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Early height growth and regeneration : applicability of prognosis components to the southern interior.. Boisvenue, Céline R.P.F. 2000

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E A R L Y HEIGHT G R O W T H A N D R E G E N E R A T I O N : APPLICABILITY O F P R O G N O S I S C O M P O N E N T S T O T H E S O U T H E R N INTERIOR O F BRITISH C O L U M B I A  By  C E L I N E B O I S V E N U E , R.P.F.  B.Sc.F., Laval University, 1993  A T H E S I S S U B M I T T E D IN PARTIAL F U L F I L M E N T O F THE REQUIREMENTS FOR THE D E G R E E O F M A S T E R O F S C I E N C E IN F O R E S T R Y  In  THE FACULTY OF G R A D U A T E STUDIES DEPARTMENT OF FORESTRY Department of Forest Resources Management  W e accept this thesis as conforming totoe/Requiredstarxjard  J¥(E UNIVERSITY O F BRITISH C O L U M B I A November 1999  © Celine Boisvenue, 1999  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my w r i t t e n p e r m i s s i o n .  Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada  Abstract Predicting stand structure through time is a challenge in all aspects of forest management. Predictions in multi-cohort, mixed species, or spatially varied stands have mostly been based on field experience 1  and have not been clearly quantified. The main hypothesis of this thesis is that the regeneration and small tree height growth components of the Northern Idaho variant of the growth and yield model Prognosis, can be calibrated for use in the stands of the southern interior of BC. The original equation forms of Prognosis Nl were applied to data collected in stands of the Columbia Shuswap moist warm Interior Cedar - Hemlock variant of the Interior Cedar Hemlock moist warm subzone (ICHmw2 of the Biogeoclimatic Ecosystem Classification system of BC) in the vicinity of Nelson, BC. The same forms were then re-fitted with the Nelson data, and finally, other model forms were applied to Nelson data. In all cases, the original fitted equations in Prognosis Nl were outperformed by either the refitted equations or equations with Nelson-based variables. The equations presented in this thesis are a valid start to the calibration of regeneration and small tree height growth models Prognosis . Some issues need to be addressed for model improvements. 80  Prognosis Nl was not developed in BC and it uses a different ecosystem classification. Although correspondences have been made between BC site series and Idaho habitat types, the two systems are different, and so are the sites. These differences contribute to errors in model predictions. The data set used for developing Prognosis Nl was much larger than the data collected around Nelson. Some data categories used in the Prognosis model had more predictor variables than the number of observations in the corresponding data category in the Nelson data set. This lack of data resulted in non-robust models.  :|  "  Despite these issues, the equations resulting from this calibration process improve the estimates of small tree height growth and regeneration in multi-cohort or mixed species stands in the southern interior of BC. Prior to the calibration efforts of Prognosis , no quantitative tools were in place to aid 60  silviculturists for predictions in these stands. Although these predictions are not completely accurate, they can serve as guidelines, to supplement field experience, for making predictions.  Table of Contents Abstract List of Tables List of Figures Acknowledgements  _  ii v vi viii  _ _  Chapter 1: Introduction  1  Chapter 2: Background 2.1 ICHmw2 Stands Around Nelson 2.2 Regeneration 2.3 Early Height Growth 2.4 Prognosis _  3 3 7 9 12  _ _ _  Chapter 3: Methods 3.1 Field Methods 3.2 Analytical Methods  _  Chapter 4: Results 4.1 Data Summaries 4.1.1 Regeneration 4.1.2 Small Trees 4.2 Equations 4.2.1 Probability of Stocking 4.2.2 Number of Stems per Hectare 4.2.3 Number of Species * ; 4.2.4 Species Composition 4.2.5 Germination Delays 4.2.6 Seedling Height  _  _  llllZZll^... _  . _  _  Chapter 6: Conclusion Literature Cited _  _  30 34 __ 39 _ 41 43 44 4 7  50 _  ' ;.  16 19  26 _ 28  111.  4.3 Small Tree Height Growth Chapter 5: Discussion  _  _ _  _ _  58 61  APPENDIX 1: Number of trees per stocked plot. Using Ordinary Least Squares (OLS) for linear estimation of scale and shape parameters of the Weibull function.  66  APPENDIX 2:Number of tree per stocked plot. Comparing residuals from non-linear estimates of B and C and non-linear estimates of the Weibull function.  70  APPENDIX 3:Number of tree per stocked plot. Comparing non-linear estimates within the Weibull, of linear coefficients of B and C t o actual C D F and C D F with predicted 6 and C.  79  APPENDIX 4: Probabilities of advanced, subsequent and excess regeneration. Re-fitted coefficients of the logit function in the logistic equation form used in Prognosis for probability predictions of advanced subsequent and excess regeneration. Coefficients of the logit functions for logistic equation resulting from a variable selection procedure for prediction of probabilities of advanced subsequent and excess regeneration.  107  APPENDIX 5: Heights of advanced and subsequent regeneration. Germination-delay. Non-linear least square estimates of data-fitted Weibull parameters, for the prediction of delay to germination for species with above 20 observations.  117  APPENDIX 6: Heights of advanced and subsequent regeneration. Germination-delay Comparison of actual delay to germination cumulative density, cumulative density of delay to germination estimated using Prognosis models and cumulative density of delay to germination estimated with a data-fitted model.  124  APPENDIX 7: Heights of advanced and subsequent regeneration. Graphs comparing residuals from Prognosis models, re-fitted prognosis models and variable selection model for height predictions of advanced and subsequent regeneration, against actual heights.  131  APPENDIX 8: Heights of advanced and subsequent regeneration. Coefficients and analysis of variance for the re-fitted Prognosis model form and the variable selection resulting model for height prediction of advanced and subsequent regeneration.  141  APPENDIX 9: Heights of advanced and subsequent. Graph comparing the number of trees per height class in the Nelson data set to linearly predicted heights, heights predicted with the Prognosis model and height predicted with data-fitted Weibull function.  158  APPENDIX 10: Height of excess regeneration. Coefficients and model forms used for linear predictions of height of excess regeneration, heights predictions using Prognosis models, and height predictions with data-fitted Weibull functions by species.  163  APPENDIX 11: Small tree height growth. Plotted residuals of small tree height growth predictions with Prognosis models and re-fitted Prognosis for Cw, Fd, Hw, PI, tolerant, intermediate tolerance and intolerant species.  178  APPENDIX 12: Small tree height growth. Plotted residuals of small tree height growth predictions with variable selection model with Prognosis variables and a variable selection model with all site variables for tolerant, intermediate tolerance and intolerant species.  184  APPENDIX 13: Small tree height growth. Variable selection model coefficients - Prognosis variables and variable selection model - all site variables for tolerant, intermediate tolerance and intolerant species.  187  iv  List of Tables Table 3.1 Idaho and Montana habitat type correspondence to BC BEC site series. Table 3.2 Definition of variables used in the Prognosis Nl and Nelson-based calibrations of Prognosis regeneration establishment and small tree height growth equations.  19 80  20  Table 4.1 Summary of number of plots by class for independent variables used to in the regeneration prediction models.  26  Table 4.2 Count of natural regeneration by species (Sps), height class (Ht) and site series.  27  Table 4.3 Average 5-year height growth by species and site series.  28  Table 4.4 Average 5-year height growth by crown competition factor (CCF) and site series.  29  Table 4.5 Average 5-year height growth of small trees by aspect, slope class and grouped site series.  30  Table 4.6 Results of Prognosis Nl equation (Model 1) and of linear model (Model 3-1) for predictions of probability of stocking for the Nelson data set.  31  Table 4.7 Summary statistics for the non-linear models for predicting the probability of plot being stocked.  33  Table 4.8 Number of observations of stocked plots and definitions of data categories in the Nelson data set.  35  Table 4.9 Performance observations comparing the NLLS and SUR estimates.  36  Table 4.10 Coefficients from Model 3-1 and Model 3-2 for estimating of Weibull parameters 6 and C. Table 4.11 Coefficients for the re-fitted Prognosis Nl equation models (Model 2) and for the best performing variable subset from a variable selection procedure in a logistic equation (Model 3), for predicting probability of number of species on a plot. Table 4.12 Summary statistics for predicting the number of species per plot with the Prognosis Nl refitted equation (Model 2) and a different subset of site variables (Model 3).  37  40  Table 4.13 Actual and predicted cumulative values for advanced, subsequent and excess regeneration.  42  Table 4.14 Number of observations of height for advanced and subsequent regeneration by species.  45  Table 4.15 Summary of predictions of height growth of small trees by species and tolerance groups using Model 1 (Prognosis Nl equation) and Model 2 (re-fitted Prognosis Nl equation) for species with more than 25 observations and species by tolerance groups.  47  Table 4.16 Summary of the resulting height growth model form and fit from Model 3.  48  Table 4.17 Proportion of advanced regeneration in the small tree height growth observations.  49  Table 6.1 Summary of equations and recommendations for use in Prognosis  59  60  v  39  List of Figures Figure 2.1 Biogeoclimatic zones of the Nelson Forest Region.  3  Figure 2.2 Species-disturbances interaction adapted to Oliver and Larson's development stages.  6  Figure 2.3 Logical organization of Prognosis.  13  Figure 2.4 Illustration of the modelling approach used in the regeneration component of Prognosis Nl.  14  Figure 3.1 Plot layout for sampling large tree, small tree and regeneration sub-populations.  18  Figure 4.1 Residuals using Model 1 plotted against predicted probability for habitat types THPL and TSHE. Figure 4.2 Residuals using Model 1 plotted against predicted probability for PSMEA/ACA habitat  31  type. Figure 4.3 Residuals using Model 3-1 plotted against predicted probability for all species. Figure 4.4 Residuals using Model 3-1 plotted model against predicted probability for habitat types THPL and TSHE. Figure 4.5 Residuals using Model 3-1 plotted against predicted probability for habitat type PSMEA/ACA.  31 32 32 32  Figure 4.6 Residuals from Model 2 against predicted probability for habitat types THPL and TSHE. Figure 4.7 Residuals from Model 2 against predicted probability for habitat type PSMEA/ACA.  34 34  Figure 4.8 Residuals from Model 3-2 against predicted probability for THPL and TSHE habitat types. Figure 4.9 Residuals from Model 3-2 against predicted probability for habitat type PSMEA/ACA.  34 34  Figure 4.10 Cumulative density functions based on B and C estimates and actual cumulative density for category 2, the youngest (<7 years), east facing, PSME habitat type.  36  Figure 4.11 Cumulative density and estimated cumulative density for data category 1.  38  Figure 4.12 Cumulative density and estimated cumulative density for data category 5.  38  Figure 4.13 Cumulative density and estimated cumulative density for data category 7.  39  Figure 4.14 Cumulative density and estimated cumulative density for data category 11.  39  Figure 4.15 Cumulative density and estimated cumulative density for data category 22.  39  Figure 4.16 Cumulative density and estimated cumulative density for data category 23.  39  Figure 4.17 Cumulative density and estimated cumulative density for data category 24.  39  Figure 4.18 Comparison of frequencies of number of species per plot estimated with three modeling approaches to the actual number of species per plot.  40  Figure 4.19 Probabilities of advanced, subsequent, and excess regeneration of Douglas-fir.  41  Figure 4.20 Probabilities of advanced, subsequent, and excess regeneration for western redcedar.  42  vi  List of Figures (cont ) 1  Figure 4.21 Cumulative density distribution of actual, Prognosis Nl equation (Model 1), and the refitted Prognosis Nl equation (Model 2) estimates of germination delay of Douglas-fir.  44  Figure 4.22 Comparison of height predictions residuals from Model 1 (Prognosis Nl equation), Model 2 (re-fitted Prognosis Nl equation), and Model 3 (a linear equation with site variables) for advanced redcedar regeneration.  45  Figure 4.23 Comparison of models of height of excess regeneration for western redcedar.  47  vii  Acknowledgements  This research was funded by the Resource Inventory Branch, Research Branch, and Forest Practices Branch of the BC Ministry of Forests via FRBC funding. I would like to thank the licensees that willingly volunteered their time, local knowledge, and maps for the field component of this thesis, including: Slocan Group - Slocan Division, Kalesnikoff Lumber, Small Business Forest Enterprise programs of the Ministry of Forests in Arrow and Kootenay Lake Districts, Atco Lumber, Meadow Creek Cedar and Selkirk College. Thank you to the field crews that patiently collected this data with me, Cornel Lencar, John MacLeod, Deb MacKillop and Christine Clay. For their assistance and patience in the analytical and field components, I would like to thank Dr. Abdel-Azim Zumrawi and Barry Snowdon of the Ministry of Forests; and also Kokanee Forests Consulting, in Nelson, for their technical support.  I am especially grateful to my supervisor Dr. Peter Marshall, and to Dr. Val LeMay. Their support, patience, and guidance have made this work possible and more importantly enjoyable. I have learnt much more than biometrics from their example.  viii  Chapter 1: Introduction Smith (1992) refers to predictions of stand responses in terms of structure and composition as an integral part of silviculture. He states that silviculture is based on a working understanding of how the forest got established and how it grows through its rotation; it is based on our ability to create constructive disturbances from which we can predict the outcome. Understanding stand dynamics is, therefore, necessary for effective manipulation of stand structure. Predicting the outcome of stand growth, with or without manipulation, is a challenge even in the most homogeneous forest stand. Tools have been developed for predicting management outcomes, but have mostly limited to even-aged monospecific stand types. Even the classic reversed J-shape curve (negative exponential) used to describe growth in uneven-aged stands can be viewed as the sum of a series of small, pure, singlecanopied stands representing a series of evenly spaced age classes, each occupying an equal area. A perfect reversed J-shaped diameter distribution can also result from even-aged stratified mixed stands (Smith 1992).  The complexity of forest ecosystems and the need for bookkeeping may justify this narrowed approach to management; however, the dynamics of multi-cohort and/or mixed species stands needs to be quantified in order to generate effective growth and yield tools. The province of British Columbia (BC) has adopted the United States (U.S.) northern Idaho version of the growth and yield model Prognosis (Prognosis Nl, Stage 1973) as a growth and yield tool for multi-cohort and/or mixed species stands in the southern interior of BC. Prognosis is a distance-independent growth and yield model; it grows stands based on the interaction among trees. The individual tree is the basic unit of projection and most combinations of species and age classes can be accommodated within the model architecture.  Efforts are presently underway to calibrate the Idaho version of Prognosis Nl for the southern interior of BC (Prognosis ). The research described in this thesis is limited to the west Kootenays, around the 60  city of Nelson. The Nelson region and the northern part of Idaho have similar species and stand structures. Stands of the Kootenays are among the most complex mix of species in BC (Braumandal and Curran 1992). In the West Kootenays, even-aged monospecific stands are practically nonexistent. Stands commonly have four to six commercial tree species, and may have up to 13 tree species in total. Growth and stand dynamics are further compounded by a variety of disturbance agents. Hence,  in these highly productive ecosystems, our understanding of stand dynamics and our predictive abilities are very limited.  The main objective of this thesis was to calibrate the basic equations of the early height growth and regeneration component of Prognosis Nl for managed stands in the Columbia-Shuswap moist warm Interior Cedar Hemlock variant of the Interior Cedar Hemlock, moist warm subzone (ICHmw2, Braumandal and Curran 1992), in the vicinity of Nelson, BC. These equations are for use in the Southern Interior version of Prognosis . The hypothesis is that the early height growth and 60  regeneration components of Prognosis Nl can be calibrated for the southern interior of BC. Each component was tested in its original form for applicability to Nelson sites. The same form was then refit to the data collected on Nelson sites and finally, other model forms were applied. The specific objectives were the following: 1) to calibrate the basic equations of the regeneration component of Prognosis Nl for ICHmw2 managed stands in the vicinity of Nelson; 2) to calibrate the basic equations for the early height growth component of PrognosisNI for ICHmw2 managed stands in the vicinity of Nelson; and 3) to improve the understanding of stand dynamics in early development stages for these stands.  Prognosis equations that will be dealt with in this project apply to early height growth and regeneration only. Chapter 2 gives background information on stand dynamics of targeted sites, their regeneration, and early height growth, and describes Prognosis Nl. Methods for the 1998 data collection phase, and a compendium of analytical methods are provided in Chapter 3. Data summaries, and results for each individual equation are included in Chapter 4. Chapter 5 contains a discussion on the calibration of these equations and how they relate to management of these stands. Finally, conclusions are presented in Chapter 6.  2  Chapter 2: Background 2.1 ICHmw2 Stands Around Nelson The Interior Cedar Hemlock (ICH) biogeoclimatic zone around Nelson is dominated byfire-originstands. Fires were both of natural origin and of human origin resulting from mining activities at the turn of the century. Deposits leading to the present soils are fluvial at the valley bottoms, glacial-fluvial in low to midslope positions, and mainly colluvial on mid to high slope positions, with occasional morainal deposits. Mixing of the organic material through pedogenesis processes have created rich soils, especially on lower slopes and valley bottoms. The ICHmw2 occurs on these rich soils. Figure 2.1 illustrates the distribution of ICH in the West Kootenays. ICHmw2 sits at 1200m to 1450m elevation above the Interior Cedar Hemlock dry warm subzone (ICHdw) in the southern reaches of Kootenay Lake and Arrow Forest Districts where ICHdw occurs; it replaces ICHdw with increasing latitude, starting as low as 500 m where ICHdw does not occur (Braumandl and Curran 1992).  Figure 2.1 Biogeoclimatic zones of the Nelson Forest Region (Braumandl and Curran 1992).  3  The type of naturally occurring disturbance associated with these sites is believed to be wildfires of moderate size (20 to 1000 ha), with a mean return interval of 200 years (Ministry of Environment of BC 1995a). These fires follow topographic and fuel distributions, forming finger-like patterns usually running up ridges, leaving trees in wetter gullies standing or partly standing, and islands of standing timber behind sheltering terrain features, in high moisture zones, or randomly (Ministry of Environment of BC 1995a). Veteran fire-adapted tree species often survive fires, and are found singly or in groups throughout the landscape.  Tree species in the ICHmw2 include (Braumandl and Curran 1992): grand fir (Bg - Abies grandis (Dougl.) Lindl.)  lodgepole pine (PI - Pinus contorta Dougl. var.  subalpine fir (Bl - Abies lasiocarpa (Hook.) Nutt.)  latifolia)  western redcedar (Cw - Thuja plicata Donn),  western white pine (Pw - Pinus monticola Dougl.),  Douglas-fir (Fd - Pseudotsuga  hybrid spruce (Sx - Picea engelmannii Parry x  1  glauca  menziesiivar.  (Beissn.) Franco),  western hemlock (Hw - Tsuga heterophylla  glauca  (Raf.)  (Moench) Voss),  paper birch (Ep - Betula papyrifera Marsh.),  Sarg.),  trembling aspen (At - Populus  western larch (Lw - Larix occidentalis Nutt.),  Michx.),  tremuloides  black cottonwood (Act - Populus balsamifera trichocarpa  Torr. & Gray)  This intricate mix of tree species is accompanied by an array of shrubs, herbaceous plants, mosses, lichens, and liverworts.  The number of species and the extent and intensity of fire alone provide many possible regeneration scenarios following fire; however, fire is not the only disturbance occurring in this subzone. Slides, avalanches, and windthrow are also common in the steep mountains of the Kootenays, as well as pest outbreaks. Pest outbreaks are increasingly apparent in the ICH. They are the most prominent disturbance in the BC and Canadian landscapes (Anonymous 1998, Voller and Harrison 1998). Pests occurring in the ICH target stands include:  Tree species code follows the British Columbia Ministry of Forests, Inventory Branch standards.  4  ssp.  • laminated root rot (Phellinus • armillaria. root rot (Armillaria • tomentosus root rot {Inonotus • white pine blister rust • mountain pine beetle  spruce leader weevil  weirii) ostoyae) tomentosus)  (Cronartium  ribicola)  Cooley spruce gall adelgid (Aldeges hemlock sawfly  (Neodiprion  Douglas-fir beetle  Indian paint fungus (Dendroctonus  • dwarf mistletoe (Aceuthobium  terminalis) cooleyi)  tsugae)  (Dendroctonus  pseudotsugae)  (Dendroctonus  ponderosae)  • spruce beetle  {Pissodes  (Echinidintium  tinctorium)  rufipennis) americanum)  • dwarf larch mistletoe (Arceuthobium  lands)  Figure 2.2 is a visualization of tree species-disturbance interactions in the ICHmw2. This figure is based on an adaptation of Oliver and Larson's (1996) stand development stages and a working knowledge of these stands. Disturbances illustrated in Figure 2.2 can occur at any stand development stage and are not restricted to their position in this diagram. While identifying the disturbance agents is generally not difficult, disturbance regimes (intensity and frequency) vary according to regional climate and biophysical conditions (Bergeron and Harvey 1997); they are not distinct categories, and hence, are difficult to identify. Only extremes of the range of size and intensity of fires (small low intensity fire versus large intense fire) are traced in Figure 2.2 with their corresponding tree-species pathways. The array of possible tree-species regenerating and surviving post disturbances vary with the intensity and size of disturbance. .  ....  Stand development stages are continuous; they are not discrete categories and are often hard to determine (Oliver,and Larson 1996). Throughout stand development, a disturbance can bring a stand back to the initiation stage (Oliver and Larson 1996). Stands go through changes at different temporal rates and can be maintained in specific states by disturbance or growth factors for varying time periods. Age does not necessarily define the development stage of a stand (Oliver and Larson 1996), as each species has its own growth rate, varying by the site and structure under which the individual tree has developed. Various stages of development can be observed in stands in the Kootenays. Post-fire compositional variation in regeneration is not only confounded by other disturbances, but also overlaid by fire suppression and human interventions.  5  I species maybe present in the canopy. Main canopy breaking-up (snags, coarse woody debris accumulation) Lower shade -tolerance species are less likely to regenerate (PI, Lw, Ep, Ate, At)  Understory species growth in canopy gaps. Regeneration of PI, Lw, Ep, Ate, At is not likeiyunless gaps are large PI and/or Fd and/or Lw and/or Ep breaking up; Cw, Hw, Bg, Bl and Sx coming up through canopy  Varying mixed canopy, likely includes PI, Fd, Lw and occasional At, Ate, Ep Regeneration of PI, Lw, Ep, Ate, At is not likely  Figure 2.2 Species-disturbance interactions on ICHmw2 sites, adapted to Oliver and Larson's (1996) development stages.  6  2.2 Regeneration Trees can regenerate by sexual and asexual methods. Establishment of tree species is a function of the presence and distance of seed source, microsite conditions and adequate seedbed, species shade tolerance, presence of competing tree or vegetation, and can also be a function of physiological adaptations to disturbance (e.g., logdepole pine fire adaptation).  Regeneration is not static. In monospecific, homogeneous stands, most regeneration is thought to occur concentrated in a post-disturbance period. However, in multi-cohort and/or mixed species stands, regeneration is apt to occur at any time through stand development. Results from a study in the forests of the American intermountain west showed stocking by natural regeneration to be a gradual accumulation process in spruce-fir forests (McCaughey  etal.  1991). Ishikawa and Ito (1989) studied  the regeneration process in a mixed forest in central Hokkaido, Japan, and found that almost all species regenerated intermittently. Regeneration is a function of simultaneous favorable site conditions and presence of a seed source of a species suited to those conditions.  Where disturbance events primarily affect individuals or small groups in the main canopy, advanced regeneration can play a crucial role in determining the structure of the next stand. Kneeshaw and Bergeron (1996) found time since disturbance and parent trees to be highly correlated to the amount and type of regeneration in the boreal mixedwoods of Quebec. The species composition of the canopy was an important factor, not only because of the presence of parent trees, but also because of its effect on seedbed characteristics. The same factors might play an important role in the ICR The number of species and regeneration dynamics in smaller, less severe, disturbances create a mix of vegetation presently not quantified or even well understood.  Looking at the influences of topographic position and evergreen-shrub understory on seedling establishment in the'Appalachians, Clinton  etal.  (1994) found seedling density to be significantly  correlated with percent slope and with age of the gap they were establishing in. The number of species decreased over time and increased with gap size. Species establishment was a function of gap area, gap age, topographic position, and cover of competing vegetation. In addition, they found species of varying tolerance levels to be capable of establishing in small to medium sized canopy openings, in the absence of an evergreen-shrub understory. These results indicate that composition of tree and  7  competing vegetation species can have an effect on establishment of lower shade tolerance trees in smaller openings.  Knowledge of species shade tolerance and germination requirements is crucial to determining regeneration. Shade-tolerance levels vary among the species in the ICHmw2. Larch, aspen, black cottonwood, and birch are very shade intolerant tree species, lodgepole pine, Douglas-fir, and white pine are intolerant; spruce and grand fir are tolerant and western hemlock, subalpine fir, and western redcedar are the most shade tolerant species (Daniel et al. 1979). Douglas-fir can have bivalent coastal/interior autoecology and may be more shade tolerant on certain aspects, slopes or site conditions (Minore 1979). Larch, lodgepole pine, and black cottonwood require exposed mineral soil or burnt forest floor for germination, while aspen and birch are more polyvalent in terms of their seedbed requirements (Klinka era/. 1998). Aspen commonly regenerates by root suckering, which is encouraged by soil disturbance. Like lodgepole pine and larch, spruce germination requires an exposed mineral seedbed or a burnt forest floor to regenerate; western redcedar and hemlock can regenerate on open forest floor (Klinka  etal.  1998). Subalpine fir and grand fir regenerate on all  substrates, while Douglas-fir and white pine regenerate better after wildfires (Klinka  etal.  1998).  Clearly, management manipulations, like site preparation, size of opening, and species left on site have a great influence on the amount and variety of species regenerating.  The main challenge to modelling regeneration is accounting for all influential factors. Most regeneration models predict the occurrence of new trees only in response to disturbance such as harvesting and site preparation (Monserud 1987). At a regional or forest level, such models may provide acceptable predictions and are valuable tools. Management of mixed species stands in the Kootenays and elsewhere has traditionally implied planting sites after clearcutting. However, these planted sites still do not conform to the general regeneration models; plantations are invaded by a considerable amount of natural regeneration, sometimes outcompeting planted stock. Therefore, regeneration scenarios from these models do not apply. The increasing use of silvicultural systems with partial cutting adds advanced regeneration and regeneration of tolerant species to the already uncertain predictions. A model that assumes regeneration occurring only as a result of a stand management treatment will produce biased predictions of future multi-cohort and/or mixed species stands (Monserud 1987).  8  In a study of the distribution of regeneration, Frohlich and Quednan (1995) found no pronounced deviation from a random distribution when studying regeneration distribution in mixed mountain forest stands in Germany. They detected only small amounts of aggregation.  Favrichon's (1998) modelling efforts in tropical forests were based on a matrix model approach. He estimated density-dependence of ingrowth using non-linear regression. Models using a successional approach to modelling regeneration have also been developed. The successional approach uses ecological site factors and/or processes to predict regeneration (e.g., JABOWA (Botkin et al. 1972)). Kimmins (1993) uses a combination of both in his model FORCYTE-11 (FORest nutrient Cycling and Yield Trend Evaluator). FORCYTE-11, or the newer version FORECAST, is a deterministic model based on species characteristics, present vegetation, and site characteristics. These models were not driven by the need to provide accurate timber yields, and thus are not particularly useful in doing so.  Many growth models simply start with trees at 1.3 metres in height, or with 20-year-old stands, avoiding the issue of regeneration or small tree growth (e.g., STEMS (Belcher ef al. 1982). Ek (1974) improved on this by using an empirical approach, where ingrowth was predicted in the smallest diameter classes as function of residual density and site index in northern hardwood stands.  Modelling establishment and subsequent growth and structures of these stands can be seen as a probability problem: the probability of the right species being present, under the right site conditions, the site being unoccupied, with the appropriate climatic and environmental conditions. Prognosis Nl uses a probability approach to modelling regeneration. The details of the analytical approach to modelling regeneration used in Prognosis Nl are presented later in this chapter.  2.3 Early Height Growth Different species have characteristically different height growth patterns (Carmean 1970b). Oliver and Larson (1996) identified four types of height-growth patterns for individual trees: asymptotic, sigmoid, linear fast, and linear slow. However, tree growth patterns alone do not determine height growth. The time required for a species to reach a given height varies with site and species. The stand structure and accumulated volume of individual trees seem to be more closely correlated with tree height than with age (Oliver and Larson 1996). Therefore, the rate of growth is dependent on the competition and site quality context. Differences among species height growth rates influence crown expansion and strongly influence the ability of a species to compete under different situations (Oliver and Larson 9  1996). Height growth patterns, together with information from shade tolerance tables (Baker 1950), can be used to place species into groupings with similar developmental characteristics (Ashton 1992). Palik and Pregitzer (1991 in Cobb etal. 1993) attributed the importance of establishment time and attribute stratification of northern hardwoods more to establishment patterns than to differences between- or within-species height growth rates in a single-cohort stand. Establishment time then becomes an important factor for competitive advantage; first-established species, with faster growth characteristics, monopolize resources and have an advantage over later established species  However, the species first established will not necessarily be dominant later in the stand development. Stand structure varies over time, changing with each growing tree and simultaneously, changing the conditions in which trees are growing. Interspecific differences in height growth and shade tolerance promote stratification of the species into layers (Cameron 1996). Species with height growth at the time when growing conditions are favorable within the growing season have competitive advantages over other species. Lodgepole pine and western larch, for example, initiate rapid height growth early in the growing season, while non-determinant species like western hemlock and western redcedar grow in height at various time throughout the year, when conditions are favorable (Schmidt etal. 1980). Schuler and Smith (1988) suggested that different rooting habits and response to drought is responsible for the higher size/density relations and the greater stand leaf area in their study of a pinyon  (Pinus edulis  Engelm.) and juniper {Juniperus  monosperma  (Engelm.) and  J.  osteosperma  Torr.) Little) stand.  Most of the variation in growth in harsh environments has been found to be due to species differentiation, although site and aspect are also contributing factors (Peterson and Peterson 1994). Certain species can compete more effectively within certain species combinations (Cobb 1993) and some have shown variable growth rates due to the effects of surrounding species (LePage 1997). Contrary to Palik and Pegitzer's (1991 in Cobb ef al. 1993) scenario that first established species becoming the dominant canopy species later in stand development, some species have been found to positively influence each other's growth (Kneeshaw and Bergeron 1996). Height distributions of major regenerating species were skewed away from small (<15cm) height classes when competing vegetation was high (>50% cover of competing vegetation) in a study in the southern Appalachians (Clinton  etal.  1994), implying the importance of competing vegetation. Oliver and Larson (1996)  identified crown position as an important factor in the growth rate potential realized by each tree. All  10  these observations reinforce the difficulty in attributing species height differences to site quality, competition, genetics, age, or other factors.  In certain species, understory or advanced regeneration can vary growth patterns drastically. Understory trees often exhibit periodically lower and higher rates of height growth as they are released and later suppressed again by temporary overstory gaps. During times of unsuppressed growth, small trees, which were formerly suppressed by high shade, can grow at a rate which parallels height growth of dominant trees which were never suppressed on the same site (Oliver and Larson 1996).  In the ICHmw2, height growth patterns can be associated with shade tolerance levels. In general, pioneer species (PI, Lw, Ep, At, Ate) tend to have more asymptotic height growth, forging their way to the main canopy for light (Oliver and Larson 1996). Other species are less differentiated in their growth patterns. Tolerant species (Cw, Hw, Sx, Bg, Bl) display mostly slow linear growth patterns, while the semi-tolerant species (Fd, Pw) tend to have sigmoidal height growth patterns (Oliver and Larson 1996). However, early height growth represents only a portion of the overall height growth patterns. Site conditions and species physiological traits still drive height growth, and most early height growth seems to follow a fast or slow linear model. Cameron (pers. com. in Nigh 1995) identifies three different ways that a mixture of species in the upper strata of canopy can occur in even-aged stands: uniform stratification (together from beginning to end), consistent stratification (one species is consistently above the other), and inconsistent stratification (species dominance varies through the development of the stand). This stratification could be applied to ICHmw2 stands in early stages of development or applied to layers under the main canopy.  Efforts to model early height growth in the United States have been fueled by the need to model young stands not dealt with in plantation growth and yield models such as OREGANON (Hann etal. 1993) and CACTOS (Wensel etal. 1986). SYSTUM-1 (Ritchie and Powers 1993) models small tree height for input into OREGANON or CASTOS using linear regression. It requires heights, species, expansion factor, and breast-height diameters, and if available, crown ratio. Competitive vegetation information is also required. Zhang ef al. (1996) predicted height and diameter for juvenile loblolly pine  {Pinus  taeda  L.) plantations based on a transformation of von Bertalanffy's differential equation for growth and nonlinear regression methods. In Europe, Golser and Hasenauer (1997) predicted the height increment for a given tree by adjusting a given site potential by the tree's crown ratio, which represents the growing conditions in the past, and a competition index, which accounts for the current competition the tree  ii  experiences. Height increment potentials were determined by the change in dominant tree height over time using regional index functions. Such functions were developed based on large and extensive databases developed over time, which are not commonly available outside Europe. A sigmoidal decay function was used to predict small tree height increments. Favrichon (1998) chose to separate trees into groups based on growth behavior, with reference to shade tolerance and maximum potential size, to model early height growth in mixed tropical forests.  2.4 Prognosis Prognosis was developed in the United States in the early 1970s (Stage 1973). It is an analytical tool built to aid natural resource managers in projecting the development of forest stands under varying management options. It has evolved into a suite of computer programs, continually being improved, dealing with stand level changes, pests in forest stands, habitat modelling, vegetation competition modelling, and landscape level management (Teck etal. 1997).  Prognosis uses a compendium of mathematical models to represent tree and stand development. It was designed to use inventory data and adapt to all types of stands and stand conditions. It is not a static model; information can be incorporated, as it becomes available. Links to other biotic and hydrologic components of the ecosystem and economic analysis procedures for selecting the most appropriate management regimes are also included in Prognosis. Figure 2.3 present a low-resolution diagram showing the logical organization of the Prognosis model (Stage 1973). The base model starts by describing initial stand conditions based on available data. Silvicultural actions scheduled for the cycle are then applied to the compiled information. Periodic diameter and height increments, mortality rates, and crown ratios are then projected for each tree record. Volumes for each tree are calculated based on updated tree attributes and stand conditions are compiled (Stage 1973). All values produced by Prognosis are given at the stand level on a per acre basis.  12  (jtarT)  CONTROL INSTRUCTIONS  Input  Simulate management  INVENTORY RECORD Calibrated models  Compile Inventory  Predict stand development  Yks DISPLAY  More Growth cycles? No  STAND DEVELOPMENT THROUGH TIME NEXT STAND Figure 2.3 Logical organization of the Prognosis model. The growth equations within Prognosis Nl, contrary to many growth and yield models, used in North America, do not use site index as a predictor variable. Most equations in Prognosis Nl are based on site variables and the habitat types developed for Idaho and Montana (Cooper etal. 1991, Pfister  etal.  1977, Steele etal. 1981).  The regeneration model of Prognosis Nl is separate from the base model. Information on regeneration is also produced by acre at a stand level. The information produced and used by the regeneration model is compiled with the "compiled inventory" information (see Figure 2.3). The technique used for developing the regeneration equations in Prognosis Nl follows Hamilton and Brickell's (1983) two-state system: each sampled plot is either stocked or not stocked. Each plot in the sample is in one of two states: stocked with at least one established seedling or non-stocked. All plots are used to develop equations predicting the probability of a plot being stocked. Only stocked plots are used to estimate the number of trees on a plot, number of species, species composition, and seedling heights (Ferguson and Carlson 1993).  13  The probability of stocking f years after disturbance is estimated by a logistic equation within the interval [0,1]. After the attributes of a stocked plot are estimated, the probability of stocking is used to scale stocked plot attribute to a per-acre basis. Four attributes are estimated for each plot: probability of regeneration, number of stems per hectare, species composition, and seedling height. Figure 2.4 illustrates the approach to regeneration modelling used in Prognosis Nl.  r+Plotnot stocked Probability of a plot being stocked  J—Plot stocked  Number of trees per plot tree age Number of species per plot)  advanced regeneration  trees that willlZ contribute to "• canopy ("best trees")  I  •  height  subsequent regeneration  Probability of each species  germination delay height  excess trees  height  Figure 2.4 Illustration of the modelling approach used in the regeneration component of Prognosis Nl (Note: boxes represent equations). When modelling regeneration, it is sometimes more important to reproduce the distribution as opposed to the mean. Prediction of mean values with linear regression would not properly simulate the natural occurrence of regeneration, because categories with high probabilities of occurrence may not be predicted. Weibull cumulative density distributions are used in Prognosis Nl to reproduce cumulative distributions for the number of trees in stocked plots, for the number of years to germination for subsequent regeneration and the age of advanced regeneration trees, both necessary in the seedling height model.  14  In Prognosis Nl, seedlings are divided into excess or best trees. Best trees play a key role in Prognosis Nl model. The idea of best trees follows from the fact that many more trees reproduce than will exist in the mature forest (Wellner 1940). By selecting a few trees on each stocked plot, attention is focused on the growing stock that will contribute to yield. Best trees of each plot are then divided into advanced or subsequent regeneration. Advanced regeneration is defined as the "best" trees that germinated more than three years prior to harvest. Subsequent regeneration is defined as the best trees up to three years old at the time of harvest and excess regeneration is all the regeneration not chosen as best trees. Each has an attached probability, predicted with logistic equations. Equations for advanced, subsequent, and excess regeneration types were developed by species. Separate equations for predicting the height of regeneration were developed for each type of regeneration, in Prognosis Nl. Height of advanced and subsequent regeneration is predicted with a linearized exponential function of site variables and site variable transformations. These equations use tree age as one of the predictor variables. This necessitates determining tree age at the end of the Prognosis cycle. Weibull functions were developed that represent the distribution of the number of years from harvest to germination, which in turn enables the determination of tree-age at the end of the Prognosis cycle. In Prognosis Nl, Weibull functions for age of advanced regeneration were developed for six different categories: three categories of the number of budworm years in the 5 years prior to disturbance (BWB4) and two categories of basal area (BAA, 0-5.97 m /ha (0-25 ft /acre) and 5.97+ 2  2  m /ha (26+ ft /acre)). Separate functions were developed to estimate germination delay of subsequent 2  2  regeneration for six data categories: three categories of years since disturbance (TIME) and two categories of the number of years since the last budworm disturbance (BWAF). The data set used in building Prognosis Nl did not include age of excess regeneration. The cumulative height of excess regeneration was estimated with Weibull distributions for which the shape and form parameters were based on the Weibull cumulative distribution of the heights of the best trees of the same species. Tree height predictions in Prognosis Nl are done with linear or linearized functions. Two models predict height growth of trees: a small tree model and a large tree model. To prevent discontinuity in predictions between these models, a function is employed for the transition from one model to the other. This function applies to trees between small and large trees (2 - ,10 inches/ 5.08 - 25.4 cm diameter outside bark at breast height (dbh)). The large tree predictions for each tree is given a weight of HWT, and the small tree prediction is given a weight of (1-HWT), avoiding discontinuity in the response surface.  15  Chapter 3: Methods  3.1 Field Methods During the summer of 1998, data were collected for purposes of calibrating the regeneration and small tree height growth components of Prognosis Nl, for incorporation into Prognosis . The targeted 60  population was the ICHmw2 stands around Nelson, BC, that had been disturbed in the last two to 20 years. Physically locating all the recently disturbed sites in the ICHmw2 for random selection of a sample was practically impossible. Tracking systems and databases are scattered between licensees and government agencies, and existing records do not necessarily correspond with the actual field status of sites. Resources necessary for locating all possible sites were not available for this project. Furthermore, providing access to field crews in operating areas may be a liability for the operating licensee in the steep terrain of the Kootenays.  A sample population was identified by contacting seven local licensees who agreed to provide access and information on recently disturbed sites on their respective license (Slocan Group - Slocan Division, Kalesnikoff Lumber, Small Business Forest Enterprise Programs in Arrow and Kootenay Lake Districts, Atco Lumber, Meadow Creek Cedar, and Selkirk College on their Woodlot license). A sample of these sites was randomly selected from a list of the sample population.  The sampling phase served two purposes: to provide a representative sample of the sampled population and to allow the Prognosis Nl small tree height growth and regeneration components to be calibrated for use in Prognosis . The sampled population is not the same as the targeted population. 60  However, the assumption is made that most of the bias between the sampled population and the targeted population is removed by the BEC system and site variables used in the equations. The regeneration components of Prognosis Nl (Ferguson and Crookston 1991) and the Prognosis Nl tree growth model (Wykoff  etal.  1982) stratify sites into habitat types (parallel to the BEC classification), site  preparation method, regeneration method, level of overstory retention, aspect, and elevation. The same stratification was used for sampling sites. Sites in the sampled population are expected to cover the same range of explanatory variables as those in the targeted population.  Model calibration is better served using purposive sampling, usually spaced throughout the population, including the extremes, in order to make the model applicable to the population as a whole. It is more  16  important to encompass the range of conditions present than it is to sample the area proportionally to the representation of the conditions when calibrating models (Demaerschalk and Kozak 1974, Demaerschalk and Kozak 1975). To capture the range of sites in these strata, there was an effort to stratify remaining sites at the mid-point in the data collection phase. Sites were then randomly selected from strata not yet covered by the data collection. For proper calibration, all these strata should be sampled. However, the calibration of Prognosis is only one objective of this project and a stratified 60  random sample was necessary to allow linking the results to the population.  Once sites were selected, plots were randomly located on each selected site. The number of plots per site was dependent on the variability of the site in terms of the stratification criteria and its size; the larger and the more variable the site in terms of the stratification characteristics, the more plots were established. A minimum of two plots were established per site. At each plot location, stands were divided into sub-populations: large trees, small trees, and regeneration. All three sub-populations were sampled at each plot, without any overlap, describing the stand as a whole when combined. Large trees were defined as trees with a dbh above 7.5 cm, small trees were defined as trees with a dbh between 7.5 cm and 2 cm, and the regeneration layer was defined as trees below 2 cm in dbh down to two-year-old seedlings.  Large trees were sampled with a fixed area plot of 11.28 rh in diameter (0.04 ha). Information collected for large trees was only used in this project for identifying residual basal area per hectare (ba/ha), crown competition factor, and overstory species.  Small trees were sampled using a fixed area plot of 3.99 m (0.005 ha) with the same centre as the large-tree plot. In the small-tree plot, dbh and height of all small trees were recorded, and trees were selected for sub-sampling age and height increment. To avoid bias in the field, selection rules were preestablished for selecting sub-sampled trees: • if two or less trees were present per species, both trees were measured for height and age. • if more than two trees per species were present, then two were randomly selected on site using the tree numbers.  Each tree selected in the sub-sample was felled for measurement of total age, and the last 5-year height growth increment (measured up to the 1997 growth) was recorded. For non-determinant  17  species, like western redcedar and western hemlock, trees were sectioned until the 5-year height increment period could be established and measured.  The regeneration sub-population was also sampled with a fixed area plot, with the same plot centre as the large-tree and small-tree plots. The Prognosis Nl model used 1/300 acre plots (0.00135 ha). For ease of calibration, the same size of plot was used in this study. The Prognosis Nl regeneration model also requires information on stocking probability. This information was acquired using four satellite plots located in the cardinal directions from the centre plot on the 11.28 m circumference line (large-plot boundary). In the regeneration-centre-plot, count per species was recorded with a sub-sample of height and ground age for the best trees. Best trees were selected in each regeneration centre-plot. 2  In satellite plots, only count per species was recorded. Plot layout is illustrated in Figure 3.1.  Large tree plot  Small tree fixed area plot  Satellite plots  Regeneration plot  Figure 3.1 Plot layout for sampling large tree, small tree and regeneration sub-populations.  Site information was collected on each plot. This included BEC site series, slope (%), aspect (degrees), elevation (m), retention level defined as the basal area (ba/ha) and species left on site postharvest, stand condition observations in terms of forest health, and any other relevant or peculiar attribute about the site.  2  Refer to section 2.4 p.14 for definition of best tree.  18  3.2 Analytical Methods The  Prognosis Nl does not include any broadleaved species. Based on the timber production  orientation for which Prognosis is being developed (Annual Allowable Cut (AAC) 60  calculation) and  following consultation with local forest licensees, the calibration in this thesis excludes hardwoods also. Tree species included in this project are: • grand fir  • western larch  • subalpine fir  • lodgepole pine  • western redcedar  • western white pine  • Douglas-fir  • hybrid spruce  • western hemlock  Ponderosa pine (Pinus  ponderosa)  is part of Prognosis Nl, but was not present in the Nelson small tree  and regeneration data, and was therefore not included in any of the analyses.  The  habitat types used in Prognosis Nl (Idaho and Montana habitat types) have been equated to the  site series (ss) of the BEC  system of BC (Robinson 1997); the systems are interchanged throughout  this thesis. Correspondence between habitat types and ss in the ICHmw2 for the Nelson Forest Region are outlined in Table 3.1.  Table 3.1 Idaho and Montana habitat type correspondence to BC BEC 1997), , ICHmw2 ss  03 04 01 05 06  site series (from Robinson  Corresponding habitat type  PSME/VACA TSHE/CLUN TSHE/CLUN THPL/OPHO THPL/OPHO -  Pseudostuga menziesii/Vaccinium Tsuga heterophylla/Clintonia uniflora Tsuga heterophylla/Clintonia uniflora Thuja plicata/Opoplanax horridus • Thuja plicata/Opoplanax horridus  caespitosum  Prognosis Nl regeneration and small tree height growth equations require many variables and variable transformations that may differ slightly from the data in the Nelson database. Variables are defined in Table 3.2 for both the regeneration sub-models and the small tree height growth model.  19  Table 3.2 Definition of variables used in the Prognosis Nl and Nelson-based Prognosis regeneration establishment and small tree height growth equations. 60  Variable  ASP/Asp_r SLG7 Slope TIME/Yrsince REGT BWAF BWB4 SQREGT/ Time 1/2  SQBWAF ELEV BAA/ ba_ha NONE/ Dsitepl -2 MECH/ Dsitepl-2 BURN/ Dsitepl-2 ROAD TPSP/ Ntrees OVER BOTTOM/ Dpos1-3 LOWER MIDSLOPE UPPER RIDGE AGE Hab/ Dhab  Definition  Plot aspect converted to radians. Plot slope tangent (slope percent/100). Number of years since last disturbance. Number of years since last disturbance without budworm; no budworm information was collected, value of 0 for the Nelson data. Number of years since last disturbance with budworm; value of 0 for the Nelson data. Number of budworm years in the 5 years before last disturbance. Sum of square roots for each year without budworm. Represented as square root of time in the Nelson data. Sum of square roots for each year with budworm; in the Nelson data, the number of years with budworm=0, therefore variable is always equal to 0. Stand elevation expressed in feet or in metres Plot residual basal area (ft/acre or in m2/ha). Class variable if no site preparation; dummy variables (0,0). 3  2  Class variable for mechanical disturbance; dummy variables (0,1) Class variable for burn disturbance in plot; dummy variables (1,0). Class variable that occurs on road cuts, road fills, or unmaintained roadbeds; no data was collected on or near road in Nelson data collection effort. Number of regeneration-size trees on the plot, conditional to the plot having at least one established seedling. Class variable for the presence of the same species in the large tree plot. Class variable for the bottom topographic positions; dummy variable for topographic position (1,0,0). Class variable for the lower topographic position; dummy variable for topographic position (1,0,0). Class variable for the midslope topographic positions; dummy variable for topographic position (0,1,0). Class variable for the upper topographic position; dummy variable for topographic position (0,1,0). Class variable for the ridge topographic positions; dummy variable for topographic position (0,0,1). Tree age at ground line. Class variable for habitat types;' dummy variable for habitat types and corresponding site series, Dhab=0 when ss=3 otherwise Dhad=1.  "Budworm defoliation was not collected. Such defoliations are not as prominent in the west Kootenays as in Idaho and Montana and occurrence in the Nelson sites was negligible. The calibration of the regeneration equations only used natural regeneration data from the Nelson data set. No calibration was performed for planted stock. b  Each equation in Prognosis Nl regeneration and small tree height components were applied to the data collected on Nelson sites. Throughout the analyses Prognosis Nl equations with their original coefficients are referred to as Model 1 for all equations. The coefficients of the same equations were then re-fitted to the Nelson data and are referred to as Model 2. Finally, other variable subsets or Model forms were applied to the Nelson data to see if predictions could be improved. These are referred to as Model 3 and Model 3-1,3-2, etc., when more than one alternative was tested.  20  Concordant to the approach used in Prognosis Nl presented in Figure 2.3, the regeneration sub-model analyses of the number of trees per plot, the number of species per plot, and the species composition were only performed on stocked plots. In the case of stocking levels, number of species and species composition, binary response variables were predicted using a logistic equation. The general equation form is:  P=(1+e-< ii>)- + ei I/3 x  [1]  1  where: P is the probability of success; e is the base of natural logarithms; # is a vector of regression coefficients; x is a vector of independent variables; and e\ is the error term.  For all binary response variables, the response surface is sigmoid-shaped and will have non-normal error terms, with non-constant error variances (Neter  etal.  1996). The binary responses are  constrained to a [0,1] interval. Consequently, using non-linear least squares (NLLS) to estimate coefficients would yield biased variance estimates for these coefficients. Maximum likelihood methods were used to estimate of the vector of regression coefficients in all these equations. Variable subsets with best maximum likelihood estimates were selected from backward and forward variable selection methods to develop Model 3. Models using maximum likelihood estimates were evaluated with: the coefficient of determination (R ), the log likelihood expressed as -2*log L (smaller values are better), the 2  probability value (p) of the Score statistic which tests the significance of the explanatory variables in the model (if p<0.05 variables in model are significant), and two statistics (Akaike Information Criterion AIC and Schwartz Criterion - SC) primarily used for comparing models (lower values are better). These statistics were calculated with the Statistical Analysis System (SAS - version 6.12 -1996). Estimated probabilities were compared to calculated stocking levels, number of species, and proportions of advanced, subsequent, and excess regeneration in plots.  Probability of stocking was determined in Prognosis Nl with separate logistic equations for each of the four habitat type groupings. Each of these equations was applied to the Nelson data using the Prognosis Nl coefficients (Model 1). Site series were grouped accordingly to their corresponding habitat types. Only two habitat type groupings encompassed all site series in the Nelson data - THPL and TSHE were calibrated together in Prognosis Nl. For each habitat type, the coefficients of the Prognosis Nl equation were re-fitted to Nelson data (Model 2). Probabilities were also estimated using linear models and ordinary least squares (OLS) estimated coefficients, the resulting model is referred to  21  as Model 3-1. Logistic equations with different subsets of variables were also evaluated for best predictor variables when using non-linear maximum likelihood coefficients estimates for both habitat types. The best resulting equation is referred to as Model 3-2.  The Prognosis Nl regeneration model uses a logistic equation to predict the probability of having a particular number of species present in a stocked plot. In the data used to build Prognosis Nl, the number of species present had a reversed J-shaped distribution, with one species having the highest probability and six or more having the least. Separate equations were developed for the probability of 1,2,3,4,5, and 6+ species occurring on a plot. These six probabilities were estimated with the Prognosis Nl equation (model 1), re-fitted coefficients of the Prognosis Nl equation (model 2), and a new subset of variables (model 3). To compare predictions from Model 1, Model 2 and Model 3, a uniform pseudo-random number was used to make discrete but unbiased, choices of the number of species on a plot. To do so, the probabilities of each number of species (1 through 6) within each model group (1,2, and 3) were calculated and totaled within each model group. For each model, the total was then divided back into the probabilities so that the sum of the adjusted probabilities was accumulated within the interval [0,1]. A uniformly distributed pseudo-random number was compared to the accumulated probabilities; the number of species on a plot was determined as the number at which the accumulated probabilities (probability of 1 to probability of 6+) first exceeded the pseudo-random number. The process was executed 1000 times for each of the three models. The frequency of each number of species was averaged and compared per model group to the actual frequency of the number of species per plot in the original data.  Seedlings in the Nelson data were divided into advanced, subsequent, and excess regeneration in the same way as in Prognosis Nl. Best trees were selected and aged only in the centre regeneration subplot; therefore, only the centre plots were used in the seedling height calibration process. As in the Prognosis Nl approach, probabilities for each regeneration type were modelled by species. Probabilities were first calculated using Prognosis Nl equations (Model 1); a second set of probabilities were calculated with the Prognosis Nl equation with re-fitted coefficients (Model 2); and a third set of probabilities were calculated with the best variable subset from a variable-selection process (Model 3). As in the Prognosis Nl approach illustrated in Figure 2.3, probabilities of excess regeneration and occurrence of best trees were cumulated within a [0,1] interval, and probabilities of advanced and subsequent regeneration (which together represent the best trees) were also cumulated within a [0,1] interval for each of the three models. A pseudo-random number was used to determine which type of regeneration by species occurred in the plot. This process was repeated 100 times and the results  22  (  averaged for each model. Results from all three models were graphically compared to the actual proportions of advanced, subsequent, and excess regeneration.  A Weibull cumulative density distribution was used to reproduce the distribution of the number of trees in stocked plots and of the number of years to germination. The Weibull equation is of the form: F(x) = 1-EXP[-(y/e)ci + ei  [2]  where: F(x) is the Cumulative Density Function (CDF); y is the number of trees per stocked plot or the number of years to germination; B is the Weibull scale parameter; and C is the Weibull shape parameter. The area bounded by the Weibull equation is within the interval [0,1]. Equation [2] is sometimes used in the form (Bailey and Dell 1973): y = 6[-ln(1-F(x))] %e 1  i  [3]  Parameters of this function were estimated using NLLS.  The number of trees per stocked plot was determined with a two-step analysis as in Prognosis Nl. In Prognosis Nl, the data were categorized by habitat series, four aspects, three time periods, and two budworm defoliation histories. Correspondences in the Nelson data included two habitat series, four aspects, and three time periods, resulting in 24 possible data categories. The 6 and C Weibull parameters were estimated using NLLS for each of the data groupings with above 20 observations. Site variables for each category were averaged and all categories were pooled to create a data set of 16 observations of B and C (16 out of 24 categories had above 20 observations). These parameters were then linearly predicted using different approaches. The first approach used the Prognosis Nl linear equation and coefficients to predict Weibull parameters B and C. The predicted parameters were then applied to the Weibull to estimate a CDF for each data-category (Model 1). The second approach used re-fitted OLS estimated coefficients in the Prognosis Nl linear equations for Sand Cpredictions. Those estimates of 6 and C produced a second CDF for each data-category (Model 2). A third CDF was produced using linear estimates of B and C with the best subset of variables from a backwards-stepwise variable selection procedure (Model 3). Of these three CDF, the closest to the actual cumulative density was selected for coefficient estimation using two other methods: a joint-generalized least-square approach (referred to as Seemingly Unrelated Regressions - SUR (Model 3-1)) (Kmenta 1986) and a NLLS estimation of linear model coefficients directly substituted in the Weibull function (Model 3-2). The model producing a cumulative density distribution closest to the actual cumulative density of the  23  number of trees per stocked plot was then further tested on the remaining data categories with less than 20 observations. Predictions of delay to germination also used a Weibull cumulative density function. As in Prognosis Nl, cumulative germination delays were predicted for advanced or subsequent regeneration of each species. Only three out of nine species in the advanced regeneration had more than 20 observations, and eight out of nine subsequent regeneration categories had more than 20 observations. For each of these categories, Weibull density functions were estimated using NLLS for determination of cumulative germination delays. Model 1 refers to the Prognosis Nl equation and Model 2 refers to the same equation, re-fitted. No other model forms were tested as no site variables are used in modelling germination delay in Prognosis Nl. Height of excess regeneration was not available in the database used to develop Prognosis Nl; however, it was collected on Nelson sites. Prognosis Nl predicts the cumulative height of excess regeneration with a Weibull function. For comparative purposes, the Prognosis Nl shape and form parameter for a Weibull distribution of cumulative height of excess regeneration were applied to the Nelson data set. A uniform pseudo-random number was used to assign height to trees for all species (Model 1) regardless of the number of observations. For species with more than 10 observations of heights, the Weibull parameters were fitted with NLLS and a uniform random number used to assign a height to each tree (Model 2). As per height of advanced and subsequent regeneration in Prognosis Nl, a best variable-subset based on OLS estimated coefficients in linearized equations was used to predict height of excess regeneration by species (Model 3 for excess regeneration). The number of trees in each height class were compared for all models by species. Heights of advanced and subsequent regeneration were predicted by species with the same equation used in Prognosis Nl (Model 1). This model was then re-fitted with the Nelson data to yield a second set of height predictions for advanced and subsequent regeneration (Model 2). Finally, best variablesubsets based on OLS estimated coefficients in linearized equations estimates, were applied to predict the natural log of height for advanced and subsequent regeneration per species (Model 3). Resulting height predictions were divided into height classes for each model and compared to the actual 3  frequencies of trees per height class. The Prognosis Nl equation for predicting small tree height growth is a linearized function (Wykoff 1986) The model form is:  24  LN HTG = HAB + LOC + 0.22157*SL*cos(ASP) - 0.12432*SL*sin(ASP) - 0.10987*SL + bi In(HT) + b *CCF + b (BAL/100) + ei [4] 2  3  where HTG is the height increment predictions for small trees, HAB is a constant that is dependent on habitat type, LOC is a constant that is dependent on location, ASP is stand aspect (degrees), SL is stand slope ratio (/100), Ht is tree height (ft or in metres), CCF is crown competition factor, BAL is the basal area in larger trees (ft /acre or in m /ha), bi, b2, b3 are species-specific regression coefficients 2  2  and d is an error term. This equation was applied to the Nelson data for species more than 25 observations and for shade-tolerance groups (Model 1). The location constant for the Clearwater and Nezperce National US Forests was used for the Nelson model run as per the calibration of the largetree growth model of Prognosis for the Nelson Region (Zumrawi 1998, pers. comm.) . The Prognosis 60  4  model was then re-fit using OLS for the same data categories (Model 2). Both height (Model 3-1) and natural log of height (Model 3-2) were predicted using the best subset of variables from a variable selection procedure fit using OLS.  Linear and linearized models used for height of regeneration and small tree height growth predictions were evaluated by comparing residuals, R square values, I square values (1-(SSres/SStot)), the standard error of the estimate (SEE = VSSres/n) and residual variance. In all analyses, results were evaluated at the 0.05 significance level with a few exceptions: • acceptance levels when a forward variable selection procedure was used was 0.01. This was reduced to allow for more flexibility for variable entry in models. • non-significant variables were eliminated from equations except: (1) those that were part of a group of dummy variables that represented a single class variable, (2) when variables were part of the Prognosis Nl model form being re-fitted or evaluated, or (3) when no model form was significant, in which case the least problematic model was chosen.  All statistical analyses were performed with SAS (SAS - version 6.12 -1996) while spreadsheet manipulations and graphs were produced with Excel (Microsoft 1997).  3  4  Height classes were: (1) < 50cm; (2) 50 - 100cm; (3) 100 - 130cm and (4) 130cm+; Dr. Abdel-Azim Zumrawi, Research Scientist, Research Branch, B.C. Ministry of Forests.  25  Chapter 4: Results  4.1 Data Summaries 4.1.1 Regeneration Data were collected on a total of 186 regeneration plots. Table 4.1 shows the range of data for classes of some independent variables used in the models. Table 4.1 Summary of number of plots by class for independent variables used in the regeneration predictions models. Class No. plots Years since last disturbance 0 15 2 15 3 17 4 8 5 9 6 23 7 12 8 8 9 5 10 19 11 13 12 11 14 3 16 3 17 2 18 6 19 11 20 2 21 2 29 2 Aspect E F N NE NW S SE SW W  30 16 12 18 12 27 19 36 16  Class No. plots Site preparation method brushing 3 burn 22 mechanic 7 none 134 Spot burn 14 Spot mech. 6 Elevation (100 m) 5 6 7 8 9 10 11 12 13 14  4 22 8 20 20 34 12 34 30 2  Slope percent 0-10 10-20 20-30 30-40 40-50 50-60 60+  23 45 46 26 23 16 7  Class Residual basal area (m/ha) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 90  No. plots  2  Topographic position crest depression level lower middle toe top upper (blank)  60 28 21 22 11 6 6 8 7 5 3 4 2 1 1 1  7 4 10 21 111 5 1 25 2  More than half of the plots (53%) were disturbed in the last 7 years. A large portion of plots (76%) had less then 20 m /ha of residual basal area, and 62% were on slopes that are 20% or less. Plots seem to be well 2  26  distributed over the elevation range, and most (60%) are in a middle topographic position. The majority of plots (72%) has had no site preparation.  The amount of natural regeneration, summed by species, height class, and site series is given in Table 4.2. Western hemlock had the highest count of all species, followed by Douglas-fir and western redcedar. The high counts of western hemlock regeneration occurred on site series 01 and 04. Site series 03 had the highest count of Douglas-fir regeneration, followed by site series 04. Site series 04 had the highest count of western redcedar regeneration, followed by site series 03 and 01. Cautious interpretations accompany this table as count of species, per height class, and site series, represent only a narrow view of a multivariate space. Distribution of sampling can skew interpretation of abundance species per site series and height class.  Table 4.2 Abundance of natural regeneration by species (Sps), height class (Ht ) and site series. a  Sps  Ht  Bg  1 2 3  a  1  01  97 13 1  "'111 Cw 1 379 2 97 3' 30 4 25 Total 23 Hw 1 1861 2 287 3 72 4 53 Total 2273 PI 18 T" 2 6 3 3 4 3 30 I Sx IS {7 2 3 3 4 20 IBSotal total •9  03  35 7  Site series 04 05  37 7 3  06  18 5 2 7 32 251 15 85 8 20 5 24 6 13 3 125 16 33 6 ' 1 7 171 17 j 9 1 1  1 43 47 221 634 179 200 66. 57 69 50 81. 189 254 1658 79 173 38 51 38 63 409 "1945 197 110 96 39 48 3 46 6 387 158 11 96 13* 21 7 15 3 1 4 2 6 Y21V 18 '29  Sps  Ht  Bl  1 2 3 4  21 1 1  Total  Fd  1 2 3 4  Total  Lw  1 2 3 4  Total  Pw  1 2 3 4  [" Total  6 3  01  a  23 193 34 4 8 239 7 7 1  03  Site series 04 05  42 20 13 6 759  208 105 120 1192 92 54 32 42 15 220 78" 107 49 13 4 22 1 14 " 96 192  141 23 21 4 189 636 117 45 56 854 101 35 9 6 151 213 78 10 5 306  6 4 2 1 13 . 85 16 5 7 113 9 6  06  2 1 3 2  2  6 21 19 6 2 27vj  9  Height classes are: (l) < 50cm; (2) 50 -100cm; (3) 100 -130cm and (4) 130cm+;  27  4.1.2 Small Trees The data collection phase of this project yielded 266 height growth measurements of small trees. Only four out of nine species (Cw, Fd, Hw, PI) had more than 25 observations of height growth. Tolerance groups are believed to share similar height growth patterns (Oliver and Larson 1996). To provide sufficient observations for model calibration and testing, species were grouped according to shade-tolerance levels. Tolerant species included grand fir, subalpine fir, western redcedar, hemlock, and spruce; semi-tolerant species included Douglas-fir and white pine; intolerant species included lodgepole pine and larch. Table 4.3 shows the average 5-year height growth of small trees by species and site series.  Table 4.3 Average 5-year height growth (in metres) by species and site series. Tolerant Species Site Series  Bg  BI  Cw  0.910 1.328 0.445 1.217  0.206 0.546 0.271 0.383 0.160 0.355  01 03 04 05 06 Average  0.040 0.040  1.076  01 03 04, 05 V Average  1.220 1.145 1.202 -1.265 1.185  0.968 1.210 1.632 1.430 1.347  Semi-tolerant species Site Series Fd  Intolerant Species Site Series Lw  '•01 03 04 ' ' 05 Average 1  0.320 '•' v 2.348 , 2.010 2.635 2.124  Pw  PI  1.743 1.424' " 2.130 1.698  Hw  0.471 0.951 0.425 0.575 0.675 0.586  Sx  0.445 0.370 0.393 0.407  Average  0.354 0.724 0.331 0.535 0.366 0.474  Average  1.076 1.151 1.305 1.298 1.221 Average  0.320 1.824 1.483 2.332 1.772  The intolerant species had the largest height growth, followed by the semi-tolerant, and the tolerant species. The poorest height growth was found for grand fir; however, this was based on very few observations of trees growing at the elevational limit to their distribution. The next smallest height growth was found for western redcedar. However, these species were on different sites with varying retention levels. Height growth differences between species may just be a consequence of site characteristics and not a statistically valid difference.  28-  Five-year height growth is averaged by classes of Crown Competition Factors (CCF) and site series in Table 4.4. It can be seen that height growth decreased with increasing CCF class. Tolerant species were found in plots across all of the CCF classes, semi-tolerant species were present in plots across the first 5 classes, and intolerant species were only found in plots that fell into the first two classes. The higher height growth in tolerant species under CCF 4 is attributed to those trees being advance regeneration (ages between 37 and 82 years).  Table 4.4 Average 5-year height growth (in metres) by CCF and site series. Tolerant species CCF Class 4 0.288  3  Site series 1 , 1 0.990., • 3 0.977 4- ,'• 0.524 • 5 1.144 6 Average 0.903 Semi-tolerant species 1  1  ;  2 0.514 0.589 0.322 0.255  3 0.214 0.165 0.217  0.428  0.207  0.277  3  CCF Class 4  0.273  5 0.324 0.090 0.400 0.216 0.366 0.292  6 0.255 0.255  Average 0.354 0.724 0.331 0.535 0.366 0.474  b  Site Series 1 1 1.468 3 1.765 4 1.686 5 2.043 Average 1.717 Intolerant species  2 0.590 0.552 0.576 0.180 0.510  0.427 0.360  0.160  0.410  0.160  5 0.535  6  0.360 0.448  CCF Class" Site Series 1 2 3 4 5 6 1 0.320 3 1.857 0.850 4 1.483 i 5 2.537 2.025 Average 1.782 1.633 The CCF classes.are: (1)<50; (2) 50-100; (3)100-150; (4) 150-200; (5)200--300; and (6) 300--400.. <  Average 1.076 1.151 1.305 1.298 1.221 Average 0.320 1.824 1.483 2.332 1.772  a  Average 5-year height growth of the small trees by aspect, slope class and grouped site series is given in Table 5  4.5. It is apparent that trees on site series 03 had the best average height growth, followed by site series 05, 04, 01, and 06. There is no clear pattern among slope and aspect classes.  5  Sites series are grouped by corresponding habitat types.  29  Table 4.5 Average 5-year height growth of small trees by aspect, slope class , grouped by site series. 3  Grouped site series 03  Average 05-06  Average 01-04  Aspect Slope Class 0 1 2 3 4 5 6 Missing 3  E 0.708 0.765  0 1 2 3 4 R D  6 Missing  0.805 1.388  0.490 0.194 0.421 0.325 0.785 0.709  N  NE  2.010 2.356  1.920 1.302 0.160 0.373 0.971  0 1 2 3 4 6 Missing  F 0.805  1.388 0.333  2.266  2.573 2.413 1.514 0.080 1.093 2.010 0.385  NW  S  SE  0.770  1.632 1.175 1.205  1.215 2.080 1.300  1.763 1.807  1.415  1.792 0.140 1.017 1.066  0.770  0.530 0.413  0.120  2.267  1.079  0.551 0.290  0.155 0.198  1.383  0.661 0.380  1.434 1.015  1.349  0.640 2.767  0.505 0.352 0.610  1.828  0.687  0.445  Sw 1.727 0.720 1.655 0.700 1.172 0.090  W 0.287  1.417  0.790  Average 0.861 1.494 1.710 1.304 1.224 1.129 1.035 2.486 1.423 1.835 1.212 1.223 1.037 0.594 0.397 0.910 1.323 0.623 2.011 0.731 0.198 1.470  1.047  0.712  1.031  1.620 1.153 1.580 0.760 0.040  1.985 0.850 1.700 1.054 1.470  0.310  0.085  2.202  0.125 1.018  1.417 0.790  0.198 1.491  0.333  0.183  1.383  0.290  0.445  0.902  1.172  1.564  0.898  0.308  1.345  0.705  Average 1.491,, 0.333 0.183 1.383 0.290 0.445 . 1.417 0.790 1.470 The slope classes are: (0) flat; (1) < 20%; (2) 20-30%; (3) 30 - 40%; (4) 40 - 50%; (5) 50 - 60%; and (6) 60%+.  4.2 Equations Results of all equations are separated in three model groups. Model 1 is the Prognosis Nl equations with original coefficients, model 2 is the Prognosis Nl equations with re-fitted coefficients and model 3 is an equation using a different subset of variables than that used in Prognosis Nl. Model group 3 is separated into model 3-1, 3-2, etc., when more than one variable subset or modelling approach was used. All models, including those in model group 3, are outlined section 3.2, Analytical Methods.  4.2.1 Probability of stocking Probability of stocking was calculated for each of the 186 plots in the Nelson data set. As outlined in the field sampling methods, each plot had a total of five regeneration subplots (930 regeneration subplots in total). A stocked subplot was defined as a subplot containing at least one seedling, more than two years old. Each  30  stocked subplot contributed 0.2 stocking to the plot. Summary statistics for the probability of stocking predicted with model 1 and model 3-1 are presented in Table 4.6.  Table 4.6 Results of Prognosis Nl equation (model 1) and of linear model (model 3-1) for predictions of probability of stocking for the Nelson data set. Species grouping PSME/VACA  No. of obs. 73  THPLandTSHE  112  Model group  Variables in model G(x1, x2, x3, sqregt, elev, Mech, Burn) G(x1, x2, x3, sqregt, elev, BAA, BAA, x5, x6, Mech, Burn) F(asp_r, slo, elec_c, dsitepl - 4, x2, x3, ba_ha) F(yrsince, slo, x4, x5, ba2) F(asp_r, slo, elev_c, dsitepl-4, x3, ba_ha) 3  model 1  2  All species  186  2  PSME/VACA THPL and TSHE  73 112  model 3-1  2  • x1 is defined as  COSfASPl'SLO^TIME  14  x2 is defnied as SIN(ASP)*SLO *TIME B  x3  K  SLO**TIME  K  I  2  SEE  0.461607  0.202961  0.755008  0.131367  0.27772520  0.217529  0.26171285  0.180401  0.40313690  0.219036  x4 is defined as TIME*  x5 is defined as COS(ASP)*SLO" BAA x6 is defined as  SIN(ASP)*SLO" BAA  Figures 4.1 and 4.2 show plots of predicted probability values and residuals (measured - predicted) for model 1. Figures 4.3,4.4 and 4.5 show the same plots for model 3-1 for all species, the PSME habitat type and the THPL/TSHE habitat types respectively.  Predicted Probability of Stocking Predicted Probability oJ Stocking  Figure 4.1 Residuals using model 1 plotted against predicted probability of stocking for habitat types THPL and TSHE.  Figure 4.2 Residuals using model 1 plotted against predicted probability of stocking for habitat type PSME/VACA.  31  Predicted Probability of Stocking  Figure 4.3 Residuals using model 3-1 plotted against predicted probability of stocking for all species.  Predicted Probability of Stocking  Figure 4.4 Residuals using model 3-1 plotted against predicted probability of stocking forTHPLandTSHE.  Prediaed Probability of Stocking  Figure 4.5 Residuals using model 3-1 plotted against predicted probability of stocking for habitat type PSME/VACA. Based on the SEE, model 1 did not fit the data as well as model 3-1 (Table 4.6). This also seems evident from comparing the residuals displayed in Figures 4.1 and 4.2 with those displayed in Figure 4.4 and 4.5. In general, model 1 underestimated the proportion of stocked plots (positive residuals), especially for the PSME habitat type. Model 3-1 appeared to fit the data better. Separating the data into habitat series for model 3-1 reduced the SEE for the PSME habitat type (Figure 4.5) over the combined model (Figure 4.3). The THPL and TSHE habitat types had a similar SEE to the combined model. However, caution should be used when interpreting the linear models, since using OLS regression for a restricted dependent variable, like the proportion of stocked plots [0,1], will result in biased and inconsistent estimates of the equation coefficients (LeMay  etal.  1993, Tobin 1958).  Further, binary response variables are known to have a sigmoidal-shaped response-surface; a linear model, therefore, does not fit the data (Figures 4.3,4.4 and 4.5 show predictions above 1).  32  Summary statistics for the non-linear models (model 2 and model 3-2) are presented in Table 4.7. Model 2 appeared to function best overall in terms of having the highest R square for the PSMEA/ACA habitat type. Model 3-2 was the best predictor for the THPL and TSHE habitat types, with a slightly higher R square than model 2. The AlC and SC were always higher for the THPL and TSHE habitat type than for the PSME habitat type. The probability values (p values) of the Score statistic shows all variables in the models to be significant at a 0.05 probability level (p<0.05).  Table 4.7 Summary statistics for the non-linear models for predicting the probability of plot being stocked. Grouping PSME  Model Variables in model R group model 2 G(x1, x2, x3, sqregt, elev, Mech, Burn) i^0."162.-^ model 3-2 • G(yrsince, slo, elev_c, x3, elev2) 0.133 G(x1, x2, x3, sqregt, elev, BAA, BAA, model 2 0.194 x5, x6, Mech, Burn) G(dsitep1-5, dss1-3, asp_r, baha, slo, model 3-2 0219 elev_c, x3, x4, elev2) 3  2  AlC  SC  227.9 220.4  290.3 243.8  Score p values 0.0001 0.0008  423.2  483.8  0.0001  405.9  466.4  0.0001  2  THPL and TSHE a  x1 is defined a s C O S f A S P r S L O ^ T I M E *  X 4 is defined a s TIME"  x2 is defnied a s  x5 is defined a s COS(ASP)*SLO" BAA  SIN(ASP)*SLO'<**TIME*  x3 SLO»*TIME»  x6 is defined as SIN(ASP)*SLO' BAA  The residuals for model 2 and model 3-2 are shown in Figures 4.6 to 4.9 for each habitat type grouping. The residuals appear to be centered on zero for each of the habitat types and models, with considerable'variation present. The variability decreased as predicted stocking approached 100 percent for model 2. The decrease in variability was less obvious for model 3-2. The residuals are truncated due to the restricted response variable. Residuals appear in five distinct lines due to of the five possible responses of the measured probabilities (0.2,0.4,0.6,0.8,1.0).  •  i.  33  a: -u.t  <r -0.6 Predicted Probability of Stocking  Predicted Probability of Stocking  Figure 4.6 Residuals from model 2 against predicted probability of stocking for habitat types THPL and TSHE.  Figure 4.7 Residuals from model 2 against predicted probability of stocking for habitat type PSMEA/ACA.  f o* I  0.4  |  02  Q.  12  1°  •  02  <0.4  **0.6  •  *«0.8  12  £, S.  co  CO  1 -0.4 c/>  § -0.6  "D CO  04 03 02 Ol 32 0  Predicted Probability of Stocking  •at  •  02  •02 •03 •Q4  aa  06'  OBN  12  •06  r£ -0.8  Figure 4.8 Residuals from model 3-2 plotted against predicted probability of stocking for habitat types THPL and TSHE.  Figure 4.9 Residuals from model 3-2 against its predicted probability for habitat type PSME/VACA.  4.2.2 Number of Stems per Hectare Only the stocked plots were used in the analysis of the number of stems per hectare; the Nelson data set has 809 stocked plots (n = 809). The number of observations per category and the data category definitions are presented in Table 4.8. Categories with more than 20 observations are shaded. Only these categories were used for estimating the number of trees per hectare.  34  Table 4.8 Number of observations of stocked plots and definition of data categories for the number of trees per stocked plot in the Nelson data set. <7 yrs  Douglas-fir habitat series 812 Category 12+ yrs number yrs  Category number  North 9 1 15 East [" T4~7~ 2 v20 South 79 , * 3 16 West | 30 4-, . 36  l  5 1  30 24 15 0  Hemlock and cedar habitat series 812 Category 12+ Category number number yrs  Category number  <7 yrs  Category number  9 10 if 12  38 111 67 32  13 14 .15 16  v  r  s  55 50 20 40  17 18 " 19 20  34 7 10 17  21* 22 23 24  Model 3-1 (SUR) and model 3-2 (NLLS) out-performed both model 1 and model 2 in all data-categories. The coefficients resulting from model 3-1 were compared to those from model 3-2. Both models performed similarly and varied only slightly per data category. Model 3-1 and model 3-2, are comparatively assessed in Table 4.9. Observations in Table 4.9 are based on plotted residuals (Appendix 2). Model 1 did not perform well for estimating B and C in any data category; re-fitting the coefficients of this equation in model 2 improved the predictions in all data categories. Model 3 yielded values closer to the actual cumulative density function than either model 1 or model 2. The linear coefficients in model 3 for predicting the B and C Weibull parameters, re-fitted with the SUR method (model 3-1), only slightly changed the coefficients of the equation for predicting B and did not change the coefficients of the equation for predicting C. Models 3 and 3-2 produced identical, predictions but model 3-2 yielded smaller standard errors of coefficients for the B predictions than model 3 (see Appendix 1). Consequently, the CDFs were almost identical for all data categories. Estimations from model 3-2, in the case of data category 2, produced a slightly better estimate than model 3-1 or model 3. Trees per stocked plot were better estimated in categories 2,4,15,16 and 20 using model 3-2, while categories 3,6,8,9,10,14,17,18, and 21 were better estimated using model 3-1. Coefficients and variables for estimating Weibull parameters B and C with model group 3,3-1,3-2 are presented in Table 4.10.  35  Table 4.9 Performance observations comparing the NLLS (model 3-2) and SUR estimates (model 3-1). Data Category  Model 3-2 Non-linear estimations*  Model 3-1 SUR estimations*  2 Slightly closer to actual CDF** Slightly farther form the CDF than the model 3-2** 3 Over-estimates the number of trees Better performance 4 Better performance Under-estimates the number of trees 6 Over-estimates Even spread of residuals - better performance 8 Spread of residuals not as even Even spread of residuals - better performance 9 Slightly larger spread of residuals Less spread of residuals - better performance 10 Slightly larger spread of residuals Less spread of residuals - better performance 13 Very similar in performance Slightly more even spread of residuals 14 Even spread of residuals - better performance Slightly larger spread of residuals 15 Even spread of residuals - better performance Slightly larger spread of residuals 16 Even spread of residuals - better performance Slightly larger spread of residuals 17 Slightly larger spread of residuals Even spread of residuals - better performance 18 Spread of residuals not as even Even spread of residuals - better performance 19 Spread of residuals not as even Even spread of residuals - better performance 20 Even spread of residuals - better performance Spread of residuals not as even 21 Spread of residuals not as even Even spread of residuals - better performance •Comments are based on only slight differences observables in plotted residuals in Appendix 1 or in Graph 1 in the case of data category 2. "See Figure 4.10.  Results are graphically presented for data category 2 (<7 years since disturbance, east facing, PSME habitat type) in Figure 4.10. CDFs for other categories with more than 20 observations are presented in Appendix 3.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  Number of trees - Classes Note: X-axis are classes, continous lines are for ease of comparison.  Figure 4.10 Cumulative density functions based on Sand Cestimates and actual cumulative density for data category 2, the youngest (<7 years), east facing, PSME habitat type.  36  Table 4.10 Coefficients from model 3-1 and model 3-2 for estimation of Weibull parameters B and C.  Averaged Variables  Model for estimating In B Coefficients SUR Non-linear Model 3-2 Model 3-2  Intercept yrs asp slope elev baha Dsitepl Dsitep2 Dsitep3 Dsitep4 CCF x1 x2 x3 dhab  -9.901183 -0.977419 -0.012110 -0.032616 0.001949 -0.211891 0.183233 -0.026616 0.224771 0.008995 0.070659 -1.219738 -1.387640 6.518250 -0.694315  3  a  x1 defined asCOS(ASP)*SLO x2 defined asSIN(ASP)*SLO x3 defined asSLOniME*  -9.763019127 -0.967698367 -0.006687009 -0.024796689 0.001129709 -0.204691320 0.104535875 0.014233875 -0.047052693 0.002097957 0.071202997 -1.180844480 -1.315628313 6.559046730 -0.651511515  Model for estimating In C Coefficients SUR Non-linear Model 3-2 Model 3-2  1.528704 0.024540 -0.019524 -0.023624 0.276545 -0.143244 -0.118227 -0.622592 0.036588 -0.063547 0.060206  1.477677808 0.021577838 -0.019852181 -0.023708609 0.274641333 -0.129351792 -0.114367373 -0.607056208 -0.013474713 -0.063269804 0.058293902  Models were tested on the remaining categories with less than 20 observations. Categories were paired in terms of site characteristics and the best performing models from the calibration categories were applied to their pairs. Out of seven data categories with less than 20 observations, only two showed CDFs with similar trends; however, none of the estimates performed well. Figure 4.11 through 4.16 show the cumulative density of each of the seven categories, along with an estimate of their cumulative percentages.  • •  CDFcategoryl  CDF estimated for category 1  Figure 4.11 Cumulative density and estimated cumulative density for data category 1.  CDF estimated for category 5  Figure 4.12 Cumulative density and estimated cumulative density for data category 5.  37  12  1  a  CD  §  0.8  D O  <x> ° - 0.6 -  £  0.4 .  1  0.4  02  •  CDF category 7  2  CDF estimated for category 7 •  Figure 4.13 Cumulative density and estimated cumulative density for data category 7.  3  4  CDFcategory11  6 7 8 9 10 CDF estimated fa category 11  Figure 4.14 Cumulative density and estimated cumulative density for category 11.  S  0.8  E 0.4 r  O  -  CCF category 22  •  CDF estimated for category 22  Figure 4.15 Cumulative density and estimated cumulative density for data category 22.  2  3 4 CDF category 24  CDFcategory23  CDF estimated for category 23  Figure 4.16 Cumulative density and estimated cumulative density for data category 23.  5  6 7 CDF estimated for category 24  Figure 4.17 Cumulative density and estimated cumulative density for data category 24.  38  4.2.3 Number of Species As outlined in the analytical methods, six separate probability functions were used estimate the probability of 1,2,3,4,5, and 6+ species. All models were closer to the actual number of species than Model 1. Coefficients for Models 2 and 3 are presented in Table 4.11. Table 4.12 summarizes the statistics for Model 2 and Model 3. Table 4.11 Coefficients for the re-fitted Prognosis Nl equation Models (Model 2) and for the best performing variable subset from a variable selection procedure in a logistic equation (Model 3), for predicting probability of number of species on a plot.  Variables  3  Intercept  1 species Model 2 3 1.4390  -2.0056  2 species Model 2 3 0.7185  3 species Model 2 3  -0.5933  0.3683  -0.5891  4 species Model 2 3 0.7668  x1  0.1697  -0.2103  -0.2920  x2  0.3234  0.0382  -0.1919  Slo  0.4410  -0.2295  1.1166  -0.0317  -0.0102  0.0074  -0.0366  -0.0189  Asp_r -0.0133  Elev*  0.0135  0.0012  bajia"  0.0976  0.0951  Ntrees -0.2843  CC'F  -0.3344 -0.0184  Dsitep Ln_TPSP  -3.8174  -3.5384  6 species Model 2 3 -3.7486  -11.597  -0.0101  Yrsince  Dhab  -1.2727  5 species Model 2 3  -0.007  0.0436 -0.001  0.0564  0.1425  0.0188  0.0037  -0.0584 -0.0255  0.0004  0.0023  -0.042  0.0349  0.2748  0.0085  0.0038 1.4502  0.0125 -0.5248  2.5333  -1.7757  TPSP2  > i • •" ' ' -0.0408 -0.1641 'Elevation for the Prognosismbdel form was in feet, while it was in meters for the BW model.' "Basal area in the Prognosis model-form was in square feet per acre while it was in square meters per hectare in the BW model. x1 is defined as COS(ASP)'SLO x2 is defined as SIN(ASP)*SLO ;  a  The statistics indicate that Model 2 is performing slightly better than Model 3 for the probability of one ;  species, two and three species. Differences between Model 2 and Model 3 were less evident for the probability of four and five species, but Model 2 still seemed to perform better slightly better than Model 3. The probability of six species seemed to be better predicted with Model 3. The significance of the variables in both model forms were different; all variables in the variable selection set of equations were significant (a = 0.05) while variables in the Prognosis model form were not all significant.  39  Table 4.12 Summary statistics for predicting the number of species per plot with the Prognosis Nl fitted equation (Model 2) and a different subset of site variables (Model 3). Predicted Model form R2 -2*log L AIC SC probability Model 2 0.37 620.161 632.161 660.336 1 species Model 3 0.0454 956.317 966.317 989.796 Model 2 0.041 922.636 938.636 976.202 2 species Model 3 0.02 940.155 950.155 973.634 Model 2 0.243 819.836 837.836 880.098 3 species Model 3 0.0048 835.891 839.891 849.282 Model 2 0.2027 448.802 464.802 502.369 4 species Model 3 0.0152 619.678 625.678 639.765 Model 2 0.0127 296.308 302.308 316.395 5 species Model 3 0.0056 302.029 306.029 315.421 Model 2 0.0015 186.267 190.267 199.659 6 species Model 3 0.0506 145.531 155.531 179.010  reScore  3  0.0001 0.0001 0.0037 0.0030 0.0078 0.0488 0.0001 0.0018 0.0001 0.0254 0.1194 0.001  Variables for each models are presented in Table 4.11 with model coefficients.  Figure 4.18 shows the averaged frequencies of number of species for all three model groups after 1000 iterations using a pseudo-random number, compared to the actual frequency distribution of the number of species per plot. Model 1 had the poorest performance in terms of approaching the actual frequency of number of species per plot. Despite the results in Table 4.11, Figure 4.18 shows that the closest frequencies to the actual frequencies were generated by Model 2.  Figure 4.18 Comparison of frequencies of the number of species per plot estimated with three modeling approaches to the actual number of species per plot.  40  4.2.4 Species Composition Figures 4.19 and 4.20 compare the results of the Prognosis Nl equations (Model 1), the re-fitted Prognosis Nl equations (Model 2), and a logistic equation using site variables (Model 3) for estimation of probabilities for each type of regeneration (advanced, subsequent and excess) of Douglas-fir and western redcedar, respectively. These species were chosen as they had the highest number of observations of all the tree species present. They also have different shade tolerances and were expected to display different regeneration patterns. None of the modeling approaches produced distributions similar to the actual distributions. Model 2 and Model 3 presented the same trends in both cases. The cumulated excess regeneration values from Models 2 and 3 were similar to the actual cumulative values for both species. The cumulated values for the Douglas-fir advanced regeneration resulting from Model 3 were close to the actual values. Douglas-fir subsequent and redcedar advanced and subsequent values were not close to the actual cumulative values for any equations. In all cases, Model 1 produced cumulative values furthest from the actual cumulative values. Cumulative values of actual regeneration and predictions from all three model groups are compared in Table 4.13 for all species.  Figure 4.19 Probabilities of advanced, subsequent, and excess regeneration of Douglas-fir.  41  Figure 4.20 Probabilities of advanced, subsequent, and excess regeneration for western redcedar.  Table 4.13 Actual and predicted cumulative values for advanced, subsequent and excess regeneration. Species Regen. Actual Prognosis Nl Re-fitted Prognosis Logistic equation Type regeneration equation Nl equation with other variables Model 1 Model 2 Model 3 Advanced 0.02381 0.36803 0.167373 0 Bi Subsequent 0.190476 0.63197 0.832627 1 0.785714 Excess 0.379231 0.636923 0.779231  Bg  Advanced Subsequent Excess  0.179245 0.132075 0.669811  0.484405 0.515595 0.022857  0.497904 0.502096 0.431905  0.423394 0.576606 0.421905  Cw  Advanced Subsequent Excess  0.252022 0.066038 0.652291  0.634292 0.365708 0.164255  0.563271 0.436729 0.629255  0.539769 0.460231 0.630851  Fd  Advanced Subsequent Excess  0.086116 0.240773 0.667838  0.301563 0.698437 0.106757  0.028613 0.971387 0.628468  0.077668 0.922332 0.609099  Hw  Advanced Subsequent Excess  0.071357 0.064322 0.81407  0.530483 0.469517 0.201169  0.342021 0.657979 0.636234  0.34266 0.65734 0.642597  Lw  Advanced Subsequent Excess  0.030303 0.353535 0.616162  0.227875 0.772125 0.010345  0.081451 0.918549 0.496207  0 1 0.496552  PI  Advanced Subsequent Excess  0.033333 0.361111 0.511111  0.004693 0.995307 0.005556  0.003315 0.996685 0.329556  0 1 0.432889  Pw  Advanced Subsequent Excess  0.054187 0.197044 0.650246  0.004652 0.995348 0.04463  0.092098 0.907902 0.646111  0.032465 0.967535 0.595  Sx  Advanced Subsequent Excess  0.047619 0.166667 0.571429  0.207931 0.792069 0.0165  0.446281 0.553719 0.516  0.025988 0.974012 0.519  42  The coefficients for Model group 2 and Model group 3 are presented in Appendix 4. The maximum likelihood estimates for some of the Prognosis Nl re-fitted equations (Model group 2) presented complete or quasi-complete separation of data point (equations for grand fir, advanced subalpine fir, hemlock, advanced white pine, and spruce). Estimates using Model group 3 presented the same problem for the advanced regeneration of grand fir. Even a one-variable model, with any site-variable or transformation of site variable, resulted in separation of the grand fir data. Table 4.13 presents all the estimates for Model group 2, regardless of data separation. A two-variable model was used for grand fir advanced regeneration that resulted in quasi-complete separation of data points.  The non-conclusive analysis of maximum likelihood estimates for certain data categories for both Model group 2 and Model group 3 could result from the low number of observations for regeneration of certain species. For example, grand fir had only 13 observations. A backward stepwise variable selection procedure with 11 variables results in an over-specified model; hence, non-conclusive maximum likelihood estimates.  4.2.5 Germination Delays Predictions of germination delay using the Model 2 were consistently closer to the actual cumulative density of germination delay than the cumulative density of germination delay estimated using Model 1. Figure 4.21 shows the actual cumulative density of germination delay and the predictions for advanced Douglas-fir regeneration using Model 1 and Model 2. Advanced Douglas-fir was chosen because it had cumulative delay predictions using Model 1 closest to the actual cumulative distribution for all regeneration categories (advanced or subsequent). The NLLS estimates of Weibull parameters (Model 2), for each species with more than 20 observations, are presented in Appendix 5. Graphs comparing actual cumulative densities, cumulative densities predicted with Model 1, and cumulative densities predicted with Model 2, for all species, are presented in Appendix 6.  43  1.2  =•  0.6 0.4  1  A  0.2  10  15  20  30  25  35  Germination-delay (years) •  ACTUAL  A Model 2  Model 1  Figure 4.21 Cumulative density distribution of actual, Prognosis Nl equation (Model 1), and re-fitted Prognosis Nl equations (Model 2)estimates of germination delay for Douglas-fir regeneration. Cumulative density distributions of germination delay estimates for data categories with above 20 observations (11 categories) were all very close to the actual density distribution using Model 2. These include germination delay for advanced and subsequent western redcedar, advanced and subsequent Douglas-fir, advanced and subsequent hemlock, subsequent subalpine fir, subsequent larch, subsequent lodgepole pine, subsequent white pine, and subsequent spruce.  4.2.6 Seedling Height Some tree species for which height prediction equations were developed in Prognosis Nl had only a few observations in the Nelson data set. Despite this lack of data, the Prognosis Nl equations (Model 1) were applied and re-fitted (Model 2) for each species for subsequent and advanced regeneration. Model 3 was only applied to categories with more than 10 observations (13 out of 18 categories). The number of observations per species and regeneration type, are presented in Table 4.14.  44  Table 4.14 Number of observations of height for advanced and subsequent regeneration by species. Number of Observations  Species Bg BI Cw Fd Hw Lw PI Pw Sx  Advanced 1 10 80 31 50 2 6 6 6  Subsequent 16 24 83 140 76 36 72 69 26  Figure 4.22 shows the residuals (actual - predicted) for height predictions against the actual heights of advanced western redcedar; predictions were made using Model 1,2, and 3. Graphs comparing residuals for all other species are presented in Appendix 7.  600  40C  200  U 5C"  100  150  200  250  300  350  400  450  SfO  -200  I  -400  *  -600  -aoo -1000 Actual Heights  •  Model 1  •  Model 2  Figure 4.22 Comparison of height prediction residuals from Model 1 (Prognosis Nl equation), Model 2 (re-fitted Prognosis Nl equations) and Model 3 (linear equation with site variables) of advanced western redcedar. Height predictions from Model 2 for data categories with less then 10 observations were all closer to the actual heights than those from Model 1. These categories were advanced regeneration of grand fir, larch, white pine, lodgepole pine, and spruce. Model 2 for advanced grand fir, larch, and white pine did not present unique solutions for least squares estimates. Of the categories with above 10 observations, only Model 2 for subalpine fir advanced regeneration did not present unique solutions to OLS estimates. Height predictions for advanced and subsequent subalpine fir, advanced hemlock,  45  subsequent lodgepole pine, and subsequent larch were closer to the actual heights when Model 3 was used. Height predictions using Model 3 were only slightly closer to the actual heights than those from Model 2 for advanced western redcedar, advanced Douglas-fir, subsequent grand fir, subsequent western redcedar, Douglas-fir, hemlock, and spruce. The coefficients and analysis of variance for Model 2 and Model 3 are given in Appendix 8.  Prediction models for height of excess regeneration were compared based on the number of trees in each height class. Figure 4.23 shows a comparison of the number of trees per height class for excess regeneration for western redcedar. Graphs for other species are given in Appendix 9. Coefficients for height of excess regeneration equations are presented in Appendix 10.  Height Classes  Figure 4.23 Comparison of models of height of excess regeneration for western redcedar.  Figure 4.23 indicates that Model 2 (re-fitted Prognosis Nl equation) is closer to the actual distribution of heights than Model 3 (linear model) or 1 (Prognosis Nl equation). Model 1 had the poorest performance of all models for all 9 species. The western redcedar, Douglas-fir, hemlock, larch, and white pine distributions using Model 2 were all closer to the actual height distribution than Model 3. Model 3 predictions were closer to the actual heights for grand fir, subalpine fir, lodgepole pine, and spruce.  46  4.3 Small Tree Height Growth Variables and statistics presented in Table 4.15, and Table 4.16 are in metres unless otherwise indicated. Natural logarithm units are indicated with an asterisk. Table 4.15 summarizes the statistics for Model 1 and Model 2 predictions. Table 4.15 Summary of predictions of height growth of small trees by species and tolerance groups using Model 1 (Prognosis Nl equation) and Model 2 (re-fitted Prognosis Nl equation) for species with more than 25 observations and species by tolerance groups. Species or Group n Model R* I SEE (m) 2  Cw  93  Fd  44  Hw  55  PI  38  Tolerant  165  Intermediate  57  Intolerant  44  "Values are in natural logarithmic units (In m).  Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2  N/A 0.320* N/A 0.526* N/A 0.211* N/A 0.493* N/A 0.266* N/A 0.531* N/A 0.462*  2  -1.194 0.241 -0.204 0.372 -0.207 0.226 -0.912 0.429 -0.202 0.193 -0.680 0.775 -0.756 0.834  0.450 0.265 1.006 0.727 0.586 0.469 1.116 0.610 0.480 0.393 0.967 0.414 1.093 0.336  The negative I values associated with all of the species and tolerance groups for the equations with 2  Model 1 are a strong indicator of poor performance (Table 4.15). Negative I square values result from the variance of the residuals being larger than the variance of the total regression for that equation. Graphs of the residuals by species and tolerance group (Appendix 11) for Model 1, indicate that 5-year height growth was overestimated on average. Refitting these equations to the data (Model 2) improved equation performance. • Graphs of the residuals by habitat series (Appendix 11) for Model 2 show that 5-year height growth estimates were no longer biased. SEE from Model 2 were lower and the I values 2  higher for Model 2. With the exception of western redcedar, which had the most height growth observations of any single species, the standard errors of the re-fitted equations for the tolerance groups were lower than those of the single species. This may be due, at least in part, to the larger number of observations for each of the tolerance groups. However, the fits obtained remained moderate at best, with R square values ranging from slightly above 0.2 to slightly above 0.5.  Summary statistics and variables used in Models 3-1 and 3-2 are presented in Table 4.16. Models 3-1 and 3-2 have different dependent variables (In(HTG) and HTG) and different predictor variables for  47  each tolerance class. For the tolerant and intermediate groups, the fit obtained was slightly better than that of Model 3, based on lower standard errors of estimate. For the intolerant group, the fit was slightly poorer than that obtained using Model 2. Prognosis Nl does not use tree age nor years since disturbance of site as predictor variables. When these were removed (i.e., Prognosis Nl variables only were used) predictions were not as good for all three tolerance groups. Residuals for models using the natural logarithm of height growth as a dependent variable are presented in Appendix 12. Coefficients for the models in Table 4.16 are presented in Appendix 13. Height growth used as a dependent variable yielded better values for I or R for tolerant and intolerant species, while using the natural 2  2  logarithm of height growth as a dependent variable had a higher I value intermediate shade tolerance 2  group. Table 4.16 Summary of the resulting height growth model form and fit from Model 3. Group  n  Model form Prognosis variables  Ln(HTG) = F(dspp1-2-3-4, dbh, ln(ht), dss1-2-3-4, elev., slope, ccf) HGT = F(dspp1 -2-3-4, dbh, ln(ht), dss1-2-3-4, aspect, elev., slope, ccf, sin_asp) Tolerant  165  R*  I  SEE (m)  0.491*  -0.357  0.522  2  2  0.502  0.317  Site variables  Ln(HTG) = F(dspp1-2-3-4, ln(ht), age, dss1-2-3-4, elev., slope, ccf) HTG = F(dspp1-2-3-4, ln(ht), age, time, dss1-2-3, slope, ccf, cos_asp)  0.622*  0.582  0.617  0.290 0.280  Prognosis variables  Intermediate  57  Ln(HTG) = F(dspp1, dhb, dss1-2-3, aspect, slope, ccf, cos_asp, sin_asp, x1, x2) HTG = F(dspp1, dbh, aspect, ccf, sin_asp, x2)  0.702*  -16.4975  0.601  Site variables  0.880* Ln(HTG) = F(dbh, age, time, ccf) HTG = F(dbh, age, time, aspect, elev., ccf, cos_asp, 0.821 sin_asp)  3.57907 0.551  0.861  0.319 0.377  Prognosis variables ••  Intolerant  44  Ln(HTG) = F(dbh, dss1-2-3, aspect, slope, ccf, cos_asp, sin_asp, BALM-100, x2) HTG = F( dbh, dss1 -2-3, aspect, slope, ccf, cos_asp, sin_asp, BALM-100, x2)  0.790*  0.654  0.499  0 7 Q 4  0 4 8 2  Site variables  Ln(HGT) = F(dbh, age, dss1-2-3, cos_asp, sin_asp, x2) HGT = F(dbh, age, dss1-2-3, cos_asp, sin_asp, ln(ht), x2)  'Values are in natural logarithm of metres units.  0.858* 0 7 6 9  ;  0.741  0.456 '  0 4 2  48  i  A substantial proportion of the height growth data was obtained from advanced regeneration. Table 4.17 outlines the proportion of advanced regeneration by species and species groupings.  Table 4.17 Proportion of advance regeneration in the small tree height growth observations. Species or Tolerance Group Cw Fd Hw PI Tolerant Intermediate Intolerant  n 93 44 55 38 165 57 44  Proportion of Advanced Regeneration 0.989 0.568 0.964 0.263 0.952 0.526 0.295  49  Chapter 5: Discussion The regeneration components and the small tree height growth equation of Prognosis Nl were tested against the Nelson data. In all cases, the refitted equations or other tested model forms performed better than the Prognosis Nl equations. Performances varied with the equation. Identifiable errors and biases in the calibration efforts stem from two major sources: the nature of Prognosis Nl itself and the Nelson data set. As previously stated, Prognosis Nl was developed in the United States and uses a site classification system to represent site productivity. The true factors determining site productivity are poorly understood and very difficult to measure (Monserud 1987). The use of a classification system is just one method of estimating site quality. Like all other methods, it has advantages and disadvantages (Carmean 1975, Klinka 1990, Thrower and Willing 1996). To date, no method perfectly estimates site productivity. They all may transmit errors to the growth models that use them. Further, the equations presented here use correspondences between BC site series and Idaho habitat types (Robinson 1997). Despite the similarities in the classification systems, the criteria for site classifications in both systems result in different sites (i.e., no site referred to as a specific habitat series corresponds exactly with a site series in the BEC). For example, the ICHmw2 site series of the BEC system ranges from dry to wet in the following order: 02,03,04,01,05,06, 07 and 08. The mesic site series were targeted in this project: 03 to 05. These correspond to three different habitat types, which in Prognosis Nl, are grouped in two habitat series. This separates the Nelson into two data sets, one per habitat series., One habitat series (the Douglas-fir habitat series) contains one sampled site series (ICHmw2 03), while the other (the Cedar-Hemlock habitat series) encompasses all other sampled site series (ICHmw2 04,01,05,06). The site classification system is used in Prognosis Nl to represent site productivity. The separation employed does not adequately represent the variance in site productivity between site series. The variability between site series 03 and site series 04 is not greater than the variability between site series 04 and 01 nor greater than the difference between site series 05 and site series 06. Introducing dummy variables into the Prognosis Nl equations to represent site series might have captured some of the residual variability. However, site series and habitat series correspondence needs to be perfected if site productivity is to be represented by a site classification system. The Nelson data set is much smaller and slightly different than the data set used in calibrating the original Prognosis Nl equations. One hundred and eighty six plots were established over the data  50  collection phase. The data set used for developing Prognosis Nl was comprised of large inventory databases. Height of advanced regeneration equations, for example, were developed with 5,649 observations of advanced regeneration seedlings, while the Nelson database had only 192. Some data categories used in the Prognosis Nl model had more predictor variables than the number of observations in the corresponding data category in the Nelson data set. This lack of data sometimes caused non-conclusive least squares or maximum likelihood estimates or non-robust models. Plots were selected from sites that had been disturbed in the last two to 20 years. Most of the sites were partially cut. These sites do not represent all possibilities of regeneration for the species modelled. Partially cut sites imply a certain amount of shade; regeneration of shade intolerant species was probably not well represented. Further, management practices might have contributed to changing the probability of establishment of some species. For example, larch regeneration is purposely not encouraged in areas infected with larch mistletoe. Partial cutting is a relatively recent method of harvesting in the Nelson region; 53% of plots had been disturbed in the last seven years. This might result in a low representation of tolerant species with slow ingress. Partially cut areas are not site-prepared because of possible damage to the residual stems when using common site preparation methods. Representation of species that require disturbed seedbeds for germination is therefore low. Partial cutting systems are harder to implement on steep slopes, consequently most of the plots were on slopes between 20% and 60% and were in a mid-slope topographic position. This plot distribution did not include steep drier slopes or wetter site sites. The species that would have captured growing space on these sites were more than likely not well represented in the data set. Similarly, the range of small trees measured does not represent the complete range of conditions. Individual tree growth varies with the species mixture and is influenced by the structure of the residual stand. For example, seven-year-old sites do not necessarily represent the mix of small trees on a 20year-old site.  ,  Hardwoods are' an important part of stand dynamics and may even have a more important role in mixed species stands than in stands dominated by one or a few tree species (Clinton et al. 1994, Ishiwaka and Ito 1989, Kneeshaw and Bergeron 1996, Monserud 1987, Oliver and Larson 1996). However, hardwoods were not modelled in Prognosis Nl or Prognosis . This decision was mostly founded on 60  the fact that hardwoods are not currently considered as commercial species in most of British Columbia and Prognosis is being developed for timber supply purposes. Balsam fir (Abies 80  balsamea)  was  51  considered a weed species in eastern north America as little as 30 years ago and is now one of the leading commercial species in eastern Canada; hardwoods in western Canada might become important as commercial species with the diminishing supply in old growth timber. Further, adequate representation of how multi-cohort and/or mixed species stands develop cannot be complete or accurate without all the factors, including hardwood trees. Despite these general concerns, the equations resulting from this calibration process provide better estimates of small tree height growth and regeneration in multi-cohort and mixed species stands in the southern interior of BC than were previously available. Equations for both components are examined individually in the following text. The regeneration modelling approach of Prognosis Nl predicts the probability of stocking, the number of trees per plot, the species composition, the probability of regeneration types (advanced, subsequent, and excess), the germination delay for subsequent regeneration, the age of advanced regeneration, and the height of seedlings. In this study, data were modelled by habitat series, or by a combination of habitat series and species or site characteristics, similar to the approach used in Prognosis Nl. Regeneration was very variable. Many factors drive regeneration; each of the factors are themselves variable (Clinton  etal.  1994, Kneeshaw and Bergeron 1996, Oliver and Larson 1996). Only a very  large and wide-ranging data set can capture that variability and produce an accurate regeneration model. The major problem for all regeneration equations presented here is the lack of data. Nevertheless, some of the regeneration equations performed adequately. The probability of stocking was best predicted using a logistic equation with a subset of site variables resulting from a maximum likelihood variable selection procedure, for both habitat type groupings. Although linear model forms were tested, the actual distribution of a binary response for the probability of stocking is expected to follow more closely a logistic form than a linear form. The best logistic equation for predicting stocking differed for each habitat type grouping. The probability of stocking for habitat type,PSME, which only includes site series 03 of the BEC system, was better predicted with the re-fitted Prognosis Nl equation. The probability of stocking for the THPL/TSHE habitat types, which include all other site series in the Nelson data set, was better predicted with the logistic equation using a subset of site variables resulting from a variable selection procedure. These models were the best of the calibrated models, and did not seem to include any biases. However, the better fit of the Prognosis Nl equation for site series 03 might just be a reflection of better correspondence between a PSME habitat type and a site series 03 than THPL/THSE habitat types and site series 04, 01, 05, and 06.  52  The number of trees per stocked plot was modelled using a cumulative density distribution (Weibull function). The Prognosis Nl equation had the poorest performance in terms of approaching the actual frequency of number of species per plot. Weibull parameters, determined with NLLS, were predicted for data categories that had more than 20 observations using linear models (16 out of 24 possible data categories). The best performing model for the B and C Weibull parameters resulted from a linear model using a subset of site variables from a variable selection procedure. These models were then re-fitted using two approaches: a joint-generalized least squares approach and by NLLS within the Weibull function. When compared to the actual cumulative density of trees per stocked plot, both models were acceptable for all 16 data categories tested. However, none of the models performed well for categories with less than 20 observations. Extrapolation of these models to data-categories with less than 20 observations did not produce cumulative density functions comparable to the actual cumulative density functions in any of the seven data categories with less than 20 observations. Although this might be a result of insufficient observations or non-representative sampling, extrapolation of these refitted models have not proven suitable other than for data categories used in calibration. The number of species was determined with separate probability equations for one, two, three, four, five, and six or more species. Separately predicting these probabilities might not reflect the real probability of a certain number of species occurring on a plot. The probability of two species occurring on one plot is dependent on one species occurring, the probability of three species is dependent on the probability on two, and so on. Other modelling approaches, like discriminant analysis, might take into account the multivariate context in which the number of species per plot occurs. Consequently they may better reflect the real probability of a particular number of species occurring. Logistic equations, with a subset of site variables from a maximum likelihood variable selection process, estimated the number of species very closely to the actual cumulative distribution of the number of species. The performance of the refitted Prognosis Nl equations was also very close to the actual cumulative distribution. However, this was not evident when the statistics were examined. For predicting the probabilities of all three types of regeneration, the refitted Prognosis Nl equation and the logistic equation using site variables showed similar trends; both performed better than the Prognosis Nl equation. However, none of the equations produced distributions similar to the actual distribution, especially with respect to subsequent regeneration, where all the approaches overestimated the probability. The situations where predicted cumulative values approached the actual cumulative values were for the probability of advanced regeneration of Fd/Pw/Sx, the probability of  53  excess regeneration of Bg/Hw/BI/Lw/PI/Sx predicted with a logistic equation using site variables, and the probability of advanced regeneration of Lw and probability of excess regeneration of Cw/Pw/Fd predicted with the re-fitted Prognosis Nl logistic equation. This might be the result of insufficient observations or sampling. Advanced and excess regeneration might also simply be better represented in stands disturbed within the last two to seven years than subsequent regeneration. Predictions of germination delay for subsequent regeneration and age of advanced regeneration were made using a cumulative density function (Weibull function) for data categories with more than 20 observations. For these 11 categories, the cumulative density for the germination delay from the refitted Prognosis Nl functions were all close to the actual density distribution. The best height predictions for most advanced and subsequent regeneration resulted from a linear model that used a subset of site variables resulting from a variable selection procedure. Height of excess regeneration was estimated equally well by the refitted Weibull function used in Prognosis Nl and the linear equation with a subset of site variables, depending on the species. The re-fitted Prognosis Nl equation for predicting height of advanced and subsequent regeneration did not present unique solutions for the ordinary least squares estimates when there were insufficient data for the number of variables in the Prognosis Nl model form. These models should be re-fitted with more data and compared to the height estimates from the linear equation with a subset of site variables. The Weibull functions, used in Prognosis Nl for predicting of height of excess regeneration, did not result in adequate predictions for estimating of heights of excess regeneration for the Nelson sites. Weibull functions fitted to the data performed better for height estimates of excess regeneration of western redcedar, Douglas-fir, hemlock, larch, and white pine. The linear model resulting from the variable selection process is recommended for estimating height of excess regeneration of grand fir, subalpine fir, and spruce. The early stages of height growth, when many species occur on a plot, might be more accurately represented in a multivariate context, since all species present have an affect on the height growth of the neighbouring species. The height growth of all species could be predicted simultaneously, in a multivariate approach, from site factors. Versions of Prognosis used in other areas of the U.S. have proven effective for modelling regeneration (Stage 1998, pers. comm.) . Different model forms might also improve predictions of regeneration and 6  early growth in multi-cohort, mixed species stands (Clinton  etal.  1994, Frohlich and Quenan 1995,  Ishiwakaand Ito 1989). 6  Dr. Albert Stage, Retired Scientist, USDA For. Serv., Interm. Res. Sta., Moscow, ID. 54  The small tree height growth equation in Prognosis Nl is a linear prediction of the natural logarithm of height growth. In the Nelson data, only four species had more than 25 observations of small tree height growth. Therefore, small trees were grouped by shade tolerance levels to provide sufficient data for model calibration. For the tolerant and intermediate tolerance groups, the best performing height growth equation for small trees was a linear model using a subset of site variables from an OLS variable selection procedure. The re-fit Prognosis Nl equation performed best for the intolerant group. In the small tree height growth equations for the Prognosis Nl equation, tree age and time since disturbance are not used for height predictions since this information was not available in the inventory data used for model building. Linear equations using both the same set of variables as Prognosis Nl, and a set of variables including tree age and time since disturbance, were compared. Even though the resulting model using age and time since disturbance performed better the one that did not include these variables, Prognosis might be accessible to a larger audience if the equation without these 80  variables was used. These variables are not easily measured in the field and most existing data sets will not contain this information. Improvements to the fit of the small tree height growth equations could possibly be obtained if more data were available. Different forms of prediction equations could also yield better height predictions. The variable selection procedure did this, but only for linear equation forms. A possible confounding factor specific to the small tree height growth data is that a substantial proportion of the height growth observations in the test data was obtained from advanced regeneration. The data set for the small tree height growth equation in Prognosis Nl consisted mainly of height growth of trees that regenerated following a disturbance (Stage 1998, pers. comm.). Although predictions were poor, the re-fit Prognosis Nl equation performed the best for the shade intolerant group, where 70% of the trees regenerated subsequent to disturbance. This suggests that the Prognosis Nl model form applied to other species groupings might still produce good height estimates for trees regenerated post-disturbance. It may also be beneficial to estimate the height growth of small tree advanced regeneration separately from that of trees that regenerated post-disturbance. Not enough data from trees that regenerated subsequent to disturbance were available to examine this possibility. Using height growth as a dependent variable seemed to produce a better model than using the natural logarithm of height growth for tolerant and intolerant species groupings, while the natural logarithm of height growth seemed to produce a better model for the intermediate tolerance species. This can be attributed to the different height growth patterns of different species even in the early stages of height growth. Most  55  height growth models use the natural logarithm of height as a dependent variable (Goesler and Hasenauer 1997, Froese 1997). However, for both the small tree height growth equations and the height of regeneration equations, only a portion of the total growth of tree height is being modelled. A number of non-linear height growth functions are described in the literature (Favrichon 1998, Gosler and Hasenauer 1997, Huang and Titus 1994 and 1992) and some of these may prove more applicable to these data than the equations that were tested. Existing models, like Prognosis Nl, are important tools as they present a quantified view of how forests or stands grow and change. However, models are rarely mobile; they produce relatively accurate scenarios only for the area in which they were developed and within the range of data used for calibration. The more growth factors encompassed by variables within the model, the more plausible the application of the model to another geographic area. If the target area is very similar to the original area, the model might be applicable in its original form. However, as the case was in this paper, forest and stands are variable and rarely have the exact same growth pattern in different geographic areas. The refitting of a model then becomes the best option. However, growth patterns, species dynamics, and sometimes management approaches need to be similar between stands for which the model was originally developed and the stands that the model is being calibrated for. If this similarity does not exist, perhaps a completely different modelling approach would be more suitable. Like any model, Prognosis should be used with caution; it is only a model and should be used for 80  guiding decisions, not of making decisions. The context in which the.model is calibrated and the logical foundation of the model need to be well understood by users. The data used for calibration also indicate the limitations of the model. The Prognosis Nl equations presented in this thesis were calibrated to the Nelson data. These equations were not tested on independent data. More testing is required to validate the Prognosis equations. The level of testing required is dependent on how 80  comfortable the users are with the state of the Prognosis equations. Goudie (1997) states that a 80  model is never validated; it is only tested to a level at which users are comfortable with. Validating a model would imply that it is a perfect representation of reality, and models are not. Monserud (1987) did not view modelling regeneration as a conceptually difficult problem. The, processes leading to the establishment of a seedling are fairly well understood. The main hurdle is that a good regeneration or early growth model would require an excellent and extensive experimental design that results in sampling the full range of important factors.. In irregular stands, this simple statement becomes a difficult achievement. Various species in these stands usually respond differently to a given set of environmental factors. Species-specific models that consider both time and space of  56  individual trees would be ideal. However, the problem of site productivity estimation remains. The solution might reside in changing the way we see the forest. In reality, each tree has its own growth curve dependent on species, site, stand structure, and where the tree sits in terms of that stand's particular structure. It comes down to finding a suitable balance for each stand, within each landscape, for each region, based on each tree as an individual production unit. Ecosystems are dynamic through these individual production units and through disturbances. Humans are part of this ecosystem, contributing another source of change. Thus, regeneration modeling is difficult to do well in practice despite its simplicity in concept. Processes like the calibration of Prognosis Nl for the southern interior of BC permits a better understanding of the dynamics of that ecosystem. Through data collection and observation of the growth of individual trees in various conditions in this ecosystem, more is learned about changes through time and space of these stands. Prognosis Nl was built on many years of experience, involving a pooling of expert and professional minds. It has proven effective for modelling irregular stands. The logic it uses is an important source of information for building other models elsewhere. However, geographically displacing a model may introduce many errors and biases. Perhaps using the same logical paths, but re-developing equations based on systems and data available to the targeted area, would avoid many of these errors and biases. This approach would yield a model that encounters local problems to which local solutions could possibly be applied. Another approach is to apply the site classification system used in Prognosis Nl (Cooper etal. 1975, Pfister etal. 1977) as the site classification in the targeted area, so that data are collected in units of land that are comparable to those for which the model was developed. Prognosis Nl was developed with a high dependency on the Idaho and Montana site classification. Successfully transferring that dependency to other geographical areas classified on different variables, especially when site productivity estimates within the model depend on land classification, is difficult.  57  C h a p t e r 6:  Conclusion  The main objective of this thesis was to calibrate the basic equations of the early height growth and regeneration components of Prognosis Nl for the southern interior of BC. In summary, all the small tree height growth equations performed poorly for the individual tree species modelled (Cw, Fd, Hw, PI); pooling of species in tolerant groups improved the performance of the re-fitted model. The best performing model was the re-fitted Prognosis Nl equation for the intolerant species grouping; this is also the group were most trees were regenerated subsequent to disturbance. Advanced regeneration did not seem to respond as the Prognosis model implied; their height growth was better predicted with an equations based on selected variables.  The regeneration equations need to be validated with more data. The plot distribution does not present the entire range of site series or species mixtures. Some of the calibrated equations presented in this thesis are appropriate for use in Prognosis and others are not. Table 6.1 briefly outlines equations 80  that were assessed as best performing based on this research, and comments on the use of these equations in Prognosis . 80  Table 6.1 Summary of equations and recommendations for use in Prognosis . 80  Prognosis Component  Data grouping Tolerant Small tree height growth Intermediate Intolerant Regeneration Probability of stocking 03 site series other site series No. of trees per plot  Best performing Use in Prognosis model Variable selection - OLS Acceptable within the range of sampled sites and trees Variable selection - OLS Acceptable within the range of sampled sites and trees Prognosis refitted Acceptable within the targeted population 80  a  Prognosis refitted Acceptable within the targeted population Variable selection logistic model - maximum likelihood Acceptable within the targeted population  groupings > , Weibull parameters with 20 obs. SUR estimations or NLLS estimates within the Weibull Acceptable within the range of data groupings calibrated Others No model performed adequately No model tested was acceptable b  No. of species per plot  Prob. of 1 to Prob. of 6  Variable selection logistic model - maximum likelihood Acceptable within the range of sampled sites  58  Table 6.1 Summary of equations and recommendations for use in Prognosis (continued). 60  Prognosis Component Probability of advanced regeneration  Data grouping Fd/Pw/Sx  Lw Others Probability of excess regeneration  Bg/Hw/BI/Lw/ Pl/Sx Cw/Pw/Fd  Probability of subsequent regeneration Germination delay  All species  Best performing model Variable selection logistic model - maximum likelihood refitted Prognosis No model performed adequately  Height of subsequent regeneration  Height of excess regeneration  a  b c  60  Acceptable within the range of sampled sites Acceptable within the range of sampled sites No model tested was acceptable  Variable selection logistic model - maximum likelihood refitted Prognosis  Acceptable within the range of sampled sites Acceptable within the range of sampled sites  No model performed adequately  No model tested was acceptable  11 combinations of Sps/regen. type  refitted Prognosis (Weibull) - NLLS  Bl/Hw/Cw/Fd  Variable selection - OLS  Others  No model performed adequately  Acceptable within the range of sampled sites  c  Height of advanced regeneration  Use in Prognosis  Acceptable within the range of sampled sites  Bl/Hw/Cw/Fd PI/Lw/BI/Sx Others Cw/Fd/Hw/Lw Pw Bg/BI/Sx Others  No model tested was acceptable  Variable selection - OLS Acceptable within the range of sampled sites No model performed adequately refitted Prognosis (Weibull) - NLLS Variable selection - OLS No model performed adequately  No model tested was acceptable Acceptable within the range of sampled sites Acceptable within the range of sampled sites No model tested was acceptable  Extrapolation to the targeted population are based on the assumptions that sites of with the same ecosystem classification designation and site characteristics will respond in the same way. See table 4.8, shade categories See p44.  The regeneration and small tree height growth approaches incorporated in Prognosis Nl appear to be applicable to sites in the ICHmw2 in the southern interior of BC. However, these approaches will need to be tested (and possibly recalibrated) using a much broader data base before Prognosis will be 60  ready for use on all southern interior sites.  59  The same caution applies to Prognosis as to any model: they are only models of reality and carry 80  biases. Nevertheless, once it is available, Prognosis  80  should improve predictions for multi-cohort  and/or mixed species stands in the southern interior of BC. Prior to the calibration efforts for Prognosis , no quantitative tools were in place to aid silviculturists in their predictions in these irregular 80  stands. More data and localization of some of the equations should remedy most errors and biases discussed in this text. 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CACTOS user's guide: The California Conifer Timber Output Simulator. Version 3.3. University of California Division of Agriculture and Natural Resources, Berkeley. 9pp.  64  Wykoff, W. R. 1986. Supplement to the user's guide for the stand Prognosis model _ Version 5.0. Gen. Tech. Rep. INT - 208. USDA Forest Service. Intermt Res. Stn, Ogden, UT. 36pp. Wykoff, W. R., N. L. Crookston, and A. R. Stage. 1982. User's guide to the Stand Prognosis Model. Gen. Tech. Rep. INT -133. USDA Forest Service. Intermt For. and Range Exp. Stn, Ogden, Utah. 86pp. Zhang, S., H. E. Burkart, and R. L. Amateis. 1996. Modeling juvenile loblolly pine plantations. Forest Ecology and Management 89:157-172.  65  APPENDIX 1: Number of trees per stocked plot. Using Ordinary Least Squares (OLS) for linear estimation of scale and shape parameters of the Weibull function.  66  o o o o o o o o o o o o o o o O O O O O O O O O O J O O O O O o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  c  o  o  o  o  o  o  o  o  o  o  o  o  o  CO CO 0) O) » - c \ i i n o ) C M T - o ) i o n n i - o ) n ( o n COIDIOiOIOKtOtDO^CVICOCOSCO (DOlOlrffllfl'-Ol'tCOOOl'-COlO  noTW'tdcoajO'-'jtonno  N ( D O U l W - - ( D I D i n N u O O O O r O l O l l D O l l D S W O S N I O W N r l D t o o ) O o m r ~ ~ L f ) c n - i — Oi— O J O C O C M T O O O I O O C M O O J ' - O ' - C O C D ' '  O  CO O  + J  -  o  O  o  O  o  O  O  •  o o o  o  O  O  O  o  O  o  o  O  O  o  O  o  o  O  O  O  o  o  o  CMOlSiDiOi-COVOW'-iOitO)'(0(0lDOrf0J0)fl)(0t0C0'-«3CgCD in(DOIflO)03NCOiMOSi3)(M01ii) 0)lf)INW--CJt-(D(D0)OVNU)0) r*-r^i-cooT-coojCNJor-.t-cooco C  O  O  I  O  O  O  C  N  T  -  O  C  M  O  O  C  M  C  0 ) 0 0 0 0 0 0 0 0 0 0  O  W  C  O  CO  i- w n * o_ o_ a. a.  I — I — I u_ 111 (a: /) w Q. (On u J t oJ—o — o t n t n w Q ^ N o h  J  o 0  J  —  - - s xx x  o o o o 0 0 0 0 0  O  O  O  O  o  o  o  O  o  O  o  O  o  O O CM O CO 0 0 0 0 * O  o  T  o  -  o  O  O  o  o  O) i-  tnmomcvjvmcor^cohWS0)r-C)N(O(DC0(DO) a j i O N c o o i n t o m i - o ) ' -  ^  O IDi-SOllOCOONOJPJlfl o c o m c o o T f f - i - ^ - c o m T-T-T-T-rrcOCOOCMCOO C O O O O ^ ' t i - T f W O W  «H  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  W cr  o f C M O j f f c v i m c o ^ o  siniflio.nMwifiifiifiw cotointDntoojtono (MWi-WS*i-WCO(D(D  II  W  O  O  T  -  O  O  O  C  M  O  I  O  -  O  T  -  C  O  O  O  O  O  O  O  O  O  — O LC LO i  5 T co — I 0) O c_  ' • W W * 0-0-0-0HI G O Q.  c5? +j ^ C O "55 C  ZD O  O Q  Q Q O S  X  o  00  OOOOOOOCOOCOOOOOO o o o o o o o o o o o o o o o OOOOOOOOOOOOOOO o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  i D S O ) * w o j i n s c D ( o n o s ( » o o a w n i - c o n n i - n m s N i t n  oi-omooh-h-cMOOcy^cotNrs o c m n o cj © »j- o m TJ-  I T -  o f o i ' - f i n o j T - i n o w c o a i ' r c M COCOOlCOt-lOCOtDNtOSSCOWCO (DIOIOO)>n(00)OIONOOIIo(Oi-  oojoocor-Tro)»-OT-coo>cri<N rTOOOmOCMOC\|T-OT-C\JCOiT - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • • • o o o o o o o o o o o o o o o  o  o o i o ( D O ) - - o ( o ^ i f l O ) c o o o i n > - S W W  -  - C O ( 0 ' ) ' a 3 0 0 1 M D *  or^T-cooT-cocMcyoh-T-coT-ai 0)0)OOOOJt-OWOOCMC0ir>(0 1  1  0 ) 0 0 0 0 0 0 0 0 0 0  Q_ Q_ •H  LU CO Q_ Z  >  i-H S  < S  O  T-  t£>  O  a. a_  CO U J CQ CO C O C O C O O T - C M C J X S S E O O Q O E X X X O  o o o o o O O C M O CO O O O O O O O O O O O O O O O O O O O _  _  O O O O O O O O O O O  ifliiioiiflWTj-iniDsios  NN0)i-(0N(0(00)1001 COiONIDnNMCOIOOlO) COCDSC0O)C0(OU)i-O)r-  coi-r-.o>iocooKcocow co cn co co to CO K) CO r i" N O CO O CM CO O T- *f TO T-  OOOOOOOOOOO OOOOOOOOOOO 1 O ' t ' t U l T N C \ I C 0 S ( D  SS IS O UJ  CO^OimCDCOCOWCOCOO NCIr-WNtr-NCOCOID i n o o o c \ i T - T - ( D o o o T  -  O  O  O  O  O  O  O  O  O  O  o c  LU  CO  LU r N n t O L U Q. a. d Q, CC Q. < UJ til LU LU L U C 0 Q _ O X l - t - l - l - l h CC (D J < H H  CO •  2 CC  a, E 3 CO  3 i  _t c a>  C O C Q C O C O C O C O O T -  0\  s i-*-CD w in OO CO CM * CO v CO ( D i- i- cO o *CO co N -<J-C w in t>o-CD V CM * Ol N (O f r c\l V M 0 !D (O COOCM ) Ol (CD Dn itcn- N (O i - 0 )0 0 1 oio m f ico- > - w U I N iO CO O J to CO CO o t c in CM n f( O o s O V O 0) to hto CM c n CM in e n m • —*— m cm io N m N s o Ifl C O ( M O ) i ID- CO t- t- CO m t n c n T oj c o 0 1 r t N r uuWN»-COSCO* C O C O O C D PI < D ( O S CM S O O O TT y~ in s n c m * _ . ^oiron- -ccooooo c ooT^r T - Toj - o -o T o v"*" N ~~ rC "Mc " i «ooisnnooNcooooiofolos o To- T o -on o OOOOOOCMO O O O O O > % C O in ~o < H l t co m co to CM in in T in ico If) 1 CO 0) N CO S CD OCO N »• coCOs * oO in C O O COC OCMt t N(O N o (OOcoNcnOf O) SOCM ooCOOco C inCOo o CO t CO o N t C CM CM M NCM CMco* cC O O CO c o o m O CM CM I-- O ) C M C O O C c oCM cn Tc—oMCMr-C coo CMin cn _l tCM^ CM rC O S C O C M O C O N C M C O N C O C M O C O O T— m CO O O l tcoi-i-tcnr^inrt* N CO T -CMli) W (D «Ns m NO ) CC OCMN >- CiO- to O O CDcoCOo ••m N 'o- m o f t o i n ^ C O f D w CM ^ co co o ccoo in CM m C m y~ CM CM s s o N T- C O o o o cn in cin om O CD O T- O O O o o CM CO o o oo o o o o o o o o o o O O CM CO T— S  i- yai cyocoo> in i* c_ cn C3 3 * cnco co co cr m O COinCM CM CO  O  O  CO CM O S * CM  CO  *  CM CO  S  N  CM  ^  u _C O •* oo Q CM (O01 sCMs y-  CO CO CD CO in CM c n oco*t cM-o (OOOTtCONCMOO)COC!)<-inCOC\l ro*- CD cncn i-y-m o m o Iin m * ^J" h- CO CO CON cocnm^roaomcoocoococo^j-TTt D T O ) CO i- CM S CO cocMoico--o)i-com T - tt o c o r-.cn r*. to co co CMO _CD. CM N CO CMfOIOmi-VCOO) <- O stcftcoscocDincoinN o co s CM co in coo cTo tSo CO ccoa CO in T-lfl--TfCMtCOCDO)COr TvO'tmcMOcocoioino) O ) CO CM CO O CO CO CO CO CO r(OCOCMCMOlOOCOSCOtO - - - - - - -o COoCOo o. o _ CM * O O O O O O CO O O CM y- y- O )c o i - c o m T - o o o o * Ifl CO CM O ) N mo oo oo o o o oo CMOOOOOOOOOOOO o o CM cn CM o N Ol OCICOC5c in OCO CO CO y- O )C CM OO CO Tf Co O CM oOi n i n c o N N o n o C MS COo rCO O O )COf-C o *co totor-»cn CO N CM r S sCO in cro rcoCO ?-O o co --y-C O cC CMSNOlinOCOr c o c o r-. c o yCM c * c n oD cococotOOTcn^J-eo rc n c o c n C M * in cso Nm m N '-mcOCMNCM'tCO o O COCOCOT iI*- y~ c)o co N oicocoinootCM * CO CM OCM CO N e n nr * T— o o * O ) t N CDinCMOOCMCOCO oOm m in N CM T - CM S CM O *•O * CM I - C CO CO O CM O O* y- rt*. O _O y-otoo m CMi-OOOOi-CO m co v o o o C o r- o> o o o C O O T - O O O O O O O O O O 0)0000000000iO  S  C CO T3 O 4-> CD •H . H O ino c3o tc_ o c oo a > c O "D t _ t_ t3 _ taC _O > -H «O o otO CO CO c o co cc rr d -  y- CM CO < <<< <  i- CM CO * CMCO^inCONCOOl <<<<<00000000000 O  APPENDIX 2:Number of tree per stocked plot. Comparing residuals from non-linear estimates of B and Cand non-linear estimates of the Weibull function.  70  CO CO •H •H CO "O T3 CO c>» _ CD CD CO CO 3 3 — • I i-l CD O O j= .Q •H E 3 >. >* O CO CO CO  ons  >.  unc der <7  -H •M  CO 4-n_  lin jmulal  C_ CO CD  est to  ad  CO CO CD O CD LU CO J= •H UJ LU •H L. cr UJ CO CD i•z. — cr 4CO iO * z " O z Ui Ll_ cr CD Z> CO•H £ CD CO _l z CO CO I •H CO CO •i-i CO LU UJ o QC CC «+- 4-  —I o  i-i o  •H >•  rH  V  „ o  » O 10 +^ m fli  + * + *  t; t S != EE <U o "i Z (O C .j J M M •H p J O) (!)  .h, it" Q ui uj - cn a: <  J  c 5 o £ • H— fc ^ o o  +^4-Jj5 ^o+Jo r-i < H w  Q- Q_  lO o  to  UJ  CC  d  o o  2 S2 7 3  m  -l  - 5a «  03  -H 05  Si w to •H UI UJ •3 *= . . L- c UJ  » ",5 2 co  5  4J  B  <0  o o «= t  a> - g g >^ > i, > i, H  -H  4_,  tu  m  S £ £  »-i CC — I 3 ' i-i i-i m *^ rn • Z to tr tr  o o i-l <H 0_ 0.  "°  A  H  I CO•H C•H ^ C O U UJJ cr  ° 5"  I-  •o z  •CO UJ ui  0> 01 ~ n co +7 n f i  •H C- V- V, "°  ~  ^  C O •O U UII C UJ ^_ C C O D  C 3CO •D  2 « ig. Z  OI m CO UJ CU M  O O D- Q-  O O  ° Z » £™  C UO J - r*- UJcr  a> «- £  » „_ 0  L >° E to £ >, 5H «m- °> j<  w  o= > ° r  c  0) ° +J i.  d o j ^CO^ 1 5^ «> i c '"l C O .J LC UO 11 1 M = < = ° 4- *-  HH  » - 2 2 > -2 -2  CU O CO• UJ L U CO ,_ " L uO _C O >H A  A  ° C 3 »3 O  CD ^  z  m  o o fH iH a. a.  LO  o  o d  o  in  o  CO £ 1-  rV>*-TC3UiD UiCD  -  S .5  *  •H O O zj n n -H O E E CO >i >i X CO CO  •H i-H 4J M CO «1 -l C (1O 1 1 +2 ^  LO H H TCO w I LU LU  CO CD CO  o  CO • LU CO LU LU  CO ^_ iH  E  « s S fl I  J I CO CO LU LU  + * + *  + * +  W T3 "D • CO CO  1-  s  •H > — I •H CO » CD °O +J 3  CO H— H—  O o o  7 1  "H H ^ o o  W  o £ CO • UJ CO UJ UJ  EE  u_ CO — ' I • CD CO S C D O frt ~- H  2 -< " s i  ""i co co  *^ +J IIUIJ IIUIJ 1  1  CO UJ cr  T-  o  r-  line lulai  unc der  ns  C O CO co c_ >, -H •H o CM T3 •o •H CO CD CD •H CO CO CO 3 3 H •-I co H— ^_ CD j= o f-O +J .Q JZl C_ ZJ E E CO o >> >. CO CO CO CO CD co" CD LU CO •rA C_ LU LU •H CD cr UJ CO CO cr 4z »O z O z * CO CC CD 3 _J z> •H CD X z CO C O E •H +•* O) I cr CC •H 4;at  es  S3  ES  ad  sz  4-> ° o 5  •3 ffi s o "  a  s  £ » co co  I LU LU  + + I +  oo  + *  CO CO -rH •H TJ "D o £ CD CD CO C 3O r — t— t1 CO £ O nO E .Q e >< >» . .5 -H co CO CO UJ CO 0= LU +-* ^_ CO h- CC z* 1*z CC _ I ill ^ z CO -H CO rC O L J cU rU CC 4- S — oo •H •H O I-o H. < H -. Q o o o  -1-1  IX"  O £ 4-1 * ^» Ti > rn m  if- CO S  — I cc3 CO S „ 5 I CCLUCC LU ^ I-I  CO  —  O t- *" < J  o o o o  APPENDIX 3:Number of tree per stocked plot. Comparing non-linear estimates within the Weibull, of linear coefficients of B and C To actual CDF and CDF with predicted B and C.  o  oo  tCO D CO CD >i C N3 3V) 3 CO  O iD O HO r- D 3 >i >i >i O CO CO CO 3: S •O Z CC Z •• * H w CC Z J CO UJ + z |1 Q Q  o •D -o H- a. r-Hoo_ — I I  w in in  5 co co co  Q °" L • 50_LUQ L U •H CO CO cu o  O O O  a. a. a.  CO a. * CO  CO * 3  3  3  CO 3 £ +-< •H O  CD O o O T3 CD CD C -a  •H  JZ  Q  O O O rQ H _ Q-H _ Qr_-(  Q. CO  * a.  co  CN 00  o c  •H  T3  CC Z J UJ * z .0-00  d) o o o "a  0 ) CO c  •H  (0 a * C O C O C O  "a  <a a. * co a. a> u_ a +-* o  JZ  a. co a. co Q. co  c  H  H  ^ Z CC z CC Z J 1  «s  i  o  LU ^jIoDE 0-C CQ.L CU C 1  o o o  + + -H J  J  o o o 0-0.0-  3 33 • H M -H CO  3: S  LU O C CC U J— zI h Z- CL CU z* CC *Z ZJ ILU O- Q Q  oo  CO Q00 Qcd Q. * CO  0) o o O "D  co a * CO CO CO  U orJ — t rH  JV'  uCO j UJ  3 C C  C D CO. CO CO (O CO c H T3 CM T3 T3 T3 CO D D D -C O HOD O f D O C C>i O >i CO >iCO CO -H CO •HOO C "O •H £.= C 00 •H U_ 5o 3:  J=  S U_  cCo O COztr •o3cr zh crUJz* C XC UCJ *O -J zO N E3L U J LU »I OCCQ- CCCX. UJ UJ UJ  • *  S  — I  — I Z  CO COCD COCO* CO (0 CO CO CO CO CO  '  DC z CC *h- z^ Z  *  UJ CL 1 S I  Z  Q LU  —J  • LU  =3 CC CC  o o_ o_  co co a. co  00  CO 3 3 3 "HH H JZ o o o +> - J3 JD JD 3 E E E O >> >> >t  n) co co co  WCCUJLUCOZh S h LU * o z cr z S cc z _i T cn LU * Z o- o  a  o oo o +-» -H -H  o o o  Q. L 0.  CD CQ c CD O -H TJ - CD  C  T-  T3  z cr z CC Z — I  CO CO Q. Q. * CO CO CO Q. CL -I  oo  c u o o O T3 as c  CD  H  T3  O  H T3 0)  -  CC LU D.  Z * Q  -J Z •  Q. CO Q- CO  ,  *T!  D  cc tr ft.  6.  00 00  L) T3 -03 - CQ to <fl wZJ  oo  Q_ 0_  E> E> lilCO C O CO LU LU CO D C UJ UC — IC LU O z tr i-h H S OLUCO  3 CC CC  00  a > c> o ca o  •O  T3  o — tr UJo CO LU  CO LU  cu  2 D O  D  O C3 UJ O CC CC Q_ D_  •H T3•HOC CD 4V  en a u Q) to coCO rj  -  L U CO ZD C UJ O — hC LU CO  o  OS  L D CU CUJC LO UCOU H p_ O o 2 LU O  3 C- C C_ O C0 D  CO Q,  CL CO Q.  C L T3 •H CD U_  CO  C D  Ui  Q. H | O C> O O DC O  CN ON  CO Q. 1 CO CO •H -H " O T3 i C CO O J: CO O C C  oo n e e  JD  c  >i >» L  CO CO 1  CO Q.  CO  Q_  z r • *c cz t— c LU * O.  O.  £  LU •  O  D_  =5 cr > *- O fO +-» +-> o o r0. H 0.i-H , 1  CO C T Q. -H |  CO + Q. CO  (0 * Q. CO *  LU CO Z) or o h- CO LU  LU  o o o o_ o_ o_  cn OS  CO  CO  =1  o o o  H Q_ HQ_H0_  CO  cr Ui o  o_  n - +J •H co OJ  * I  L O CO CO C — i a. -h |  T-  CO  Q.  CO  TJ CD CO  TJ CD CO  O E  O O .Q JO E E C / ) >> > LU ( A CO LU  LU CC I-  CO LU LU  UJ CC I-  CO LU UJ  2 cr  D O CO  ZJ) U CO  *H  Q. Q. CO CO Q. Q. * cO cO co co co a co co co  a a co  co  « a u  - - Qc)o CO CO •  CO CL CO Q. * in a  co z:  CC LU U LU  — I  LU  CO  Q. OOCD D CC OC 0-C Q £  LU  (0 CO 3 "O O (1"O ) -DO (OWE 3 3 >,  LoU CO D l—r Co O LU LU  a. a o 2 LU o D CC OC CC L. C u  OO O rH rH 0-0.0r-i  co a. * CO * D. C*L o. (0 ca COCL Q. CO  a. o o E ZSUJCCCC= >  old)  r_  — i I i-H •—i  0_ Q O  S LU C  3 C_ = > O CD C O  ON  (HQ.*  (73 CO U) •H -H T3 T3 T3 CD CD CD tfl CO CO D D 3  D _ UQ S J O DC DU O 0_D CO  o o o r0 H _rH0_rH 0_  t- 'H ffl  o o o  0. o_-iHo_ r-H r-i  00  (B H >  o o o 1  i-H1 i-H1 i—I  3 3 3 CO  CL CO  O- O O 2 LU H =3 or =3 O 1 05  Q.  CL *  CO Q. *  CO Q.  CO  o o o  o_ o. o_  a at (0  CL  +  On  z cc z * j- * C C Z SD L U • D £*3 CC O S_ LU DUD O D_ CO  rO Q. *  www • H -H -H  hut:  0) 0) a> www 3 33 < H — i I  — i1  o o o Ij j3 n  CL. O £ LU D OC CC L.  T3 O W O CH) L-H L"O • COLO L (0C ffl iCL H |  U C C DO C  o o  in 3 v CD .a  L (S Of U0)  a> c  H  cc z uj *  E 3  Q- a o  a. co co *  (0 to  C O  cd a a cu  CM O  i- CO n u n CD • H JD H C O C D EH S+ • -LJ= fU fl)£* ) O CD -H C C C C- O+-» - CO C CD HCZ*HCD  02 D U  O O L U CC U D CL V)  cO co cO co  CO  CO  CO  CO  o  T 3 T3 TJ CD CD CD  CO CO CO 3 3 3  0-0 0C E LU 3 C_ C 3CO O C0 a. CO * Q. CO  CO inCOco CLq «a * CO CO CO COQ. CO Q. *CO CO  o  .c  •—i — •t  • — I  o r0o H- 0 o r-i r-t - 0 -  o  or z E UJ * Z3 O- o o  o o o  — i I < — I 1 — I D _ D _ Q_  03 00 00 + 00 Q. CO Q. c0 * CO Q. * * CO Q. 00 * * a n a  C O CL C O CL C O CL Z CC  *C hZ - * L * 1S 3O L Q) J C CU L O cu « x: 2  *- c  O O O 1  1  1  C O CL C O CL C O CL C O  APPENDIX 4: Probabilities of advance, subsequent and excess regeneration. Re-fitted coefficients of the logit function in the logistic equation form used in Prognosis for probability predictions of advance subsequent and excess regeneration. Coefficients of the logit functions for logistic equation resulting from a variable selection procedure for prediction of probabilities of advance subsequent and excess regeneration.  107  o o O  O  r*  CO  UJ  co o co co T— co o cn  to too  Ti  •H .-I  ui i_ co cr co  o  O  r- co o o o  O  O  C  D  O)  CM  cn  CO  CM CO CM ID CO t CO t  10 o  o  CO O ) K N. O  ID  CM CM  CD  ^ s co cn cn r*cn co  to * N CO CO ID IiD- C D O C CO O) O) OC)\JCO OD) N  "O C CO  ID L CD O  TI  3 cn *  o  O  CO  CO O  o  CO  CM O  CO CM CM  T-  T-  O  O  O  O  O  O  co r- cn CM co T - co N O r i - CO Ol TTT-  CO  O  O  r- co  i-  CM CO  O  O  O  O  O  O  O  O  1O  O  O  O  O  o  -H  T3 e  .  o  o  w <r  CM CO N CM in T—  CO  O  X  CO O O  +J  T3 0)  o  CO  O  O  O  O  *  cn o  CM CO  CO O  o  CM  CM  O  CO  CO  2  D  O  CC - H  A  -H  C0 .Q  T-  CO  CO  CO  CO  •  C H(-LU O t- h-LU> L C C< CO [_O U C OC_CSZi-CMJWWJ>I 0_>t-iXXC/)OOLUOO H  J  CC >  X  CD CD ID  o cn cn O CO o cn T -  CO  — i  o •H +-*  O) O) O) O O o  CO CO CO o  O CO  z  8  ~ c  —  £  O OC •  X •H CD CO CO  LU  ID N I D CO S * CM CO CD CM CM  »- CM O) C O T - CO CO COi-tOtCMCOCO CO C D NO'tOCOCDCDO CONSCOfflCDCDCO o3 cC n oC O Oo o o o o o o o o o  CO CO  >  O -H +-> t- o  CC  C 3  u_  c  OllOOiDCOCOOt t N t i D N i D ^ O om in m co CD  T-  CM  O  T  O C O C D C O O C zO n•—It~> _J O (0 id 0) " DZ• H -H Q O Oc (_ C .C  T-  •—I  CD  *  't  O CD N- CD CO  CO CO O ) N CM CO CD  T-  CM  T3 E CO N  •  00  "D  CO CD T - O) ID CM  co in co co i CM O O O O  CO  C O  C O  C_ CO  LL  (DCMOlOOlDCMTt'^CMlDCOO o m CM CO CD T- co o r*. in * O CM i - O CM CM • LO i -  CO O S (O CM  ro a a -H to 3E=^ C o cc or z —i  ^  to to e c CO CD 3 H O CC CC Z -I •H CO  CO  (0 CD CO  -H  "O  QC CO a  CM ID CM  c o o o o o o o o o o o o  ° O) — 3 cc 1 .a ? © 3 cc  r-  C  JD  CO CO CO • • -H -H W C L L X) O  CD CD O CO 10 C_ O O CD a Q. 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CO 0 It  W  O  O CC  • c- t_  C  n  o  -H -H  L  cfl T3 CO CL -M CO CO CD a cc  O  o o  O  CO CO CO CO  w  CD CD t_ CO CO  "D D O C O " " cd co  CD o E 3 Z  ^ C H  —I  i~ +->  CO O  CO CO CO  CO  CO CM U I CO CO O ) LO CO • *  r  O  CM O O  r  m  n  •osococoN'-nw  COinN^'tONi-V  cu •H  ifito^cooiT-nm coiO'tTfomncoin  CO  N T - C D O I L O ^ S C O C M  E  C O O C M T - C O N O t r  o o o o o o o o o  •M  "D CD L L CO CO T3 3 C D " CO CO  i - CM CM r ID TT O - * O O  L  Ii)  C D t t - t r c o 3 - t  o  cr  O C n C M C O O C J C M ^ J ' C O L O C O o j o o m o o c r O C J ) O T - O  5 5-  Cfl CD CD -H rH  n  z  ja  CD  CO CO  5  0_ —•  CO c O Cfl C L -H CO CD CD O CC  CL  CC  c 0-  O  O  c_  n E 3 Z  +•*  r-t  r-  LO  CM ^ CM 3 ^ - c o o r - c o a o c n c o i o c M  cr  C O O C O O C O C M C D ' - J - O N . f -  C O O t r - C O C O - l - C M C O C O L C O n O ) N O ) C \ i n C M O ( D  "D  r-4 O C O i - N i - T - C M C O i n T - . O ) CO  C - C D O N C O O O O r ^ C O O  o o o o z  n n c_ L  CD  c t_ O CD CO CD  *f  CO >  co ||=  C- LO  co 'tr  •o e  » - - ^ i O C O i O o o i o 0 O C M  c o o o o o o o o o o o  CD 1^ 3  ) i O C O i M i O L  L  D  O  O  co a) a •H CO CO CD a cc  a n CO E 0) 3 cc z  o o o o  JC C -rH -i  "O C cd  CO CO i— CM CO O . CO CO O CM CO CO CO O ) O ) CO ID E T O O O O O rH -H  (0  'H  CO CM  CO  TJ- CD  CM  - tr- n -<± th- N•«*•  o o o  LU O.  h  O  h  O  O  i-  T-  o o o  0_ X CO _ I C O C O C O < C C C L C L C L C 0 Q Q < C D > Q O O  APPENDIX 5: Heights of advance and subsequent regeneration. Germination-delay Non-linear least square estimates of data-fitted Weibull parameters, for the prediction of delay to germination for species with above 20 observations.  117  a> => o> m 1  inO)coO)iOtO<J)Ocn o i c o m N C D C i n c o n  rH  co eo  CT> co CO CM O CO CO o CO T O y-  HWCOIflOlT-T-rT-  crcjcncoooooo CO  c o t n o o o o o o  * o  O O CO i _J eo o  o o t o o t o m o c n i o > - n ^ n o T - T cvjocoi-eoeocnco  CO»--«J-CMOCOOO  u-CMCMCocncnoo  CJ  r-. in in s ir» r~-  CM CO  o<-ovioinif)in'in  CO eo ycn CO CM O  < CO  CO  f co CO v  o  CO  V  T- O  O t O C M W O t O ' -  •f-CMiOT-tOCMCOCn  IC3>-t\ICMi-i-^r*~ O T~ T~ T - , -  . O t - c s j o v w c o r * -  c  CC CC ZD  I  -—  o o o o o o o  C S CD o o eo —I  O  O  Tr-  CO  C M O <-> T m Ifl C\l N  cMiococvitrincDco^coco  S(D[OncOIO--i-(M<MW  CO  cooitD'-'J'CoscicncjJO)  CO CM CO cn CM -rs  I * - C D V T - C O V V V T ^ - ^ -  co co I-  vr If) N CO CO  T-  O) CM T  >r V CO  <  to  tO T -  o" o  CO O CO  V  O -<t ^ CM co V co cn  C  O  f-  CM  I  •T m CO  N  CO Ol  o  c  CC CC 3  —  CO CO CO CM eo rv h- CO o> to  T  -  O  O  O  O  O  O  O  i-  CCO i O -j  O i  CO CO CO >f O) ro r- v to If)  •r- O co • -  CM co o to o if) IT O  i co eo co co O. 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O CTCMCOOOOOO 1-  O  O  O  O  O  O  O  O  t* CO CO Q) tO CO * tO CM O CO CO  O  N CO O) O)  in (3> Oi Ol to *r to o co to co CM CM O  CL O  O  CO CM T- CM  co co co in m m m O CO T- CM CM CM CM CM O •  tO tO  CO to r-ito t o CO to o  to m cn eo  T  T  r-in v s f co co co f O O Co o  cn in CM  cn m o CO . CO  w *->  >. TJ < CO  CO to TT CO  cn m cn CM LO'-CMCO'TinCO  c  :  ii)c*)--i--fcotfsconc*)n  I CCD'fOlT-CDlDtOtOCOCDiD i  o - m o * * - ( N o o o o o o o  «  cu m co C S C OC M O O C O U JC Oo CO  I -~*-000000000  H—  en L  o  co o  o  in  r- O CO  o •  m o in  TT  < CO  o - - n o - t ( o - - c o o s p j  <MOCOOJ01COT-00<M»-  S<BSU)OIOO)(J1010)ffl  MoimiDnnnnnnn i»i(Din--o)Oododd  C M C O CO m T-  co m  o I.  O  •-  CM O  '  EE  3  —  m > o. CM co cn L a s co  CMO>0<0-*-|-*-f'  r - r - c f j T - O O O O O O T  -  O  O  O  O  O  O  O  O  c  5 i - co  O O CO O'-nionO'-Nifl BOIOSSCMrrO OOCOCOCOCOCDOO •3'r-COCOOi-O'---  -• O) O Ol S N  fflvoncowncon D. O  i-  O  CO CM CD  •  o • E  o o o i --  O t- iC-OI*- o • 4o"-HO (- mN-oCO c Va. LU CDtoCO E  >. -a  CD  TI- cn m Ta-  in v co < C O C hM - O) a> in co + > C O CO <tj I- o co in • H O O T-  O  O  i *r t o o co co i  s  E  t  CO N CO C ESS •-  t  3  CM  O  T-  T  CN  lO > Q . « - CD cn t_ a . co CM 01 3 OJ O O 4-J N N  tamcococMcncncncncno) s c o t D c o o m m i n i n m u ) C T C O < O O J * - O O O O O O CO T  -  O  O  O  O  O  O  O  O  O  L. (0 3 cr  CO e c >. co  tf  N rcn co * o) CM  CL O  CO  CO  <  -H *-  •  r-  c 5 o o o o -cr co o —i s w CO o  5  ' "* T  CO T  -— s c  tu  CO O)  »- co 0 01 CM co co ( t cn  SCOCDCMCMCMCMCMCMOJ  Q . O * - O O T - » - T - * - - - T - - -  N CD CO N CM m cn io m TT o o o o  rm m *-  co +-> <  <i> CO « £ a.  o v o c o m w - t r o c o i n !CO<TC0CJN-ti-t»)r- r m c g c o - r - f i n c c a o ) o i o - f o c o c o v n n n c M s n t s w n n o n i n T - l O t M O V ' T V ' f ' t >co(0(ocMco*na>a)ciio)cn  I B f  CO  in CD I  cr cr o  —  o)  L a in co o u co m O CM CO •H C O CO 4-i CO CO O CO T•H CU CO CO M  co CM v co to m m i.. CCtOOinmCMCMCMCMOJ 3ioiom-r-r-i--i-(--)C r - e r - e l - T - O O O O O O CO O O O O O O O O O  in T CJ> CD CO CO  : -H *-  c  O T ' * N 1 0 B ( J ) > - C O -cj-r-tMO-crCMOmr(OOiOCOSCOCOCOCO T - c o n - f i o i o m i n i n  o  i CD N N S N S  -_ O H Ct) co * J  o  -  O TT-  C O S HI CO  o  «-  CC C 3  —  S  ' If) CD r- CO C  c_ o> TCO *•* CM s in CM  O) t-  Q. V T-  : cocnocncMococococo I 0) CM co r-.'in CO S S S N : I i <  i s N o i n N i D i o c o c o c o 3N<rmCMCMC\ir<ICMCM  D cr cn  oiC ToC oO oC MoOoO oO OoOoO oO  C  O co *•rf"-  CO T-  T 0) CM  O X ' - O C O C B ' - C O O ) - - x r - r - o o o o o "1 CO f~- CM t T <?  C L O C M ^ - O O O O O O O  CD to CO ^ 0.  o CD co O CD if)  o v o i c M o n o n c o ( D o m N n c M r - i n i B l O O l r l O n r r r mcMcov-cMOiaoo) ( O W C O n r r i - r r ncocMOi-'-'-'-'-  >OllOlOCDSCOCQCOCOC0  W 0) CO  o  -H  CC  CC 3  —  o CO rCO CO O  APPENDIX 6: Heights of advance and subsequent regeneration. Germination-delay Comparison of actual delay to germination cumulative density, cumulative density of delay to germination estimated using Prognosis models and cumulative density of delay to germination estimated with a datafitted model  124  B O *  &hccs UJ  • CC 0,  vi  o > <  S  O C3  2^1  i  £ 3  ££8  I— cc 1— 0- o •H LU U- J O  -« =3 _J r— CO LU  S8 I s = § 5 E § < CO  1  ,S 3  s  3  : cc LU • fX Q  8  CO O. UJ cr • cc o. >• 0. 5  5^3 o. a. o  3 ^ ^ £ CC Q  O  S O r. -H CO Q Q IU  o z  n o  I>  g= S < CO Cv U.  3 8° o u  z  >  g5  o z  « CO Z  °~  LU  UJ +J Q O H UJ >  UJ Q  5P 3 5 o a U J > J— IO t Z  UJ  D  D  a  i—i =»  ) \— CD LU  3a§ 3  S  a  cr LU a. a  m CM  To  ACTUAL CU UULATIVE DENSITY F lot of CM PER*DELAY. S ymbol us ESTIMATED WEIBULL CUUULATIV E DENSITY Plot of PREC •DELAY. PROGNOSIS UODEL CU UULATIVE DENSITY P] Ot Of CUUPRCK3-DELAY.  S3  THE COUPARING FITTED-WEIBUl L PREDICT)ONS PROGNOSIS UODEL PR EDICTIONS AND ACTU/VL CUMULATIVE DENSITI DELAY OF ADVANCED WESTERN I EULOCK RE(SENERATION CTUAL CUUULATIV! DENSITY f lot Of CUMPE R-DELAY. £ ymbol usee is STIUATED WEIBUL1 CUMULATE E DENSITY PI ot of PRE!•DELAY. S) ROGNOSIS UODEL C UUULATIVE DENSITY P101 of CUMPRCK3*DELAY. Symbi  HE COMPARING FI1 TED-WEIBUl L PREDICTS S ROGNOSIS MODEL 1 REDICTION! AND ACTUAL CUMULATIVE DENSITIES FOfl ELAY OF SUBSEQUI NT SUBALP] NE FIR REGEr ERATION  MO CM  • 01 3 CO =3 -i  CO 3 r  •o >*  m o  a  s if) « © O O O O O O O O O O O © ! o o o o o o o o  COUPARING FI1rTED-WEIBULL PREIJICTIONS J  u, X CC LU r-Q.rO  8 5  E w V Ifl  PROGNO SIS UODEL (JUUULATIV E DENSI!fY Plot of CUUPRC)G*DELAY.  ffl O  ESTIUA TED WEIBUL1. CUUULAT IVE DEN!ilTY Plo t of PREII'DELAY.  — I CO LJJ VE DENSIT  UENT DOUG Q. > > 1- >  LL CUUULA  «-<=>_) l  CUUULATI  CTIONS  cc  DENSIT]  PREI •DELAY.  ENSITY Plot 01 CUUPRCiG'DELAY.  DENSI TY Plot of  ot of CUUPER"!JELAY. 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M  >  t - CO — I O Z rH  UJ CO +-> a o ui  z  rH LU O. 0- >  t• —s— iai —  S o w  a a  to 3 z a  amto  CD z o a  s e °-  =J  LU 3 Q o LU J  < _J 3  a  O  > _l rH 3 _l  ,85  5 3  • cc a. >- Q. a CC  5 _§ Cl_ O_ a 3 > I  Q  UJ  <  O  -J  cc Q  a  °" 5 £  0  H D-  Q UJ >  Z  UJ O  t s a  6o  :  1§5 UJ  Q  3  o  LU J > _J rH 3  h <  J 3  a 3  UJ  g  5  X CC UJ r- 0.  a  m rH ui s O  3  a  3 O  _J UJ Q  o a  CO  cr to uj co o. co  COMPARING F ITTED-WEIBU!.L PREDICT][ONS iNOSIS MODEL PREDICTIONS> AND ACTU/VL CUMULATIVEI DENSITI \Y OF SUBSEQ UENT WHITE 1'INE REGENi[RATION IAL CUMULATI VE DENSITY f>lot Of CUIIPER'DELAY. SSymbol us [MATED WEIBU LL CUMULATEIE DENSITY Plot ot PRE!)*DELAY. iNOSIS MODEL CUMULATIVE DENSITY P]Lot of CUMPRCX3*DELAY.  CN  •o >.  Ul S 3 X CC UJ  y co cr  UJ h  -§3 0} If) S  E B • >  8  UJ J  > H  — r CO LU  O  _J D  CO  33d 13 UJ O  O * • LU cc CC CL a, a  8  3 ,- 3  3P 33  o  3S § 3*" 3 O (A  J  O  CO E  <  UJ  < S g ? 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Coefficients and analysis of variance for the re-fitted Prognosis model form and the variable selection resulting model for height prediction of advance and subsequent regeneration.  141  CN  tcd  c_ o  •  T-  C_  CO  O  -H  CD  CD "D O E  CD Q  Z  U>  T-  T"  o o  o o  o o  o o  O O  CD CD LO •M -H T CD CO S E e -rCU - H t- f (O  O CD CO Tt 00 't  CO  CO  o o  •D C_ O O O O c C_ o o o o CO LU O O O O •H O O O O o o o  o o o  o o o  CO T-  O) o ^ (D  O o  o o o  o o o  00 O) C\|  CO 't LO LO O)  T-  O) S  UJ LU —I CO o c r Lu > UJ o CC  <  > Q <  O  CC 0_  CD  c_ D  Q.  CD T3  O L  O h  +-< O  CO 2  CO £_ CD -H CD EE CO c_ CO Q.  LL. A JD O C_ Q_  TJ C (0 o> u c  CD JZ •H  >  T3 co  c_ O 4-  CD  O  cz rH CO CO CD E  i •  JC U) "55  >  CO  c o -H  u_  o c o  o o o o  4O  CD t_ CO -3 B =3 c r CO CO  -#—•  <J TJ  £. 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CO If)  O i S i n n o s N W N i n m  H— o o i n T C M O i o i D t ^ n n  CT CO N O) N (/) D Ifl N O  C  O CM CO CO CO o i - CM CO  CO CM O  co  N  CO O oo CO  a>  • CO Ol O > I • CO O LO • *<t co t r i - co CO LO i - CO . . o • s  t- o o  a>  c\j  1-  o  f H LU O a) -Q o tEl CO CC CJ T LL  a  L  o  L L n c o s s s n n i o c D c o c a a c n i r i u ) J  O  O  C  M  C  M  O  O  O  O  O  O  O  O  C  O  O  J  O  O  o o o o o o o o o o o o o o o o  c Q . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T-  CO CO CO CO  o ooo o nnioiooominoioioco O O C O C O C M C M O O C O C O t ^ C O C O W C M  O T I N C M O O I I O I O I - I -  i D O i - T - n n i o m o o L n T I N N O O r - r . r O W C M W r - i - O O C O a ) .=>000'-T--r--r-00 t r c O t ^ L O L O C M C M ^ T - LcOoLnOnO J O J C c o i MOi OONL- oh n O  CD c LO CO CO D n c r co co CD CO C CM (0 CO  O) CD >o co t CM O O  eo O cn co n OJ CM CO  CO  O  O  O  O  O  O  O  O  T  -  I  -  CO O in rCO O)  CM CM CM CM  CO CM O CO LO co N co CO O CD i -  CM CO co co O -r-  LO  LO  T-  T-  CM CM i - Tr- s . CO CO  M C M O O C M C M O  O  O  O  O  O  CO  CO CO CO  T-  T - T - O C M C O C O C M L O O  CO CC O C J O  O  I-  O  CO O  O  D_  O  S  CM  - H  A O O J O J T - r- CO CO J Z I O C O C O C O C O T - T o o o o o o o o t_ Q . O O O O O O O  in  co co co co m  LO  V ) T - C 0 C 0 T - T - C 0 C O  L  0  )  0  0  -  -  ^  n  n  o  o  . n f c n c o c o c o c o c n o ) i30r»r-tOtOiOiO CTrTttCOnLflLf) W C O t - r - O C O r r O O O CO CO O) O) Ho o i n c o o j o i t f  CT t r CD  CO  CM O ) CO CO  H- W t  CO CM CO  c  c  TT-  !>.  T-  O  LO LO O O  A  O  o  o  o  o  o  Q  -  0  0  0  r  N N  o 0  V  0  t  0  ^  COCOLOCMCMi-CO^t ^•tfOOCOOT-ro o o o O O O O O O C O C D o o o o O O O O O O C D i 0  0  0  0  0  0  0  0  0  0  0  0  0  T-  T - T - CO CO I O  O o o  O O N N © © o o o o o o o o o o o o  0  0  0  0  .  T—  « CD L CO D CT CO  fO  S CO O CO CM  o  LO CM C0 N O) T - N. CO CO O ) CM (O O)  C O LU tl-  CO  C 0 C 0 O O S S C \ I C \ i n c 0 n i n s N c n c i i ' - r - t r N M C M i O L O O O C O C O O r - . n C O C O O O N N f f l t D O I D C 0 r - r - O ) O ) ( D C O L O i O C 0 t CO  CO CO CD tr  T-  o rLO co  irLO CM CO  CO  CO CO o O •* rCM rco  •»r  CO CO N CM o co  o  o  r-  *r -*  CO LO  o  O iO ID ^ cn CM co CM t r CM 1  CO IO CM CM CD CD Z  o rH  LU O LU  LU  LU  H  O  cr - i  CO rH co c r =3  <  or O  CM CM  CM CO  CO  o  lO ^r o>  O  O CO  w w n n r - v T t i o o s o f f l w o  CL 3  o  co cc o  L  o cs — i  0  COCOCDCDOCDCMCOCOOOOJCM — — COOJCOCOLOCMCMOOCOCO ._ _ r-0)i-%— h - C D O O O T T C O l D r - t O L n C O C O ' - r - c n c T ) COlOCDCOOCMCMi-T-CMCM ( o n o w o i o o t ' ' f c o c o o c o o c o o c o c o c n o c o c o tTCMOlOOCOCOCMCMCOCO  C O C O - t r ^ i - i - C O C O C O C O C M C M O J C O  ) 0 ) \ N o n n i O O O C l l f l n 0 C 0 C 0  0  «t  O O l O l f l r - r - N N f f l O ) T - T - o o t r t c M C M t r t r . 0 o O U C  0  LO  < C M '  — J  CO L 1  Q. L  y-  CM  L L r - i - ^ T - f - w c v c n o ) A O ^ ^ r i - O O W W IlOCDIOTfl-OONCM O O O O O O O O O O l_ 0 - 0 0 0 0 0 0 0 0 0  o n n T T t j T f i n i f i  WNtOCOCOCOCMCMTtt C ) ( I ) f T ' - r - - - > - 0 0  Ol CO LO T -  l U ' - O O ' - r C D t C O O n w w i n n n i D t Q i n i n Tfinioeococotococo  co o j o  co o  o o_ 3 i-H UJ O O C_ (d cc O  CM CO CO O O  o o o o o o o o o o  LL L O  LO 00 CD CO  Tt rCD CD ^JN TJ" CO  yf n nD S N CD C CM CD i-» CM T-  N  CM CD CD C D r-- O O CM i- ihS  N  CO CM CM CD  CD L CO = > cr CO  Tf ^ CO f N  S  C D  CO CM CO CO CN CO L O CD  lO CM O CM  CM CM CM LO O •  </l  co in O co O iS C CO CO CM L  co =) cr CO  CC  I-  LO CO CO  CO  LO  ^— CO CD  ^— CO CD  CD  CD  CO  s in co T CO o o rOi O) w CO N CO T—  ifl  -I UJ CO  CO cc => CO >  <  CM  in in  CD  w in  CO  to  ro  s to  y- f-  co co  o o o o o o o o o o o o o o o o o o o o o o o o TJ-  c o c o c o c o c o c o c o o s  o i n m t M C M O O c o c M c o o o c o c o c o o j o i c o c o c o n s m i n ' - t ' J I O C O C D C D ' r O C D O l O ) roinincocoT-r-oitOt-r^ • ^ • T l - C O C 0 ^ - T f ^ N . 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O CO CO CO CD CO "T 3 CM S O t-  CM CM  CO CO CO 00 CO  *  in  CO CM LO O CC -I  CO CM  CM  TTCO T-  rO O  o o  LO  co  O  o  L  1  1  L  MO  to  r-  a> as  T- t r ^ o O o o t_ Q -  tr  o o o eo CD  t f T f O O O O O O C O o o o o o o o o o o  0  0  0  0  0  0  0  0  0  0  o  0  o  0  O O O O O O o o o o o 0  0  0  0  0  0  n O O N N i n i O N N l f l N O ' t l C O C O C O C O « 0 ) 0 ) I J ) a O ) C i ) r - r - i O ( D C O C O C O C O C O ( D C O t o c o c o c v j c v j o o c o o o e o c D c o i - ^ - c o c o ^ r t f o r - r ^ c o c o i n i n o o c o c n c M o o o o o c o c o I f l f f l C O ' - ' - N N r - i - O t O C O C D t D N S N N • i - i o u i n n o o c o n w N ' f o i O / C M C j i o i o r - O O W C M ( O C O C O a ) 0 ' - ! f l ( D ( O S N * *  ininiofflffloooaiconossioi/i  CO  UJ  CO  co  co oo t r CO CO 3 CO L O « c r eo oo CM CO O O l o CM CO O) f- K) N C M t_  CO  CM r » CO CM CO ID  cn  o  00 ID o o  < cr UJ  z o  Q  01 O  Ol t C M C O C O C O ^ ^ I O O O l C M  i - C O C O r r O O O l O ) "CMCMIOIOCOCOIOCO COCO'r-T-CMCMOOCO o o o o o o o o o  o o  o  s  o  o  o  o  f  t  o  o  n  c  o  o  o  c  o  M  o o eoo oo  o  N  o  N  o  w  o  m  n  To-  v-  o  n  Oi-r-WCMVI-OOCOSTTCOCO CDCOCOCOCOLOIOCOCOLO'^-OIOID  C- o> CO CO CM O) 3 m o) CT LO O  CM CM i co n t CO CO O T - T - LO CD CO CO S N oi CO to LO ^ ^ CI  CO o i L O L O c o t r r ^ c o c o f CM C M O O O O L D C O O I O O O O) o i c n c n i o o o c o c o LO CO ( D N S I - r - l / j L O l O CO lOi n C M C M O O O C M C M a CM C M C M C M C O O L O t r t f LO t - T - o o c o i o o e o e o O) CO O ^ f l O N O C D C O  CMCMCM1 t*"tCMCMCOCOCO*'5,  tCO ZJ CT co  o hCO CO o  E  C M  t r TT L O CM CO CM 00 CM r - co  CO o  •o t_ -p  IO CO  CU CU  CD  O)  (0  o  e e (0 H C +J  CO CO D_ L U  c o T - o ^ c M c o o r - o o o T - T - o  CC CD O O  in  ifl  T-  O  [_  •  I" LO  •  O  O -  O O  O  L O O ) O ) N CO T - •T- T - O O in co co O O O O O C O O —  O  O  O  O  O  O  CO o o O O O O  O  O  S C O C O t t C D O N * r -  C O r - W C J C D C O N r - C U W ' - r l l ) L O < - t C O n O V r - ( D * 1  I  L C O C D C D C O C O T - C O I - C O O O  ffl^CJCJO)OJLOr-C0'r-0 )CO 3 1 0 N S N N C 0 C M C a c M o o o ) 0 ) t c o o ) ( C O W t O I O N S N C O f f l l C O O O r - T - N O C D C 4l  0 N O O o c o c o O C O C O O C O C O  c f i o i o o i o t o f f l i n T r *  cn co s  3 CT • * OJ CD CO O i ^ o O) T 1- CO LO O)  r-  Q  CO T  O T p j m O ' - W ' - o  C- rH  E  APPENDIX 9: Heights of advance and subsequent. Graph comparing the number of trees per height class in the Nelson data set to linearly predicted heights, heights predicted with the Prognosis model and height predicted with data-fitted Weibull function.  158  Height of excess regeneration comparison of models for grand fir  Height of excess regeneration comparison of models for Douglas fir  Height Classes  Height of excess regeneration comparison of models for western larch  Height Classes  Height of excess regeneration comparison of models for white pine  Height Classes  4  160  Height of excess regeneration comparison of models for subalpine fir  Height of excess regeneration comparison of models for hemlock  Height Classes  Height of excess regeneration comparison of models for lodgepole pine  Height of excess regeneration comparison of models for spruce  Height Classes  APPENDIX 10: Height of excess regeneration. Coefficients and model forms used for linear predictions of height of excess regeneration, heights predictions using Prognosis models, and height predictions with data-fitted Weibull functions by species.  163  LL O i - i - 00 C O A O O C O C O -O O f- T- C O C O O O C O C O i-  (_  0-  o o o o o m o o t  f  C O C O TO O O O r r (O C O L CO C O C M C M C O C O C O C O iN SD C M 1OI O) i CT CO 01 CU O O C O C O C O C O o o co cn cn LO in co co o oi  cr ^ co cn cn co CM  .  a) i n ^ o L T- C M T crj r^. 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IV OI CO CD co CM • CO  0_ 2 3 O  > C  CO CD O T— CM i-i T - * - CM CM CM  hO 2  >i TJ  CO <  +J CO  CM LO CD O O  O  CO  CO  co co O  CO C O CD O CO  CM CM CM CM CM •<*• •»* *t *t 1 "  CD  O CM O <t  C  CD C_ CO  LO O O CM CM O O O O  CD CO S S CM  4-* CD  O O 0) l_ C_ 3 CO O CD CO CC  TJ * < — I CO CU H  L L O — ( O O C O 3 •—-  m m CM  r-t  APPENDIX 11: Small tree height growth. Plotted residuals of small tree height growth predictions with Prognosis models and re-fitted Prognosis for Cw, Fd, Hw, PI, tolerant, intermediate tolerance and intolerant species.  178  • •  •o o E  •  u> 'S o  o L.  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Plotted residuals of small tree height growth predictions with variable selection model with Prognosis variables and a variable selection model with all site variables for tolerant, intermediate tolerance and intolerant species.  184  00  CD CO  • •  CO CO > (A  O ~ —  tri cu  O  T3  •  If •o q  •  •  • O) O  • •  «> CD Cfl as  C <  •••  d  co >  (6o|!)ue-6}i|) sienpjssy  •  • •  •  •  •4*  cn CM  in •  o  c  <  (6o|ijtifi-6m) sienpisay  APPENDIX13: Small tree height growth. 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