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 fulfillment
of the requirements for the degree of
Master of Science
University of Washington
2007
Program Authorized to Offer Degree: College of Forest Resources
University of Washington
Graduate School
This is to certify that I have examined this copy of a master’s 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:
In presenting this thesis in partial fulfillment of the requirements for a master’s
degree at the University of Washington, I agree that the Library shall make its copies
freely 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 the
U.S. Copyright Law. Any other reproduction for any purpose or by any means shall
not be allowed without my written permission.
Signature
Date
University of Washington
Abstract
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
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 vegetation 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
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.
TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Chapter 1:
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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:
Introduction
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 Growth
of Young Plantations of Pacific Northwest Douglas-fir Incorporating 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
i
Chapter 4:
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.1 Height and Diameter Increment . . . . . . . . . . . . . . . . . . . . .
75
4.2 Secondary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.3 Potential Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.4 Height Model Coefficients . . . . . . . . . . . . . . . . . . . . . . . .
83
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
ii
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. . . . . .
33
2.11 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
iii
3.6 Bootstrap distributions of the parameters of the full height growth model. 68
3.7 Scatterplot matrix of the bootstrap coefficient estimates. . . . . . . .
iv
69
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 equation
2.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 through
3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.3 Effects of changing the model coefficients within the bootstrap distribution limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.4 Sensitivity of the height growth model to parameter changes. . . . . .
70
v
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.
vi
1
Chapter 1
INTRODUCTION
1.1
Growth and Yield Models
Management decisions in forestry have the potential to impact greatly both the financial 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 harvests 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 conditions 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
2
harvest scheduling decision without consulting the output from at least one growth
model. Models are even being used in the formulation of policy concerning forest
resources (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). Examples 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
3
explanation for this trend is that large increases in computing power are more readily
available. Creating and using individual-tree models can be computationally intensive. 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 adaptable 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. Most
individual-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 for
individual-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
4
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, the
basal 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 mixedspecies 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 estimate 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
5
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, Zeide
1994). 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 composition. 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 Technology Initiative 2006). This has led to shorter rotations and an increase in young
stand 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 thinning treatments are considered and when decisions about harvest age are made. However, 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
6
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 manager’s 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
7
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 the
stand 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, CONIFERS (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-
8
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. The
final 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.
9
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 densities: 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 installations 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-
10
RVMM Coastal
RVMM Cascade
SMC Type III
Figure 2.1: Locations of the study tree plots within the Pacific Northwest. Plots
from 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 combination of the two, with planting stock chosen by the landowner. Site preparation
was typically performed to match with the landowner’s current “best management
11
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 measurement 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 discussion), 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 desired. Almost 25 percent of the usable vegetation measurements occurred between
tree measurements. Vegetation was classified by life form, either Shrub, Forb, Grass
or Fern.
12
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 Alternative 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, information 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 (labeled
as “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 and
height 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
13
or not, was recorded for each tree.
PMPs were remeasured only once, typically after two years. Some tree measurements 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 treatment 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 experiment, hardwoods in the SMC installations were removed if they reached half the
height of surrounding conifers. Hardwoods below this height were treated as vegetation 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
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).
15
2.1.2 Computed Variables
Stand-level variables were computed from the individual tree data at each tree measurement. 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
0.005454154
X
d2i
n
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 stands
where 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.
16
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 young
as 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 cover
of 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.
17
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
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
dΥ
dt
remains constant between the beginning and end measurements.
∆Υ =
Υi+n − Υi
n
(2.1)
where:
∆Υ
is the average change per year of tree dimension Υ (DBH, Height, etc)
over the remeasurement period,
Υi
is the value of tree dimension Υ at the beginning of a n-year growth
period, and
Υi+n
is the value of tree dimension Υ at the end of a n-year growth
period
The actual change in tree dimension Υ 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 Υ 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
6000
2000
3000
4000
5000
SMC
RVMM
0
0
1000
2000
3000
4000
Number of Tree−measurements
5000
SMC
RVMM
1000
Number of Tree−measurements
6000
19
10
18
26
34
42
50
58
1
2
3
4
5
6
7
8
9
10
Initial Height (ft.)
Initial Breast Height Diameter (in.)
(a)
(b)
12
70
100
2
SMC
RVMM
80
60
0
0
10
20
40
Number of Plot−measurements
50
40
30
20
Total Number of Plots
60
SMC
RVMM
30
40
50
60
70
80
90
100
1
3
5
7
9
11
15
19
Site Index (ft. at base age 30)
Total Stand Age (yrs. from seed)
(c)
(d)
23
Figure 2.2: Histograms of (a) initial tree heights in feet, (b) initial tree breast-height
diameters in inches, (c) site indexes for the given plots, and (d) total stand ages
in years from seed germination. In parts (a) and (b), trees count once for each
measurement, and in (d) each plot counts once for each measurement.
20
changed to the value expected in this middle part of the remeasurement period, as
shown in equation 2.2 (Clutter 1963).
Xcent =
n − 1 Xi+n − Xi
×
+ Xi
2
n
(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 throughout 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 should
be 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. Coefficient estimates were obtained using the nls function in the R statistical program (R
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.
dij = f (H(0)ij ) =
∆H
1
h4
h1 + h2 H(0)ij + h3 H(0)ij
(2.3)
where:
dij
∆H
is the predicted one-year change in total height in feet of tree j on
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.
plot i,
c2 for this model was 0.412. R
c2 in this and all following cases is used to symbolize
R
the 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,
c2 can be used to compare fits between
as long as the same data is used in each model, R
models. 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, but
dij ), where ∆Hij and ∆H
dij are the observed
instead of using the residuals (∆Hij - ∆H
and predicted mean annual height growth of tree j in plot i during the observed
dij ).
growing period, it is helpful to display these errors in terms of a ratio (∆Hij / ∆H
These “error ratios” were used to investigate the inclusion of further predictors into
the height growth model.
22
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-parametric
smoothing 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
2
1
0
−1
^
Error Ratio(∆H ∆H)
3
4
displayed in figure 2.3.
40
60
80
100
Site Index (ft. at base age 30)
Figure 2.3: Error ratios from model 2.3 plotted against stand site index with a loess
line overlaid.
A sigmoidal form for g(Si) is suggested by the loess line in figure 2.3. This sig-
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.
Rd
(E)ij = g(Si ) =
c1 Sic2
cc32 + Sic2
(2.4)
where:
Rd
(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 of f (H0 ) and g(S) resulted in an overparameterized function that
was slow to converge even after removing redundant parameters. In order to alleviate
this 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 (Nelder
c2 of 0.536.
1966). Model 2.5 needed only 5 parameters and fit the data with an R
dij = b(H(0)ij , Si ) =
∆H
where:
b1 +
−1
b3 H(0)ij
Si−b2
1
b2 −1
+ b4 Si1−b2 + b5 H(0)ij
(2.5)
24
dij
∆H
is the predicted one-year change in total height in feet of tree j on
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.
plot i,
dij over the space encompassing the range
Figure 2.4 shows a contour plot of ∆H
of 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 loess
80
60
40
Site Index (ft. at base age 30)
100
trend line overlaid.
0
10
20
30
40
50
60
Initial Height (ft.)
Figure 2.4: A contour plot of the height growth prediction surface from model 2.5
over ranges of initial height and site index representative of the data.
4
2
0
−4
−8
Height Growth Residual (ft.)
6
25
0
10
20
30
40
50
60
Initial Height (ft.)
4
2
0
−4
−8
Height Growth Residual (ft.)
6
(a)
40
60
80
100
Site Index (ft. at base age 30)
(b)
Figure 2.5: Height growth residual plots from model 2.5. Panel (a) shows residual
against initial height and panel (b) shows residual against site index.
26
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.
h2
r(H(0)ij , H(top)i ) = exp(h1 ∗ exp(H(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 inner
exponential, 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
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 relative
1.0
0.5
Top height = 1
Top height = 10
Top height = 25
Top height = 50
0.0
r(H, Htop)
1.5
height is displayed in figure 2.6.
0.0
0.5
1.0
1.5
Relative Height(H Htop)
Figure 2.6: Values of the relative height modifying function in model 2.6 versus
relative 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
28
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 is
shown for four of the top height bins in figure 2.7.
50
100
1.6
1.4
1.2
1.0
Plot Mean Error Ratio
1.4
1.2
1.0
0
0.8
1.6
Top Height: 9.4 to 12.8 feet
0.8
Plot Mean Error Ratio
Top Height: 2.2 to 9.4 feet
150
20
40
60
Vegetation Cover
Vegetation Cover
(c)
200
1.6
1.4
1.2
1.0
Plot Mean Error Ratio
150
0.8
1.6
1.4
1.2
1.0
Plot Mean Error Ratio
Top Height: 27.6 to 47.2 feet
0.8
100
120
(b)
Top Height: 20.4 to 25.8 feet
50
100
Vegetation Cover
(a)
0
80
0
50
100
150
Vegetation Cover
(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 12.8
feet, (c) 20.4 to 25.8, and (d) is 27.6 to 47.2. These four top height intervals are not
consecutive.
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 increasing to a flat or slightly positive slope as top height increases. This is illustrated in
29
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 modifier
function 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 + ν1 ∗ (H(top)i − ν2 ) + ν3 ∗ (H(top)i − ν2 ) ∗ Vi
(2.7)
where:
H(top)i
is as defined above,
Vi
is the mean vegetation cover for plot i, and
ν1 , ν2 and ν3
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 predictions over the linear version in 2.7. Lines for predicted slope and intercept appear as
dotted lines in figure 2.8.
´ ´
´
³
³
³
ν2
ν2
v(H(top)i , Vi) = 1 + ν1 exp H(top)i + ν3 exp H(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
0.004
+14
+57
+53
+54
+60
+52
0.9
+26
+26
10
20
30
Htop Bin Center (ft.)
(a)
+10
Slope of RE~V
0.000
0.002
+25
−0.004
Intercept of RE~V
1.0
1.1
1.2
1.3
30
+7 +10
40
50
+7
+26
+52
+26
+60
+54
+57
+53
+14
+25
10
20
30
Htop Bin Center (ft.)
40
50
(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 ). The
regressions were performed on data from plots with a top height within the interval
with the given center (See figure 2.7). Dashed lines show the predictions from model
2.7 and dotted lines show the predictions from model 2.8. The number of plot measurements contained in each interval is displayed next to the data points. Error bars
represent one standard error of the intercept or slope coefficient.
31
1 for all top heights.
³
³
´ ´
ν2
v(H(top)i , Vi) = 1 + ν1 exp H(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(ν1 + ν2 ∗ (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 νn 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(ν1 + ν2 ∗ (H(top)i ))
p
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.
0.5
Veg. Cover = 0
Veg. Cover = 25
Veg. Cover = 75
Veg. Cover = 150
−0.5
0.0
v(Htop, V)
1.0
1.5
32
0
10
20
30
40
Stand Top Height (ft.)
Figure 2.9: A plot of the value of the vegetation modifier in model 2.9 against top
height for several levels of vegetative cover. The value of the modifier becomes negative when top height is small and vegetative cover is high.
0.5
Veg. Cover = 0
Veg. Cover = 25
Veg. Cover = 75
Veg. Cover = 150
−0.5
0.0
v(Htop, V)
1.0
1.5
33
0
10
20
30
40
Stand Top Height (ft.)
Figure 2.10: A plot of the value of the vegetation modifier in model 2.10 against top
height for several levels of vegetative cover. In contrast to figure 2.9, the value of the
modifier stays positive for all values of top height and vegetative cover.
1.20
+20
+6
+21
+4
Slope of RE~D
0e+00
+6
Intercept of RE~D
1.00
1.10
+17
+41
+18
+38+43 +43+39+44
+4
+21+37
10
+8
20
30
Htop Bin Center (ft.)
40
+37
+44
+8
+38+43+41+43+39
+18
50
+6
+17
+20
−4e−04
0.90
4e−04
34
+6
10
20
30
Htop Bin Center (ft.)
(a)
40
50
(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 ). Dashed
lines are predictions from model 2.12, and dotted lines are predictions from model
2.13. The number of plot measurements contained in each interval is displayed next
to the data points. Error bars represent one standard error of the intercept or slope
coefficient.
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 functions 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),
35
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. Adding this modifier function improved the fit of the
overall 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 of d2 ,
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 growth
from 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),
36
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 this
index 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. Because 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 ) =
b1 +
−1
b3 H(0)ij
Si−1.5
1
0.5
+ b4 Si−0.5 + b5 H(0)ij
(2.15)
Multiplied together, the base function and the three modifier functions yield equac2 of 0.611.
tion 2.16. This model fit with an R
dij = b(H(0)ij , Si ) × r(H(0)ij , H(top)i ) × v(H(top)i , Vi) × d(H(top)i , Ti )
∆H
(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
c2 )
The coefficient values and model fit statistics (sum of squared residuals and R
37
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 increment
model. SSR is sum of squared residuals.
Model:
b1
b2
b3
b4
b5
h1
h2
ν1
ν2
d1
d2
SSR
c2
R
2.5
2.6
2.11
2.16 w/ b2
2.16
−3.410e−01 −5.058e−01 −2.158e−01 −2.047e−01 −2.571e−01
1.383e+00
1.292e+00
1.542e+00
1.552e+00
−
9.195e+02
5.760e+02
9.299e+02
9.953e+02
8.147e+02
2.000e+00
1.685e+00
3.299e+00
3.614e+00
3.192e+00
5.753e−02
1.004e−01
2.602e−02
2.162e−02
2.875e−02
−
8.000e−03
3.155e−02
3.713e−02
3.682e−02
−
4.153e−01
3.284e−01
3.064e−01
3.072e−01
−
− −2.337e+00 −2.345e+00 −2.355e+00
−
− −1.080e−01 −1.188e−01 −1.178e−01
−
−
− −6.038e−06 −6.047e−06
−
−
−
3.116e+01
3.111e+01
9.871e+03
8.693e+03
8.416e+03
8.287e+03
8.287e+03
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 to
grow 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
38
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 is
that 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
c2 values. Basal area
terms to the model significantly improved the fit based on R
per acre and diameter-height ratio were added as terms in the base function as well
because of similar improvements in fit.
d2 = b(D
∆D
(0)ij , H(0)ij , Si , H(top)i , Bi ) =
ij
D
H
(0)ij
b2
(0)ij
+ b8 H(top)i
+ b9 H(top)i )
b1 D(0)ij
exp(b3 D(0)ij + b4 Bi + b7 H(0)ij
(1 + exp(b5 − b6 Si ))
where:
d2
∆D
ij
(2.17)
is the predicted one-year change in squared breast height diameter in
inches 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
39
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 on
diameter 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.
d2 = b(D(0)ij , H(0)ij , Si , H(top)i , Bi ) × d(H(top)i , Ti )
∆D
ij
(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
c2 )
The coefficient values and model fit statistics (sum of squared residuals and R
for the base function alone and the base function with the density modifier are shown
in 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. To
account 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 breastheight 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.
0.5
0.0
−1.0
−2.0
DBH growth Residual (in.)
1.0
1.5
40
0.0
0.2
0.4
0.6
0.8
1.0
Predicted DBH growth (in.)
Figure 2.12: Residuals versus predicted values from model 2.18. Both predicted
growth and residual were converted from squared units to untransformed scale (inches)
before plotting.
41
Table 2.2: Coefficients and fit statistics for the different stages of the squared diameter
increment model. SSR is sum of squared residuals.
Model:
b1
b2
b3
b4
b5
b6
b7
b8
b9
d1
d2
SSR
c2
R
2.17
7.023e+00
1.422e+00
1.468e−01
−6.109e−03
1.907e+00
2.466e−02
−1.791e+00
−5.580e−01
−4.801e−02
−
−
1.999e+04
0.836
2.18
7.969e+00
1.494e+00
1.381e−01
−4.286e−03
1.825e+00
2.443e−02
−2.261e+00
−6.156e−01
−5.221e−02
−7.350e−06
1.189e+01
1.982e+04
0.837
Table 2.3 gives the estimated parameters for the initial DBH model, equation (2.19).
The negative values for β2 and β3 indicate that for a given tree height, an increase in
density of vegetation cover will decrease the predicted diameter. This is explainable
because, in general, trees trying to outgrow competition put more effort into height
growth (Oliver and Larson 1996).
dij ) = β0 + β1 Hij + β2 Ti + β3 Vi
ln(D
(2.19)
where:
dij )
ln(D
is the predicted natural log of the breast-height diameter of tree j
in plot i,
Hij
is the height of tree j in plot i,
Ti
is the stems per acre in plot i,
Vi
is the vegetation cover in plot i, and
β0 to β3
are parameters estimated by the R function lm
42
Table 2.3: Coefficients estimated for the model described in equation 2.19.
β0
β1
β2
β3
Estimate
−3.79E+00
4.93E−01
−1.15E−04
−1.64E−03
Std. Error
t value
Pr(> |t|)
2.8340E−02 −133.59 <0.000001
4.4135E−03
111.67 <0.000001
7.3536E−06 −15.69 <0.000001
1.1623E−04 −14.08 <0.000001
Model 2.19 predicted initial DBH with a residual squared error (in log scale) of
c2 of this model was 0.7263. There appears, from a plot of residuals
0.2615. The R
against predicted values (figure 2.13) to be no problems with heteroskedasticity. This
is likely due to the log transformation of the response variable and the narrow range of
the response variable allowed in the fitting dataset. The distribution of the residuals
from this model had heavy tails, but appeared to be effectively unskewed.
2.2.4 A Vegetation Dynamics Model
Incorporating vegetation cover in the tree height growth model requires that the
amount of change in vegetation cover be modeled as well. The vegetation cover data
is highly variable with a coefficient of variation at the quadrant level within plots
ranging from about 0.2 to 0.7. The coefficient of variation decreases as the mean
vegetation cover increases. Also, it was normal for the quadrants within a plot to
change vegetation cover in opposite directions during a given period. The plot mean
vegetation cover should increase from stand establishment to a peak and decline
when the stand nears canopy closure. The same effect should be noticeable in terms
of stand top height and stand basal area. Figure 2.14 shows stand mean vegetation
cover plotted against both stand top height and stand basal area. The peak is visible
in the loess lines fit to the data in both panels of the figure.
A linear function was used to model the vegetation dynamics. The response
variable in this model is the annual change in vegetation cover and it is predicted by
a function of initial vegetation cover, stems per acre, basal area per acre, site index,
1.0
0.5
−0.5
0.0
Residual (in.)
1.5
2.0
43
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Predicted DBH (in.)
Figure 2.13: Residuals versus predicted values from model 2.19. The predicted DBH
was transformed to the original scale from the log scale before computing residuals.
150
100
0
50
Plot Mean Vegetation Cover
100
50
0
Plot Mean Vegetation Cover
150
44
0
10
20
30
40
0
20
40
60
80
100
Stand Top Height (ft.)
Stand Basal area (sq. ft. / ac.)
(a)
(b)
120
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.
45
and top height. Because the plot level tree data was computed for some plots by
interpolating between measurements, the centering technique of estimating one-year
coefficients 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.
di = β0 +β1 V(0)i +β2 Bi +β3 Ti +β4 Si +β5 H(top)i +β6 Vi Ti +β7 Vi H(top)i +β8 Bi Si (2.20)
∆V
where:
di
∆V
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, and
H(top)i
is the stand top height of plot i
β0 to β8
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 α = 0.05.
However, all three interaction terms in the model had slopes statistically different from
0. The insignificant first order terms were left in the model to maintain its hierarchical
c2 of
nature. A check of model assumptions revealed no noticeable problems. The R
this 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.
46
Table 2.4: Coefficients estimated for the model described in equation 2.20.
β0
β1
β2
β3
β4
β5
β6
β7
β8
Estimate
1.23e+00
−1.46e−01
1.29e−01
−5.93e−04
7.31e−02
6.30e−01
1.01e−04
−5.38e−03
−5.31e−03
Std. Error t value
6.4379e+00
0.19
5.8846e−02 −2.48
1.6200e−01
0.80
3.0242e−03 −0.20
6.4017e−02
1.14
1.8634e−01
3.38
5.3689e−05
1.89
2.3947e−03 −2.25
1.9780e−03 −2.68
Pr(> |t|)
8.481e−01
1.386e−02
4.272e−01
8.447e−01
2.544e−01
8.402e−04
6.052e−02
2.562e−02
7.774e−03
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 a
binomial random variable depends on the mean, the observations are iteratively reweighted 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
47
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 then
be calculated as the square root of the two-year predicted survival.
ηij = β0 + β1 H(0)ij + β2 H(top)i + β3 Ti + β4 H(top)i H(0)ij
(2.21)
where:
ηij
is log(b
µij ) − log(1 − µ
bij ), the log of the predicted odds ratio of tree j
in plot i,
µ
bij
is the predicted probability of survival over the two-year period of
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
β0 to β4
are parameters estimated by the R function glm
tree j in plot i,
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 actual p-values should be
only slightly larger than those displayed in the table.
To apply the predicted probability of survival to the rest of the growth model,
the expansion factor for each tree, or the number of trees per unit area each tree
represents, is multiplied by the predicted survival at each one-year time step. Each
tree in the tree list will then represent fewer trees at each time step.
2.2.6 Annualizing the Model
As mentioned previously, the height growth and diameter growth models described
above are built to predict one year worth of growth. For investigating model forms,
48
Table 2.5: Coefficients estimated for the model described in equation 2.21.
β0
β1
β2
β3
β4
Estimate Std. Error z value
Pr(> |z|)
3.93e+00 8.0877e−02
48.60 <0.000001
1.19e−01 8.2448e−03
14.44 <0.000001
−5.02e−02 5.0829e−03 −9.88 <0.000001
−4.80e−04 5.1830e−05 −9.27 <0.000001
−2.01e−03 2.1242e−04 −9.49 <0.000001
the growth period centering technique, described on page 18, was used to reduce the
bias involved in estimating one-year growth from variable remeasurement intervals.
This section describes an iterative fitting technique used to avoid the assumption that
the mean growth rate occurs in the center of the remeasurement period. This timing
is likely to change at different values of the predictor variables.
McDill and Amateis (1993) describe two techniques to fit annualized growth equations from multiple year growth periods. One technique is to use a recursive growth
function, as shown in equation 2.22. Here the total growth during the growing period
is the sum of several one-year periods. None of these one-year growth allotments are
observed, and must be predicted at the same time as predicting the values of the
coefficients, β.
Yd
i+n = Yi + f (Yi , X, β) + f (Yi+1 , X, β) + . . . + f (Yi+n−1 , X, β)
Yd
i+n
= Yi + f (Yi, X, β) +
f (f (Yi, X, β), X, β) +
...+
f (. . . f (f (Yi, X, β), X, β) . . . , X, β)
where:
(2.22)
49
Yi
is the yield as measured at year i,
Yd
i+n
is predicted yield after a n-year growth period,
X
is an array of predictor variable observations,
β
is an array of model coefficients, and
f ()
is a function with inputs Yi , X, and β and output Yi+1 − Yi
This method is an ideal way to fit a one-year growth model to the data, but suffers
from two large drawbacks. One is that it can be difficult to code such a model into a
statistical package. The more important drawback is that the predictor variables, X,
will be properties of the tree or stand which would be expected to change at each time
step. In the case of this model, top height and relative height, as well as vegetative
cover and density would need to be updated at each time step. This makes coding
such a model into a statistical package extremely difficult, if not impossible.
The second of two methods described by McDill and Amateis (1993), also used
by Cao et al. (2002), is much simpler to implement into a fitting procedure. Both
techniques were shown to give very similar results. This second method is an iterative
procedure that predicts the first year’s growth of a tree as a multiple of the total period
growth. This is shown in mathematical notation in equation 2.23.
qb × (Yi+n − Yi ) = f (Yi, Xi , β)
where:
Yi
is the yield as measured at year i,
Yi+n
is yield after a n-year growth period,
Xi
is an array of predictor variable observations measured at year i,
β
is an array of model coefficients, and
f ()
is a function with inputs Yi , X, and β and output Yi+1 − Yi, and
qb
is the predicted proportion of the total period growth occurring in
the first year (for a given tree).
(2.23)
50
The model is fit in three steps:
1. Before the first iteration qb, the proportion of total period growth occurring in
the first year, is estimated to be the total period growth divided by the number
of years, n, in the period.
2. The model of form 2.23 is fit. The response is the observed n-year period growth
multiplied by qb
3. Each tree is grown for n years with the current coefficient estimates, and q is
computed as the predicted first year growth divided by the predicted n-year
growth.
Step one is performed once, and steps two and three are repeated until convergence
is met. Convergence can be determined by checking the qb values or the model coef-
ficients for change at each time step. Here, convergence was met when no coefficient
changed in the first five significant digits.
Computing qb for each tree took a large amount of time. In order to reduce the time
needed to perform this method on both models, the height and DBH growth models
were fit at the same time using this method. This differed from the steps outlined
above only in that two values qbh and qbd were needed to represent the proportion of
height and diameter growth, respectively, occurring in the first year of the period.
At each iteration, the change in vegetation was predicted with model 2.20, and the
amount of mortality was predicted with model 2.21. This allowed the stems per acre,
basal area per acre, and vegetation cover to be updated for each year of the n-year
growth projections. The gains in using this technique on the mortality model 2.21 and
the vegetation cover model 2.20 were assumed to be too small to be worth the extra
effort. Therefore only the height and DBH growth models were fit in this iterative
manner.
51
Convergence occurred in 8 iterations. The coefficients produced by this procedure
were different than those shown in tables 2.1 and 2.2, which was expected. After fitting
the diameter growth model (2.18) in this manner, coefficient d2 became negative and
was no longer significantly different from 0. The value of d1 kept the same sign and
remained significant. Under the current formulation of the density modifier function,
this implies that the effect of increasing density is a decrease in expected growth
for all top heights. This also implies that, when defined as in the height growth
function, no “crossover” point was evident in the DBH data. The density modifier
function was removed from the DBH growth model, leaving just the base function.
The iterative fitting process was repeated to get estimates for the reduced diameter
growth function. Table 2.6 shows the final height growth model coefficients, and table
c2 was 0.603 for the height
2.7 shows the final DBH growth model coefficients. The R
increment model, and 0.784 for the squared diameter increment model. Respective
root mean squared errors were 0.6746 and 1.1852. Correlations among the parameters
from each model are shown in tables 2.8 and 2.9.
Table 2.6: Coefficients from the full height growth model in equation 2.16. These
parameters are predicted using the annualization procedure laid out in equation 2.23.
b1
b3
b4
b5
h1
h2
ν1
ν2
d1
d2
Estimate
−2.96e−01
9.59e+02
3.39e+00
3.13e−02
1.86e−02
3.51e−01
−2.58e+00
−1.04e−01
−6.47e−06
2.88e+01
Std. Error
1.2235e−02
3.8243e+01
1.4277e−01
1.1571e−03
2.7878e−03
1.2946e−02
8.9522e−02
7.4247e−03
5.8943e−07
1.0651e+00
t value
−24.18
25.08
23.72
27.08
6.68
27.12
−28.83
−14.07
−10.98
27.08
Pr(> |t|)
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
<0.000001
52
Table 2.7: Coefficients from the full squared diameter growth model in equation 2.18.
These parameters are predicted using the annualization procedure laid out in equation
2.23.
Estimate Std. Error t value
Pr(> |t|)
b1
1.52e+01 2.8171e+00
5.40 <0.000001
b2
1.63e+00 3.5725e−02
45.72 <0.000001
b3
1.78e−01 7.6053e−03
23.36 <0.000001
b4 −6.08e−03 1.6589e−04 −36.62 <0.000001
b5
1.85e+00 5.3037e−02
34.94 <0.000001
b6
2.30e−02 3.3403e−03
6.87 <0.000001
b7 −5.01e+00 3.7253e−01 −13.44 <0.000001
b8 −9.16e−01 6.6063e−02 −13.87 <0.000001
b9 −5.72e−02 2.3830e−03 −23.99 <0.000001
Table 2.8: Correlations among the parameters of the annual height increment model
2.16 fit using the iterative procedure.
b3
b4
b5
h1
h2
ν1
ν2
d1
d2
b1
0.46
−0.88
−0.03
−0.41
0.38
−0.47
0.39
−0.17
−0.22
b3
b4
b5
h1
h2
ν1
ν2
−0.70
0.47 −0.44
−0.74
0.49 −0.10
0.69 −0.47
0.13 −0.98
−0.83
0.60 −0.21
0.68 −0.62
0.64 −0.62
0.40 −0.54
0.52 −0.83
−0.16
0.12
0.15
0.02
0.05
0.14 −0.13
−0.13
0.23
0.02
0.08 −0.03
0.15 −0.25
d1
0.68
53
Table 2.9: Correlations among the parameters of the annual squared diameter increment model 2.17 fit using the iterative procedure.
b2
b3
b4
b5
b6
b7
b8
b9
b1
0.57
0.50
0.06
0.42
−0.79
−0.59
−0.61
−0.61
b2
0.55
0.12
−0.01
−0.07
−0.87
−0.88
−0.87
b3
b4
b5
b6
b7
b8
0.12
0.00
0.02
0.03
0.03 −0.38
−0.82 −0.06
0.02 0.02
−0.81 −0.12 −0.01 0.04 0.91
−0.85 −0.29 −0.03 0.04 0.92 0.93
54
Chapter 3
AN INDIVIDUAL-TREE MODEL TO PREDICT THE
ANNUAL HEIGHT GROWTH OF YOUNG
PLANTATIONS OF PACIFIC NORTHWEST
DOUGLAS-FIR INCORPORATING THE EFFECTS OF
DENSITY AND VEGETATIVE COMPETITION.
3.1
Introduction
The rotation lengths of Douglas-fir (Pseudotsuga menziesii (Mirbel) Franco) stands
in the Pacific Northwest have become short in comparison to a couple of decades
ago (The Rural Technology Initiative 2006). At the same time, the number of mills
that can process larger diameter logs has diminished and advances in technology have
increased product recovery from smaller diameter logs. Accordingly, interest in young
stand management has increased.
Estimating the expected benefit to be realized from activities such as site preparation, fertilization and density management has been in the forefront of research activity for some time (Oliver et al. 1986). Models that accurately model the response
to these activities are critical to those planning and implementing early silvicultural
treatments.
The two most common growth models for the area, ORGANON (Hann 2003) and
FVS (Dixon 2002), work best on stands after they near canopy closure. Managers use
these models to choose from several treatment options to find the most profitable, or
the one that most meets management objectives. Although FVS includes a component
to estimate natural and planted regeneration establishment (Ferguson and Crookston
55
1991, Donnelly 1997), it has a 10-year time step which prevents detailed evaluation
of management regimes. Young stand growth models are needed that are capable
of aiding managers make fine-tuned decisions about treatment options available at
planting and through the time when crown closure begins to occur.
In this manuscript, a one-year height growth model for young stands of Douglasfir in the Pacific Northwest is described. This height growth model was developed
as part of a complete growth modeling project with the objective of simulating the
growth of a Douglas-fir plantation until the age when the stand nears crown closure.
Once the trees reach this age the data should be transferred to a model designed for
growing well-established stands. The height growth equation takes into account the
density of a stand and the average amount of competing vegetation cover measured in
a stand. The quality and repeatability of the coefficient estimates for this model are
evaluated using a data resampling technique known as a bootstrap. The sensitivity
of the model to changes in these coefficient estimates is analyzed to determine what
effect errors in coefficient estimates would have.
3.2
Methods
3.2.1 The Data
Modeling data were provided by the Stand Management Cooperative (SMC) and the
Regional Vegetation Management Model (RVMM) project. The SMC is a consortium
of landowners in the Pacific Northwest established in 1985 to pool resources in order
to provide high quality information on the long-term effects of silvicultural treatments
(Maguire et al. 1991). The SMC data used for this project are a product of a planting
density trial (the “Type III” installations). There are 34 installations with 3 to 23
plots, each from 0.2 to 0.5 acres (0.04 to 0.20 hectares) in size (Silviculture Project
TAC 1991). The RVMM project was started in the early 1990’s for the purpose of
developing a growth model to weigh various vegetation management options. The
56
RVMM dataset contains data from 98 research plots, each 0.1 acres (0.04 hectares) in
size, in each of the Coastal and Cascade mountain ranges (Shula 1998). The locations
of the SMC installations and the RVMM plots are shown in figure 3.1.
At each measurement, the total height and diameter of each tree in the plot
were observed and recorded. Diameter was measured at ground level until the tree
reached breast height, after which only breast-height diameter (DBH) was measured.
A subsample of trees were measured for crown width and height to crown base in the
RVMM plots. On the SMC plots these variables were measured on a subsample of
trees in the plot. Remeasurement interval was 2 years in the RVMM plots and 2 to 4
years in the SMC plots.
Vegetation was sampled on four 100th acre subplots within each tree plot. Each
subplot was split into four equal-sized quadrants and ocular estimates of cover were
recorded for each species in the quadrant. These 16 quadrant samples were averaged
to get a mean plot level value of each species. The cover for all species classified as a
fern or as a shrub was summed to create a single number for vegetation cover. The
unit of this number is percent, but it ranges from 0 to 212 when many overlapping
layers of species were noted.
3.2.2 The Height Growth Function
The largest proportion of the total variation in height growth was explained with a
function of only two predictor variables, initial height and site index. Site index in
this was case computed from the data with the curves of Flewelling et al. (2001),
which were designed for use with younger plantations of Douglas-fir. This non-linear
function, shown as equation 3.1, will be referred to as the base function.
dij = b(H(0)ij , Si ) =
∆H
b1 +
−1
b3 H(0)ij
Si−1.5
1
0.5
+ b4 Si−0.5 + b5 H(0)ij
(3.1)
57
RVMM Coastal
RVMM Cascade
SMC Type III
Figure 3.1: Locations of the study tree plots within the Pacific Northwest. Plots
from 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 ◦.
58
where:
dij
∆H
is the one-year change in total height in feet of tree j in plot i,
H(0)ij
is the initial total height in feet of tree j in plot i,
Si
is the site index (base age 30 years) of plot i, and
b1 through b5
are parameters estimated by the R function nls.
The growth estimates from this base function are modified by a series of three multiplier functions, shown in equations 3.2 to 3.4. All three of theses modifier functions
are affected by stand top height (H(top)i ), or the mean height of the largest 40 trees per
acre. The largest trees were determined by total height until all trees reached breast
height. Subsequently, the largest trees were determined by breast height diameter
(DBH).
h2
r(H(0)ij , H(top)i ) = exp(h1 ∗ exp(H(top)i
) ∗ log(H(0)ij /H(top)i ))
√
v(H(top)i , Vi )
= exp(−exp(ν1 + ν2 ∗ (H(top)i )) Vi )
d(H(top)i , Ti )
= 1 + d1 ∗ (H(top)i − d2 ) ∗ (Ti − 300)
where:
r(H(0)ij , H(top)i ) is the relative height modifier function,
v(H(top)i , Vi)
is the vegetation cover modifier function,
d(H(top)i , Ti )
is the density (stems per acre) modifier function,
H(0)ij
is the initial total height in feet of tree j in plot i,
H(top)i
is the top height of plot i,
H(0)ij /H(top)i
is the relative height of tree j in plot i,
Vi
is the mean vegetation (shrub and fern) cover in plot i,
Ti
is the stems per acre in plot i,
h1,2 , ν1,2 , d1,2
are model parameters.
(3.2)
(3.3)
(3.4)
59
The relative height modifier function in equation 3.2 gives the taller trees in a
stand a faster growth rate and the smaller trees a slower growth rate. The amount of
increase or reduction depends on the current top height of a given stand. Younger,
shorter stands do not see as much inter-tree competition as older, taller stands and
this function reflects this observation.
The vegetation modifier function in equation 3.3 also changes as the stand gets
taller. In this case the amount of vegetation in a stand will have decreasing impact
on the individual tree growth rates as the stand climbs out of the influence of the
vegetation competition. The value of ν2 in this function dictates the rate at which
vegetation effects diminish with increasing top height.
Equation 3.4 adjusts growth rates for the number of living trees per acre in the
stand. Previous work has shown that higher densities actually increase the early
height growth of the stand (Scott et al. 1998, Turnblom 1998, Woodruff et al. 2002).
This phenomenon was observed in the datasets used to build this model. The initial
increase in height growth at higher densities diminishes with increasing top height
until reaching the “crossover” point. At this point, the effects of stand density revert
to the usual pattern of reduced height growth at higher densities. The location of
this crossover point is given by the parameter d2 .
Coefficients for one-year increment in height were estimated using the iterative procedure described by McDill and Amateis (1993). In the first iteration, the response is
the total period height growth divided by the number of years in the growing period.
In subsequent iterations, the current coefficient estimates were used to calculate the
proportion of the predicted period growth that occurs in the first year of the period.
This proportion is multiplied by the observed period growth to create the new response variable. Iteration continues until no coefficients change in any of the first five
significant digits. A mortality model was used to update the stems per acre and basal
area per acre and a vegetation cover change model was used to predict yearly change
60
in vegetation at each time step during the iterations. Coefficients at each time step
were estimated using the nls function in the R statistical program (R Development
Core Team 2006).
3.2.3 Bootstrap Validation and Sensitivity Analysis
There are several methods to assess the behavior of a growth and yield model Vanclay
and Skovsgaard (1997). More traditional techniques are to use multiple datasets for
fitting and benchmarking. A similar, but more general technique is to use crossvalidation. Recent advances in computing speed have made it quite simple to use
data resampling techniques such as the bootstrap and jackknife procedures (Efron
and Tibshirani 1993). To test the stability of the height growth model coefficients, the
distribution associated with the coefficient estimates is quantified using a bootstrap
procedure. An analysis of the sensitivity of the model to changes in the coefficient
estimates is performed to assess the stability of the model under different coefficient
values from the bootstrap distributions.
The bootstrap procedure consists of repeatedly fitting the model to independent
datasets resampled from the original dataset. For each of 1056 bootstrap runs, records
were randomly selected, with replacement, from the original dataset to create a resampled dataset of the same size. Because of the grouped structure of the dataset,
resampling was done at the plot level. If a plot was chosen, then all trees at every
measurement of the plot were included in the bootstrap dataset. Data from plots selected n times in a given resample were duplicated n times in the bootstrap dataset.
The model was re-fit to each of these bootstrap datasets and the coefficient estimates
were recorded. This procedure is expected to produce an approximation of the true
distribution of each model coefficient. The number of bootstrap runs, 1056, represents
a compromise between computing time and large sample size.
Resampling techniques for model testing are dependent on a representative sample
61
of data. More than 20,000 tree growth records from 388 plot measurements were used
in the fitting of the height growth model. The data come from plots that span most of
western Washington and Oregon. It is ideal that a dataset contain information about
all growing conditions occurring in the area of application. While this is practically
impossible, this dataset does encompass a wide range of growing conditions over a
period of 13 years (1991 to 2004).
While sensitivity can be assessed by looking at derivatives, this can be a complicated analysis with ten parameters and five predictor variables. A simpler method
is to look at the predictions of the model in a more qualitative manner. The sensitivity of the model to a given parameter depends on the values of the predictor
variables (H(0)ij , H(top)i , Vi , etc.). Therefore, the sensitivity should be analyzed at
several combinations of these predictor variables. To accomplish this, 15 of the 243
possible combinations three levels of each predictor variable were randomly selected
(three levels of relative height, rather than three levels of total height were selected to
prevent unlikely combinations of height and top height). The selected sets of predictors are given in table 3.1. For each of these fifteen sets of predictor values, the model
was evaluated for each of the 1056 coefficient sets from the bootstrap procedure. This
process created a table of 15840 (15x1056) model predictions.
Table 3.1: Fifteen sets of predictor values selected for use in the analysis of model
sensitivity.
Set
Si
Ti H(top)i H(0)ij Vi Set
Si
Ti H(top)i H(0)ij
1 104 100
15 12.75 0
9 84 100
40 24.00
2 84 100
40 24.00 0 10 104 600
15 12.75
3 84 600
40 34.00 30 11 104 300
4
3.40
4 104 600
40 44.00 0 12 64 600
40 24.00
40 34.00
5 104 300
15 16.50 30 13 84 300
6 64 100
40 24.00 0 14 104 300
15 12.75
7 84 300
15 12.75 0 15 64 100
40 34.00
8 64 300
4
3.40 90
Vi
30
0
30
30
90
0
0
62
For each model coefficient, under each set of predictor variables, a scatterplot of
model prediction against the studentized bootstrap values of the given coefficient was
created. Because these scatterplots suggested a linear relationship was appropriate,
the slopes from each of these regressions should be roughly equivalent to the sensitivity
of the model to the given coefficient at the given set of predictor values. The slope in
each regression represents the average change in the model prediction in feet caused by
a change in the given coefficient by one standard error (estimated from the bootstrap
results).
3.3
Results
The estimated coefficient values for the height growth model fit to the original dataset
are shown in table 3.2. Approximate standard errors derived from linear approximations to the response surface are given. The model fit with an R2 (1- residual
SS/corrected total SS) of 0.603 and the mean squared error was 0.6745 feet.
Table 3.2: Coefficients from the full height growth model in equations 3.1 through 3.4
with the estimated and bootstrap standard errors.
b1
b3
b4
b5
h1
h2
v1
v2
d1
d2
Estimate Estimated Std. Error Bootstrap Std. Error
−2.96e−01
1.2235e−02
5.7627e−02
9.59e+02
3.8243e+01
1.9501e+02
3.39e+00
1.4277e−01
6.3889e−01
3.13e−02
1.1571e−03
4.5429e−03
1.86e−02
2.7878e−03
4.1219e−01
3.51e−01
1.2946e−02
3.0645e−02
−2.58e+00
8.9522e−02
2.1271e−06
−1.04e−01
7.4247e−03
7.3742e+00
−6.47e−06
5.8943e−07
9.9214e−03
2.88e+01
1.0651e+00
3.9400−02
Figure 3.2 shows contours of the surface of predicted height growth using the base
function in equation 3.1. This contour plot represents actual model prediction only
when the value of modifier functions 3.2 through 3.4 are all equal to 1.
63
The output of the modifier functions across a range of top height, and several
levels of relative height, vegetation cover, and density are shown in figures 3.3 through
3.5. The change in the effects of the predictor variables are noticeable as top height
increases. Relative height has an increasingly important effect as the stand height
increases (figure 3.3). Vegetation cover has a negligible effect on height growth by the
time the stand reaches about 25 feet (figure 3.4). Density changes from increasing to
80
60
40
Site Index (ft. at base age 30)
100
decreasing height growth at a stand top height of 29 feet (figure 3.5).
0
10
20
30
40
50
60
Initial Height (ft.)
Figure 3.2: A contour plot of the height growth prediction surface from the base
function of the height growth model, 3.1, over ranges of initial height and site index
representative of the data.
The number of tree observations in the bootstrap datasets ranged from 16420
to 24470. The mean square errors of the bootstrap fits ranged from 0.63 to 0.73.
0.5
0.0
Rel. height = 0.3
Rel. height = 0.6
Rel. height = 1
Rel. height = 1.3
−1.0
−0.5
r(H(0)ij, H(top)i)
1.0
1.5
64
0
10
20
30
40
50
60
Top Height (ft.)
Figure 3.3: Values of the relative height modifying function in model 3.2 for a range
of top height and several relative heights.
0.5
Veg. Cover = 0
Veg. Cover = 25
Veg. Cover = 75
Veg. Cover = 150
−0.5
0.0
v(H(top)i, Vi)
1.0
1.5
65
0
10
20
30
40
50
60
Top Height (ft.)
Figure 3.4: Values of the vegetation cover modifying function in model 3.3 for a range
of top height and several vegetation cover levels.
0.5
Stems per Acre = 100
Stems per Acre = 300
Stems per Acre = 600
Stems per Acre = 1200
−0.5
0.0
d(H(top)i, Di)
1.0
1.5
66
0
10
20
30
40
50
60
Top Height (ft.)
Figure 3.5: Values of the density modifying function in model 3.4 for a range of top
height and several planting densities.
67
Ninety five percent of these mean squared errors were within the range 0.64 to 0.71.
The model needed help converging with eight of the bootstrap datasets. This help
consisted of either a shift in the starting values for the coefficients or more allowed
iterations (the default limit of the nls function is 50).
Figure 3.6 shows the bootstrap distributions of all ten coefficients. A vertical
dotted line is placed at 0 for reference, and the approximate distributions from normal
theory are overlaid with a dashed gray line. Table 3.2 gives the bootstrap estimates of
the coefficient standard errors. Neither the estimated distributions nor the estimated
standard errors had much resemblance to the bootstrap prediction for all coefficients.
Despite these differences, no decisions about included parameters are changed.
Figure 3.7 is a scatterplot matrix of the coefficient estimates. The upper triangle
of the matrix contains the correlation coefficient of the values of one coefficient against
another. These linear correlations are only for reference given that definite non-linear
relationships are obvious between a few of the parameters.
Table 3.3 shows the maximum and minimum predicted height growth, over all
bootstrap coefficient sets, for each predictor variable set described in table 3.1. The
total change in predicted value over all 1056 bootstrap coefficient sets is shown. This
change is never more than 15 percent of the lowest value. As these are from the
extreme values of the coefficient distribution, actual errors in the coefficient estimates
are not likely to be large enough to cause a bias this large in the model predictions.
Table 3.4 shows summary statistics for the slopes from the regressions of model
change against studentized coefficient value change. The predictor value sets (from
table 3.1) are listed in ascending order according to the size of the regression slope
calculated with the given predictor values. This is to show trends among predictor
variables and model sensitivity. For example, if we use the predictor values for set 10
(which has higher site index and density values), increasing b1 by one standard error
( 0.058 from table 3.2) would be expected to decrease the model prediction by 0.077
68
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
0.1
−500
0
500
0
1
2
3
4
5
−0.01
0.00
0.01
0.02
−4
−3
−2
−1
0
1
−0.20
−0.15
−0.10
v1
−1e−05
0.00
0.01
−5e−06
0.02
0.03
h1
0.04
0.05
−0.05
0.00
0.05
v2
0e+00
−10
0
10
d1
−0.01
0.03
b5
b4
−5
1500
b3
b1
−1
1000
20
30
40
50
60
d2
0.04
0.05
0.06
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
h2
Figure 3.6: Bootstrap distributions of the full height growth model (3.1 - 3.4) parameters. Theoretical distributions are overlaid (scaled to fit in same window), with the
fitted value of the coefficient shown with a solid vertical bar. Zero is indicated with
a dotted line.
69
b1
0.44
−0.91
−0.15
−0.47
0.43
−0.43
−0.39
−0.46
0.46
b3
−0.68
0.45
−0.81
0.49
−0.14
−0.14
−0.86
0.86
b4
−0.26
0.59
−0.57
0.33
0.35
0.64
−0.66
b5
−0.11
0.17
0.35
0.19
−0.29
0.33
v1
−0.78
0.26
0.22
0.75
−0.79
v2
−0.38
−0.36
−0.52
0.58
d1
0.7
0.21
−0.18
d2
0.21
−0.18
h1
−0.96
h2
Figure 3.7: A scatterplot matrix of the 1056 bootstrap coefficient estimates. The
upper portion contains the linear correlation between the coefficients.
70
Table 3.3: Minimum and maximum predicted one-year height growth in feet for all
15 sets of predictor variables across all 1056 bootstrap coefficient sets.
dij -min ∆H
ˆ ij -max
ˆ ij -min ∆H
ˆ ij -max
Set ∆H
diff. Set ∆H
diff.
1
3.859
4.973 1.114
9
2.113
2.918 0.804
2
2.120
3.121 1.001 10
4.037
5.179 1.143
3
2.758
3.189 0.431 11
1.732
2.479 0.747
4
3.643
4.650 1.007 12
1.648
2.142 0.494
5
4.037
4.604 0.567 13
2.726
3.271 0.545
6
1.627
2.429 0.802 14
3.930
5.056 1.126
7
3.073
3.945 0.873 15
2.109
3.113 1.003
8
0.648
1.042 0.394
feet.
Table 3.4: The sensitivity (slope) of the height growth model predictions to changes in
the coefficients. The minimum, median, and maximum sensitivity for all 15 predictor
sets (from table 3.1)are given. The predictor value sets are listed in ascending order
according to the size of the regression slope calculated with the given predictor values.
Coeff.
b1
b3
b4
b5
h1
h2
ν1
ν2
d1
d2
3.4
Min
−0.077
−0.082
−0.101
−0.077
−0.068
−0.010
−0.094
−0.080
−0.050
−0.059
Med
0.014
0.002
−0.020
−0.023
−0.016
0.040
−0.019
−0.016
−0.002
−0.009
Max Range
0.113 0.190
0.045 0.127
0.066 0.167
0.055 0.132
0.039 0.107
0.088 0.098
0.021 0.115
0.014 0.094
0.062 0.113
0.050 0.109
Order
10,5,14,1,11,4,7,8,3,12,13,9,2,6,15
11,10,14,1,5,7,4,8,13,12,3,9,2,6,15
15,6,2,9,12,13,3,7,8,4,1,14,10,5,11
4,2,9,13,6,15,3,11,12,8,5,7,10,14,1
15,2,6,4,8,9,3,13,12,7,5,11,1,14,10
5,11,12,13,3,8,9,4,10,14,7,1,6,2,15
15,2,9,6,13,4,12,7,3,8,10,14,1,11,5
15,2,6,9,13,7,12,3,1,14,4,8,10,11,5
15,6,2,8,13,3,4,9,12,7,5,1,14,10,11
11,10,5,14,1,7,12,9,13,3,2,4,8,6,15
Discussion
Many of the bootstrap distributions were skewed, and standard errors were far different from those estimated from normal distribution theory. This is attributable to the
grouping structure of the data. The standard errors were estimated assuming that
data are independently distributed. Another bootstrap procedure, in which obser-
71
vations were resampled at the tree-measurement level (trees were randomly selected
regardless of plot or measurement number association) resulted in distributions very
close to those predicted. Despite the underestimated variances, the coefficient estimates should be unbiased, and the bootstrap distributions give no evidence to the
contrary.
The observed stability of the height growth model coefficients permits some confidence in the model form. The moderate changes observed in the coefficient values
during the bootstrap procedure had only a small effect on the output of the model.
The largest change in model prediction caused by shifting a parameter by one standard error was 0.113 feet. This occurred when the b1 parameter was perturbed. The
stability of coefficients in the base function, b1 through b4 , is of particular importance
as the base function alone accounts for more than fifty percent of the variation in
one-year height growth.
The vegetation modifier function, equation 3.3, was well-behaved during the bootstrap process. A good degree of certainty exists around the values of the coefficients
ν1 and ν2 . The particular model form allowed for data from both younger and older
stands. The lack of vegetative effect in taller trees did not interfere with the prediction of such an effect on shorter trees. Accordingly, this function heavily decreases
expected height growth in shorter stands with much vegetation cover. As the stand
top height increases, this effect rapidly shrinks. Because the effect is based on the
top height and not age, the effects will last longer on stands of low site index, further
widening the gap in height between them and more productive stands.
The d2 coefficient in the density modifier function seemed fairly stable around
30, though some much higher values occurred. The estimate of 29 feet for d2 , the
coefficient that determines when the density effect “crossover” occurs, was slightly
higher than expected. The lowest estimate from the bootstrap procedure was 20 feet.
However, Woodruff et al. (2002) show for a few plantations that larger densities start
72
to reduce growth around age 7. The earliest that a tree should be expected to reach 29
feet is about 10 years, on poorer sites, this age is much older. Flewelling et al. (2001)
found that, while this crossover point to vary by density, the earliest crossover was
at about 15 years with a density of about 1600 trees per acre. Given that Woodruff
et al. (2002) used a statistically sound randomized block design, this incompatibility is
troubling. More investigation will need to be done to fully understand this relationship
and more accurately model the effects of density.
From table 3.3, the most change in the height growth prediction that any of the
changes of coefficients caused was about a 1.1 feet. While this is a large discrepancy
on a per-year time scale, it should be considered an extreme. Eighty percent of the
predictions using the bootstrap coefficient sets (the bootstrap predictions) are within
five percent of the predictions using the coefficients in table 3.2. Ninety-seven percent
of the same bootstrap predictions are within ten percent of the final model predictions.
The probability that errors in the coefficient estimates are large enough to cause such
extreme errors in the height growth estimates should be very slim.
Because many parameters were correlated (figure 3.7), shifting one parameter
would likely result in shifts in one or more other parameters. Incorporating this
correlations among model coefficients when analyzing model sensitivity makes interpretation more difficult. The effect of changing one given coefficient within the model
is a mixture of the importance of the given coefficient and the importance of the
coefficients with which the given coefficient is correlated. Therefore, investigating the
fluctuations of one coefficient alone, while holding the other coefficients fixed, might
provide inaccurate sensitivity values. This may lead to a belief that one parameter is
more or less meaningful than it truly is to the model outcome.
Table 3.4 shows that the model sensitivity to the h1 coefficient is moderately
negative for predictor sets 15, 2 and 6. For the same sets, the model sensitivity
is highly positive for coefficient h2 . These 3 sets all have the highest value for top
73
height and the lowest values for density. Because an increase in h1 is nearly perfectly
associated with a decrease in h2 the effect of changing either of these coefficients
is largely mirrored by the associated change in the opposite direction of the other
coefficient. In this case it was assumed that the coefficients are highly correlated
because the shape of relative height modifier function is very similar under many
values of the two coefficients. Changing only one of these coefficients should then
change the model prediction more than changing one coefficient and letting the other
respond appropriately. Therefore, the sensitivity reported in an analysis that did not
account for for the coefficient correlation might report too high a sensitivity.
It is interesting to note that the sensitivity of the model to each coefficient ranged
from negative to positive values for all coefficients (table 3.4). The model sensitivity
to some coefficients was most negative in many cases where the sensitivity to other
coefficients was most positive. For example, parameter sets 15 and 5 appear as the
minimum or maximum several times in table 3.4. This may be another instance where
the sensitivity of the model to one coefficient may be mitigated by the subsequent
changes in the other coefficients. In such a situation, looking at the sensitivity of the
model to changes in only one coefficient could produce values that are either larger
or smaller than what would be actually experienced under different datasets.
The b3 coefficient seems to have the weakest effect of all the coefficients in the
base function (3.1). This suggests that the term accounting for the site index-initial
height interaction plays only a small part. The rest of the coefficients have a stronger
effect, which suggests that the base function plays a very large part in the model
outcome. The two coefficients from the vegetation modifier (3.3) as well as those
from the density modifier (3.4) have a wide range, and can therefore play a very
important role in the model outcome. Getting good estimates of these coefficients is
then crucial. Looking at the bootstrap distributions of these coefficients is one way
to measure the precision of these estimates.
74
3.5
Conclusion
The model described in this paper seems able to predict height growth accurately
while taking account of some of the effects of planting density and vegetation cover.
These abilities were examined using a bootstrap procedure, which proved to be a
valuable tool to test a non-linear height growth model. Through this procedure, the
distributions of the model coefficients and residual squared error were approximated.
While bootstrap standard errors of the coefficients were much higher than the estimated standard errors, the coefficients seemed fairly stable overall. Little effect on
the expected height growth could be seen by changing the coefficients within their
ranges under the bootstrap estimate of their multivariate distribution.
75
Chapter 4
DISCUSSION
4.1
Height and Diameter Increment
The fitting of the height and diameter increment models as one-year increment models
added much complexity to the modeling process. Two methods to obtain one-year
coefficients, centering and iterative refitting, were used to fit both models. While the
two methods produced similar coefficients in most cases, the fit statistics were not
directly comparable. The response variables were different even though they represent
the same concept, one-year growth. This difference led to a slightly greater sum of
residuals squared using the iterative method for both models. Also, the optimal model
with one method was not be the best under the second method. This was true for the
diameter growth model, which lost significance of the density modifier function. This
may be a result of the dependence of the diameter growth model on the height growth
model. At each iteration of the iterative method for fitting one-year increment, the
trees were grown using both the diameter growth and the height growth models in
their current state. Because both initial height and top height are predicted using
the height growth model, perhaps some of the effects of the density modifier in the
height growth model migrated into the diameter growth model.
The full height growth model described by equation 2.16 performed fairly well
within the range of data available with an R2 of 0.611. These results are comparable
to those of Ritchie and Hamann (2007) who were able to predict two-year height
increment of young Douglas-fir with an R2 of 0.65. Because the base function 2.15
in this model produced an R2 of 0.536 alone, this implies that initial height and site
76
index are the most important predictors of height increment. The base function of the
height growth model (equation 2.15) produces a typical height-age curve. This curve
has a quickly increasing slope until some inflection point is reached, thereafter the
slope slowly decreases. The timing of this maximum growth decreases as site index
increases. While this curve should be well defined for younger trees, it may be poorly
defined for heights of about 30 feet due to a small number of observations available
(figure 2.2a).
The vegetation modifier function was found to significantly improve the predictive
abilities of the overall height growth model. It produced strong effects for high levels of
vegetation cover in very young stands. A similar effect of early vegetation competition
on seedling shoot biomass can be seen in Tesch and Hobbs (1989) and Morris et al.
(1993). The effects of vegetation competition decrease exponentially (figure 2.10) until
the stand outgrows the vegetation competition. With only plot level vegetation data,
this model has a necessarily different form for incorporating vegetation competition
than that of CONIFERS (Ritchie 2006). In CONIFERS, each shrub was measured
and modeled as equivalent to a tree. The crown area of competing vegetation in
CONIFERS is summed from individual plant crown measurements and enters the
height growth model in a similar manner as the crown area of trees. Such methods
prevent the need for a separate plot level vegetation cover dynamics model, such as
equation 2.20.
Vegetation competition, as collected in this study, did not seem to affect diameter
growth enough that it was a useful predictor. This was unexpected because previous
research has shown otherwise. Watt et al. (2003) found that vegetation presence
significantly reduced basal area growth, and was able to model this effect. Ritchie
and Hamann (2007) found crown area of taller shrubs to affect diameter growth
significantly in the CONIFERS model.
The effect of inter-tree competition was modeled in two parts, the effect on all trees
77
of the density in a stand and the effects of individual-tree competitive position. The
former was modeled by the density modifier function. Previous research has shown
the effect of density to be different in young Douglas-fir stands than in established
stands (Scott et al. 1998). This modifier function has a positive slope with respect to
density for stand with top height of less than about 28 feet, the value of coefficient
d2 . This point in time is referred to as the ”crossover point”. In taller stands, this
slope becomes increasingly negative. For stands planted and maintained at 300 trees
per acre, the value of this modifier will always be 1.
Woodruff et al. (2002) found the crossover point to be somewhere around 7 years
from planting. Turnblom (1998) found that, in 7 to 9 year old stands, the largest 100
trees per acre were still exhibiting the cumulative effect of higher density at planting.
However, Flewelling et al. (2001) found this crossover point can vary by density and
that it occurred much later. For all studied densities, the crossover occurred at least
15 years from seed, with the crossover point of lowest densities postponed until past
age 35. Because the crossover point in the height growth model is based on top height,
the age of predicted crossover will vary by both density and site quality.
As a measure of stand density, basal area per acre had a stronger effect on squared
diameter growth than stems per acre. This variable entered the model in the base
function. Basal area was found to have similar effect for all levels of top height, so
using it as a predictor in the density modifier function did not yield any benefit. With
basal area in the model, the density modifier function, with stems per acre, was only
of marginal help. This finding was even more clear when fitting with the iterative
method.
The other component in the effect of in inter-tree competition, tree position, was
modeled as a function of stand top height and individual tree height in the relative
height modifier. Many trees in the original dataset had missing observations for at
least one important variable. Therefore, other measures of tree position, such as
78
crown area in taller trees used in CONIFERS (Ritchie and Hamann 2007), would
need to be computed from interpolated heights, diameters and crown area. Relative
height needs only the individual height and a good estimate of top height to compute.
Not all trees in the plot need to have measured height to compute a top height.
As expected, at low stand top height, the function results in almost no effect of
tree position. As top height increases, the effect of the height of a tree relative to the
stand top height has an increasingly larger effect. Trees taller than the top height
will get a boost in growth and trees at a very low position will grow far less. The
effect of relative height was expected to be related to site index because higher quality
sites are able to support more trees. Investigations of the data left this expectation
unconfirmed, so the relative height modifier does not include site index as a predictor.
The relative height modifier was not included in the diameter growth model because incorporating relative height and top height as terms in the base function worked
just as well and was much simpler. Both relative height and top height had strongly
significant negative slopes in the base function. This indicates that, with all other
variables held constant, higher relative heights result in smaller predictions of squared
diameter growth. This result was unexpected, and hard to explain, but likely is related to predictor covariance. A bootstrap validation similar to that done for the
height growth model would be useful in understanding this coefficient.
4.2
Secondary Models
The height and diameter growth models were of primary interest. However, some of
the predictor variables in these two models require updating at each time step during
simulation. These variables include initial diameter, vegetation cover, basal area and
stems per acre.
The initial diameter model was necessary because trees will be input into the
model that are too short to have a breast-height diameter. Because of the form of
79
the diameter growth model base function (equation 2.17), very small initial diameters
yield small predictions. One concern for the initial diameter model was that the
predicted initial diameter will be very close to 0. However, the initial diameter model
will not predict an initial diameter much smaller than about 0.10 inches, which is
the value predicted by the model if initial height is 4.5 feet and values beyond the
maximum values observed in the data are input for the other predictors. Therefore,
extremely small initial diameters should cause little problem unless they are input by
the user.
Vegetation was a significant predictor of initial diameter even though it was not
for diameter growth. It was found to have a negative effect on predicted initial
height. This suggests that perhaps much of the effect of vegetation competition on
diameter growth occurs very early. Increases of either density or site index also
reduced the predicted initial height value. The residual plot (figure 2.13) indicates a
slight parabolic trend in predicted initial diameter residual.
Mortality was predicted as the probability of two-year tree survival. The binary
form of the response variable prevented the same iterative technique from being used
to predict one-year mortality. In order to get annual survival rates, an assumption
was made that the probability of mortality during any two year period will not change
much between years. This assumption was not tested, but did not seem overly bold,
and should not raise much concern.
The model of change in vegetation cover was the weakest link in the entire project.
Because it was a plot level model and not all plots had vegetation measurements at
multiple times, very few observations were available for the fitting of this model. The
iterative procedure was not used to fit this model because it should produce little
or no benefit due to the high variation in the response. Regardless, the model does
typically predict increases in vegetation at low top height and decreases at large top
height. This leads to the expected rise followed by a decline in vegetation cover as the
80
stand ages. However, for some predictor values the model will predict decline even
if initial vegetation cover is 0, resulting in negative values for vegetation cover. This
happened in the data only when top height was greater than 35 feet. By this point
vegetation should have no real effect on height growth.
4.3
Potential Concerns
Vegetation cover, as quantified by the combined cover of ferns and shrubs, did play
a significant role in height prediction. The cover of grasses and forbs were not found
to add to this role. Previous research has shown a reduction in seedling growth
with grass competition (Ball et al. 2002). The cover of grasses and forbs is most
likely higher on plots with little shrub or fern cover, and perhaps is present in too
little quantities to have a detectable impact. Ninety-five percent of all plots have
less than 21.5 percent cover of grasses. The same percentage of plots had less than
43.1 percent cover of forbs. In contrast, the same percentage of plots had less than 82
percent cover of shrubs. If grass competition does reduce tree growth, then this model
will likely overpredict the height growth for trees with little or no shrub competition
but abundant grass competition. Grass cover may need to be incorporated into the
model for this reason.
Some extreme predictions of height growth can occur with the current specification
of the relative height modifier function. For large top heights, a large value for relative
height can lead to large multipliers of height growth. This is most likely to occur if
the model is applied to a stand in which trees from a previous stand were left, such
as the result of a shelterwood or green-tree retention cut. The number of remaining
trees that can be left to provide underplanted seedlings with enough light can be quite
low. The likely result of this is a sample plot with one remaining tree and quite a
few seedlings. The top height calculated from the tree list would be inflated, and the
relative height of the remaining tree would be very extreme. The data used to fit this
model included no such stands, and contain only previously clearcut plantations. A
81
user would not be advised to apply this model to such a stand for the above reason
alone. However, the inevitability of such an event prompts preventative measures.
One such preventative measure would be to fit an alternative model form that would
taper off at higher relative heights. This model would likely take a sigmoidal shape
when plotted out as in figure 2.6. A loss of some fit quality would be acceptable in
order to have the modifier behave well outside of the range of relative heights found in
the modeling data. Another possibility would be to include some actual or expected
data from older stands into the dataset in order to coerce the model to fit through
these points (Buchman et al. 1983). Such a technique may affect the fit of the model
only slightly.
A similar problem exists with the diameter growth equation (2.18). At some
point, the expected basal area growth of a given tree should reach a maximum. This
maximum is probably not reached for any tree in the modeling dataset. With all other
predictors held constant, the positive estimate for the b4 coefficient in equation 2.18
would lead to continually increasing basal area increment as initial diameter increases
with time. With older trees in the dataset, the value of this coefficient would most
likely be negative. Because other predictors are correlated with initial diameter, the
degree of this problem is indecipherable, but could be studied with simulation. The
negative b9 also indicates a similar correlation problem, and would result in decreases
in diameter growth with increasing site index. Solutions to both coefficient problems
could be the same as those described in the preceding paragraph. Alternatively, b4 or
b9 can be fixed at reasonable values while the other coefficients are estimated.
The height growth model 2.16 currently provides no mechanism for trees to change
rank during a simulation of stand growth. The tallest trees at planting will remain the
tallest at the end of the simulation. The addition of a random variate to the growth
of each tree at each time step may allow some change in position as the trees age in
the simulations. Also, as the trees grow, there is currently no mechanism to allow for
82
tree damage. In the first few years of stand establishment, seedlings are susceptible
to many forms of damage. This model was fit with data trimmed of such damaged
trees, and this may result in overpredictions of growth. A damage rate as well as the
potential affect of damage can be estimated from the data to adjust growth rates to
account for this overprediction.
While the model will predict the growth of only Douglas-fir, many stands will
contain seedlings of other species, planted or not. This limits the potential use of this
model unless equations are included for the growth of these other species. However,
because plantation of purely Douglas-fir are fairly common in the Pacific Northwest,
this model should find plenty of applications.
Because the model was built from two combined datasets, the compatibility of
these datasets may affect the fit of the models. Significant differences were found
between the mean and variance of the height model residuals by study, as tested with
a t-test and F-test, respectively. There are several plausible explanations for this difference. The RVMM and SMC had similar protocols, but the treatment of hardwood
species in both studies was not alike. The addition of hardwood competition may
be better modeled by the use of crown cover measurements as indicators of density
rather than stems per acre or basal area. Also the protocols of the SMC study were
designed to reduce variance within treatments. However, many of the RVMM plots
were installed in existing plantations that were not necessarily associated with the
“best management practices.” Differences in stand management could lead to more
variation growth. Despite higher variation, it was important that data from such
stands were included because the effort put into the management of study plots is
likely to be higher than that put into the care of purely commercial stands.
83
4.4
Height Model Coefficients
The grouping structure of the data was not accounted for during the model fitting.
However, after performing the bootstrap validation, the significance of this grouping
structure became clear. Bootstrap estimates of the distributions of the height model
coefficients were much wider than those from linear approximations produced in the
statistical model output. This likely had much to do with the covariance structure
of the data. Data were grouped into plots, but the nonlinear regression model did
not account for this when estimating the variance. The coefficient estimates turned
out to be highly related to which plots were resampled at each bootstrap iteration.
Despite this higher than expected coefficient variation, the output of the model was
surprisingly stable when the different coefficient values were used to predict height
growth. Future work will be done to include a random plot effect into the model.
Better estimates of the tree-level variation can be used to introduce some stochasticity
into the model.
The high correlation between several model parameters also led to interesting effects when analyzing model sensitivity. The sensitivity of the model to the parameters
of the density modifier function and the vegetation cover modifier function is smaller
than one would be led to believe by investigating either of these parameters alone.
The sensitivity to the relative height modifier parameters, on the other hand, would
be underestimated by looking at h1 or h2 alone. Regardless of the model sensitivity
to any one parameter, most changes in the coefficients caused only minimal change
in the model predictions. Even if some bias exists in the coefficient estimates, this
should cause only slight bias in the model predictions.
4.5
Conclusions
The growth equations and static equations presented make up a nearly complete
stand growth simulator for young plantations of Douglas-fir. The height growth was
84
significantly affected by plot vegetation cover and stand density. A growth model
that incorporates these factors can be used to evaluate different density and vegetation
management regimes as means to meet stand management objectives. The coefficients
of the height growth model were found to be fairly stable and unbiased using a
bootstrap model validation procedure. Changes in the model predictions under most
of the coefficient distributions were minor. This should imply that, as long as the
model form is approximately correct, height growth predictions of the model should
be accurate.
While the modeling dataset was large, the range of applications for this model are
relatively small. Care should be taken not to use this model in applications for which
it was not designed. Because most of the data was from stands younger than 17 years
from seed, serious bias could result if projections are allowed to go beyond this age.
85
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