 13

1.04.02 PipeQual Model for Jack Pine and Model Forecasts .............................................................. 14

1.04.03 Analyses ................................................................................................................................. 14

1.05 Results and Discussion ................................................................................................................... 17

1.05.01 Model Implementation ............................................................................................................. 17

1.05.02 Stand-level Predictions ........................................................................................................... 18

1.05.03 Tree-level Predictions ............................................................................................................. 21

1.06 Conclusion ....................................................................................................................................... 27

Article 2. Connecting theory to forest decision support tools: hybridization of a process-based productivity

model with empirical height-yield curves ........................................................................................................... 29

2.01 Abstract............................................................................................................................................ 29

2.02 Résumé ........................................................................................................................................... 31

2.03 Introduction ...................................................................................................................................... 33

2.04 Methods ........................................................................................................................................... 37

2.04.01 CroBas Model ......................................................................................................................... 37

2.04.02 Sensitivity Analysis.................................................................................................................. 39

2.04.03 Calibration ............................................................................................................................... 40

2.04.04 Validation ................................................................................................................................ 42

2.05 Results ............................................................................................................................................. 47

2.05.01 Sensitivity Analysis.................................................................................................................. 47

2.05.02 Calibration ............................................................................................................................... 49

2.05.03 Validation ................................................................................................................................ 54

2.06 Discussion ....................................................................................................................................... 57

2.07 Conclusion ....................................................................................................................................... 63

General Conclusion ........................................................................................................................................... 65

References ........................................................................................................................................................ 67

Appendix 1 ........................................................................................................................................................ 75

Appendix 2 ........................................................................................................................................................ 77

Appendix 3 ........................................................................................................................................................ 79

viii

Appendix 4 ........................................................................................................................................................ 81

Appendix 5 ........................................................................................................................................................ 83

Background ................................................................................................................................................... 83

Methods and Results .................................................................................................................................... 83

k ................................................................................................................................................................ 83

an .............................................................................................................................................................. 84

Conclusion .................................................................................................................................................... 84

Appendix 6 ........................................................................................................................................................ 85

ix

List of Tables

Table 1: Summary of Inputs used to run PipeQual for qualitative analyses.* .................................................... 17

Table 2: CroBas parameters and their initial default values. ............................................................................ 38

Table 3: CroBas inputs used to represent the twelve Pothier and Savard (1998) yield curves at 20 years for the

stand height sensitivity analysis. ....................................................................................................................... 40

Table 4: Netdown results of the plot database. ................................................................................................. 43

Table 5: Summary of plot data by measurement (time) period.** ...................................................................... 43

Table 6: Summary of plot PAI data by measurement (time) period.** ............................................................... 44

Table 7: Summary of the Québec PSP data used as inputs to initiate CroBas.* ............................................... 44

Table 8: Results of the sensitivity calculations: the contribution (%) of the parameters and inputs to the overall

uncertainty of the CroBas model outputs. Numbers in bold indicate the parameters that contribute to the top

90% of the variability for that specific output. .................................................................................................... 48

Table 9: P0 and aσ values, by SI, as predicted by the empirical relationships calibrated within the final BYSM

routine. .............................................................................................................................................................. 51

Table 10: Summary of CroBas predictions by measurement (time) period.* ..................................................... 55

Table 11: Summary of CroBas-predicted increment data by measurement (time) period.* .............................. 55

Table 12: Summary of attribute differences (actual – predicted) by measurement (time) period.** .................. 55

Table 13: Summary of attribute increment differences (actual – predicted) by measurement (time) period.* ... 56

xi

List of Figures

Figure 1: Stand-level height (a), SPH (b), G (c) and total volume (d) predictions vs. stand age after running

PipeQual for 100 years, using inputs from 20 Québec PSPs. ........................................................................... 19

Figure 2: Individual tree heights as predicted by CroBas over 100 years for twenty Québec permanent sample

plots. Red, bold lines indicate the stand dominant heights predicted by Pothier and Savard (1998). Black dots

indicate tree measurements used to initiate PipeQual. Blue dots indicate subsequent plot measurements. ... 26

Figure 3: Histograms of CroBas’ outputs from the Monte Carlo Simulation. Histograms correspond to the

predicted height errors at stand ages 50 and 100. ............................................................................................ 48

Figure 4: Prior distribution (red), and posterior distributions (black) of P0 estimates obtained by calibrating each

stand independently. Posterior distributions, in this case, are grouped by SI-class. ........................................ 51

Figure 5: Graphical convergence diagnostic diagrams for the new parameters of the empirical relationship

between P0 and SI (βP0,1 and βP0,2). Trace plots on the left demonstrate chain mixing and stationarity. Density

plots on the right are the posterior distributions of the parameter estimates, and demonstrate a unimodal

distribution of parameter estimates. .................................................................................................................. 52

Figure 6: Graphical convergence diagnostic diagrams for the new parameters of the empirical relationship

between aσ and SI (βaσ,1 and βaσ,2). Trace plots on the left demonstrate chain mixing and stationarity. Density

plots on the right are the posterior distributions of the parameter estimates, and demonstrate a unimodal

distribution of parameter estimates. .................................................................................................................. 53

Figure 7: Graphical representation of CroBas-predicted heights (dashed lines) after updating the model to

reflect BYSM results. Solid lines are the Pothier and Savard (1998) height curves. Colour-codes for the

dashed lines are black = low density, red = med density, and blue = high density. .......................................... 54

xiii

Foreword

For this research, Stephanie Ewen, Frédéric Raulier, and Valerie LeMay took part in the conception and

design of this study. S. Ewen conducted the analytical, simulation and calibration work, and wrote and edited

the manuscript under the supervision of F. Raulier and V. LeMay.

Neither of the two articles included have been submitted for publication, nor published. However, it is the

intent of S. Ewen to submit Article 2 for publication with F. Raulier as a co-author.

1

General Introduction

Growth and yield models are used in forest management to predict forest growth under a variety of forest

interventions. Also, these models are used by researchers and practitioners to examine growth processes.

This information is often used operationally to make forest management decisions. These growth and yield

models can be broadly separated into two types: empirical versus process-based models (Sievänen and Burk

1993; Robinson and Ek 2003; Weiskittel et al. 2011; Burkhart and Tomé 2012; Garcia 2012). Empirical

models may be tree-, stand-, or mixed tree and stand-level models that are created by fitting model sub-

components using available historical data along with data from experiments. Process-based models may be

at the tree- (CroBas; Mäkelä 1997, for example) or forest-level (3-PG; Landsberg and Waring 1997, for

example) and are mathematical representations of the biological processes that contribute to tree or forest

growth. Both types of models have advantages and applications for which they are best suited. As a result, a

number of authors have proposed hybrid models that combine empirical and process models to improve

accuracy of predictions while extending the amplitude of possible applications (Peng et al. 2002; Radtke and

Robinson 2006; Raulier et al. 2003; Robinson and Ek 2003; Valentine and Mäkelä 2005). This general

introduction provides the impetus for the method of creating a hybrid model of a process-based tree-growth

model and empirical yield curves presented in this research.

Empirical yield models, particularly stand-level models, have been extensively used in large-scale planning

applications because they often have been rigorously tested and have been shown to accurately predict stand-

level attributes in applications for which they are designed. These characteristics give users a high level of

confidence that predictions are reasonable and can form the basis of forest management policy or investment

decision. However, these models do not represent biological processes and may not be accurate outside of

the conditions that exist within the data used to fit these models (Sievänen and Burk 1993; Pothier and Savard

1998; Robinson and Ek 2003; Weiskittel et al. 2011; Burkhart and Tomé 2012; Garcia 2012).

Process-based tree- and forest-growth models are commonly used in research settings to study the main

drivers of tree growth, how the processes of tree growth interact, and the expected response of growth to

changes under those processes. These models are developed to comprehensively describe forest and tree

growth processes mathematically with a system of causal relationships and interactions. Parameterization and

calibration of these models has historically been very difficult, and tree-level process-models require inputs not

commonly available from forest inventories. Therefore, they are not commonly used for large-scale planning

(Sievänen and Burk 1993; Mäkelä et al. 2000; Robinson and Ek 2003; Burkhart and Tomé 2012; Garcia 2012).

2

Recently, the Canadian federal government has identified that the current approach to managing public forests

is no longer contributing to an economically sustainable or globally competitive forest industry in Canada (NRC

2012). A federal investment strategy in forest research has been developed to focus on reviving and re-

invigorating the industry to regain value and global competitiveness (NRC 2012). While there is substantial

investment in the research of gaining value from new products and new processing technologies, recognition

has also been given to the potential value to be gained by adjusting the resource supply paradigm (NRC

2012). Historically, technology and policy limitations have dictated that the distribution of Canadian forest

products be primarily resource-driven rather than consumer-driven. Advancements in computing efficiency

and support from government research agencies have provided the environment and motivation necessary to

develop methods for the forest industry to respond efficiently to consumer demands (Jerbi et al. 2012). The

development of modelling frameworks that represent the forest supply chain and support forest management

decision-making that optimizes value within that chain have been the major area of focus for changing the

forest resource distribution paradigm (Jerbi et al. 2012; NRC 2012). Forest growth and yield models have a

role within these modelling frameworks to predict the growth and value production attributes of the forest

landbase (Cloutier n.d.).

ForValueNet is a National Science and Engineering Research Council (NSERC) funded strategic network that

was established in 2008 as one of the initiatives created to focus on value optimization of wood manufacturing

processes, harvesting decisions, and silvicultural activities in the Canadian Boreal Forest by developing an

integrated modelling decision-support system (DSS) that represents the forest-wood supply chain (Cloutier

n.d.; NRC 2012). In the context of a DSS, a growth and yield model is needed to predict volume growth and

stem characteristics that are related to wood quality, end-use and value. The model predictions are expected

to be sensitive to silvicultural and harvesting activities in order to aid forest managers in making investment

decisions that will maximize forest product value.

CroBas is a process-based tree growth model that was developed to provide a comprehensive description of

tree growth and biomass allocation of the average tree within a stand. Specifically, CroBas was designed to

study how those processes contribute to wood quality by predicting the live crown length, total cross-sectional

area of branches, sapwood area at the base of the live crown and shape of an average tree in response to

different stand conditions (Mäkelä 1997; Mäkelä et al. 1997). Subsequently, additional WHORL and BRANCH

modules were added to CroBas (Mäkelä 2002; Mäkelä and Mäkinen 2003), becoming PipeQual. The WHORL

module predicts the vertical structure of the stem and of branches along the stem based on pipe model theory

(Mäkelä 2002). CroBas is modified to act as the growth engine (the TREE module) and runs in parallel with

the WHORL module, each providing feedback to the other. The BRANCH module uses carbon balance and

allocation predictions from both the TREE and WHORL modules to predict the number, size, compass and

3

insertion angles, and self-pruning rates of branches within each whorl through empirical relationships (Mäkelä

and Mäkinen 2003). A further modification included within PipeQual is the ability to model multiple tree size

classes within a single stand (Mäkelä 2002; Mäkelä and Mäkinen 2003). The advantage of using CroBas or

PipeQual (CroBas-PipeQual) within a DSS is its capacity to predict tree characteristics that describe crown

structure, an attribute critical to tree growth and economic value related to wood characteristics. Further,

diverse and changing stand conditions can be simulated and only input variables are needed (Mäkelä 1997;

Schneider et al. 2011b).

As part of a forest-level DSS, CroBas-PipeQual would be used to predict individual-tree shape and size

attributes for forest stands of common coniferous species across Canada’s Boreal Forest to inform harvest

scheduling, silvicultural investment and infrastructure development decisions. To fulfill these requirements,

CroBas-PipeQual must provide reliable forecasts. The primary outputs of interest from CroBas for integration

within a forest-level DSS are stand-level growth and yield attributes including volume per hectare and average

tree size; secondary to stand-level attributes, detailed tree lists can provide information needed to predict value

(Martin Simard, Alexis Achim and Normand Paradis, pers. comm.). The driving objective of the research

presented in this thesis was to evaluate the use of CroBas-PipeQual, particularly for jack pine (Pinus

banksiana Lamb.) since the model has been initially calibrated for this common Canadian Boreal Forest

species. Chapter 1 provides a qualitative review of stand-level attributes predicted with the jack pine version

of PipeQual for use as a large-scale forest planning tool within a forest DSS. The results of Chapter 1

indicated that CroBas-PipeQual requires some improvement to build confidence in its stand-level predictions

over long time periods. Thus, Chapter 2 presents and evaluates an approach that uses Bayesian melding to

create a hybrid model of the simpler 1997 version of CroBas with an existing empirical yield model. The

general conclusion provides a reflection on the overall utility of CroBas-PipeQual for use as a forest planning

tool within a forest DSS, the specific results of this research, and suggested areas of future research.

5

Article 1. Qualitative assessment of a process-

based forest growth model for use in a value-

optimization decision support tool

1.01 Abstract

Within the context of forest supply chain optimization, and to best understand forest development and raw

timber value over time, a forest growth and yield model should be accurate, represent a large variety of

management interventions, and be easily and transparently implemented. Further, tree-level attributes that

relate to wood quality and value must be forecasted, and this is not commonly the case with existing empirical

growth and yield models. PipeQual (Mäkelä and Mäkinen 2003) is a process-based tree-level growth model

that has demonstrated strengths in many of these aspects, particularly in being able to forecast growth under

alternative forest management interventions, and in predicting important tree-level attributes related to quality

such as live crown length, knot size and frequency, and bole shape.

In this research, I applied qualitative methods to assess PipeQual, as calibrated for the common Canadian

Boreal Forest species, jack pine (Pinus banksiana Lamb.), as the potential growth and yield model component

of a supply chain model. Using real data from permanent sample plots (PSPs), PipeQual was used to

simulate 100 years of growth. Stand- and tree-level outputs were graphed and qualitatively assessed, based

on local and biological knowledge of tree growth for the PSPs.

Qualitative analyses identified that the predicted range of basic stand attributes including height, basal area

and volume is outside the range of those actually observed in Québec jack pine stands. Also, the predicted

trajectories of these attributes are not biologically appropriate. Although the version of PipeQual used is still

under development, I did identify areas of research that would increase the applicability of PipeQual in a

supply chain or forest-level decision support model (DSS).

7

1.02 Résumé

Dans le contexte de l’optimisation de la chaîne d’approvisionnement forestier et pour mieux comprendre la

dynamique de croissance en forêt et celle de la valeur du bois comme matière première, un modèle de

croissance et de rendement devrait être précis et capable d’émuler une large variété d’interventions en forêt. Il

devrait être facile et transparent à l’usage. De plus, les attributs à l’échelle de l’arbre qui sont corrélés à la

qualité du bois et à sa valeur doivent pouvoir être fournis, ce qui n’est pas chose fréquente avec les modèles

de croissance et de rendement empiriques. PipeQual (Mäkelä et Mäkinen 2003) est un modèle de croissance

d’arbre basé sur les processus qui a démontré son utilité pour plusieurs de ces aspects, particulièrement

grâce à sa capacité de prévoir la croissance suite à différentes interventions sylvicoles et de fournir des

attributs à l’échelle de l’arbre reliés à la qualité du bois, tels que la longueur de la cime, la fréquence et la taille

des noeuds et le défilement de la tige.

Cette recherche utilise des analyses qualitatives pour évaluer l'application de PipeQual comme modèle de

croissance, une fois calibré pour le pin gris (Pinus banksiana Lamb.), une espèce commune de la forêt boréale

canadienne. À l’aide de données observées dans des placettes-échantillons permanentes (PEP), 100 ans de

croissance ont été simulés avec PipeQual. Les résultats des simulations ont été évalués de façon qualitative à

l’échelle des arbres et des placettes, sur la base de connaissances générales sur la biologie de la croissance

des arbres.

L’analyse qualitative a identifié que les valeurs prédites à l’échelle des placettes pour la hauteur, la surface

terrière et le volume sortent de la gamme de valeurs observées pour le pin gris au Québec. En outre, les

trajectoires simulées par PipeQual pour ces attributs ne sont pas biologiquement conformes. La

paramétrisation actuelle de PipeQual ne permet pas actuellement d’intégrer ce modèle dans un outil

décisionnel, même s’il a des caractéristiques désirables pour estimer la valeur et la qualité des arbres.

9

1.03 Introduction

ForValueNet is a strategic network that was established in 2008 with the intent of broadening the

understanding of value-added manufacturing and intensive forest management within the context of the

Canadian Boreal Forest (Réseau stratégique ForêtValeur 2008). The overall objective of this strategic network

is to develop an integrated decision-support system (DSS) that represents the forest-wood value chain to be

used for the value optimization of wood manufacturing processes, harvesting decisions, and silvicultural

activities (Cloutier n.d.). Within this DSS, the forest growth model is expected to predict volume growth and

stem characteristics used to predict wood quality, end-use and value. The model predictions are expected to

be sensitive to silvicultural and harvesting activities in order to aid forest managers in making decisions that

will maximize forest product value.

PipeQual is a process-based tree growth model that was developed for Scots pine (Pinus sylvestrus L.) in

Finland, that uses functional relationships and carbon balance modelling to describe tree growth and biomass

allocation (Mäkelä and Mäkinen 2003). Stand-level attributes are derived from a combination of cumulative

tree-level attributes and tree mortality predicted with respect to stand conditions. The benefit of PipeQual lies

in its capacity to predict crown structure over time in diverse and changing stand conditions. Crown structure

prediction is critical for predicting tree growth, available wood products and economic value (Schneider et al.

2011b). Additionally, because PipeQual is a process-based model that represents biological growth

processes, as opposed to an empirical model fitted using observations resulting from historic management and

climatic conditions, it may be more accurate under new, innovative forest practices and changes in climatic

conditions. Although currently only calibrated for use with single-species, even-aged stands, given its ability to

predict tree structure and growth in response to inter-tree competition for light resources, PipeQual is expected

to be adaptable to mixed stands. PipeQual and is simpler predecessor, CroBas, have been in use as research

tools in Finland for several years, and have formed the basis of several subsequent modelling endeavours

(Mäkelä 1997; Mäkelä et al. 1997; Mäkelä 2002; Robinson and Ek 2003; Valentine and Mäkelä 2005; Raulier

2006; Dykstra and Monserud 2009; Coll et al. 2011). As a result, users have a sense of reliability and

acceptability, particularly because of the model structure. For these reasons, PipeQual was selected as the

forest growth and yield model to be calibrated for coniferous pure-species stands of Canada’s Boreal Forests,

and integrated into the forest-supply-chain modelling network.

Within the integrated modelling network, PipeQual is expected to be applicable at a variety of scales providing

also a variety of outputs. At the tree-level, attributes of interest include the shape and sizes of every tree in a

stand (Cloutier n.d.). However, the overall goal of the integrated modelling network is to have a series of

national-scale tools that will enhance the global competitiveness of Canadian Boreal Forest products by

10

improving management decision-making with respect to overall forest value (Cloutier n.d.). To date, the tree-

level functional relationships for jack pine (Pinus banksiana Lamb.), including an additional sub-model that

accounts for inter-nodal branching, have been calibrated (Schneider et al. 2011a & 2011b; Goudiaby 2011).

Calibration of these tree-level relationships has been done using data from destructively-sampled trees in

Québec. However, little attention has been paid to ensure stand-level attributes are accurate and realistic

(Robert Schneider, pers.comm.).

Further, some concern does exist with respect to the use of PipeQual as the growth and yield model in a

forest-level DSS. As currently calibrated for jack pine, site quality is a model input, but not for most of the

growth processes. Site index (SI), stand top-height often referenced at 50 years breast height age, is

commonly used to index site productivity in simple-structured stands (i.e., even-aged, and few species). As a

measure of height, SI is closely linked to height and volume growth. As a result, empirical growth and yield

curves for simple-structured stands require site index as an input to predict growth and yield over time.

PipeQual currently does require site index (reference 20 years total age) as an input (Schneider et al. 2011b,

Robert Schneider, pers.comm). However, this is only used to estimate the occurrence of inter-nodal branches,

not growth and yield over time. Another area of concern is that the inputs required to initiate PipeQual are not

commonly available in forest inventories, as with other tree-level growth and yield models.

With these concerns in mind, permanent sample plot (PSP) data from the province of Québec was used to

initiate PipeQual, and then the behaviours and predictions over a 100-yr forecast period were observed.

Although the PSP data used do not cover the entire 100-yr management period (MRNF n.d.), examination of

this 100-yr period was considered to be an essential step in evaluating whether model behaviours were

realistic given first principles of growth. The 100-yr interval may be considered reflective of an approximate

biological rotation period or a reasonable length of time for a strategic management plan. Also, predictions of

state variables are highly dependent on the input state variables over short prediction periods; a 100-yr interval

will provide a better opportunity to observe model behaviour rather than artefacts of model inputs. Qualitative

assessments were used to understand and observe trends that indicate whether growth predictions are

expected to be reasonable over the entire length of a planning horizon.

One of the underlying assumptions of process-based models is that they can be extended to make predictions

under conditions that have not yet been observed in those systems (Sievänen and Burk 1993; Mäkelä et al.

2000; Robinson and Ek 2003; Garcia 2012). If sub-models are assumed to adequately represent the

biological processes occurring, accurate predictions should occur. Qualitative analyses will provide an

indication of whether the general behaviour of the model is biologically appropriate, testing that particular

assumption of the model.

11

Finally, implementing PipeQual using PSP data as a proxy for forest inventory plot data provided insights into

the model’s utility with respect to the ease and transparency of implementation on a large scale over long

planning horizons. Also, the qualitative analyses provided an aid to understanding how well forest inventory

data (here PSP data) might represent the average tree used as the basis of growth in CroBas and PipeQual

(Mäkelä 1997; Mäkelä and Mäkinen 2003). The overall objective of this research in this paper was to use

these qualitative analyses to characterize the applicability and limitations of integrating PipeQual, as calibrated

for jack pine, into a DSS for forest supply chain decision optimization modelling in the province of Québec.

13

1.04 Methods

1.04.01 PSP Data

A province-wide network of PSPs has been established and maintained in the province of Québec by the

Ministry of Natural Resources and Wildlife (MRNF) since 1970 (MRNF n.d.; MRNF 2009). These PSPs are

used to monitor physical, dendrometric, and ecological changes in the forest over time. Plots have been re-

measured on average every 10 years up to five times. Within each 400 m2 plot, species and diameter at breast

height (DBH) were recorded for every tree with a DBH larger than 9 cm. Heights and age (ring count at 1 m)

were measured on one to 13 randomly selected trees. Saplings taller than 1.3 m with a DBH smaller than 9 cm

were counted by DBH-classes of 2 cm within a subplot of 40 m2. Further information on plot sampling can be

found in the field sampling handbook (MRNF n.d.; MRNF 2009).

From the 11,805 PSPs in the database, 156 pure pine PSPs were selected, which were 100% jack pine by

basal area per hectare at the first measurement. Before using the data for initiating PipeQual, the selected

PSP data were explored and any data anomalies were corrected when possible, including correcting tree

numbers that changed between subsequent plot visits.

The 1997 version of CroBas is initiated with and used to predict average-tree- and stand-level attributes based

on a small number of inputs, specifically: average tree height, average tree crown length, and number of trees

per hectare. However, PipeQual allows for multiple size classes as inputs, thereby preventing the need for an

average tree to represent the entire stand (or plot) (Mäkela and Mäkinen 2003). Rather than creating size

classes, each tree or sapling DBH-class represented its own size class; therefore, each tree was forecast

using this version of PipeQual.

To obtain the model initiation inputs, the first measurements of each pine-leading PSP were used. However,

first, any unmeasured heights were imputed using Equation 1 from Pothier and Savard (1998). Although these

height models were not designed for use on trees less than 9 cm at DBH, this was the best available model to

infer the average height of saplings within each DBH class. Crown lengths are not available within the Québec

PSP dataset; the crown length model of Holdaway (1986) was assumed to provide sufficiently accurate

estimates for this variable. This is a potential concern because crown height is used directly within the model

to predict many tree-level attributes and drive growth (Mäkelä 1997). Once any missing heights and crown

lengths were estimated, averages were calculated.

A number of variables were calculated on each PSP to compare to PipeQual outputs. First, DBH and height

provided input to the Québec tree volume models of Perron (2003) to calculate individual tree volumes..

These volumes were summarized to obtain volume per hectare for each measurement of each PSP. Site

14

indices (SIs) and ages are useful in plot classification and are used as inputs to empirical models. Also, a 20-

yr SI value is used in this version of PipeQual to predict inter-nodal branching of jack pine. Stand age and SI

for each PSP were calculated as outlined in Pothier and Savard (1998). Plots that did not have age, site index

or a height for every tree (measured or imputed) were removed from further analyses.

1.04.02 PipeQual Model for Jack Pine and Model Forecasts

The intent was to implement PipeQual, as parameterized for jack pine by Schneider et al. (2011a; 2011b;

Goudiaby 2011), as a “black box” and comment on the results from using the model without alterations:, there

was minimal investigation of the actual fitted models or process flow of the model.

The CAPSIS (computer-aided projection of strategies in silvicuture; a forest-growth modelling platform)

graphical user interface was used as the programming platform for PipeQual for jack pine within ForValueNet

and allows for manual adjustment of the parameter values and batch processing of stands (Dufour-Kowalski et

al. 2012). However, because only jack pine PSPs were forecast, and model had been calibrated for jack pine

(100% by basal area), default parameters were used to manually forecast each PSP. Schneider et al. (2011a;

2011b; Goudiaby 2011) did not use Mäkelä’s (1997) mortality model for the CAPSIS version of PipeQual.

Rather, the mortality model from the local Québec forest growth model, Artémis-2009 (Fortin and Langevin

2010), was used in its place (Robert Schneider pers. comm.).

Using this model, each PSP was forecast for a 100-year period and a text file of the model predictions for each

of the 100 years was output. The outputs included, but were not limited to: tree DBH, tree height, tree

merchantable and total volumes, and number of stems per hectare (SPH) representing the average tree in

each tree size class. Stand-level attributes were derived from a combination of cumulative tree-level

attributes, and tree mortality predicted with respect to stand conditions.

Once each PSP was forecast, the PipeQual model outputs were compared to measured PSP variables.

However, as mentioned, the PSPs had not been measured for 100 years. As a result, the PipeQual model

outputs were also compared to forecasts using the Pothier and Savard (1998) yield models where the same

model inputs were used in the forecast. Although PipeQual’s behaviour or predictions were not expected to be

identical to the Pothier and Savard (1998) yield curves, the behaviour and scale of predictions was expected to

be similar.

1.04.03 Analyses

Difficulties were encountered with many of the pure pine plots, particularly that the raw plot information would

not load into CAPSIS, or PipeQual would not progress forward to grow the plot if more than one tree in the list

of trees input to the model had the same height or crown length (termed “duplicate” heights and crown lengths

15

here; Venceslas Goudiaby, pers.comm.). Therefore, to proceed, plots that had duplicate heights and crown

lengths were removed from the analysis leaving 20 plots.

For the remaining 20 plots that were forecasted using PipeQual, qualitative analyses consisted of graphing

forecasts of:

1. Total stand basal area (G) over average plot (i.e., stand) age;

2. Total stand volume over stand age;

3. Total stand density over stand age;

4. Average stand height over stand age; and

5. Individual tree heights over stand age.

For the graphs showing individual tree heights, the Pothier and Savard (1998)-predicted dominant stand height

curve and also PSP height measurements were overlaid with the PipeQual model predictions. All graphs were

visually assessed for plausible growth behaviours. As noted, trajectories using the same inputs and the

Pothier and Savard (1998) yield models served as a reference for assessing the scale of predictions for all

attributes.

17

1.05 Results and Discussion

1.05.01 Model Implementation

Several important observations regarding the applicability of PipeQual were made in the process of

implementing the model. First, the inability to automate the model poses a barrier to broad-scale application.

Not only is running each stand through the CAPSIS graphical user interface inefficient, this implementation

method is also susceptible to user-induced errors such as mislabeling files or importing incorrect input files

when performed repetitively. As a research tool, the number of graphical displays and outputs available is

incredibly useful, allowing users to test growth behaviour of a stand in different management scenarios.

However, the processing time of each PSP was quite long, often taking 30 minutes or more for one stand to be

projected for 100 years. Presumably, creating a batch version of PipeQual, without the additional graphical

outputs would be faster and more appropriate for use at a large scale.

In order for PipeQual to be applicable to a wide variety of site conditions, some of the parameter values would

have to vary to reflect changing site and climate conditions (Mäkelä et. al. 1997; Raulier 2006). Currently, to

adjust the parameter values, they must be manually adjusted in the graphical user interface. Parameter

changes are not tracked or recorded as an output. Again, this makes the CAPSIS version of PipeQual

susceptible to user-error in entering parameter values incorrectly, or not maintaining a record of parameter

values associated with specific runs of the model for future use or analysis.

These particular implementation impediments are minor logistical issues when considering the research

question of whether PipeQual is appropriate for integration into a DSS. There are several other forest growth

and yield models currently being run within the CAPSIS modelling platform for landscape-level applications (de

Coligny 2011). Presumably, this indicates that the software could be easily developed to bring that same ease

of application to PipeQual.

Table 1: Summary of Inputs used to run PipeQual for qualitative analyses.*

Attribute Mean SD Min Max

SPH 1396 2184 25 9300

DBH (cm) 11.1 3.3 5.4 18.8

Height (m) 7.7 2.4 4.8 14.0

Crown Length (m) 5.8 1.3 3.8 8.2

Age (yrs) 44 26 18 106

* SPH = stems per hectare; DBH = diameter at breast height (1.3 m); SD = standard deviation of the input values; Min = Minimum input value used; Max = maximum input value used.

A more important finding in this research with respect to model implementation was the large number of PSPs

that could not be loaded or forecasted. This finding indicates that the application of this version of PipeQual to

a variety of stand types is very limited. In this research, it could only be applied to 0.2% of all Québec PSPs,

18

and 3% of all pine-leading PSPs (≥ 75% jack pine, by basal area; Pothier and Savard 1998), which is very

restrictive. Table 1 summarizes the inputs used to run PipeQual for the 20 pure-pine PSPs that were

successfully loaded and projected. By using more general stand-averaged forest inventory data (i.e., tree

cohorts instead of individual trees), some of the problems encountered could be avoided. Specifically, the

processing errors associated with “duplicate” input values could be eliminated by either using single stand-

average values or actually creating size classes to allow similarly sized trees to be grouped together.

Unfortunately, using single stand-average values as the inputs does diminish PipeQual’s ability to provide

information on multiple tree size classes. Furthermore, although the basic stand-level CroBas model is based

on the concept of growing the average tree in a stand, the concept is poorly described. Even when multiple

size classes are used, each size class is represented by a theoretical average tree that is not well defined in

the papers describing the model (Mäkelä 1997; Mäkelä et. al. 1997; Mäkelä 2002; Mäkelä and Mäkinen 2003).

In this research, each individual tree measured in the PSP was used as an independent size class rather than

pooling the trees into size classes and calculating the attributes of the average tree representing each size

class. The details of what the average tree is will likely be important when initiating stands that are already

well established, are showing much more stem differentiation than a new plantation, and have limited

information available compared to the data available for PSPs. Basic forest inventory data in Canada is

typically based on the dominant forest canopy layer that is visible aerially for forest interpreters. Smaller trees

are often not included, despite the role they play in stand development and space occupation. Further

research and model testing is required to understand if the exclusion of small trees will change model

predictions unfavourably.

1.05.02 Stand-level Predictions

Figure 1 depicts the stand-level yield predictions of height, density, G, and total volume resulting from running

PipeQual for 100 years, using the inputs calculated from 20 Québec PSPs. Stand age on the x-axis is based

on the average stand age at the initial measurement, calculated as per Pothier and Savard (1998).

Immediately, when observing the graphics of the stand-level attributes, the range of predictions for height, G

and volume appear to be outside the range of values that would be expected in pine stands of the Boreal

Forest in Québec (Figure 1 and Table 1 Pothier and Savard 1998; Tables 1 and 12 Pothier and Auger 2011).

The Boreal Forest is characterized by a cold climate with short growing seasons in which forest growth rates

are relatively slow (Kimball et al. 2000). Some approximate maximum values for dominant stand height, G and

total volume reported for shade-intolerant softwoods (including jack pine) in Québec are 30 m, 35 m2/ha and

325 m3/ha (Pothier and Savard 1998, Pothier and Auger 2011). PipeQual predicted that after 100 years,

several stands would surpass these values; the trend is for these values to be over-predicted.

19

(a)

(b)

(c)

(d)

Figure 1: Stand-level height (a), SPH (b), G (c) and total volume (d) predictions vs. stand age after running PipeQual for 100 years, using inputs from 20 Québec PSPs.

Not only are the heights predicted to be greater than expected, trees are also predicted to continue growing at

much higher rates and for longer time frames than expected. Typically, the height of jack pine stands in the

Boreal Forest is observed to plateau at a stand age between 80 and 100 years, depending on site quality

(Pothier and Savard 1998). Although heights are predicted to plateau in a few stands, in most the heights are

predicted continue to grow well beyond the expected plateau age (Figure 1a). One stand in particular appears

to have dramatic height growth in one year. In this case, the dramatic increase in predicted height is the result

of PipeQual predicting mortality of an entire, relatively short size class.

For the stands that are eventually predicted to plateau in height, that plateau does not necessarily appear to

occur on a biologically relevant time scale. Height curves generally have a sigmoidal shape, related to the

stand establishment, stem exclusion (rapid growth), understory reinitiation (maturation, slowing growth), and

20

old growth (over-mature, stand break-up, little to no growth) stages of stand development (Oliver 1981).

However, the sigmoidal shapes of the height curves generated by PipeQual do not appear to be representative

of those four stages of development. For instance, several of the PipeQual-generated height curves displayed

a lag period of relatively slow growth upon initiation of the model. This would be reasonable if the inputs were

created for a new stand or bare ground still in the stand establishment development stage. However, based

on the starting points of the curves, the input data represent stands that are likely either in the stem exclusion

or understory reinitiation stages already. In general, the height predictions and growth rates produced by

PipeQual do not appear to be biologically appropriate in either behaviour or scale.

Given that CroBas and PipeQual do not model ingrowth of new stems, but do model both natural and density-

dependent mortality, the general declining shape of the stand density curves observed is what might be

expected (Figure 1b). These curves are more difficult to assess qualitatively, particularly for the rate of

mortality and the timing of mortality, than the height growth curves because the actual attribute that the curves

represent is unclear. Trees between 2 and 9 cm in DBH were included in the input estimate of total stand

density without any consideration of crown class or relative size in the stand. In correspondence with this

assumption about the inputs, the stand density predicted by PipeQual likely represents the total stand density

as well. Most stand density information commonly available for mature stands is related to the density of the

merchantable stems or the co-dominant / dominant stratum of trees. Therefore, there is difficulty in forming a

baseline of what is a biologically realistic scale of density and biologically realistic mortality rate. Qualitatively,

there is no initial indication that there is reason to be concerned about the density predictions.

Jack pine is a short-lived, early-successional species that typically exhibits a sigmoidal G yield curve that

reaches a maximum during the understory reinitiation stage and then starts declining as the stand reaches old-

growth and breaks up (Pothier and Savard 1998). Again, the characteristics of the G yield curves are

associated with certain stages of stand development (Alder 1980; Oliver 1981). There are some PipeQual-

predicted G yield curves which do not exhibit this characteristic shape at all; G is predicted to continue growing

well beyond biologically appropriate values (Figure 1c). The curves that do display appropriate overall shape

and scale are, again, in many cases not representing the development stages that the stands are expected to

be in. There are four stands that only reach a maximum G of 4 m2/ha before G decline begins (Figure 1c). In

all four cases, there was only one tree recorded in the PSP at the time of the first measurement. This is likely

due to the minimum size requirement for measurement paired with no small-tree tally data in the PSPs used to

create these inputs rather than a realistic scenario. Despite the sigmoidal shape of these four G yield curves,

the behaviour and scale is again not what is expected for jack pine in Québec. For PSPs with only one tree

available as input, the total stand density without mortality, is 25 stems per hectare. To reach a maximum total

stand G of 4 m2/ha, each tree would have to have a G of 0.16 m2/ha and a DBH of approximately 45 cm. This

21

is not a commonly-observed DBH in pine stands of the Canadian Boreal Forest, so for this value to be

predicted for 20% of the stands modelled in this research is not appropriate (Raulier et al. 2003; Chen et al.

2008). In general, the basal area predictions and growth rates produced by PipeQual do not appear to be

biologically appropriate in either behaviour or scale.

Total stand volume is highly dependant on stand height and total basal area (Alder 1980). Given that there

are concerns about PipeQual’s scale of predictions and prediction behaviour for both stand height and total

basal area, the observation of this same problem in total volume prediction was expected. Again, volume yield

curves have a sigmoidal shape that represents distinct stages of stand development (Alder 1980; Oliver 1981).

In all cases, the initial behaviour of the PipeQual-predicted yield curves indicates that stands are in the stand

establishment stage of development. However, according to the PSP data, all stands used to generate model

inputs should be in the stem exclusion or understory reinitiation stages of development (Figure 1d). Height

and basal-area over-prediction in many stands by the end of the 100-year simulation compounded to create

volumes that are also far over-predicted at the end of the 100-year simulation. The greater concern is the

behaviour of the volume yield predictions generated by PipeQual. Often, in forest management, the optimal

time to harvest a stand is believed to be at the point when the stand reaches maximum mean annual

increment (MAI): the year that the stand experiences maximum volume growth rate. Harvesting at this time

maximizes the productivity of the ground that the stand is growing on (FOPER n.d.). Many strategic timber

supply models will estimate the year of the maximum MAI from the inflection point on the yield curves. Without

appropriate yield prediction behaviour, strategic decisions based on MAI information will be poorly founded.

Thus, PipeQual’s volume predictions and growth rates produced do not appear to be biologically appropriate in

either behaviour or scale.

The results presented above indicate that PipeQual does not adequately predict height, basal area or volume.

However, it should be noted that assumptions were made with respect to interpolating tree heights, and

calculating crown length that confound the results. The errors associated with the model were not explicitly

separated from the errors associated with these assumptions. PipeQual is not advised to be used, in the state

and with the assumptions tested for this research, as a productivity model for jack pine in the province of

Québec and integration into a supply chain modelling network.

1.05.03 Tree-level Predictions

PipeQual’s appeal as a process-based forest-growth model is related to its ability to predict the tree-level

attributes that are indicative of wood quality. Few of the tree-level attributes of interest, such as crown length

or branch size, are available in the PSP dataset to use for analysis. Before assessing PipeQual’s predictive

ability for those attributes, individual tree heights were assessed, as they are more readily available in the

22

database for future quantitative comparison and are more commonly researched and documented. The

results of graphing predicted individual tree heights over time for each plot are given in Figure 2. As with the

stand height, independent of age or existing state when input into PipeQual, most trees were predicted to have

a period of slow growth following model initiation before reaching a predicted period of increasing growth rates

(Figure 2). This behaviour is not what is anticipated based on existing knowledge of tree growth (Assman

1970). PipeQual-predicted tree height of almost all trees surpasses the yield-table predicted dominant stand

height by the end of the 100-yr simulation and do not plateau as expected. Clearly, individual tree heights and

height growth rates are positively biased.

One trend identified in the graphs of individual tree heights over time (Figure 2) that could not be identified by

looking at stand height alone was the observation that in some cases, the canopy position of trees within a

stand becomes inverted (e.g. Figure 2: Plot ID = 7209401902). Although not unique to a single stand type in

this research, this trend is particularly noticeable in stands that have a large number of size classes. For this

predicted behaviour to be true, shorter trees in the stand would need to be more productive than tall trees.

However, because jack pine is a shade-intolerant species, this is not likely.

In some cases, the tallest trees are predicted to have little or no growth (e.g. Figure 2: Plot ID = 9003502601).

In many of those cases, this is what is expected of the trees based on historic observations (Pothier and

Savard 1998). This observation indicates that perhaps there is not much work required to improve the

behaviour of PipeQual for all tree sizes.

23

24

25

26

Figure 2: Individual tree heights as predicted by CroBas over 100 years for twenty Québec permanent sample plots. Red, bold lines indicate the stand dominant heights predicted by Pothier and Savard (1998). Black dots

indicate tree measurements used to initiate PipeQual. Blue dots indicate subsequent plot measurements.

27

1.06 Conclusion

The observations made indicate that PipeQual, as calibrated for jack pine in Québec, is not yet ready to be

incorporated into a large-scale, decision-support model to aid in optimizing forest value. Qualitative analyses

identified that basic stand attributes including height, basal area and volume are being predicted to reach

values outside the range of those actually observed in Québec jack pine stands. Curve trajectories are not as

expected; the shape of most PipeQual-predicted yield curves suggest that stands are in a stand initiation

development phase, when in reality most of the PSPs should be in stands in the stem exclusion, understory

reinitiation or even old growth stage of development.

Before using CroBas or PipeQual to predict quality-related tree-level attributes for coniferous species in the

Boreal Forest of Canada, the stand-level predictions and overall behaviour must be improved. The difficulties

encountered with running PipeQual for many stands and with automating the model runs must also be

investigated to allow for more efficient model implementation. Calibrating PipeQual within the CAPSIS

modelling platform is not suggested as the model is difficult to manipulate efficiently. Further evaluation of

PipeQual through quantitative analyses is not required as qualitative analyses were sufficient to demonstrate

that, as currently calibrated, it is not applicable for integration into a forest supply-chain decision-optimization

modelling network in the province of Québec.

CroBas and PipeQual have strength as process-based models that predict quality-related tree-level attributes.

Although this research has identified, through qualitative analyses, shortcomings in PipeQual’s scale of

predictions and predictive behaviour, if these issues can be ameliorated, CroBas and / or PipeQual may still be

an incredibly useful decision-making tool when the goal is to optimize forest value. To improve stand-level

behaviour and predictions, the simpler tree-growth version of PipeQual, CroBas, could be used to predict only

stand-level attributes based on the development of a single average tree in a more easily-manipulated

modelling platform and re-calibrated to stand-level measurements.

29

Article 2. Connecting theory to forest decision

support tools: hybridization of a process-based

productivity model with empirical height-yield

curves

2.01 Abstract

Process-based tree and forest models are mathematical representations of the biological processes that

contribute to tree and forest growth. They are meant to provide a greater understanding of the main drivers of

tree growth, how the processes of tree growth interact, and the expected response of growth to changes in

those processes, such as climate changes or human intervention. CroBas is particularly designed to study

how those growth processes contribute to wood quality by predicting branching patterns, knot sizes, live crown

length, and tree shape. Process-based models are inherently difficult to use for forest-scale planning and

management decisions. Often, insufficient information exists to parameterize the processes that they

represent. Thus, they are commonly used in a research setting where the data and scenarios are highly

controlled rather than a forest management application with large volumes of forest inventory data, and a

common goal of predicting sustainable volume harvest levels. In this research, we used Bayesian melding to

calibrate the stand-level version of the CroBas model for jack pine (Pinus banksiana Lamb.) in Québec,

Canada using empirical height yield models commonly used in the province. The overall goal was that this

hybrid model could be used for forest-level planning. This objective was met by:

1. Identifying CroBas’ key parameters for height prediction in a sensitivity analysis;

2. Calibrating the key parameters identified in the sensitivity analysis using Bayesian melding of CroBas

and empirical height yield curves; and

3. Testing the re-calibrated version of CroBas against Québec PSP data.

Key findings include:

1. The input values and parameters related to net photosynthesis had the greatest effect on height

predictions;

2. To improve model performance, at least two of the parameters related to net productivity should vary

with site quality: the maximum rate of canopy photosynthesis per unit area (P0), and the reduction of

photosynthetic rate per unit of crown length (aσ); and

3. The model performance still needs improvement before being integrated into a forest-supply-chain-

decision-optimization modeling network as a large-scale growth-prediction tool.

31

2.02 Résumé

Les modèles basés sur les processus sont généralement utilisés en recherche en raison du fait qu’ils restent

difficiles à paramétrer et à utiliser pour la gestion ou la planification forestière. Toutefois, l'hybridation de

modèles basés sur les processus avec des modèles empiriques est une approche qui pourrait permettre de

résoudre cette problématique. Dans cette recherche, CroBas a été hybridé avec les tables de production

forestières utilisées au Québec pour améliorer leur applicabilité à la prise de décision en gestion forestière.

Cet objectif a été atteint en:

1. Identifiant les paramètres les plus sensibles pour prédire la croissance de hauteur, à l’aide d’une

analyse de sensibilité,

2. Utilisant la « fusion bayesienne » pour hybrider CroBas avec les tables de production forestière, et

3. En évaluant les prévisions de hauteur de CroBas avec des mesures faites à l’aide de placettes

échantillon permanentes.

Les conclusions générales sont que:

1. Les prédictions de hauteur sont les plus sensibles aux valeurs des intrants et des paramètres liés à la

photosynthèse;

2. La performance de CroBas est améliorée quand nous incluons les relations entre deux paramètres de

productivité nette et l'indice de qualité de station; et

3. La performance de CroBas a besoin d’être améliorée avant son utilisation à des fins de planification

forestière.

33

2.03 Introduction

Growth and yield models are used in forest management and forest research as a means to gain

understanding about forest growth and the effects of different forms of forest intervention. Often, they are used

to make forest management decisions such as forecasting landscape-level sustainable harvest rates over long

time scales. In research, they are commonly used to study the response of forest stands to specific

interventions. These two general applications of growth and yield models are served by a wide variety of

models that can be broadly divided into two types: empirical versus process-based models (Sievänen and

Burk 1993; Robinson and Ek 2003; Weiskittel et al. 2011; Burkhart and Tomé 2012; Garcia 2012). Empirical

models are those that are created by using statistical methods to fit model sub-components using available

historical data along with data from experiments. Process-based forest models, however, are mathematical

representations of the biological processes that contribute to tree or forest growth. Both category of model has

advantages and applications that they are best suited to. As a result, a number of authors have proposed

hybrid models that combine empirical and process models to improve accuracy of predictions while extending

the amplitude of possible applications (Peng et al. 2002; Radtke and Robinson 2006; Raulier et al. 2003;

Robinson and Ek 2003; Valentine and Mäkelä 2005).

Empirical yield models, particularly stand-level models, have been extensively used in large-scale planning

applications because they often have been rigorously tested and have been shown to accurately predict stand-

level attributes in applications for which they are designed. These characteristics give users a high level of

confidence that predictions are reasonable and can form the basis of forest management policy or investment

decision. However, these models do not represent biological processes and may not be accurate outside of

the conditions that exist within the data used to fit these models (Sievänen and Burk 1993; Pothier and Savard

1998; Robinson and Ek 2003; Weiskittel et al. 2011; Burkhart and Tomé 2012; Garcia 2012).

Process-based tree and forest growth models are commonly used in research settings to study the main

drivers of tree growth, how the processes of tree growth interact, and the expected response of growth to

changes under those processes. These models are developed to comprehensively describe forest and tree

growth processes mathematically with a system of causal relationships and interactions. Some examples of

process-based forest growth models are 3-PG (Landsberg and Waring 1997), JABOWA (Botkin et al. 1972),

FOREST 5 (Robinson and Ek 2003), PipeQual (Mäkelä 2002), FORUG (Verbeeck et al. 2006), and TRIPLEX

(Peng et al. 2002). Parameterization and calibration of these models has historically been very difficult, and

tree-level process-models require inputs not commonly available from forest inventories. Therefore, they are

not commonly used for large-scale planning (Sievänen and Burk 1993; Mäkelä et al. 2000; Robinson and Ek

2003; Burkhart and Tomé 2012; Garcia 2012).

34

PipeQual is a process-based tree-growth model that was developed in Finland for even-aged Scots pine

(Pinus sylvestris L.; Mäkelä and Mäkinen 2003), then re-parameterized for jack pine (Pinus banksiana Lamb.)

in Canada’s Boreal Forest (Goudiaby 2011). PipeQual is specifically designed to study how tree and forest

growth processes contribute to wood quality by predicting branching patterns, knot sizes, live crown length,

and tree shape (Mäkelä and Mäkinen 2003). It was selected as the forest growth model to be used within an

integrated forest decision support system (DSS) for the Boreal Forest of Canada because of its ability to

predict such quality- and value-related attributes (Cloutier n.d.). To date, the allometric, tree-level equations

have been the focus of calibration, and little attention has been paid to ensure stand-level attributes remain

reasonable and realistic (Robert Schneider, pers.comm.). Not surprisingly, the result is that PipeQual, as

calibrated for jack pine, does not sufficiently predict the stand-level attributes of height, basal area (G), density

or volume to be used in a forest planning context (Ewen Chapter 1; Shcherbinina 2012).

CroBas is a simplified version of PipeQual and is itself a process-based carbon allocation model that

calculates net photosynthesis of the average tree in a forest stand, based on its theoretical foliage weight,

photosynthetic capacity and respiratory losses. It then distributes the net assimilated carbon amongst carbon

pools via functional and allometric relationships. Within CroBas, self-pruning and stand mortality are driven by

stand density and crown closure. It does not support multiple size classes, prediction of branching patterns, or

produce any graphical outputs as does the PipeQual version previously evaluated (Ewen Chapter 1;

Shcherbinina 2012). CroBas is of interest for use within a Boreal Forest DSS because of its ability to predict

quality-related tree attributes such as sapwood ratios, live crown length and total branch basal area (Schneider

et al. 2011b).

As mentioned, calibration of process-based models is inherently challenging as a result of the large number of

parameters and correlated processes that often exist for the purpose of comprehensive system portrayal. As

observed with CroBas, process-based models are not typically as reliable or applicable in large-scale forest

management applications as empirical yield tables have been. To address this disparity between the two

model types, there exists a variety of methods to hybridize process-based models with empirical models.

Hybridization is the combination of statistical and mechanistic approaches to obtain parameter values and

model relationships that will yield model-predicted results that are similar in accuracy to those of empirical

models under similar conditions while being based in physiological processes as much as possible (Weiskittel

et al. 2011). The resulting hybridized models share the strengths of both model types in that they are more

reliable quantitatively, and also present a reasonable biological representation of the processes they are

modeled after. Thus, they can potentially be applicable as a management tool in changing forest management

paradigms, or climate conditions (Mäkelä et al. 2000; Robinson and Ek 2003; Radtke and Robinson 2006;

Weiskittel et al. 2011; Burkhart and Tomé 2012).

35

Radtke and Robinson (2006) suggested the use of Bayesian melding (BYSM) to hybridize process-based

forest growth models with empirical forest yield tables. Complex model structure does not inhibit the

application of this particular method, and BYSM can be used to gain a greater understanding of the correlation

structure of such models. One characteristic of BYSM is the capability to include prior information from

numerous sources in the formation of prior parameter distributions for parameter values. This allows

parameter values to be constrained to remain within biologically-relevant limits if enough information about the

parameters is available to do so. The likelihood functions and data-inputs can be modified within the BYSM

framework, once created, to reflect new or different research questions, emphasize different model

characteristics, or incorporate new data when working towards continuous model improvement. Unfortunately,

the trade-offs for these benefits are that BYSM has high computer processing requirements with long

processing times, and analysis outputs are not always useful or may be difficult to interpret. The strength and

weakness of the BYSM results are in the model structure itself, as this method assumes that model structure is

fixed and appropriate, and finds the most likely combination of model parameters under the assumption that

the data was produced from the model being calibrated. Finally, despite the advantage of being able to

produce results for complex models, a combination of correlated parameters, parallel model processes and /

or vague prior distributions due to lack of parameter-specific information may prevent the iterative search from

converging on a single, stable and identifiable combination of parameter values as has been reported in many

attempts to apply statistical techniques to process-based models (Kéry 2010; Bolker 2007). The BYSM

method has been identified as a potentially useful method for calibrating the jack pine height predictions of

CroBas to the jack pine yield curves of Québec (Pothier and Savard 1998). CroBas was chosen over

PipeQual for this calibration because the general growth processes for carbon allocation, coarse tree growth

and the mathematical equations that describe them are unchanged between the two models, and it predicts

the stand-level attributes of interest in a more simplified format than PipeQual.

To avoid the potential risk of not obtaining useful results when using BYSM for the calibration of complex,

process-based models, this research simplifies calibration by first identifying the critical parameters related to

height prediction with a sensitivity analysis. Once identified, literature review provides the values of some

critical parameters that have been well-researched and documented. Remaining critical parameter values will

be estimated using BYSM of CroBas to empirical yield curve outputs.

Height is an attribute that has been extensively researched and documented; height growth is generally

believed to be well-represented by empirical height-age curves (Pothier and Savard 1998; Skovsgaard and

Vanclay 2008; Pothier and Auger 2011; Weiskittel et al. 2011; Burkhart and Tomé 2012), and is the primary

driver of the tree growth and structural description in CroBas as the means for crown length expansion

36

(Mäkelä 1997). Height growth is responsive to and reflective of site and climate conditions; being able to

model this with CroBas in a wide range of site conditions should improve the applicability of CroBas.

The objective of this research was to demonstrate the use of BYSM to improve the utility of CroBas by

calibrating the height prediction of the jack pine stand-level version of the model to the empirical height yield

curves of Pothier and Savard (1998) in the province of Québec. Specifically, this was achieved by:

1. Identifying CroBas’ key parameters for height prediction in a sensitivity analysis;

2. Calibrating the key parameters identified in the sensitivity analysis using Bayesian melding of CroBas

and empirical height yield curves; and

3. Testing the re-calibrated version of CroBas against Québec PSP data.

37

2.04 Methods

2.04.01 CroBas Model

The basic 1997 average tree based stand-level version of CroBas described by Mäkelä was re-written in the R

programming language (The R Core Team 2012) and updated with the jack pine-specific parameter values

obtained through literature review and from personal communication with Robert Schneider(Goudiaby 2011).

Specifically, k (light extinction coefficient), an (specific leaf area), and z (fractal dimension of the crown / 2)

were updated to reflect published information (Table 2). CroBas contains 41 parameters; their default values

and descriptions can be found in Table 2.Osawa (1995) calculated the fractal dimension of the crown of jack

pine to be 2.12, thus, z, was updated to 1.06. The crown fractal dimension is a parameter of the

photosynthetic process, and other parameter values of that process will be conditioned on its value. Chen and

Black (1992) suggested that leaf area index (LAI) reporting be standardized to represent the hemi-surface LAI

of non-flat leaves as opposed to the projected LAI. To update CroBas with a hemi-surface LAI, the measured

projected specific leaf area, an, was converted to a hemi-surface value of 7.74. The light extinction coefficient,

k, was also updated to 0.43 to correspond to the hemi-surface LAI. Appendix 5 outlines the methods used to

obtain these values. There is evidence in the literature that the light extinction coefficient should not remain

constant as the stand develops (Smith et al. 1991; Sampson and Smith 1993; Aubin et al. 2000), but allowing

parameters to vary with time was not implemented.

38

Table 2: CroBas parameters and their initial default values.

Parameter Name

Value Parameter Description Process Source

z 1.06 ½ the fractal dimension of foliage in crown Photosynthesis Osawa 1995

ξ 0.05 Surface area density of foliage Photosynthesis Mäkelä 1997

cb 0.2473 Ratio of crown radius to crown length Photosynthesis Schneider unpubl.

ct 1 Ratio of transport root length to stem length Allocation Mäkelä 1997

αs 0.002263 Sapwood area in stem: foliage weight Allocation Schneider unpubl.

αb 0.003546 Sapwood area in branches: foliage weight Allocation Schneider unpubl.

αt 0.00042 Sapwood area in transport roots: foliage weight Allocation Mäkelä 1997

αr 0.59 Fine root: foliage weight Allocation Schneider unpubl.

φs 1 Form factor of sapwood in stem below crown Allocation Mäkelä 1997

φc 0.75 Form factor of sapwood in stem within crown Allocation Mäkelä 1997

φ’b 0.75 Form factor of sapwood in branches Allocation Mäkelä 1997

φ’t 1 Form factor of sapwood in transport roots Allocation Mäkelä 1997

φb 0.185475 Used to calculate φ’b φt 1 Used to calculate φ’t ρs 421 Wood density of the stem Allocation Schneider unpubl.

ρb 421 Wood density of the branches Allocation Schneider unpubl.

ρt 421 Wood density of the transport roots Allocation Schneider unpubl.

r1 0.2 Specific maintenance respiration rate of foliage and fine roots

Net Growth Mäkelä 1997

r2 0.015 Specific maintenance respiration rate of wood Net Growth Schneider unpubl.

aq 1 Parameter related to self-pruning Self-pruning Schneider unpubl.

q 8 Degree of control by crown coverage of self-pruning Self-pruning Schneider unpubl.

m0 0.001 Specific mortality rate independent of density Mortality Mäkelä 1997

m1 0.01 Density-dependant mortality parameter Mortality Mäkelä 1997

p 5 Degree of control by crown coverage of mortality Mortality Mäkelä 1997

an 7.74 Specific leaf area Photosynthesis Appendix 5

P0 2.4 Maximum rate of canopy photosynthesis per unit area Photosynthesis Mäkelä 1997

k 0.43 Extinction coefficient Photosynthesis Appendix 5

aσ 0.06 Decrease of photosynthesis per unit crown length Photosynthesis Schneider unpubl.

Y 0.65 Carbon use efficiency Net Growth Mäkelä 1997

sf 0.25 Specific senescence rate of foliage Senescence Mäkelä 1997

sr 1 Specific senescence rate of fine roots Senescence Mäkelä 1997

ds0 1 Specific stem sapwood area turnover rate per unit relative pruning

Senescence Mäkelä 1997

db0 1 Specific branch sapwood area turnover rate per unit relative pruning

Senescence Mäkelä 1997

dt0 1 Specific transport root sapwood area turnover rate per unit relative pruning

Senescence Mäkelä 1997

ds1 0.015 Specific stem sapwood area turnover rate in the case of no pruning

Senescence Schneider unpubl.

db1 0.015 Specific branch sapwood area turnover rate in the case of no pruning

Senescence Schneider unpubl.

dt1 0.02 Specific transport root sapwood area turnover rate in the case of no pruning

Senescence Schneider unpubl.

ψs 1 Form factor of senescent sapwood in stem below crown Senescence Mäkelä 1997

ψc 0.5 Form factor of senescent sapwood in stem inside crown Senescence Mäkelä 1997

ψb 1 Form factor of senescent sapwood in branches Senescence Schneider unpubl.

ψt 1 Form factor of senescent sapwood in transport roots Senescence Schneider unpubl.

39

2.04.02 Sensitivity Analysis

Before attempting to use BYSM to calibrate CroBas to the local Québec empirical yield curves, a sensitivity

analysis was used to identify the parameters and inputs that contribute to at least 90% of the variability in

CroBas’ height growth predictions. This served multiple purposes in that key processes were identified within

CroBas for predicting height and the number of parameters to calibrate was reduced. This information also

provided a focus for future research. The goal was that, by reducing the number of parameters of interest for

the BYSM calibration routine, BYSM processing times would be reduced, and the BYSM routine should

converge on single, stable, independent parameter distributions for the parameters of interest (Kéry 2010).

A Monte Carlo simulation approach was used to assess the sensitivity of CroBas’ height predictions to

incremental changes in parameter values and rank parameter importance, as outlined and demonstrated by

Verbeeck et al. (2006). In this procedure, random parameter values are selected from a uniform distribution

covering the range of the default parameter value +/- 10%. These parameters are then used to run a model

simulation, and model outputs can then be used to quantify sensitivity of outputs to incremental changes in

parameter values. The nature of using small increments allows for the variability of the outputs to be related to

the random parameter values through a first-order Taylor Series approximation to quantify sensitivity. The

specific procedure used was as follows:

1. Select parameter values for the Monte Carlo simulation from a uniform distribution in the range of the

default parameter value +/- 10%.1

2. Generate and replicate twelve sets of fixed inputs 1000 times each (Table 3).

a. Heights correspond to those predicted with the Pothier and Savard (1998) height curves for

each of the four site classes, corresponding to site index (SI) values of 9, 12, 15, and 18.

b. Initial densities were arbitrarily chosen using knowledge of Québec stocking standards and

natural regeneration patterns of jack pine (Doucet 2000; MRNFP 2003) to represent the total

stand density, as Pothier and Savard (1998) only provide the density of merchantable stems.

c. Peng et al.’s 2001 Chapman-Richards function was used to predict the diameter at breast

height (DBH) from the stand height. This was used as input, along with height and density,

to calculate crown length using Holdaway’s 1986 relationship.

3. Run the Monte Carlo simulation of CroBas with 12,000 iterations, 1000 iterations per set of inputs,

using the parameter and input values selected above. All stands are projected for 80 years beyond

the starting point to represent growth from age 20 to age 100.

4. For each iteration, extract the predicted heights at 30 and 80 years past the starting point, which

represent the predicted heights at stand ages 50 and 100.

5. Calculate the height prediction error at stand ages 50 and 100 by subtracting the CroBas-predicted

height from the Pothier and Savard (1998) height.

1 Parameters φb and φ’b, and φt and φ’t are related to each other within CroBas, however, that relationship was ignored when generating their random value for this sensitivity analysis.

40

6. Center all inputs used to initiate the simulations by subtracting the mean value of each input from the

individual input values.

7. Center all parameters used in the simulations by subtracting the default value of each parameter from

the individual, randomly selected parameter values.

Table 3: CroBas inputs used to represent the twelve Pothier and Savard (1998) yield curves at 20 years for the stand height sensitivity analysis.

Scenario Initial Height (m) Initial Crown Length (m) Initial Density (SPH)

SI 9 - density low 5.5 3.8 1000

SI 9 - density med 5.5 3.3 2500

SI 9 - density high 5.5 2.7 5000

SI 12 - density low 7.4 5.1 1000

SI 12 - density med 7.4 4.1 2500

SI 12 - density high 7.4 3.3 5000

SI 15 - density low 9.4 6.2 1000

SI 15 - density med 9.4 4.8 2500

SI 15 - density high 9.4 3.8 5000

SI 18 - density low 11.4 7.1 1000

SI 18 - density med 11.4 5.4 2500

SI 18 - density high 11.4 4.2 5000 * SI = site index = dominant stand height at a breast height age of 50 yrs.

8. Perform four multiple linear regressions (MLRs; Verbeeck et al. 2006), using the centred inputs and

parameters of the simulation as the independent variables, and the

a. height at stand age 50;

b. height at stand age 100;

c. height prediction error at stand age 50; and

d. height prediction error at stand age 100 as the dependent variables.

9. Use results of the MLRs to characterize the influence of the inputs and parameters on CroBas’ height

predictions, and to rank the parameters by importance. Specifically, use the coefficient estimates for

the parameters and inputs, and the variance of the predicted response variables to calculated CroBas’

sensitivity to each independent variable as per the methodology discussed in Verbeeck et al. (2006).

10. Identify the inputs and parameters that contribute to 90% of the variability of height predictions as the

parameters of interest for calibration.

2.04.03 Calibration

Radtke and Robinson (2006) proposed a Bayesian method of hybridizing a process-based model with well-

tested and trusted empirical models, using an approach called Bayesian melding. BYSM uses the foundations

of Bayesian statistics to optimize process-based-model predictions using empirical model predictions in lieu of

data (Radke and Robinson 2006).

The empirical yield tables for jack pine in Québec, produced by Pothier and Savard (1998) are based on data

from 3,811 temporary sample plots that cover a range of site and stand conditions. Their annual volume yield

predictions have been independently evaluated using data from the Québec permanent sample plots (PSPs) in

41

which both the average difference between measured and model-estimated values and the root mean square

error (RMSE) were considered reasonable (0.0 m3/ha and 1.8 m3/ha; respectively) (Pothier and Auger 2011).

These empirical growth curves were used in lieu of data for the Bayesian melding procedure. One caveat to

their use is that predictions become unreliable beyond a stand age of 100 years, when stand senescence

begins (Pothier and Savard 1998); thus, these yield models were not used beyond this limit.

The Bayesian melding procedure allows the use of prior information about the parameters to inform the

parameter estimates produced. In most cases, however, there is little information available about the

parameter distributions; in such cases, flat, uninformative prior distributions ensure that posterior distributions

of parameter values are driven by the yield curve information rather than inappropriate priors (Kéry 2010).

When specifying the model in a Bayesian framework, some parameters may be entered as fixed values while

values for others are estimated so as to produce model outputs that maximize the likelihood that the yield

curve data was produced by the model (Kéry 2010). The parameters that were estimated using BYSM were

identified based on results from the sensitivity analysis, preliminary iterations of the BYSM procedure and

literature review. Parameters to which CroBas is insensitive or which have been updated through literature

review were fixed at their current default value (Table 2). Remaining parameters to which CroBas’ height

predictions are sensitive were estimated.

The inputs required for the yield curves included relative density and SI, which are not required inputs for

CroBas. However, CroBas’ predictions could be improved if relationships between those inputs and some

parameter values are established (Raulier 2006). The inclusion of such relationships into CroBas was

explored in the specification of the model in the Bayesian framework.

There were several iterations of the BYSM procedure performed to observe how results varied with different

variations of the method. These variations included calibrating parameter values for:

1. each simulated stand in 12 independent BYSM procedures (Table 3);

2. each SI group of simulated stands in four independent BYSM procedures;

3. all simulated stands in a single BYSM procedure;

4. each SI group of simulated stands in a single BYSM procedure; and

5. new parameters describing the relationship between SI and CroBas’ parameters.

The specific BYSM procedure followed to obtain the final results reported is detailed below:

1. Generate height values for 12 theoretical stands from Pothier and Savard’s (1998) equations without

adding stochastic error, representing a combination of four SI, three relative density classes, and

stand ages 20 to 100.

42

2. Assign total starting densities to each of the 12 theoretical stands, in correspondence to the three

relative density classes as follows: high = 5,000, medium = 2,500, and low = 1,000 stems per hectare

(SPH).

3. Generate uninformative priors for the parameters being calibrated.

4. Set a narrow prior for the standard deviation of height predictions to minimize the difference between

CroBas-predicted height and empirical heights for every year.

5. Run BYSM procedures allowing only the parameters being calibrated to vary, while all other

parameters are fixed to the default values.

a. CroBas inputs are the height and density as calculated in (1 and 2), and the Holdaway

(1986) crown length calculated using Pothier and Savard’s (1998) heights, Peng et al.’s

(2001) corresponding DBH and the arbitrary total densities for the first year (age = 20).

Although there are 12 theoretical stands, there are only four height curves and four DBH

curves. The only inputs that change between stands of the same SI is input density and

crown length.

b. Six Monte Carlo Markov Chain (MCMC) chains of 250,000 iterations and a thinning interval

of five were used. The first 5,000 iterations were dropped as “burn-in” iterations (Kéry

2010).

6. Observe BYSM procedure outputs for convergence and stationarity of the MCMC chains (Kéry 2010).

7. Run the Metropolis-Hastings rejection rate, Geweke stationarity, Gelman convergence, Raftery-Lewis

accuracy, and Heidelburger-Welsh stationarity diagnostics (Appendix C of Ntzoufras 2009) to confirm

that the assumptions of the BYSM have been satisfied and posterior distributions are stable and

stationary.

8. Report and use new parameter values to re-run CroBas using the same inputs as used for the

calibration. Then, visually assess the results.

2.04.04 Validation

The Québec PSP database used to validate CroBas after calibration was last updated on June 10, 2009 and

includes data from 12,409 PSPs across the entire productive forest of the province of Québec. Plots have

been established and re-measured since 1970 by the Ministère des Ressources Naturelles du Québec (MRNF

n.d.). PSPs have been re-measured up to five times with re-measurement intervals varying from 2 to 32

years. The sampling manual for the PSPs (MRNF 2009) outlines in detail the information collected and

available in the database. Specifically, the data used for validation included DBH (available for all trees and

used to calculate G), species, height (not available for all trees), age (not available for all trees or at all time

intervals; averaged for the plot), disturbance and mortality information. Measurements from this database

were used to calculate SI, according to the methods outlined in “Actualisation des tables de production”

(Pothier and Savard 1998).

Only plots that fit the following criteria were used:

1. pure jack pine (≥ 75% jack pine, based on total basal area of trees ≥ 9.0 cm in DBH);

2. sufficient age and height information to calculate SI; and

43

3. at least two subsequent measurements without a major disturbance (Table 4).

Major disturbances were identified as all instances where there was a stand-originating event or disturbance

identified in combination with a negative basal area growth rate (MRNF 2009).

Table 4: Netdown results of the plot database.

Criteria # of plots % of total

Total 12,409 100

Pure jack pine 529 4.3

Dominant height information available 474 3.8

Site Index information available 219 1.8

Measured at least twice without disturbance 160 1.3

Table 5 summarizes the data from the 160 PSPs used in the analysis. Although all plots used had age, height

and SI information available for at least one measurement period, those attributes were not necessarily

populated for every measurement period. Missing dominant heights were not interpolated. Missing ages were

calculated using the earliest record of age for the plot and the cumulative time interval between plot

measurements. A single, average SI value was calculated and used for every measurement period of every

plot using data from each measurement. Basal area (G), stems per hectare (SPH) and volume data were

available for every PSP at every time period. Volume is the estimated gross merchantable volume of

merchantable trees, which is all wood volume above a 15 cm stump height up to a 9 cm merchantable limit

(Perron 2003). Basal area (G) is the total G of merchantable trees, meaning trees that are greater than 9.0 cm

in DBH. Merchantable density also includes all merchantable trees. Total density is the density of all trees

greater than 1.0 cm in DBH: calculated using sapling tally information and merchantable trees.

Table 5: Summary of plot data by measurement (time) period.**

Time n Age (yrs)

(sd) SI (m)

(sd) Ht (m)*

(sd) G (m2/ha)

(sd) SPH (merch)

(sd) SPH (total)

(sd) Volume (m3/ha)

(sd)

1 160 43 18 15.4 3.2 14.0 3.3 15.8 9.3 1054 574 2833 2247 87.7 69.9

2 160 52 18 15.4 3.2 15.1 3.4 17.9 9.6 1080 529 2738 2130 109.5 80.9

3 82 61 17 15.0 3.1 15.7 3.2 18.2 8.7 1029 514 2800 2502 114.0 73.9

4 29 68 17 14.0 2.7 15.2 3.0 17.5 8.1 997 535 3196 2986 101.9 68.3

5 2 74 3 10.9 0.6 12.7 2.6 12.8 12.0 1163 972 5288 972 52.6 56.0

* Not all plots had a dominant height prediction for each measurement, so the sample size (n) does not reflect the sample size used to calculate the average dominant height. Sample sizes for height are: 156, 159, 80, 29, and 2. ** n = sample size; sd = standard deviation of the values measured in the PSPs; SI = site index = dominant stand height at a breast height age of 50 yrs; Ht = height; G = basal area; SPH = stems per hectare.

For each measurement period, periodic annual increments (PAI) of height, G and volume were calculated by

taking the difference between the attribute value at the given measurement period and the first measurement,

then dividing the difference by the cumulative time interval (Table 6).

44

Table 6: Summary of plot PAI data by measurement (time) period.**

Time n Ht PAI* (m/yr)

(sd) G PAI (m2/ha/yr)

(sd) Volume PAI (m3/ha/yr)

(sd)

2 160 0.13 0.13 0.23 0.42 2.40 2.96

3 82 0.11 0.09 0.20 0.25 1.92 1.91

4 29 0.10 0.09 0.22 0.22 1.66 1.36

5 2 0.09 0.10 0.30 0.34 1.34 1.65 * The sample size (n) does not reflect the sample size used to calculate the average dominant height increment. Sample sizes for height increments are: 156, 77, 27, and 2. ** n = sample size; Ht = height; PAI = periodic annual increment; sd = standard deviation of the increment values measured in the PSPs; G = basal area.

For 156 of the plots of interest, data from the first measurement was used to construct the inputs necessary to

run CroBas. The four remaining plots did not have height measurements available at the first measurement

because at that time, trees were not yet large enough to be measured. One of those four plots had only one

height estimate, and, therefore, was removed from this analysis. For the remaining three plots, rather than

estimate an initial height for each plot, CroBas inputs were constructed using the earliest measurement with

height information available (Table 7). The density of the merchantable trees best corresponds to the height

and DBH values used and will be used as the density input value. The average DBH used to construct the

inputs is the arithmetic mean DBH of live, merchantable trees in the plot calculated using real DBH

measurements. This average DBH value was used to estimate crown length, as per Holdaway (1986).

Table 7: Summary of the Québec PSP data used as inputs to initiate CroBas.*

Input Mean SD

SI 15.4 3.2

# yrs to project 17 9

Ht (m) 13.9 3.4

SPH (merchantable) 1067 568

Crown Length 7.9 1.2

* SI = site index = dominant stand height at a breast height age of 50 yrs; Ht = height; SPH = stems per hectare

Using the inputs summarized in Table 7, the re-calibrated stand-level version of CroBas for jack pine was run

for the appropriate number of years necessary to obtain predictions that correspond to the subsequent

measurement periods of the plot. For each measurement period, predicted PAI for height, G and volume were

calculated by taking the difference between the predicted attribute values at the given measurement period

and the initial measurement period, then dividing the difference by the time interval to obtain the annual growth

rate.

Prediction error of CroBas (i.e., the actual (PSP) – predicted (CroBas) differences) was calculated for the

height, DBH, G, density, volume, height PAI, DBH PAI, G PAI and volume PAI of each plot. Note that height

was the primary attribute of interest. Prediction errors were averaged and provided with RMSE by

45

measurement period, despite there being multiple measurements per plot and the differences being correlated.

To further characterize CroBas’ validity, the prediction error of each attribute was graphed over time for each

PSP.

47

2.05 Results

2.05.01 Sensitivity Analysis

Histograms of the CroBas model outputs from the Monte Carlo simulations (height error at stand age 50, and

height error at stand age 100) are provided in Figure 3. The range of prediction errors (Figure 3) confirm that

CroBas is not performing well with height predictions. The majority of the prediction errors are positive,

suggesting that CroBas under-predicts height, relative to Pothier and Savard (1998), for most random

combinations of parameter values and input values simulated for these analyses. Furthermore, the range of

prediction error increases for the heights predicted at a stand age of 100-yrs suggesting that the prediction

error increases over time.

As described in Verbeeck et al. (2006), centered model inputs and parameters were used as the regressors in

linear models to predict the four model outputs described in the Methods section and rank the regressors for

their relative importance to the model output (Table 8). Note that the R2 values ranged from .912 to .985,

indicating that the linear regressions accounted for up to 98.5% of the total variability in the model output,

depending on whether the output was height or height prediction error and the time scale of interest. The

coefficient values from the linear models were used from the analysis to estimate model sensitivity (Table 8) as

per Equation 6 in Verbeeck et al. (2006). For each of the four outputs, the parameters that accounted for the

top 90% of the variability of that specific output were identified (Table 8). Note that for each specific output,

CroBas was sensitive to a slightly different combination of parameters. In all cases, the inputs caused more

variability in the height predictions than the model parameters. The summary information from each of the four

multiple linear regression (MLR) analyses performed using the stats package in R version 2.14.1 (The R Core

Team 2012) is available in Appendix 1 to 4.

48

Figure 3: Histograms of CroBas’ outputs from the Monte Carlo Simulation. Histograms correspond to the predicted height errors at stand ages 50 and 100.

Table 8: Results of the sensitivity calculations: the contribution (%) of the parameters and inputs to the overall uncertainty of the CroBas model outputs. Numbers in bold indicate the parameters that contribute to the top

90% of the variability for that specific output.

Parameter Symbol Description Process Contribution to Overall Uncertainty (%)

Height 50 Height 100 Error 50 Error 100

height.ini Input tree height 64.3 17.0 74.5 75.3

crown.ht Input crown length

9.0 1.5 6.6 0.4

density.ini Input stand density

1.9 26.7 1.3 8.0

P0 P0 Maximum rate of canopy photosynthesis per unit area

Photosynthesis 6.7 14.5 4.9 4.3

An an Specific leaf area Photosynthesis 5.1 11.2 3.6 3.4

K k Extinction coefficient

Photosynthesis 5.1 11.2 3.6 3.3

Y Y Carbon use efficiency

Total Growth 2.7 5.7 1.9 1.7

aSigma aσ Decrease of photosynthesis per unit crown length

Photosynthesis 1.1 2.5 0.8 0.8

Alphar αr Sapwood area:foliage weight ratio in fine roots

Allocation 1.0 2.2 0.7 0.7

49

2.05.02 Calibration

Before starting this research, literature review identified existing information and research supporting updated

hemi-surface estimates of an and k (7.74 m2/kg and 0.43, respectively; Appendix 5; Chen and Black 1992).

These parameters were therefore not calibrated with BYSM.

When αr was added to the BYSM calibration routine, it was estimated to be zero. However, the existence of

fine roots indicates that this parameter should be greater than zero, and that this estimate is not biologically

realistic. Thus, the default value (0.59) was assumed to be more appropriate than zero, and αr was not

calibrated with BYSM.

Because of the close relationship between Y and P0, the two could not be re-parameterized simultaneously

(equation 8; Mäkelä 1997). The literature supporting the default value of 0.65 used by Mäkelä (1997) was

corroborated and the default was not changed. The key parameters remaining to calibrate with BYSM were P0

and aσ.

In most initial iterations of the BYSM calibration, the P0 increased with increasing SI, and had little variability

between density classes within one SI-group. Figure 4 shows the broad prior distribution provided for P0 at the

beginning of the calibration in red, with the narrow posterior distributions of P0 for each stand resulting from the

calibration in black. Note that the posterior distributions were clumped by SI class while the posterior

distributions for each density class within a SI class overlap. Specifically, P0 seemed to increase

approximately exponentially with SI. As a result of these observations, the following empirical equation was

included in the model being calibrated by BYSM:

P0 = βP0,1 + e βP0,2* SI (Equation 1)

Pothier et al. (1988) presented evidence that hydraulic conductivity is higher on poor sites than on rich sites,

but that there is no significant difference in hydraulic conductivity between thinned and control stands. This

reported trend corresponds to the aσ value increasing with SI, and not varying with density class. Therefore, aσ

was constrained to increase with increasing SI only. An empirical relationship between aσ and SI was included

in the calibration routine:

aσ = βaσ,1 + βaσ,2 * SI (Equation 2)

The values of the new parameters used to explain the variability in P0 and aσ with SI are -0.639 +/- 0.0002,

0.031 +/- 0.0000, 0.003 +/- 0.0000 and 0.001 +/- 0.0000 for βP0,1, βP0,2, βaσ,1 and βaσ,2 respectively. These

values were obtained after dropping two out of six chains from the calibration results that did not reach

50

stationarity, and two out of six chains that converged on a set of parameter values (-1.005 and 0 for βP0,1 and

βP0,2, respectively) that produced a single negative P0 value (-0.005 kgcarbon/m2yr) for all SI groups that is not

biologically reasonable. Figure 5 and 5 display the graphical convergence diagnostic diagrams for the new

parameters that resulted from removing four of six MCMC chains. Trace plots on the left demonstrate chain

mixing and stationarity. Density plots on the right are the posterior distributions of the parameter estimates,

and demonstrate a unimodal distribution of parameter estimates. Despite these improvements to the

calibration outputs, the results failed the Raftery-Lewis diagnostic test. Dependence factors were much

greater than five, indicating there may be serial correlation in the MCMC chains, and thus unreliable estimates

of percentiles from the posterior distributions (Appendix C of Ntzoufras 2009). The Raftery-Lewis diagnostic

does not test the convergence of the mean parameter estimate: the characteristic of the posterior distributions

that is of greatest interest in this research. Thus, the parameter value point estimates were expected to be

reasonable. The P0 and aσ values calculated using the new empirical relationships with SI are presented in

Table 9.

51

Figure 4: Prior distribution (red), and posterior distributions (black) of P0 estimates obtained by calibrating each stand independently. Posterior distributions, in this case, are grouped by SI-class.

Table 9: P0 and aσ values, by SI, as predicted by the empirical relationships calibrated within the final BYSM routine.

SI P0 aσ

9 0.683 0.012

12 0.811 0.015

15 0.953 0.018

18 1.108 0.021 * SI = site index = dominant stand height at a breast height age of 50 yrs; P0 = Maximum rate of canopy photosynthesis per unit area; aσ = Decrease of photosynthesis per unit crown length

52

new

.P0.1

new

.P0.2

Figure 5: Graphical convergence diagnostic diagrams for the new parameters of the empirical relationship between P0 and SI (βP0,1 and βP0,2). Trace plots on the left demonstrate chain mixing and stationarity. Density

plots on the right are the posterior distributions of the parameter estimates, and demonstrate a unimodal distribution of parameter estimates.

53

new

.aS

igm

a.1

new

.aS

igm

a.2

Figure 6: Graphical convergence diagnostic diagrams for the new parameters of the empirical relationship between aσ and SI (βaσ,1 and βaσ,2). Trace plots on the left demonstrate chain mixing and stationarity. Density

plots on the right are the posterior distributions of the parameter estimates, and demonstrate a unimodal distribution of parameter estimates.

Figure 7 shows the heights predicted by CroBas for each of the 12 simulated stands (dashed lines) after being

updated to reflect the final results of the BYSM calibration. Solid lines represent the Pothier and Savard

(1998) height curves to which CroBas was calibrated. The range of height predictions was greatly improved

over the observations of Ewen (Chapter 1). However, the CroBas-predicted heights did not reach a plateau as

the Pothier and Savard-predicted (1998) heights did.

54

Figure 7: Graphical representation of CroBas-predicted heights (dashed lines) after updating the model to reflect BYSM results. Solid lines are the Pothier and Savard (1998) height curves. Colour-codes for the dashed lines

are black = low density, red = med density, and blue = high density.

2.05.03 Validation

Table 10 summarizes the CroBas predictions for the 159 PSPs that were used in the validation. The height,

density and total volume are direct outputs from CroBas, while G is calculated from DBH and density. DBH is

not an input into CroBas, and Mäkelä’s 1997 model does not specify how to calculate a starting DBH value.

Therefore, initial DBH was calculated with Peng et al.’s (2001) height-DBH relationship. Subsequent DBH

predictions are the result of cumulating DBH growth values onto this initial value. Sample size was not 159 for

measurement one because not all 159 plots were initiated with data from the first measurement. There were

no missing values in the outputs from CroBas.

55

Density in CroBas is meant to reflect the total stand density (Mäkelä 1997). However, in this case, the density

of merchantable-sized trees was used as the input to be consistent with the other inputs used. Volume

predicted by CroBas is the total stem volume of all trees represented by this stand density.

Table 10: Summary of CroBas predictions by measurement (time) period.*

Time n Height (m) (sd) G (m2/ha) (sd) Density (SPH) (sd) Volume (m3/ha) (sd)

1 156 14.0 3.3 22.1 18.1 1078 560 175.8 163.3

2 158 14.1 3.0 12.7 7.2 665 321 99.5 64.2

3 82 14.2 2.6 12.2 6.2 602 294 100.9 57.5

4 29 13.6 2.4 10.6 5.0 559 307 85.4 45.5

5 2 11.7 0.3 5.1 1.0 288 66 38.1 7.7

* n = sample size; sd = standard deviation of the CroBas-predicted values; G = basal area.

Table 11 summarizes the CroBas-predicted periodic annual increments resulting from running CroBas with the

PSP data as input. Note that the sample sizes in Table 11 are not the same as in Table 10 because

increments are only available for those measurement periods following the initial measurement, and not all

stands are initiated in the first measurement period.

Table 11: Summary of CroBas-predicted increment data by measurement (time) period.*

Time n Ht PAI (m/yr)

(sd) G PAI (m2/ha/yr)

(sd) Volume PAI (m3/ha/yr)

(sd)

2 156 0.02 0.09 -1.15 2.03 -9.32 17.93

3 81 0.04 0.07 -0.28 0.51 -1.78 4.17

4 29 0.05 0.06 -0.07 0.33 -0.21 2.69

5 2 0.06 0.01 0.09 0.02 0.83 0.19

* n = sample size; Ht = height; PAI = periodic annual increment; sd = standard deviation of the CroBas-predicted values; G = basal area.

The outputs summarized in Tables 10 and 11 were compared to the measured PSP data, summarized in

Table 5 and 6, by calculating the average differences and the RMSEs (Table 12 and 13).

Table 12: Summary of attribute differences (actual – predicted) by measurement (time) period.**

Time n Volume (m3/ha)

(RMSE) Height (m)*

(RMSE) G (m2/ha)

(RMSE) DBH (cm)

(RMSE) Density (SPH)

(RMSE)

1 156 -85.9 131.9 0.0 0.0 -5.9 11.7 -1.1 3.5 0 0

2 158 11.1 62.8 1.0 1.5 5.4 9.5 -1.3 3.0 425 630

3 82 13.1 45.5 1.5 2.2 6.0 8.4 -1.1 2.6 426 582

4 29 16.6 42.2 1.6 2.5 6.9 8.8 -1.1 2.7 439 611

5 2 14.5 37.1 1.1 2.3 7.7 10.9 -3.8 3.9 874 1084

* Not all plots had a dominant height prediction for each measurement, so the sample size (n) does not reflect the sample size used to calculate the average height differences. Sample sizes for height are: 156, 158, 80, 29, and 2. ** n = sample size; RMSE = root mean-squared error; G = basal area; DBH = diameter at breast height; SPH = stems per hectare.

56

Table 13: Summary of attribute increment differences (actual – predicted) by measurement (time) period.*

Time n Volume (m3/ha/yr) (RMSE) Height (m/yr)* (RMSE) G (m2/ha/yr) (RMSE) DBH (cm/yr) (RMSE)

2 156 11.8 22.1 0.11 0.16 1.4 2.5 -0.03 0.13

3 81 3.7 5.9 0.07 0.11 0.5 0.7 -0.04 0.13

4 29 2.0 3.5 0.06 0.09 0.3 0.4 -0.05 0.12

5 2 0.5 1.1 0.03 0.07 0.2 0.3 -0.15 0.15

* n = sample size; RMSE = root mean-squared error; G = basal area; DBH = diameter at breast height.

Appendix 6 contains graphs of the individual plot-level differences over time. The most outstanding result from

this validation was the differences between actual and predicted values when CroBas is first initiated.

Height increments were smaller than those measured in the field, but statistical and operational importance of

this difference is yet to be determined.

The range of height predictions was greatly improved over the observations of Ewen (Chapter 1), and having

key parameters vary with SI improved the scale of predictions for larger ecological amplitude.

57

2.06 Discussion

To best understand CroBas’ behaviour, the most interesting outputs from the Monte Carlo simulations

performed in the sensitivity analyses are the predicted heights not the prediction error of the heights.

Generally in forest management of structurally simple stands, as with these jack pine stands, the height at 50

years (the SI of the stand) is an important output. In the sensitivity analysis, however, the height at age 50

was most influenced by the input height at age 20. Over time, the input heights became less important to

CroBas’ height predictions, while model parameters and processes became more important. Therefore,

predicted height at age 50 does not provide as much information about the CroBas processes that dominate

height growth as do the sensitivity analysis results from the predicted height at age 100.

The sensitivity analysis results showed that the net photosynthesis of the tree, carbon-use efficiency, and

below-ground allocation are the dominant processes representing height growth. Parameters related to

photosynthesis are the most important in all cases, and parameters related to net growth above ground

became more important with time. Also of interest is that, while crown length is an important input in the

prediction of height at age 50, it is not at age 100, and the opposite trend is observed for density. This may be

misleading because these two inputs are implicitly linked by using density to compute the inputs for crown

length estimates (Holdaway 1986).

All results from the sensitivity analyses identified the inputs as the most critical values to predicting height; a

characteristic that has been observed in other process-based models (Matala et al. 2003). This reflects that

CroBas is sensitive to the state that the stand is in when the model is implemented and would be sensitive to

interventions that changed the input values. Using the sensitivity rankings from all four analyses, six

parameters were identified as creating the greatest variability in height prediction. These parameters are P0,

an, k, Y, aσ and αr. Of particular interest is that the parameter that was identified as creating the greatest

variability in CroBas’ predicted heights was P0, which represents the maximum rate of canopy photosynthesis

per unit area. P0 is reported to change with both climate and site quality, but not density (Gower et. al. 1993,

Tan and Hogan 1995). Parameter correlation was not accounted for within the sensitivity analysis because

neither correlation structure nor correlation coefficients of the parameter values from various independent

sources were known (Table 2). However, because only small deviations from the default parameter values

were used (maximum 10%) within the sensitivity analysis, we expected that their correlation would not affect

the sensitivity analysis results (Verbeeck et al. 2006).

Correlation was observed between parameters during model calibration when the BYSM search function was

allowed to search a much greater parameter space for all six of the critical parameters identified in the

sensitivity analysis, and these correlations prevented the BYSM routine to converge on a single, identifiable

58

solution for parameter values. This challenge to calibration is a common occurrence when using few inputs to

estimate many parameter values for which several parameter combinations will produce the same model

outputs, referred to as equifinality (Beven and Freer 2001, Duchemin et al. 2008, Jakeman and Hornberger

1993). Characteristically, this observation suggests that CroBas is too complex to be calibrated for or used on

a large spatial scale with such few (3) inputs (Jakeman and Hornberger 1993). One proposed method of

approaching equifinality within a calibration routine includes the addition of further complexity to the process-

based model of interest, describing the inter-relationships of parameters and relationships of those parameters

spatially-variable factors (Duchemin et al. 2008). This approach both changes the model structure, thus

acknowledging the insufficiency of the model itself, and necessitates the use of more inputs, often unavailable

or expensive to acquire. Another solution to the equifinality problem is to fix biologically relevant parameters to

known, measured values to constrain the number of combinations of parameters possible (Duchemin et al.

2008). This restricts the calibration process to find parameter combinations that maintain the biological

relevance of the fixed parameter values, but also limits the results of the calibration to be applicable only in

scenarios where it is reasonable to assume those parameter values and the model structure are correct. To

maintain the objective of this research of testing CroBas as a growth model for use within a DSS, the approach

of fixing some of the six critical parameters to biologically-founded values was used.

Of the six critical parameters identified, an and k were updated based on literature review. Specifically, they

were updated to be consistent with the current reporting standard of hemi-surface leaf area index (LAI) values

(Chen and Black 2002; Appendix 5). There was some indication in the literature that k should vary with LAI as

the stand develops (Smith et al. 1991; Sampson and Smith 1993; Aubin et al. 2000). Including this

functionality into CroBas would require the model structure itself to be altered to include a relationship between

LAI and k, and was outside of the scope of this research. However, further investigation on including this

information may contribute to improving height prediction behaviour in response to the observation in

subsequent calibration and validation routines that CroBas’ predicted height growth did not plateau as

expected. Specifically, for the EcoLEAP program, Raulier et al. (1999; pers. comm.; 0) demonstrated a

negative correlation between k and LAI in jack pine. This type of relationship would allow the specific rate of

carbon assimilation per unit area to be higher in young, open stands that have yet to reach canopy closure and

achieve a constant LAI (equation 27 Mäkelä 1997), thus ameliorating the height growth predictions relative to

what is biologically expected and observed.

Carbon allocation to fine roots, fine root weight and the ratio of fine roots to foliage weight are difficult to

quantify because data collection is time- and labour-intensive (Vanninen and Mäkelä 1999). Existing studies

and information is difficult to interpret because data-collection methods and definitions of fine roots are

inconsistent and identification of fine root species, parent tree and living / dead status can be subjective

59

(Vanninen and Mäkelä 1999). However, the existence of fine roots indicates that this parameter should be

greater than zero, as calibration results indicated. There are some existing theories that this parameter should

vary with site and stand conditions or with stand development over time (Vanninen and Mäkelä 1999).

However, these theories are not quantified with actual field measurements for jack pine. Similar to k, allowing

αr to vary with stand development may improve the overall predictive behaviour of CroBas. More information

about this parameter is required before its value can be improved, and the default value (0.59) was assumed

to be sufficient in the absence of better information. Note that, due to parameter and process correlation within

CroBas, all calibration results are conditional on the default αr value.

According to Pothier et al. (1988), sapwood conductance of jack pine decreases more rapidly with age in

good-quality sites (SI = 20 m, for example) than in poor-quality sites (SI = 9 m, for example). Their observed

trends are congruent with the result of this research that aσ increases with SI. They also suggest that on

stagnant sites, hydraulic conductance may increase with time, which was outside of the range of conditions

simulated for this calibration. One discrepancy between this research and that of Pothier et al. (1988) was that

the BYSM, when performed for each of the 12 simulated stands independently, suggests that aσ varies with

density class, while Pothier et al. (1988) noted no significant difference in conductivity between density

classes. This initially observed trend was removed from the BYSM calibration by only allowing aσ to vary with

SI. The result of the BYSM calibration in this case did produce reasonable diagnostic statistics, and improved

the height predictions of CroBas over having a single aσ for all simulated stands. The trend observed between

aσ and density class may be a real trend, or indicative of poor model representation of density-dependent

processes. Furthermore, in CroBas, aσ directly reduces the photosynthetic capacity relative to the crown

length, and crown length was calculated in all cases using stand density (Holdaway 1986). This may have

contributed to the detection of a relationship between aσ and density. If further research confirms that there is

no true relationship between hydraulic conductivity and density, then perhaps aσ should decrease the

photosynthetic capacity by unit of stem length, which is density independent, rather than crown length. Similar

to the k and αr values, investigation of the change in aσ with stand development over time that was noted by

Pothier et al. (1988) may ameliorate the growth predictions relative to what is biologically expected and

observed.

The BYSM calibration was able to identify and quantify an exponential relationship between P0 and SI. Using

SI to predict P0 did improve the value of the height estimates, but perhaps could be further improved by using

more covariates such as climate information. The identification of parameter relationships to SI highlights that

CroBas is essentially a structural growth model and that the interaction between growth and environmental

constraints is missing (Mäkelä 2003; Weiskittel et al. 2011). Rather than add complexity to CroBas itself by

creating and quantifying relationships between parameter values and environmental variables, its predictive

60

capabilities could be improved in a hierarchical, multi-scale modelling approach. This approach would

integrate the strength and research invested in large-scale modelling of photosynthetic rates or net growth in

response to environmental variables (as in 3-PG for example; Landsberg and Waring 1997; Weiskittel et al.

2011) with the wood-quality and stem characteristic predictive strengths that exist within CroBas.

By putting the empirical equations relating P0 and aσ to SI directly into the calibration process, the parameter

values are constrained to fit those relationships and no further estimation is required for the values of the new

parameters. Constraining the parameter values in this way did not appear to diminish the predictive ability of

CroBas: results obtained were similar to those obtained when CroBas was calibrated for a single P0 and aσ for

each SI-group without any implicit relationship between these values. CroBas would not necessarily need to

include these formulas; rather, they could be used externally to provide the model with site-specific P0 and aσ

values as inputs. Therefore, the process-based nature of the model is maintained. However, as identified by

Radtke and Robinson (2006), the empirical relationships calibrated with the BYSM method remain limited in

application to the range of SI values to which they were calibrated, and the application of CroBas itself is

therefore limited to this range in the absence of information about parameter values outside of this SI range.

Furthermore, although testing the model adequacy was not performed with the BYSM calibration, requiring

parameter values to change in order to reduce the prediction error may suggest that the model form is still

inadequate for the intended application of predicting jack pine growth across the Boreal Forest in Québec

(Radtke and Robinson 2006).

Challenges were encountered when generating model inputs from the PSP data for model validation. Input

values are meant to represent the average tree in the stand, yet the notion itself of the average tree is poorly

defined (Mäkelä 1997). Mäkelä (pers. comm.) has indicated that the average tree should represent the total

population of trees in the stand, yet PSP and inventory data is typically only available for merchantable-sized

trees or trees in the co-dominant canopy layer. In uniform, mature stands, there may not be a large variation

in the definition of an average tree. However, for stands where most trees are below the merchantable limit or

have an unreported component in the intermediate and suppressed canopy layers what the average tree

represents can be unclear. Specifically for this validation, the average stand age was 43 years with an

average SI of 15 for the CroBas inputs generated from PSP data. Within these stands, we expect there is a

potential for a number of trees to exist below the threshold merchantability DBH limit of 9.5 cm, but that

contribute greatly to stand basal area, volume and growing conditions. A clear definition of what the average

tree actually represents, and how to properly interpret the input requirements and predicted values needs to be

understood, especially once complexity is added to the model to include multiple size classes, each

represented by an average tree for the size class.

61

Despite using real field data to initiate CroBas for the validation of the BYSM results, CroBas predicted a much

higher stand volume and basal area than that actually observed in the field. Initially, basal area appears to

possibly be over-predicted as a result of the initial prediction of the average DBH (Peng et al. 2001) being

biased and then scaled up by stand density. This explanation is also applicable to the volume, as initial tree-

level volume is predicted as per Lambert et al.’s (2005) biomass equations which predict stem biomass using

DBH and height. Allometric relationships within CroBas can be used to calculate DBH from the active and

disused pipe area when the crown length and size of trees allow the user to assume that there is no disused

pipe area at model initiation. Currently, there is no way to calculate the disused pipe area at breast height

upon model initiation for trees which are assumed to have this. Thus, the assumption was made to use

relationships from Peng et al. (2001) and Lambert et al. (2005) to generate starting values for DBH and

volume, respectively. This was necessary in the absence of a defined method for obtaining such values in

Mäkelä’s (1997) description of CroBas if the model inputs were to be limited to height, crown length, and

density. In review, the parameters of the relationship between initial DBH and initial height could have been

re-estimated within the BYSM framework to minimize the error attributed to this assumption.

Validation results revealed that predicted basal area and volume growth rates are negative. This is likely the

result of assumed mortality with no in-growth. If the total density (instead of the density of merchantable

stems) was used to run CroBas, the initial annual mortality rate and the rate of decrease in basal area and

volume would be larger than predicted in this research (equation 32, Mäkelä 1997). There are several plots

that lose basal area and volume shortly after being initiated in CroBas. From previous observations of CroBas,

this is the result of large mortality post-initiation if the inputs do not reflect an expected combination of values

within CroBas. Thus, there is a high rate of simulated mortality to allow the model to adjust itself to fit its own

expected inputs. Further research is required for how to best use forest inventory information as input into the

model.

Given that the goal of this research was to calibrate CroBas for height growth, the basal area and volume

differences observed were not thoroughly investigated. Looking to the height curves generated after the

calibration (Figure 7), the lower height growth rates predicted by CroBas over the time frame analyzed in this

validation could have been expected. Generally, the height differences observed are assumed to be the result

of slow initial height growth estimates that do not plateau as much as biologically expected, or as much as

observed in the data used to generate the empirical height yield curves (Pothier and Savard 1998). The scale

of predictions is greatly improved over those observed by Ewen (Chapter 1). As mentioned above, the

behaviour of the height predictions is still not identical to the behaviour of the Pothier and Savard (1998)

predictions, and may be improved by allowing some parameters to vary with stand development.

62

In contrast to these calibration and validation results, Goudiaby (2011) was able to produce PipeQual

predictions that were very similar to Pothier and Savard’s (1998) stand-level predictions for both height and

DBH. The differences between his research and the results presented here could be attributed to several

variables, and the conclusions may indeed not be comparable. Primarily, in his research, PipeQual was

initiated from a bare-ground state for stand with a SI of 18 when compared to Pothier and Savard (1998). This

eliminated the need to rely on as many assumptions as were required to utilize PSP data of various ages, site

conditions and development states. Furthermore, Goudiaby (2011) had parameter values from unpublished

data, adjusted some parameters to site conditions using visual checks and also estimated some parameter

values through trial and error. One goal of this research was to use an automated approach (BYSM) to

estimate parameter values, which didn’t allow the same level of “fine-tuning” as Goudiaby (2011) achieved.

The differences between the results of these two research projects further highlights the need to better

understand how to use forest data to create CroBas inputs, and how site factors influence parameter values.

The BYSM method selected to hybridize CroBas with the outputs of empirical height curves revealed

relationships between biologically-based parameters and SI that are supported by previous research. Despite

such positive outcomes, the BYSM method is not an ideal solution independently for hybridizing process and

empirical models. To have meaningful results, many assumptions were made about what parameter values

remained fixed at default values as many parameter values could not be obtained simultaneously due to

correlation of parameters and overall relationships within CroBas. The BYSM method could be improved

within this application if the correlation structure of the model was identified, and defined within the calibration

routine.

63

2.07 Conclusion

The objective of this research was to demonstrate the use of BYSM to improve the utility of CroBas by

calibrating the height prediction of the jack pine stand-level version of the model to the empirical height yield

curves of Pothier and Savard (1998) in the province of Québec. Specifically, this was sufficiently achieved by:

1. Identifying CroBas’ key parameters for height prediction in a sensitivity analysis;

2. Calibrating the key parameters identified in the sensitivity analysis using Bayesian melding of CroBas

and empirical height yield curves; and

3. Testing the re-calibrated version of CroBas against Québec PSP data.

The result of the sensitivity analysis is to recommend a closer look at the annual photosynthesis and net

growth processes in CroBas and the default parameter values for P0, an, k, Y, aσ and αr. Re-parameterization

of those values may possibly be able to produce more realistic height curves for stands of jack pine in the

province of Québec.

Fine root ratio, αr, remains as a parameter for which there is little information on either its relationship to site

conditions or its expected values. The BYSM method used in this research did not prove to adequately re-

calibrate this difficult and costly parameter to measure in the field. Further research is required to find a more

appropriate method to assign a value to this parameter.

Previous to this research, the version of CroBas calibrated for use within a Boreal Forest DSS was not

explicitly linked to site quality, despite findings that its predictive ability is improved by doing so (Raulier 2006).

The results presented here provide a means to do this by calibrating empirical relationships of P0 and aσ with

SI, thus expanding the range of applicability of CroBas, and providing a focus for future research to improve

this relationship.

Stand height predictions were improved following the BYSM calibration; however, growth rates and stand-level

attribute predictions of other key growth and yield attributes are still poorly represented by CroBas. The

validation results clearly show that CroBas is still not sufficient to be used as a landscape-level forest-planning

tool.

65

General Conclusion

The driving objective of the research presented in this thesis was to evaluate the use of the process-based

forest growth model, CroBas-PipeQual, as the primary forest growth model in a forest supply-chain value

optimization modelling network.

The results of initial qualitative analyses of PipeQual for jack pine indicated that the model, as currently

calibrated, is not yet ready to be incorporated into a large-scale, decision-support model to aid in optimizing

forest value. Basic stand attributes including height, basal area and volume are being predicted to reach

values outside the range of those actually observed in Québec jack pine stands. Prediction behaviour and

model trajectories were not as expected; the shape of most PipeQual-predicted stand yield curves suggested

that stands are in a stand initiation development phase, when in reality most of the permanent sample plots

(PSPs) used to create model inputs for this analysis should be in stands in the stem exclusion, understory re-

initiation or old growth stage of development (Oliver 1981).

Qualitative analyses identified shortcomings in PipeQual’s scale of predictions and predictive behaviour. To

ameliorate these observed issues, the basic carbon allocation module of PipeQual, CroBas, was used to

predict only stand-level attributes, based on the development of a single average tree, in a more easily-

manipulated modelling platform. This version of CroBas for jack pine was re-calibrated for height predictions

by:

1. Identifying CroBas’ key parameters for height prediction in a sensitivity analysis; and

2. Calibrating the key parameters identified in the sensitivity analysis using Bayesian melding (BYSM) of

CroBas and empirical height yield curves for Québec.

The result of the sensitivity analysis is to recommend a closer look at the annual photosynthesis and net

growth processes in CroBas and the default parameter values for P0, an, k, Y, aσ and αr. Re-parameterization

of those values may possibly produce more realistic height curves for stands of jack pine in the province of

Québec.

Fine root ratio, αr, remains as a parameter for which there is little information on either its relationship to site

conditions or its expected values. The BYSM method used in this research did not prove to adequately re-

calibrate this difficult and costly parameter to measure in the field. Further research is required to find a more

appropriate method to assign a value to this parameter.

66

A literature review of parameter values indicated that some parameter values should be changing as stands

develop over time. Further research is required to determine how this knowledge can be included within

CroBas. Doing so may improve the shape of yield curves and behaviour of growth prediction.

Challenges were encountered when generating model inputs. Input values are meant to represent the

average tree in the stand, yet the average tree is poorly defined (Mäkelä 1997). A clear definition of what the

average tree actually represents, and how to properly interpret the input requirements and predicted values

needs to be understood, especially once complexity is added to the model to include multiple size classes,

each represented by an average tree for the size class.

Previous to this research, the version of CroBas-PipeQual calibrated for use in a Boreal Forest DSS was not

explicitly linked to site quality, despite findings that its predictive ability is improved by doing so (Raulier 2006).

The results of the calibration provide a means to do this by parameterizing empirical relationships of P0 and aσ

with SI, thus expanding the range of applicability of CroBas. Other site-related covariates, such as climate

variables, may further improve this estimation of these model parameters. Alternatively, forest growth, yield

and quality modelling could be improved overall by combining CroBas with other models that are more suited

to modelling net productivity, such as 3-PG (Landsberg and Waring 1997; Robinson and Ek 2003).

The use of a sensitivity analysis to identify key model parameters to height prediction was an effective

approach to simplify the calibration process. Although the BYSM technique, in this case, was useful for

hybridizing CroBas with empirical yield curves, it may not be the ultimate solution to the calibration of process-

based models in the absence of information on parameter values. As identified in this research, BYSM is

unable to validate the overall structure of the model, and maximizes the likelihood of the data with the

assumption that the model structure is correct. Also, if there is not enough information available to refine prior

distributions of priors or there are correlated processes and parameters, there may not be one single,

identifiable combination of parameters that maximizes the likelihood function.

Stand height predictions were improved following the BYSM calibration; however, growth rates and stand-level

attribute predictions of other key growth and yield attributes were still poorly represented by CroBas.

Qualitative analysis following calibration clearly shows that CroBas is still not adequate to be used as a

landscape-level forest-planning tool.

67

References

Alder, D. 1980. FAO Forestry paper: Forest volume estimation and yield prediction (vol.2). Rome:

Food and Agriculture Organization of the United Nations. 196 pp.

Assman, E. 1970. The principles of forest yield study: Studies of the organic production, structure,

increment and yield of forest stands. (English translation). Oxford, New York, Toronto,

Sydney, Braunschweig: Pergamon Press. 506 pp.

Aubin, I., Beaudet, M., and Messier, C. 2000. Light extinction coefficients specific to the understory

vegetation of the southern Boreal Forest, Québec. Canadian Journal of Forest Research.

30: 168-177.

Beven, K. and Freer, J. 2001. Equifinality, data assimilation, and uncertainty estimation in

mechanistic modelling of complex environmental systems using the GLUE methodology. J.

of Hydrology. 249: 11-29.

Bolker, B. 2007. Ecological models and data in R. Princeton & Oxford: Princeton University Press.

Bond-Lamberty, B., Wang, C., and Gower, S.T. 2003. The use of multiple measurement techniques

to refine estimates of conifer needle geometry. Canadian Journal of Forest Research. 33:

101-105.

Botkin, D.B., Janak, J.F., and Wallis, J.R. 1972. Some ecological consequences of a computer

model of forest growth. Journal of Ecology. 60(3): 849-872.

Burkhart, H.E., and Tomé, M. 2012. Modeling of forest trees and stands. Dordrecht: Springer

Science+Business Media. 457 pp.

Chen, J.M., and Black, T.A. 1992. Defining leaf area index for non-flat leaves. Plant, Cell &

Environment. 15(4): 421-429.

Chen, H.Y.H., Fu, S., Monserud, R.A., and Gillies, I.C. 2008. Relative size and stand age determine

Pinus banksiana mortality. Forest Ecology and Management. 255: 3980-3984.

Cloutier, A. n.d. ForValueNet: Development of integrated forest management and wood

manufacturing decision support systems for a value-added forest industry. Unpublished

research proposal. 56 pp.

Coll, L., Schneider, R., Berninger, F., Domenicano, S., and Messier, C. 2011. Quantifying the effect

of nitrogen-induced physiological and structural changes on poplar growth using a carbon-

balance model. Tree Physiology. 31: 381-390.

68

de Coligny, F. 2011. CAPSIS: Computer-aided projection of strategies in silviculture.

http://capsis.cirad.fr/capsis/home [Accessed Nov 28, 2011]

Coops, N.C., Waring, R.H., and Landsberg, J.J. 1998. Assessing forest productivity in Australia and

New Zealand using a physiologically-based model driven with averaged monthly weather

data and satellite-derived estimates of canopy photosynthetic capacity. Forest Ecology and

Management. 104: 113-127.

Doucet, R. 2000. Note de recherche forestierè no. 100: Régénération du pin gris au moyen des

semences contenues dans les déchets de coupe. Direction le la recherche forestiers, Forêt

Québec, Ministère des Ressources Naturelles. 6 pp.

Duchemin, B., Maisongrande, P., Boulet, G. and Benhadj, I. 2008. A simple algorithm for yield

estimates: Evaluation for semi-arid irrigated winter wheat monitored with green leaf area

index. Environmental Modelling & Software. 23: 876-892.

Dufour-Kowalski, S., Courbaud, B., Dreyfus, P., Meredieu, C., and de Coligny, F. 2012. Capsis: an

open software framework and community for forest growth modelling. Annals of Forest

Science. 69:221–233.

Dykstra, D.P. and Monserud, R.A. (eds). 2009. Forest growth and timber quality: Crown models and

simulation methods for sustainable forest management. Proceedings of an International

Conference, August 7-10, 2007, Portland, OR. U.S. Department of Agriculture, Forest

Service. General Technical Report PNW-GTR-791. 282 pp.

FOPER. n.d. Optimal Rotation (Part II). http://foper.unu.edu/course/?page_id=166 [accessed Jan

4, 2013]

Fortin, M., and Langevin, L. 2010. Artémis-2009: un modèle de croissance basé sur une approche

par tiges individuelles pour les forêts du Québec. Gouvernement du Québec, Ministère des

Ressources naturelles et de la Faune, Direction de la recherche forestière. 68 pp.

Garcia, O. 2012. IUFRO 4.01.05 wiki: Knowledge base: Stand modelling.

https://sites.google.com/site/iufro40105/knowledge-base/stand-modelling. [accessed Jan 2,

2012]

Girardin, M.-P., Raulier, F., Bernier, P.Y., and Tardif, J.C. 2008. Response of tree growth to a

changing climate in boreal central Canada: A comparison of empirical, process-based, and

hybrid modelling approaches. Ecological Modelling. 213: 209-228.

Goudiaby, V.C.A. 2011. Réponse du pin gris et de l’épinette noire à l’éclaircie commerciale en forêt

boréale. Thèse du doctorat en sciences de l’environnement. Université du Québec en

Abitibi-Témiscaminque, Rouyn-Noranda, QC.

69

Goudiaby, V., Brais, S., Grenier, Y., and Berninger, F. 2011. Thinning effects on jack pine and black

spruce photosynthesis in eastern Boreal Forests of Canada. Silva Fennica. 45(4): 595-609.

Gower, S.T., Reich, P.B., and Son, Y. 1993. Canopy dynamics and aboveground production of five

tree species with different leaf longevities. Tree Physiology. 12: 327-345.

Gower, S.T., Vogel, J.G., Norman, J.M., Kucharik, C.J., Steele, S.J., and Stow, T.K. 1997. Carbon

distribution and aboveground net primary production in aspen, jack pine, and black spruce

stands in Saskatchewan and Manitoba. Canada Journal of Geophysical Research. 102: 29-

41.

Hall, R.J., Raulier, F., Price, D.T., Arsenault, E., Bernier, P.Y., Case, B.S., and Guo, X. 2006.

Integrating remote sensing and climate data with process-based models to map forest

productivity within West-Central Alberta's Boreal Forest: Ecoleap-West. Forestry Chronicle.

82(2): 159-176.

Holdaway, M. 1986. Modelling tree crown ratio. The Forestry Chronicle. 62(5): 451-455.

Jakeman, A.J. and Hornberger, G.M. 1993. How much complexity is warranted in a rainfall-runoff

model? Water Resources Research. 29(8): 2637-2649.

Jerbi, W., Gaudreault, J., D’Amours, S., Nourelfath, M., Lemieux, S., Marier, P., and Bouchard, M.

2012. Optimization/simulation-based framework for the evaluation of supply chain

management policies in the forest products industry. Proceedings of the 2012 IEEE

International Conference on Systems, Man, and Cybernetics. October 14-17, 2012. Seoul,

Korea.

Kéry, M. 2010. Introduction to WinBUGS for ecologists. Burlington, MA: Elsevier Inc.

Kimball, J.S., Keyser, A.R., Running, S.W., and Saatchi, S.S. 2000. Regional assessment of Boreal

Forest productivity using an ecological process model and remote sensing parameter maps.

Tree Physiology. 20: 761-775.

Lambert, M.-C., Ung, C.-H., and Raulier, F. 2005. Canadian national aboveground biomass

equations. Can J. For. Res. 35(8): 1996-2018.

Landsberg, J.J., and Waring, R.H., 1997. A generalized model of forest productivity using simplified

concepts of radiation-use efficiency, carbon balance and partitioning. Forest Ecology and

Management. 95(3): 209-228.

Mäkelä, A. 1997. A carbon balance model of growth and self-pruning in trees based on structural

relationships. Forest Science. 43(1): 7-24.

70

Mäkelä, A., Vanninen, P., and Ikonen, V.-P. 1997. An application of process-based modelling to the

development of branchiness in Scots pine. Silva Fennica. 31(3): 369-380.

Mäkelä, A., Landsberg, J., Ek, A.R., Burk, T.E., Ter-Mikaelian, M., Ågren, G.I., Oliver, C.D., and

Puttonen, P. 2000. Process-based models for forest ecosystem management: current state

of the art and challenges for practical implementation. Tree Physiology. 20: 289-298.

Mäkelä, A. 2002. Derivation of stem taper from the pipe theory in a carbon balance framework.

Tree Physiology. 22: 891-905.

Mäkelä, A. 2003. Process-based modelling of tree and stand growth: towards a hierarchical

treatment of multi-scale processes. Can. J. For. Res. 33: 398-409.

Mäkelä, A. and Mäkinen, H. 2003. Generating 3D sawlogs with a process-based growth model.

Forest Ecology and Management. 184: 337-354.

Matala, J., Hynynen, J., Miina, J., Ojansuu, R., Peltola, H., Sievänen, R., Väisänen, H. and Kellomäki,

S. 2003. Comparison of a physiological model and a statistical model for prediction of

growth and yield in boreal forests. Ecological Modelling. 161: 95-116.

MRNF. n.d. Les placettes-échantillons. Ministère des Ressources naturelles et Faune, Québec.

http://www.mrnf.gouv.qc.ca/forets/connaissances/connaissances-inventaire-placettes.jsp

[Accessed Nov 28, 2011]

MRNF. 2009. Normes d’inventaire forestier: placettes-échantillons permanents: Édition 2009.

Gouvernement du Québec, Ministère des Ressources naturelles et de la Faune, Direction

des inventaires forestiers, Forêt Québec. 255 pp.

MRNFP. 2003. Manuel d'aménagement forestier, 4ème édition. Dir. Prog. For., Min. Ress. Nat.

Faune Parcs, Gouv. Québec, Québec.

NRC. 2009. The atlas of Canada: Land. Natural Resources Canada.

http://atlas.nrcan.gc.ca/auth/english/maps/environment/land [Accessed Sept, 2012]

NRC. 2012. Squeezing more value from trees. Natural Resources Canada.

http://cfs.nrcan.gc.ca/pages/182 [Accessed Jan 15, 2013]

NRC. 2013. The state of Canada’s forests: Annual report 2012. Canada: Her Majesty the Queen in

Right of Canada. 56 pp.

Ntzoufras, I. 2009. Bayesian modeling using WinBUGS. Hoboken, New Jersey: John Wiley & Sons.

Oliver, C.D. 1981. Forest development in North America following major disturbances. Forest

Ecology and Management. 3: 153-168.

71

Osawa, A. 1995. Inverse relationship of crown fractal dimension to self-thinning exponent of tree

populations: a hypothesis. Can. J. For. Res. 25: 1608-1617.

Peng, C., Zhang, L., and Liu, J. Developing and validating nonlinear height-diameter models for

major tree species of Ontario’s boreal forests. N. J. App. For. 18(3): 87-94.

Peng, C., Liu, J., Dang, Q., Apps, M.J., and Jiang, H. 2002. TRIPLEX: a generic hybrid model for

predicting forest growth and carbon and nitrogen dynamics. Ecological Modelling. 153: 109-

130.

Perron, J.-Y. 2003. Tarif de cubage general: Volume marchand brut. Ministère des Ressources

naturelles, de la faune et des parcs: Forêt Québec. 3e publication. 60 pp.

Pothier, D., Margolis, H.A., and Waring, R.H. 1988. Patterns of change or sapwood permeability and

sapwood conductance with stand development. Canadian Journal of Forest Research. 19:

432-439.

Pothier, D., and Savard, F. 1998. Actualisation des tables de production pour les principales

espèces forestières du Québec. Québec, QC: Gouvernement du Québec. 183 pp.

Pothier, D., and Auger, I. 2011. NATURA-2009: un modèle de prévision de la croissance à l’échelle

du peuplement pour les forêts du Québec. Mémoire de recherche forestière no. 163.

Gouvernement du Québec, Ministère des ressources naturelles, Direction de la recherche

forestière. 76 pp.

The R Core Team. 2012. R: A language and environment for statistical computing. Vienna, Austria:

R Foundation for Statistical Computing. http://R-project.org [Accessed Jan. 12, 2013]

Radtke, P.J., and Robinson, A. P. 2006. A Bayesian strategy for combining predictions from

empirical and process models. Ecological Modelling. 190: 287-298.

Raulier, F., Bernier, P.Y. and Ung, C.-H. Canopy photosynthesis of sugar maple (Acer saccharum):

comparing big-leaf and multilayer extrapolations of leaf-level measurements. Tree Phys.

19(7): 407-420.

Raulier, F., Bernier, P.Y., and Ung, C.-H. 2000. Modelling the influence of temperature on monthly

gross primary productivity of sugar maple stands. Tree Physiology. 20: 333-345.

Raulier, F., Lambert, M.-C., Pothier, D., and Ung, C.-H. 2003. Impact of dominant tree dynamics on

site index curves. Forest Ecology and Management. 184: 65-78.

Raulier, F. 2006. Prévision du rendement de l’éclaircie commerciale dans les sapinières denses de

la Réserve Faunique des Laurentides. Rapport d’industrie. 22 pp.

72

Réseau stratégique ForêtValeur. 2008. NSERC strategic network on forest management for value-

added products. http://www.forvaluenet-foretvaleur.ca/index.php?lg=enandid=1 [accessed

May, 2011]

Robinson, A., and Ek, A. 2003. Description and validation of a hybrid model of forest growth and

stand dynamics for the Great Lakes region. Ecological Modelling. 170: 73-104.

Robinson, A., and Froese, R. 2004. Model validation using equivalence tests. Ecological Modelling.

176: 349-358.

Sampson, D.A., and Smith, F.W. 1993. Influence of canopy architecture on light penetration in

lodgepole pine (Pinus contorta var. latifolia) forests. Agricultural and Forest Meteorology.

64: 63-79.

Schneider, R., Berninger, F., Ung, C.-H., Mäkelä, A., Swift, D.E., and Zhang, S.Y. 2011a. Within

crown variation in the relationship between foliage biomass and sapwood area in jack pine.

Tree Physiology. 31(1): 22-29.

Schneider, R., Fortin, M., Berninger, F., Ung, C.-H., Swift, D.E., and Zhang, S.Y. 2011b. Modelling

jack pine (Pinus banksiana Lamb.) foliage density distribution. Forest Science. 57(3): 180-

188.

Shcherbinina, A. 2012. Validation of the jack pine version of CroBas-PipeQual. MSc thesis.

University of British Columbia.

https://circle.ubc.ca/bitstream/handle/2429/43368/ubc_2012_fall_shcherbinina_anna.pdf?seq

uence=1 [Accessed Dec 10, 2012]

Sievänen, R., and Burk, T.E. 1993. Adjusting a process-based growth model for varying site

conditions through parameter estimation. Canadian Journal of Forest Research. 23: 1837-

1851.

Skovsgaard, J.P., and Vanclay, J.K. 2008. Forest site productivity: a review of the evolution of

dendrometric concepts for even-aged stands. Forestry. 81(1): 13-31.

Smith, F.W., Sampson, D.A., and Long, J.N. 1991. Comparison of leaf area index estimates from

tree allometrics and measured light interception. Forest Science. 37(6): 1682-1688.

Tan, W., and Hogan, G.D. 1995. Limitations to net photosynthesis as affected by nitrogen status in

jack pine (Pinus banksiana Lamb.) seedlings. Journal of Experimental Botany. 46(285):

407-413.

Valentine, H.T., and Mäkelä, A. 2005. Bridging process-based and empirical approaches to

modelling tree growth. Tree Physiology. 25: 769-779.

73

Vanninen, P. and Mäkelä, A. 1999. Fine root biomass of Scots pine stands differing in age and soil

fertility in southern Finland. Tree Physiology. 19: 823-830.

Verbeeck, H., Samson, R., Verdonck, F., and Lemeur, R. 2006. Parameter sensitivity and

uncertainty of the forest carbon flux model FORUG: a Monte Carlo analysis. Tree

Physiology. 26(6): 807-817.

Weiskittel, A.R., Hann, D.W., Kershaw, J.A., and Vanclay, J.K. 2011. Forest growth and yield

modeling. Chichester, West Sussex: John Wiley & Sons.

75

Appendix 1

Multiple linear regression results for the predicted height at stand age 50

lm(formula = to.graph[, o] ~ ., data = data.subset, x = TRUE) Residuals: Min 1Q Median 3Q Max -1.8060 -0.1864 -0.0199 0.1643 2.0935 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 9.470e+00 2.947e-03 3213.471 < 2e-16 *** height.ini 6.893e-01 3.612e-03 190.850 < 2e-16 *** crown.ht -4.635e-01 8.688e-03 -53.354 < 2e-16 *** density.ini 1.584e-04 4.673e-06 33.894 < 2e-16 *** z -3.027e+00 4.794e-02 -63.147 < 2e-16 *** zeta -1.818e+01 1.022e+00 -17.788 < 2e-16 *** cb 6.230e+00 2.057e-01 30.286 < 2e-16 *** ct -2.162e-02 5.075e-02 -0.426 0.670151 alphas -6.941e+02 2.256e+01 -30.774 < 2e-16 *** alphab -2.264e+02 1.439e+01 -15.734 < 2e-16 *** alphat -1.092e+03 1.213e+02 -9.001 < 2e-16 *** alphar -5.579e+00 8.648e-02 -64.517 < 2e-16 *** phis -5.906e-01 5.117e-02 -11.542 < 2e-16 *** phic -6.030e-01 6.833e-02 -8.825 < 2e-16 *** phibprime -5.760e-02 6.793e-02 -0.848 0.396541 phitprime -4.991e-02 5.101e-02 -0.978 0.327903 phib -1.141e+00 2.758e-01 -4.136 3.56e-05 *** phit -2.629e-01 5.099e-02 -5.155 2.57e-07 *** raus -3.682e-03 1.213e-04 -30.349 < 2e-16 *** raub -1.890e-03 1.212e-04 -15.601 < 2e-16 *** raut -8.976e-04 1.206e-04 -7.441 1.06e-13 *** r1 -1.036e+01 2.558e-01 -40.489 < 2e-16 *** r2 -7.246e+01 3.393e+00 -21.354 < 2e-16 *** aq 7.511e-01 5.116e-02 14.682 < 2e-16 *** q -4.395e-03 6.336e-03 -0.694 0.487937 m0 4.495e+01 5.102e+01 0.881 0.378357 m1 -1.264e+00 5.073e+00 -0.249 0.803315 p 1.880e-03 1.023e-02 0.184 0.854187 an 1.253e+00 8.544e-03 146.627 < 2e-16 *** P0 3.537e+00 2.117e-02 167.098 < 2e-16 *** k 5.326e+01 3.640e-01 146.329 < 2e-16 *** aSigma -5.619e+01 8.454e-01 -66.472 < 2e-16 *** Y -8.375e+00 7.896e-02 -106.065 < 2e-16 *** sf -4.314e+00 2.029e-01 -21.256 < 2e-16 *** sr -2.615e+00 5.052e-02 -51.753 < 2e-16 *** ds0 -1.753e-01 5.130e-02 -3.416 0.000637 *** db0 -1.072e-01 5.101e-02 -2.102 0.035549 * dt0 2.890e-03 5.145e-02 0.056 0.955206 ds1 -2.860e+01 3.402e+00 -8.408 < 2e-16 *** db1 -2.835e+01 3.409e+00 -8.315 < 2e-16 *** dt1 -7.808e+00 2.535e+00 -3.080 0.002076 ** psis -4.253e-01 5.096e-02 -8.345 < 2e-16 *** psic -2.933e-01 1.021e-01 -2.871 0.004096 ** psib -5.352e-01 5.106e-02 -10.482 < 2e-16 *** psit -1.950e-01 5.131e-02 -3.800 0.000145 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3222 on 11955 degrees of freedom Multiple R-squared: 0.962, Adjusted R-squared: 0.9618 F-statistic: 6872 on 44 and 11955 DF, p-value: < 2.2e-16

77

Appendix 2

Multiple linear regression results for the predicted height at stand age 100

lm(formula = to.graph[, o] ~ ., data = data.subset, x = TRUE) Residuals: Min 1Q Median 3Q Max -2.3997 -0.4058 -0.0527 0.3336 3.8034 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.026e+01 5.857e-03 1751.257 < 2e-16 *** height.ini 3.802e-01 7.178e-03 52.958 < 2e-16 *** crown.ht -2.053e-01 1.727e-02 -11.886 < 2e-16 *** density.ini 6.399e-04 9.288e-06 68.900 < 2e-16 *** z -4.507e+00 9.529e-02 -47.297 < 2e-16 *** zeta -2.636e+01 2.031e+00 -12.976 < 2e-16 *** cb 1.340e+01 4.088e-01 32.782 < 2e-16 *** ct -4.861e-02 1.009e-01 -0.482 0.629943 alphas -1.117e+03 4.483e+01 -24.912 < 2e-16 *** alphab -3.547e+02 2.860e+01 -12.402 < 2e-16 *** alphat -1.580e+03 2.411e+02 -6.554 5.83e-11 *** alphar -8.918e+00 1.719e-01 -51.881 < 2e-16 *** phis -1.035e+00 1.017e-01 -10.173 < 2e-16 *** phic -8.683e-01 1.358e-01 -6.394 1.68e-10 *** phibprime -9.907e-02 1.350e-01 -0.734 0.463129 phitprime -1.287e-01 1.014e-01 -1.270 0.204184 phib -1.975e+00 5.483e-01 -3.602 0.000317 *** phit -3.840e-01 1.013e-01 -3.788 0.000152 *** raus -5.938e-03 2.411e-04 -24.630 < 2e-16 *** raub -2.831e-03 2.408e-04 -11.757 < 2e-16 *** raut -1.502e-03 2.397e-04 -6.267 3.81e-10 *** r1 -1.622e+01 5.085e-01 -31.897 < 2e-16 *** r2 -1.232e+02 6.745e+00 -18.268 < 2e-16 *** aq 2.561e+00 1.017e-01 25.185 < 2e-16 *** q -2.337e-02 1.259e-02 -1.855 0.063561 . m0 8.223e+01 1.014e+02 0.811 0.417459 m1 -1.731e+01 1.008e+01 -1.717 0.086063 . p 2.012e-03 2.033e-02 0.099 0.921150 an 1.990e+00 1.698e-02 117.169 < 2e-16 *** P0 5.584e+00 4.207e-02 132.722 < 2e-16 *** k 8.456e+01 7.235e-01 116.883 < 2e-16 *** aSigma -9.235e+01 1.680e+00 -54.960 < 2e-16 *** Y -1.303e+01 1.569e-01 -83.000 < 2e-16 *** sf -6.608e+00 4.034e-01 -16.383 < 2e-16 *** sr -4.219e+00 1.004e-01 -42.017 < 2e-16 *** ds0 -2.028e-01 1.020e-01 -1.988 0.046784 * db0 -1.175e-01 1.014e-01 -1.159 0.246349 dt0 3.188e-02 1.023e-01 0.312 0.755249 ds1 -5.222e+01 6.761e+00 -7.724 1.22e-14 *** db1 -4.452e+01 6.777e+00 -6.570 5.25e-11 *** dt1 -1.162e+01 5.039e+00 -2.306 0.021107 * psis -7.050e-01 1.013e-01 -6.959 3.60e-12 *** psic -5.439e-01 2.030e-01 -2.679 0.007394 ** psib -8.721e-01 1.015e-01 -8.593 < 2e-16 *** psit -3.564e-01 1.020e-01 -3.495 0.000477 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6404 on 11955 degrees of freedom Multiple R-squared: 0.9118, Adjusted R-squared: 0.9115 F-statistic: 2808 on 44 and 11955 DF, p-value: < 2.2e-16

79

Appendix 3

Multiple linear regression results for the predicted height error at stand age 50

lm(formula = to.graph[, o] ~ ., data = data.subset, x = TRUE) Residuals: Min 1Q Median 3Q Max -2.10279 -0.16243 0.02078 0.18685 1.79630 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.985e+00 2.931e-03 1359.919 < 2e-16 *** height.ini 8.742e-01 3.592e-03 243.396 < 2e-16 *** crown.ht 4.680e-01 8.640e-03 54.169 < 2e-16 *** density.ini -1.562e-04 4.647e-06 -33.603 < 2e-16 *** z 3.027e+00 4.768e-02 63.482 < 2e-16 *** zeta 1.817e+01 1.016e+00 17.874 < 2e-16 *** cb -6.232e+00 2.046e-01 -30.467 < 2e-16 *** ct 2.223e-02 5.047e-02 0.441 0.659564 alphas 6.942e+02 2.243e+01 30.947 < 2e-16 *** alphab 2.266e+02 1.431e+01 15.837 < 2e-16 *** alphat 1.092e+03 1.206e+02 9.048 < 2e-16 *** alphar 5.582e+00 8.600e-02 64.906 < 2e-16 *** phis 5.915e-01 5.089e-02 11.623 < 2e-16 *** phic 6.010e-01 6.795e-02 8.845 < 2e-16 *** phibprime 5.666e-02 6.756e-02 0.839 0.401665 phitprime 4.931e-02 5.073e-02 0.972 0.331006 phib 1.138e+00 2.743e-01 4.148 3.37e-05 *** phit 2.610e-01 5.071e-02 5.147 2.69e-07 *** raus 3.685e-03 1.206e-04 30.543 < 2e-16 *** raub 1.889e-03 1.205e-04 15.673 < 2e-16 *** raut 9.005e-04 1.200e-04 7.507 6.48e-14 *** r1 1.037e+01 2.544e-01 40.745 < 2e-16 *** r2 7.236e+01 3.375e+00 21.442 < 2e-16 *** aq -7.533e-01 5.088e-02 -14.807 < 2e-16 *** q 4.281e-03 6.301e-03 0.679 0.496886 m0 -4.414e+01 5.074e+01 -0.870 0.384384 m1 1.312e+00 5.045e+00 0.260 0.794899 p -1.642e-03 1.017e-02 -0.161 0.871738 an -1.253e+00 8.497e-03 -147.418 < 2e-16 *** P0 -3.537e+00 2.105e-02 -168.019 < 2e-16 *** k -5.327e+01 3.620e-01 -147.159 < 2e-16 *** aSigma 5.618e+01 8.407e-01 66.825 < 2e-16 *** Y 8.371e+00 7.852e-02 106.606 < 2e-16 *** sf 4.318e+00 2.018e-01 21.394 < 2e-16 *** sr 2.615e+00 5.025e-02 52.042 < 2e-16 *** ds0 1.760e-01 5.102e-02 3.449 0.000565 *** db0 1.082e-01 5.073e-02 2.133 0.032927 * dt0 -3.677e-03 5.117e-02 -0.072 0.942706 ds1 2.861e+01 3.383e+00 8.458 < 2e-16 *** db1 2.841e+01 3.391e+00 8.380 < 2e-16 *** dt1 7.756e+00 2.521e+00 3.076 0.002100 ** psis 4.271e-01 5.068e-02 8.427 < 2e-16 *** psic 2.989e-01 1.016e-01 2.943 0.003260 ** psib 5.349e-01 5.078e-02 10.534 < 2e-16 *** psit 1.950e-01 5.103e-02 3.821 0.000133 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3204 on 11955 degrees of freedom Multiple R-squared: 0.9852, Adjusted R-squared: 0.9851 F-statistic: 1.803e+04 on 44 and 11955 DF, p-value: < 2.2e-16

81

Appendix 4

Multiple linear regression results for the predicted height error at stand age 100

lm(formula = to.graph[, o] ~ ., data = data.subset, x = TRUE) Residuals: Min 1Q Median 3Q Max -3.7893 -0.3318 0.0547 0.4055 2.3848 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.692e+00 5.844e-03 974.014 < 2e-16 *** height.ini 1.458e+00 7.162e-03 203.654 < 2e-16 *** crown.ht 2.124e-01 1.723e-02 12.331 < 2e-16 *** density.ini -6.364e-04 9.266e-06 -68.675 < 2e-16 *** z 4.506e+00 9.507e-02 47.395 < 2e-16 *** zeta 2.634e+01 2.027e+00 12.996 < 2e-16 *** cb -1.341e+01 4.079e-01 -32.869 < 2e-16 *** ct 4.959e-02 1.006e-01 0.493 0.622190 alphas 1.117e+03 4.473e+01 24.972 < 2e-16 *** alphab 3.551e+02 2.854e+01 12.443 < 2e-16 *** alphat 1.580e+03 2.406e+02 6.567 5.36e-11 *** alphar 8.922e+00 1.715e-01 52.027 < 2e-16 *** phis 1.036e+00 1.015e-01 10.210 < 2e-16 *** phic 8.652e-01 1.355e-01 6.386 1.77e-10 *** phibprime 9.757e-02 1.347e-01 0.724 0.468905 phitprime 1.278e-01 1.012e-01 1.263 0.206487 phib 1.970e+00 5.470e-01 3.602 0.000317 *** phit 3.809e-01 1.011e-01 3.767 0.000166 *** raus 5.943e-03 2.405e-04 24.708 < 2e-16 *** raub 2.829e-03 2.403e-04 11.773 < 2e-16 *** raut 1.507e-03 2.392e-04 6.301 3.06e-10 *** r1 1.623e+01 5.073e-01 31.996 < 2e-16 *** r2 1.230e+02 6.729e+00 18.286 < 2e-16 *** aq -2.564e+00 1.014e-01 -25.279 < 2e-16 *** q 2.319e-02 1.256e-02 1.845 0.065026 . m0 -8.093e+01 1.012e+02 -0.800 0.423768 m1 1.739e+01 1.006e+01 1.728 0.083955 . p -1.632e-03 2.028e-02 -0.080 0.935879 an -1.989e+00 1.694e-02 -117.424 < 2e-16 *** P0 -5.584e+00 4.198e-02 -133.026 < 2e-16 *** k -8.457e+01 7.218e-01 -117.169 < 2e-16 *** aSigma 9.233e+01 1.676e+00 55.076 < 2e-16 *** Y 1.302e+01 1.566e-01 83.155 < 2e-16 *** sf 6.615e+00 4.024e-01 16.437 < 2e-16 *** sr 4.220e+00 1.002e-01 42.116 < 2e-16 *** ds0 2.039e-01 1.017e-01 2.004 0.045058 * db0 1.191e-01 1.011e-01 1.178 0.239019 dt0 -3.314e-02 1.020e-01 -0.325 0.745318 ds1 5.224e+01 6.745e+00 7.745 1.03e-14 *** db1 4.462e+01 6.761e+00 6.600 4.29e-11 *** dt1 1.154e+01 5.027e+00 2.295 0.021737 * psis 7.079e-01 1.011e-01 7.004 2.61e-12 *** psic 5.529e-01 2.025e-01 2.730 0.006345 ** psib 8.716e-01 1.013e-01 8.608 < 2e-16 *** psit 3.564e-01 1.018e-01 3.503 0.000462 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6389 on 11955 degrees of freedom Multiple R-squared: 0.9743, Adjusted R-squared: 0.9742 F-statistic: 1.029e+04 on 44 and 11955 DF, p-value: < 2.2e-16

83

Appendix 5

Background

In the sensitivity analysis performed, most of the parameters that had the biggest influence on CroBas’ volume and

height predictive ability were those related to calculating the net photosynthesis. This includes the k parameter or the

average light extinction coefficient. The k parameter, in combination with the leaf area index (LAI) represents the

amount of incoming light absorbed by the forest canopy per unit ground area and is applied using the Beer-Lambert

model to increase the net photosynthesis of the stand as LAI increases. According to Mäkelä (1997), the Beer-

Lambert model was chosen because it provides “a good approximation of a very detailed shading model.” The k-

value of 0.18 used by Mäkelä was selected to force the Beer-Lambert model to approximate the more detailed

shading model of Oker-Blom et.al. (1989).

Biologically, the k-value should be in agreement with the units of LAI being used. In the case of CroBas, the specific

leaf area (an) is representative of the projected area of a needle per unit dry weight (Goudiaby et al. 2011). The

specific leaf area is then scaled up using stand density and foliage weight to obtain total stand LAI (equation 28;

Makela, 1997). Therefore, the LAI used within CroBas is a projected LAI as opposed to a hemi-surface or total LAI

as defined in Chen and Black (1992). This is no longer the standardized format for reporting LAI, and an should be

updated to represent the hemi-surface specific leaf area so the LAI calculated within CroBas will also be the hemi-

surface LAI.

Methods and Results

k

For use in the StandLEAP model of the ECOLEAP project (described in Hall et al. 2006 and Girardin et al. 2008),

Raulier et al. (1999) developed an empirical relationship between stand-level hemi-surface LAI (HSLA) and an

average light extinction coefficient (equation A5). This relationship was parameterized using detailed calculations of

diffuse (equation A2) and direct (equation A3) light extinction coefficients based on data found in Gower et al. (1997)

for jack pine (Pinus banksiana Lamb.). The result is that, for jack pine, the average k value changes with HSLA as

per (Frédéric Raulier, pers. comm..):

k = 0.47413025 – 0.04414448 * ln(HSLA)

Gower et al. (1997) reported regional-average HSLA values for jack pine in southern boreal sites to be 2.4 and 2.8

for old (65-yrs) and young (25-yrs) stands, respectively. Kimball et al. (2000) confirm this value and report a regional-

average HSLA for jack pine-dominated dry coniferous forests of 2.5 based on biomass maps of their study site in the

southern Boreal Forest and allometric relationships (Table 3). Gower et al. (1997) did report slightly lower LAI values

84

for northern sites where growing conditions were influenced by permafrost, and soils were dominated by glacial

lacustrine deposits. The conditions in the province of Québec, however, are not influenced by such soils (NRC 2009)

and are likely best represented by the southern boreal sites of the study (Gower et al. 1997).

An average HSLA of 2.5 is representative of Québec jack pine stands for the purpose of obtaining a single average

light extinction coefficient for the study area. The corresponding k value is 0.4336811.

an

The current default value for the projected specific leaf area is consistent with values measured and reported in

literature (for example see Goudiaby et al. 2011) and will not be changed. For needles having a random orientation

when measured for projected specific leaf area, as was the case in determining an in CroBas, Bond-Lamberty et al.

(2003) suggest that the HSLA can be obtained by:

HSLA = (1.29 m2 HSLA / m2 PLA) * PLA

Using the above conversion factor, and the default an value of 6 m2 PLA / kg the specific leaf area, expressed in units

of hemi-surface leaf area the most appropriate for use in CroBas to represent jack pine in Québec is:

HSLA = (1.29 m2 HSLA / m2 PLA) * 6 m2 PLA / kg = 7.74 m2 HSLA / kg

Conclusion

Fix the default k value of CroBas to 0.43 and the an value to 7.74 m2 HSLA / kg.

85

Appendix 6

The following graphs show the differences calculated for the validation portion of this research between

CroBas predictions and PSP measurements for each plot over time for (a) height, (b) height increment, (c)

DBH, (d) DBH increment, (e) BA, (f) BA increment, (g) volume, (h) volume increment, and (i) density.

(a)

(b)

(c)

(d)

86

(e)

(f)

(g)

(h)

(i)