an individual-tree model to predict the annual growth of young stands of douglas-fir (pseudotsuga...
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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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University of Washington
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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University of Washington
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
2.4 Coefficients estimated for model 2.20. . . . . . . . . . . . . . . . . . . 46
2.5 Coefficients estimated for model 2.21. . . . . . . . . . . . . . . . . . . 48
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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1
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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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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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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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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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
Site Index (ft. at base age 30)
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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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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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.
Initial Height (ft.)
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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0 10 20 30 40 50 60
8
4
0
2
4
6
Initial Height (ft.)
He
ightGrow
thRes
idua
l(ft.)
(a)
40 60 80 100
8
4
0
2
4
6
Site Index (ft. at base age 30)
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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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
Top Height: 9.4 to 12.8 feet
(a) (b)
0 50 100 150 200
0.8
1.0
1.2
1.4
1.6
Vegetation Cover
PlotMeanEr
rorRatio
Top Height: 20.4 to 25.8 feet
0 50 100 150
0.8
1.0
1.2
1.4
1.6
Vegetation Cover
PlotMeanEr
rorRatio
Top Height: 27.6 to 47.2 feet
(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
Htop Bin Center (ft.)
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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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
Stand Top Height (ft.)
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
Htop Bin Center (ft.)
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
Htop Bin Center (ft.)
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
Stand Top Height (ft.)
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