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 BIBLIOGRAPHY Ball, M. C., Egerton, J., Lutze, J. L., Gutschick, V. P., and Cunningham, R. B. (2002). 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