An Individual-tree Model to Predict the Annual Growth of Young Stands of Douglas-ﬁr (Pseudotsuga menziesii (Mirbel) Franco) in the Paciﬁc Northwest

8/6/2019 An Individual-tree Model to Predict the Annual Growth of Young Stands of Douglas-fir (Pseudotsuga menziesii (Mirbe

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An Individual-tree Model to Predict the Annual Growth of

Young Stands of Douglas-fir (Pseudotsuga menziesii (Mirbel)Franco) in the Pacific Northwest

Nicholas Vaughn

A thesis submitted in partial fulfillmentof the requirements for the degree of

Master of Science

University of Washington

2007

Program Authorized to Offer Degree: College of Forest Resources


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Graduate School

This is to certify that I have examined this copy of a masters thesis by

Nicholas Vaughn

and have found that it is complete and satisfactory in all respects,

and that any and all revisions required by the final

examining committee have been made.

Committee Members:

Eric C. Turnblom

David G. Briggs

James D. Flewelling

David D. Marshall

Martin W. Ritchie

Date:


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In presenting this thesis in partial fulfillment of the requirements for a mastersdegree at the University of Washington, I agree that the Library shall make its copiesfreely available for inspection. I further agree that extensive copying of this thesis is

allowable only for scholarly purposes, consistent with fair use as prescribed in theU.S. Copyright Law. Any other reproduction for any purpose or by any means shallnot be allowed without my written permission.

Signature

Date


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Abstract

An Individual-tree Model to Predict the Annual Growth of YoungStands of Douglas-fir (Pseudotsuga menziesii (Mirbel) Franco) in the

Pacific Northwest

Nicholas Vaughn

Chair of the Supervisory Committee:

Associate Professor Eric C. Turnblom

College of Forest Resources

Individual-tree equations for the one-year height and breast-height diameter growth

of young plantations of Douglas-fir (Pseudotsuga menziesii (Mirbel) Franco) in the

Pacific Northwest are presented and analyzed. The height growth equation accounts

for percent cover of competing vegetation, and for the seedling density. Relative

height, the ratio of tree height to stand top height (mean height of the 40 largest

trees per acre) is used as an index of tree position. The dynamic effects of competing

vegetation, density and relative height were modeled to change with stand top height.

Models to predict initial height for trees passing breast-height, the change in vegeta-

tion cover, and the probability of tree survival are presented as well. Height growth is

predicted with an R2 of 0.60. Diameter growth, modeled as squared-diameter growth,

was predicted with an R2 of 0.78.

Supporting models are presented as well. A model to predict the initial diameter

of a tree crossing breast height fit very well with an R2 of 0.73. A model to predict

annual change in vegetation cover was not strong, though it did produce the expected


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response as the stand height increases. Finally the two-year probability of survival of

a given tree was found to be related to stand height, tree height, and stand density.

A bootstrap procedure enabled the diagnosis of the height growth model coefficient

distributions. The sensitivity of model predictions to changes within these predicted

coefficient distributions is presented. Despite larger than expected standard errors

of the coefficients, model predictions were insensitive to small fluctuations in the

coefficients. Correlations among the coefficient estimates may explain the relatively

small changes in model predictions.


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TABLE OF CONTENTS

Page

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Growth and Yield Models . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Existing Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Young Stand Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2: Development of the Model . . . . . . . . . . . . . . . . . . . . . 9

2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Model design and selection . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 3: An Individual-tree Model to Predict the Annual Height Growthof Young Plantations of Pacific Northwest Douglas-fir Incorpo-rating the Effects of Density and Vegetative Competition. . . . 54

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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Chapter 4: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 Height and Diameter Increment . . . . . . . . . . . . . . . . . . . . . 754.2 Secondary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Potential Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Height Model Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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LIST OF FIGURES

Figure Number Page

2.1 Location map of study plots. . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Histograms of certain dataset values mentioned in the text. . . . . . . 19

2.3 Model errors against stand site index. . . . . . . . . . . . . . . . . . . 22

2.4 A contour plot of the height growth prediction surface from model 2.5. 24

2.5 Height growth residual plots from model 2.5 . . . . . . . . . . . . . . 25

2.6 Values of the relative height modifying function in model 2.6. . . . . . 27

2.7 Scatter plots of mean plot error ratio against mean plot vegetation. . 28

2.8 The intercept and slope of error ratio against vegetation cover. . . . . 30

2.9 A plot of the value of the vegetation modifier in model 2.9. . . . . . . 32

2.10 A plot of the value of the vegetation modifier in model 2.10. . . . . . 332.11 The intercept and slope of error ratio against stems per acre . . . . . 34

2.12 Residual plot for model 2.18 . . . . . . . . . . . . . . . . . . . . . . . 40

2.13 Residual plot for model 2.19 . . . . . . . . . . . . . . . . . . . . . . . 43

2.14 Vegetation cover across stand top height and basal area. . . . . . . . 44

3.1 Location map of study plots. . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 A contour plot of the height growth prediction surface (in feet) from

equation 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Values of the relative height modifying function in model 3.2. . . . . . 64

3.4 Values of the vegetation cover modifying function in model 3.3. . . . 65

3.5 Values of the density modifying function in model 3.4. . . . . . . . . . 66

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3.6 Bootstrap distributions of the parameters of the full height growth model. 68

3.7 Scatterplot matrix of the bootstrap coefficient estimates. . . . . . . . 69

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LIST OF TABLES

Table Number Page

2.1 Coefficients of the height increment model. . . . . . . . . . . . . . . . 37

2.2 Coefficients of the squared diameter increment model. . . . . . . . . . 41

2.3 Coefficients estimated for model 2.19. . . . . . . . . . . . . . . . . . . 42



2.6 Coefficients from the full height growth model in equation 2.16. . . . 51

2.7 Coefficients from the full squared diameter growth model in equation2.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.8 Correlations among the parameters of model 2.16. . . . . . . . . . . . 52

2.9 Correlations among the parameters of model 2.17. . . . . . . . . . . . 53

3.1 Predictor values used in the model sensitivity analysis. . . . . . . . . 61

3.2 Coefficients from the full height growth model in equations 3.1 through3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Effects of changing the model coefficients within the bootstrap distri-bution limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 Sensitivity of the height growth model to parameter changes. . . . . . 70

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ACKNOWLEDGMENTS

In a project like this, first acknowledgment should go to the sources of funding.

Therefore, I would like to thank members and supporters of the Agenda 2020 program,

the US Forest Service and the Stand Management Cooperative. Without them this

project would never have begun.

My committee was very kind to provide encouragement and wonderful feedback

along the way. They did not even laugh much at my amateur mistakes. I feel that

my committee provided a far more useful education than did my coursework. The

amount of knowledge they have absorbed from their experience in this field is beyond

impressive. I can only hope to retain half as much information as any of my committee

members have shown is possible.

I also benefited from the experience and help of many others. An incomplete list

would include all members of the Stand Management Cooperative, especially Randal

Greggs, Larry Raynes, Greg Johnson, Mark Hanus and Jeff Madsen. From the RVMM

project, Bob Shula, Steven Radosevich, and Steve Knowe for providing a large amount

of data for this project.

Last but not least, I would like to thank my friends and family for their constant

support. I would not have survived without it. This is especially true for my wife,

Jilleen, for her support even though she was a stressed out student as well. I forget

many things (as she knows too well), but I will never forget what this means to me.

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Chapter 1

INTRODUCTION

1.1 Growth and Yield Models

Management decisions in forestry have the potential to impact greatly both the fi-

nancial well-being of the forest owner and the future ecological condition resulting

from timber management activities. Understanding the growth and yield potential

of a given stand of trees is vital to optimizing these decisions to meet the goals at

hand. The idea of a yield model first appeared in the western world in the late 18th

century (Vanclay 1994). Long before the availability of an electronic computer, these

early models took the form of yield tables indexed by site quality and age (Hann and

Riitters 1982). Users of these tables could read off the expected stand volume at a

given age. This information, along with other factors, could be used to schedule har-

vests based on expected returns at the stand level. Since then the abilities of growth

and yield models (where growth is the periodic change, and yield is the accumulation

of growth) have steadily advanced. However, while the internal processes of growth

models have changed, the output of many modern computer growth and yield models

is still formatted as a stand yield table (Curtis et al. 1982). Given the current con-

ditions input by the user, modern growth models will still output the predicted yield

attributes at the end of multiple growth periods.

The modern growth model is quite a testament to the important role of science in

natural resources. The role of such models in decision making for such management

regimes has become very significant. It is rare that any manager would make a


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harvest scheduling decision without consulting the output from at least one growth

model. Models are even being used in the formulation of policy concerning forestresources (Meadows and Robinson 2002). While local management decisions can

affect a significant area, the formulation of policy using information from an inaccurate

model can have longer lasting negative impacts on an even larger area and a greater

number of people. It is a long-term goal of growth and yield modeling to build models

of greater accuracy over larger domains of applicability.

1.2 Existing Growth Models

Growth models can be loosely separated into groups based upon two main differences:

1) the modeling resolution grown and 2) the employment of spatial data (Munro 1974).

However, a given model may partially fit into several of these groups concurrently.

The first classification, modeling resolution, segregates models that grow whole stands,

diameter classes, and individual trees. Whole stand models grow stand-level variables

such as basal area and stand volume, while more complex models project either

distributions of diameters or individual-tree diameters. Many models are built with

components from both of the above types. These models are hard to classify into any

one of the above pure categories (Vanclay 1994). For instance, the detailed tree-level

information needed for some purposes can be disaggregated from whole-stand growth

(Ritchie and Hann 1997).

There are numerous growth and yield models designed for trees in the Pacific

Northwest. Examples of whole-stand models are DFSIM (Curtis et al. 1981), PPSIM

(DeMars and Barrett 1987), and TREELAB (Pittman and Turnblom 2003). Ex-

amples of individual tree models include ORGANON (Hann 2003) and FVS (Dixon

2002), a model consisting of localized versions of Prognosis (Stage 1973, Wykoff et al.

1982). Several additional examples are available (Ritchie 1999).

There has been a trend towards increased usage of individual tree modeling. One


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explanation for this trend is that large increases in computing power are more readily

available. Creating and using individual-tree models can be computationally inten-sive. Another reason is a desire for more wood quality and piece size distribution

detail in model output. A third reason is that individual-tree models are adapt-

able to multiple-age and multiple-species stands and can more easily model complex

silvicultural treatment designs. For individual tree models, the second model type

classification distinguishes between models that use inter-tree spatial relationships

to estimate the effects of competition and those that do not. These are known,

respectively, as distance-dependent and distance-independent growth models. Mostindividual-tree models are distance-independent because the large amount of in-field

data collection required to build and use distance-dependent models is not practical

for many users in a management setting. However, distance-dependent models can

be very useful for researchers attempting to more fully understand the dynamics of

inter-tree competition.

While every model is unique with regard to the exact form used, the types of

predictor variables used to build individual-tree growth models are fairly standard.

Because the growth of individual trees is highly influenced by the size and distance of

surrounding trees, individual-tree models typically include at least some expressions

of competition. If the locations of each tree in relation to the others are known,

distance dependent competition measures can be used. Tome and Burkhart (1989)

and Biging (1992) provide a synopsis of these measures, which typically take into

account the size relationship and the distance between trees. Opie (1968) reviewed

the strength of several definitions of surrounding basal area as growth predictors forindividual-tree growth.

Absent the tree location data, stand-level measures of density can be used. There

are several such measures (Bickford 1957). The number of trees per unit area is a

simple measure of density. This number is directly related to the average spacing


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between trees. Basal area per unit area combines tree spacing and the quadratic

mean diameter at breast height. For stands with many trees below breast height, thebasal area is very small. Fei et al. (2006) found the sum of tree heights on a per unit

area basis, aggregate height, worked well as a measure of density for young mixed

hardwood stands. Measures of stand-level crown cover, such as crown competition

factor (Krajicek et al. 1961), have been used with success (Wykoff 1990). There is

debate whether or not using spatially explicit density measures is beneficial. Martin

and Ek (1984) found that, for plantations of red pine (Pinus resinosa Ait.), stand

level basal area worked as well as indexes incorporating inter-tree distances. In mixed-species stands of Scots pine (Pinus sylvestris L.) and Norway spruce (Picea abies (L.)

Karst.), Pukkala et al. (1994) found density-dependent competition metrics to be

superior.

Along with stand-level measures of density, indexes of the competitive position

of an individual tree are commonly used. The crown cover of the stand at a given

multiple of the height of an individual tree was used by Hann and Hanus (2002b) to es-

timate height increment of several conifers in southwest Oregon. Wykoff et al. (1982)

and (Hann and Hanus 2002a) used the basal area in larger trees (BAL) to predict

diameter growth. Ritchie and Hamann (2007) found that crown area of taller trees

improved predictions of height and diameter growth of young Douglas-fir. Ritchie

and Hann (1986) used the ratio of tree height over stand top height interpolated from

site index curves with some success to predict height growth.

Site productivity can also be an important predictor, but it is slightly more difficult

to measure. Indexes have been built using several methods. One, site index, is a

measure of the expected height of a stand at a given age, usually 50 or 100 years. In the

Pacific Northwest, two commonly used site index curves for Douglas-fir (Pseudotsuga

menziesii (Mirbel) Franco) are King (1966) and Bruce (1981). The pervasive nature of

site index in forestry ensures that it is widely understood. As a cumulation of expected


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height growth, site index is also convenient to use when modeling the height growth

of trees. However, there currently is some debate about its use (Monserud 1987, Zeide1994). For instance, the early growth of Douglas-fir is influenced by factors unrelated

to inherent productivity, such as site preparation and planting density (Scott et al.

1998). Fluctuations in weather patterns can last for several years, and may highly

influence the height growth of young stands (Villalba et al. 1992). This can lead to

erroneous estimates of the site index of a stand. Additionally, site index does not

apply well to mixed species or multiple aged stands. For this reason, Stage (1973)

did not include site index as a predictor in the Prognosis growth model.

To bypass the problems with site index, other site productivity indexes are usually

built from soil properties, topographical features, climate statistics, and species com-

position. These are correlated fairly well with site index (Steinbrenner 1979, Klinka

and Carter 1990), and have the advantage of being species independent. However,

these productivity indexes have a few problems of their own. For instance, on sites

with high variability in soil quality, these indexes will depend highly on where soil

data is collected. Also, it can be relatively expensive to gather the required data for

these indexes.

1.3 Young Stand Modeling

Harvests on public lands have decreased in response to public demand. Concurrently,

industrial forestry in the Pacific Northwest has become more intensive, because of

increased demand for wood products and a loss of harvestable area (The Rural Tech-

nology Initiative 2006). This has led to shorter rotations and an increase in youngstand research. Most models are designed to grow stands of established trees from

slightly before the beginning of the stem exclusion phase. This is the age when thin-

ning treatments are considered and when decisions about harvest age are made. How-

ever, some important decisions are made earlier in the life of a stand. These include

planting density and competing vegetation control, as well as early pre-commercial


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thinning operations.

Growth and yield models built to predict tree and stand growth during this early

period are not as common as those for older stands, however several examples do exist.

Westfall et al. (2004) predicted size distributions of loblolly pine (Pinus taeda L.) in

the Southeast, looking mainly at the effects of site preparation and fertilization levels.

Zhang et al. (1996) grew juvenile loblolly pine to look at the effects of various density

levels. Mason et al. (1997) and Mason (2001) developed a model of young Monterrey

pine (Pinus radiata D. Don) in New Zealand to link with older models and look at

effects of site preparation and seedling handling on tree growth and survival. Watt

et al. (2003) looked at the effects of weed competition using a more process-based

model. For the Pacific Northwest, a model for younger plantations of Douglas-fir,

called RVMM, is described by Shula (1998) and Knowe et al. (1992; 2005).

A considerable amount of work has been performed attempting to understand

the dynamics between young trees and surrounding vegetation (Tesch and Hobbs

1989, Morris et al. 1993, Knowe et al. 1997, Cole et al. 2003, Nilsson and Allen

2003, Watt et al. 2003, Comeau and Rose 2006). The long term effects of early

vegetation competition on ponderosa pine (Pinus ponderosa P. & C. Lawson) growth

is presented in Zhang. et al. (2006). It is generally believed that the amount of

vegetation surrounding a seedling heavily influences the rate of height growth of the

seedling (Oliver and Larson 1996). It is much to the managers benefit to understand

how much the residual vegetation in a plantation will decrease the growth of the

planted seedlings. A typical goal of plantation management is to optimize the volume

growth of a given stand with minimal cost. Being able to estimate crop tree response

to the different vegetation management options is vital to this goal.

In addition to vegetation management, density management is an additional tool

to redistribute growth. It is well known that the density of a stand can severely affect

the growth rate of individual trees (Oliver and Larson 1996). This is especially true


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as denser stands reach crown closure earlier, and light becomes a limited resource

earlier. Denser stands will have slower tree growth at this stage, and beyond until thestand self-thins enough to reduce the overall competition. Some research has shown

that, at least in the Pacific Northwest, density seems to have the opposite effect in

much younger stands of Douglas-fir (Scott et al. 1998, Turnblom 1998, Woodruff et al.

2002). In such stands, higher densities result in faster growth. As the stand ages,

this effect crosses back at some point to the more expected decreased growth with

increased density. The causes of this cross-over effect are unknown, but assumed

to be related to canopy closure (Turnblom 1998).

1.4 Objectives

This thesis describes the creation of individual-tree, young stand growth equations

for Douglas-fir plantations in the Pacific Northwest. The specific objectives of these

equations are:

1. To develop predictive equations for early (through age 15) individual-tree height

growth and diameter growth.

2. To incorporate the impacts of competing vegetation and spacing/thinning on

early stand growth in young Douglas-fir plantations.

The growth equations produced will be merged into the existing simulator, CONIF-

ERS (Ritchie 2006). CONIFERS is an individual-plant growth and yield simulator for

young mixed-conifer stands in southern Oregon and northern California. Users will

be able to choose which region, and therefore which growth equations, the simulator

will use to project tree growth. Differences in available data prevented the refitting

of the existing growth equations in CONIFERS (Ritchie and Hamann 2006; 2007).

This first chapter in this thesis acts as a general introduction. Chapter two presents

a detailed description of the data and methodology used to create the growth equa-


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tions. The third chapter, meant to stand alone as a journal-submittable manuscript,

describes the use of bootstrap methods to examine the height growth equation. Thefinal chapter presents the overall discussion and conclusion. Every attempt was made

to fully disclose the weaknesses of this model, and the situations in which it would

be valid to use this model. Of particular interest is the extent of the data used in

building this model, as applications that are beyond the range of modeling data are

not advised.


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Chapter 2

DEVELOPMENT OF THE MODEL

2.1 Data

2.1.1 The Sources of Data

The data for this project comes from two sources, both of which contain surveys

from an array of plots scattered throughout the Pacific Northwest north of Roseburg,

Oregon (about 43 north latitude) and west of Mount Rainier (about 121.5 west

longitude). These sources are described in detail below.

The Stand Management Cooperative

The Stand Management Cooperative (SMC) is a consortium of landowners in the

Pacific Northwest established in 1985 to pool resources in order to provide high quality

data on the long-term effects of silvicultural treatments (Maguire et al. 1991). The

data for this project is a product of a planting density trial, the experimental units

of which are known within the SMC as Type III installations. There are 34 such

installations with sufficient data for this project. The locations of these installations

are shown in figure 2.1.

The installations contain six planting plots containing trees planted at the densi-ties: 100 (21x21 foot spacing), 200 (15x15), 300 (12x12), 440 (10x10), 680 (8x8), and

1210 (6x6) trees per acre (Silviculture Project TAC 1991). The plots at some instal-

lations were split to retain one subplot with the original density. The other subplots

were assigned pruning and thinning treatments. This resulted in 3 to 23 subplots,

each from 0.2 to 0.5 acres (0.08 to 0.20 hectares) in size per installation. Each in-


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RVMM CoastalRVMM CascadeSMC Type III

Figure 2.1: Locations of the study tree plots within the Pacific Northwest. Plotsfrom the two datasets of the RVMM project are noted with a + (Coastal) or a (Cascade). SMC Installations, which contain multiple tree plots, are noted with a

.

stallation was planted with seedlings of species Douglas-fir (Pseudotsuga menziesii

(Mirbel) Franco), western hemlock (Tsuga heterophylla [Raf.] Sarg.), or a 50:50 com-

bination of the two, with planting stock chosen by the landowner. Site preparation

was typically performed to match with the landowners current best management


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practices. Information on the type and amount of site preparation was not collected

in an standardized manner and is, in many cases, missing altogether. Each measure-ment plot was installed within an area of similar treatment at least 1 acre in size to

allow for a buffer between measurement plots.

Within each measurement subplot (referred to as a tree plot in future discus-

sion), every living tree of the conifer species of interest was tagged and measured for

at least one of diameter and height. Basal diameter (15 cm from base) was typically

measured until the first or second measurement after the trees reached breast height.

Thereafter, breast-height (4.5 feet from base) diameter (DBH) was the sole measure

of bole diameter. Total height and height to live crown base were measured on a

subsample of trees on the plot. Size units were metric on Canadian installations and

English in the United States. The remeasurement interval for tree plots was typically

two years early in the development of the stand and every four years when the stand

reached 30 feet in height.

Within the tree plot, a cluster of four circular 1/100th acre plots (referred to as

a vegetation plot in future discussion) were installed in the four quadrants of the

tree plot to measure competing vegetation. Within each circular vegetation plot,

vegetation percent cover and average height were estimated ocularly by species for

each of the four quadrants of the circle. These measurements typically took place

before the first tree plot measurement and during the first growing season following

a tree measurement. However, not all vegetation measurements occurred when de-

sired. Almost 25 percent of the usable vegetation measurements occurred between

tree measurements. Vegetation was classified by life form, either Shrub, Forb, Grass

or Fern.


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Regional Vegetation Management Model

The Regional Vegetation Management Model (RVMM) was a US Forest Service

funded project initiated by Oregon State University to model young stand growth

(Shula 1998, Knowe et al. 1992; 2005). The RVMM project was an observational

study. Plots were located based on a desire to fill gaps in the ranges of several

stand-level variables, including age, location, and vegetation abundance. Many plots

were established on the remnant plots of the CRAFTS (Coordinated Research on Al-

ternative Forestry Treatments and Systems) study (CRAFTS Experimental Design

Subcommittee 1981). There are 196 RVMM plots, 98 in the Coastal Range of western

Washington and Oregon, and 98 in the Cascades (Figure 2.1).

The treatment history the plots was not always well documented, however, infor-

mation on the date of the last thinning treatment and the last vegetation reduction

is recorded in most cases. The site preparation type was recorded for each plot.

The tree plot setup differed slightly from that of SMC. The tree plots (labeled as

PMP) are smaller, at 0.1 acre (0.04 hectares), and contain four subplots (labeledas CMP) of 0.01 acres (0.004 hectares) each. Diameter was measured on all trees.

Basal diameter was measured on trees smaller than 4.5 feet, otherwise DBH was

measured. Conifers within the CMPs were tagged and height, crown width, and

height to the base of live crown (height to lowest whorl with 3/4 live branches) were

recorded. A subsample of the conifers outside the CMPs were tagged and measured at

this intensity in order to bring the total number of tagged trees to about 30 percent

of the total number of trees on the PMP. For every tree of any species, DBH andheight were measured in metric units. Trees with multiple stems, as is common with

some hardwoods, were tagged as one tree, but all individual stems up to five stems

(the largest five, otherwise) were measured and recorded. Four hardwood trees were

intensively measured in each CMP. Each stem on multiple-stemmed trees share the

same height, crown width and height to crown base. The number of stems, measured


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or not, was recorded for each tree.

PMPs were remeasured only once, typically after two years. Some tree measure-

ments took place during the growing season, but in most cases, the remeasurement

took place during the same part of the season. Only PMPs with a remeasurement

within 3 weeks of the first measurement were used.

Vegetation was measured in the same year as the trees, and these measurements

occurred within the CMPs. Identical to the SMC surveys, each CMP was split into

four quadrants. Percent cover and average height were ocularly estimated by species

within these quadrants. Unlike the SMC surveys, surveys were also done using a line

transect technique at the same time.

Complications

The main complication in attempting to combine the datasets was the differing treat-

ment of hardwood competition. On the RVMM plots, hardwoods were measured and

even tagged along with the conifers in the tree plots. However, as part of the ex-

periment, hardwoods in the SMC installations were removed if they reached half the

height of surrounding conifers. Hardwoods below this height were treated as vege-

tation and measured for percent cover only in the vegetation plots. However, in the

RVMM plots, percent cover measurements were not performed on any tree species.

Because hardwoods were not measured as vegetation and as trees on any single plot,

there was no way to convert from one estimate of hardwood competition cover to the

other.

Because the RVMM vegetation data did not come with a species list, the SMC

species list was assumed to use the same coding system for all species. After a

merging of the datasets, some species codes were not found in the species list. After

investigation, it appears that many arbitrary codes were recorded in the field, when

a species was not identifiable. While the intention was to replace these codes in the


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14

database later, they were never replaced. These codes made up a small portion of the

dataset. Thus, unidentified species were labeled as Shrubs if they exceeded 1.5 feet,otherwise they were labeled as Forbs. The RVMM vegetation transect data was not

used because no such data exists from the SMC project. High correlation was found

between the optical and transect data, indicating that little would be gained by using

the transect data.

As previously mentioned, initial hopes of incorporating site preparation methods

as a predictor of tree growth were diminished when it was realized that the information

recorded was not consistent enough between the two projects to use. In order for

such information to be included, an idea of the type, intensity and timing of the

site preparation treatments would be necessary. However, it is assumed that at least

some of the site preparation is expressed through the realized vegetation cover at a

later date. Users of the growth model produced by this project will therefore only be

able to assess the effects of early vegetation control without respect to the particular

method used.

A last complication is the difference in time between SMC tree plot and vegetation

plot measurements. For this project, a simple linear interpolation was used to estimate

the percent cover of vegetation only when vegetation measurements were done both

before and after a tree measurement. A linear interpolation is reasonable because little

is known about the nature of the increases or decreases in vegetation cover between

measurements. The true trajectory curves could be either concave or convex, and a

linear interpolation assumes neither. Limiting the dataset to those plots which have

associated vegetation measurements reduced the number of usable plot measurements

by more than half, from 1033 to 438 (from 87659 to 31902 tree-growth observations).


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15

2.1.2 Computed Variables

Stand-level variables were computed from the individual tree data at each tree mea-

surement. Four of these variables, described below, were incorporated as predictor

variables in the growth model.

Basal Area Per Acre

Basal area per acre, BA, was calculated as

0.005454154n

d2i

A

where di is the DBH, in inches, of living tree i in a plot of size A acres which contains

n trees. Plots with no trees above breast height were assigned a basal area of 0. This

value was used as a measure of stand density.

Stems Per Acre

The total number of living stems of all species (including multiple stems of individual

trees) divided by the plot size in acres, SPA, was used as another measure of density.

Top Height

Top height, Htop, is defined for the purposes of this project as the average height of the

40 largest trees per acre, based on diameter at breast height. In the younger standswhere few trees have even reached breast height, two options were available. One

would be to use basal diameter, and the other would be to take the average height

of the 40 tallest trees per acre. At this point in the development of the trees little

difference was found between the top heights produced by both options. Therefore,

the second option, averaging the heights of the 40 tallest trees per acre.


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Site Index

Construction of a soil and weather based index of site productivity was unsuccessful

because sufficient datasets for the entire study area were not found. Site index was

then the best option to incorporate some index of site productivity into this analysis.

Because of the heavy influence of density on young stand growth, the site index

curves selected were those created by Flewelling et al. (2001) for plantation-grown

Douglas fir. These curves were created for younger stands than previous curves by

King (1966) and Bruce (1981), and can be very stable at plantation ages as youngas 10 years from seed. Furthermore, they were built to account for early effects

of planting density. A base age of 30 years from seed was used to describe the

expected top height of each stand in the dataset in comparison with the other stands.

Study sites which did not have a measurement later than 7 years from birth were left

out of the dataset. After this reduction, 19 RVMM plots and 3 SMC plots from 1

installation were removed from the dataset. The estimated site index taken from the

last measurement of each plot was used.

Vegetation Cover

Average plot vegetation cover for each species of shrub or fern (the two most influential

of the vegetation classes) was summed to create a plot-level variable describing the

competing vegetation cover on a given study plot at a given measurement. Shrubs

and ferns were found to have a strong impact on height growth, while adding the coverof grasses and forbs did not noticeably add any information to the model. Percent

cover of trees was not used in calculations of this variable because of major differences

between the data sources in this area. This number is in percent units, though the

cover values for all included species can sum to numbers greater than 100. This is

largely an effect of overlapping layers of vegetation.


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2.1.3 Data Cleaning

Several observations were flagged as probable errors throughout any work with the

data. These observations were checked, and in many cases removed when values were

decided as clearly data recording or entry errors. Negative changes in DBH or height

on young, undamaged trees were suspect, and were likely due to measurement error

in most cases. It is much more difficult to define a removal criterion for large positive

changes. To avoid biasing results by removing more negative errors than positive

errors, no such criterion was used. Only in cases where it was obvious that either the

wrong tree was measured or a mistype occurred during data entry, were observations

removed prior to model fitting.

No techniques were used to fill in missing values. The size of the combined datasets

is large enough that such actions are unnecessary. Tree observations with any missing

values of variables used in the model were removed prior to model fitting. Also, trees

noted as damaged by the survey crews were not used in the model fitting. Such trees

were noted as having any code signifying broken or damaged tops or diseases. The

SMC condition codes were much more detailed than those of the RVMM, but both

contained ample information for this purpose.

In the unfiltered dataset, 8.8 percent of the Douglas-fir were trees marked as

either sick, damaged or broken. About 6 percent of the trees died, and 4 percent

were removed during thinning operations. Less than 1 percent of trees were marked

as forked above or below breast height. No such trees were used in the modeling

dataset.

2.1.4 Variable Summaries

Figure 2.2 shows the distribution of several variables in both datasets. SMC data is

represented with gray bars and RVMM is represented in black. It should be noted

that data for ages greater than 17 years was very slim, as was data for heights greater


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18

than 45 feet. Using this model to grow stands to ages or heights past this range would

likely result in growth estimates of unknown certainty.

2.2 Model design and selection

2.2.1 Annual growth model - centered growing period

In order to build a growth model that works on an annual basis from data with variable

remeasurement periods ranging from 2 to 4 years, some special measures need to be

taken. The response variable for each equation needs to represent, without bias, the

one year growth of a given tree. Simply using the average growth per year for the

remeasurement, as shown in equation 2.1, will not meet this requirement. McDill

and Amateis (1993) give a good explanation of why this is so. Briefly, it results from

assuming ddt

remains constant between the beginning and end measurements.

=i+n i

n(2.1)

where:

is the average change per year of tree dimension (DBH, Height, etc)

over the remeasurement period,

i is the value of tree dimension at the beginning of a n-year growth

period, and

i+n is the value of tree dimension at the end of a n-year growth

period

The actual change in tree dimension is likely to be curvilinear, thus resulting in

bias. However, per the mean value theorem, at some point between year i and year

i + n, the change in will equal the average change. Typically, this point is assumed

to be the middle of the remeasurement period. To use this property advantageously,

the response variable can be that described in equation 2.1, while the predictors are


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2 10 18 26 34 42 50 58

Initial Height (ft.)

NumberofTreemeasurements

0

1000

2000

3000

4000

5000

6000

SMCRVMM

1 2 3 4 5 6 7 8 9 10 12

Initial Breast Height Diameter (in.)

NumberofTreemeasurements

0

1000

2000

3000

4000

5000

6000

SMCRVMM

(a) (b)

30 40 50 60 70 80 90 100

Site Index (ft. at base age 30)

TotalNumberofPlots

0

10

20

30

40

50

60

70

SMCRVMM

1 3 5 7 9 11 15 19 23

Total Stand Age (yrs. from seed)

NumberofPlotmeasurements

0

20

40

60

80

100

SMCRVMM

(c) (d)

Figure 2.2: Histograms of (a) initial tree heights in feet, (b) initial tree breast-heightdiameters in inches, (c) site indexes for the given plots, and (d) total stand agesin years from seed germination. In parts (a) and (b), trees count once for eachmeasurement, and in (d) each plot counts once for each measurement.


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changed to the value expected in this middle part of the remeasurement period, as

shown in equation 2.2 (Clutter 1963).

Xcent =n 1

2 Xi+n Xi

n+ Xi (2.2)

where:

Xi is the value of covariate X at the beginning of a n-year growth

period,

Xi+n is the value of covariate X at the end of a n-year growth

period,

Xcent is the expected value of covariate X at the beginning of a one-year

growth period centered in the actual n-year remeasurement period,

During the model-building process, the one-year coefficients were fit using the

centered growing period technique of equation 2.2. When all parts of the model form

were defined, a different technique, described in section 2.2.6, was used to obtain

improved coefficient estimates for all dimensions of tree growth.

2.2.2 Height growth

Measuring crews measured the height, unlike DBH or basal diameter, on trees through-

out the study period. For this reason, the height growth is the driving variable in

this model. The two most significant predictors of one-year Douglas-fir height growth

were initial height of the tree and site index. The predicted tree height growth shouldbe constrained as a positive function. In order to model this, a multiplicative model

is useful. This can be accomplished by transforming the response in a linear model,

commonly with a log function, or by using a nonlinear model. In this case, the latter

was chosen in order to keep residuals normally distributed in a familiar scale. Coeffi-

cient estimates were obtained using the nls function in the R statistical program (R


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21

Development Core Team 2006).

Base function

The base model was fit in steps. In the initial step, height growth was expressed as

a function of initial height alone (the strongest predictor). This function is shown in

equation 2.3.

Hij = f(H(0)ij) = 1h1 + h2H(0)ij + h3H

h4(0)ij

(2.3)

where:Hij is the predicted one-year change in total height in feet of tree j onplot i,

H(0)ij is the initial total height in feet of tree j in plot i, and

h1 to h4 are parameters estimated by the R function nls.

R2 for this model was 0.412. R2 in this and all following cases is used to symbolizethe ratio of sum of squares model over corrected total sum of squares. This number is

analogous to, but not the same as that used for summarizing linear models. However,

as long as the same data is used in each model,R2 can be used to compare fits betweenmodels. This was how decisions were made about increasing the complexity of the

model.

If the form of the base function was additive, plots of the residuals against several

additional predictors would normally be a good way to start looking into additional

terms to add into the model. This can still be done with a multiplicative model, butinstead of using the residuals (Hij - Hij), where Hij and Hij are the observedand predicted mean annual height growth of tree j in plot i during the observed

growing period, it is helpful to display these errors in terms of a ratio (Hij / Hij).These error ratios were used to investigate the inclusion of further predictors into

the height growth model.


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To investigate the additional effect of site index on the height growth, the error

ratios from model 1 were plotted against site index in figure 2.3. A a non-parametricsmoothing line called a loess (Cleveland 1979) line, was added to show trend. The

trend in the middle range of site index is clear, however what happens at the extremes

is dictated by relatively little data. A function fit through this data, g(Si) multiplied

by the function in model 2.3, f(H(0)ij), creates a potential function f(H(0)ij)g(Si) for

predicting height growth from initial height and site index together. The function

g(Si) was defined to produce reasonable behavior beyond the range of site index

displayed in figure 2.3.

40 60 80 100

1

0

1

2

3

4


Error

Ratio(H

H)

Figure 2.3: Error ratios from model 2.3 plotted against stand site index with a loessline overlaid.

A sigmoidal form for g(Si) is suggested by the loess line in figure 2.3. This sig-


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23

moidal function would level out at high and low values of site index. An equation of

the form in 2.4 was used to model this behavior.

R(E)ij = g(Si) = c1Sc2icc23 + S

c2i

(2.4)

where:

R(E)ij is the predicted error ratio from model 2.3 for tree j in plot i,Si is the stand-level site index associated with plot j

c1

to c3

are parameters estimated by the R function nls.

The combination off(H0) and g(S) resulted in an overparameterized function that

was slow to converge even after removing redundant parameters. In order to alleviatethis problem, a substitute model that could be very flexible, yet stable would needed

to be found. The best of several candidate functions that could produce a similar

prediction surface with fewer parameters is shown in equation 2.5. This function, is

essentially an inverse polynomial function with the integer power restriction relaxed.

To minimize paramter correlation, a restriction was placed on the powers on the

individual terms in the denominator. Inverse polynomial functions have been used to

model tree height in the past (King 1966), and can be very useful in general (Nelder1966). Model 2.5 needed only 5 parameters and fit the data with anR2 of 0.536.

Hij = b(H(0)ij , Si) = 1b1 + b3H

1(0)ijS

b2i + b4S

1b2i + b5H

b21(0)ij

(2.5)

where:


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24

Hij is the predicted one-year change in total height in feet of tree j onplot i,

H(0)ij is the initial total height in feet of tree j in plot i,

Si as defined above, and

b1 to b5 are parameters estimated by the R function nls.

Figure 2.4 shows a contour plot of Hij over the space encompassing the rangeof initial height and site index in the dataset. Figure 2.5 shows plots of the height

growth residuals from model 2.5 against initial height and site index with a loesstrend line overlaid.


Site

Index

(ft.a

tba

seage

30)

0 10 20 30 40 50 60

40

60

80

100

Figure 2.4: A contour plot of the height growth prediction surface from model 2.5over ranges of initial height and site index representative of the data.


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25

0 10 20 30 40 50 60

8

4

0

2

4

6


He

ightGrow

thRes

idua

l(ft.)

(a)

40 60 80 100

8

4

0

2

4

6


He

ightGrow

thRes

idu

al(ft.)

(b)

Figure 2.5: Height growth residual plots from model 2.5. Panel (a) shows residualagainst initial height and panel (b) shows residual against site index.


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Relative height modifier

As the stand ages, competition for resources becomes more intensive. These resources

usually include photosynthetically active light as well as soil moisture and nutrients

(Oliver and Larson 1996). When this is the case, the size of a tree, relative to the

other trees in the stand, has a great impact on the height growth. Because of this

relative height was used to create a height growth modifying function. Relative height

is defined in this project as the height of a tree divided by the top height of the stand

(as defined on page 15). Ninety-five percent of the relative height values in the dataset

occurred in the range 0.278 to 1.107.

This modifying function took the form in equation 2.6, which ensures that if a

tree has height equal to the stand top height, the function will take the value 1. The

effect of relative height increases exponentially as the stand top height increases, so,

for a given value of relative height, the function will take a value closer to 1 in shorter

stands than in taller stands.

r(H(0)ij , H(top)i) = exp(h1 exp(Hh2(top)i) log(H(0)ij/H(top)i)) (2.6)

where:

H(0)ij is as defined above,

H(top)i is the top height of the trees in plot i, and

h1 and h2 are parameters estimated by the R function nls.

This function initially included stems per acre (SPA) as a term inside the innerexponential, however this term added little to the predictability of the function. The

modifying function and the base model were fit at the same time in the same step,

so all parameters in r(H(0)ij , H(top)i) and b(H(0)ij , Si) were allowed to vary during the

search for a minimum residual squared error. Coefficient estimates for the relative

height modifier and updated estimates for the base function are shown in the second


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27

column of table 2.1. A plot of the value produced by this function, using the coefficient

values from the second column in table 2.1, for several values of top height and relativeheight is displayed in figure 2.6.

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

Relative Height(H Htop)

r(H,

Htop

)

Top height = 1Top height = 10Top height = 25Top height = 50

Figure 2.6: Values of the relative height modifying function in model 2.6 versusrelative height for several values of top height.

Vegetation modifier

The amount of vegetation on the plot should have an effect on the rate of height

growth. Furthermore, this effect should not remain constant as the trees grew taller,

and the effect of vegetation on tree height growth should decrease as time increases.

To investigate this relationship, the error ratios were binned into 3-foot top height

intervals. Concern was given only to the error ratios from stands with a top height in


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one given interval at a time. For each bin, a simple linear regression was performed

with error ratio as the response and total vegetation cover as the predictor. This isshown for four of the top height bins in figure 2.7.

0 50 100 150

0.8

1.0

1.2

1.4

1.6

Vegetation Cover

P

lotMeanErrorRatio

Top Height: 2.2 to 9.4 feet

20 40 60 80 100 120

0.8

1.0

1.2

1.4

1.6

Vegetation Cover

P

lotMeanErrorRatio


(a) (b)

0 50 100 150 200

0.8

1.0

1.2

1.4

1.6

Vegetation Cover

PlotMeanEr

rorRatio


0 50 100 150

0.8

1.0

1.2

1.4

1.6

Vegetation Cover

PlotMeanEr

rorRatio


(c) (d)

Figure 2.7: Scatter plots and linear regression lines of mean plot error ratio against

mean plot vegetation for the top height intervals: (a) 2.8 to 9.4 feet, (b) 9.4 to 12.8feet, (c) 20.4 to 25.8, and (d) is 27.6 to 47.2. These four top height intervals are notconsecutive.

The slope and intercept for each bin were plotted against the center of the top

height bin. There is a clear trend from a highly negative slope at low top height in-

creasing to a flat or slightly positive slope as top height increases. This is illustrated in


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figure 2.8, in which the intercept and slope from each bin regression are plotted against

the bin centers. This trend was incorporated into the model by adding a new modifierfunction described in equation 2.7. The lines predicted by this model are shown as

dashed lines in figure 2.8. Both modifying functions and the base model were fit at the

same time in the same step, so all parameters in v(H(top)i, Vi), r(H(0)ij , H(top)i), and

b(H(0)ij , Si) were allowed to vary during the search for a minimum residual squared

error.

v(H(top)i, Vi) = 1 + 1 (H(top)i 2) + 3 (H(top)i 2) Vi (2.7)

where:

H(top)i is as defined above,

Vi is the mean vegetation cover for plot i, and

1, 2 and 3 are parameters estimated by the R function nls.

Because there was no known reason why increasing vegetation would have an

increasingly positive effect on height growth as the stand gets taller, the slope was

assumed to have a horizontal asymptote at a slope of 0. Likewise, the intercept should

have a horizontal asymptote at one. An attempt was made at fitting the exponential

function in equation 2.8 through the data. This model had slightly improved predic-

tions over the linear version in 2.7. Lines for predicted slope and intercept appear as

dotted lines in figure 2.8.

v(H(top)i, Vi) =

1 + 1expH2(top)i

+ 3exp

H2(top)i

Vi

(2.8)

Model 2.8 was slow to converge, indicating possible over-parametrization. A simplified

version of this function, shown in equation 2.9, converged quickly and resulted in little

loss of predictive ability. This modification sets the intercept term equal to a constant


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+

+

+

++

+

++

+

+ +

10 20 30 40 50

0.9

1.0

1.1

1.2

1.3

25

54

52

57

53

60

26

26

14

7 10

Htop Bin Center (ft.)

InterceptofRE~V

+

+

+

++

++

+

+

+

+

10 20 30 40 50

0.004

0.000

0.00

2

0.004

25

54

52

5753

60

2626

14

7

10


SlopeofRE~V

(a) (b)Figure 2.8: The intercept, (a), and slope, (b), of the regression of error ratio (R(E)ij)against vegetation cover (Vi) for several 3-foot intervals of top height (H(top)i). Theregressions were performed on data from plots with a top height within the intervalwith the given center (See figure 2.7). Dashed lines show the predictions from model2.7 and dotted lines show the predictions from model 2.8. The number of plot mea-surements contained in each interval is displayed next to the data points. Error barsrepresent one standard error of the intercept or slope coefficient.


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31

1 for all top heights.

v(H(top)i, Vi) =

1 + 1expH2(top)i

Vi

(2.9)

However, a new problem emerged when using model 2.9. The value of the modifier

function is negative at low top height and high vegetation levels as shown in figure 2.9.

This combination of top height and vegetation cover is not represented in the data,

however, it may occur in nature. The vegetation modifier function was wrapped inside

an exponential (equation 2.10) to limit the possible values of the modifier function

between 0 and 1. The value of this modifier function against top height for selected

vegetation cover levels is shown in figure 2.10.

v(H(top)i, Vi) = exp(exp(1 + 2 (H(top)i)) Vi) (2.10)

Lastly, it was found that using the square root transformation of vegetation cover

term in the modifier function, shown in equation 2.11, improved the fit slightly and

decreased the estimated standard errors of the n parameters. Equation 2.11 is the

final form of the vegetation modifier function. Updated coefficient estimates after

including this modifier function are shown in the third column of table 2.1

v(H(top)i, Vi) = exp(exp(1 + 2 (H(top)i))

Vi) (2.11)

Density modifier

A similar process as that used to build a vegetation modifier was used to investigate

the potential for a stand density modifier. Data were binned by top height intervals,

and regressions of mean error-ratio against density, as stems per acre, were performed

on the data within each bin. In figure 2.11 the slope and intercept of these regressions

are plotted against the center of the top height interval with which they are associated.


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0 10 20 30 40

0.5

0.0

0.5

1

.0

1.5

Stand Top Height (ft.)

v(Htop,

V)

Veg. Cover = 0Veg. Cover = 25Veg. Cover = 75Veg. Cover = 150

Figure 2.9: A plot of the value of the vegetation modifier in model 2.9 against topheight for several levels of vegetative cover. The value of the modifier becomes nega-tive when top height is small and vegetative cover is high.


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0 10 20 30 40

0.5

0.0

0.5

1

.0

1.5


v(Htop,

V)

Veg. Cover = 0Veg. Cover = 25Veg. Cover = 75Veg. Cover = 150

Figure 2.10: A plot of the value of the vegetation modifier in model 2.10 against topheight for several levels of vegetative cover. In contrast to figure 2.9, the value of themodifier stays positive for all values of top height and vegetative cover.


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+ +

+ ++

+ + ++

+

++

+

+

+

10 20 30 40 50

0.90

1.00

1.10

1

.20

21 37

38 4341

4339

44

18

17

206

4

8

6


InterceptofRE~D

+

++ + + + +

+

+

+

+

+

+

+

+

10 20 30 40 50

4e04

0e+00

4e04

21

3738 43 41 43 39

44

18

17

20

6

4

8

6


SlopeofRE~D

(a) (b)

Figure 2.11: The intercept, (a), and slope, (b), of the regression of error ratio (R(E)ij)against stems per acre (Ti) for several 3-foot intervals of top height (H(top)i). Dashedlines are predictions from model 2.12, and dotted lines are predictions from model2.13. The number of plot measurements contained in each interval is displayed nextto the data points. Error bars represent one standard error of the intercept or slopecoefficient.

As shown in figure 2.11, for the first eight bins, up to a top height of about 30 feet,

all slopes are positive and all intercepts are negative. This implies that the model

incorporating the base function and both the relative height and vegetation modifiers

under-predicted the actual growth for higher densities. The next four bins, slopes

are negative and intercepts are positive or near 1, indicating overprediction. While

results past this point are unclear, most likely because of insufficient data, increasing

density in older stands should result in further decreases in growth. Little is known

about this relationship across the full spectrum of top height, so a simple linear fit

through the points in figure 2.11 seems adequate to fit the data.

Equation 2.12 describes the density modifier function. It is a linear function of

stems per acre (Ti), where the intercept and slope parameters are themselves func-

tions of top height (H(top)i). All three modifying functions and the base model were

fit at the same time in the same step, so all parameters in r(H(top)i, Ti), v(H(top)i, Vi),


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r(H(0)ij , H(top)i), and b(H(0)ij , Si) were allowed to vary during the search for a min-

imum residual squared error. Adding this modifier function improved the fit of theoverall model.

d(H(top)i, Ti) = 1 + d1 (H(top)i d2) + d3 (H(top)i d2) Ti (2.12)

where:

H(top)i is as defined above,

Ti is the stems per acre for plot i, and

d1, d2 and d2 are parameters estimated by the R function nls.

Function 2.12 appears in both parts of figure 2.11 as a dashed line. The value ofd2,

the top height at which the effect of increasing density goes from positive to negative

(crossover point) was around 31 feet. The d1 term was found to be significantly

different from 0, but with a relatively high p-value of nearly 0.05. Therefore, d1 was

removed from the model leaving model 2.13. Here d3 was renamed to d1. Removing

d1 parameter resulted, as expected, in very little difference in the fit of the model.

d(H(top)i, Ti) = 1 + d1 (H(top)i d2) Ti (2.13)

The value of the d2 crossover parameter changed to 32 feet. Attempts to allow

this crossover point to vary with site index proved futile, as no trends were found.

Equation 2.13 will take the value of 1, implying no change in the expected growthfrom the rest of the model, at a top height of about 32 feet (H(top)i = d2) and at a

density of zero stems per acre (Ti = 0). The latter makes little sense as no forests are

planted at zero stems per acre. In order for the density modifier function to take the

value of 1 at a more typical planting density, the variable Ti was shifted by 300 stems

per acre (equation 2.14). This matches the value given by Flewelling et al. (2001),


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and represents a tree spacing at planting of about 12 feet. This would mean that the

density modifier function would have no effect unless the stand deviated from thisindex density. This shift had no effect on the fit of the overall model.

d(H(top)i, Ti) = 1 + d1 (H(top)i d2) (Ti 300) (2.14)

The full height growth model

The b2 parameter in the base function had very high correlation with all other base

function parameters, suggesting that the model may still be overparameterized. Be-

cause d2 was consistently estimated to be in the range 1.3 to 1.8, during the model

building, this parameter was fixed at a value of 1.5. The final base function is shown

in equation 2.15.

b(H(0)ij , Si) =1

b1 + b3H1(0)ijS

1.5i + b4S

0.5i + b5H

0.5(0)ij

(2.15)

Multiplied together, the base function and the three modifier functions yield equa-

tion 2.16. This model fit with anR2 of 0.611.

Hij = b(H(0)ij , Si) r(H(0)ij , H(top)i) v(H(top)i, Vi) d(H(top)i, Ti) (2.16)

where:

b(H(0)ij , Si) is as described in equation 2.15

r(H(0)ij , H(top)i) is as described in equation 2.6

v(H(top)i, Vi) is as described in equation 2.11

d(H(top)i, Ti) is as described in equation 2.14

The coefficient values and model fit statistics (sum of squared residuals and R2)


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at different steps in the building of the height increment model (2.16) are shown in

table 2.1.Table 2.1: Coefficients and fit statistics for the different stages of the height incrementmodel. SSR is sum of squared residuals.

Model: 2.5 2.6 2.11 2.16 w/ b2 2.16b1 3.410e01 5.058e01 2.158e01 2.047e01 2.571e01b2 1.383e+00 1.292e+00 1.542e+00 1.552e+00 b3 9.195e+02 5.760e+02 9.299e+02 9.953e+02 8.147e+02b4 2.000e+00 1.685e+00 3.299e+00 3.614e+00 3.192e+00b5 5.753e02 1.004e01 2.602e02 2.162e02 2.875e02h1

8.000e

03 3.155e

02 3.713e

02 3.682e

02

h2 4.153e01 3.284e01 3.064e01 3.072e011 2.337e+00 2.345e+00 2.355e+002 1.080e01 1.188e01 1.178e01d1 6.038e06 6.047e06d2 3.116e+01 3.111e+01

SSR 9.871e+03 8.693e+03 8.416e+03 8.287e+03 8.287e+03R2 0.536 0.592 0.605 0.611 0.611

2.2.3 Diameter growth

Modeling diameter growth of young trees is complicated by the fact that diameter is

typically measured at breast height (4.5 feet). Difficulties arise when at least some of

the trees on a plot have not yet reached breast height. While basal diameter (taken at

near ground level) growth would be simpler to model through this period, sufficient

data were not collected to produce such a model. Two models are thus needed to

simulate the breast height diameter of young stands. One is a model to get an initial

diameter for the tree when it crosses the breast height threshold, and the other togrow the diameter from this point. Both of these models are presented in this section.

Breast Height Diameter Growth

DBH is grown as an increase in squared DBH because the yearly increment of area

should be more constant, across initial DBH, than yearly increment in radius. The


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base function of the DBH growth model in equation 2.17, is similar to the diameter

increment function in CONIFERS (Ritchie and Hamann 2007). The difference isthat a logistic curve is built in to model the effects of site index. The base function

includes many more predictors than that of the height growth base function. Attempts

at fitting a dynamic modifier function for relative height, such as equation 2.6, were

unsuccessful. Therefore, top height and relative height were included as terms in

the base function. This implies that the effect (slope) of relative height on diameter

growth remained more constant as the stands increased in top height. Adding these

terms to the model significantly improved the fit based on R2 values. Basal areaper acre and diameter-height ratio were added as terms in the base function as well

because of similar improvements in fit.

D2ij = b(D(0)ij , H(0)ij , Si, H(top)i, Bi) =b1D

b2(0)ijexp(b3D(0)ij + b4Bi + b7

D(0)ij

H(0)ij+ b8

H(0)ij

H(top)i+ b9H(top)i)

(1 + exp(b5 b6Si))(2.17)

where:

D2ij is the predicted one-year change in squared breast height diameter ininches of tree j in plot i,

H(0)ij is the initial total height in feet of tree j in plot i,

D(0)ij is the initial breast height diameter in inches of tree j in plot i,

Si is the stand-level site index associated with plot i

Bi is the stand-level basal area per acre of plot i

b1 to b9 are parameters estimated by the R function nls, and

other variables are as defined previously.

Neither the vegetative competition modifier nor the relative height modifier were

found to significantly improve the DBH growth predictions. It was assumed that


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vegetation would have a similar effect on DBH growth as it had on height growth.

This turned out not to be the case,suggesting that much of the vegetative effect ondiameter is subsiding by the time the tree reaches breast height. In fact, simply

adding vegetation cover to the base function was of little help. The density modifier

was marginally effective in improving DBH growth estimates, and has the same form

as it did for the height growth model (Equation 2.14). The overall function for

squared DBH growth is given in equation 2.18. As with the height growth model,

both functions were fit simultaneously to the data.

D2ij = b(D(0)ij , H(0)ij , Si, H(top)i, Bi) d(H(top)i, Ti) (2.18)

where:

b(D(0)ij , H(0)ij , Si, H(top)i, Bi) is as described in equation 2.17 and

d(H(top)i, Ti) is as described in equation 2.14

The coefficient values and model fit statistics (sum of squared residuals and R2)for the base function alone and the base function with the density modifier are shownin table 2.2. Residuals are plotted against predicted values, both in diameter scale,

in figure 2.12.

Initial Breast Height Diameter model

When a tree crosses the breast-height threshold in the model, a function is needed to

assign an initial DBH to the tree. This initial DBH depends heavily on how far the

height of the tree is projected to be past breast-height at the end of the season. Toaccount for this, a static linear function of height, density (as stems per acre), and

plot vegetation cover was fit to the data. All Douglas-fir trees in the dataset with

a height between breast-height and 7.5 feet (the largest a tree shorter than breast-

height was expected to be able to grow to in one year) were included in the data for

this function. A log transformation was performed on the response variable, DBH.


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0.0 0.2 0.4 0.6 0.8 1.0

2

.0

1

.0

0.0

0.5

1.0

1.5

Predicted DBH growth (in.)

DBH

grow

thRes

idua

l(in.

)

Figure 2.12: Residuals versus predicted values from model 2.18. Both predictedgrowth and residual were converted from squared units to untransformed scale (inches)before plotting.


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Table 2.3: Coefficients estimated for the model described in equation 2.19.

Estimate Std. Error t value Pr(> |t|)0 3.79E+00 2.8340E02 133.59


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0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

.5

0.0

0.5

1.0

1.5

2.0

Predicted DBH (in.)

Res

idua

l(in.)

Figure 2.13: Residuals versus predicted values from model 2.19. The predicted DBHwas transformed to the original scale from the log scale before computing residuals.


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0 10 20 30 40

0

50

100

1

50


PlotMeanVegetationCover

0 20 40 60 80 100 120

0

50

100

1

50

Stand Basal area (sq. ft. / ac.)

PlotMeanVegetationCover

(a) (b)

Figure 2.14: Plot level vegetation cover (sum of percents for individual species) against(a) stand top height in feet and (b) stand basal area in square feet per acre.


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and top height. Because the plot level tree data was computed for some plots by

interpolating between measurements, the centering technique of estimating one-yearcoefficients was not used. To do so would not likely reduce bias in the estimates. A

stepwise procedure helped determine which variables (from all squared and interaction

terms) should be included in the model. The final form of this function is displayed

as equation 2.20.

Vi = 0+1V(0)i+2Bi+3Ti+4Si+5H(top)i+6ViTi+7ViH(top)i+8BiSi (2.20)where:

Vi is the predicted one-year change in vegetation cover (%) at plot i,V(0)i is the initial vegetation cover at plot i,

Bi is the basal area per acre in plot i,

Ti is the stems per acre in plot i,

Si is the site index of plot i, andH(top)i is the stand top height of plot i

0 to 8 are parameters estimated by the R function lm

Values of the coefficients in model 2.20 are displayed in table 2.4. Of the five first

order predictors, only top height had a slope significantly different from 0 at = 0.05.

However, all three interaction terms in the model had slopes statistically different from0. The insignificant first order terms were left in the model to maintain its hierarchical

nature. A check of model assumptions revealed no noticeable problems. The R2 ofthis model is 0.3004 and the residual squared error is relatively high at 10.53. Model

2.20 is not a strong model, but with the high variability in the response, the results

were somewhat better than expected.


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Table 2.4: Coefficients estimated for the model described in equation 2.20.

Estimate Std. Error t value Pr(> |t|)0 1.23e+00 6.4379e+00 0.19 8.481e011 1.46e01 5.8846e02 2.48 1.386e022 1.29e01 1.6200e01 0.80 4.272e013 5.93e04 3.0242e03 0.20 8.447e014 7.31e02 6.4017e02 1.14 2.544e015 6.30e01 1.8634e01 3.38 8.402e046 1.01e04 5.3689e05 1.89 6.052e027 5.38e03 2.3947e03 2.25 2.562e028 5.31e03 1.9780e03 2.68 7.774e03

2.2.5 Mortality

The amount of mortality expected of a stand is important to predict. Failure to do

so can lead to significant overestimates of future yield. The probability of mortality

is dependent on the position of the tree in the hierarchy of the stand. Larger trees

are in a better position to compete for needed resources. Logistic regression models

allow a linear function to be fit to the logit of the probability of an event. The logit

function transforms a sigmoidal curve with range of only 0 to 1 into an approximately

linear continuous function without implicit range limits.

A generalized linear model was fit to the survival (0 or 1) of each tree that was

measured at both ends of a two-year period and was not marked as dead at the

beginning of the period. A binomial distribution was assumed around the predicted

mean for each tree. The model was fit using the R function glm, which uses a maximum

likelihood approach (R Development Core Team 2006). Because the variance of abinomial random variable depends on the mean, the observations are iteratively re-

weighted until convergence in fit is achieved. Variables were added or dropped from

the model based on the AIC value. Since AIC is a function of the deviance and

the number of parameters, adding unnecessary variables to the model and removing

helpful variables both inflate the AIC value. The final form of this model is displayed


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as equation 2.21. The probability of survival was assumed to be nearly equal for each

year of any two-year period. The predicted one-year probability of survival can thenbe calculated as the square root of the two-year predicted survival.

ij = 0 + 1H(0)ij + 2H(top)i + 3Ti + 4H(top)iH(0)ij (2.21)

where:

ij is log(ij) log(1 ij), the log of the predicted odds ratio of tree jin plot i,ij is the predicted probability of survival over the two-year period oftree j in plot i,

H(0)ij is the initial height of tree j in plot i,

H(top)i is the stand top height of plot i

Ti is the stems per acre in plot i, and

0 to 4 are parameters estimated by the R function glm

The estimated values for the coefficients in model 2.21 are shown in table 2.5.

Again a high number of trees results in incredibly low p-values, where instead degrees

of freedom should be based on the number of plots. The

Documents

An Individual-tree Model to Predict the Annual Growth of Young Stands of Douglas-ﬁr (Pseudotsuga menziesii (Mirbel) Franco) in the Paciﬁc Northwest