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Assessing Uncertainty in Forest Dynamic Models

Permanent Link: http://ufdc.ufl.edu/UFE0022389/00001

Material Information

Title: Assessing Uncertainty in Forest Dynamic Models A Case Study Using SYMFOR
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Valle, Denis
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: amazon, forest, multimodel, simulation, uncertainty, variance
Forest Resources and Conservation -- Dissertations, Academic -- UF
Genre: Forest Resources and Conservation thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Despite its local, regional and global importance, the Amazon forest faces multiple threats. Forest dynamic models have been widely used to evaluate the impact of a number of anthropogenic impacts on the forest, such as timber logging and climate change. I hypothesize that these models, however, have usually failed to report the full uncertainty associated with their projections. I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium assumption and maximum size assumption. I then quantified four sources of model uncertainty using the tropical forest simulation model SIMFLORA: model stochasticity, parameter uncertainty, starting condition effect, and modeling assumptions. My results suggest that modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect than other sources of uncertainty that are more commonly taken into account, such as parameter uncertainty, particularly when assumptions are used to deal with sub-model extrapolations. Also, to reduce assumption uncertainty in particular, and overall model uncertainty in general, it is of fundamental importance to use the available data to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process). Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid the use of some of these assumptions. Using SIMFLORA as a case study, my results indicate that the overall modeling uncertainty is likely to be underestimated if all four sources listed above are not simultaneously considered. Finally, the method developed in this thesis to partition overall variance of the mean into different uncertainty sources can be applied to quantify the uncertainty of other models, not restricted to forest dynamic models.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Denis Valle.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Staudhammer, Christina Lynn.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022389:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022389/00001

Material Information

Title: Assessing Uncertainty in Forest Dynamic Models A Case Study Using SYMFOR
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Valle, Denis
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: amazon, forest, multimodel, simulation, uncertainty, variance
Forest Resources and Conservation -- Dissertations, Academic -- UF
Genre: Forest Resources and Conservation thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Despite its local, regional and global importance, the Amazon forest faces multiple threats. Forest dynamic models have been widely used to evaluate the impact of a number of anthropogenic impacts on the forest, such as timber logging and climate change. I hypothesize that these models, however, have usually failed to report the full uncertainty associated with their projections. I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium assumption and maximum size assumption. I then quantified four sources of model uncertainty using the tropical forest simulation model SIMFLORA: model stochasticity, parameter uncertainty, starting condition effect, and modeling assumptions. My results suggest that modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect than other sources of uncertainty that are more commonly taken into account, such as parameter uncertainty, particularly when assumptions are used to deal with sub-model extrapolations. Also, to reduce assumption uncertainty in particular, and overall model uncertainty in general, it is of fundamental importance to use the available data to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process). Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid the use of some of these assumptions. Using SIMFLORA as a case study, my results indicate that the overall modeling uncertainty is likely to be underestimated if all four sources listed above are not simultaneously considered. Finally, the method developed in this thesis to partition overall variance of the mean into different uncertainty sources can be applied to quantify the uncertainty of other models, not restricted to forest dynamic models.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Denis Valle.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Staudhammer, Christina Lynn.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022389:00001


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ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS:
A CASE STUDY USING SYMFOR




















By

DENIS RIBEIRO DO VALLE


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008


































2008 Denis Ribeiro do Valle

































To my family, in particular to my wife Natercia Moura do Valle.









ACKNOWLEDGMENTS

I thank my advisor Christina Staudhammer and the committee members who have given

me the freedom to pursue my research ideas and have greatly supported my search for the best

methods to address these ideas.









TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ..............................................................................................................4

LIST OF TABLES ......... ..... .... ....................................................6

LIST OF FIGURES .................................. .. ..... ..... ................. .7

ABSTRAC T ..........................................................................................

CHAPTER

1 INTRODUCTION ............... ................. ........... .............................. 10

O v erv iew ................... ...................1...................0..........
O bje ctiv e s ................... ...................1...................1..........

2 M E T H O D S .......................................................................................................13

M odel Description .............................. ............ ...... ........... 13
T ap ajo s D ata set ................................................................................................13
M modeling A ssum options A nalyzed...................................................................... ...............15
Dynamic Equilibrium Assumption (DEA) ....................................... ...............15
M aximum Size A ssum ption (M SA)...................................................... ..... .......... 17
M modifications to SIM FLO R A .............................................. ..................... ...............18
O ne H hundred Y ear Sim ulations.............................................................................. ............20
D ata A n a ly sis ................. ................................................................................. ............... 2 1
Variance Component Analysis .......................................................... ............... 22
Probability of Each Model Given the Data (r) ......................................................23

3 R E SU L T S ...........................................................................................32

Description of the Baseline Simulation Set for the Unlogged Forest............... ................ 32
M ean M modeling R results ............... ........ ...................... ... ................ .... ..32
Probability of Each Model and Comparison of Uncertainty Sources.................................33

4 D ISC U S SIO N ..............................................................................................42

Types of A ssum options ................................ .. ........ .. ........................ ................. 42
C om prison of U uncertainty Sources.......................................................................... ... ... 43

5 C O N C L U SIO N S ................. ......................................... ........ ........ ..... .... ...... .. 49

L IST O F R E F E R E N C E S .............................................................................. ...........................50

B IO G R A PH IC A L SK E T C H .............................................................................. .....................57









LIST OF TABLES


Table page

2-1. Examples of assumptions contained in a sample of forest dynamic models ...................26

2-2. Original growth, recruitment" and mortalitytt sub-model parameters...........................27

2-3. Approaches and modified parameters used to implement assumptions ..........................28

2-4. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine
T uning m odel. .............................................................................. 29

2-5. Analysis of Variance used to determine uncertainty due to plots and model
sto c h a stic ity ................................................... ..................... ................ 3 0

2-6. Analysis of Variance used to determine parameter uncertainty for the baseline
m odel ........... ................................................. .................. ........ ...... 30

3-1. Percentage change in the unlogged forest...................................... ........................ 36

3-2. Estimated posterior probability of each model, given the recruitment data, the
m ortality data, and both datasets com bined ............................................ ............... 36









LIST OF FIGURES


Figure pe

2-1. Nested design used for simulations....................... ..... ............................. 31

3-1. Comparison of observed versus simulated stand level data .............................................37

3-2. Mean simulation results from different models (baseline + 4 assumptions) over a
tim e w window of 100 years ................................................................................. ..... ..38

3-3. Overall variance of the mean when data are not taken into account in estimating
model probabilities (i.e., assumptions are not evaluated in light of the data),
partitioned between parameter uncertainty, starting conditions effect, model
stochasticity, and assum options effect ........................................ ........................... 39

3-4. Overall variance of the mean when data are taken into account in estimating the
probability of each model (i.e., assumptions are evaluated in light of the data),
partitioned between parameter uncertainty, starting conditions effect, model
stochasticity, and assum options effect ........................................ ........................... 40

3-5. Multimodel average (continuous line) and 95% confidence interval (dashed line),
shown when data are not used (grey) and when data are used (black) to estimate the
probability of each m odel. ......................................... ........................... 41

4-1. Combinations of competition index and DBH contained in the data used to calibrate
the growth sub-model for species-group 10............................................ ............... 47

4-2. Overall variance of the mean when data are taken into account in estimating the
probability of each model (i.e., assumptions are evaluated in light of the data),
including only the simulation sets from the baseline model and the model with the
modified growth sub-model, partitioned between parameter uncertainty, starting
conditions effect, model stochasticity, and assumptions effect......................................48









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS:
A CASE STUDY USING SYMFOR

By

Denis Ribeiro do Valle

August, 2008

Chair: Christina Staudhammer
Major: Forest Resources and Conservation

Despite its local, regional and global importance, the Amazon forest faces multiple threats.

Forest dynamic models have been widely used to evaluate the impact of a number of

anthropogenic impacts on the forest, such as timber logging and climate change. I hypothesize

that these models, however, have usually failed to report the full uncertainty associated with their

projections.

I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium

assumption and maximum size assumption. I then quantified four sources of model uncertainty

using the tropical forest simulation model SIMFLORA: model stochasticity, parameter

uncertainty, starting condition effect, and modeling assumptions. My results suggest that

modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect

than other sources of uncertainty that are more commonly taken into account, such as parameter

uncertainty, particularly when assumptions are used to deal with sub-model extrapolations. Also,

to reduce assumption uncertainty in particular, and overall model uncertainty in general, it is of

fundamental importance to use the available data to determine the probability of each model (i.e.,

data are used to evaluate the different assumptions adopted in the modeling process).

Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid









the use of some of these assumptions. Using SIMFLORA as a case study, my results indicate that

the overall modeling uncertainty is likely to be underestimated if all four sources listed above are

not simultaneously considered. Finally, the method developed in this thesis to partition overall

variance of the mean into different uncertainty sources can be applied to quantify the uncertainty

of other models, not restricted to forest dynamic models.









CHAPTER 1
INTRODUCTION

Overview

The Brazilian Amazon contains about 40% of the world's remaining tropical rainforest

and its vital role in global biodiversity, terrestrial carbon storage, regional hydrology and climate

has been widely recognized (Nepstad et al. 1999, Laurance et al. 2001, Malhi et al. 2002,

Nepstad et al. 2002). At the same time the use of its natural resources is essential for millions of

rural Amazonians' health and livelihoods. For instance, many medicinal plant species found in

the forest are the sole health care option for many rural poor (Shanley and Luz 2003), game is

frequently the most important source of protein and fat to these people (Redford 1992) and, on a

regional scale, timber logging is one of the most important rural activities for the economy

(Lentini et al. 2003).

Despite the local, regional and global importance of Amazonia's natural resources, the

Brazilian Amazon faces multiple threats. In 2003, the total deforested area was estimated at 15%

of the original area (Soares-Filho et al. 2006), and this area is increasing at one of the highest

rates in the world, with a mean annual increase of 18,100 km2 yr-1 (Malhi et al. 2008).

Furthermore, the remaining forest is not intact. Generally uncontrolled, selective logging affects

an annual area ranging from 10,000 20,000 km2 (Nepstad et al. 1999, Asner et al. 2005), which

can greatly increase fire risk, one of the greatest threats to the forests of Amaz6nia (Nepstad et

al. 2001, Cochrane 2003). By 2010 approximately a tenth of the Brazilian Amazon is planned to

be designated as forest concession area, where selective logging is to take place (Verissimo et al.

2002). In this context, sustainable use of natural resources and the balance between satisfying

immediate human needs and maintaining other ecosystem functions will require quantitative









knowledge about the ecosystem's present and future responses (Clark et al. 2001, DeFries et al.

2004).

Numerous forest dynamic models have been developed to try to make reliable long-term

and large-scale prediction using available short-term and small-scale empirical data (Pacala et al.

1996, Kammesheidt et al. 2001). There has also been a growing awareness of the importance of

quantifying modeling uncertainties, with some leading science journals (e.g., Ecology,

Ecological Modeling, Global Environmental Change) devoting issues solely to this theme (Clark

2003, Dessai et al. 2007, Lek 2007). Nevertheless, when uncertainties from forest dynamic

model projections are presented, they either refer to model stochasticity (Gourlet-Fleury et al.

2005, Degen et al. 2006), effect of starting conditions (Phillips et al. 2004, van Gardingen et al.

2006), or parameter uncertainty (Pacala et al. 1996). A fourth source of uncertainty refers to the

assumptions used when designing the model (e.g., the choice of equations to represent ecological

processes); this can be a key source of uncertainty (Varis and Kuikka 1999, Qian et al. 2003,

Brugnach 2005, der Lee et al. 2006). I do not know of any article that reports model uncertainties

due to these assumptions and that analyzes all these sources of uncertainty jointly. As a

consequence, I expect that the uncertainty in model forecasts has, in general, been under-

estimated.

Objectives

The objectives of this thesis, based on a commonly used forest dynamic model, are to: a)

quantify model uncertainty derived from model stochasticity, parameter estimation, starting

conditions and modeling assumptions; and, b) compare these sources of uncertainty in order to

evaluate which sources contribute the most to the overall model uncertainty.

Using SIMFLORA as a case study, the main hypotheses I will test are: a) forest dynamic

model uncertainty has been underestimated by not simultaneously including the sources of









uncertainty identified above; and b) model assumptions are the greatest source of overall model

uncertainty. As a result, I intend to propose a new method to determine overall model

uncertainty.









CHAPTER 2
METHODS

Model Description

The Silviculture and Yield Management for Tropical Forests (SYMFOR) is a modeling

framework that combines a management model, which allows the user to specify silvicultural

activities in mixed tropical forest (e.g., harvest, thinning, poisoning, enrichment planting), with

an empirical spatially explicit individual tree-based ecological model, which simulates the

natural processes of recruitment, growth and mortality. The SYMFOR model, originally

designed for use in Indonesia (Phillips et al., 2003), was adapted for use in Guyana (Phillips et

al. 2002b) and then for the Brazilian Amazon (Phillips et al. 2004). In Brazil, the model was

further adapted by incorporating new management options and translating the user interface into

Portuguese to become the model SIMFLORA. To date, three studies within the Brazilian

Amazon have been published using SIMFLORA, two of them based on the Tapaj6s dataset

(Phillips et al. 2004, van Gardingen et al. 2006) and one of them based on the Paragominas

dataset (Valle et al. 2007). Other SYMFOR articles and reports can be found at

www.symfor.org. All results presented in this manuscript are assessed in relation to

SIMFLORA's overall (all trees with diameter at breast height, DBH > 5 cm) and commercial

basal area projections based on the Tapaj6s dataset. Commercial basal area is defined throughout

this thesis as the basal area of trees from commercial species with DBH greater than the legal

minimum logging diameter for the Brazilian Amazon region (i.e., 45 cm).

Tapajos Dataset

The series of plots at Tapaj6s km 114 comprise 60 Permanent Sample Plots (PSPs), each

of 0.25 ha, initially measured (all trees with DBH > 5 cm) in 1981 in unlogged primary forest.

Twelve of these plots were left unlogged (region 14) while a silvicultural experiment with a









randomized block design was installed in the remaining 48 plots (region 11). In this silvicultural

experiment, all treatments were logged for timber in 1982 with similar logging intensities across

treatments but with different thinning intensities applied in 1993-1994. The series of plots at

Tapaj6s km 67 (region 12) comprise 36 Permanent Sample Plots (PSPs), each of 0.25 ha. This

stand was logged in 1979 but permanent plots were installed only in 1981. Detailed description

of the forest and these experiments can be found elsewhere (Silva et al. 1995, Silva et al. 1996,

Alder and Silva 2000, Phillips et al. 2004, Oliveira 2005).

To initialize the model for the 100-yr simulations, I needed census data (xy-coordinates,

species group, and diameter from each tree within the plot) from unlogged plots in order to be

able to simulate stand dynamics in two scenarios: an unlogged scenario and a simulated logging

scenario. Therefore, I only used the 1981 pre-logging data from region 11 and 14. Because the

model requires 1 ha plots, it was necessary to join 4 plots to create a composite 1 ha plot,

generating a total of 15 plots. Only plots from the same experimental block (prior to the

experimental logging treatments) were joined together, in an effort to avoid within-plot

variability and to increase between-plot variability.

Another set of simulations were conducted to compare the observed with the simulated

data. To compare how well SYMFOR simulates stand dynamics without having to worry about

how well it simulates the logging and thinning treatments effectively applied in the field,

simulations were initialized with the 1981 unlogged forest data for those plots that were not

logged (region 14) and with the 1981 and 1983 logged forest data from region 12 and region 11,

respectively. For this comparison, the observed data consisted of the time series from regions 11,

12, and 14 prior to the thinning treatment. The same procedure to create composite 1 ha plots

was used and, although this procedure mixed different treatments into a single composite plot, it









does not affect the simulations as experimental treatments only differed in terms of thinning

treatments applied in 1993-1994 and only pre-thinning data were used.

Modeling Assumptions Analyzed

Based on a literature review of forest stand dynamic models applied in the Amazon Basin,

I identified several major modeling assumptions, from which two of the most common

assumptions were chosen for the present analysis. A brief description of these assumptions is

given below.

Dynamic Equilibrium Assumption (DEA)

The dynamic equilibrium assumption is a common forest dynamics modeling assumption

(Kammesheidt et al. 2001, Porte and Bartelink 2002), generally being interpreted as a stable

basal area and/or tree density on species group and/or stand level for an undisturbed forest. This

assumption is implemented by adjusting the mortality and/or the recruitment sub-models in such

a way as to force the model to exhibit this equilibrium. The pragmatic justification for this

procedure is that recruitment and mortality data are notoriously noisy and therefore empirical

parameters are likely to be poorly estimated and need to be adjusted. Indeed, long-term forest

monitoring studies in general are not well suited to collect extensive data on tree mortality: large

sample sizes are required, mortality causes are not easily determined and errors on plot

measurement (e.g., trees that were missed during measurement, lost their numbers or were

harvested without record) can have a significant impact on mortality estimates (Alder and

Synnott 1992, Alder and Silva 2000, Alder 2002). Also, recruitment data are plagued by the

difficulty of species identification of seedlings and small trees and are generally highly stochastic

(Vanclay 1994). As a consequence, in a recent review of mixed forest models, recruitment was

often found to be poorly modeled (Porte and Bartelink 2002).









To force the model to exhibit a dynamic equilibrium, two approaches have usually been

adopted (Table 2-1):

* Approach la: In the first approach (DEA Recruitment) every tree that dies, either due to
natural mortality or logging, is replaced by a newly recruited tree with the minimum
diameter of measurement.

* Approach lb: The second approach (DEA Fine Tuning) uses an iterative method to fine-
tune (modify) parameters derived from the data so that the model exhibits the desired
behavior (i.e., the dynamic equilibrium for an undisturbed forest). My approach was to
modify these parameters within their confidence intervals. Tuning of the model is
commonly done in many existing multi-component forest growth models, both empirical
and mechanistic (Gertner et al. 1995).

The dynamic equilibrium assumption has a long tradition in fisheries, forestry and ecology.

For instance, the concept of maximum sustainable yield, both in fisheries and forestry, is based

on the idea that, in the long term, the number of individuals tends to remain constant when it is

equal to the carrying capacity of the ecosystem. In ecology, the assumption that in the long-term

an undisturbed forest exhibits a dynamic equilibrium is wide-spread. For instance, the

assumption that the forest is in the steady state has been used for the metabolic theory of ecology

(Brown et al. 2004), to generate corrections of recruitment rate (Sheil and May 1996) and net

primary productivity (Malhi et al. 2004), to determine instantaneous decomposition rates (Palace

et al. 2008), and to derive the expected steady-state diameter distribution based on demographic

rates (Coomes et al. 2003, Kohyama et al. 2003, Muller-Landau et al. 2006).

Despite its wide use, the dynamic equilibrium assumption has been recently contested

based on empirical findings that reveal that undisturbed tropical forests have been accumulating

biomass and have shown increased turnover rates (Phillips and Gentry 1994, Phillips et al. 1998,

Baker et al. 2004, Lewis et al. 2004, Phillips et al. 2008).









Maximum Size Assumption (MSA)

Although large trees comprise a major fraction of above-ground forest biomass (Clark and

Clark 1996, Chambers et al. 1998, Chambers et al. 2001, Keller et al. 2001, Chave et al. 2003),

the relative scarcity of these individuals and difficulty of measurement limits data collection for

maximum tree size, large tree growth and mortality rate. As a consequence, simulation models

often create trees considered to be unrealistically large or old (Porte and Bartelink 2002). Two

approaches have generally been adopted to prevent trees from growing to unrealistic sizes (Table

2-1):

* Approach 2a: The first approach (MSA Mortality) is to arbitrarily enhance mortality
probability over a given diameter threshold. My approach was to increase the mortality rate
to 100% for trees that reached the maximum diameter (based on the Tapajos dataset) of its
species group.

* Approach 2b: The second approach (MSA Growth) is to fit a diameter increment function
in which increment tends to zero as tree diameter tends towards the species maximum size
or simply to assume (as I have done) that increment drops to zero after the tree reaches the
species maximum size.

There is mixed empirical evidence for these assumptions. For instance, a low mortality and

a continued diameter growth was observed for large individuals (>70 cm DBH) in a tropical

forest in Costa Rica (Clark and Clark 1996, Clark and Clark 1999). In the Brazilian Amazon, the

largest trees were observed to have the highest growth rates (Vieira et al. 2004).

A final approach would be to estimate the mortality rate based on species maximum size

(or age) and mean diameter increment or diameter distribution (Chave 1999, Mailly et al. 2000,

Alder et al. 2002, Kohler et al. 2003, Degen et al. 2006). This approach was not analyzed

because it is generally adopted only when no data on mortality are available.

I limited my analyses to these assumptions in order to keep my simulations, results and

discussions more concise. However, I do acknowledge that there are numerous other

assumptions within forest dynamic models, such as: a) assuming independent annual diameter









growth and/or recruitment rate (i.e., no serial correlation); b) using coarse estimates of gap sizes

and gap formation frequency, sometimes based on "best guesses" (Phillips et al. 2004); c)

assuming that all trees above the minimum felling diameter from medium-sized and large, mid-

and late successional species are harvestable, instead of identifying which trees are from

commercial species (Kammesheidt et al. 2001); and d) assuming no harvest loss or a fixed

harvest loss (e.g., due to hollowed trees, logs not found, stumps cut too high), which, depending

on the model, can vary from 30 to 60% (Huth and Ditzer 2001, Kammesheidt et al. 2001,

Phillips et al. 2004).

Modifications to SIMFLORA

Two of the uncertainty sources (model stochasticity and effect of starting conditions) are

already simulated by SIMFLORA, and hence could be assessed directly. In contrast, the

uncertainty derived from modeling assumptions had to be assessed by either changing initial

parameter values (see original parameter estimates in Table 2-2) or by modifying SIMFLORA's

source code. Modifications due to modeling assumptions are briefly described in Table 2-3 and

the modified parameters are shown in Table 2-4.

Uncertainty due to parameter estimation was assessed by allowing parameters of the main

simulated ecological processes (i.e., growth, recruitment and mortality sub-models) to vary. This

uncertainty arises due to unmeasured covariates (termed "process error") or errors in

measurement (termed "observation error" or "measurement error") (Ellner and Fieberg 2003,

Clark and Bjornstad 2004). There are many sub-categories of process error and each implies

different strategies to simulate uncertainty in parameter estimates. For instance, variation among

individuals might be simulated by randomly drawing parameter estimates for each tree at the

beginning of the simulation or when it is recruited and keeping them fixed throughout the

simulation, while variation due to climate year-to-year differences might be simulated by









randomly drawing parameter estimates for each year and keeping them fixed for all trees during

that given year. Although it would be interesting to assess the differences among these

alternative strategies, I chose to simulate uncertainty in parameter estimates as if it were solely

due to measurement error. Therefore, I randomly drew parameter estimates in the beginning of

each run, keeping them constant throughout the run, similar to the error analyses conducted by

others (Pacala et al. 1996, Wisdom et al. 2000).

The data required to assess parameter estimation uncertainty were the parameter estimates

and their covariance matrices within a given sub-model. Estimation of the original non-linear

regression parameters (using PROC NLIN; SAS Institute Inc. 2000) for the recruitment model

(F = rle r + rI + r4, in which parameter rs was set to zero) failed to converge for two species

groups in the original SIMFLORA calibration, yielding high standard errors. Therefore, for these

two species groups, parameters rl and r2 were set to zero and parameters r3 and r4 and their

respective covariance matrix were re-estimated after the model successfully converged (Table 2-

2).

The original growth model was calibrated without taking into account the repeated

measures nature of the diameter increment data. The justification for this was that parameter

estimates were unbiased and biased standard errors of the parameters would not be used. I

modified the covariance matrices provided by PROC NLIN by re-estimating the variances of the

mean parameters, assuming that the number of independent data was equal to the number of

trees, arguably biasing high the parameter variances. Overall parameter uncertainty is expected

to be lower than what will be reported throughout this manuscript.

Once the covariance matrix for each equation was obtained, a vector of n random

multivariate normal parameter estimates (X) was generated by using a vector of n (the number of









correlated parameters) independent standard normal numbers (Z), a vector of the mean parameter

estimates (u) and the lower triangular Cholesky matrix (A), given by eqn. 2-1:

X = u + AZ [2-1]

Given that each sub-model for each species group was calibrated separately, I assumed that

parameters between sub-models or between species groups were not correlated. To avoid

biologically unrealistic growth and recruitment, diameter increment and recruitment rates were

constrained by imposing an upper limit equal to the observed (species group specific) 99th

percentile of the diameter increment and recruitment rate, respectively.

One Hundred Year Simulations

I simulated stand dynamics for two extreme scenarios: i) an undisturbed forest; and ii) a

heavily logged-over forest (where a simulated logging extracted all trees >45 cm DBH from

commercial species resulting in a mean logging intensity of 75 6 m3 ha-1 [mean 95%

confidence interval]). The logging was simulated in the beginning of the run and was exactly the

same for all simulations in order to ensure an identical starting point for all subsequent stand

projections. These two extreme scenarios were chosen so that the potential range of the

assumption effects on projected overall and commercial basal area could be assessed.

Five sets of simulations (baseline + one set for each assumption) were generated, each

consisting of twenty 100-yr simulations for each plot and scenario (undisturbed and heavily

logged-over forest; Fig. 2-1A). The baseline simulation for my study used the parameter set

estimated directly from the data (Table 2-2; P. Phillips unpublished manuscript) and, despite not

being completely free from assumptions, it is the simulation that most accurately reflects the data

used for calibration.









One extra set of simulations was run solely to determine uncertainty due to parameter

estimation, consisting of 500 100-yr simulations for each plot and scenario. Parameters were

drawn randomly every two repetitions, resulting in a nested experimental design (individual runs

nested within parameter sets nested within plots; Fig. 2-1B) which allowed my statistical analysis

to disentangle uncertainty due to model stochasticity, parameter estimates and the effect of

starting conditions (details are given in the Data Analysis section). I chose to determine

parameter estimation uncertainty with a separate set of simulations because mean model

projections when parameters were allowed to vary differed markedly from the baseline

simulation (with parameters fixed at their mean values). This is a consequence of Jensen's

inequality acting on the non-linear sub-models used by SIMFLORA (examples and discussion

regarding Jensen's inequality effect can be found in Pacala et al. 1996, Ruel and Ayres 1999).

Data Analysis

Let L = {S,..., S, } be a finite set of model alternatives, x be the data, y be the response

variable, A, and Co be the expected value and the variance, respectively, of the response variable

given the data and the ith model alternative (i.e., E(y I x, S) = ,U; Var(y I x, S) = o2 ), and the

probability of the ith model given the data be z, (i.e., P(S, I x) = z, ). Draper (1995) showed that

m m
Var(yI x,L) = ,oC-2 + (j, ( -/)2 In other words, the variance of the response variable is
1 1

the sum of the within model variance and the between model variance, both weighted by the

probability of each model given the data.

m m
Using similar arguments, it can be shown that Var( | x, L)= Co + (/, /)2 ,
1=1 1Ti

where C 2; is the variance of the mean of the ith model alternative. This equation can be further









expanded by decomposing the within model variance of the mean into the variances of the mean

due to model stochasticity, due to plots (i.e., different starting conditions), and due to parameter

uncertainty (o o p,, and o,,, respectively). The uncertainty due to modeling


m
assumptions is defined here as the variance between models, given by a-z (/, /)2
1=1

Therefore, the key equation in our study that allowed me to partition the overall variance of the

mean into the different uncertainty sources is given by eqn. 2-2:


~Var~ \--(, L) / C 2" + C2 + C,) (' U)2 [2-2]
1=1 1=1

The elements of the within model variance of the mean (o- o2-p and C2 ) in eqn.
y,ms,l y,p,l y, pu,

2-2 were estimated by dividing the corresponding variance component (determined by the

variance component analysis detailed below) by the appropriate number of observations (n). The

estimation the probability of each model given the data (;ri) is also detailed below.

Variance Component Analysis

The variance due to starting conditions and due to model stochasticity (oC2 m, and C ,

respectively) were determined using the expected means squares from an Analysis of Variance

(ANOVA) with one random effect (Table 2-5) estimated for every 10-yr time step. Using the

simulation set in which parameters were allowed to vary, I determined the uncertainty due to

parameter estimation ( 2,,, ) by using the variance components analysis summarized in Table

2-6. This table was also estimated separately for every 10-yr time step. The variance due to

parameter uncertainty was the only result used from this set of simulations. Due to the

prohibitively large amount of computer time necessary to determine o,2 for all models and









since I was interested in the magnitude and not the exact value of this parameter, I determined

o2, only for the baseline model and assumed it to be the same for all of the analyzed models.

Probability of Each Model Given the Data (z,)

Bayes rule was used to combine the information from the mortality, growth, and

recruitment datasets into a single number, namely the posterior probability of each model

(onwards referred to simply as probability of each model). The use of Bayes rule for model 1

(Mi), for example, is given by eqn. 2-3:

L(D Mi) x L(D, I MI) x L(Dg I ,) x Zr(M,)
z(M I, Dr, ZL(D IM,)x L(D, IM,)x L(Dg M,) x (M,) [2-3]


where L is the likelihood, zi(M,) and zi(M, I D,,D,, Dg) are the prior and posterior

probabilities, respectively, of model M,; and Di, D,, and Dg denote the mortality, recruitment,

and growth datasets, respectively.

Equal priors were assigned to each model (i.e., i(M,) = 1/5). The likelihood of each

dataset given each model (L(D M,),L(D, M,), and L(Dg I M,)) was not easily estimated

since the data within each dataset were not independent. To adequately determine the likelihood

of each dataset given each model, I modeled each dataset as a function of SYMFOR's equations

and inserted random effects to circumvent the lack of independence of individual observations,

as described below:

* Recruitment of trees: Let RJsk" be the number of recruited trees from species group s, in
10 x 10 m subplot j, within plot k, in time interval n (with duration of tn years). I assumed

that Rskn Poisson(t, *, where F is the annual recruitment rate. The annual
recruitment rate F is a function of the annual recruitment rate estimated by SYMFOR's
F
recruitment equation ( SYMFOR, recruitment equation shown in Table 2-2), and subplot and
plot random effects (Esubplot and EPot, respectively), given by









F FSYMFOR exp(Esplot + EP1) This equation was built so that F would always be

greater or equal to zero and, when the random effects were equal to zero, F would be
equal to SYMFOR

S Mortality of trees: Let Yhkn denote if tree h, from species group s, subplot, and plot k,
died (1) or stayed alive (0) in time interval n (with duration of tn years). I assumed that
Yjkn ~ Bernoulli(p), where p is the probability of dying. This probability was estimated
as = 1- (1 -M)'", where AM is the annual mortality rate given by
SMSYMFOR/-MSYMFOR) Xp(Esubplot )
M = exp(EPl This equation can be summarized as
1 + [SYMFOR /(1 MSYIFOR )] exp(Ebplot )
logit( ) = logit(MSYOR) + Esubpot Similar to the notation used before, MSYMoR is the
annual mortality rate as estimated by SYMFOR's mortality sub-model (Table 2-2), and
Esubplot is the random subplot effect. This equation was built so that AM would always be
between zero and one and, when the random effect was equal to zero, AM would be equal to
M SYMFOR '

Growth data were not used to estimate the probability of each model because all models

shared the same growth sub-model (except for the MSA Growth model, where the growth data

do not contribute to discerning this model from the others). All random effects were assumed to

come from a normal distribution with mean zero and variance to be estimated. The decision of

which random effects to include was based on the number of levels of each random effect (a

small number of levels would not allow a good estimation of the variance associated with it;

Bolker et al., in prep.) and a preliminary analysis of the correlation structure of the residuals. Flat

priors (uniform between 0 and 10,000) were given to the precisions of the normal distributions.

Gibbs sampling through WinBUGS 1.4 (Bayesian inference Using Gibbs Sampling for

Windows; Spiegelhalter et al. 1996) was used twice: first to estimate the modes of the variances

of the random effects, second to estimate the mean likelihood of each model (with variances

fixed at their estimated modes). The mean likelihood of each model is an approximation to the

marginal likelihood, averaged over all possible values of the random effects. For the first step, 3









chains (with different initial values) were run to check for convergence (using Gelman and

Rubin's convergence statistic, as modified by Brooks and Gelman (1998), a value lower than 1.2

was taken as indicative of convergence), each with 5,000 iterations. To estimate the mean

likelihood of each model, given the data, a single chain was used with 5,000 iterations.









Table 2-1. Examples of assumptions contained in a sample of forest dynamic models (see main text for a description of these
assumptions)
Assumptions
Models DEA DEA MSA MSA References
Recruit. F. Tuning Mortal. Growth


SELVA


GEMFORM
MYRLIN
SYMFOR-
SIMFLORA
TROLL
FORMIND 2.0
CAFOGROM


S (Gourlet-Fleury and Houllier 2000, Gourlet-Fleury et al. 2005)
S(Chambers et al. 2001, Chambers et al. 2004)
(Alder unpublished manuscript, Alder 2002, Marshall et al.
2002)
(Alder 2002, Alder et al. 2002, Nicol et al. 2002)
i (Phillips et al. 2003, Phillips et al. 2004, van Gardingen et al.
2006, Valle et al. 2007)
S (Chave 1999)
S (Kammesheidt et al. 2001, Kohler et al. 2003)
S (Alder and Silva 2000)
(Favrichon 1998)









Table 2-2. Original growth, recruitmentft and mortalityftt sub-model parameters
Species Growth model Recruitment model Mortality model
groups ao al a2 as a4 rl r2 r3 r4 bd mo mi
1 0.003 0.008 0.045 -0.001 0.117 0.029 -3.047 0.000 0.006 7.5 2.6 1.8
2 0.005 -0.011 0.050 0.000 0.114 0.045 0.000 0.000 0.000 3.8 2.9 3.0
3 0.745 -0.739 0.000 -0.001 0.195 0.000 0.000 0.044 0.014 15.0 2.0 1.1
4 0.004 0.021 0.159 -0.001 0.103 0.301 -0.825 0.000 -0.249 5.0 3.0 2.4
5 0.001 0.022 0.029 0.000 0.002 0.701 -0.020 0.000 -0.696 7.5 2.3 1.2
6 -0.051 0.081 0.005 -0.001 0.152 0.008 -2.806 0.000 -0.001 7.5 2.5 1.6
7 2.263 -2.246 0.000 -0.002 0.368 0.074 -2.293 0.000 -0.004 7.5 4.5 3.3
8 -0.003 -0.175 0.163 -0.001 0.600 0.000 -6.906 0.000 0.003 15.0 2.5 0.9
9 0.009 0.333 0.078 -0.005 -0.428 0.059 -0.788 0.000 -0.051 5.0 8.9 3.6
10 0.007 0.081 0.029 -0.001 -0.135 0.000 0.000 0.002 0.004 15.0 4.3 2.8
Parameters were estimated by P. Phillips (unpublished manuscript) except for bolded parameters (see main text for detailed
S explanation). tGrowth sub-model: I =D(ao + ae "')+ ac+aa4 ttRecruitment sub-model:F = rle' + l +'1. tttMortality sub-model:
mo mifD n if bd +5 < D. D is the diameter at breast height (cm), C is the diameter-independent competition index, F is the recruitment rate in a
10 x 10 m square (trees yr-1), Iis the diameter increment (cm yr-1), Mis the mortality rate (%), bdis the upper limit of the first
diameter class (cm), and ao, al, a2, a3, a4, rl, r2, r3, r4, mo, and mi are the estimated parameters. For further details regarding variable
definitions and equation forms, refer to Phillips et al. (2002a, 2004).









Table 2-3. Approaches and modified parameters used to implement assumptions
Assumption Approaches Description Modified parameters
Recruitment model modified in order to replace any killed or
DEA naturally dead tree by a new recruited tree. This new tree has Only recruitment model structure
Recruitment the same tree attributes from the dead tree except for the DBH, modified in source code.
Undisturbed forests which was set to 5 cm.
are in a dynamic Simulation of an undisturbed forest was run for 100 years and
equilibrium DEA model parameters were fine-tuned (modified iteratively within
Fe T g their 95% confidence interval) to exhibit a constant species Table 2-4
Fine Tuning
composition (constant basal area and tree density per species
group) during this time window.
MSA T Only mortality model structure
Tree species have a MSA Mortality was increased to 100% for trees with DBH > Dmax. Only mortality model structure
maximum attainable Mortality modified in source code
m axim um attainable ------------------------ -----
MSA A Only growth model structure
size MSA Growth was reduced to zero for trees with DBH > Dmax. n ro mode srucur
Growth modified in source code.
DEA and MSA stand for Dynamic Equilibrium Assumption and Maximum Size Assumption, respectively. Dmax is the maximum
diameter (cm) of a species group, based on the available data.









Table 2-4. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model (modified parameters
in bold)
Species groups characteristics DEA Fine Tuning
Group name (reference) D95 G ri r4 mo m1
1 Slow growing mid-canopy 41.8 0.21 0.029 0.006 2.3 2.0
2 Slow growing understory 15.9 0.09 0.045 0.000 2.3 2.9
3 Medium growing mid-canopy 57.2 0.29 0.000 0.013 1.9 1.4
4 Slow growing lower canopy 27.7 0.18 0.301 -0.238 2.9 2.7
5 Medium growing upper canopy 72.5 0.26 0.701 -0.696 1.9 1.2
6 Fast growing upper canopy 76.0 0.54 0.001 -0.008 3.3 2.0
7 Fast growing pioneers 35.8 0.54 0.040 -0.100 5.0 3.8
8 Emergents climax 104.0 0.37 0.000 0.005 3.0 0.9
9 Very fast growing pioneers 38.7 1.26 0.030 -0.100 10.4 4.9
10 Very fast growing upper canopy 78.2 0.94 0.000 0.004 4.3 3.9
D95 is the 95th percentile of the cumulative diameter frequency distribution (cm), G is the average growth rate (cm yr1), rl and r4 are
parameters from the recruitment sub-model, and mo and mi are parameters from the mortality sub-model.









Table 2-5. Analysis of Variance used to determine uncertainty due to plots and model stochasticityt
Source of variation DF E(MS)
Plot npiot-l 2 +n -2
y,ms, rep y,p,
Error nplot *((nep-) a2
y,ms,z
Total nplot'"n .,-1
t These parameters were estimated separately for each model (baseline model plus one model for each of the four assumptions), for
every logging scenario (logged and unlogged forest), and for every 10-yr time step. This analysis corresponds to the nested design
shown in Fig. 2-1A. O2,. and 2 are the variances due to model stochasticity and due to plots (i.e., due to different starting
conditions), respectively, for the ith model; nplot and nep are the number of plots (i.e., 15) and number of repetitions per plot (i.e., 20),
respectively.

Table 2-6. Analysis of Variance used to determine parameter uncertainty for the baseline model
Source of variation DF E(MS)
Plot npo-1 ,ms,, + np ypu, + nrps lps y,p,l
Par. uncertainty lplot*( n -1) 2, + nl, S 2ps
M. stochast. nplot* npn*(nrps-1) 2
Total nplot* 11 '"I -1
tThis parameter was estimated separately for each logging scenario (logged and unlogged forest), and for every 10-yr time step. This
analysis corresponds to the nested design shown in Fig. 2-1B. 2 ,,,, 2, ,,, and ,2 are the variances due to plots (i.e., due to different
starting conditions), due to parameter uncertainty, and due to model stochasticity, respectively, for the ith model; nEiot, n and n' are
the number of plots (i.e., 15), number of repetitions per parameter set (i.e., 2), and the number of randomly drawn parameter sets per
plot (i.e., 250), respectively.









































Figure 2-1. Nested design used for simulations. R and PS stand for repetitions and parameter
sets, respectively. A) Nested design used to determine uncertainty due to
assumptions, to starting conditions effect and to model stochasticity. This design was
used for the 5 models evaluated (baseline + 4 assumptions); B) Nested design used to
determine parameter uncertainty. Only the baseline model was used for these
simulations.









CHAPTER 3
RESULTS

Description of the Baseline Simulation Set for the Unlogged Forest

Simulations of the unlogged forest from region 11 and region 14, without any of the

dynamic equilibrium or maximum size assumptions (i.e., baseline model), predicted drastic

changes over a period of 100 years in species group composition, both in basal area and tree

density, particularly for the pioneer species groups (species groups 7 and 9). Also, tree density

and basal area of big trees (DBH > D95 [the 95th percentile of the diameter distribution in the

beginning of the simulation]) increased for nearly all species groups (Table 3-1).

These somewhat counterintuitive results for an undisturbed forest are shown to indicate

why a forest modeler might want to adoptpost hoc assumptions that force the model to behave

as expected. The effect of these assumptions is analyzed below.

Mean Modeling Results

Models with different assumptions did not predict drastically different stand dynamics for

short simulation lengths, within the time span of available field data (Fig. 3-1). Furthermore, a

visual comparison of the observed versus the simulated stand level results does not allow a clear

cut determination of which assumptions best fitted the observed data, which might mislead one

to conclude that uncertainty due to assumptions is negligible.

Nevertheless, the average long-term (i.e., 100 years) simulation result changed drastically

according to the adopted assumptions, particularly for the dynamic equilibrium assumptions,

suggesting that assumptions are indeed a major source of uncertainty in model projections (Fig.

3-2). Unfortunately, empirical data covering this time span do not generally exist and therefore it

is not possible to discern which of the simulation results best reflects reality. Again, as shown

previously, the baseline scenario clearly indicates a mean trend of overall basal area that is









outside the natural variability currently found within the area (black circles in Fig. 3-2), both for

the unlogged and logged scenarios. One might assume that the basal area of the unlogged

simulations after 100 years should be within the range of the recently measured basal area (as

often assumed in the forest modeling literature), in which case one would judge the model with

the DEA Tuning assumption to be more biologically sound when compared to other models (i.e.,

baseline, MSA Growth, MSA Mortality, and DEA Recruitment).

Another important aspect is that the effect of adding a given assumption to the baseline

model may depend on the management scenario being simulated (see Fig. 3-2). For instance,

when the projections of the DEA Recruitment model are compared to the projections from the

baseline model, it is clear that the difference in their projections is larger in the logged scenario

then in the unlogged scenario from year 10 to year 60. As a consequence, even if all simulations

were carried out with a single set of assumptions (e.g., DEA Recruitment), it would not

necessarily follow that the comparison of logging scenarios (e.g., different cutting cycles,

logging intensities, and/or harvesting systems) would remain unchanged since this assumption

might have different effects depending on the specific logging scenario being simulated.

Probability of Each Model and Comparison of Uncertainty Sources

The dynamic equilibrium assumption (DEA) and the maximum size assumption (MSA) are

often adopted after the model has been calibrated and some of its outputs assessed, typically as a

way to avoid what is judged to be unrealistic model behavior. Also, field data are generally not

used to evaluate these assumptions, particularly because model modifications due to assumptions

are often seen as "minor" changes in parameters. Thus, to assess the uncertainty of adopting

these assumptions without using field data to evaluate them, equal weights were set for all

models (;, = 1/5).









The effects of all sources of uncertainty tended to increase with time except for the effect

of starting conditions. Also, the joint effect of all of the chosen assumptions (i.e., MSA Growth,

MSA Mortality, DEA Recruitment, DEA Fine Tuning) was the greatest source of uncertainty

(except in relation to commercial basal area in the unlogged scenario), probably due to the fact

that, in contrast to the other sources of uncertainty, between model variance does not decrease

with an increased number of simulations or plots measured. In general, the starting condition

effect was the second greatest source of uncertainty, possibly due to the small number (i.e., 15)

of 1 ha plots being simulated. Parameter uncertainty and model stochasticity were of similar

magnitude and considerably smaller than assumption and starting conditions effects. Clearly,

overall model uncertainty is often dominated by the uncertainty due to assumption effects (Fig.

3-3).

When the data are used to estimate the probability of each model (which implies that the

assumptions are evaluated based on the field data), these results drastically changed,

substantially reducing the uncertainty due to assumptions. As expected, the sub-model data

equally supported the baseline, MSA Growth and MSA Mortality models (i.e., these three

models had the same posterior model probability; Table 3-2). Although commercial basal area

was best predicted by the DEA Fine Tuning model (data not shown) and this assumption was

implemented by carefully changing parameter values within their confidence intervals, the sub-

model data clearly conflicted with the DEA models, yielding an approximately zero probability

for this model. With equal weights for the MSA Growth, MSA Mortality and baseline models,

the greatest source of uncertainty became the starting conditions effect while all the remaining

sources of uncertainty were of similar magnitudes (Fig. 3-4).









It is important to note that, over the 100-yr simulation, using data to estimate the

probability of each model resulted in a 13-fold decrease in overall variance of the mean for the

stand basal area, both for logged and unlogged simulations, and a 5-fold and 2-fold decrease in

overall variance for the commercial basal area in the logged and unlogged simulations,

respectively (note the different scales used in Fig. 3-3 when compared to Fig. 3-4). Furthermore,

the multimodel average (average of the projections from all models, weighted by the probability

of each model) also substantially changed when data were used to estimate model probabilities

(Fig. 3-5). The reduction in overall variance of the mean and the change in the multimodel

average have practical relevance from a forest management perspective. For instance, when data

are taken into account in estimating the probability of each model, the overall basal area in the

logged forest is expected to return to its pre-logging levels within 20 to 30 years, instead of

within 20 to > 100 years, and the post-logging recovery of commercial basal area after 100 years

is 46-59%, instead of 37-67%.









Table 3-1. Percentage change in the unlogged forest
Species All trees Large trees (DBH > D95)
groups Basal area Tree density Basal area Tree density
1 -10 -22 46 30
2 -33 -25 -25 -40
3 18 7 200 182
4 -20 -23 69 18
5 -10 -30 53 33
6 173 89 407 324
7 444 202 1092 836
8 -7 -45 18 25
9 1015 987 2209 2015
10 70 57 415 175
Total 29 2 141 82
Percentage change was determined as (100*[Yloo-Yo]/Yo), where Y, is the plot characteristic in
year i and D95 is the 95th percentile of the diameter distribution in the beginning of the
simulation.


Table 3-2. Posterior probabilityT of each model, given the recruitment data, the mortality data,
and both datasets combined
Recruitment Mortality Both datasets
Model .
data data combined
Baseline 0.333 0.25 0.333
MSA mortality 0.333 0.25 0.333
MSA growth 0.333 0.25 0.333
DEA recruitment 0.000 0.25 0.000
DEA tuning 0.000 0.00 0.000
SThe posterior probability was estimated with Bayes theorem by combining the prior probability
(each model had an equal prior probability, 1/5) and the likelihood (estimated using WinBUGS).
Recruitment and mortality data came from logged and unlogged forests (regions 11, 12, and 14).




















o DoEA Recrlt
ACalendar ye, I











C
"pA 8 M oAG .





















1980 1985 1990 1995
Calendar year
1982 1984 1986 1988 1990 1992
Calendar year
-. -. -


9 0

1982 I I I














1982 1984 1986 1988 1990
Calendar year


, 8




I I I I
S0)







z D
1982 1984 1986 1988 1990 1992
Calendar year






oo 0









,a I_





(0 0
1980 1985 1990 1995

Calendar year



---F-


Figure 3-1. Comparison of observed versus simulated stand level data. Observed stand level
results are shown in black (empty black circles are 1-ha plots and black line is their
average) and average simulation results from models with different assumptions, over
the time window of the observed data, are shown by grey lines. Overall basal area is
shown in left panels (A,C,E) and commercial basal area is shown in right panels
(B,D,F). Top panels (A and B) show region 11 (logged forest), middle panels (C and
D) show region 12 (logged forest), and lower panels (E and F) show region 14
(unlogged forest).
















-A B

L0


E o
0 0
g


0 Baseline
SDEA Recruit.
o DEA Tuning
Co4 MSA Mortal.
MSA Growth

oC 0D 0


S O 0 0
(0

E -



E
CO




0
C D)


0 20 40 60 80 100 0 20 40 60 80 100

Time (years) Time (years)


Figure 3-2. Mean simulation results from different models (baseline + 4 assumptions) over a
time window of 100 years. Simulations were initiated with 15 one-hectare plots from
regions 11 and 14, and logging (in logged scenario) occurred in year 0. Initial
observed overall and commercial basal area values for each of the 15 plots are
represented by black circles placed in year 100 to illustrate how mean simulation
results, except from the DEA Tuning model, can extrapolate beyond the original data
range. Left panels (A and C) show results from logged forest and right panels (B and
D) show results from unlogged forest; upper panels (A and B) show overall basal area
and lower panels (C and D) show commercial basal area.















SA 0 B



(D C-
> .. -.


Cl lI o .. I

0 10 20 30 40 50 D8 70 80 9 100 0 10 20 30 40 50 S0 70 80 90 100





O' C Param. uncertainty D
5 [ Starting cond. effect -
-E lo 0 Model Stochasticity
| cC [ -Assump. effect ; ,;

a l l l l l l l l li l I
: : 'l /"{





N I n I I IE I^a I I I I I o-
0 1 30 3 40 0 0 0 70 80 90 100 0 10 20 30 40 50 0S 70 80 90 100
Time (years) Time (years)

Figure 3-3. Overall variance of the mean when data are not taken into account in estimating
model probabilities (i.e., assumptions are not evaluated in light of the data),
partitioned between parameter uncertainty, starting conditions effect, model
stochasticity, and assumptions effect. Results from logged forest are shown in left
panels (A and C) and results from unlogged forest are shown in right panels (B and
D); upper panels (A and B) show results regarding overall basal area and lower panels
(C and D) show results regarding commercial basal area.















Ca-


M I IO





















p e Sctarting condi effect mod








(C m a D ashow rescts arears)
Co CDVww v m o |i | i || | |
W3 .3 3

































partitioned between parameter uncertainty, starting conditions effect, model
stochasticity, and assumptions effect. Results from logged forest are shown in left
panels (A and C) and results from unlogged forest are shown in right panels (B and
0 0

0 10 20 30 40 60 60 70 80 90 100 0 10 20 30 40 50 00 70 80 90 100







Cu *- Param uncertainty m D
S Starting cond. effect
Model Stochasticity ii
W Cj Assump. effect CD





















D); upper panels (A and B) show results regarding overall basal area and lower panels
(C and D) show results regarding commercial basal area.













Unlogged


A





/-



Without data
Ij With data


C






.-- -


20 40 60
20 40 60


B


r z


D


100
100 0


Time (years) Time (years)


Figure 3-5. Multimodel average (continuous line) and 95% confidence interval (dashed line),
shown when data are not used (grey) and when data are used (black) to estimate the
probability of each model. Results for unlogged forest are shown in left panels (A and
C) and results for logged forest are shown in right panels (B and D). Upper panels (A
and B) show overall basal area and lower panels (C and D) show commercial basal
area.


Logged









CHAPTER 4
DISCUSSION

Types of Assumptions

Many alternative assumptions can be considered biologically reasonable. Model

projections, however, are greatly influenced by the specific set of assumptions that are chosen,

particularly when these assumptions are not evaluated in light of the data (i.e., data are not used

to estimate the probabilities of each model). For instance, data are not generally used to compare

the baseline model to the fine tuned model. Furthermore, the same assumption can often be

implemented in many different ways, potentially resulting in very different long-term

projections, as shown for the DEA models. As a consequence, it is crucial that the uncertainty

associated with these assumptions be adequately reported.

The use of the data to estimate the probability of each model revealed a clear distinction

between the analyzed assumptions. The Dynamic Equilibrium Assumption (DEA, as

implemented in this thesis) forced the model towards the desired model behavior, either by

completely ignoring the recruitment data (as in the DEA Recruitment model), or by slightly

changing the fitted parameters from the recruitment and mortality sub-models (as in the DEA

Fine Tuning model). Both DEA approaches directly conflicted with the data. Similarly, the

Maximum Size Assumption (MSA) was originally implemented by fine tuning the mortality

parameter of large trees so that the undisturbed forest would keep a somewhat constant large tree

density per species group after 100 years of simulation. Despite the highly variable and small

dataset for large trees, this assumption (as previously implemented) nevertheless strongly

conflicted with the mortality data.

In contrast to the Dynamic Equilibrium Assumption (DEA), the Maximum Size

Assumption (MSA, as implemented in this thesis) was used to fill a knowledge gap that exists in









the calibration dataset, therefore not conflicting with the calibration data. Indeed, a multi-

component iterative model can easily start extrapolating outside the data range without an

obvious indication to the user that this extrapolation is occurring. Trees with DBH > Dmax are

one of the most obvious model extrapolations and assumptions regarding the dynamics of large

trees are needed. Other extrapolations are far more subtle and frequently go unrecognized. Fig.

4-1, for instance, depicts the combinations of diameter and competition index for which the

growth sub-model of species group 10 would extrapolate outside the range of field observations.

Knowledge gaps within the calibration dataset are indeed numerous and therefore assumptions

must be used to decide what the model should do in these situations. When not acknowledged,

the implicit assumption is that the statistical relationship between the dependent variable and the

covariates can extrapolate correctly outside the data range.

Thus, two types of assumptions are recognized in my study: the first type of assumption

changes the model despite its conflict with the empirical data (as in the dynamic equilibrium

assumption), while the second type does not conflict with the data since the changes to the

original sub-model (calibrated with the data) only result in different sub-model extrapolations (as

the maximum size assumption). I believe that only the second type is justified. Nevertheless, if

the first type of assumption is to be used, then the uncertainty associated with it should be

properly reported. For instance, if the data are ignored in estimating the probability of each

model, uncertainty due to this assumption can be estimated as the squared difference between the

average result from the projections with and without the assumption (i.e., this assumes both

models have the same probability).

Comparison of Uncertainty Sources

The field of statistics has traditionally acknowledged parametric uncertainty once a

particular model form has been chosen. The exclusion of model construction and selection









uncertainty has been shown, however, to result in over-optimistic predictive or inferential

uncertainty, which can have serious implications and result in over-confident decision making

(Draper 1995, Hoeting et al. 1999). Likewise, probably the most studied source of uncertainty in

the ecological modeling literature has been parameter estimate uncertainty, either with a local or

global sensitivity analysis (Saltelli et al. 2000, Ellner and Fieberg 2003).

The problem of ignoring model uncertainty is likely to be exacerbated in situations where

model extrapolations from available data are needed for decision making given that models that

are very different mathematically can have similar fits to the data but wildly different predictions

outside the data range (Chatfield 1995, Draper 1995). I have shown here that assumptions used

when sub-models are extrapolated (i.e., MSA) can have an effect of similar magnitude to other

more traditional sources of uncertainty, such as parameter uncertainty. The uncertainty

associated with the maximum size assumption, however, was relatively small. This observation

requires further investigation, in that it begs another question: Could the effect of assumptions,

when related to sub-model extrapolation, be the largest source of uncertainty? To compare the

effect of adopting different strategies when the growth sub-model extrapolated, I modified the

growth sub-model so that the species-group mean diameter increment was used whenever the

combination of covariates (diameter and competition index) extrapolated the data range (as

depicted in Fig. 4-1). Simulations with this modified growth sub-model were then compared to

simulations with the baseline model (which assumes that diameter increment is correctly

estimated by the growth sub-model when it is extrapolated). This comparison revealed that

assumption effect, even when used only to deal with sub-model extrapolations, can indeed be the

largest source of uncertainty (Fig. 4-2).









Given that models are, by definition, simplifications of reality containing numerous

assumptions, how can all these assumptions be taken into account? One might argue that even if

the simulations are biased, the comparison of different logging scenarios (e.g., different cutting

cycles and harvest intensity combinations) simulated with the same set of assumptions would

generally be unbiased (Phillips et al. 2003). Unfortunately, my results revealed that this might

not always be true given that the assumption effect may greatly depend on the scenario being

simulated (e.g., logged vs. unlogged forest; Fig. 3-2 and Fig. 4-2).

To take these assumptions into account, I believe that models should be built with many

redundant components (e.g., different equations to represent a given sub-model or alternative

assumptions when sub-models are extrapolated) and simulations performed using different

subsets of these components. Logically, this approach is only possible for models built in a

modular fashion. Then, different data sources could be used to assign updated probabilities to

each model structure. Also, model averaging has been increasingly suggested as a method that

can incorporate uncertainty due to model construction and selection (Wintle et al. 2003, Ellison

2004). However, to my knowledge, model averaging has rarely been extended to more complex

iterative models, such as forest dynamic models.

Uncertainty due to assumptions can only be reduced with carefully planned experiments or

targeted collection of observational data. For instance, thinning experiments might help to

determine growth patterns for combinations of diameter and competition index outside the range

of the data currently available. Likewise, the usual data collection for tropical forest dynamics,

through the monitoring of one hectare permanent plots (Alder and Synnott 1992), are unlikely to

generate sufficient data on large tree dynamics, due to their low density. In the absence of such a

data, model projections that integrate the results from models with different assumptions are









likely to be much more robust than projections based on a single model (and consequently a

single set of assumptions).
















0






CA 0


oo

20 40 60 80










DBH (cm)


Figure 4-1. Combinations of competition index and DBH contained in the data used to calibrate
the growth sub-model for species-group 10. This figure reveals the regions (indicated
by the question marks), in independent variable space, that will require an
extrapolation of the growth sub-model of species-group 10.























L0 W
CO CD
SA A sm B



o c



I I I I I I I I I I o I I I I I I I I I I I

S 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100





moe C I Param. uncertainty o D
] Starting cond. effect
t Model Stochasticity m ,t
c are Assump. effect ) 701


E o


> co 1 11

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Time (years) Time (years)


Figure 4-2. Overall variance of the mean when data are taken into account in estimating the
probability of each model (i.e., assumptions are evaluated in light of the data),
including only the simulation sets from the baseline model and the model with the
modified growth sub-model, partitioned between parameter uncertainty, starting
conditions effect, model stochasticity, and assumptions effect. Results from logged
forest are shown in left panels (A and C) and results from unlogged forest are shown
in right panels (B and D); upper panels (A and B) show results regarding overall basal
area and lower panels (C and D) show results regarding commercial basal area.









CHAPTER 5
CONCLUSIONS

* It is crucial that available data be used to determine the probability of each model (i.e., data
are used to evaluate the different assumptions adopted in the modeling process), to reduce
assumption uncertainty in particular, and overall model uncertainty in general.

* Uncertainty due to modeling assumptions can be of greater or similar magnitude when
compared to the other sources of uncertainty that are more commonly assessed.

* Targeted experimental studies are needed to reduce the need to extrapolate sub-model
results and to reduce uncertainty due to modeling assumptions.

* Overall modeling uncertainty is likely to be underestimated when these four sources of
uncertainty (i.e., model stochasticity, parameter estimation, starting conditions and
modeling assumptions) are not jointly taken into account.

I have shown how the overall uncertainty of the mean can be partitioned among the

different sources of uncertainty, with particular emphasis on the uncertainty that arises due to the

use of models with different assumptions. This method can potentially be applied to other types

of models.

Forest dynamic models have and will continue to be used to predict the outcomes of direct

or indirect human-induced changes (e.g., logging, burning, fragmentation or carbon

accumulation in the atmosphere), sometimes with millennium-long time windows (Chambers et

al., 2001). Nevertheless, given that the information content of modeling projections is inversely

proportional to their uncertainty and failure to adequately report uncertainty associated with

these projections can mislead decisions (Clark et al., 2001), it is critical to report the overall

uncertainty of model projections.









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BIOGRAPHICAL SKETCH

Denis Ribeiro do Valle was born in Sao Paulo, Brazil, having completed his undergraduate

studies in forestry at ESALQ, the Agriculture School of the University of Sao Paulo. During his

undergraduate studies, his experience was mostly with even-aged forestry, with internships in

cellulose companies such as International Paper and Arauco Forestal. Before finishing his

undergraduate studies, however, he participated on an experimental logging of mahogany in the

state of Acre, becoming fascinated with the challenges of uneven-aged forestry.

In August 2003, Denis started working as an assistant researcher at a non-governmental

organization (NGO) called Amazon Institute for the People and the Environment (IMAZON),

within the Dendrogene Project (a cooperation between IMAZON, Embrapa Amazonia Oriental,

Brazil, and the University of Edinburgh, Scotland). Most of his work was related to forest

dynamic modeling, focusing on questions regarding sustainable timber logging. In August 2006,

Denis started his master's studies at the University of Florida.





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1 ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By DENIS RIBEIRO DO VALLE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008

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2 2008 Denis Ribeiro do Valle

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3 To my family, in particular to my wife Natercia Moura do Valle.

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4 ACKNOWLEDGMENTS I thank m y advisor Christina Staudhammer and the committee members who have given me the freedom to pursue my research ideas and have greatly supported my search for the best methods to address these ideas.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4LIST OF TABLES................................................................................................................. ..........6LIST OF FIGURES.........................................................................................................................7ABSTRACT.....................................................................................................................................8CHAPTER 1 INTRODUCTION..................................................................................................................10Overview....................................................................................................................... ..........10Objectives...............................................................................................................................112 METHODS.............................................................................................................................13Model Description..................................................................................................................13Tapajos Dataset.......................................................................................................................13Modeling Assumptions Analyzed........................................................................................... 15Dynamic Equilibrium Assumption (DEA)...................................................................... 15Maximum Size Assumption (MSA)................................................................................ 17Modifications to SIMFLORA.................................................................................................18One Hundred Year Simulations.............................................................................................. 20Data Analysis..........................................................................................................................21Variance Component Analysis........................................................................................ 22Probability of Each Model Given the Data (i ).............................................................233 RESULTS...............................................................................................................................32Description of the Baseline Simu lation Set for the Unlogged Forest..................................... 32Mean Modeling Results..........................................................................................................32Probability of Each Model and Comparison of Uncertainty Sources..................................... 334 DISCUSSION.........................................................................................................................42Types of Assumptions............................................................................................................42Comparison of Uncertainty Sources....................................................................................... 435 CONCLUSIONS.................................................................................................................... 49LIST OF REFERENCES...............................................................................................................50BIOGRAPHICAL SKETCH.........................................................................................................57

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6 LIST OF TABLES Table page 2-1. Examples of assumptions containe d in a sam ple of forest dynamic models..................... 262-2. Original growth, recruitment and mortality sub-model parameters........................... 272-3. Approaches and modified parame ters used to implement assumptions............................282-4. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model................................................................................................................... ..292-5. Analysis of Variance used to determine uncertainty due to plots and model stochasticity........................................................................................................................302-6. Analysis of Variance used to determine parameter uncertainty for the baseline model..................................................................................................................................303-1. Percentage change in the unlogged forest..........................................................................363-2. Estimated posterior probability of e ach model, given the recruitment data, the mortality data, and both datasets combined....................................................................... 36

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7 LIST OF FIGURES Figure page 2-1. Nested design used for simulations.................................................................................... 313-1. Comparison of observed vers us simulated stand level data............................................... 373-2. Mean simulation results from differ ent models (baseline + 4 assumptions) over a time window of 100 years..................................................................................................383-3. Overall variance of the mean when da ta are not taken into account in estimating model probabilities (i.e., assumptions are not evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect.................................................................................393-4. Overall variance of the mean when da ta are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect.................................................................................403-5. Multimodel average (continuous line) a nd 95% confidence interval (dashed line), shown when data are not used (grey) and wh en data are used (black) to estimate the probability of each model.................................................................................................. 414-1. Combinations of compe tition index and DBH contained in the data used to calibrate the growth sub-model for species-group 10....................................................................... 474-2. Overall variance of the mean when da ta are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from th e baseline model and the model with the modified growth sub-model, partitione d between parameter uncertainty, starting conditions effect, model stochas ticity, and assumptions effect.........................................48

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8 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By Denis Ribeiro do Valle August, 2008 Chair: Christina Staudhammer Major: Forest Resources and Conservation Despite its local, regional and global importanc e, the Amazon forest faces multiple threats. Forest dynamic models have been widely used to evaluate the impact of a number of anthropogenic impacts on the forest, such as ti mber logging and climate change. I hypothesize that these models, however, have usually failed to report the full uncertainty associated with their projections. I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium assumption and maximum size assumption. I then qua ntified four sources of model uncertainty using the tropical forest simulation mode l SIMFLORA: model stochasticity, parameter uncertainty, starting condition effect, and mode ling assumptions. My results suggest that modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect than other sources of uncertainty that are more commonly taken into account, such as parameter uncertainty, particularly when a ssumptions are used to deal with sub-model extrapolations. Also, to reduce assumption uncertainty in particular, and overall model uncer tainty in general, it is of fundamental importance to use the available data to determine the probability of each model (i.e., data are used to evaluate the different a ssumptions adopted in the modeling process). Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid

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9 the use of some of these assumptions. Using SIMF LORA as a case study, my results indicate that the overall modeling uncertainty is li kely to be underestimated if all four sources listed above are not simultaneously considered. Finally, the method developed in this thesis to partition overall variance of the mean into different uncertainty s ources can be applied to quantify the uncertainty of other models, not restricted to forest dynamic models.

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10 CHAPTER 1 INTRODUCTION Overview The Brazilian Amazon contains about 40% of the world's remaining tropical rainforest and its vital role in global biodiversity, terres trial carbon storage, regional hydrology and climate has been widely recognized (Nepstad et al. 1999, Laurance et al 2001, Malhi et al. 2002, Nepstad et al. 2002). At the same time the use of it s natural resources is essential for millions of rural Amazonians' health and livelihoods. For instance, many medicinal plant species found in the forest are the sole health care option for many rural poor (Shanley and Luz 2003), game is frequently the most important source of protei n and fat to these people (Redford 1992) and, on a regional scale, timber logging is one of the mo st important rural activities for the economy (Lentini et al. 2003). Despite the local, regional and global impor tance of Amazonias na tural resources, the Brazilian Amazon faces multiple threats. In 2003, the total deforested area was estimated at 15% of the original area (Soares-Fil ho et al. 2006), and this area is in creasing at one of the highest rates in the world, with a m ean annual increase of 18,100 km2 yr-1 (Malhi et al. 2008). Furthermore, the remaining forest is not intact Generally uncontrolled, se lective logging affects an annual area ranging from 10,000 20,000 km2 (Nepstad et al. 1999, As ner et al. 2005), which can greatly increase fire risk, one of the greatest threats to the forests of Amaznia (Nepstad et al. 2001, Cochrane 2003). By 2010 approximately a tenth of the Brazilian Amazon is planned to be designated as forest concession area, where select ive logging is to take place (Verissimo et al. 2002). In this context, sustaina ble use of natural resources an d the balance between satisfying immediate human needs and maintaining other ecosystem functions wi ll require quantitative

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11 knowledge about the ecosystems present and future responses (Clark et al. 2001, DeFries et al. 2004). Numerous forest dynamic models have been de veloped to try to make reliable long-term and large-scale prediction using av ailable short-term and small-scale empirical data (Pacala et al. 1996, Kammesheidt et al. 2001). There has also been a growing awareness of the importance of quantifying modeling uncertainti es, with some leading scie nce journals (e.g., Ecology, Ecological Modeling, Global Environm ental Change) devoting issues so lely to this theme (Clark 2003, Dessai et al. 2007, Lek 2007). Nevertheless, when uncertainties from forest dynamic model projections are presented, th ey either refer to model stochasticity (Gourlet-Fleury et al. 2005, Degen et al. 2006), effect of starting conditions (Phillips et al. 2004, van Gardingen et al. 2006), or parameter uncertainty (Pac ala et al. 1996). A fourth source of uncertainty refers to the assumptions used when designing the model (e.g., th e choice of equations to represent ecological processes); this can be a key source of uncer tainty (Varis and Kuikka 1999, Qian et al. 2003, Brugnach 2005, der Lee et al. 2006). I do not know of any article that reports model uncertainties due to these assumptions and that analyzes a ll these sources of uncertainty jointly. As a consequence, I expect that the uncertainty in model forecasts has, in general, been underestimated. Objectives The objectives of this thesis, based on a commonly used fore st dynam ic model, are to: a) quantify model uncertainty derived from mode l stochasticity, parameter estimation, starting conditions and modeling assumptions; and, b) compar e these sources of uncertainty in order to evaluate which sources cont ribute the most to the overall model uncertainty. Using SIMFLORA as a case study, the main hypot heses I will test ar e: a) forest dynamic model uncertainty has been underestimated by not simultaneously including the sources of

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12 uncertainty identified above; and b) model assumptions are the gr eatest source of overall model uncertainty. As a result, I intend to propos e a new method to determine overall model uncertainty.

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13 CHAPTER 2 METHODS Model Description The Silviculture and Yield Management for Tropical Forests (SYMFOR) is a modeling framework that combines a management model, wh ich allows the user to specify silvicultural activities in mixed tropical forest (e.g., harvest, thinning, pois oning, enrichment planting), with an empirical spatially explicit individual treebased ecological model, which simulates the natural processes of recruitment, growth a nd mortality. The SYMFOR model, originally designed for use in Indonesia (Phillips et al., 2003), was adapted for use in Guyana (Phillips et al. 2002b) and then for the Brazilian Amazon (Phill ips et al. 2004). In Brazil, the model was further adapted by incorporating new management options and translating the user interface into Portuguese to become the model SIMFLORA. To date, three studies within the Brazilian Amazon have been published using SIMFLORA, two of them based on the Tapajs dataset (Phillips et al. 2004, van Gardingen et al. 2006) and one of them based on the Paragominas dataset (Valle et al. 2007). Other SYMFOR articles and reports can be found at www.symfor.org All results presented in this manuscript are assessed in relation to SIMFLORAs overall (all trees with diameter at breast height, DBH 5 cm) and commercial basal area projections based on the Tapajs data set. Commercial basal area is defined throughout this thesis as the basal area of trees from comm ercial species with DBH greater than the legal minimum logging diameter for the Br azilian Amazon region (i.e., 45 cm). Tapajos Dataset The series of plots at Tapajs km 114 comprise 60 Permanent Sample Plots (PSPs), each of 0.25 ha, initially measur ed (all trees with DBH 5 cm) in 1981 in unlogged primary forest. Twelve of these plots were left unlogged (regi on 14) while a silvicultu ral experiment with a

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14 randomized block design was installed in the remain ing 48 plots (region 11). In this silvicultural experiment, all treatments were logged for timber in 1982 with similar logging intensities across treatments but with different thinning intensit ies applied in 1993-1994. Th e series of plots at Tapajs km 67 (region 12) comprise 36 Permanent Sample Plots (PSPs), each of 0.25 ha. This stand was logged in 1979 but permanent plots were installed only in 1981. Detailed description of the forest and these experi ments can be found elsewhere (Sil va et al. 1995, Silva et al. 1996, Alder and Silva 2000, Phillips et al. 2004, Oliveira 2005). To initialize the model for the 100-yr simulati ons, I needed census data (xy-coordinates, species group, and diameter from each tree with in the plot) from unlogged plots in order to be able to simulate stand dynamics in two scenar ios: an unlogged scenario and a simulated logging scenario. Therefore, I only used the 1981 pre-logging data from region 11 and 14. Because the model requires 1 ha plots, it was necessary to jo in 4 plots to create a composite 1 ha plot, generating a total of 15 plots. Only plots from the same experimental block (prior to the experimental logging treatments) were joined together, in an effort to avoid within-plot variability and to increase between-plot variability. Another set of simulations were conducted to compare the observed with the simulated data. To compare how well SYMFOR simulates stand dynamics without having to worry about how well it simulates the logging and thinning treatments effec tively applied in the field, simulations were initialized w ith the 1981 unlogged forest data fo r those plots that were not logged (region 14) and with the 1981 and 1983 l ogged forest data from region 12 and region 11, respectively. For this comparison, the observed data consisted of the time series from regions 11, 12, and 14 prior to the thinning treatment. The sa me procedure to create composite 1 ha plots was used and, although this procedure mixed differen t treatments into a single composite plot, it

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15 does not affect the simulations as experimental treatments only differed in terms of thinning treatments applied in 1993-1994 and only pre-thinning data were used. Modeling Assumptions Analyzed Based on a literature review of forest sta nd dynam ic models applie d in the Amazon Basin, I identified several major modeling assumpti ons, from which two of the most common assumptions were chosen for the present analysis A brief description of these assumptions is given below. Dynamic Equilibrium Assumption (DEA) The dynam ic equilibrium assumption is a co mmon forest dynamics modeling assumption (Kammesheidt et al. 2001, Porte and Bartelink 2002), generally be ing interpreted as a stable basal area and/or tree density on species group and/or stand level for an undisturbed forest. This assumption is implemented by adjusting the mortality and/or the recruitment sub-models in such a way as to force the model to exhibit this e quilibrium. The pragmatic justification for this procedure is that recruitment and mortality data are notoriously noisy and therefore empirical parameters are likely to be poorly estimated and need to be adjusted. Indeed, long-term forest monitoring studies in general are not well suited to collect extens ive data on tree mortality: large sample sizes are required, mortality causes are not easily determined and errors on plot measurement (e.g., trees that were missed during measurement, lost their numbers or were harvested without record) can have a signifi cant impact on mortality estimates (Alder and Synnott 1992, Alder and Silva 2000, Alder 2002). Als o, recruitment data are plagued by the difficulty of species identification of seedlings and small trees and are generally highly stochastic (Vanclay 1994). As a consequence, in a recent re view of mixed forest models, recruitment was often found to be poorly modele d (Porte and Bartelink 2002).

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16 To force the model to exhibit a dynamic equ ilibrium, two approaches have usually been adopted (Table 2-1): Approach 1a : In the first approach (DEA Recruitmen t) every tree that dies, either due to natural mortality or logging, is replaced by a newly recruited tree with the minimum diameter of measurement. Approach 1b: The second approach (DEA Fine Tuning) uses an iterative method to finetune (modify) parameters derived from the da ta so that the model exhibits the desired behavior (i.e., the dynamic equilibrium for an undisturbed forest). My approach was to modify these parameters within their conf idence intervals. Tuni ng of the model is commonly done in many existing multi-component forest growth models, both empirical and mechanistic (Ger tner et al. 1995). The dynamic equilibrium assumption has a long tr adition in fisheries, forestry and ecology. For instance, the concept of maximum sustainable yield, both in fisheries and forestry, is based on the idea that, in the long term, the number of individuals tends to remain constant when it is equal to the carrying capacity of the ecosystem. In ecology, the a ssumption that in the long-term an undisturbed forest exhibits a dynamic equ ilibrium is wide-spread. For instance, the assumption that the forest is in the steady state has been used for the metabolic theory of ecology (Brown et al. 2004), to generate corrections of recruitment rate (Sheil and May 1996) and net primary productivity (Malhi et al. 2004), to dete rmine instantaneous decomposition rates (Palace et al. 2008), and to derive the expected steadystate diameter distribut ion based on demographic rates (Coomes et al. 2003, Kohyama et al. 2003, Muller-Landau et al. 2006). Despite its wide use, the dynamic equilibrium assumption has been recently contested based on empirical findings that reveal that undi sturbed tropical forests have been accumulating biomass and have shown increased turnover rate s (Phillips and Gentry 1994, Phillips et al. 1998, Baker et al. 2004, Lewis et al. 2004, Phillips et al. 2008).

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17 Maximum Size Assumption (MSA) Although large trees comprise a m ajor fraction of above-ground forest biomass (Clark and Clark 1996, Chambers et al. 1998, Chambers et al. 2001, Keller et al. 200 1, Chave et al. 2003), the relative scarcity of these in dividuals and difficulty of measurement limits data collection for maximum tree size, large tree growth and morta lity rate. As a consequence, simulation models often create trees considered to be unrealistically large or old (Porte and Bartelink 2002). Two approaches have generally been adopted to prevent trees from gr owing to unrealistic sizes (Table 2-1): Approach 2a : The first approach (MSA Mortality) is to arbitrarily enhance mortality probability over a given diameter threshold. My a pproach was to increase the mortality rate to 100% for trees that reached the maximum diameter (based on the Ta pajos dataset) of its species group. Approach 2b: The second approach (MSA Growth) is to fit a diameter increment function in which increment tends to zero as tree diam eter tends towards the species maximum size or simply to assume (as I have done) that incr ement drops to zero af ter the tree reaches the species maximum size. There is mixed empirical evidence for these assumptions. For instance, a low mortality and a continued diameter growth was observed for large individuals (>70 cm DBH) in a tropical forest in Costa Rica (Clark a nd Clark 1996, Clark and Clark 1999) In the Brazilian Amazon, the largest trees were observed to have the highest growth rates (Vieira et al. 2004). A final approach would be to estimate the mo rtality rate based on species maximum size (or age) and mean diameter increment or diam eter distribution (Chave 1999, Mailly et al. 2000, Alder et al. 2002, Kohler et al 2003, Degen et al. 2006). This approach was not analyzed because it is generally a dopted only when no data on mortality are available. I limited my analyses to these assumptions in order to keep my simulations, results and discussions more concise. However, I do acknowledge that there are numerous other assumptions within forest dynamic models, such as: a) assuming indepe ndent annual diameter

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18 growth and/or recruitment rate (i .e., no serial correlation); b) usi ng coarse estimates of gap sizes and gap formation frequency, sometimes based on best guesses (Phillips et al. 2004); c) assuming that all trees above the minimum felling diameter from medium-sized and large, midand late successional species are harvestable, instead of identifying which trees are from commercial species (Kammesheidt et al. 2001); and d) assuming no harvest loss or a fixed harvest loss (e.g., due to hollowed trees, logs not found, stumps cut too high), which, depending on the model, can vary from 30 to 60% (Huth and Ditzer 2001, Kammesheidt et al. 2001, Phillips et al. 2004). Modifications to SIMFLORA Two of the uncertainty sources (m odel stocha sticity and effect of starting conditions) are already simulated by SIMFLORA, and hence coul d be assessed directly. In contrast, the uncertainty derived from modeling assumptions ha d to be assessed by either changing initial parameter values (see original parameter estima tes in Table 2-2) or by modifying SIMFLORAs source code. Modifications due to modeling assump tions are briefly descri bed in Table 2-3 and the modified parameters are shown in Table 2-4. Uncertainty due to parameter estimation was assessed by allowing parameters of the main simulated ecological processes (i.e ., growth, recruitment and mortal ity sub-models) to vary. This uncertainty arises due to unmeasured covari ates (termed "process error") or errors in measurement (termed "observation error" or "measurement error") (Ellner and Fieberg 2003, Clark and Bjornstad 2004). There are many sub-categories of process error and each implies different strategies to simulate uncertainty in parameter estimates. For instance, variation among individuals might be simulated by randomly draw ing parameter estimates for each tree at the beginning of the simulation or when it is recruited and keeping them fixed throughout the simulation, while variation due to climate year-to-year differences might be simulated by

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19 randomly drawing parameter estimat es for each year and keeping them fixed for all trees during that given year. Although it would be intere sting to assess the differences among these alternative strategies, I chose to simulate uncertainty in parameter estimates as if it were solely due to measurement error. Therefore, I randomly drew parameter estimates in the beginning of each run, keeping them constant throughout the r un, similar to the error analyses conducted by others (Pacala et al. 199 6, Wisdom et al. 2000). The data required to assess parameter estima tion uncertainty were the parameter estimates and their covariance matrices within a given su b-model. Estimation of the original non-linear regression parameters (using PR OC NLIN; SAS Institute Inc. 2000) for the recruitment model (43 12rIrerFIr, in which parameter r3 was set to zero) failed to converge for two species groups in the original SIMFLORA calibration, yielding high standard errors. Therefore, for these two species groups, parameters r1 and r2 were set to zero and parameters r3 and r4 and their respective covariance matrix were re-estimated after the m odel successfully converged (Table 22). The original growth model was calibrated without taking into account the repeated measures nature of the diameter increment data. The justification for this was that parameter estimates were unbiased and biased standard e rrors of the parameters would not be used. I modified the covariance matrices provided by PROC NLIN by re-estimating the variances of the mean parameters, assuming that the number of independent data was eq ual to the number of trees, arguably biasing high the parameter variances. Overall parame ter uncertainty is expected to be lower than what will be reported throughout this manuscript. Once the covariance matrix for each equation was obtained, a vector of n random multivariate normal parameter estimates (X) was generated by using a vector of n (the number of

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20 correlated parameters) independent standard normal numbers (Z), a vector of the mean parameter estimates () and the lower triangular Cholesky matrix (A), given by eqn. 2-1: AZX [2-1] Given that each sub-model for each species gr oup was calibrated separately, I assumed that parameters between sub-models or between sp ecies groups were not correlated. To avoid biologically unrealistic growth a nd recruitment, diameter increment and recruitment rates were constrained by imposing an upper limit equal to the observed (species group specific) 99th percentile of the diameter increment and recruitment rate, respectively. One Hundred Year Simulations I sim ulated stand dynamics for two extreme scen arios: i) an undisturbed forest; and ii) a heavily logged-over forest (where a simulated logging extracted all trees 45 cm DBH from commercial species resulting in a mean logging intensity of 75 6 m3 ha-1 [mean 95% confidence interval]). The logging was simulated in the beginning of the run and was exactly the same for all simulations in order to ensure an identical starting point for all subsequent stand projections. These two extreme scenarios were chosen so that the potential range of the assumption effects on projected overall and comm ercial basal area could be assessed. Five sets of simulations (baseline + one se t for each assumption) were generated, each consisting of twenty 100-yr simulations for each plot and scenario (undisturbed and heavily logged-over forest; Fig. 2-1A). The baseline si mulation for my study used the parameter set estimated directly from the data (Table 2-2; P. Phillips unpublished manuscript) and, despite not being completely free from assumptions, it is the simulation that most accurately reflects the data used for calibration.

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21 One extra set of simulations was run solely to determine uncertainty due to parameter estimation, consisting of 500 100-yr simulations fo r each plot and scenario. Parameters were drawn randomly every two repetitions, resulting in a nested experi mental design (individual runs nested within parameter sets nested within plots; Fig. 2-1B) which allowed my statistical analysis to disentangle uncertainty due to model stocha sticity, parameter estimates and the effect of starting conditions (details are given in the Da ta Analysis section). I chose to determine parameter estimation uncertainty with a separa te set of simulations because mean model projections when parameters were allowed to vary differed markedly from the baseline simulation (with parameters fixed at their mean values). This is a consequence of Jensens inequality acting on the non-linear sub-models used by SIMFLORA (examples and discussion regarding Jensen's inequality effect can be found in Pacala et al. 1996, Ruel and Ayres 1999). Data Analysis Let } ,...,{1mSSL be a finite set of model alternatives, x be the data, y be the response variable, i and 2ibe the expected value and the variance, respectively, of the response variable given the data and the ith model alternative (i.e.,iiSxyE ),| (; 2),|(iiSxyVar), and the probability of the ith model given the data be i (i.e., i ixSP )| (). Draper (1995) showed that m i ii m i iiLxyVar1 2 1 2)( ),|(. In other words, the variance of the response variable is the sum of the within model variance and the between model variance, both weighted by the probability of each model given the data. Using similar arguments, it can be shown that m i ii m i iyiLxyVar1 2 1 2 ,)( ),|(, where 2 iyis the variance of the mean of the i th model alternative. This equation can be further

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22 expanded by decomposing the within model variance of the mean in to the variances of the mean due to model stochasticity, due to plots (i.e., di fferent starting conditions), and due to parameter uncertainty ( 2 ,, imsy, 2 ,, ipy, and 2 ,, ipuy, respectively). The uncer tainty due to modeling assumptions is defined here as th e variance between models, given by m i ii 1 2)(. Therefore, the key equation in our study that allo wed me to partition the overall variance of the mean into the different uncertainty sources is given by eqn. 2-2: m i ii m i ipuyipyimsyiLxyVar1 2 1 2 ,, 2 ,, 2 ,,)() (),|( [2-2] The elements of the within model variance of the mean ( 2 ,, imsy, 2 ,, ipy, and 2 ,, ipuy) in eqn. 2-2 were estimated by dividing the correspond ing variance component (determined by the variance component analysis de tailed below) by the appropriate number of observations ( n). The estimation the probability of each model given the data (i ) is also detailed below. Variance Component Analysis The variance due to starting conditions and due to model stochasticity (2 ,, imsyand 2 ,, ipy, respectively) were determined using the expected means squares from an Analysis of Variance (ANOVA) with one random effect (Table 2-5) estimated for every 10-yr time step. Using the simulation set in which parameters were allowed to vary, I determined the uncertainty due to parameter estimation (2 ,, ipuy) by using the variance components analysis summarized in Table 2-6. This table was also estimated separately for every 10-yr time step. The variance due to parameter uncertainty was the only result used from this set of simulations. Due to the prohibitively large amount of comp uter time necessary to determine 2 ,, ipuy for all models and

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23 since I was interested in the ma gnitude and not the exact value of this parameter, I determined 2 ,, ipuyonly for the baseline model and assumed it to be the same for all of the analyzed models. Probability of Each Model Given the Data (i ) Bayes rule was used to combine the information from the mortality, growth, and recruitment datasets into a single number, na mely the posterior probability of each model (onwards referred to simply as probability of each model). The use of Bayes rule for model 1 ( M1), for example, is given by eqn. 2-3: )()|()|()|( )()|()|()|( ),,|(1 1 1 1 1 1 i g ir im g r m grmMMDLMDLMDL MMDLMDLMDL DDDM [2-3] where L is the likelihood, ) (iM and ),,|(grmiDDDM are the prior and posterior probabilities, respectively, of model Mi; and Dm, Dr, and Dg denote the mortality, recruitment, and growth datasets, respectively. Equal priors were assigned to each model (i.e., 5 /1)( iM ). The likelihood of each dataset given each model (), |(),|(ir imMDLMDL and ) |(igMDL ) was not easily estimated since the data within each data set were not independent. To ad equately determine the likelihood of each dataset given each model, I modeled eac h dataset as a function of SYMFORs equations and inserted random effects to circumvent the lack of indepe ndence of individual observations, as described below: Recruitment of trees: Let sjknR be the number of recruited trees from species group s, in 10 x 10 m subplot j, within plot k, in time inte rval n (with duration of tn years). I assumed that ) *(~ FtPoisson Rn sjkn, where F is the annual recruitment rate. The annual recruitment rate F is a function of the annual recru itment rate estimated by SYMFORs recruitment equation (SYMFORF recruitment equation shown in Table 2-2), and subplot and plot random effects (subplotE andplotE respectively), given by

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24 ) exp(* plot subplot SYMFOREE FF This equation was built so that F would always be greater or equal to zero and, when the random effects were equal to zero, F would be equal to SYMFORF Mortality of trees: Let hsjknY denote if tree h, from species group s, subplot j, and plot k, died (1) or stayed alive (0) in time interval n (with duration of tn years). I assumed that ) (~pBernoulli Yhsjkn, where p is the probability of dying. This probability was estimated asntM p) 1(1 where M is the annual mortality rate given by )exp(*)] 1/( [1 )exp(*)] 1/( [ subplot SYMFOR SYMFOR subplot SYMFOR SYMFORE M M E M M M This equation can be summarized as logit( M ) = logit(SYMFORM ) + subplotE. Similar to the notation used before, SYMFORM is the annual mortality rate as estimated by SYMFORs mortality sub-model (Table 2-2), and subplotE is the random subplot effect. This equation was built so that M would always be between zero and one and, when the random effect was equal to zero, M would be equal to SYMFORM Growth data were not used to estimate the probability of each mode l because all models shared the same growth sub-model (except for the MSA Growth model, where the growth data do not contribute to discerning this model from the others). All random effects were assumed to come from a normal distribution with mean zero and variance to be estimated. The decision of which random effects to include was based on the number of leve ls of each random effect (a small number of levels would not allow a good es timation of the variance associated with it; Bolker et al., in prep.) and a preliminary analysis of the correlation structur e of the residuals. Flat priors (uniform between 0 and 10,000) were given to the precisions of the normal distributions. Gibbs sampling through WinBUGS 1.4 (Bayesian inference Using Gibbs Sampling for Windows; Spiegelhalter et al. 1996) was used twice: first to estim ate the modes of the variances of the random effects, second to estimate the mean likelihood of each model (with variances fixed at their estimated modes). The mean like lihood of each model is an approximation to the marginal likelihood, averaged over all possible values of the random effects. For the first step, 3

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25 chains (with different initial values) were run to check for convergence (using Gelman and Rubins convergence statistic, as modified by Brooks and Gelman (1998), a value lower than 1.2 was taken as indicative of convergence), each with 5,000 itera tions. To estimate the mean likelihood of each model, given the data, a si ngle chain was used with 5,000 iterations.

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26Table 2-1. Examples of assumptions contai ned in a sample of forest dynamic models (see main text for a description of these assumptions) Assumptions Models DEA Recruit. DEA F. Tuning MSA Mortal. MSA Growth References SELVA (Gourlet-Fleury and Houllier 2000, Gourlet-Fleury et al. 2005) ----------(Chambers et al. 2001, Chambers et al. 2004) GEMFORM (Alder unpublished manuscript, Alder 2002, Marshall et al. 2002) MYRLIN (Alder 2002, Alder et al. 2002, Nicol et al. 2002) SYMFORSIMFLORA (Phillips et al. 2003, Phillips et al. 2004, van Gardingen et al. 2006, Valle et al. 2007) TROLL (Chave 1999) FORMIND 2.0 (Kammesheidt et al. 200 1, Kohler et al. 2003) CAFOGROM (Alder and Silva 2000) ----------(Favrichon 1998)

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27Table 2-2. Original growt h, recruitment and mortal ity sub-model parameters Growth model Recruitment model Mortality model Species groups a0 a1 a2 a3 a4 r1 r2 r3 r4 bd m0 m1 1 0.003 0.008 0.045 -0.001 0.117 0.029 -3.047 0.000 0.006 7.5 2.6 1.8 2 0.005 -0.011 0.050 0.000 0.114 0.045 0.000 0.000 0.000 3.8 2.9 3.0 3 0.745 -0.739 0.000 -0.001 0.195 0.000 0.000 0.044 0.014 15.0 2.0 1.1 4 0.004 0.021 0.159 -0.001 0.103 0.301 -0.825 0.000 -0.249 5.0 3.0 2.4 5 0.001 0.022 0.029 0.000 0.002 0.701 -0.020 0.000 -0.696 7.5 2.3 1.2 6 -0.051 0.081 0.005 -0.001 0.152 0.008 -2.806 0.000 -0.001 7.5 2.5 1.6 7 2.263 -2.246 0.000 -0.002 0.368 0.074 -2.293 0.000 -0.004 7.5 4.5 3.3 8 -0.003 -0.175 0.163 -0.001 0.600 0.000 -6.906 0.000 0.003 15.0 2.5 0.9 9 0.009 0.333 0.078 -0.005 -0.428 0.059 -0.788 0.000 -0.051 5.0 8.9 3.6 10 0.007 0.081 0.029 -0.001 -0.135 0.000 0.000 0.002 0.004 15.0 4.3 2.8 Parameters were estimated by P. Phillips (unpublished manuscrip t) except for bolded parameters (see main text for detailed explanation). Growth sub-model: 43 10) (2aCaeaaDIDa. Recruitment sub-model:43 12e rIrrFIr. Mortality sub-model: Db bD if if m m M 5 5d d 1 0. D is the diameter at breast height (cm), C is the diameter-independent competition index, F is the recruitment rate in a 10 x 10 m square (trees yr-1), I is the diameter increment (cm yr-1), M is the mortality rate (%), bd is the upper limit of the first diameter class (cm), and a0, a1, a2, a3, a4, r1, r2, r3, r4, m0, and m1 are the estimated parameters. For further details regarding variable definitions and equation forms, re fer to Phillips et al. (2002a, 2004).

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28Table 2-3. Approaches and modified pa rameters used to implement assumptions Assumption ApproachesDescription Modified parameters DEA Recruitment Recruitment model modified in order to replace any killed or naturally dead tree by a new recruited tree. This new tree has the same tree attributes from the dead tree except for the DBH, which was set to 5 cm. Only recruitment model structure modified in source code. Undisturbed forests are in a dynamic equilibrium DEA Fine Tuning Simulation of an undisturbed fo rest was run for 100 years and model parameters were fine-tune d (modified iteratively within their 95% confidence interval) to exhibit a constant species composition (constant basal area and tree density per species group) during this time window. Table 2-4 MSA Mortality Mortality was increased to 100% for trees with DBH Dmax. Only mortality model structure modified in source code Tree species have a maximum attainable size MSA Growth Growth was reduced to zero for trees with DBH Dmax. Only growth model structure modified in source code. DEA and MSA stand for Dynamic Equilibrium Assump tion and Maximum Size Assumption, respectively. Dmax is the maximum diameter (cm) of a species group, based on the available data.

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29Table 2-4. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model (modified parameters in bold) Species groups characteristics DEA Fine Tuning Group name (reference) D95 G r1 r4 m0 m1 1 Slow growing mid-canopy 41.8 0.21 0.0290.006 2.3 2.0 2 Slow growing understory 15.9 0.09 0.0450.000 2.3 2.9 3 Medium growing mid-canopy 57.2 0.29 0.0000.013 1.9 1.4 4 Slow growing lower canopy 27.7 0.18 0.301-0.2382.9 2.7 5 Medium growing upper canopy 72.5 0.26 0.701-0.6961.9 1.2 6 Fast growing upper canopy 76.0 0.54 0.001-0.0083.3 2.0 7 Fast growing pioneers 35.8 0.54 0.040-0.1005.0 3.8 8 Emergents climax 104.00.37 0.0000.005 3.0 0.9 9 Very fast growing pioneers 38.7 1.26 0.030-0.10010.44.9 10 Very fast growing upper canopy 78.2 0.94 0.0000.004 4.3 3.9 D95 is the 95th percentile of the cumulative diameter frequency distribution (cm), G is the average growth rate (cm yr-1), r1 and r4 are parameters from the recruitment sub-model, and m0 and m1 are parameters from the mortality sub-model.

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30Table 2-5. Analysis of Variance used to determin e uncertainty due to plots and model stochasticity Source of variation DF E(MS) Plot nplot-1 2 ,, 2 ,, ipyrepimsyn Error nplot *(nrep-1) 2 ,, imsy Total nplot*nrep-1 These parameters were estimated separately for each model (bas eline model plus one model for each of the four assumptions), f or every logging scenario (logged and unlogged fore st), and for every 10-yr time step. This analysis corresponds to the nested des ign shown in Fig. 2-1A. 2 ,, imsy and 2 ,, ipyare the variances due to model stochasticity an d due to plots (i.e., due to different starting conditions), respectively, for the ith model; nplot and nrep are the number of plots (i.e., 15) and num ber of repetitions per plot (i.e., 20), respectively. Table 2-6. Analysis of Variance used to dete rmine parameter uncertainty for the baseline model Source of variation DF E(MS) Plot nplot-1 ipy psrps ipuy rpsimsynn n,, 2 ,, 2 2 ,, Par. uncertainty nplot*( nps-1) ipuy rpsimsyn,, 2 2 ,, M. stochast. nplot* nps*(nrps-1) 2 ,, imsy Total nplot* nps*nrps-1 This parameter was estimated separately for each logging scenario (logged and unlogged forest), and for every 10-yr time step. This analysis corresponds to the nested design shown in Fig. 2-1B. ipy ,, 2, ipuy ,, 2, and 2 ,, imsy are the variances due to pl ots (i.e., due to different starting conditions), due to parameter uncertainty, and due to model stochasticity, respectively, for the ith model; nplot, nrps, and nps are the number of plots (i.e., 15), number of repetitions per parame ter set (i.e., 2), and the number of randomly drawn parameter s ets per plot (i.e., 250), respectively.

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31 Figure 2-1. Nested design used for simulations R and PS stand for repetitions and parameter sets, respectively. A) Nested design used to determine uncertainty due to assumptions, to starting conditi ons effect and to model stochasticity. This design was used for the 5 models evaluated (baseline + 4 assumptions); B) Ne sted design used to determine parameter uncertainty. Only the baseline model was used for these simulations.

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32 CHAPTER 3 RESULTS Description of the Baseline Simula tion Set for the Unlogged Forest Simulations of the unlogged forest from regi on 11 and region 14, without any of the dynamic equilibrium or maximum size assumpti ons (i.e., baseline model), predicted drastic changes over a period of 100 years in species gr oup composition, both in basal area and tree density, particularly for the pioneer species gr oups (species groups 7 and 9). Also, tree density and basal area of big trees (DBH D95 [the 95th percentile of the diam eter distribution in the beginning of the simulation]) increased for nearly all species groups (Table 3-1). These somewhat counterintuitive results for an undisturbed forest are shown to indicate why a forest modeler might want to adopt post hoc assumptions that force the model to behave as expected. The effect of these assumptions is analyzed below. Mean Modeling Results Models with different assumptions did not pr edict drastically differe nt stand dynamics for short simulation lengths, within the time span of available field data (Fig. 3-1). Furthermore, a visual comparison of the observed versus the simulated stand level results does not allow a clear cut determination of which assumptions best fitted the observed data, which might mislead one to conclude that uncertainty due to assumptions is negligible. Nevertheless, the average long-term (i.e., 100 ye ars) simulation result changed drastically according to the adopted assumptions, particul arly for the dynamic equilibrium assumptions, suggesting that assumptions are indeed a major s ource of uncertainty in model projections (Fig. 3-2). Unfortunately, empirical data covering this time span do not ge nerally exist and therefore it is not possible to discern which of the simulation results best reflects real ity. Again, as shown previously, the baseline scenario clearly indica tes a mean trend of overall basal area that is

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33 outside the natural variability cu rrently found within the area (black circles in Fig. 3-2), both for the unlogged and logged scenarios. One might assume that the basal area of the unlogged simulations after 100 years should be within the range of the r ecently measured basal area (as often assumed in the forest modeling literature), in which case one woul d judge the model with the DEA Tuning assumption to be more biologically sound when compared to other models (i.e., baseline, MSA Growth, MSA Mortality, and DEA Recruitment). Another important aspect is that the effect of adding a given assumption to the baseline model may depend on the management scenario be ing simulated (see Fig. 3-2). For instance, when the projections of the DEA Recruitment m odel are compared to the projections from the baseline model, it is clear that the difference in their projections is larger in the logged scenario then in the unlogged scenario from year 10 to year 60. As a consequence, even if all simulations were carried out with a single set of assumptions (e.g., DEA Recruitment), it would not necessarily follow that the comparison of loggi ng scenarios (e.g., different cutting cycles, logging intensities, and/or harv esting systems) would remain unchanged since this assumption might have different effects depending on the specific logging scenario being simulated. Probability of Each Model and Comparison of Uncertainty Sources The dynamic equilibrium assumption (DEA) a nd the maximum size assumption (MSA) are often adopted after the model has been calibrated a nd some of its outputs assessed, typically as a way to avoid what is judged to be unrealistic model behavior. Also, field data are generally not used to evaluate these assumptions, particularly because model modifications due to assumptions are often seen as minor changes in paramete rs. Thus, to assess the uncertainty of adopting these assumptions without using field data to evaluate them, equal weights were set for all models (5 /1i ).

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34 The effects of all sources of uncertainty tended to increase with time except for the effect of starting conditions. Also, the joint effect of all of the chosen assumptions (i.e., MSA Growth, MSA Mortality, DEA Recruitment, DEA Fine Tuni ng) was the greatest source of uncertainty (except in relation to commercial basal area in the unlogge d scenario), probably due to the fact that, in contrast to the other sources of uncertainty, between model variance does not decrease with an increased number of simulations or plots measured. In genera l, the starting condition effect was the second greatest s ource of uncertainty, possibly du e to the small number (i.e., 15) of 1 ha plots being simulated. Parameter uncerta inty and model stochasticity were of similar magnitude and considerably smaller than assu mption and starting conditions effects. Clearly, overall model uncertainty is often dominated by the uncertainty due to assumption effects (Fig. 3-3). When the data are used to estimate the probabi lity of each model (which implies that the assumptions are evaluated based on the field data), these results drastically changed, substantially reducing th e uncertainty due to assumptions. As expected, the sub-model data equally supported the baseline, MSA Growth and MSA Mortality models (i.e., these three models had the same posterior model probabi lity; Table 3-2). Although commercial basal area was best predicted by the DEA Fine Tuning mode l (data not shown) and this assumption was implemented by carefully changing parameter values within their confidence intervals, the submodel data clearly conflicted with the DEA mo dels, yielding an approximately zero probability for this model. With equal weights for the MS A Growth, MSA Mortality and baseline models, the greatest source of uncertainty became the st arting conditions effect while all the remaining sources of uncertainty were of similar magnitudes (Fig. 3-4).

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35 It is important to note that, over the 100-yr simulation, using data to estimate the probability of each model resulted in a 13-fold decrease in overall variance of the mean for the stand basal area, both for logged and unlogged simulations, and a 5fold and 2-fold decrease in overall variance for the commercial basal area in the logged and unlogged simulations, respectively (note the different scales used in Fig. 3-3 when compared to Fig. 3-4). Furthermore, the multimodel average (average of the projectio ns from all models, weighted by the probability of each model) also substantially changed when da ta were used to estimate model probabilities (Fig. 3-5). The reduction in overall variance of the mean and the change in the multimodel average have practical relevance from a forest ma nagement perspective. For instance, when data are taken into account in estimating the probability of each model, the ov erall basal area in the logged forest is expected to re turn to its pre-logging levels wi thin 20 to 30 years, instead of within 20 to > 100 years, and th e post-logging recovery of comme rcial basal area after 100 years is 46-59%, instead of 37-67%.

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36 Table 3-1. Percentage change in the unlogged forest All trees Large trees (DBH D95) Species groups Basal area Tree density Basal area Tree density 1 -10 -22 46 30 2 -33 -25 -25 -40 3 18 7 200 182 4 -20 -23 69 18 5 -10 -30 53 33 6 173 89 407 324 7 444 202 1092 836 8 -7 -45 18 25 9 1015 987 2209 2015 10 70 57 415 175 Total 29 2 141 82 Percentage change was determined as (100*[Y100-Y0]/Y0), where Yi is the plot characteristic in year i and D95 is the 95th percentile of the diameter di stribution in the beginning of the simulation. Table 3-2. Posterior probability of each model, given the recruitment data, the mortality data, and both datasets combined Model Recruitment data Mortality data Both datasets combined Baseline 0.333 0.25 0.333 MSA mortality 0.333 0.25 0.333 MSA growth 0.333 0.25 0.333 DEA recruitment 0.000 0.25 0.000 DEA tuning 0.000 0.00 0.000 The posterior probability was estimated with Bayes theorem by combining the prior probability (each model had an equal prior probability, 1/5) and the likelihood (estim ated using WinBUGS). Recruitment and mortality data came from logged and unlogged forests (regions 11, 12, and 14).

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37 Figure 3-1. Comparison of obser ved versus simulated stand leve l data. Observed stand level results are shown in black (empty black circ les are 1-ha plots and black line is their average) and average simulation results from models with differe nt assumptions, over the time window of the observed data, are shown by grey lines. Overall basal area is shown in left panels (A,C,E) and commerc ial basal area is shown in right panels (B,D,F). Top panels (A and B) show regi on 11 (logged forest), middle panels (C and D) show region 12 (logged forest), and lower panels (E and F) show region 14 (unlogged forest).

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38 Figure 3-2. Mean simulation re sults from different models (b aseline + 4 assumptions) over a time window of 100 years. Simulations were in itiated with 15 one-hectare plots from regions 11 and 14, and logging (in logged s cenario) occurred in year 0. Initial observed overall and commercial basal area values for each of the 15 plots are represented by black circles placed in y ear 100 to illustrate how mean simulation results, except from the DEA Tuning model, can extrapolate beyond the original data range. Left panels (A and C) show results from logged forest and right panels (B and D) show results from unlogged forest; upper panels (A and B) show overall basal area and lower panels (C and D) show commercial basal area.

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39 Figure 3-3. Overall variance of the mean when data are not taken into account in estimating model probabilities (i.e., assu mptions are not evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results re garding overall basal area and lower panels (C and D) show results rega rding commercial basal area.

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40 Figure 3-4. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results re garding overall basal area and lower panels (C and D) show results rega rding commercial basal area.

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41 Figure 3-5. Multimodel average (continuous line ) and 95% confidence interval (dashed line), shown when data are not used (grey) and wh en data are used (black) to estimate the probability of each model. Results for unlogged forest are shown in left panels (A and C) and results for logged forest are shown in right panels (B and D). Upper panels (A and B) show overall basal area and lower panels (C and D) s how commercial basal area.

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42 CHAPTER 4 DISCUSSION Types of Assumptions Many alternative assumptions can be cons idered biologically reasonable. Model projections, however, are greatly influenced by the specific set of assumptions that are chosen, particularly when these assumptions are not evaluate d in light of the data (i.e., data are not used to estimate the probabilities of each model). For instance, data are not generally used to compare the baseline model to the fine tuned model. Fu rthermore, the same assumption can often be implemented in many different ways, potential ly resulting in very different long-term projections, as shown for the DEA models. As a c onsequence, it is crucial that the uncertainty associated with these assumptions be adequately reported. The use of the data to estimate the probability of each model revealed a clear distinction between the analyzed assumptions. The D ynamic Equilibrium Assumption (DEA, as implemented in this thesis) forced the model towards the desired model behavior, either by completely ignoring the recruitm ent data (as in the DEA Recruitment model), or by slightly changing the fitted parameters from the recruitment and mortality sub-models (as in the DEA Fine Tuning model). Both DEA a pproaches directly conflicted with the data. Similarly, the Maximum Size Assumption (MSA) was originally implemented by fine tuning the mortality parameter of large trees so that the undisturbed forest would keep a somewhat constant large tree density per species group after 100 years of simu lation. Despite the high ly variable and small dataset for large trees, this assumption (as pr eviously implemented) nevertheless strongly conflicted with the mortality data. In contrast to the Dynamic Equilibri um Assumption (DEA), the Maximum Size Assumption (MSA, as implemented in this thesis) was used to fill a knowledge gap that exists in

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43 the calibration dataset, therefor e not conflicting with the cali bration data. Indeed, a multicomponent iterative model can easily start extr apolating outside the da ta range without an obvious indication to the user that this extr apolation is occurring. Trees with DBH > Dmax are one of the most obvious model extrapolations an d assumptions regarding the dynamics of large trees are needed. Other extrapolat ions are far more subtle and frequently go unrecognized. Fig. 4-1, for instance, depicts the combinations of diameter and competition index for which the growth sub-model of species gr oup 10 would extrapolate outside th e range of field observations. Knowledge gaps within the calibration dataset ar e indeed numerous and therefore assumptions must be used to decide what the model should do in these situations. When not acknowledged, the implicit assumption is that the statistical re lationship between the depe ndent variable and the covariates can extrapolate corr ectly outside the data range. Thus, two types of assumptions are recognized in my study: the firs t type of assumption changes the model despite its conflict with the empirical data (as in the dynamic equilibrium assumption), while the second type does not conf lict with the data since the changes to the original sub-model (calibrated with the data) only result in different sub-model extrapolations (as the maximum size assumption). I believe that only the second type is justified. Nevertheless, if the first type of assumption is to be used, then the uncertainty associat ed with it should be properly reported. For instance, if the data ar e ignored in estimating the probability of each model, uncertainty due to this assumption can be estimated as the squared difference between the average result from the projections with and wi thout the assumption (i.e ., this assumes both models have the same probability). Comparison of Uncertainty Sources The field of statistics has traditionally acknowledged parametric uncertainty once a particular model form has been chosen. The exclusion of model cons truction and selection

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44 uncertainty has been shown, however, to result in over-optimistic predictive or inferential uncertainty, which can have serious implicati ons and result in over-c onfident decision making (Draper 1995, Hoeting et al. 1999). Likewise, probably the most st udied source of uncertainty in the ecological modeling literature has been paramete r estimate uncertainty, ei ther with a local or global sensitivity analysis (Saltelli et al. 2000, Ellner and Fieberg 2003). The problem of ignoring model uncertainty is likely to be exacerbated in situations where model extrapolations from availa ble data are needed for decision making given that models that are very different mathematically can have similar fits to the data but wildly different predictions outside the data range (Chatfield 1995, Draper 1995). I have shown here that assumptions used when sub-models are extrapolated (i.e., MSA) can have an effect of similar magnitude to other more traditional sources of uncertainty, such as parameter uncertainty. The uncertainty associated with the maximum size assumption, however, was relatively small. This observation requires further investigation, in that it begs another question: Could the effect of assumptions, when related to sub-model extrapolation, be th e largest source of uncertainty? To compare the effect of adopting different stra tegies when the growth sub-mode l extrapolated, I modified the growth sub-model so that the species-group mean diameter incr ement was used whenever the combination of covariates (diameter and compe tition index) extrapolated the data range (as depicted in Fig. 4-1). Simulations with this m odified growth sub-model were then compared to simulations with the baseline model (which a ssumes that diameter increment is correctly estimated by the growth sub-model when it is extrapolated). This comparison revealed that assumption effect, even when used only to deal with sub-model extrapolations, can indeed be the largest source of uncertainty (Fig. 4-2).

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45 Given that models are, by definition, simplif ications of reality containing numerous assumptions, how can all these assumptions be take n into account? One might argue that even if the simulations are biased, the comparison of di fferent logging scenarios (e.g., different cutting cycles and harvest intensity combinations) simu lated with the same set of assumptions would generally be unbiased (Phillips et al. 2003). Unfort unately, my results rev ealed that this might not always be true given that the assumption effect may great ly depend on the scenario being simulated (e.g., logged vs. unlogged fo rest; Fig. 3-2 and Fig. 4-2). To take these assumptions into account, I be lieve that models should be built with many redundant components (e.g., different equations to represent a given sub-model or alternative assumptions when sub-models are extrapolated ) and simulations performed using different subsets of these components. Logically, this ap proach is only possible for models built in a modular fashion. Then, different data sources could be used to assign updated pr obabilities to each model structure. Also, model averaging has been increasingly suggested as a method that can incorporate uncertainty due to model constr uction and selection (Wintle et al. 2003, Ellison 2004). However, to my knowledge, model averaging has rarely been extended to more complex iterative models, such as forest dynamic models. Uncertainty due to assumptions can only be re duced with carefully planned experiments or targeted collection of observat ional data. For instance, thinni ng experiments might help to determine growth patterns for combinations of diameter and competition index outside the range of the data currently available. Likewise, the us ual data collection for tr opical forest dynamics, through the monitoring of one h ectare permanent plots (Alder and Synnott 1992), are unlikely to generate sufficient data on large tree dynamics, due to their low density. In the absence of such a data, model projections that inte grate the results from models with different assumptions are

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46 likely to be much more robust than projecti ons based on a single model (and consequently a single set of assumptions).

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47 Figure 4-1. Combinations of co mpetition index and DBH contained in the data used to calibrate the growth sub-model for sp ecies-group 10. This figure re veals the regions (indicated by the question marks), in independent variable space, that will require an extrapolation of the growth sub-model of species-group 10.

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48 Figure 4-2. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from the baseline model and the model with the modified growth sub-model, partitioned between parameter uncertainty, starting conditions effect, model stoc hasticity, and assumptions e ffect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); uppe r panels (A and B) show re sults regarding overall basal area and lower panels (C and D) show results regarding commercial basal area.

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49 CHAPTER 5 CONCLUSIONS It is crucial that available data be used to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process), to reduce assumption uncertainty in particular, a nd overall model uncertainty in general. Uncertainty due to modeling assumptions can be of greater or similar magnitude when compared to the other sources of uncerta inty that are more commonly assessed. Targeted experimental studies are needed to reduce the need to extrapolate sub-model results and to reduce uncertainty due to modeling assumptions. Overall modeling uncertainty is likely to be underestimated when th ese four sources of uncertainty (i.e., model stochasticity, para meter estimation, starting conditions and modeling assumptions) are not jointly taken into account. I have shown how the overall uncertainty of the mean can be partitioned among the different sources of uncertainty, with particular emphasis on the uncertainty that arises due to the use of models with different assumptions. This method can potentially be applied to other types of models. Forest dynamic models have and wi ll continue to be used to pr edict the outcomes of direct or indirect human-induced changes (e.g., logging, bur ning, fragmentation or carbon accumulation in the atmosphere), sometime s with millennium-long time windows (Chambers et al., 2001). Nevertheless, given that the information content of modeling projections is inversely proportional to their uncertainty and failure to ad equately report uncertainty associated with these projections can mislead decisions (Clark et al., 2001), it is critical to report the overall uncertainty of model projections.

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50 LIST OF REFERENCES Alder, D. 2002. Sim ple diameter class and cohort modelling methods for practical forest management in ITTO Workshop Proceedings on Growth and Yield, Kuala Lumpur, Malaysia. Alder, D., F. Oavika, M. Sanchez, J. N. M. Silv a, P. van der Hout, and H. L. Wright. 2002. A comparison of species growth rates from f our moist tropical fo rest regions using increment-size ordination. Internat ional Forestry Review 4:196-205. Alder, D., and J. N. M. Silva. 2000. An empiri cal cohort model for management of Terra Firme forests in the Brazilian Amazon. Forest Ecology and Management 130:141-157. Alder, D., and T. J. Synnott. 1992. Permanent Sample Plot Techniques for Mixed Tropical Forest. Oxford Forestry Institute, Department of Plant Sciences, University of Oxford, Oxford, UK. Asner, G. P., D. E. Knapp, E. N. Broadbent, P. J. C. Oliveira, M. Keller, and J. N. Silva. 2005. Selective logging in the Brazili an Amazon. Science 310:480-482. Baker, T. R., O. L. Phillips, Y. Malhi, S. Almeida, L. Arroyo, A. Di Fiore, T. Erwin, N. Higuchi, T. J. Killeen, S. G. Laurance, W. F. Lauran ce, S. Lewis, A. Monteagudo, D. A. Neill, P. N. Vargas, N. C. A. Pitman, J. N. M. Silva, and R. V. Martinez. 2004. Increasing biomass in Amazonian forest plots. Phil.Trans.R.Soc.Lond.B 359:353-365. Brooks, S. P., and A. Gelman. 1998. General met hods for monitoring convergence of iterative simulations. Journal of Computationa l and Graphical Statistics 7:434-455. Brown, J. H., J. F. Gillooly, A. P. Allen, V. M. Savage, and G. B. West. 2004. Toward a metabolic theory of eco logy. Ecology 85:1771-1789. Brugnach, M. 2005. Process level sensitivity analys is for complex ecologi cal models. Ecological Modelling 187:99-120. Chambers, J. Q., N. Higuchi, and J. P. Schimel. 1998. Ancient trees in Amazonia. Nature 391:135-136. Chambers, J. Q., N. Higuchi, L. M. Teixeira, J. Santos, S. G. Laurance, and S. E. Trumbore. 2004. Response of tree biomass and wood litter to disturbance in a Central Amazon Forest. Oecologia 141:596-611. Chambers, J. Q., N. Higuchi, E. S. Tribuzy, and S. E. Trumbore. 2001. Carbon sink for a century. Nature 410:429-429. Chatfield, C. 1995. Model uncertainty, data mining and statistical inference. Journal of the Royal Statistical Society. Series A (S tatistics in Society) 158:419-466.

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57 BIOGRAPHICAL SKETCH Denis Ribeiro do Valle was born in So Paulo, Brazil, having com pleted his undergraduate studies in forestry at ESALQ, the Agriculture Sc hool of the University of So Paulo. During his undergraduate studies, his experien ce was mostly with even-aged forestry, with internships in cellulose companies such as International Paper and Arauco Forestal. Before finishing his undergraduate studies, however, he participated on an experimental logging of mahogany in the state of Acre, becoming fascinated with the challenges of uneven-aged forestry. In August 2003, Denis started working as an assistant researcher at a non-governmental organization (NGO) called Amazon Institute fo r the People and the Environment (IMAZON), within the Dendrogene Project (a cooperation between IMAZON, Embrapa Amazonia Oriental, Brazil, and the University of Edinburgh, Scotland ). Most of his work was related to forest dynamic modeling, focusing on questions regardi ng sustainable timber logging. In August 2006, Denis started his masters studies at the University of Florida.