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Validation of the Jack pine version of CroBas-PipeQual Shcherbinina, Anna 2012

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  VALIDATION OF THE JACK PINE VERSION OF CROBAS-PIPEQUAL  by  ANNA SHCHERBININA B.Sc., Moscow State Forest University, Russia, 2001 Ph.D., Moscow State Forest University, Russia, 2004   A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Forestry)     THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2012 © Anna Shcherbinina, 2012   ii  ABSTRACT  The objective of this study was to validate the performance of the jack pine (Pinus banksiana L.) version of the growth simulator CroBas-PipeQual. A data-based approach and a simulation approach were applied. For the data-based validation, Ontario permanent sample plots (PSPs) were used as input to the CroBas-PipeQual simulator to obtain predicted values for growth of dbh, height, basal area per ha, changes in crown height and number of trees per ha. The corresponding growth (change) in these variables was compared with the measured growth (change) on the PSPs. For the simulation studies, I assessed the impact of different initial stand densities, different initial heights, and different values of the model parameters on the output variables and compared the simulated response to biological expectations.  CroBas-PipeQual simulator generally underestimated growth/change of all variables at the stand and single tree level using both default and adjusted values of alpha-r. Running the simulator with the adjusted values of alpha-r did not show marked improvements in the simulator´s performance.  The simulator generally produced logical results based on the simulations conducted. In denser stands, trees had a smaller average dbh at a given age, and basal area growth and change in number of trees were higher. Initial density had almost no influence on height and crown height growth. The number of stems per ha remained essentially the same on sites with different productivities, indicating that there may be problems with estimating mortality in the simulator. Varying the initial heights had almost no impact on the number of stems per ha, dbh growth and basal area growth.  Overall, the simulator performed reasonably. However, it requires further testing. More testing with suitable data will allow assessing the simulator´s applicability to other site types and geographical locations. Another area that needs attention is modelling mortality. Recalibrating some of the equation parameters internal to the jack pine version of CrosBas-PipeQual is recommended, followed by further testing. Finally, the accuracy of predictions of variables not evaluated in this study (e.g., branch sizes, numbers, and locations) should be assessed before these components of the simulator are used operationally.   iii  TABLE OF CONTENTS  ABSTRACT .................................................................................................................................... ii TABLE OF CONTENTS ............................................................................................................... iii LIST OF TABLES .......................................................................................................................... v LIST OF FIGURES ...................................................................................................................... vii ACKNOWLEDGMENTS ............................................................................................................. ix 1. Introduction ................................................................................................................................. 1 1.1 Purpose and Objectives ....................................................................................................................... 1 1.2 Background ......................................................................................................................................... 2 1.2.1 Growth and Yield Models ............................................................................................................ 2 1.2.2 Validation Techniques ................................................................................................................. 4 1.2.3 Structure of CroBas-PipeQual ..................................................................................................... 8 2. Methods..................................................................................................................................... 13 2.1 Data ................................................................................................................................................... 13 2.2 Validation Using Independent Data .................................................................................................. 15 2.3 Validation Using Simulations ........................................................................................................... 16 3. Results ....................................................................................................................................... 18 3.1 Validation Using Independent Data (Default Alpha-r) ..................................................................... 18 3.2 Validation Using Independent Data (Adjusted Alpha-r) .................................................................. 21 3.3 Equivalence Testing (Default Alpha-r) ............................................................................................. 22 3.4 Equivalence Testing (Adjusted Alpha-r) .......................................................................................... 32 3.5 Validation Using Simulations ........................................................................................................... 37 3.5.1 Changing Values of Alpha-r ...................................................................................................... 38 3.5.2 Changing the Initial Density ...................................................................................................... 39 3.5.3 Changing the Initial Height ........................................................................................................ 46 3.6 Summary ........................................................................................................................................... 54 4. Discussion ................................................................................................................................. 56 4.1 Validation Using Independent Data (Default and Adjusted Alpha-r) ............................................... 56 4.2 Validation Using Simulations ........................................................................................................... 59 4.3 Changing Values of Alpha-r ............................................................................................................. 60 5. Conclusion ................................................................................................................................ 62 5.1 Summary of Findings ........................................................................................................................ 62   iv  5.2 Suggestions for Further Work ........................................................................................................... 63 5.3 Final Statements ................................................................................................................................ 63 References ..................................................................................................................................... 65 APPENDIX ................................................................................................................................... 72                       v  LIST OF TABLES Table 1. Summary of the growing conditions for ecozones of Ontario Shield. ....................... 13 Table 2. Number of jack pine PSPs by category ..................................................................... 15 Table 3. List of plots used in the alpha-r simulations .............................................................. 17 Table 4. Number of PSPs for which growth/change was underestimated by  CroBas-PipeQual over the total number of PSPs in a category (for the default  value of alpha-r (0.59). ..................................................................................................... 19 Table 5. Actual average growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown  height, basal area, and number of stems/ha at the stand level for low productivity  sites (for the default value of alpha-r = 0.59). Positive growth/change differences highlighted. ....................................................................................................................... 19 Table 6. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown  height, basal area, and number of stems/ha at the stand level for high productivity  sites (for the default value of alpha-r = 0.59). Positive growth/change differences highlighted. ....................................................................................................................... 20 Table 7. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, and crown height at the individual-tree level for low productivity sites (for the default value of alpha-r = 0.59). Positive growth/changes differences highlighted. ............................................................ 20 Table 8. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, and crown height at the individual tree level for high productivity sites (for the default value of alpha-r = 0.59). ................................................................................................................................. 21 Table 9. Number of PSPs in which growth/change was underestimated by CroBas-  PipeQual over the total number of PSPs in a category using adjusted values of  alpha-r = 0.35 (low productivity stands), and alpha-r = 0.42 (high productivity  stands) at the stand level. .................................................................................................. 22 Table 10. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown  height, basal area, and number of stems/ha at the stand level for low productivity  sites (value of alpha-r = 0.35). Positive growth/change differences are highlighted. ...... 23 Table 11. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown  height, basal area, and number of stems/ha at the stand level for high productivity  sites (value of alpha-r =0.42). Positive growth/change differences are highlighted. ....... 23   vi  Table 12. Results of conducting equivalence tests for the intercept (default value of  alpha-r = 0.59) at the stand level. ..................................................................................... 25 Table 13. Results of conducting equivalence tests for the slope (default value of  alpha-r = 0.59) at the stand level. ..................................................................................... 28 Table 14. Results of conducting equivalence tests for the intercept (default value of  alpha-r = 0.59) at the individual tree level........................................................................ 29 Table 15. Results of conducting equivalence tests for the slope (default value of  alpha-r = 0.59) at the individual tree level........................................................................ 31 Table 16. Results of conducting equivalence tests for the intercept (default value of  alpha-r = 0.59) at the individual tree level (all trees combined, n = 1012). ..................... 33 Table 17. Results of conducting equivalence tests for the slope (default value  of alpha-r = 0.59) at the individual tree level (all trees combined, n = 1012). ................. 33 Table 18. Results of conducting equivalence tests for the intercept (adjusted values  of alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at  the stand level. .................................................................................................................. 34 Table 19. Results of conducting equivalence tests for the slope (adjusted values of  alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at the  stand level. ........................................................................................................................ 35 Table 20. Comparison of stand characteristics of the data used for model calibration  and model validation, different ecological zones. ............................................................ 58                  vii  LIST OF FIGURES Figure 1. Ecozones, ecoregions, and ecodistricts of Ontario (from Old Growth Policy for Ontario´s Crown Forests/version 1, 2003). ........................................................................... 14 Figure 2. (a) Measured change in crown height vs. predicted change in crown height and  (b) measured height growth vs. predicted height growth. Both graphs are for moderately- aged and older PSPs on high productivity sites with the default value for alpha-r (0.59). ... 24 Figure 3. Measured single tree height growth vs. predicted height growth on medium sites  for the default value of alpha-r (0.59).................................................................................... 32 Figure 4. Measured height growth vs. predicted height growth, all trees combined, for the  default value of alpha-r (0.59). .............................................................................................. 33 Figure 5. Effect of different values of alpha-r on output variables: (a) quadratic mean dbh;  (b) mean stand height; (c) basal area per ha; and (d) stems per ha. ....................................... 38 Figure 6. Effect of density on PSP NP9203 with alpha-r = 0.15: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 40 Figure 7. Effect of density on NP9203 with alpha-r = 0.59 (default value): (a) quadratic  mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal  area per ha; and (e) stems per ha. .......................................................................................... 41 Figure 8.  Effect of density on NP9203 with alpha-r = 1.03): (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 42 Figure 9. Effect of density on TIM9308 with alpha-r = 0.15: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 43 Figure 10. Effect of density on TIM9308 with alpha-r = 0.59 (default value): (a) quadratic  mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal  area per ha; and (e) stems per ha. .......................................................................................... 44 Figure 11. Effect of density on TIM9308 with alpha-r = 1.03: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 45 Figure 12. Jack pine density management diagram. Vertical lines represent simulated plot development through time (1- NP9203, 2 – TIM9308) (The density management  diagram was taken from Archibald and Bowling, 1994). ...................................................... 47 Figure 13. Effect of height on NP9203 with alpha-r = 0.15: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 48 Figure 14. Effect of height on NP9203 with alpha-r = 0.59: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 49   viii  Figure 15. Effect of height on NP9203 with alpha-r = 1.03: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 50 Figure 16. Effect of height on TIM9308 with alpha-r = 0.15: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 51 Figure 17. Effect of height on TIM9308 with alpha-r = 0.59: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 52 Figure 18. Effect of height on TIM9308 with alpha-r = 1.03: (a) quadratic mean dbh;  (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha;  and (e) stems per ha. .............................................................................................................. 53    ix  ACKNOWLEDGMENTS I am sincerely grateful to my supervisor Dr. Peter Marshall for his intellectual support and guidance throughout the thesis writing. I would also like to thank my committee members Dr. Abdel-Azim Zumrawi, Dr. Bruce Larson, Dr. Valerie LeMay, and Dr. Robert Guy for their help throughout this thesis.  It is a great pleasure to thank Dr. Robert Schneider and Dr. Venceslas-Claude Alexander Goudiaby from the University of Quebec for their support, Dr. Phil Comeau for the opportunity to visit the University of Helsinki (2011) and the annual ForValueNet meeting in Edmonton (2011), Dr. Annikki Mäkelä and Dr. Frank Berninger from the University of Helsinki for collaboration and ideas. This thesis would not have been possible without funds from ForValueNet project. This support is gratefully acknowledged. Finally I would like to thank my parents Vera Shcherbinina and Anatolii Shcherbinin for their life-long encouragement, and my friends for their kindness.              1  1. Introduction Many models of natural systems have been developed for management and research over the past several decades. Often, such models are used to forecast the future conditions of these systems. In forestry, growth and yield models are commonly used for this purpose at the scale of single trees or stands.  One of the most important components of model development and application is model validation. Model validation provides model users with information on the potential precision and accuracy of model projections, as well as providing information for subsequent model improvements. Validation may be defined as a procedure that helps to determine how useful and accurate the model predictions are (Marcot et al., 1983). In other words, validation is necessary to decide if “the model mimics the real world well enough for its stated purpose” (Giere, 1991, cited by Rykiel, 1996). Validation means that the model provides a match of its predictions and measurements for a specific application (Brand and Holdaway, 1983; Van Horn, 1971, cited by Power, 1993; Rykiel, 1996). Validation can take place during the process of building a model; however, it can also occur after the model has been developed (Goulding, 1979).   1.1 Purpose and Objectives  This research is a part of Theme 1 of the ForValueNet NSERC Strategic Network1. The objective of this network is to develop new models to simulate and optimize the forest-wood value chain. The project includes five themes. Themes 1 to 4 aim to develop models that predict stand and tree growth as well as wood quality. The last theme focuses on integrating the new models into a decision support system to maximize the value of a single tree using its external stem characteristics to predict internal wood quality. The ForValueNet is focused on four dominant tree species found in Canada's boreal forests: black spruce (Picea mariana (Mill.) BSP), white spruce (Picea glauca (Moench) Voss), jack pine (Pinus banksiana Lamb.) and trembling aspen (Populus tremuloides Michx.). Research conducted within Theme 1 is aimed at the development of a model which will simulate the tree/crown growth for the species considered in the Network. CroBas-PipeQual (Mäkelä, 1986, 1997) was the model architecture chosen by the Theme 1 scientists. My objective in this thesis is to conduct a validation of the jack pine version of CroBas-PipeQual developed by Robert Schneider at the University of Quebec at Montreal (Schneider et al., 2008a, 2008b, 2011a, 2011b; Beaulieu et al., 2011). This validation will provide a template that can be followed for validating other versions of CroBas-PipeQual that may be developed within this project.                                                            1  See http://www.forvaluenet-foretvaleur.ca for more information on ForValueNet.   2  Background information about growth and yield modelling, validation techniques and the structure of the CroBas-PipeQual can be found in the remainder of this chapter. Chapter 2 describes the data and methods used for validation. Results are presented in Chapter 3. Chapter 4 contains a discussion, and conclusions are presented in Chapter 5.  1.2 Background 1.2.1 Growth and Yield Models Growth and yield models are used by forest managers to predict future stand conditions based on changes in forest dynamics (growth, mortality, and reproduction) (Peng, 2000). According to Burkhart (1990; cited by Peng, 2000) growth and yield models can be classified into two categories based on variables that describe the growth process: whole stand and individual tree models. Whole stand models (e.g., yield curves and tables) are generally developed using data from permanent and temporary sample plots. They are constructed based on stand-level attributes and site parameters (basal area, age, site index, stocking, and stand volume). They generally provide accurate predictions for even-aged stands with a single species; however, their application to mixed-species forests is limited due to the difficulty describing the structure of mixed forests using a few stand-level variables (Vanclay, 1994). One of the advantages of whole stand models is that the input information is relatively easy to obtain. However, they do not provide detailed information about individual trees in a stand. Individual tree growth models are generally more accurate than whole stand models in predicting growth on different sites and for different conditions, allow more flexibility, provide detailed information about a tree (internal characteristics, stem quality), and can better describe the effect of diseases and insect attacks on stands, as well as stand responses to different treatments. However, individual tree models require more detailed input information, they are more expensive to develop, and they can introduce greater error (Weiskittel et al., 2011).  Individual tree models may be further separated into distance-independent models and distance- dependent models. Single-tree distance-dependent models require information about the spatial distribution of stems. Mapped stand data are generally expensive to obtain, and application/development of these models may be limited by lack of these data. Single-tree distance-independent models are generally precise and provide high resolution (Weiskittel et al., 2011).  Growth and yield models can also be divided into empirical, process-based (mechanistic), and hybrid groups based on their structure and type of processes that drive the simulation system (Kimmins, 1990; Mohren et al., 1994).     3  Empirical models are often constructed using inventory data and generally require simple input information. They translate measured data into mathematical relationships/functions (Le Roux et al., 2001). Empirical models are useful in summarizing large datasets. They can be used for various types of analyses, including exploring the impact of different treatments and regimes using simulations. However, they are not robust when predicting the influence of climate changes or environmental stress (Kimmins, 1990; Shugart et al., 1992; Korzukhin et al., 1996; Peng, 2000; Pretzsch et al., 2006, cited by Weiskittel et al., 2011).   Process-based models represent plant responses to environmental changes and human manipulations, such as fertilization and pruning, based on physiological processes (Kimmins, 1999; Peng, 2000). They describe major physiological attributes (e.g., photosynthesis, respiration, carbon allocation within a tree) using mathematical equations. Consequently, they are applicable to modeling the impact of changing conditions over the long term (Le Roux et al., 2001; Weiskittel et al., 2011). Photosynthesis is often considered to be the main growth process which is treated as a basis of carbon balance or as an independent variable. “Modeling forest growth in terms of carbon balance involves calculating assimilation of carbon and its distribution at different levels of organization in the stand.” (Mäkelä et al., 2000). Process-based models have some challenges. Among them there are initialization, parameterization, scale, and sensitivity (Weiskittel et al., 2011). This type of model usually requires site-specific information such as detailed climate and soil data, the accuracy and resolution of which might influence process-based model predictions. Equation parameters must be adjusted for each new species and site. It is necessary to use specialized equipment for measuring physiological attributes, which is expensive and may be time consuming. Data are usually obtained from specific sites, such as eddy-flux towers. Such data do not cover the range of variability of a particular ecosystem and it limits the application of process-based models to larger scales. Finally, this type of model is usually highly sensitive to parameters that control respiration, light interception, and photosynthesis (Weiskittel et al., 2011).  Hybrid models are a combination of statistical and process-based approaches to modeling growth (Kimmins, 1999; Kimmins et al., 2008; Mäkelä, 2009). They attempt to compensate for the limitations of both these approaches. Hybrid models can be applied to predict growth in short and long term (Peng, 2000). This approach allows more flexibility when considering the variability of climate and geographical differences, uses simpler input data than process-based models, and allows functions to be mechanistically represented (Weiskittel et al., 2011). Using the classification systems described above, CrosBas-PipeQual can be described as a hybrid, single-tree, distance-independent growth and yield model.    4  1.2.2 Validation Techniques Validation techniques can be divided into four main groups: (1) subjective assessment; (2) visual techniques; (3) deviance measures; and (4) statistical tests (Mayer and Butler, 1993; Vanclay and Skovsgaard, 1997). Sensitivity analysis and equivalence testing are also widely used for validation purposes. 1.2.2.1 Subjective Assessment This approach includes using the Turing test where experts are given both real-world and simulated data and are asked to distinguish between them. However, the results of this test can be easily misinterpreted. If the real and simulated data are sufficiently alike to offer a realistic test, they should be amenable to statistical testing which avoids potential difficulties with personal bias. If the data are unsuited to statistical testing, it is likely that they will contain certain identifiable features which may make the distinction between the two sets easy (Law and Kelton, 1992, cited by Mayer and Butler, 1993). 1.2.2.2 Visual Techniques This group of techniques includes plotting of simulated and empirical data against a common independent variable. The simulated data are usually represented by lines and the empirical data by points. Such data are most commonly presented as time series plots.  1.2.2.3 Deviance Measures Deviance measures use the differences between the simulated and measured values. Two measures are frequently used for numerical data – mean absolute error (MAE) and mean absolute percent error (MA%E) (Mayer and Butler, 1993). MAE =       ̂    and MA%E=       ̂           , where    are observed values,  ̂  are simulated values, and n is the number of pairs of observed and simulated values. Using the root mean square error (RMSE) is an alternative to using absolute differences, where RMSE=√ (    ̂ )   . 1.2.2.4 Statistical Tests Many statistical tests can be used to evaluate model performance. Usually several simple tests are combined to examine different aspects of model behavior (Vanclay and Skovsgaard, 1997). The type of statistical test that might be appropriate depends on the type of data available. Overall population tests (e.g., unpaired t-test) might be applied in the case of measuring the sample population (Conover, 1980, cited by Mayer and Butler, 1993). The paired t-test, regression analysis, and non-parametric sign test are applicable when paired samples are available.    5  The coefficient of determination R2 is used to compare model predictions to measurements; it indicates the degree of fit, and helps to evaluate model performance (Vanclay and Skovsgaard, 1997). Along with R2, model efficiency (EF), which directly compares predictions with measurements, may be used: EF=1-  (    ̂ )   (    ̅ )  . This statistic uses a relative scale to indicate goodness of fit or to evaluate performance, with the value equal to 1 for a “perfect” fit, 0 for the model behaving as an average, and a negative value indicating a worse fit than using the average (Mayer and Butler, 1993). The paired t-test provides a null hypothesis of mean difference between two samples (mean prediction error) equal to 0. Larger t values indicate larger mean prediction errors. t=  ̅ √  (    ) ̅̅ ̅  (   ) , where    is the difference between a measurement and a model prediction at one point, and  ̅ is the average of the differences. The t-test is widely used in practice; however, it has been shown to be inappropriate for model validation. Freese (1960) pointed out that the paired t-test “uses one form of accuracy (precision) to test for the other form (freedom from bias), frequently with anomalous results”.  The    test is an alternative to the paired t-test for testing the accuracy of growth models (Freese, 1960). However, the    test has some disadvantages: it rejects valid models too often, and it has subjectively stated accuracy (Yang et al., 2004). Different forms of the    test can be used (Yang et al., 2004): (1) the standard    test with degrees of freedom = n:    =  (    ̂ )    , where   =     , is an acceptable accuracy specified in the form of hypothesized variance or a critical error, E is an allowable error in units of the observed y, and   is a standard normal deviate at the specified probability level; (2) the bias free    test with degrees of freedom = n-1 (assuming a constant bias):       =   (    ̂ )        , where B =  (    ̂ )   (bias); and (3) the bias-free    test with degrees of freedom = n-2 (assuming the bias is a linear function of   ):      =        ̂   , where     is the observed values of ith observation,  ̂  is the predicted value of the ith observation, and SSE is the error sum of squares of the simple linear regression of          ̂ .      Evaluating a simple linear regression between observed and measured values (  =  +   ̂ ) is another widely used method for validating a model. The intercept   =0 and slope   =1 are tested separately using a t-test (Montgomery and Peck, 1992, cited by Yang et al., 2004).  The simultaneous F test is considered to be one of the most powerful tests for model validation. It is able to correctly reject invalid models and accept valid ones when prediction errors are independent of each other (Mayer et al., 1993). The test evaluates the regression by testing   6  intercept   =0 and slope   =1 simultaneously (Montgomery and Peck, 1992, cited by Yang et al., 2004): F=  (    )      ̂ (    )(    )   ̂  (    )    (    ̃ )  (   ) , where   ̃  is the predicted value from   =  +   ̂ . This test statistic has degrees of freedom = 2 for the numerator and degrees of freedom = n-2 for the denominator. The adequacy of validating the model by testing if the regression line goes through the origin with a 45º slope was shown by Kleijnen et al. (1998, cited by Yang et al., 2004).  The novel test was suggested for this purpose. It involves regressing the prediction errors on the sum of the measured and predicted values using simple linear regression:         (    ̂ ) and jointly testing the intercept and the slope (       ) using the extra sum of squares method: F= (   )(         )      , where     =∑    is the reduced sum of squared errors,     =∑(    ̂ )   is the full sum of squared errors. The numerator has two degrees of freedom and the denominator has n-2 degrees of freedom. The nonparametric Brown-Mood test was proposed as an alternative to the simultaneous F test and the novel test (Daniel, 1990, cited by Yang et al., 2004). It jointly tests whether the intercept and the slope of the regression line are equal to some hypothesized values. The regression line   =  +   ̂  is evaluated by testing whether   =0 and   =1. To calculate the test statistics, the data points are plotted as a scattered diagram, then a regression line   =  +   ̂  and a vertical line through the median of x-values are drawn. The test statistic is computed as   =   ×(     ⁄ )  (     ⁄ )  , where    is the number of data points above the hypothesized line   = ̂  and to the left of the vertical line, and    is the number of data points above the hypothesized line   = ̂  and to the right of the vertical line. The distribution of calculated test statistics follows approximately a    distribution with two degrees of freedom. The nonparametric Kolmogorov-Smirnov (KS) test allows two samples to be compared by calculating the distance between their empirical distribution functions. It can be also used to test normality of distribution as well as to evaluate if the prediction errors have a normal distribution with a mean of zero (Yang et al., 2004). Stephens (1974) developed a modified KS test (  ) which assumes normal distribution but unknown true mean and variance:   =D×(√           √ ⁄ ), where D  is the maximum vertical difference between the theoretical and sample cumulative distribution functions.  The sign test may be used to test the hypothesis that the median of prediction errors is zero. Numbers of positive and negative differences are calculated. Under the null hypothesis, the number of positive differences should be approximately the same as the number of negative differences. The test statistic follows a binomial distribution with probability p = 0.5 for a sign to be positive and probability q = 0.5 for a sign to be negative.   7  The Wilcoxon signed-rank test is used to compare paired data that comes from the same population.  It tests the null hypothesis that the median difference between the pairs is zero. Pairs are ranked and test statistic is calculated: T=         √         , where    is the rank of the i th observation with the corresponding sign and    is the number of observations with that sign. 1.2.2.5 Sensitivity Analysis Simulation exercises are often used to carry out sensitivity analyses to observe how the changing values of components or parameters might change the corresponding output (Soares et al., 1995; Breierova and Choudhari, 1996; Vanclay et al., 1996). Sensitivity analysis methods can be divided into three groups: mathematical, statistical and graphical (Frey and Patil, 2002). Mathematical methods assess sensitivity of a model output to the range of variation of an input. These methods also can be used for verification and validation, and identify inputs that require further data acquisition or research. They include nominal range sensitivity analysis, break-even analysis, difference in log-odds ratio, and automatic differentiation (Wotawa et al., 1997, cited by Frey and Patil, 2002). Statistical methods involve running simulations in which inputs are assigned probability distributions and assessing the effect of variance in inputs on the output distribution. Statistical methods allow identifying the effect of interactions among multiple inputs. These methods include Monte-Carlo simulation, Latin hypercube sampling, regression analysis, analysis of variance, response surface method, and the Fourier amplitude sensitivity test. Graphical methods are used to give visual indication of how an output is affected by variation in inputs (Frey and Patil, 2002). Sensitivity approaches have been used to validate such models such growth and yield models as STEM for red pine (Pinus resinosa Ait.) stands in USA (Gertner, 1987), LS-FVC for stands located in Ontario (Lacerte et al., 2004), PrognosisBC for Douglas-fir (Pseudotsuga menziesii var. glauca (Beissn.) Franco) stands in interior British Columbia (Marshall et al., 2008). 1.2.2.6 Equivalence Testing Equivalence testing is derived from bioequivalence testing (Robinson et al., 2005), which is used to compare the performance of generic drugs with established drugs. Recently, equivalence tests have been applied in psychological research, natural science research and for model validation (Robinson and Froese, 2004). The equivalence test looks for similarities between model predictions and measured data. The test assumes that the populations being compared are different and uses the data to prove otherwise. Equivalence testing includes the following steps: (1) choosing a test statistic and a size (α); (2) choosing a region of equivalence (I) which is close enough to the hypothesized value that the   8  difference is practically irrelevant. The mean of the difference is computed, and an upper and a lower one-sided 1 – α confidence interval constructed. The null hypothesis of dissimilarity is rejected if the two one-sided confidence intervals around the mean difference are entirely contained within the interval of equivalence. I chose to use equivalence testing for the data-based component of the validation I conducted on CroBas-PipeQual, supplemented by sensitivity analyses to assess its behavior under a range of different input conditions.  1.2.3 Structure of CroBas-PipeQual CroBas-PipeQual is an individual tree growth simulator based on three structural theories that govern the allocation of growth among different carbon pools (Mäkelä, 1986, 1997, Mäkelä and Hari, 1986): 1. the pipe model theory (Shinozaki et al., 1964a, b); 2. the principal of functional balance (Brouwer, 1962, cited by Mäkelä, 1986); and 3. an allometric relationship between crown surface area and foliage mass (Zeide and Pfeifer, 1991). A constant ratio between crown length and average branch length was also assumed for Scots pine. Crown diameter of trees was considered to be proportional to crown length   =  *  , where    is crown radius,    is crown length, and    is the ratio of crown radius to crown length (Mäkelä, 1997).  The pipe model theory was first formulated by Shinozaki et al. (1964). It describes a tree stem as consisting of a system of unit pipes. These pipes are connected to foliage thus providing water and nutrients as well as physical support to the crown. The theory states a constant ratio exists between sapwood area at the base of crown or branches and foliage mass. The pipe model theory can also be applied to coarse roots. Vaninen and Mäkelä (1999) found an approximately constant ratio between fine root biomass and coarse root sapwood for Scots pine (Pinus sylvestris L.) stands. Pipe model theory indicates that the sapwood area at some height and the foliage weight above this height are connected through a constant ratio: A=α*  ; where  α is an empirical constant, and   is foliage biomass (Mäkelä, 1986). White (1935, cited by Mäkelä, 1986) was the first to establish the principal of functional balance; it was later mathematically described by Brouwer (1962, cited by Mäkelä, 1986) and Davidson (1969a, 1969b). This principle postulates a balance between metabolic activities of roots and foliage. In other words, if    is foliage biomass,    is root biomass,    is specific photosynthetic activity, and    is specific root activity, then the ratio between carbon and nitrogen can be expressed as   *  *  =  *  , where    is an empirical parameter.    9  An allometric relationship between crown surface area and foliage mass means the balance between self-shading and consumption of carbon in the formation of branches is   =  *   , where z is the allometric exponent, 1 ≤z ≤ 1.5,    is the lateral crown surface area,    is a constant (Zeide and Pfeifer, 1992). The CroBas simulator2 (crown base) was originally constructed for Scots pine stands in Southern Finland (Mäkelä, 1997).  It included two structural modules (TREE and WHORL) and was used as a basis for developing a timber quality model called PipeQual (pipe model for wood quality predictions) (Mäkelä et al., 2000). Hereafter, the simulator will be referred to as CrosBas- PipeQual, to reflect the two components. CroBas-PipeQual uses five state variables for biomass (e.g., foliage, fine roots, branch sapwood, stem sapwood, and transport root sapwood) and three state variables for dimensions (e.g., crown length, height of crown base and sapwood area of crown base) to describe tree growth (Mäkelä, 1997). Dimensional variables can be converted to biomass ratios using constant form factors for sapwood and heartwood in stem, branches and coarse roots. The simulator is applied to the life span of a tree and has a time resolution of one year.  The simulator incorporates self-thinning at the stand level and self-pruning at the single tree level. However, it does not consider the effect of changes in site quality and climate, nor multiple age classes. Further, it cannot simulate disturbances such as diseases, fires, and wind throw. Also, regeneration (ingrowth) is not included in the simulator. Site productivity is indirectly provided through the site specific parameter alpha-r (  ) that represents the below ground to above ground biomass ratio. A modular approach is used within the simulator where the three modules (TREE, WHORL, and BRANCH) interact through sharing inputs and outputs (Mäkelä and Makinen, 2003). The TREE module calculates biomass and dimensional variables that are used by the WHORL module to determine the distributions of foliage biomass, sapwood area, branch length over whorls, as well as heartwood area (Mäkelä, 2002). Growth is derived from the carbon balance based on photosynthesis, respiration, and resource allocation between and within trees (Mäkelä, 2009). Individual trees are combined to form a stand. The stand is represented by several tree size classes (n ≤ 10), with each class represented by a mean tree.  TREE calculates annual tree growth as the difference between photosynthesis (P) and maintenance respiration(  ): G=    =   (P-  ), where Y is a factor that converts units of photosynthesis to units of dry weight and includes carbon losses due to growth respiration (Mäkelä, 1997). Photosynthesis depends on the foliage weight/biomass of the tree but self- shading and shading by neighbors reduces the leaf-specific photosynthesis rate (  ): P=                                                               2  The term “simulator” will be used for CrosBas-PipeQual the remainder of this thesis to allow easier differentiation between the overall model (the simulator) and its component equations (individual models).   10  (Mäkelä, 1990, cited by Mäkelä, 2002). Maintenance respiration is assumed proportional to the dry weight of the live parts of a plant (foliage, fine roots, and sapwood):    =   (  +  ) +   (  +  +  ), where    and    are empirical parameters, and   ,   ,   ,   ,    are the biomass of foliage, fine roots, stem, branches, and transport roots respectively (Mäkelä and Hari, 1986; Mäkelä, 1997;  Mäkelä, 2002).  The net growth of each biomass compartment is calculated as the difference between gross growth    and the rate of senescence           =   -  ; (Mäkelä, 1999). The rate of senescence is assumed to be proportional to the dry weight of certain components:   =   , i=f, r, b, where f is foliage, r is fine roots, and b is branches. The dimensional variables (crown length,   ; crown radius,   ; and the transport root system radius,   ) are converted to biomass variables using structural parameters (Mäkelä and Albrektson, 1992; Mäkelä, 2002). The following equations are used to obtain sapwood weights of stem (  ) , branches  (  ) , and transport roots(  ) :    = (     +     )      and    =         , i = b, t, where    is the density of wood,    and    are empirical form constants, and    is the sapwood area at the crown base. The WHORL module controls vertical structure of the stem, and is updated every year based on the pipe model theory, assuming a fixed shape for the vertical foliage distribution, using information from the TREE module (Mäkelä, 2002). TREE calculates annual height growth through computing growth rates of the crown and stem. Annual height growth is used to determine the location of whorls. The active and disused pipe area of the stem and branches in the whorl, internode length, mean branch length for the whorl, and foliage attached to the whorl are used to describe the whorl. Total tree foliage obtained from the TREE module is used to calculate the foliage in each whorl. A constant relative distribution of total foliage between the top of the tree and the crown base is assumed. Pipe model theory provides the basis for calculating stem and branch sapwood areas. The area of disused pipes is updated each year by adding the area of wood that is no longer connected to live foliage. As the foliage moves upwards with crown rise, the disused pipe area increases (Mäkelä, 2002; Mäkelä and Mäkinen, 2003).  The simulator uses a β-function to describe the vertical foliage biomass distribution, which is determined by two parameters p and q. Foliage density at relative crown depth x is calculated as:   (x)=    (   ) ∫   (   )     , where   is the total foliage mass obtained from the TREE module. In each whorl, branch active pipe area is assumed proportional to foliage mass in that whorl:     =       , where     is the branch pipe area,    is foliage mass, and    is the coefficient of proportionality (Mäkelä and Vanninen, 2001, cited by Mäkelä, 2002).    11  The active pipe area of the stem at the base of a whorl is assumed proportional to foliage mass above the whorl:     =   (  )        , where    is the coefficient of proportionality for stems, and     is the active pipe area of the stem at the base of whorl. Mean branch length at whorl i,    , is calculated as     =        , where     and z are parameters (z < 1), and     (m) is canopy depth at whorl i (Mäkelä, 2002).  The BRANCH module calculates the initial number, size distribution, compass angle, and insertion angle of branches in the new whorl, the rate of branch death within the crown, the change of the insertion angle and size distribution over time, and the rate of self-pruning of branches below the live crown (Mäkelä and Mäkinen, 2003). The BRANCH module uses information from the TREE (tree height, height of crown base, stem diameter at breast height, and annual height increment) and the WHORL (the total basal area of branches in the whorl, whorl age) modules as input. Compass angle and height in the stem are used to describe the location of a branch. Each branch is initiated from the pith at the bottom of the stem internode, with the initial diameter of zero. The insertion angle and branch diameter are determined each year, to give a curvature and size to the branch. When a branch dies, its diameter growth ceases, and a dead knot is formed.  The total basal area of live branches and the active-disused pipe ratio in each whorl calculated by the WHORL module are distributed between individual branches in whorls by the BRANCH module. The following equation is used to calculate the initial number of branches in a whorl:    = 2.3757+1.0496 ln (Δh) -        (     )      +   , where Δh is the height increment (cm), H is the tree height (m),      is breast height diameter (cm), and    is a normally distributed random variable for individual whorls with mean 0 and variance 1.0302. The insertion angle, α(  ) in degrees of branches is calculated as: ln (α(   )) = 3.8690-0.0050     +0.1772 ln(w)- 0.0019    ln(w)+  +  , where w is the whorl age,    is the variance component for individual trees with mean 0 and variance 0.0070, and    is a normally distributed random number with mean 0 and variance 0.0380 (Mäkelä, 2002).  The WHORL module calculates the total basal area of the live branches in a whorl     , which are used along with the number of live branches in the whorl (  ) to obtain the mean basal area,      . Relative branch sizes (    ) are computed from a uniform distribution as       =        (1+  ),          is a uniform random number from the range (-1, 1) (Mäkelä, 2002).   CroBas-PipeQual was calibrated for jack pine in Eastern Canada (Schneider et al., 2008, 2010). Jack pine has both nodal and inter-nodal whorls, and foliage distribution differs between them. The beta function was applied to predict foliage biomass distribution depending on the foliage type. Tree-level and stand-level variables influenced the inter-nodal foliage distribution, whereas the nodal foliage distribution only was affected by tree-level variables.    12   The validation undertaken in this thesis is for the jack pine version of the CroBas-PipeQual simulator and focuses on the output from the TREE module.  Predicted branch characteristics were not assessed.    13  2. Methods I used two principal approaches to evaluate the jack pine version of the simulator: data-based validation and simulation-based validation. To conduct the data-based validation, data from Ontario permanent sample plots (PSPs) were used as input to the CroBas-PipeQual simulator to obtain predicted values for several variables (growth of dbh, height, crown height, basal area per ha, and changes in number of trees per ha). The corresponding growth (change) in these variables was compared with the measured growth (change) on the PSPs. For the simulation studies, I assessed the impact of different initial stand densities, different initial heights, and different values of alpha-r on the output variables and compared the simulated responses to biological expectations.   2.1 Data The PSPs used for validation were located across Northern Ontario in the following ecoregions: 3E, 3S, 3W, 4S, and 4W (Figure 1). Table 1 contains a summary of the growing conditions for the various ecoregions.  The PSPs consist of three circular growth subplots that were arranged within a bigger mortality plot (6398 m2) on radii spaced 1200 apart. The first radius was selected randomly. Mortality plots were used to measure tree mortality and downed woody debris. Individual tree characteristics were sampled on the growth plots. Each growth plot had a radius of 11.28 m (400 m2) and contained three circular shrub plots to sample shrub composition and percent cover. Three PSPs in the dataset contained growth plots with radii of 7.98 m (200 m2) and one PSP contained growth plots with radii of 9.77 m (300 m2). Data on tree seedlings, tree regeneration, tree saplings, low shrubs and ground vegetation were collected on the shrub plots or on smaller regeneration plots arranged within the shrub plot (Hayden et al., 1995). Only data obtained from the growth plots were used in the validation.  Table 1. Summary of the growing conditions for ecozones of Ontario Shield. Ecozone # of plots in each ecozone Annual mean daily temperature, ⁰C Total precipitation, mm Growing degree days above 5⁰C 3E 38 -1 800 1350 3W 7 0 800 1300 3S 7 -0.5 675 1450 4S 19 1.5 700 1600 4W 2 2 750 1550   14    Figure 1. Ecozones, ecoregions, and ecodistricts of Ontario (from Old Growth Policy for Ontario´s Crown Forests/version 1, 2003).   15  The PSPs were established in stands that originated between 1944 and 1987. These stands were naturally regenerated, seeded, or planted. Most PSPs were measured twice (with five or ten years between measurements); some of them were measured three times (with five years between measurements). In the latter case I only used first and last measurements. Species, origin (natural, seeded, planted), diameter at breast height outside bark (dbh), status (live/dead), and crown class were available for each tree in the sample plots. For selected trees, the total height of the tree and height to the base of the live crown were also available.  The dataset included 208 clusters of subplots with jack pine as the leading species. Other tree species present in the PSPs were white birch (Betula papyrifera Marshall), trembling aspen (Populus tremuloides Michx.), black spruce (Picea mariana Mill.) and balsam fir (Abies balsamea (L.) Mill.). Ninety PSP clusters were selected that contained at least 80 percent jack pine by basal area. Subsequently, 17 of these PSPs with negative basal area growth (i.e., abnormally high levels of mortality) were eliminated. Most of those plots were of moderate or older age and most of them were found on high productivity sites. The reminder of the PSPs (73 in total) were divided into 18 groups according to stand density classes (low, medium, high), age classes (young – 0 to 40 years, moderate – 41 to 80 years, old – 81 years and more) and site productivity classes (low and high). See Appendix 1 for a list of PSPs by category and Table 2 for numbers of plots by category.  Table 2. Number of jack pine PSPs by category.  Low productivity High productivity     Age Dens. Young  Moderate  Old  Young  Moderate  Old  Low 4  1 - 16 - - Med. 1 2 2 16 7 2 High 3 6 - 1 9 3  2.2 Validation Using Independent Data CroBas-PipeQual was run for each of the PSPs in the test dataset using starting data from the first measurement to obtain predictions of conditions at the second measurement (generally a prediction period of 5 or 10 years). Starting values were subtracted from the predicted values to obtain predicted change (growth) in the selected variables.   16  For assessing single-tree dbh (cm), height (m), and crown height (m) growth predictions, trees that were measured for height and crown height at both the first and second occasion in a PSP were selected as representative trees within the PSP. The number of trees per ha represented by each of these trees was determined from a stand table using 5 cm dbh classes. For example, if there were y trees per ha in a dbh class and x trees with measured heights, each measured tree represented y/x trees per ha.  For assessing stand-level growth predictions, growth in quadratic mean dbh (cm), average height (m), average height to the base of live crown (m), number of trees per ha, and basal area per ha (m2) were predicted. Plot data at the time of the first and second measurement were first expressed in a stand table using 5 cm dbh classes. A midpoint tree from each dbh class at the time of the first measurement was input into the simulator to represent all trees in that class. Height and crown height for each representative tree were predicted from dbh. The simulator was then used to predict the values of these variables at the time of the second plot measurement. Actual growth in the selected variables over the measurement period was determined from the differences in the stand tables between the first and second measurement period. Predicted growth was calculated as the difference between the values of the variables obtained from the simulator following the growth period and the original values provided as the starting condition. Values of basal area obtained from stand tables with 5 cm dbh classes were occasionally quite different (more than 1 m2) from values calculated based on all measured trees due to averaging. To avoid these differences I constructed stand tables based on 2 cm dbh classes for basal area growth, and obtained an average dbh for all trees in a particular class. Total height of trees and height to the crown base were calculated using equations based on the average dbh of the trees in a certain dbh class.  I used equivalence tests (Robinson et al., 2005) to examine the null hypotheses of dissimilarity between the predictions of growth from the simulator and the measured growth for each category of PSP and on a plot-by-plot basis using matching trees. I also performed this test using all matching trees combined without regarding site productivity and stand density. Differences among PSP categories were assessed to determine if there appeared to be any patterns (trends) in average differences among the factors that defined the categories (age, density, and site productivity). Regions of equivalence were set as follows: ±10 % for the intercept and ±25 % for the slope (after Froese and Robinson, 2007).  2.3 Validation Using Simulations The simulations were designed to compare the development of stands with different starting conditions. A range of initial stand structures was produced. Input parameters included dbh, height, crown length and age for each diameter class of trees. I evaluated the simulator for a range of stand density and height by:   17  1. changing density (number of stems per ha for two typical PSPs were increased two, four and eight times and decreased two and four times); and 2. changing height in each dbh class (decreased height for two typical PSPs by 50%, 25% and 10% as well as increased height by 50%, 25% and 10%).  Two plots with the same age at the first time of measurements (16 years) were selected to provide baseline starting conditions: one plot (TIM9308) had low productivity and low initial density; the other (NP9203) had high productivity and low density. The simulator was run for 150 years for these starting conditions and three different values of the site parameter alpha-r (    - the ratio of fine root biomass to foliage biomass: 0.15; 0.59; 1.03). The value of this parameter should be higher at low productivity sites and lower at high productivity sites. The variables of interest were graphed after growing the stands for 10, 25, 50, 75, 100, 125 and 150 years. I also ran simulations for seven young plots using a range of values of alpha-r (  ) between 0.15 and 1.03 (in total, 12 values). I selected two plots (one each with good and poor predictions from the simulator using default value of alpha-r (0.59) for the different initial conditions, except for the high density and high productivity condition where only one plot was selected, and ran these plots for 150 years. See Table 3 for a listing of the plots used in the alpha-r simulations. The predicted values of the variables were plotted against the range of values of alpha-r.    I also ran the model separately for PSPs with low and high productivity using different values of alpha-r (ranging from 0.15 to 1.03) to get the value of the total height of trees from the simulator close to the measured values. Values of alpha-r were averaged over each of the plots, and the adjusted alpha-r parameter was used to rerun all PSPs for the data-based validation.     Table 3. List of plots used in the alpha-r simulations. Density Productivity PSP with good predictions PSP with poor predictions Low Low TIM9312 BP9307 High TIM9309 CHA9303 High Low BP9333 BP9327 High KLK9206      18  3. Results In this chapter I will present results of testing the CroBas-PipeQual simulator against independent data from Ontario PSPs using both the default value of alpha-r (   = 0.59 for all groups of PSPs) and adjusted values of alpha-r (   = 0.35 for low productivity sites and    = 0.42 for high productivity sites). In addition, the long-term impacts of different starting tree heights, stand densities, and values of alpha-r on simulator output are presented.   3.1 Validation Using Independent Data (Default Alpha-r) Dbh growth, height growth, crown height change, basal area growth, and changes in number of trees were underestimated for most PSPs at the stand level using the CroBas-PipeQual simulator with the default value of alpha-r (Table 4). Dbh growth was underestimated for most of the different initial conditions. However, it was slightly overestimated for low productivity, moderately-aged PSPs with low densities and for older PSPs with medium densities, as well as for high productivity, older PSPs with medium densities (Tables 5-6). At the individual-tree level, the simulator overestimated dbh growth for several low density, moderately-aged, low productivity PSPs, but otherwise dbh growth was underestimated (Tables 7-8).     The simulator overestimated height growth at the stand level for high productivity, older PSPs of high density (Tables 5-6). At the individual tree level, the simulator overestimated height growth for low productivity, moderately-aged PSPs with low densities (Tables 7-8).  Change in the height to the base of live crown was overestimated (i.e., actual crown lift occurred more slowly) at the stand level for low density, younger PSPs of low productivity, for low and high density, moderately-aged PSPs of low productivity, for moderately-aged PSPs of high density and high productivity, and for low density, young PSPs of high productivity (Tables 5-6). At the individual tree level, the simulator overestimated crown height change for low productivity moderately-aged PSPs with low densities (Tables 8-9). Basal area growth was overestimated for low productivity, younger PSPs of high density, for low productivity, moderately-aged PSPs of low, medium and high density, for low productivity, older PSPs of medium density, as well as for high productivity moderately-aged PSPs of high density and older PSPs of medium density (Tables 5-6). The simulator underestimated the change in number of stems per ha (mortality) for all categories of PSPs, except for low productivity, moderately-aged PSPs with low densities and high productivity younger PSPs with low densities and older PSPs with high densities (Tables 5-6).   19  Note that positive differences in the tables equate to underestimating mortality since the changes in stems per ha are negative; this is an opposite interpretation from the other variables. Table 4. Number of PSPs for which growth/change was underestimated by CroBas-PipeQual over the total number of PSPs in a category (for the default value of alpha-r (0.59).   Low Productivity High Productivity     Age Dens.  Young Moderate Old Young Moderate Old      Level Var.1 Stand Tree Stand Tree Stand Tree Stand Tree Stand Tree Stand Tree Low  Dbh 4/4 3/4 0/1 0/1 - - 11/16 14/16 - - - -  Ht 4/4 4/4 1/1 0/1 - - 12/16 16/16 - - - -  Crht 1/4 1/4 0/1 0/1 - - 4/16 4/16 - - - -  Basa 4/4 - 0/1 - - - 10/16 - - - - - Number 2/4 - 1/1 - - - 15/16 - - - - - Medium  Dbh 1/1 1/1 2/2 2/2 1/2 2/2 16/16 16/16 7/7 7/7 1/2 2/2  Ht 1/1 1/1 2/2 2/2 1/2 2/2 14/16 16/16 6/7 7/7 2/2 2/2  Crht 1/1 1/1 2/2 2/2 1/2 2/2 10/16 10/16 4/7 6/7 - -  Basa 1/1 - 0/2 - 1/2 - 14/16 - 4/7 - 0/2 - Number 0/1 - 0/2 - 1/2 - 1/16 - 0/7 - 1/2 - High  Dbh 3/3 3/3 5/6 4/6 - - 1/1 1/1 9/9 9/9 2/3 3/3  Ht 3/3 3/3 3/6 5/6 - - 1/1 1/1 6/9 9/9 2/3 3/3  Crht 2/3 3/3 2/6 3/6 - - 1/1 0/1 2/9 4/9 2/3 2/3  Basa 0/3 - 1/6 - - - 1/1 - 0/9 - 2/3 - Number 0/3 - 0/6 - - - 0/1 - 0/9 - 1/3 - Overall  Dbh 8/8 7/8 7/9 6/9 1/2 2/2 28/33 31/33 16/16 16/16 3/5 5/5  Ht 8/8 8/8 6/9 7/9 1/2 2/2 27/33 33/33 12/16 16/16 4/5 5/5  Crht 4/8 5/8 4/9 5/9 1/2 2/2 15/33 14/33 6/16 10/16 2/3 2/3  Basa 5/8 - 1/9 - 1/2 - 25/33 - 4/16 - 2/5 - Number 2/8 - 1/9 - 1/2 - 16/33 - 0/16 - 2/5 -  1  Dbh represents growth in quadratic mean diameter at breast height at the stand level and growth in diameter at breast height at the tree level (cm);  Ht represents average tree height growth at the stand level and individual tree height growth at the tree level (m);  Crht represents the average change in height to the base of the live crown at the stand level and change in individual tree height to base of the live crown at the tree level (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha.  Table 5. Actual average growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown height, basal area, and number of stems/ha at the stand level for low productivity sites (for the default value of alpha-r = 0.59). Positive growth/change differences highlighted.   Dbh Growth (cm) Height Growth (m) Crown Height Change (m) Basal Area Growth (m2/ha) Change in Number of Trees Per Ha Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 5.21 -2.04 4.14 -1.70 1.97 0.38 8.74 -4.93 -214 83 Medium 2.36 -1.25 2.71 -0.63 2.65 -0.60 5.4 -3.29 -200 60 High 2.30 -1.23 2.79 -1.22 1.86 -0.25 2.54 0.65 -1019 737 Moderate Low 1.62 0.45 1.92 -0.63 -0.09 1.21 3.12 0.05 -50 -101 Medium 1.87 -0.66 1.94 -0.98 3.18 -2.30 0.81 1.39 -321 204 High 1.67 -0.48 0.89 -0.20 0.23 0.41 0.92 1.65 -450 297 Old Medium 0.89 0.42 0.59 -0.15 0.43 -0.10 0.67 0.64 -59 10 Overall Average 2.27 -0.68 2.14 -0.79 1.46 -0.18 3.17 -0.55 -330 184   20  Overall, the simulator with the default value of alpha-r underestimated dbh growth and height growth for most PSPs at the stand level. Crown height change and basal area growth were underestimated for almost half of plots, and change in number of stems was mostly underestimated. At the individual tree level, the simulator underestimated changes in the variables of interest on most PSPs.   Table 6. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown height, basal area, and number of stems/ha at the stand level for high productivity sites (for the default value of alpha-r = 0.59). Positive growth/change differences highlighted.   1 There appeared to be measurement errors for crown height in the two PSPs in this category, so no results are reported.  Table 7. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, and crown height at the individual-tree level for low productivity sites (for the default value of alpha-r = 0.59). Positive growth/changes differences highlighted.             Dbh Growth (cm) Height Growth (m) Crown Height Change (m) Basal Area Growth (m2/ha) Change in Number of Trees Per Ha Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 4.26 -0.35 3.74 -0.84 2.19 0.54 8.91 -1.63 -180 -14 Medium 3.12 -1.56 3.12 -0.68 2.47 -0.06 7.51 -2.93 -704 431 High 1.39 -0.40 2.52 -0.74 1.97 -0.23 5.27 -0.85 -967 401 Moderate Medium 2.29 -1.28 1.97 -1.07 1.07 -0.23 1.74 -0.26 -332 224 High 1.80 -0.89 1.23 -0.49 0.14 0.55 0.15 1.42 -362 240 Old Medium 1.00 0.49 1.16 -0.72 --1 --1 0.20 1.60 -83 12 High 1.27 -0.01 1.01 0.13 1.14 -0.08 2.00 -0.62 -128 -25 Overall Average 2.16 -0.57 2.11 -0.63 1.50 0.08 3.68 -0.47 -394 181  Dbh Growth (cm) Height Growth (m) Crown Height Change (m) Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 4.65 -0.79 4.12 -1.85 2.38 -0.59 Medium 2.42 -1.79 3.32 -2.12 3.04 -1.83 High 2.74 -1.77 2.52 -1.96 2.11 -1.20 Moderate Low 1.37 1.02 0.33 0.21 -0.39 0.90 Medium 1.51 -0.15 1.11 -0.70 3.28 -2.88 High 1.92 -0.83 0.93 -0.97 0.22 -0.14 Old Medium 1.46 -0.08 0.79 -0.72 1.26 -1.23 Overall Average 2.30 -0.63 1.87 -1.16 1.70 -1.00   21  3.2 Validation Using Independent Data (Adjusted Alpha-r) Adjusting the values of    for low and high productivity sites improved simulator predictions relative to the actual growth on the PSPs. The number of PSPs for which CroBas-PipeQual underestimated values of growth/change decreased overall (Table 9).  However, the number remained the same for low productivity, young PSPs with medium densities, moderately-aged PSPs with low and medium densities, and for high productivity moderately-aged PSPs with medium densities.  Table 8. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, and crown height at the individual tree level for high productivity sites (for the default value of alpha-r = 0.59).           1 There appeared to be measurement errors for crown height in the two PSPs in this category, so no results are reported.   The simulator gave better predictions in terms of absolute values of predicted growth/changes using the adjusted values of    compared to using the default value (Tables 10-11). Predictions were closer to observations for dbh growth, height growth, crown height change, and basal area growth on low productivity PSPs compared to high productivity PSPs. There were no improvements in predictions of the number of stems per ha.            Dbh Growth (cm) Height Growth (m) Crown Height Change (m) Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 5.14 -1.77 4.18 -1.75 2.53 -0.29 Medium 3.56 -2.27 3.54 -1.64 3.15 -1.26 High 1.40 -0.35 2.34 -0.67 2.00 -0.34 Moderate Medium 2.28 -1.24 1.53 -1.11 1.20 -0.79 High 1.58 -0.73 1.24 -1.01 0.44 -0.22 Old Medium 1.20 -0.24 0.33 -0.28 --1 --1 High 1.70 -0.75 0.80 -0.74 0.60 -0.55 Overall Average 2.41 -1.05 1.99 -1.03 1.65 -0.58   22  Table 9. Number of PSPs in which growth/change was underestimated by CroBas-PipeQual over the total number of PSPs in a category using adjusted values of alpha-r = 0.35 (low productivity stands), and alpha-r = 0.42 (high productivity stands) at the stand level.   Density     Variable1 Low Productivity High Productivity Young Moderate Old Young Moderate Old Low  Dbh 2/4 0/1 - 8/16 - -  Ht 3/4 1/1 - 11/16 - -  Crht 0/4 0/1 - 3/16 - -  Basa 2/4 0/1 - 9/16 - - Number 2/4 1/1 - 13/16 - - Medium  Dbh 1/1 2/2 1/2 16/16 7/7 1/2  Ht 1/1 2/2 1/2 12/16 6/7 1/2  Crht 1/1 2/2 1/2 4/16 4/7 0/2  Basa 1/1 0/2 0/2 13/16 4/7 0/2 Number 0/1 0/2 1/2 1/16 0/7 1/2 High  Dbh 3/3 5/6 - 1/1 8/9 2/3  Ht 2/3 2/6 - 1/1 6/9 0/3  Crht 1/3 2/6 - 0/1 1/9 2/3  Basa 0/3 0/6 - 0/1 0/9 2/3 Number 0/3 0/6 - 0/1 0/9 1/3 Overall  Dbh 6/8 7/9 1/2 25/33 15/16 3/5  Ht 6/8 5/9 1/2 24/33 12/16 1/5  Crht 2/8 4/9 1/2 7/33 5/16 2/5  Basa 3/8 0/9 0/2 22/33 4/16 2/5 Number 2/8 1/9 1/2 14/33 0/16 2/5  1  Dbh represents growth in quadratic mean diameter at breast height (cm);  Ht represents average tree height growth (m);  Crht represents the average change in height to the base of the live crown (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha.   3.3 Equivalence Testing (Default Alpha-r) Equivalence tests were employed for testing the null hypothesis of dissimilarity between simulator predictions and observed data (Robinson et al., 2005). The following variables were assessed: dbh growth, height growth, crown height change, basal area growth, and change in number of stems per ha at the stand level; and dbh growth, height growth, and crown height change at the individual tree level. PSPs were combined based on the productivity and density classes. I also ran the test using all measured trees combined on low and high productivity PSPs at the tree level, and all PSPs combined on low and high productivity sites at the stand level.       23  Table 10. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown height, basal area, and number of stems/ha at the stand level for low productivity sites (value of alpha-r = 0.35). Positive growth/change differences are highlighted.   Table 11. Average actual growth/change ( y ) and difference between average predicted growth/change and average actual growth/change ( yy ˆ ) for dbh, height, crown height, basal area, and number of stems/ha at the stand level for high productivity sites (value of alpha-r =0.42). Positive growth/change differences are highlighted.  1 There appeared to be measurement errors for crown height in the two PSPs in this category, so no results are reported.   Examples of output graphs are provided below for change in crown height and height growth for moderately-aged and older PSPs on high productivity sites (Figure 2). In these graphs, the black solid slanted line is the regression line of predicted values versus observed values. The grey  Dbh Growth (cm) Height Growth (m) Crown Height Change (m) Basal Area Growth (m2/ha) Change in Number of Trees Per Ha  Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 4.24 0.98 3.79 -0.43 1.97 0.80 6.25 0.24 -173 18 Medium 2.36 -1.06 2.71 -0.17 2.65 -0.14 5.4 -2.68 -200 63 High 2.30 -1.06 2.79 -0.77 1.86 0.15 2.54 1.50 -1019 741 Moderate Low 1.62 0.56 1.92 -0.38 -0.09 1.47 3.12 0.21 -50 -86 Medium 1.87 -0.56 1.94 -0.80 3.18 -2.12 0.81 1.75 -321 207 High 1.67 -0.29 0.89 -0.03 0.23 0.57 0.92 2.40 -450 299 Old Medium 0.89 0.64 0.59 -0.07 0.43 -0.02 0.67 1.05 -59 11 Overall Average 2.13 -0.11 2.09 -0.38 1.46 0.10 2.82 0.64 -325 179  Dbh Growth(cm) Height Growth (m) Crown Height Change (m) Basal Area Growth (m2/ha) Change in Number of Trees Per Ha  Age Density  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅)  ̅ ( ̂- ̅) Young Low 4.26 -0.01 3.74 -0.42 2.19 0.96 8.91 -0.69 -180 -8 Medium 3.12 -1.39 3.12 -0.29 2.47 0.33 7.51 -2.14 -704 437 High 1.39 -0.27 2.52 -0.46 1.97 0.05 5.27 0.03 -967 410 Moderate Medium 2.29 -1.20 1.97 -0.94 1.07 -0.09 1.74 0.02 -332 226 High 1.80 -0.80 1.23 -0.38 0.14 0.66 0.15 1.82 -362 241 Old Medium 1.00 0.70 1.16 -0.23 --1 --1 0.20 1.90 -83 19 High 1.27 0.13 1.01 0.25 1.14 0.06 2.00 -0.04 -128 -22 Overall Average 2.16 -0.41 2.11 -0.35 1.50 0.33 3.68 0.13 -394 186   24  vertical line3  shows the mean of predictions and its associated confidence intervals for the intercept. The grey horizontal bar represents the equivalence region for the intercept to test bias. If the vertical line falls entirely within the grey bar, then the null hypothesis of dissimilarity may be rejected. The diagonal dotted lines represent the equivalence region for the slope. The black vertical line is a confidence interval for the slope. If it falls within the angle between the dotted lines, then the slope of the regression line is close to 1 (R, Package Equivalence, 2006). These figures show that the confidence intervals for both slope and intercept for stand height growth and change in crown height did not fall within their specified region of equivalence (±10% for the intercept and ±25% for the slope). Consequently, the null hypothesis of dissimilarity was not rejected for any of these simulations. The results of the equivalence tests for all variables and groupings of plots at the stand level for both the intercept and the slope for the default value of alpha-r are given in Tables 12-13. Even though the null hypothesis of dissimilarity was not rejected for any variables, predictions were similar to measurements for changes in crown height for young PSPs with low and high productivity and for medium density PSPs. Also, the differences (biases) were smaller than the region of equivalence for dbh growth and changes in number of stems for low density stands.     (a)       (b)  Figure 2. (a) Measured change in crown height vs. predicted change in crown height and (b) measured height growth vs. predicted height growth. Both graphs are for moderately- aged and older PSPs on high productivity sites with the default value for alpha-r (0.59).                                                           3  The grey vertical line is superimposed on the black vertical line. It is visible from the grey cross bars on the vertical line. Predicted height growth(m) M ea su re d he ig ht  g ro w th (m ) -1 0 1 2 3 4 0.5 1.0 1.5 Predicted crown height growth (cm) A ct ua l c ro w n he ig ht  g ro w th  (c m ) -3 -2 -1 0 1 2 3 0.2 0.4 0.6 0.8 1.0 1.2 1.4  25  Growth/changes of all variables were more biased for older PSPs of both low and high productivity than in younger PSPs (Tables 12-13). Dbh growth, change in crown height, and basal area growth in younger PSPs with low productivity had higher percent bias than those variables in younger PSPs with high productivity. A different pattern emerged in older stands. Dbh growth, height growth, and change in crown height had higher percent bias in stands of high productivity; basal area growth was less biased in high productivity stands, and change in number of trees remained the same.  No clear trend was observed for dbh growth, height growth, change in crown height, and change in number of trees across density classes; however, bias for basal area growth increased with increasing density. The minimum rejection interval decreased for dbh growth, increased for height growth, changes in crown height, and basal area growth, and remained almost the same for change in number of trees in older PSPs of low productivity. Growth/change of all variables had bigger rejection intervals for older PSPs with high productivity than in younger PSPs. The rejection intervals were less for growth/change of all variables in younger PSPs with high productivity than in younger PSPs with low productivity. In older PSPs, dbh growth and height growth had bigger rejection intervals on high productivity sites, while change in crown height, basal area growth, and change in number of trees had bigger rejection intervals for low productivity sites. Height growth and change in crown height were less variable on less dense PSPs. However, there was no apparent relationship between stand density and dbh growth, basal area growth, and change in number of trees.  The null hypothesis of dissimilarity was not rejected for any variables when running equivalence tests for all PSPs combined into groups of low and high productivity. Only for change in crown height for low high productivity PSPs were predictions similar to measurements. Differences between predictions and measurements were less than the region of equivalence for changes in crown height for low and high productivity PSPs and for basal area for low productivity PSPs.     Change in crown height and basal area growth in high productivity PSPs had higher bias than in PSPs of low productivity. Height growth and change in number of trees had lower bias in PSPs of high productivity; dbh growth was similar for PSPs in both productivity classes. Biases for most variables and most groups of PSPs were negative, which means that predicted values of variables were smaller than the measured ones. The minimum rejection intervals decreased for high productivity PSPs for all variables.     26  Table 12. Results of conducting equivalence tests for the intercept (default value of alpha-r = 0.59) at the stand level.  Prod. Level / Density Age Variable1  ̅2  ̂3 CI4 IR5 Reject H0 Min. Rejection Interval (%) % Bias                     Low  Young  Dbh 3.28 2.85 1.49 5.07 2.57 3.14 No 78 -13  Ht 3.09 2.23 2.12 4.05 2.01 2.45 No 81 -28  Crht 2.01 1.92 1.56 2.46 1.73 2.11 No 28 -4  Basa 4.75 4.09 2.16 7.34 3.68 4.50 No 79 -14 Number -494 -204 -892 -99 -224 -184 No 337 -59 Moderate + old  Dbh 1.56 1.29 1.36 2.00 1.16 1.42 No 55 -17  Ht 1.12 0.75 0.56 1.68 0.68 0.83 No 124 -33  Crht 0.77 0.67 -0.29 1.83 0.60 0.74 No 173 -13  Basa 1.05 2.33 0.15 1.95 2.10 2.56 No 94 122 Number -319 -127 -539 -99 -140 -114 No 324 -60 High Young   Dbh 3.61 2.68 3.19 4.03 2.41 2.95 No 50 -26  Ht 3.40 2.64 3.01 3.79 2.38 2.90 No 44 -22  Crht 2.32 2.54 1.99 2.66 2.29 2.79 No 22 10  Basa 8.11 5.88 6.89 9.33 5.29 6.47 No 59 -28 Number -457 -243 -700 -214 -267 -219 No 188 -47 Moderate + old  Dbh 1.81 1.05 1.49 2.13 0.95 1.16 No 103 -42  Ht 1.44 0.82 0.92 1.96 0.74 0.90 No 139 -43  Crht 0.42 0.76 -0.24 1.08 0.68 0.84 No 132 81  Basa 0.95 1.54 0.27 1.63 1.39 1.69 No 82 62 Number -292 -117 -372 -211 -129 -105 No 218 -60 Low Density  Dbh 4.13 3.95 3.42 4.84 3.56 4.35 No 23 -4  Ht 3.66 2.79 3.08 4.24 2.51 3.07 No 52 -24  Crht 2.04 2.53 1.58 2.50 2.28 2.78 No 38 24  Basa 8.13 6.70 6.34 9.91 6.03 7.37 No 48 -18 Number -172 -185 -316 -28 -204 -166 No 85 8 Medium Density  Dbh 2.53 1.37 2.13 2.93 1.23 1.51 No 114 -46  Ht 2.46 1.70 2.00 2.92 1.53 1.87 No 72 -31  Crht 1.78 1.65 1.19 2.37 1.49 1.82 No 44 -7  Basa 4.70 3.21 3.21 6.19 2.89 3.53 No 93 -32 Number -491 -191 -738 -244 -210 -172 No 286 -61 High Density  Dbh 1.74 1.06 1.48 2.00 0.95 1.17 No 89 -39  Ht 1.31 0.95 0.89 1.73 0.86 1.05 No 82 -28  Crht 0.62 0.90 0.06 1.18 0.81 0.99 No 93 45  Basa 1.17 2.17 0.48 1.86 1.95 2.39 No 78 86 Number -471 -177 -632 -310 -195 -159 No 257 -62           27   Table 12. Results of conducting equivalence tests for the intercept (default value of alpha-r = 0.59) at the stand level (continued).  Prod. Level / Density Age Variable1  ̅2  ̂3 CI4 IR5 Reject H0 Min. Rejection Interval (%) % Bias                     Low productivity  Dbh 2.49 1.64 1.57 3.41 1.48 1.80 No 108 -34  Ht 2.02 1.31 1.33 2.71 1.18 1.44 No 107 -35  Crht 1.29 1.24 0.65 1.93 1.12 1.36 No 56 -4  Basa 3.14 2.79 1.43 4.85 2.51 3.07 No 74 11 Number -425 -160 -640 -210 -176 -144 No 300 -62 High productivity  Dbh 2.92 1.98 2.55 3.29 1.78 2.18 No 66 -32  Ht 2.64 1.95 2.24 3.04 1.76 2.15 No 56 -26  Crht 1.58 1.87 1.17 1.99 1.68 2.06 No 37 18  Basa 5.56 4.24 4.26 6.86 3.82 4.66 No 62 -24 Number -398 -207 -549 -247 -228 -186 No 165 -48   1  Dbh represents growth in quadratic mean diameter at breast height (cm);  Ht represents average tree height growth (m);  Crht represents the average change in height to the base of the live crown (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha. 2   ̅ – average measured growth/change on PSPs. 3  ̂ – average predicted growth/change using simulator. 4 Confidence interval for the intercept. 5 Indifference region.  Estimates of the slope were higher than 1 for: (1) changes in number of trees for all groups of PSPs; (2) height growth and change in crown height for moderately-aged and older PSPs with low productivity; and (3) change in crown height for moderately-aged and older PSPs with high productivity, as well as for high density PSPs (Table 13). Estimates of the slope were negative for dbh growth for moderately-aged and older PSPs of low and high productivity, as well as for high density PSPs, and for basal area growth for young PSPs with low productivity and for moderately-aged and older PSPs with high productivity.   Dbh growth, height growth, change in crown height and basal area growth had bigger rejection intervals for older PSPs of low and high productivity; the rejection interval for change in number of trees was smaller for the same PSPs. Growth/changes of all variables had smaller rejection intervals in younger, high productivity PSPs than in younger, low productivity PSPs. However, the rejection intervals for dbh growth and basal area growth were larger for older, high productivity PSPs. Height growth, change in crown height, and change in number of trees had bigger rejection intervals for low productivity PSPs. There was no apparent relationship with density.    28  Table 13. Results of conducting equivalence tests for the slope (default value of alpha-r = 0.59) at the stand level.  Prod. Level / Density Age Variable1   2 CI 3 IR4 Reject H0 Minimum Rejection Interval (%)                     Low Young  Dbh 0.41 -0.11 0.93 0.75 1.25 No 111  Ht 0.64 -0.68 1.96 0.75 1.25 No 168  Crht 0.69 -0.01 1.39 0.75 1.25 No 101  Basa -0.08 -1.24 1.08 0.75 1.25 No 124 Number 4.17 1.29 7.05 0.75 1.25 No 605 Moderate + old  Dbh -0.44 -1.72 0.83 0.75 1.25 No 172  Ht 2.03 0.85 3.23 0.75 1.25 No 223  Crht 1.99 -1.60 5.57 0.75 1.25 No 457  Basa 0.42 -0.84 1.68 0.75 1.25 No 184 Number 3.72 2.31 5.11 0.75 1.25 No 411 High Young   Dbh 0.42 0.24 0.59 0.75 1.25 No 41  Ht 0.57 0.26 0.88 0.75 1.25 No 12  Crht 0.35 0.05 0.65 0.75 1.25 No 35  Basa 0.31 -0.04 0.66 0.75 1.25 No 104 Number 3.73 2.75 4.71 0.75 1.25 No 371 Moderate + old  Dbh -0.67 -1.58 0.24 0.75 1.25 No 258  Ht 0.97 -0.43 2.37 0.75 1.25 No 143  Crht 2.41 0.80 4.02 0.75 1.25 No 302  Basa -0.001 -0.99 0.99 0.75 1.25 No 199 Number 1.90 -0.04 3.84 0.75 1.25 No 284 Low Density  Dbh 0.32 0.03 0.60 0.75 1.25 No 97  Ht 0.54 -0.06 1.13 0.75 1.25 No 106  Crht 0.64 0.21 1.06 0.75 1.25 No 79  Basa 0.12 -0.45 0.69 0.75 1.25 No 145 Number 2.33 1.20 3.46 0.75 1.25 No 246 Medium Density  Dbh 0.79 0.23 1.36 0.75 1.25 No 77  Ht 0.77 0.53 1.02 0.75 1.25 No 47  Crht 0.72 0.32 1.12 0.75 1.25 No 68  Basa 0.88 0.46 1.30 0.75 1.25 No 54 Number 3.72 2.81 4.63 0.75 1.25 No 363 High Density  Dbh -0.10 -1.06 0.85 0.75 1.25 No 206  Ht 0.87 -0.01 1.74 0.75 1.25 No 101  Crht 1.56 0.50 2.63 0.75 1.25 No 163  Basa 0.83 0.21 1.45 0.75 1.25 No 79 Number 2.45 1.54 3.37 0.75 1.25 No 237     29  Table 13. Results of conducting equivalence tests for the slope (default value of alpha-r = 0.59) at the stand level (continued).  Prod. Level / Density Age Variable1   2 CI 3 IR4 Reject H0 Minimum Rejection Interval (%)                     Low Productivity  Dbh 0.99 0.63 1.36 0.75 1.25 No 37   Ht 1.63 1.28 1.99 0.75 1.25 No 99   Crht 0.82 0.14 1.50 0.75 1.25 No 86   Basa 1.47 0.25 2.68 0.75 1.25 No 168  Number 3.43 2.44 4.42 0.75 1.25 No 342 High Productivity  Dbh 0.52 0.35 0.68 0.75 1.25 No 65   Ht 0.81 0.58 1.04 0.75 1.25 No 42   Crht 0.75 0.49 1.01 0.75 1.25 No 51   Basa 0.90 0.63 1.18 0.75 1.25 No 37  Number 2.17 1.44 2.89 0.75 1.25 No 189  1  Dbh represents growth in quadratic mean diameter at breast height (cm);  Ht represents average tree height growth (m);  Crht represents the average change in height to the base of the live crown (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha. 2 Slope of the relationship between measured and predicted growth/change. 3 Confidence interval for the slope of the relationship between measured and predicted growth/change. 4 Indifference region.   Height growth, change in crown height, basal area growth, and change in number of trees had smaller rejection intervals in stands of high productivity; the rejection intervals for dbh growth were smaller in stands of low productivity.   The results of the equivalence tests for the intercept and the slope for single-tree dbh growth, height growth, and change in crown height using the default value of alpha-r are given in Tables 14 and 15, respectively. The null hypothesis of dissimilarity could not be rejected for any of the variables. Only predictions of crown height change for low density stands were close to measured values. All growth/changes were more biased for older PSPs than younger PSPs, except for dbh growth, which had higher percent bias in younger PSPs of low productivity. Dbh growth, height growth, and change in crown height were better predicted for the younger and older PSPs with low productivity than for high productivity PSPs. Bias increased in denser PSPs for height growth and change in crown height, but there was no clear trend with density for dbh growth predictions. The minimum rejection interval was larger for older PSPs for growth/change of all variables, except for dbh. Growth/change of all variables had smaller rejection intervals in younger, high productivity PSPs than in younger, low productivity PSPs. Dbh growth had more variability (bigger rejection intervals) in older PSPs of high productivity. Height growth and change in    30  crown  height  had  bigger  rejection  intervals  in  low  productivity  PSPs.  Height growth and change in crown height were less variable on less dense PSPs. No apparent relationship existed between stand density and variability for dbh growth predictions.  Table 14. Results of conducting equivalence tests for the intercept (default value of alpha-r = 0.59) at the individual tree level.  Prod. Level / Density Age Variable1  ̅2  ̂3 CI4 IR5 Reject H0 Minimum Rejection Interval (%) % Bias                     Low  Young  Dbh 3.10 1.39 2.81 3.39 1.25 1.53 No 144 -55  Ht 3.33 1.49 3.13 3.53 1.34 1.64 No 137 -55  Crht 2.36 1.42 2.15 2.57 1.28 1.56 No 81 -40 Moderate + old  Dbh 1.99 1.24 1.70 2.28 1.12 1.36 No 84 -38  Ht 1.25 0.18 1.04 1.46 0.16 0.20 No 711 -86  Crht 1.39 0.15 1.00 1.78 0.14 0.17 No 1086 -89 High Young   Dbh 4.23 2.30 4.07 4.39 2.07 2.53 No 91 -46  Ht 3.93 2.25 3.82 4.04 2.03 2.48 No 80 -43  Crht 2.81 2.18 2.68 2.94 1.96 2.40 No 35 -22 Moderate + old  Dbh 1.79 0.92 1.65 1.92 0.83 1.01 No 109 -49  Ht 1.42 0.25 1.29 1.55 0.23 0.28 No 520 -82  Crht 0.34 0.23 0.03 0.65 0.21 0.25 No 182 -32 Low Density  Dbh 4.71 2.88 4.50 4.92 2.59 3.17 No 71 -39  Ht 4.19 2.29 4.05 4.33 2.06 2.52 No 89 -45  Crht 2.49 2.16 2.35 2.63 1.94 2.38 No 22 -13 Medium Density  Dbh 2.97 1.30 2.81 3.13 1.17 1.43 No 141 -56  Ht 2.82 1.37 2.67 2.97 1.23 1.51 No 117 -51  Crht 2.36 1.36 2.11 2.61 1.22 1.50 No 92 -42 High Density  Dbh 1.85 0.92 1.72 1.98 0.83 1.01 No 115 -50  Ht 1.52 0.42 1.40 1.64 0.38 0.46 No 290 -72  Crht 0.75 0.40 0.55 0.95 0.36 0.44 No 138 -47  1   Dbh represents single-tree dbh growth;  Ht represents single-tree height growth; and  Crht represents change in crown height. 2   ̅ – average measured growth/change on PSPs. 3  ̂ – average predicted growth/change using simulator. 4 Confidence interval for the intercept. 5 Indifference region.    31  Table 15. Results of conducting equivalence tests for the slope (default value of alpha-r = 0.59) at the individual tree level.  Prod. Level / Density Age Variable1   2 CI 3 IR4 Reject H0 Minimum Rejection Interval (%)                     Low  Young  Dbh 0.36 0.24 0.48 0.75 1.25 No 76  Ht 0.63 0.43 0.83 0.75 1.25 No 57  Crht 0.48 0.21 0.74 0.75 1.25 No 79 Moderate + old  Dbh -0.09 -0.63 0.45 0.75 1.25 No 163  Ht 2.04 1.26 2.81 0.75 1.25 No 181  Crht 3.17 1.80 4.54 0.75 1.25 No 354 High Young   Dbh 0.45 0.36 0.54 0.75 1.25 No 64  Ht 0.63 0.56 0.71 0.75 1.25 No 44  Crht 0.44 0.34 0.54 0.75 1.25 No 66 Moderate + old  Dbh 0.66 0.41 0.91 0.75 1.25 No 59  Ht 0.93 0.50 1.36 0.75 1.25 No 50  Crht 3.07 2.13 4.01 0.75 1.25 No 301 Low  Dbh 0.35 0.26 0.44 0.75 1.25 No 74  Ht 0.57 0.46 0.68 0.75 1.25 No 54  Crht 0.57 0.45 0.69 0.75 1.25 No 55 Medium  Dbh 0.31 0.11 0.50 0.75 1.25 No 89  Ht 0.70 0.80 0.94 0.75 1.25 Yes 20  Crht 0.75 0.59 0.91 0.75 1.25 No 41 High  Dbh 0.66 0.36 0.97 0.75 1.25 No 64  Ht 0.85 0.65 1.04 0.75 1.25 No 35  Crht 1.36 1.03 1.70 0.75 1.25 No 70  1   Dbh represents single-tree dbh growth;  Ht represents single-tree height growth; and  Crht represents change in crown height. 2 Slope of the relationship between measured and predicted growth/change. 3 Confidence interval for the slope of the relationship between measured and predicted growth/change. 4 Indifference region.   The null hypothesis of dissimilarity was rejected for single-tree height growth on medium density sites (Figure 3). The estimates of the slope were greater than 1 for height growth and crown height change for moderately-aged and older PSPs of low productivity, crown height change for moderately-aged and older PSPs of high productivity and crown height change for high density PSPs (without regard to site productivity and stand age). Negative estimates for the slope were observed for dbh growth for moderately-aged and older PSPs of low productivity.   32   Figure 3. Measured single tree height growth vs. predicted height growth on medium sites for the default value of alpha-r (0.59).  The minimum rejection intervals were larger for dbh growth, height growth, and change in crown height in older PSPs of low and high productivity, except for dbh growth in high productivity PSPs. Growth/change of all variables had smaller rejection intervals in high productivity PSPs than in low productivity PSPs.  Tables 16-17 show results of the equivalence tests for the three individual tree variables for both intercept and slope for the default value of alpha-r. The null hypothesis was not rejected for dbh growth and crown height change for either the intercept or the slope. However, the null hypothesis could be rejected for the slope for height growth (Figure 4). All slopes were positive but less than 1.  3.4 Equivalence Testing (Adjusted Alpha-r) The equivalence test results using the adjusted alpha-r values for low and high productivity PSPs are given in Tables 18 and 19. None of the null hypotheses of dissimilarity were rejected for any of the variables at the stand level. However, predicted dbh and basal area growth for young, low productivity PSPs were close to the observations, as were predictions of dbh and crown height growth for moderately-aged and older PSPs with low productivity, height growth for young PSPs with high productivity, and dbh, height, and basal area growth for low density PSPs.     33   Figure 4. Measured height growth vs. predicted height growth, all trees combined, for the default value of alpha-r (0.59).  Table 16. Results of conducting equivalence tests for the intercept (default value of alpha-r = 0.59) at the individual tree level (all trees combined, n = 1012). Variable 1  ̅2  ̂3 CI4 IR5 Reject H0 Minimum Rejection Interval (%) % Bias                      Dbh 3.21 1.72 3.09 3.33 1.55 1.89 No 96 -47  Ht 2.89 1.40 2.79 2.99 1.26 1.54 No 114 -52  Crht 1.95 1.35 1.82 2.08 1.22 1.49 No 54 -31  1   Dbh represents single-tree dbh growth;  Ht represents single-tree height growth; and  Crht represents change in crown height. 2   ̅ – average measured growth/change on PSPs. 3  ̂ – average predicted growth/change using simulator. 4 Confidence interval for the intercept. 5 Indifference region.  Table 17. Results of conducting equivalence tests for the slope (default value of alpha-r = 0.59) at the individual tree level (all trees combined, n = 1012). Variable   1 CI 2 IR3 Reject H0 Minimum Rejection Interval (%)                      Dbh 0.68 0.61 0.75 0.75 1.25 No 39  Ht 0.98 0.92 1.03 0.75 1.25 Yes 8  Crht 0.81 0.72 0.90 0.75 1.25 No 28  1 Slope of the relationship between measured and predicted growth/change. 2 Confidence interval for the slope of the relationship between measured and predicted growth/change. 3 Indifference region.   34  Table 18. Results of conducting equivalence tests for the intercept (adjusted values of alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at the stand level. Prod. Level / Density Age Variable1  ̅2  ̂3 CI4 IR5 Reject H0 Minimum Rejection Interval (%) % Bias                     Low  Young  Dbh 3.28 3.24 1.49 5.07 2.92 3.56 No 57 -1  Ht 3.09 2.76 2.12 4.05 2.48 3.04 No 47 -11  Crht 2.01 2.45 1.56 2.46 2.21 2.70 No 36 22  Basa 4.75 5.09 2.16 7.34 4.58 5.60 No 58 7 Number -494 -199 -892 -99 -179 -219 No 348 -60 Moderate + old  Dbh 1.56 1.47 1.36 2.00 1.32 1.62 No 36 -6  Ht 1.12 0.92 0.56 1.68 0.83 1.01 No 83 -18  Crht 0.77 0.84 -0.29 1.83 0.76 0.92 No 118 9  Basa 1.05 2.91 0.15 1.95 2.62 3.20 No 95 177 Number -319 -124 -539 -99 -136 -112 No 335 -61 High Young   Dbh 3.61 2.84 3.19 4.03 2.56 3.12 No 42 -21  Ht 3.40 3.11 3.01 3.79 2.80 3.42 No 22 -8  Crht 2.32 3.01 1.99 2.66 2.71 3.31 No 34 30  Basa 8.11 6.91 6.89 9.33 6.22 7.60 No 35 -15 Number -457 -257 -700 -214 -283 -231 No 172 -44 Moderate + old  Dbh 1.81 1.16 1.49 2.13 1.04 1.28 No 84 -36  Ht 1.44 0.99 0.92 1.96 0.89 1.09 No 98 -31  Crht 0.42 0.93 -0.24 1.08 0.84 1.02 No 126 121  Basa 0.95 1.94 0.27 1.63 1.75 2.13 No 86 104 Number -292 -115 -372 -211 -127 -103 No 223 -61 Low  Dbh 4.13 4.09 3.42 4.84 3.68 4.50 No 18 -1  Ht 3.66 3.34 3.08 4.24 3.01 3.67 No 27 -9  Crht 2.04 3.12 1.58 2.50 2.81 3.43 No 49 53  Basa 8.13 8.01 6.34 9.91 7.21 8.81 No 24 -1 Number -172 -215 -316 -28 -237 -193 No 87 25 Medium  Dbh 2.53 1.53 2.13 2.93 1.38 1.68 No 92 -40  Ht 2.46 2.02 2.00 2.92 1.82 2.22 No 45 -18  Crht 1.78 1.98 1.19 2.37 1.78 2.18 No 40 11  Basa 4.70 3.82 3.21 6.19 3.44 4.20 No 62 -19 Number -491 -187 -738 -244 -206 -168 No 295 -62 High  Dbh 1.74 1.20 1.48 2.00 1.08 1.32 No 67 -31  Ht 1.31 1.13 0.89 1.73 1.02 1.24 No 53 -14  Crht 0.62 1.08 0.06 1.18 0.97 1.19 No 94 74  Basa 1.17 2.84 0.48 1.86 2.56 3.12 No 83 143 Number -471 -180 -632 -310 -206 -162 No 251 -62            35   Table 18. Results of conducting equivalence tests for the intercept (adjusted values of alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at the stand level (continued). Prod. Level / Density Age Variable1  ̅2  ̂3 CI4 IR5 Reject H0 Minimum Rejection Interval (%) % Bias                     Low productivity  Dbh 2.49 2.10 1.57 3.41 1.89 2.31 No 62 -16  Ht 2.02 1.71 1.33 2.71 1.54 1.88 No 58 -15  Crht 1.29 1.56 0.63 1.95 1.40 1.72 No 60 21  Basa 3.14 4.03 1.44 4.84 3.63 4.43 No 64 28 Number -425 -168 -645 -205 -184 -151 No 284 140 High productivity  Dbh 2.92 2.19 2.55 3.29 1.97 2.41 No 50 -25  Ht 2.64 2.28 2.24 3.04 2.05 2.51 No 33 -14  Crht 1.58 2.20 1.17 1.99 1.98 2.42 No 47 39  Basa 5.56 4.98 4.26 6.86 4.48 5.48 No 38 -10 Number -398 -202 -549 -247 -222 -182 No 172 -49  1  Dbh represents growth in quadratic mean diameter at breast height (cm);  Ht represents average tree height growth (m);  Crht represents the average change in height to the base of the live crown (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha. 2   ̅ – average measured growth/change on PSPs. 3  ̂ – average predicted growth/change using simulator. 4 Confidence interval for the intercept. 5 Indifference region.   The minimum rejection interval and bias decreased for most PSP groups using the adjusted values of alpha-r compared to the default value. However, the minimum rejection interval increased for: (1) crown height change for young PSPs for both productivity levels; (2) basal area growth for moderately-aged and older PSPs with high productivity and high density; and (3) change in the number of stems for young, moderately-aged and older PSPs of both productivity levels, and for low and medium density stands. The bias increased for: (1) crown height change for young PSPs for both productivity levels, moderately-aged and older PSPs of high productivity, and low, medium and high density stands; (2) basal area change for moderately- aged and older PSPs of both productivity levels, and high density PSPs; and (3) change in number of stems for low density stands. Dbh growth and change in crown height for high productivity PSPs had higher bias than in stands of low productivity. Basal area growth and change in number of trees had lower bias in stands of high productivity; height growth was similar in PSPs of low and high productivity. Bias for most variables was negative, which meant that predicted values of variables were smaller than the measured ones.   36  Table 19. Results of conducting equivalence tests for the slope (adjusted values of alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at the stand level. Prod. Level / Density Age Variable1   2 CI 3 IR4 Reject H0 Minimum Rejection Interval (%)                     Low  Young  Dbh 0.38 -0.09 0.84 0.75 1.25 No 109  Ht 0.55 -0.48 1.58 0.75 1.25 No 148  Crht 0.52 -0.09 1.13 0.75 1.25 No 109  Basa -0.04 -1.03 0.94 0.75 1.25 No 103 Number 4.22 1.41 7.04 0.75 1.25 No 604 Moderate + old  Dbh -0.61 -1.75 0.52 0.75 1.25 No 275  Ht 1.65 0.59 2.70 0.75 1.25 No 170  Crht 1.39 -1.57 4.35 0.75 1.25 No 257  Basa 0.19 -0.95 1.33 0.75 1.25 No 195 Number 3.87 2.58 5.17 0.75 1.25 No 417 High Young   Dbh 0.43 0.27 0.59 0.75 1.25 No 73  Ht 0.46 0.19 0.72 0.75 1.25 No 81  Crht 0.27 0.02 0.53 0.75 1.25 No 98  Basa 0.37 0.08 0.66 0.75 1.25 No 92 Number 2.53 1.47 3.59 0.75 1.25 No 259 Moderate + old  Dbh -0.60 -1.38 0.19 0.75 1.25 No 238  Ht 0.82 -0.38 2.02 0.75 1.25 No 138  Crht 1.75 0.26 3.24 0.75 1.25 No 224  Basa -0.04 -0.96 0.88 0.75 1.25 No 196 Number 2.05 0.13 3.98 0.75 1.25 No 298 Low  Dbh 0.50 0.29 0.71 0.75 1.25 No 71  Ht 0.42 -0.06 0.91 0.75 1.25 No 106  Crht 0.47 0.10 0.84 0.75 1.25 No 90  Basa 0.36 -0.05 0.76 0.75 1.25 No 105 Number 1.03 0.26 1.81 0.75 1.25 No 81 Medium  Dbh 0.67 0.16 1.18 0.75 1.25 No 84  Ht 0.67 0.46 0.89 0.75 1.25 No 54  Crht 0.61 0.26 0.97 0.75 1.25 No 74  Basa 0.77 0.41 1.13 0.75 1.25 No 59 Number 3.82 2.88 4.76 0.75 1.25 No 376 High  Dbh -0.08 -0.88 0.72 0.75 1.25 No 188  Ht 0.80 0.06 1.54 0.75 1.25 No 94  Crht 1.35 0.44 2.25 0.75 1.25 No 125  Basa 0.75 0.25 1.24 0.75 1.25 No 75 Number 2.79 1.91 3.67 0.75 1.25 No 267            37   Table 19. Results of conducting equivalence tests for the slope (adjusted values of alpha-r = 0.42 for high productivity and 0.35 for low productivity PSPs) at the stand level (continued). Prod. Level / Density Age Variable1   2 CI 3 IR4 Reject H0 Minimum Rejection Interval (%)                     Low   Dbh 0.80 0.60 1.00 0.75 1.25 No 40  Ht 1.07 0.78 1.35 0.75 1.25 No 35  Crht 0.66 0.10 1.22 0.75 1.25 No 90  Basa 3.70 2.68 4.73 0.75 1.25 No 373 Number 0.91 0.49 1.34 0.75 1.25 No 51 High  Dbh 0.56 0.40 0.71 0.75 1.25 No 60  Ht 0.71 0.51 0.91 0.75 1.25 No 49  Crht 0.65 0.42 0.88 0.75 1.25 No 58  Basa 0.79 0.55 1.03 0.75 1.25 No 45 Number 2.26 1.51 3.01 0.75 1.25 No 201   1  Dbh represents growth in quadratic mean diameter at breast height (cm);  Ht represents average tree height growth (m);  Crht represents the average change in height to the base of the live crown (m);  Basa represents the growth in basal area per ha (m2); and  Number represents the change (reduction) in stems per ha. 1 Slope of the relationship between measured and predicted growth/change. 2 Confidence interval for the slope of the relationship between measured and predicted growth/change. 3 Indifference region.   The estimate of the slope was not close to 1 for any of the variables. It was higher than 1 for: (1) changes in number of trees for all groups of PSPs; (2) height growth for moderately-aged and older PSPs with low productivity; and (3) changes in crown height for moderately-aged and older PSPs with high productivity, as well as for high density PSPs. Estimates of the slope were negative for: (1) dbh growth for moderately-aged and older PSPs for both productivity levels, and for high density PSPs; and (2) for basal area growth of young PSPs with low productivity and for moderately-aged and older PSPs with high productivity.    3.5 Validation Using Simulations I produced a range of initial stand structures to begin simulations and ran the CrosBas-PipeQual simulator for 150 years. Input varying parameters included a range of initial densities, initial heights, and values of alpha-r.    38   3.5.1 Changing Values of Alpha-r The effect of different values of the parameter alpha-r on four variables at the stand level was investigated. Lower levels of alpha-r should reflect a higher quality site.  A similar trend existed for each PSP: as alpha-r increased, the values of the variables decreased (Figure 5). For example, diameter ranged from 21 cm to 17 cm  (Figure 5a) and height ranged from 28 m to 20 m, from the lowest value of alpha-r to the highest value, at age 100 (Figure 5b). The simulator produced high values of basal area for high productivity sites, while the number of stems per ha did not change much at all with changes in alpha-r (Figures 5c and 5d). This indicates a possible problem with estimating mortality, especially on high productivity (low alpha-r) sites.         (a)                                                                             (b)            (c)                                                                            (d)  Figure 5. Effect of different values of alpha-r on output variables: (a) quadratic mean dbh; (b) mean stand height; (c) basal area per ha; and (d) stems per ha.   0 10 20 30 40 0 1 2 P re d ic te d  h e ig h t,  m  alpha-r 25 years 50 years 100 years 0 20 40 60 80 0 1 2 P re d ic te d  b asa l a re a,  m 2  alpha-r 25 years 50 years 100 years 0 5 10 15 20 25 0 1 2 P re d ic te d  d ia m e te r,  c m  alpha-r 25 years 50 years 100 years 0 500 1000 1500 2000 2500 3000 0 2 P re d ic te d  n u m b e r o f ste m s p e r h a alpha-r 25 years 50 years 100 years  39  3.5.2 Changing the Initial Density Two PSPs were chosen to provide baseline values for the varying density simulations. These PSPs had different original densities and stand structures. The approach I used for producing a range of starting densities resulted in some quite high starting densities for some of the scenarios based on adjustments to PSP TIM9308. Both PSPs showed similar trends: diameter growth decreased as site quality changed from rich to poor (Figures 6a, 7a, 8a, 9a, 10a, 11a). Diameter growth was greatest on rich sites (alpha-r = 0.15) at low densities. For the PSP with the lower initial density (NP9203) at low densities (314 and 629 trees/ha) on rich sites (alpha-r = 0.15), changes in diameter with time were larger than at medium or high densities (Figures 6a, 7a, 8a). In denser stands, trees had a smaller average diameter at a given age. As the site became less productive (values of alpha-r changed from 0.15 to 0.59 and 1.03), diameter growth decreased. At an age of 150 years, diameter was greater than 50 cm in the least dense stands on rich sites (alpha-r = 0.15) (Figure 6a). On the poorest site (alpha-r = 1.03), diameter reached only 40 cm (Figure 8a). For TIM9308 at an age of 150 years, diameter reached 40 cm on a good quality site and it decreased to 30 cm for the poorest site and the least dense conditions. The simulator could not project the highest densities on either the medium (alpha-r = 0.59, N = 24392 stems/ha) and low productivity sites (alpha-r = 1.03, N = 12196 stems/ha and N = 24392 stems/ha) for all five variables. Overall, the change in diameter with time and stand density was reasonable: diameter increased with time and decreased from rich site to poor sites. As expected, height growth decreased with a decrease in site quality (Figures 6b, 7b, 8b, 9b, 10b, 11b). No separation occurred for height in high productivity stands of different densities for NP9203 (Figure 6b). However, there was not much difference in height between rich and poor sites. At 150 years, height was 35 m on good quality sites and 30 m on poor sites. One would expect a greater difference between the heights of stands at 150 years on good and poor sites. This might indicate that height is not sensitive to changes in alpha-r or that the adjustments of alpha-r do not reflect site difference. A different pattern for height growth occurred for TIM9308. There, height was similar at 150 years for the stands with medium (1525 and 3048 trees/ha) and high (6098 trees/ha) initial densities assuming high productivity. On low productivity sites, stands with low (762 trees/ha) and high (12196 and 24392 trees/ha) initial densities had lower predicted heights than stands with more moderate densities. The lowest density (762 stems/ha) and highest density (12196 and 24392 stems/ha) curves were separated from medium density curves. Crown height was lower on the low productivity site throughout the simulation (Figures 6c, 7c, 8c, 9c, 10c, 11c). Changes in crown height followed a pattern similar to changes in height. For NP9203, height to the base of the live crown was lower on low productivity sites (Figure 8c vs. Figure 6c). Crown height values were similar for the stands of different densities on rich sites except for the least dense stands (Figures 6c, 7c, 8c). On  the  poorest  sites  (alpha-r=1.03)  there    40    (a) (b)       (c)                                                                                       (d)   (e) Figure 6. Effect of density on PSP NP9203 with alpha-r = 0.15: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 35 40 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 20 40 60 80 100 120 0 100 200P re d ic te d  b asa l a re a,  m 2  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 2000 4000 6000 8000 10000 0 100 200 P re d ic te d  n u m b e r o f st e m s p e r h a  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 10 20 30 40 50 60 0 100 200P re d ic te d  d ia m e te r,  c m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 10 20 30 40 0 100 200 P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064  41        (a)                                                                          (b)      (c)                                                                          (d)   (e) Figure 7. Effect of density on NP9203 with alpha-r = 0.59 (default value): (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 10 20 30 40 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 20 40 60 80 100 0 100 200P re d ic te d  b asa l a re a,  m 2  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 2000 4000 6000 8000 10000 0 100 200 P re d ic te d  n u m b e r o f st e m s p e r h a  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 10 20 30 40 50 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 5 10 15 20 25 30 0 100 200 P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064  42    (a) (b)    (c)                                                                                      (d)  (e) Figure 8.  Effect of density on NP9203 with alpha-r = 1.03): (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 10 20 30 40 50 60 70 0 100 200 P re d ic te d  b asa l a re a,  m 2  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 2000 4000 6000 8000 10000 0 100 200P re d ic te d  n u m b e r o f st e m s p e r h a Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 10 20 30 40 50 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064 0 5 10 15 20 25 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=314 N=629 N=1258 N=2516 N=5032 N=10064  43   (a)                                                                                        (b)   (c) (d)  (e) Figure 9. Effect of density on TIM9308 with alpha-r = 0.15: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 10 20 30 40 50 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 N=24392 0 50 100 150 200 250 0 100 200 P re d ic te d  b asa l a re a,  m 2  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 N=24392 0 5000 10000 15000 20000 25000 0 100 200P re d ic te d  n u m b e r o f t re e s p e r h a Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 N=24392 0 10 20 30 40 50 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 N=24392 0 10 20 30 40 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 N=24392  44    (a)                                                                                    (b)          (c)                                                                                    (d)         (e) Figure 10. Effect of density on TIM9308 with alpha-r = 0.59 (default value): (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 10 20 30 40 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 0 10 20 30 40 0 100 200P re d ic te d  d ia m e te r,  c m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 0 10 20 30 40 0 100 200 P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 0 20 40 60 80 100 120 140 0 100 200P re d ic te d  b asa l a re a p e r h a Projected time, years N=762 N=1525 N=3049 N=6098 N=12196 0 2000 4000 6000 8000 10000 12000 0 100 200P re d ic te d  n u m b e r o f st e m s p e r h a  Projected time, years N=762 N=1525 N=3049 N=6098 N=12196  45                                                                                             (a)                                                                                        (b)         (c)                                                                                       (d)    (e) Figure 11. Effect of density on TIM9308 with alpha-r = 1.03: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 0 100 200 P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098 0 10 20 30 40 50 60 70 0 100 200 P re d ic te d  b asa l a re a,  m 2  Projected time, years N=762 N=1525 N=3049 N=6098 0 1000 2000 3000 4000 5000 6000 0 100 200 P re d ic te d  n u m b e r o f t re e s p e r h a Projected time, years N=762 N=1525 N=3049 N=6098 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years N=762 N=1525 N=3049 N=6098 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years N=762 N=1525 N=3049 N=6098  46  was a separation in crown height values, with the highest value of 20 m for the densest condition (Figure 8c). For the rich site with medium (1525 and 3048 trees/ha) and high (6098 trees/ha) initial densities, crown height was similar at 150 years (35 m). On low productivity sites, with low (762 trees/ha) and high (12196 and 24392 trees/ha) initial densities, the height to the crown base was lower than for the moderate densities.  Higher basal areas were found for higher density conditions at the end of the simulation period (Figures 6d, 7d, 8d, 9d, 10d, and 11d). All graphs leveled off at the same age and then stayed flat (except for low density stands). Basal area reached unreasonably high values under high density conditions on rich sites (e.g., for NP9203 79.7 m2/ha and 108.5 m2/ha for starting conditions of 5032 trees/ha and 10064 trees/ha, respectively). For TIM9308 these values were even higher (202 m2/ha for 24392 trees/ha). Even on poor sites, basal area reached 64 m2/ha for 10064 trees/ha (NP9203). This indicates that the simulator does not kill trees fast enough or that it grows trees too fast at high density levels. However, the simulator performed reasonably well using a density of 1258 trees/ha (NP9203) which was a real stand density for this PSP. For rich sites, basal area was 43 m2/ha at 150 years of age; for poor sites it reached 32 m2/ha at that age. The number of trees decreased with time (Figures 6e, 7e, 8e, 9e, 10e, 11e). There was more mortality in stands with high initial densities (5032 and 10064 trees/ha for NP9203 and 24392 and 12196 trees/ha for TIM9308) compared to those at lower densities. There was less mortality on better quality sites.  The simulated stands were plotted on the density management diagram for Jack pine stands in the boreal region of Ontario (Archibald and Bowling, 1994), using mean total tree volume and stems per ha development through time (Figure 12). All stands projected entered the zone of imminent competition mortality (ZICM). All stand trajectories, once in the ZICM, should track the 1.0 relative density line (solid line at the right edge of the density management diagram) leftwards for the rest of their life. This was the trajectory followed by the lower density stand projections illustrated in Figure 12. However, the projections of stands with medium and high density continued above the 1.0 relative density line, with little evidence of self-thinning mortality until the last few growth periods. The problem worsened as initial density increased. This result supports the findings of the data-based validation that there are problems with underestimating mortality within the CrosBas-PipeQual simulator, particularly at higher densities. 3.5.3 Changing the Initial Height I used the same PSPs that I used to simulate different initial densities to produce different starting conditions by changing the initial height of the trees. For starting conditions based on plot TIM9308, diameter growth was similar for the plots with different initial heights on good sites, where it reached 35 cm (Figure 13a). For less productive sites, the final diameter decreased to 26 cm (Figure 15a). Diameter growth was not influenced by   47  the initial tree height. For starting conditions based on PSP NP9203, which had higher initial heights, I got slightly different results. Not surprisingly, diameter increased through time, with its values decreasing from the good site (25 cm) to the poor site (20 cm) (Figure 16a, 17a, 18a). The values of diameter were similar for stands with different initial heights.  Figure 12. Jack pine density management diagram. Vertical lines represent simulated plot development through time (1- NP9203, 2 – TIM9308) (The density management diagram was taken from Archibald and Bowling, 1994).    48       (a)           (b)        (c)           (d)  (e) Figure 13. Effect of height on NP9203 with alpha-r = 0.15: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 10 20 30 40 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 60 0 100 200 P re d ic te d  b asa l a re a,  m 2  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 200 400 600 800 1000 1200 1400 0 100 200P re d ic te d  n u m b e r o f st e m s p e r h a Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  49        (a)  (b)        (c) (d)  (e) Figure 14. Effect of height on NP9203 with alpha-r = 0.59: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 35 0 100 200P re d ic te d  c ro w n  b ase , m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 0 100 200P re d ic te d  b asa l a re a,  m 2  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 200 400 600 800 1000 1200 1400 0 200P re d ic te d  n u m b e r o f t re e s p e r h a Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  50          (a) (b)         (c) (d)  (e) Figure 15. Effect of height on NP9203 with alpha-r = 1.03: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 0 100 200 P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 0 100 200P re d ic te d  b asa l a re a,  m 2  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 200 400 600 800 1000 1200 1400 0 200 P re d ic te d  n u m b e r o f st e m s p e r h a Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  51             (a)          (b)           (c)              (d)  (e)            Figure 16. Effect of height on TIM9308 with alpha-r = 0.15: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 0 100 200P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 0 100 200P re d ic te d  b asa l a re a,  m 2  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 500 1000 1500 2000 2500 3000 0 100 200 P re d ic te d  n u m b e r o f st e m s p e r h a  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  52         (a)         (b)         (c)         (d)  (e)  Figure 17. Effect of height on TIM9308 with alpha-r = 0.59: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 30 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 35 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 60 70 0 100 200 P re d ic te d  b asa l a re a p e r h a  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 500 1000 1500 2000 2500 3000 0 100 200P re d ic te d  n u m b e r o f st e m s p e r h a Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  53           (a)    (b)          (c) (d)  (e) Figure 18. Effect of height on TIM9308 with alpha-r = 1.03: (a) quadratic mean dbh; (b) mean stand height; (c) mean height to the base of the crown; (d) basal area per ha; and (e) stems per ha. 0 5 10 15 20 25 0 100 200 P re d ic te d  d ia m e te r,  c m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 30 35 0 100 200 P re d ic te d  h e ig h t,  m  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 5 10 15 20 25 0 100 200P re d ic te d  c ro w n  h e ig h t,  m  Predicted time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 10 20 30 40 50 60 0 100 200 P re d ic te d  b asa l a re a,  m 2  Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up 0 500 1000 1500 2000 2500 3000 0 100 200 P re d ic te d  n u m b e r o f st e m s p e r h a Projected time, years initial height 50% down 25% down 10% down 50% up 25% up 10% up  54  Higher values of height and crown height growth were obtained when the initial height was increased by 25 and 50% using NP9203 (Figures 13b, 14b, 15b). The final values of height changed from 42 m to 28 m between the good site and the poor site. Height and crown height were higher when the initial heights were increased by 50% on the best site using TIM9308 (Figures 16b, 17b, 18b). Final height decreased from 43 m to 29 m between the good site and the poor site. Basal area increased up to 100 years then decreased; however, the different initial heights showed the same trend of changing basal area through time (Figures 13d, 14d, 15d). Plots with initial height increased by 50% had slightly smaller values of basal area. For TIM9308, basal area values were similar for PSPs with different starting heights except for initial height decreases by 50%, which was above the rest of the curves (Figure 16d, 17d, 18d). The change in final basal area from the good site (42 m2/ha) to the poor site (30 m2/ha) appeared reasonable.  Changes in the number of stems through time were not affected by the initial height (Figures 13e, 14e, 15e, 16e, 17e, 18e). The number of stems decreased with time and ended up with about 500 stems per ha for NP9203 and about 1400 stems per ha for TIM9308.  3.6 Summary The jack pine version of the CroBas-PipeQual simulator systematically underestimated dbh growth, height growth, crown height change, basal area growth, and changes in number of trees at the stand and single tree level when the default and adjusted values of alpha-r were used. The simulator performed better at the individual tree level; however, values for most of the variables were still underestimated on most PSPs. Running the simulator with the adjusted value of alpha-r gave slightly better results at the stand level (predictions were closer to observations). The null hypothesis of dissimilarity was not rejected for any variables at the stand level when using the default value of alpha-r. Bias for most variables and most groups of PSPs was negative, which meant that predicted values of changes were smaller than the measured changes. The minimum rejection intervals were larger than the indifference intervals for all variables across all groups of PSPs. Estimates of the slope were generally higher than 1 for some variables (the changes in number of trees, the height growth and crown height growth). Estimates of the slope were negative for dbh growth and for basal area growth. The null hypothesis of dissimilarity was not rejected for any of the variables and any groups of PSPs at the individual tree level. The null hypothesis was rejected only for height growth when using grouped single tree data (i.e., without dividing the data into groups according to age, stand density, and site productivity). All slopes were positive but less than 1. I failed to reject any null hypotheses for any of the variables at the stand level with the adjusted values of alpha-r. The minimum rejection interval and bias decreased for most PSP groups   55  compared to the default value of alpha-r. The estimate of the slope was not close to 1 for any of the variables.  Running CroBas-PipeQual with a range of values of alpha-r showed that growth/changes of the variables I was tracking were lower in stands with higher alpha-r values (corresponding to poorest sites). Stands with different productivity values had similar numbers of stems per ha which could indicate a problem with modelling mortality. When running the simulator with different initial stand densities, similar trends occurred for both baseline PSPs. In denser stands, trees had a smaller average dbh at a given age, and basal area growth and change in number of trees were higher. Initial density had almost no influence on height growth and crown height change; differences between these variables were small for stands with different densities. The onset of significant density-dependent mortality in initially dense stands occurred at higher densities than expected based on a density management diagram for jack pine in Ontario. Predicted dbh growth and basal area growth were similar for both baseline PSPs with different initial heights. There was little difference in predicted number of stems per ha for stands with different initial heights. However, stands with their initial height increased 25% and 50% of the actual values had higher values of height and crown height at the end of the simulation period.   56  4. Discussion 4.1 Validation Using Independent Data (Default and Adjusted Alpha-r) The jack pine version of the CroBas-PipeQual simulator underestimated dbh growth, height growth, crown height change, basal area growth, and changes in number of trees for most PSPs at the stand and single tree level when the default and adjusted values of alpha-r were used. The simulator best predicted height growth and change in crown height at the stand level, and gave the poorest prediction for basal area growth. The biggest negative difference between predicted and measured basal area (underestimation) was observed for young PSPs of low and medium density. The biggest positive difference (overestimation) between predictions and measurements was found for basal area growth of low productivity PSPs in moderately-aged PSPs of medium and high density and for older PSPs of medium density. There are several possible explanations for the discrepancies between simulated and measured data. One of them is that the data used for validation came from a limited number of PSPs; more PSPs might give more stable results. For example, only four young PSPs of low density and one young PSP of medium density were used for validation.  Another possibility is that the simulator might not “kill” small trees fast enough to provide reasonable results (i.e., simulator underestimated actual mortality). The number of stems per ha was mostly overestimated by the simulator; this can influence growth/change in the other variables I examined in ways consistent with my observations. Density dependent mortality tends to differentially remove more small trees than large trees. More trees present will lead to lower diameter growth at both the single tree and the stand level and retention of small trees will constrain the growth of average dbh at the stand level. If more small trees persist than there should be, this would also influence height growth predictions at both the single tree and the stand levels through both higher assumed density levels and, at the stand level, through the dampening effect of small trees on change in average height. The effect of more trees on basal area growth predictions is less evident, and it is possibly neutral over a range of moderate densities. At the stand level, interaction between trees occurs through shading which influences the rate of photosynthesis, and through crown coverage, which affects mortality. CrosBas-PipeQual uses crown coverage to drive mortality (Mäkelä, 1997). Change in the height to the crown base occurs when crowns touch each other (i.e., as crown coverage increases) (Mäkelä et al., 1997).  If more trees are present than there should be, it could be that crown coverage would increase. However, it is also possible that the crowns of other trees will expand more if there are fewer trees, offsetting potential decreases in crown coverage. Height to the crown base was underestimated using the default value of alpha-r for most of the PSPs in the test data.    57  Previous validations of the simulator for tree species other than jack pine showed that it produced results that were in a good agreement with measurements. Mäkelä (1997, 2002) used CroBas to run simulations of young Scots pine stands and compared simulated results to the observed values of stems at the time of harvest (using four different initial stand densities). The simulator provided a good fit for simulated and measured trees. Robinson and Ek (2003) used CroBas as a basis for the Forest 5 simulator. They ran simulations for 150 years using four different initial stand densities and compared the results to a red pine (Pinus resinosa Sol. ex Aiton) dataset. Again, the simulation results were in a good agreement with the measurements.  CroBas-PipeQual does not consider the effect of site quality, climate, disturbances, and multiple age classes. It cannot simulate disturbances such as diseases, fires, wind throw, as well as changes in soil nutrients and moisture levels (which change with time and influence the stand structure). Regeneration (ingrowth) also is not incorporated into the model. While preparing PSPs to be brought into the model, I removed all trees that were not measured at the first measurement to remove ingrowth. CroBas-PipeQual was originally developed as a single-species single-aged model for pure Scots pine stands (Mäkelä, 1986). The version validated in this thesis was calibrated later for pure jack pine stands (Schneider et al., 2008). The model grows trees of other species (white birch, trembling aspen, black spruce, and balsam fir) as if they were jack pine trees ignoring stand dynamics and different characteristics of other species (e.g., crown architecture, whether they are fast or slow growing, and how they change with age). The presence of other species on PSPs that the simulator cannot grow might contribute to the differences between simulator projections and PSP growth/change. In the testing, I used only those PSPs with high levels of jack pine (> 80% by basal area) and assumed that all of the trees were jack pine. Since the simulation period was short (7 to 11 years) and the proportion of jack pine in the PSPs used was high, my expectation is that any effect that the presence of species other than jack pine might have on the results of my testing would be minor. Another concern when comparing the simulated growth/change of variables with observed growth/change was the relatively short period between measurements (7-11 year period). I might have been able to observe more stable patterns across density, site quality, or age if longer projection periods were available.  Equivalence tests were employed to quantify statistically the performance of the simulator against the PSP data. The null hypothesis of dissimilarity was rejected only for height growth for medium density PSPs at the individual tree level when using the default value of alpha-r. Although running the simulator with the adjusted alpha-r values gave better predictions (predictions were closer to observations), I still failed to reject the null hypothesis for all variables and groups of PSPs.    58  Failure to reject the null hypothesis might be caused by different reasons. For example, the simulator is invalid, the test is not powerful enough, and/or arbitrarily selected regions of equivalence are too strict (Froese and Robinson, 2007). I do not consider the equivalence regions I set (10%) to be too strict. A similar region was used by Froese and Robinson (2007). Although there was quite a bit of variability in growth predictions from plot to plot, lack of test power did not appear to be a major issue. Rather, there appeared to be bias present for almost all conditions and variables I checked. Another factor that may have contributed to the failure to reject most null hypothesis was the fact that the regression assumption of equal error variances was not met for most variables and most groups of PSPs. This will introduce errors into the p-values provided. One suggestion for overcoming this difficulty would be to use bootstrap equivalence tests (Robinson et al., 2005, Leites et al., 2009). The data used for validation differed from the data used for calibration of the models (functions) internal to the jack pine version of the CroBas-PipeQual simulator in terms of growing conditions and management activities. The validation data were naturally regenerated, seeded, or planted. The trees used for calibration were sampled from naturally regenerated stands and operational plantations in central Quebec and precommercially thinned stands from New Brunswick (Schneider et al., 2008). There were no treatments included in the PSPs used for validation. Only 84 trees were used for calibration. Likely, the internal functions within the simulator are not as robust as they might be if a larger calibration dataset was available. There was no information available about the initial densities of either the calibration or validation data. The data used for calibration possibly represents a wider range of geographical conditions than the data used for validation. However, the stand characteristics of the Ontario validation data mostly fell within the range of stand characteristics used for model calibration (Table 20).   Table 20. Comparison of stand characteristics of the data used for model calibration and model validation, different ecological zones.  Ontario (validation) Ontario NB Quebec QMD, cm 10.49 (3.27-21.98) 21.9 (18.5-25.2) 15.3 (13.6-17.7) 10.9 (6.0-17.7) Basal area, m2/ha 17.79 (0.76-37.91) 32.4 (26.3-39.1) 29.2 (14.1-36.2) 18.3 (8.6-31.4) Density, stems/ha 2245 (517-6375) 964 (544-1456) 1808 (575-2575) 2124 (1143-3850) Age, years 42 (9-127) 36 56 22.5 (20-37)  The slopes of the regression lines between the predicted and measured values were less than 1 for most variables and groups of PSPs. This indicates that the difference between predictions and observations become bigger as the observed values increase (over-predictions got bigger). However, estimates of the slopes were negative for some variables; in those cases the predicted values increased as the measured values of a variable decreased. Obviously those situations   59  indicate poor predictions by the simulator and tend to be found when examining conditions represented by only a few PSPs.    4.2 Validation Using Simulations Running simulations with different initial stand densities and heights showed that the simulator produced results that were biologically reasonable, with the exception of too little mortality occurring in dense stands. As was expected, diameter growth decreased as site quality changed from good to poor. Diameter growth was largest on good sites (alpha-r=0.15) at low densities. Stand density did not affect height and height to the base of live crown. Basal area increased with time and density; the denser the stand, the faster it grew, and the larger basal area it had. The number of trees decreased with time with more mortality occurring in stands with higher initial densities. There was slightly less mortality on better quality sites. For stands with high initial densities, the number of stems remained unreasonably high at 150 years (for example, 4214 stems/ha on a high productivity site and 3395 stems/ha on a low productivity site). The initial density had a smaller effect on height and crown height than on diameter, basal area, and number of stems per ha. As alpha-r increased, the values of most of the stand- and tree-level variables decreased at a given age. This is logical if higher alpha-r values are taken to represent poorer sites. However, the number of stems per ha remained almost the same for the different alpha-r values For example, the difference in number of stems per ha between poor and good sites after a simulation period of 100 years was only about 80 trees (Figures 6 and 8). This indicates a possible problem with estimating mortality. Makela (1986) conducted simulations for planted Scots pine stands with density varying from 2000 to 25000 stems/ha using CroBas. Natural mortality stabilized the density in denser stands and stands with different initial densities ended up at the same density in 75 years. However, this did not happen in case of jack pine version of the simulator. Possible reasons for this include: (1) unreasonable initial conditions given to the simulator; (2) inaccurate estimation of the parameters of the internal CrosBas-PipeQual functions; (3) problems with modelling crown coverage; and (4) not considering the influence of the changes in soil richness, light availability, etc. in the simulator. Mortality in CroBas-PipeQual, which is controlled by crown coverage, is affected by stand structure (Mäkelä, 1997). Mortality increases when crown coverage increases and the physical space between trees decreases. There are two coefficients in the simulator that define density independent and density dependent mortality. There appear to be problems with these coefficients in the jack pine version of the simulator that are affecting mortality estimation, particularly in initially dense stands.   60  Parameters related to mortality, self-pruning and crowding in the initial version of the simulator were found by trial and error using three variables – foliage weight, pruning height, and initial density that varied from 500 to 7500 seedlings/ha (Mäkelä, 1997). I used initial densities higher than 7500 stems/ha for some of the simulations. Possibly, the parameters related to mortality were calibrated just for this particular range of densities and the simulator does not work properly when the density is outside of this range.   4.3 Changing Values of Alpha-r Alpha-r represents the ratio between fine root biomass (weight) and foliage biomass and is specific to the site. It varies with the age of stands. Young fast growing stands should have higher values of alpha-r, and older established stands should have lower values of alpha-r (Vanninen et al., 1996, Mäkelä, 1997). Extensive research has been done to explain the allocation of nutrients among below and above ground plant components. Several models were developed to describe root/shoot allocation: allometric models (Pearshall, 1927; Troughton, 1956, cited by Wilson, 1988), functional equilibrium models (Brenchley, 1916; Brouwer, 1963; Davidson, 1969a, 1969b; Iwasa and Roughgarden, 1984, cited by Wilson, 1988), Thornley´s model (Thornley, 1972, cited by Wilson, 1988), and hormone models (Keeble, 1931; Luckwill, 1960, cited by Wilson, 1988).  Research has been conducted for herbaceous plants that showed that the root/shoot (or shoot/root) ratio changes with plant growth (Wilson, 1987). The ratio decreases with an increase in plant size (Monk, 1966). When one of the major nutrients (nitrogen, phosphate) is in deficit, the root/shoot ratio becomes larger. Water availability has the same effect on this ratio. The root/shoot ratio increases when conditions become less favourable and it decreases in response to improved conditions - fertilization, aeration, pest control (Harris, 1992). Deficiency of light in the canopy causes the ratio to decrease (Isawa and Roughgarden, 1984). Growth of roots and shoots changes during the life of a plant when environmental conditions change (during the growing season), so a plant adjusts the allocation of nutrients between roots and shoots (Isawa and Roughgarden, 1984). Fine root biomass changes with age. For example, on low productivity sites it increases as basal area and foliage biomass increase until 33 years in Douglas-fir stands, then it stabilizes (Vogt. et al., 1987). The fine root/foliage biomass ratio depends upon the stage of the stand development, as well as site productivity. Before canopy closure, stands have less fine root biomass than foliage biomass on both good and poor sites. The relationship between chemical composition of litter (initial content of nitrogen that controls the decomposition rate of the litter) and root biomass influences the ratio on poor sites. On good quality sites such relationships were not found and other factors such as climate might influence the decay rate of litterfall (Vogt et al., 1987).    61  Alpha-r is the one of the structural parameters within CroBas-PipeQual that can be easily varied by the user. According to the functional balance approach, plants optimize the allocation of nutrients between roots and shoots as a response to changing environmental conditions (Harris, 1992; Agren and Franklin, 2003). In trying to optimize the value of alpha-r to better predict height growth, I expected to find a lower value of this parameter for high quality sites and higher value for poor sites. However, I obtained 0.35 for low productivity sites and 0.42 for high productivity sites which contradicts the theory. This might be caused by errors in determining site index, inaccuracy in the simulator (height is not sensitive to changes in alpha-r), differences in ages of the PSPs, and small sample sizes. I used 15 PSPs for low productivity sites and 19 for high productivity. To get reasonable results it may be necessary to use PSPs of the same age and condition, and to use a larger dataset.   62  5. Conclusion The objective of this thesis was to evaluate performance of the jack pine version of CroBas- PipeQual. This was done through comparing the simulator’s predictions to actual growth on a series of jack pine dominated PSPs obtained from the Government of Ontario and by simulating long-term stand development under a range of different initial conditions.  5.1 Summary of Findings The results of testing the simulator against the independent dataset from Ontario PSPs showed that CroBas-PipeQual tended to underestimate dbh growth, height growth, crown height change, basal area growth, and changes in number of trees at the stand and single tree level with the default and adjusted values of alpha-r. Predicted values of variables were closer to measurements at the individual tree level than at the stand level. Running the simulator with the adjusted values of alpha-r did not show substantial improvements in simulator performance.  While testing the model using equivalence tests with both the default and adjusted values of alpha-r, I failed to reject the null hypothesis of dissimilarity for any of the variables and groups of PSPs at the stand and individual tree level. Most variables and most groups of PSPs had a negative bias, which meant that predicted values of changes were smaller than the measured changes. The minimum rejection intervals were larger than the indifference intervals for all variables across all groups of PSPs. Estimates of the slope were higher than 1 for the changes in number of trees, the height growth and crown height growth. Estimates of the slope were negative for dbh growth and for basal area growth. The minimum rejection interval and bias decreased for most PSP groups using the adjusted values of alpha-r compared to the default value. The estimate of the slope was not close to 1 for any of the variables. The simulator produced reasonable results based on the sensitivity analyses conducted, with the exception of predicted mortality in initially dense stands. In denser stands, trees had a smaller average dbh at a given age, and basal area growth and change in number of trees were higher. Initial density had almost no influence on height and crown height growth; differences between these variables were small for stands with different densities. Good quality sites (smaller values of alpha-r) demonstrated better growth than poor quality sites with the same starting conditions otherwise. However, the number of stems per ha remained the same on sites of different productivity which indicates that there may be problems with the estimating mortality in the simulator. Varying the initial tree heights had almost no impact on the number of stems per ha, dbh growth and basal area growth. However, stands with the initial height increased 25% and 50% of the actual had higher values of height and crown height at the end of the simulation period.   63  5.2 Suggestions for Further Work Possible further steps that might help to explain the simulation results include:  (1) decomposing the graphs produced by the simulator into different dbh classes to investigate which trees grow faster or how diameter, height, and height to the crown base change in each dbh class. It would also be helpful to identify the dbh class in which the largest mortality occurs (it should be largest in the dbh class with the smallest diameter value) as well as to detect in what dbh classes the largest deviations of predictions from measurements take place. Testing how the competitive status of a tree (as reflected by its size) might influence the simulation results might also help to understand simulator performance.  (2) doing more extensive statistical testing using other techniques. One suggestion is to use bootstrap equivalence tests (Robinson et al., 2005, Leites et al., 2009). Other techniques that are often used for validation of growth simulators models include calculating the root mean square error (RMSE), mean absolute error (MAE), goodness-of-fit values (Subedi and Sharma, 2011).  (3) comparing simulator output with that of other jack pine simulators. Several simulators capable of predicting jack pine tree and stand characteristics (although not the detailed branch characteristics provided by CrosBas-PipeQual) were developed during the last 20 years (e.g., Huang et al., 1992, Peng et al., 2001, Zhang et al., 2002, Mailly and Gaudreault, 2005, Subedi and Sharma, 2011). Comparing CroBas-PipeQual stand and tree predictions with those obtained using other models might reveal some areas for improvement. (4) testing the accuracy of predicted variables not examined in this study. This study examined tree and stand variables measured in most PSP datasets. Some of the more unique variables provided by the CrosBas-PipeQual simulator (e.g., branch size, numbers and location on individual trees) were not tested. Despite the scarcity of appropriate independent data, predictions of these variables should be assessed before these components of the simulator are widely used.  5.3 Final Statements The jack pine version of the CrosBas-PipeQual simulator requires further development and testing. One priority should be to refit some of the internal equations using a dataset that is larger and that represents a broader geographical amplitude than the data used in the initial calibration. More testing with suitable data will allow assessment of the simulator´s applicability to other site types and geographical locations. Another area that needs attention is modelling mortality. Validation results showed that the simulator might not “kill” small trees fast enough to provide biologically reasonable results. The number of stems per ha was mostly overestimated, particularly in initially dense stands, and this   64  can influence growth in other variables. The rate of mortality depends on the crown coverage (Mäkelä, 1997, Mäkelä et al., 1997). Mortality increases when the crown coverage increases. Recalibrating the internal equations that define mortality in the simulator might help to improve its performance. Overall, testing the model against the independent dataset and running simulations with different initial stand densities and heights showed that the simulator produced promising results. However, recalibrating some of the equation parameters internal to CroBas-PipeQual is recommended, followed by further testing. Finally, the accuracy of predictions of variables not evaluated in this study (e.g., branch sizes, numbers, and locations) should be assessed before these components of the simulator are used operationally.   65  References Agren, G.I., and Franklin, O. 2003. 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Young  Moderate  Old  Young  Moderate  Old  Low  BP9307 GGM9424 TIM9308 TIM9312  KLK9444  ― BP9304     NP9203 BP9308     TIM9303 BP9313     TIM9309 BP9411     TIM9318 CHA9302  TIM9325 CHA9303  TIM9331 HEA9302  TIM9350 LSJP9302  WWA9514  ― ― Med.  GGM9414 BP9334 LWP9552  BP9318 BP9520  BP9328     TBP9509 BP9335     TIM9307 BP9538     TIM9316 BP9539     TIM9320 CHA9513   TIM9335 LWP9557 WWA9503 LSJP9330  WWA9505 NP9303    WWA9518  CHA9505    NP9605 CHA9509    NP9205 CHA9511    WWA9501 LSJP9460       IFF9306 WWA9301  High  BP9322 BP9327 BP9333  BP9314 KLK9427 LSJP9306 LWP9402 LWP9551 TBP9208 ― KLK9206  CHA9523   IFF9301 LSJP9431 LSJP9435 NP9598 TIM9305 TIM9306 TIM9313 TIM9315  LSJP9325 LWP9204 TIM9317     

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