Impacts of Tree Improvement Programs on Yields of White Spruce and Hybrid Spruce in the Canadian Boreal Forest by Suborna Shekhor Ahmed B.Sc., Applied Statistics, University of Dhaka, 2006 M.Sc., Applied Statistics, University of Dhaka, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (VANCOUVER) October 2016 © Suborna Shekhor Ahmed, 2016 ii Abstract Local and regional timber shortages may be ameliorated via planting improved stocks with higher yields. In this dissertation, I addressed an important knowledge gap on the impacts of tree improvement programs on yields of white spruce (Picea glauca (Moench) Voss) and hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) plantations across the boreal and hemiboreal forests of Canada using meta-modelling approaches. In particular, I used meta-data for white and hybrid spruce provenance trials extracted from the literature to: (i) forecast provenance yields over time for broad spatial and temporal extents; (ii) model yields of provenances relative to standard stocks (termed “gain”) over time; and (iii) test alternatives for forecasting each provenance at a location using available repeated-measures data. In the first study, provenance height over time trajectories were modelled by incorporating the effects of climatic variables, provenances and site characteristics into mixed-effects nonlinear models via a random coefficients modelling approach. Height trajectories were strongly affected by planting site and provenance climates, along with planting site characteristics. The height trajectory meta-model was incorporated into an existing growth and yield model, which can be used to predict provenance yields for long temporal and large spatial extents. In the second study, the impacts of the particular gain definition (i.e., selection age, proportion of top performers) were examined using the model from the first study, and one definition was selected. A meta-model of gain as a function of plantation age, planting density, and planting site iii climate was developed. Planting site climate strongly affected these gain trajectories. The gain definition and trajectory model can be used to evaluate potential gains of using improved white and hybrid spruce stocks. Forecasts are needed to evaluate provenance (or progeny) performance at harvest, often 80 or more years from planting. In the third study, three alternative procedures (population-averaged, subject-specific, and autocorrelation) to forecast repeated measures for a particular progeny at a location were compared and evaluated by virtually removing some repeated measures. The subject-specific forecasts were best with accuracies similar to the measurement precision using standard height measurement devices given five or more prior measurements. iv Preface In this dissertation, I mainly worked with Dr. Valerie LeMay on each research chapter. The contributions of committee members to each research chapter are summarized in the tables below. Chapter 2. Categories Suborna Ahmed Valerie LeMay Gary Bull Alvin Yanchuk Peter Marshall Andrew Robinson Total Identify research problem 70 10 8 4 4 4 100 Designing research 85 10 0 3 0 2 100 Analyzing data 95 5 0 0 0 0 100 Manuscript writing 70 14 3 3 5 5 100 Chapter 3. Categories Suborna Ahmed Valerie LeMay Gary Bull Alvin Yanchuk Peter Marshall Andrew Robinson Total Identify research problem 80 10 4 2 2 2 100 Designing research 85 12 0 3 0 0 100 Analyzing data 95 5 0 0 0 0 100 Manuscript writing 75 17 2 0 3 3 100 Chapter 4. Categories Suborna Ahmed Valerie LeMay Gary Bull Alvin Yanchuk Peter Marshall Andrew Robinson Total Identify research problem 85 15 0 0 0 0 100 Designing research 85 15 0 0 0 0 100 Analyzing data 95 5 0 0 0 0 100 Manuscript writing 75 17 0 2 4 2 100 v Table of Contents Abstract ....................................................................................................................................... ii Preface ........................................................................................................................................ iv Table of Contents ........................................................................................................................ v List of Tables .............................................................................................................................. ix List of Figures ............................................................................................................................ xi Glossary .................................................................................................................................... xiii Acknowledgements ................................................................................................................... xv Dedication ............................................................................................................................... xvii 1. Introduction ............................................................................................................................. 1 1.1 Background ...................................................................................................................... 1 1.2 Importance of White and Hybrid Spruces ........................................................................ 3 1.3 Quantifying Yields of Improved Stocks........................................................................... 4 1.4 Using Meta-Modelling for Larger Temporal and Spatial Extents ................................... 7 1.5 Forecasting Individual Provenances on a Planting Site ................................................... 8 1.6 Objectives and Dissertation Structure .............................................................................. 9 2. Meta-Modelling to Quantify Height Yields of White Spruce and Hybrid Spruce Provenances in the Canadian Boreal Forest ....................................................................................................... 13 2.1 Introduction .................................................................................................................... 13 2.2 Methods .......................................................................................................................... 19 vi 2.2.1 Meta-Datasets ......................................................................................................... 19 2.2.2 Climate Data ........................................................................................................... 24 2.2.3 Meta-Models ........................................................................................................... 27 2.3 Results ............................................................................................................................ 38 2.3.1 Average Height Trajectory Models ........................................................................ 38 2.3.2 Using the Average Height Trajectory Model to Forecast Yields ........................... 43 2.4 Discussion ...................................................................................................................... 49 2.5 Conclusions .................................................................................................................... 54 3. Gain of White Spruce and Hybrid Spruce Provenances in the Canadian Boreal Forest ....... 55 3.1 Introduction .................................................................................................................... 55 3.2 Methods .......................................................................................................................... 59 3.2.1 Meta-Data and Height Trajectories ......................................................................... 59 3.2.2 Effects of Age and Definition of Top Performers .................................................. 62 3.2.3 Gain Trajectory Model ............................................................................................ 65 3.3 Results ............................................................................................................................ 67 3.3.1 Gain Definition Sensitivity Analysis ...................................................................... 67 3.3.2 Gain Trajectory Model ............................................................................................ 71 3.4 Discussion ...................................................................................................................... 79 3.5 Conclusions .................................................................................................................... 84 4. Alternative Methods to Forecast Yields of Hybrid Spruce Progenies................................... 85 vii 4.1 Introduction .................................................................................................................... 85 4.2 Methods .......................................................................................................................... 88 4.2.1 Progeny Repeated-Measures Data .......................................................................... 88 4.2.2 Height Trajectory Model......................................................................................... 91 4.2.3 Forecasting Methods ............................................................................................... 94 4.2.4 Evaluating Forecast Accuracies .............................................................................. 97 4.3 Results ............................................................................................................................ 99 4.3.1 NLMM models........................................................................................................ 99 4.3.2 Forecast Accuracies .............................................................................................. 102 4.3.3 Demonstration of Alternative Forecast Methods .................................................. 108 4.4 Discussion .................................................................................................................... 110 4.5 Conclusions .................................................................................................................. 114 5. Conclusions ......................................................................................................................... 116 5.1 Contributions to Knowledge ........................................................................................ 116 5.1.1 Summary ............................................................................................................... 116 5.1.2 Chapter 2: Research Conclusions and Implications.............................................. 117 5.1.3 Chapter 3: Research Conclusions and Implications.............................................. 118 5.1.4 Chapter 4: Research Conclusions and Implications.............................................. 119 5.2 Limitations ................................................................................................................... 120 5.3 Future Research ............................................................................................................ 121 viii References ............................................................................................................................... 123 ix List of Tables Table 2-1. Sources of meta-data for white spruce provenances (Sw meta-dataset) in the boreal forest of Canada. ........................................................................................................................... 20 Table 2-2. Variable ranges for the Sw meta-dataset (30 planting sites). ...................................... 21 Table 2-3. Sources of meta-data covering selected hybrid spruce provenances (Sw-Se meta-dataset) in British Columbia ......................................................................................................... 23 Table 2-4. Variable ranges for the Sw-Se meta-dataset (38 planting sites). ................................. 24 Table 2-5. Climatic variable ranges for the Sw meta-dataset (30 planting sites). ........................ 26 Table 2-6. Climatic variable ranges for the Sw-Se meta-dataset (38 planting sites). ................... 27 Table 2-7. Provenances at the Petawawa National Forestry Institute site (46° latitude, -77.5° longitude, 170 m elevation, 847 mm MAP, 5°C MAT and 98 days DD) in Ontario. .................. 34 Table 2-8. Geographic locations and climatic variables of provenances in the boreal and hemiboreal Forests of Canada used to illustrate height trajectories. ............................................ 36 Table 2-9. List of fitted models and fit statistics using the Sw meta-dataset................................ 39 Table 2-10. Parameter estimates (standard errors of estimates) and 95% confidence intervals for the selected model using Sw meta-dataset. ................................................................................... 40 Table 2-11. List of fitted models and fit statistics using the Sw-Se meta-dataset. ....................... 41 Table 2-12. Parameter estimates (standard errors of estimates) and 95% confidence intervals for the selected model using Sw-Se meta-dataset. ............................................................................. 42 Table 3-1. Fit statistics for a subset of height gain trajectory models. ......................................... 72 Table 3-2. Parameter estimates (and standard errors) for the selected gain trajectory model. ..... 75 x Table 4-1. Summary statistics for the three hybrid spruce progeny trials in BC. ......................... 90 Table 4-2. Selected fitted models with fit statistics. ................................................................... 100 Table 4-3. Parameter estimates (standard errors) for the selected NLMM model (Model I). .... 101 Table 4-4. Parameter estimates (standard errors) for the less accurate NLMM model (Model III)...................................................................................................................................................... 102 Table 4-5. RMSPEs (m) for plantation ages 27 and 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). ......... 103 Table 4-6. MAPEs (m) for plantation ages 27 and 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). ......... 104 Table 4-7. RMSPEs (m) for plantation 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). ............................ 105 Table 4-8. MAPEs (m) for plantation age 42 years only using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). ......... 106 Table 4-9. RMSPEs (m) for plantation age 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the less accurate NLMM model (Model III). ............. 107 Table 4-10. MAPEs (m) for plantation age 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the less accurate NLMM model (Model III). ............. 108 xi List of Figures Figure 2-1. Boreal and hemiboreal forests of Canada and USA (Brandt 2009). .......................... 17 Figure 2-2. Observed average height trajectory of selected white spruce provenances in the boreal forest of Canada (dots indicate a single measurement in time). ........................................ 22 Figure 2-3. Geographic locations of the selected planting site at the Petawawa National Forestry Institute (PNFI) in Ontario along with the 25 provenances that were tested at the site. .............. 33 Figure 2-4. Stand-level model by Prégent et al. (2010) modified for yields of different provenances and planting sites...................................................................................................... 37 Figure 2-5. Predicted average height (dashed line) and measured average height (solid line and box shape to show actual measurement time) for the 25 provenances (provenance IDs given in Table 2.7) planted at the Petawawa National Forestry Institute planting site. ............................. 44 Figure 2-6. Predicted average height trajectory of simulated white spruce provenances using the Petawawa National Forestry Institute planting site. ..................................................................... 46 Figure 2-7. Predicted quadratic mean DBH (cm) and volume (m3 U/ha) over plantation age (years) for simulated white spruce/hybrid spruce provenances using the Petawawa National Forestry Institute planting site. .................................................................................................................... 48 Figure 3-1. Geographic locations of planting sites in the meta-data set. ...................................... 60 Figure 3-2. Predicted average height trajectory of some white spruce provenances (Prov.) and planting sites (Site) of the meta-data. ........................................................................................... 62 Figure 3-3. Hypothetical predicted average height trajectory of white spruce populations showing top performing provenances (top 15%) versus all provenances (baseline 100%) at an evaluation age of 15 years. ............................................................................................................................. 63 xii Figure 3-4. Gain over time for four planting sites where the top performers (15%) were selected at various evaluation ages (baseline is 100% of provenances). Gain based on selection of top performers at each age (“changing performers”) is included for comparison. ............................. 68 Figure 3-5. Gain over time at evaluation age 15 for different definitions of top performers (i.e., top 5%, 15% or 25%) and baseline (i.e., 75% or 100%). ............................................................. 70 Figure 3-6. Gain over time by planting site. ................................................................................. 71 Figure 3-7. Predicted height gains (RH) over age using the null model and using the base model that changes with plantation age versus the actual gains for two planting sites. .......................... 73 Figure 3-8. Predicted height gain (RH) over age for two planting sites using: (i) Model VII; (ii) base model; and (iii) actual gain. .................................................................................................. 76 Figure 3-9. Changes in the predicted height gain (RH) over age for combinations of values for site degree days (DD), mean annual daily temperature (MAT), and mean annual precipitation (MAP), given a planting site elevation of 400 m and a planting density of 2,500 stems per ha. . 77 Figure 3-10. Changes in the estimated scale parameter (β�0i U) with changes in site degree days (DD), mean annual daily temperature (MAT), and mean annual precipitation (MAP) given a planting site elevation of 400 m and a planting density of 2,500 stems per ha. ........................... 79 Figure 4-1. Observed average height trajectories for of hybrid spruce progenies in BC. ............ 91 Figure 4-2. Observed (≤ plantation age 42 years) and forecasted (plantation ages 43 to 80 years) average heights for the ‘Fraser-Fort George10’ progeny at the Quesnel planting site. .............. 109 xiii Glossary AIC Akaike’s Information Criterion. ATISC Alberta Tree Improvement and Seed Centre. DD Mean degree days greater than 5°C. DDdif Degree days distance between the site and the provenance. DMAP Mean annual precipitation distance between the site and the provenance. DMAT Mean daily temperature distance between the site and the provenance. Gain multiplier Factor used to increase the yield trajectory. GDP Gross Domestic Product. Growth and Yield model A model (often consisting of a number of sub-models) that is designed to predict the growth and yield of trees or stands over time. Height gain Gain in yields for improved stocks relative to standard stocks. Height measures Field-measured heights. Height trajectory Heights over time. Height forecast Predicted heights for a future time. IDW Inverse Distance Weighting. MPE Mean Prediction Error. MAT Mean Daily Temperature. MAP Mean Annual Precipitation. xiv NBTIC New Brunswick Tree Improvement Council. NLMM Nonlinear Mixed Models. Plus trees Seeds are collected from phenotypically-superior individuals called “plus trees” from natural stands with unknown male parents. Progeny versus provenance Provenances are select trees from particular locations, whereas progenies are off-spring with known parents. Random coefficients (aka parameter prediction) Model parameters are replaced by functions of explanatory variables. Repeated measures Measurement of the same subject at several times. RH gain Relative Height gain. RMSE Root Mean Squared Error. RMSPE Root Mean Square Prediction Error. Site index Site height at a reference age, indicating the tree and stand growth and yield potential. Tree improvement Application of genomic tools to increase the value of trees compared to the regular or base materials. xv Acknowledgements Foremost, I would like to thank Dr. Valerie LeMay for her enormous effort and thoughtful suggestions to keep performing this research work. She provided a huge amount of time whenever I needed and helped me understand complex forestry insights to perform this research. Her vast knowledge in forest biometrics was extremely helpful for me in many complex situations in my dissertation. I am very grateful to Dr. Valerie LeMay for her continuous support to compete this dissertation. She is a remarkable mentor for me. Thanks to my PhD thesis co-supervisor, Dr. Gary Bull, for helping me to get in touch with researchers in tree genetics, providing helpful suggestions for this research, and providing the funding. Thanks to my thesis supervisory committee, Drs. Andrew Robinson, Peter Marshall and Alvin Yanchuk for guiding me in this journey. I am extremely thankful to Dr. Alvin Yanchuk who always encouraged and provided materials to better understand tree improvement programs and provided many helpful suggestions. Dr. Andrew Robinson provided detailed and thoughtful suggestion on my selected approaches for statistical analysis and guided me on applying applicable model evaluation techniques. Dr. Peter Marshall provided extremely helpful reviews and recommendations. Also, Dr. Peter Marshall let me teach with him in an online biometrics course and helped me in every step. Special thanks go to Dr. Yousry El-Kassaby for providing very helpful suggestions on tree genomics. xvi I thank Mr. Barry Jaquish and Dr. Alvin Yanchuk for sharing the BC spruce provenance trials. Also, thanks go to Dr. Jean Beaulieu for providing data for a further measurement of Quebec spruce provenance trials. Many thanks to the SMarTForests Project for providing funding for four years. This funds were received by Dr. Gary Bull through Genome Canada, Genome Quebec, Genome British Columbia and Genome Alberta for the SMarTForests Project (www.smartforests.ca). Thanks to my dear friends in the biometrics lab: Kyle, Jamie and Pramila for their support whenever I needed it. Also, my former lab mates helped me a lot while I first started working in Biometrics lab in Forestry: Leah, Andrew, Enrico, Terry and Anna. Thanks to all my dear Bangladeshi friends in UBC for their support and encouragement. xvii Dedication To my mother, Lata Ahmed; my husband, Ehsan Karim and my sister, Meherabun Usha, who have always cared and supported me in this long journey 1 1. Introduction 1.1 Background The worldwide demand for wood fibre is increasing with increases in the human population, rapid urbanization, increases in literacy and education levels, increases in the use of sustainable products, and the emergence of bio-products from wood fibre (Cropper and Griffiths 1994; Fenning and Gershenzon 2002; Nikolakis and Innes 2014). In Canada, 1.25% of the gross domestic product (GDP) was derived from the forest industry, but this contribution has been declining (Natural Resources Canada 2013). While the demand for wood fibre is increasing, the forest industry is facing pressure to conserve forest lands for environmental services. Forest sustainability has become a critical factor in determining the competitiveness of Canada’s forest product industry. Therefore, new approaches are needed to maintain balance between wood fibre demand and conservation efforts. Tree improvement programs can ameliorate current and future wood supply shortages (Petrinovic et al. 2009), particularly since genomic techniques have shown significant potential to expand the range of management options for shaping future forest conditions (Weng et al. 2009). Among other mediation strategies, lowering the time to harvest via faster growth rates will increase wood supply. At the same time, improvements in other tree qualities, such as a greater uniformity in the tree form, can reduce processing costs. Tree breeding (i.e., improvement) programs can contribute to increases in growth (Carlisle 1970; Zobel and Talbert 1984; White et al. 2007) and to improved wood quality (Carlisle 1970; Bendtsen 1978; Vargas-Hernandez and Adams 1991). These programs can also be used to improve other traits, such as 2 resistance to insects and diseases, adaptability to new environments (Zobel and Talbert 1984; Kitzmiller 1990), frost hardiness, and drought tolerance thereby reducing risks of plantation failures (Smith et al. 2007). Improvements towards reaching full genetic potential, particularly for improved growth and wood properties, can only be achieved when environmental constraints are reduced through silviculture and management practices (Zobel and Talbert 1984; Li et al. 1999; White et al. 2007). For example, one strategy for improved plantation success may be to produce trees that have a greater tolerance of limiting factors which cannot be overcome through forest management (Zobel and Talbert 1984). In Canada, the impacts of tree improvement programs on timber supply are uncertain due to a lack of sufficient information covering the long temporal and vast spatial scales. This is particularly important for the boreal and hemiboreal forests where harvest ages are 80 years or more and the area spans 552 million hectares (Natural Resources Canada 2015). Although program start dates, species, and processes vary from province to province, tree improvement programs in Canada mostly started in the mid to late 20th century (Farrar 1969; Fowler and Morgenstern 1990; Petrinovic et al. 2009). As a result, one major obstacle in quantifying the yields of improved stocks is that the most of the trees in provenance trials are young relative to harvest ages. Further, data for all trials in all provinces have not been amalgamated, nor summarized across trials. Finally, there is as yet no common agreement on how to measure gains obtained through use of improved over standard stocks. These issues are partly due to fact that management of forest lands is a provincial, rather than federal, mandate (Fowler and Morgenstern 1990), with the exception of the territories in the north and specifically identified federal lands and parks. As a result, each province differs with regards to the stage of their tree 3 improvement program and also the species being tested. Collectively, these issues make it difficult to assess the impacts of tree improvement programs in Canada. In this dissertation, these knowledge gaps were narrowed by examining, modelling, and forecasting yields of white spruce (Picea glauca (Moench) Voss) and hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) in provenance trials across the boreal and hemiboreal forests of Canada. 1.2 Importance of White and Hybrid Spruces White spruce occurs naturally in all provinces and territories of Canada (Sutton 1973), except Nunavut. White spruce is a very important commercial species, being the largest component of an estimated wood supply of 224 million m3 per year (Canadian Forest Service 2015). Because of this extensive spatial distribution and also because of its commercial importance, white spruce is the most widely-used species in tree breeding programs in Canada (Petrinovic et al. 2009). The Quebec tree breeding program started with seed source trials in the 1950’s and, in 1982, plantations of improved white spruce trees were established (Fowler and Morgenstern 1990). In British Columbia (BC) and Alberta, tree improvement of white spruce was initiated in 1956 and 1975, respectively. Programs in Newfoundland, Manitoba, New Brunswick and Nova Scotia were started at later dates, often with white spruce as a primary species along with other commercially important species (Fowler and Morgenstern 1990). 4 The extensive spatial distribution of white spruce is partly due to its ability to establish naturally on well-drained and fertile soils, as well on a wide variety of soils and in a wide climatic range, mostly in mixed species stands (Sutton 1973; Hosie 1978). White spruce can grow to 40 m tall and can live 300 years or longer. Also, this species can survive up to 50 years in the understory (Ontario Ministry of Natural Resources, 1995). White spruce can be infected by needle rusts and other diseases, including root diseases that are known to affect its growth (Whitney 1995). Hybrid spruce is also important in tree breeding programs since they occupy a large portion of the hemiboreal forests of Canada, especially in British Columbia (Xie and Yanchuk 2003) where it is a key commercial tree species (Xie 2003). Growth and survival of hybrid spruce are highly influenced by the soil chemical condition (Maynard and Curran 2009). Hybrid spruce can grow to 60 m tall. Mature hybrid spruce trees can be damaged by insects (e.g., Warren root collar weevil (Hylobius warreni)) and occasionally attacks by diseases (e.g., Lodgepole pine dwarf mistletoe (Arceuthobium americanum Nutt. ex Engelm.)) (Burleigh et al. 2014). Wood characteristics of white and hybrid spruces are similar (Xie and Yanchuk 2002). 1.3 Quantifying Yields of Improved Stocks A number of approaches have been used to estimate yields of improved stocks at rotation age. These approaches vary in that yields of improved stocks can be estimated directly or via estimating gain (i.e., yields relative to a standard). When gains are estimated, they are often implemented as multipliers to adjust existing growth and yield models to predict yields of 5 improved stocks. These multipliers can be: (i) constant for all ages (i.e., age-invariant) or not; and (ii) used to modify one component or more than one component of the growth and yield model. One method that has been used to estimate genetic gain (e.g., yield relative to a standard) is to use the ‘Lambeth’ approach based on the concept that there is a relationship between the gains measured at juvenile and that of mature ages (Lambeth 1980; Falconer 1989; Lambeth and Dill 2001). As a result, the age-age gain correlation is modelled. A number of authors have used an age-age correlation model to estimate age-variant multipliers used to predict yields of improved stocks at a future time. For example, Di Lucca (1999), Xie and Yanchuk (2003), and Yanchuk (2003) used the Lambeth approach in a yield forecast model called TIPSY (Table Interpolation Program for Stand Yields) to forecast genetic gains (Goudie 2003). Similarly, Moss (unpublished) developed a stand level growth simulator model termed Stand Management Financial Analyst (SMFA) in which genetic gain was incorporated, along with responses to different silvicultural treatments, and economic analysis of costs and revenues. Another alternative for estimating the yield of plantations with improved genetic stock is to use an age-invariant multiplier applied to the site index of the planting site. This increased site index approach was examined by Buford and Burkhart (1987) for loblolly pine (Pinus taeda L.) using data from 25-year-old provenance trials. They found that the height trajectory of improved stocks relative to unimproved stocks changed in level but not in shape. McInnis and Tosh (2004) projected the height of improved black spruce (Picea mariana (Mill.) B.S.P) and jack pine (Pinus banksiana Lamb.) beyond the measured data in New Brunswick using the trajectory given 6 by site index curves developed by Ker et al. (1983). The height trajectories were then compared with yields using regular stock (i.e., no change to the site index) using the STAMAN stand growth model (Roussell et al. 1993). They estimated that the merchantable volume gain was about 30% for improved black spruce and more than 30% for jack pine over 40 years of projection. Petrinovic et al. (2009) followed a similar method where site indices were increased by 10% over the yield of regular stock represented by yield tables produced by Bolghari and Bertrand (1984). To obtain this multiplier, they estimated the mean dominant height of the selected families relative to the dominant height over all populations (Petrinovic et al. 2009). Economic analyses were then performed to evaluate the benefits of using improved white spruce trees in Quebec. Ahtikoski et al. (2012) estimated the growth of genetically improved Scots pine (Pinus sylvestris L.) in Finland by increasing the site index used in the MOTTI growth simulator. To estimate the multiplier, they assessed height gain at harvest age and found a 3-15% gain by comparing improved and unimproved stocks. Ahtikoski et al. (2013) applied a 3-15% genetic gain in height of genetically improved Scots pine using the approach by Ahtikoski et al. (2012) to evaluate financial incentives for using improved materials under varying climates. As an alternative to altering the site index, age-invariant multipliers can be used to alter sub-models of growth and yield simulators to simulate genetic gain (Adams et al. 2006). Hamilton and Rehfeldt (1994) proposed a number of multipliers for ponderosa pine (Pinus ponderosa Laws.). Using the Stand Prognosis Model (Stage 1973; Wykoff et al.1982), they modified the simulated heights until they matched with the average heights of improved stocks at ages 8, 14 and 19 years; a similar approach was followed for unimproved stocks. Carson et al. (1999) estimated the gain of genetically improved radiata pine (Pinus radiata D. Don) in New Zealand 7 using the climbing select seedlot (labelled as growth form 7 (GF7)) as the baseline. Gould et al. (2008) obtained genetic gain multipliers by comparing the height and diameter growth of 15-year-old Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) improved and unimproved stocks trees. 1.4 Using Meta-Modelling for Larger Temporal and Spatial Extents Regardless of whether yields of improved stocks are estimated directly or indirectly via estimating gain, meta-modelling can be used to increase the applicable temporal and spatial extents of developed models. Meta-modelling (Gilliams et al. 2005), meta-analysis, and combined estimators (e.g., Green and Strawderman 1991) approaches use previous research and data to develop a more general model, often on a broader scale in time and/or space than any of the original studies. Previous meta-data may be at the individual (e.g., repeated measures on trees or plots) or aggregate level (e.g., model outputs). Meta-modelling approaches have been used in a limited number of studies to evaluate growth and yield responses using provenance trials (Rehfeldt et al. 1999; Newton 2003; McLane et al. 2011; Leites et al. 2012). Rehfeldt et al. (1999) combined a number of provenances of Larix sukaczewii Dylis, L. sibirica Ledeb., and L.gmelinii (Rupr.) Rupr. at four planting sites in Alberta, Canada, and used the combined data to model mean heights over time. Although Newton (2003) used the term “Systematic Review” rather than meta-modelling in his paper, he analyzed yield responses of a number of commercially important conifer species by combining information from previous tree improvement studies in a meta-dataset to estimate relative height 8 gain and yield of improved stocks. McLane et al. (2011) combined the data for several common garden experiments for lodgepole pine (Pinus contorta var. latifolia Douglas ex Louden) to estimate the basal area growth of different provenances. Leites et al. (2012) used plot-level data from several plantations of Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) and modelled height yields. 1.5 Forecasting Individual Provenances on a Planting Site To assess performance of a particular provenance (or progeny) on a planting site, an accurate forecast to likely harvest age would be beneficial. For this purpose, a developed meta-model could be used to obtain a population-averaged prediction. However, where repeated measures exist for that provenance, these measures could be used to improve the forecast using approaches previously used to forecast permanent sample plot yields. In provenance or progeny trials, measurements of each provenance are usually taken several times. Mixed effects models can be used to provide localized estimates of each provenance that can account for autocorrelation among repeated measures. In particular, nonlinear mixed models (NLMM) following Gregoire and Schabenberger (1996) have been used in a number of forestry applications (e.g., Wang et al. 2007; Meng and Huang 2009; Meng and Huang 2010; Huang et al. 2011) but not yet implemented for forecasting growth and yield improved stocks. Magnussen and Yanchuk (1994) used linear mixed-effects models for predicting breeding values and accounted autocorrelation in time. Meng and Huang (2009) used NLMMs to obtain subject-specific estimates of trees outside the model-fitting data. Meng and Huang (2010) demonstrated 9 an alternative method to obtain subject-specific parameter estimates where repeated measures for the provenance could be used in the fitted autocorrelation model to improve forecasts (Judge et al. 1988). 1.6 Objectives and Dissertation Structure The overall goal of this research was to provide a possible mechanism to estimate genetic gain (relative to a standard) as well as yield at harvest as a result of using improved spruce stocks. The intent was that this research would contribute towards evaluating the use of improved genetic stocks to ameliorate possible timber supply shortages in the boreal and hemiboreal forests of Canada. To achieve the overall goal, a number of intermediate goals were identified along with corresponding research activities. These were organized into three research chapters in the dissertation: meta-models to quantify yields of white spruce and hybrid spruce provenances across broad spatial and temporal extents (Chapter 2); meta-models to quantify yield gains (Chapter 3); and alternative methods to forecast yields of hybrid spruce provenances (Chapter 4). As part of this research, a meta-database of white spruce and hybrid spruce provenance trials in the Canadian boreal and hemiboreal forests of Canada was developed, based on an extensive literature search (as described in Chapter 2, the first research chapter). The specific objectives of these research chapters are as follows: 10 1. Meta-models to quantify yield The overall goal in this research chapter was to provide a possible mechanism to evaluate impacts of using improved spruce stocks to ameliorate timber supply shortages in the boreal and hemiboreal forests. The specific objectives were: i. To develop a height yield trajectory (i.e., cumulative height growth over time) model for white spruce provenance trials across a large range of planting sites of the Canadian boreal forest. ii. To develop a height yield trajectory model for white spruce and hybrid spruce provenance trials across a large range of planting sites of the Canadian boreal and hemiboreal forests. iii. To assess the impact of climatic, site, and provenances on height yield. iv. To incorporate a height trajectory model into an existing growth and yield model to forecast volume per ha, basal area per ha, and average diameter trajectories for provenance and planting site climates. 2. Quantify gain The overall goal in this research chapter was to estimate the gain for improved stock of white spruce and hybrid spruce in the boreal and hemiboreal forests of Canada. The specific objectives were: i. To assess the effects of the evaluation age (i.e., the age when top performers are selected), in concert with the definitions of the top performers and the baseline, on the estimated gain. 11 ii. To assess whether gain changes over time and/or with planting site using a modelling approach and based on a selected definition of the evaluation age, the top performers, and the baseline. iii. To provide a final model that estimates the changes in gain for white spruce over a wide spatial and temporal range. 3. Alternative methods to forecast provenance yields The overall goal in this research chapter was to evaluate alternative methods to forecast height yields of hybrid spruce progenies given prior repeated measures. For this purpose, I used data from progeny trials from hemiboreal forests of BC. The specific objectives were: i. To compare forecast accuracies among three forecast methods (i.e., population-averaged model; subject-specific forecasts; and autocorrelation model forecasts for progenies that were included in the data used to fit the hybrid spruce height yield model). ii. To compare forecast accuracies among three forecast methods for progenies that were not included in the model-fitting data. iii. To evaluate possible forecast accuracy improvements due to including additional prior measures. iv. To evaluate the effects of the accuracy of the fixed-effects part of the non-linear mixed model on forecast accuracies. 12 In the final chapter (Chapter 5), I highlighted the major contributions of this research towards reducing the knowledge gap on growth and yield of tree improvement programs. The most important findings are briefly summarized, and recommendations for further research are given. 13 2. Meta-Modelling to Quantify Height Yields of White Spruce and Hybrid Spruce Provenances in the Canadian Boreal Forest1 2.1 Introduction The Canadian forest industry is facing increased competition for forest land from urban expansion and from other resource sectors including agriculture, oil, and gas, resulting in a diminishing fiber supply (Kissinger and Rees 2007; Brockerhoff et al. 2008). At the same time, climate changes are expected to impact forests via changes in forest growth rates and in natural disturbances caused by fire, diseases, and insects (Dale et al. 2001; Soja et al. 2007). Tree improvement programs can provide improved seedling stocks with faster growth rates and/or better disease and insect resistances (Zobel and Talbert 1984; Kitzmiller 1990; White et al. 2007; Weiskittel et al. 2011). Using improved stock can ameliorate timber supply shortages and facilitate a more competitive forest sector, while continuing to provide a broad range of ecosystem services by supporting sustainable forest management over the entire forest land area (Petrinovic et al. 2009). In Canada, tree improvement programs began in the 1950s with the overall goal of providing improved stocks with superior traits (Fowler and Morgenstern 1990). Initially, seeds were collected from phenotypically-superior individuals called “plus trees” from natural stands with unknown male parents; performances were then assessed using provenance trials representing multiple seed sources to establish seed planning zones. Subsequent selections of seeds through both wind (i.e., natural-stand mating) or controlled-pollinated crosses of these first-generation 1 A version of this chapter is in review: Ahmed, S. S., LeMay, V., Yanchuk, A., Robinson, A., Marshall, P. and Bull, G. In Review. Meta-modelling to quantify yields of white spruce and hybrid spruce provenances in the Canadian boreal forest. 14 trees were used to establish progeny trials to assess the extent of genetic control over selection attributes, to rank parents and offspring, to establish production populations (seed orchards), and to quantify the expected genetic gain from improved stocks. However, quantifying gain from improved stocks remains a challenge for a number of reasons. First, trees are long-lived species and, as a result: (i) the time to maturity (and subsequent generations) is long, unlike most agricultural crops; and (ii) even the earliest provenance trials in Canada represent only short periods of time relative to biological rotation ages reported for natural stands. Second, growth rates can increase with subsequent selections of best performing trees (McInnis and Tosh 2004); Carson et al. (1999) termed this a “moving target”. Third, there is an environmental effect and also there may be a provenance by environment effect in that provenance performances vary with planting site characteristics (Matyas 1994; Rweyongeza et al. 2011). Finally, in quantifying improvements, a baseline for comparison must be used. This baseline could be wild stocks (e.g., Woods et al. 1995; Xie and Yanchuk 2003), seeds from the planting location (i.e., local provenance, Nance and wells 1981; Carson et al. 1999), or the average of all provenances in the trials (e.g., Magnussen and Yanchuk 1994; Newton 2003). In spite of these difficulties, a number of researchers have quantified yields of improved stocks for Canadian tree species. For example, Beaulieu (1996) compared yields of white spruce (Picea glauca (Moench) Voss) provenances and their breeding performances based on trials in Quebec. Simpson and Tosh (1997) reported the 20-year growth increase of black spruce (Picea mariana (Mill.) B.S.P.), white spruce, tamarack (Larix laricina (Du Roil) K. Koch), and jack pine (Pinus banksiana Lamb.) using provenance trials in New Brunswick. Xie and Yanchuk (2003) 15 discussed a number of ways to quantify genetic gain and described the method used to modify natural forest yields of commercial species of British Columbia. McInnis and Tosh (2004) modelled growth and yield trends of first-generation jack pine (Pinus banksiana Lamb.) and black spruce in New Brunswick using 20-year measurements. Petrinovic et al. (2009) quantified yields of improved white spruce as a means of determining the economic benefits of the tree improvement program in Quebec. Meta-modelling techniques provide a mechanism for combining results of many trials to estimate yields of improved stocks over longer temporal and larger spatial extents. Meta-modelling is related to the meta-analysis and combined estimators approaches, each of which provide more precise estimates by synthesizing data from different studies (Borenstein et al. 2009). Meta-modelling (also called “meta-regression”) generally refers to the combined use of aggregate data (termed macro-data in some papers) previously reported in separate studies (Gilliams et al. 2005), but may also incorporate individual data (termed micro-data in some papers) (Raghunathan et al. 2003). Meta-analysis differs in that it generally refers to the combination of trials with the same experimental designs (DerSimonian and Laird 1986; Koricheva et al. 1998). Finally, combined estimators, such as the James-Stein estimator, combine aggregate results from previous research (e.g., an existing model) with new data to provide an updated or localized model (e.g., Green and Strawderman 1991). The use of meta-modelling to combine information across provenance trials has been reported in a limited number of studies. Rehfeldt et al. (1999) predicted mean heights of different provenances of Larix sukaczewii Dylis, L. sibirica Ledeb., and L.gmelinii (Rupr.) Rupr. at four 16 planting sites in Alberta using simple models and climate transfer variables (i.e., differences in climate between provenance and planting site locations). Newton (2003) modelled yield responses of a number of conifer species of the boreal forest of Canada using reported results for several provenance trials from Canada and the United States. McLane et al. (2011) combined the tree-level information of several lodgepole pine (Pinus contorta var. latifolia Douglas ex Louden) common garden experiments in Washington State, USA and British Columbia, Canada. They used a random-coefficients mixed-effects modelling approach to estimate the basal area growth of different provenances (i.e. seed source location) under current and forecasted future climates. Leites et al. (2012) used a mixed-effects model and plot-level data from several plantations in the north-western USA to predict three-year height response of Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco). In this study, meta-modelling approaches were used to model yields of white spruce and hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) provenances of the boreal and hemiboreal forests of Canada. Globally, the boreal forest circles the earth across Nordic countries, Russia, and Canada (Burton et al. 2003). The Canadian boreal forest represents 28% of the world boreal forest and is the largest forest type in Canada, covering 552 million hectares (Natural Resources Canada 2015) and occurring in all 10 provinces and two of the three territories (Figure 2-1). In this forest type, trees experience dry, cold and very long winters, whereas the short summer season is cool and moist (Bonan and Shugart 1989). As a result, species diversity and growth rates are quite low (Kelty et al. 1992). The dominant tree species are aspen (Populus tremuloides Michx.), jack pine, white spruce, and black spruce. Typically, white spruce grows in mixtures with aspen, willow (Salix spp.), paper birch (Betula papyrifera 17 Marsh.) and balsam fir (Abies balsamea (L.) Mill.), but can also occur in mixed black-white spruce stands. The hemiboreal forest is to the south of boreal forest and is transitional with temperature forests, except in Alberta and British Columbia where it is generally to the south and west of the boreal forest and is transitional to mountainous forests. As a transitional forest, characteristics are similar to the boreal forest (Brandt 2009), but species from other forest types can occur including a number of hardwoods south of the boreal forest in Ontario and Quebec, and hybrid spruces and other species in Alberta and British Columbia. Figure 2-1. Boreal and hemiboreal forests of Canada and USA (Brandt 2009). White and hybrid spruces are very commercial important species and occupy a large spatial extent across the boreal and hemiboreal forests of Canada. White spruce is a shade-tolerant coniferous species that is considered a “plastic” species because of its ability to adapt to extreme climates (Burns and Honkala 1990). White spruce hybridizes with Engelmann spruce (Picea 18 engelmannii Parry ex Engelm.) at mid-elevations (Rajora and Dancik 2000) and occurs on large portions of the hemiboreal forest in British Columbia (Xie and Yanchuk 2002) and in the foothills in Alberta (Rweyongeza et al. 2011). Growth and survival of hybrid spruce is sensitive to soil nutrient conditions (Xie et al. 1998), generally growing faster on acidic soils (Maynard and Curran 2009). Whereas heights of white spruce can reach 30 metres, this hybrid can reach up to 60 m in height. The overall goal of this research is to provide a possible mechanism to evaluate impacts of using improved spruce stocks to ameliorate timber supply shortages in the boreal and hemiboreal forests. The specific objective of this study was to estimate yields over time for a wide range of provenances and planting locations. To achieve this objective, a height yield trajectory (i.e., cumulative height growth over time) model was developed using meta-data for white spruce provenance trials across a large range of planting sites of the Canadian boreal forest. Climatic variables for the provenance and planting site locations were then used to modify height trajectories following an approach similar to Leites et al. (2012). This was repeated by augmenting the white spruce meta-data with hybrid spruce provenances trials from the hemiboreal forest, thereby providing an even wider range of site and provenance climate variables. The developed average height trajectory model was then illustrated using an existing planting site with several provenances. However, since climate variables for provenance and planting site locations were used in the average height trajectory model, the model was also illustrated using a large range of provenances and planting sites not specifically represented in the meta-data. Finally, the average height trajectory model was incorporated into an existing 19 growth and yield model to forecast volume per ha, basal area per ha, and average diameter trajectories for provenance and planting site climates via altering the average height trajectories. 2.2 Methods 2.2.1 Meta-Datasets Average height trajectory models were developed for two meta-datasets separately. The first meta-dataset included white spruce provenances from the Canadian boreal forest (hereafter, the Sw meta-dataset). This was augmented with white and hybrid spruce provenances from the Canadian hemiboreal forest (hereafter, the Sw-Se meta-dataset). The Sw meta-dataset was collected from published articles, internal reports, government reports, and other sources principally using the Web of Science® (Science Citation Index expanded) search engine following an approach similar to that used by Newton (2003) and Varhola et al. (2010). In some cases, authors were contacted for more information, such as additional re-measurement data and unreported variables. Since a longer temporal extent was of interest, provenance trials2 generally included open-pollinated and first generation stocks. From this search, 16 studies were selected representing 30 planting sites in Canada with provenances from Canada and the United States (Table 2-1). For each selected study, the following variables were obtained: (i) species; (ii) planting site label; (iii) latitude and longitude of the planting site; (iv) elevation of the planting site (m); (v) plantation start date and measurement dates; (vi) initial planting density (stems per hectare); (vii) plantation age (years); (viii) average height of each 2 Data from ATISC (2011) were from progeny rather than provenance trials. 20 provenance at each measurement time (m); and (ix) latitude, longitude, and elevation of each provenance. Table 2-1. Sources of meta-data for white spruce provenances (Sw meta-dataset) in the boreal forest of Canada. Source Province Planting Site Plantation Ages (years) Number of Provenances Tested Density (Stems/ha) Elevation (metres) Beaulieu (1996) Quebec Drummondville 20 11 6,726 85 aBeaulieu (1996) and Quebec Casey 14-24 28 3,019 440 aBeaulieu (1996) Quebec Grandes-piles 14 20 3,019 370 Beaulieu (1996) and Quebec Harrington 14-20 41 6,726 210 aBeaulieu (1996) and Quebec St-Jacques-des-Piles 14-24 26 3,019 393 ATISC (2011) Alberta Calling Lake 15 6 1,600 625 ATISC (2011) Alberta Kinosis Lake 15 6 1,600 495 ATISC (2011) Alberta Wandering River 15 6 1,600 567 Morgenstern et al. (2006) Ontario Petawawa National Forestry Institute 15-44 25 3,086 170 Hall (1986) Newfoundland Gander 25 32 3,086 140 Morgenstern and Copies (1999) Ontario Chalk River 1 6 73 666 170 Morgenstern and Copies (1999) Ontario Chalk River 2 6-10 71 666 170 Morgenstern and Copies (1999) Ontario Chalk River_194D1 27-33 25 15,625 170 Morgenstern and Copies (1999) Ontario Chalk River_194M 19-31 53 3,086 170 Morgenstern and Copies (1999) Ontario Chalk River_93C 25-38 25 6,726 170 Morgenstern and Copies (1999) Ontario Dorset 25-32 25 15,625 300 Morgenstern and Copies (1999) Ontario Dryden 11-16 77 3,086 410 Morgenstern and Copies (1999) Ontario Fort Frances 11-18 66 157 340 Morgenstern and Copies (1999) Ontario Hearst 7-11 85 2,500 320 Morgenstern and Copies (1999) Ontario Kapuskasing 20-33 25 3,086 170 Morgenstern and Copies (1999) Ontario Kenora 11-18 49 157 410 Morgenstern and Copies (1999) Ontario Lake Dore 21-33 12 16,667 120 Morgenstern and Copies (1999) Ontario Nipigon 11-16 80 157 190 Morgenstern and Copies (1999) Ontario Owen Sound 6-12 64 2,500 430 Morgenstern and Copies (1999) Ontario Owen Sound_194E 25-33 24 3,086 430 Morgenstern and Copies (1999) Ontario Owen Sound_93C 30-38 25 6,726 430 Morgenstern and Copies (1999) Ontario Red Lake 6-11 79 157 370 Morgenstern and Copies (1999) Ontario Sudbury 8-13 84 157 350 Morgenstern and Copies (1999) Ontario Thunder Bay 23-31 48 3,086 300 Khalil (1979) Newfoundland North Pond 20 32 3,086 71 a Dr. Jean Beaulieu provided further measurements of the provenance trial established by the Canadian Forest Service in Quebec, Canada. 21 For many of these studies, the location information for each provenance used in the study was identified from other sources. The plantation start date, measurements dates and densities varied across planting sites (Table 2-2). Although, repeated measures were preferable for this study, some provenances had only a single time measurement reported. The average height trajectories by provenance show a relatively regular pattern (Figure 2-2). Table 2-2. Variable ranges for the Sw meta-dataset (30 planting sites). Variable Minimum Mean Maximum Planting site latitude 44.20 47.52 56.30 Planting site longitude -113.13 -82.47 -53.81 Planting site elevation (metres) 71.00 285.70 625.00 Provenance latitude 43.70 48.03 62.03 Provenance longitude -139.00 -84.59 -55.85 Provenance elevation (metres) 20.00 318.70 1,585.00 Plantation age at last measurement (years) 6.00 15.00 44.00 Initial planting density (stems/ha) 157.00 2,729.17 16,667.00 22 Figure 2-2. Observed average height trajectory of selected white spruce provenances in the boreal forest of Canada (dots indicate a single measurement in time). The Sw-Se meta-dataset was obtained by augmenting the Sw meta-dataset to include hybrid spruce provenances3 data from the hemiboreal forest of British Columbia. The same criteria for selection were used, resulting in eight additional studies (Table 2-3) and a wider spatial and elevation extent (Table 2-4). 3 Data for hybrid spruce were from progeny rather than provenance trials. 23 Table 2-3. Sources of meta-data covering selected hybrid spruce provenances (Sw-Se meta-dataset) in British Columbia Source Province Planting Site Plantation Ages (years) Number of Provenances Tested Density (Stems/ha) Elevation (metres) Xie (2003) British Columbia Central Plateau 10 6 2,500 960 Xie (2003) British Columbia Fort Nelson 10 6 2,500 520 Xie (2003) British Columbia Hudson Hope 10 6 2,500 890 Xie (2003) British Columbia McGregor 10 6 2,500 670 Xie (2003) British Columbia Mount Robson 10 6 2,500 1000 aYanchuk and Jaquish unpublished data British Columbia Aleza Lake 2-42 25 2,500 700 aYanchuk and Jaquish unpublished data British Columbia Prince George Tree Improvement Station 2-42 25 2,500 610 aYanchuk and Jaquish unpublished data British Columbia Quesnel 2-42 25 2,500 915 a Dr. Alvin Yanchuk and Barry Jaquish provided several measurements of provenances in three trials in May, 2013. Provenance trials were initiated by British Columbia Ministry of Forests under the interior spruce tree breeding program in British Columbia. 24 Table 2-4. Variable ranges for the Sw-Se meta-dataset (38 planting sites). Variable Minimum Mean Maximum Planting site latitude 44.20 49.09 58.98 Planting site longitude -124.28 -92.61 -53.81 Planting site elevation (metres) 71.00 403.06 1,000.00 Provenance latitude 43.70 49.48 62.03 Provenance longitude -139.00 -94.14 -55.85 Provenance elevation (metres) 20.00 450.62 1,585.00 Plantation age at last measurement (years) 2.00 15.00 44.00 Initial planting density (stems/ha) 157.00 2,455.00 16,667.00 2.2.2 Climate Data For each provenance at every planting site, climate information, specifically, climate normals were added to the meta-datasets. To obtain these data, the process followed was: 1. Weather station data for the 1981-2010 period were extracted from the Environment Canada climate normals database (www.climate.weatheroffice.ec.gc.ca, accessed 10th September, 2013) and from the National Oceanic and Atmospheric Administration’s (NOAA) National Climatic Data Center (NCDC) (ftp://ftp.ncdc.noaa.gov/pub/data/normals/1981-2010, accessed 11th September, 2013). The USA data were then converted to equivalent SI units. 2. For each weather station, the following climate variables were calculated for each year and then averaged over the 1981-2010 period: (i) mean daily temperature (MAT); (ii) mean 25 annual precipitation (MAP); and (iii) mean annual number of days (i.e., “degree days”) with a temperature greater than 5°C (DD). 3. These weather station climate variables were then used to interpolate climate normals for each provenance location and planting site using inverse distance weighting (IDW). Using the climate normals for each provenance and planting site, climate transfer distances were calculated for each provenance at a planting site as: 1. Mean daily temperature distance (DMAT) = site MAT – provenance MAT, where a positive DMAT occurs for a provenance from a colder area that is planted at a warmer planting site, and a negative DMAT occurs for a provenance from a warmer area that is planted at a colder planting site. 2. Mean annual precipitation distance (DMAP) = site MAP – provenance MAP, where a positive DMAP occurs when precipitation is higher at the planting site than at the provenance location, and a negative DMAP occurs when precipitation is lower at the planting site than at the provenance location. 3. Degree days distance (DDdif) = site DD – provenance DD, where a positive DDdif occurs when DD is higher at the planting site than at the provenance location, and a negative DDdif occurs when DD is lower at the planting site than at the provenance location. Summaries of the climate transfer distance variables for the two meta-data sets are presented in Tables 2-5 and 2-6. Of note, the DD for the provenances had a very wide range since this is a threshold metric. As a result, the provenances from the USA sites had much higher values for DD, since a greater number of days reached the threshold of 5°C. This also resulted in a very wide range for the DDdif as shown in Table 2-6 (see also Table 2-7 later in this dissertation). 26 Table 2-5. Climatic variable ranges for the Sw meta-dataset (30 planting sites). aVariable Minimum Mean Maximum Site MAT (°C) -0.50 3.82 6.85 Site MAP (mm) 423.40 880.79 1,165.55 Site DD (days) 46.00 81.70 119.00 Provenance MAT (°C) -4.39 3.59 7.68 Provenance MAP (mm) 277.69 863.63 1,590.78 Provenance DD (days) 28.00 78.00 203.00 DMAT (°C) -5.85 0.23 8.66 DMAP (mm) -731.00 17.16 861.89 DDdif (days) -132.00 3.00 84.00 aMAT (°C) is the mean daily temperature; MAP (mm) is the mean annual precipitation; DD (days) is the mean degree days greater than 5°C; and DMAT (°C) is mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the provenance. 27 Table 2-6. Climatic variable ranges for the Sw-Se meta-dataset (38 planting sites). aVariable Minimum Mean Maximum Site MAT (°C) -0.50 3.99 6.85 Site MAP (mm) 423.40 811.12 1,166.55 Site DD (days) 46.00 81.70 119.00 Provenance MAT (°C) -4.39 3.78 7.68 Provenance MAP (mm) 277.69 801.20 1,590.78 Provenance DD (days) 28.00 74.00 203.00 DMAT (°C) -6.83 0.21 8.66 DMAP (mm) -731.05 9.92 861.90 DDdif (days) -132.00 2.47 85.00 aMAT (°C) is the mean daily temperature; MAP (mm) is the mean annual precipitation; DD (days) is the mean degree days greater than 5°C; and DMAT (°C) is mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the provenance. 2.2.3 Meta-Models 2.2.3.1 Base model and parameter prediction The Chapman-Richard growth model (Pienaar and Turnbull 1973) was selected as the base average height trajectory model: [2.1] H𝑖𝑖𝑖𝑖𝑖𝑖 = 𝜃𝜃1 �1 − exp �1 − 𝜃𝜃2 age𝑖𝑖𝑖𝑖𝑖𝑖��� 11−𝜃𝜃3� + 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 28 where H𝑖𝑖𝑖𝑖𝑖𝑖 is the average height of planting site i and provenance j at measurement time t; age𝑖𝑖𝑖𝑖𝑖𝑖 is the plantation age; and 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 is the error term. This model has three parameters: the asymptote (𝜃𝜃1) specifies the maximum height and the other two are shape parameters (𝜃𝜃2 and 𝜃𝜃3). This base model represents the average trajectory over all provenances and sites. Although other models have been used for this average height trajectory, the Chapman-Richards model: (i) has a biological basis as a derivation of the organism growth model by Von Bertalanffy (1957); and (ii) represents yields in a simplified mathematical form with only three parameters that nonetheless reflect a wide range of curve shapes. As a result, this model has been commonly used for yield of trees and stands (e.g., Weiskittel et al. 2011). The model also reaches an asymptote which is likely for average heights of high-value plantations since these would only decline if: (i) regeneration/ingrowth trees were not removed (i.e., no stand tending to remove non-crop trees) meaning that shorter trees would be averaged in; ii) the taller trees died at a higher rate than shorter trees; or (iii) there was damage to trees and "damaged" heights were included in the average height. The first case is unlikely for higher value plantations; even if regeneration/ingrowth trees were present, often only crop-trees are included in average heights. The second case is also unlikely since little structural diversity is expected in plantations with controlled spacing and limited genetic stock. Finally, the plantation would be "compromised" due to catastrophic damage and mortality. Therefore, in the absence of any information on these plantations to harvest age and a low probability of average height deline, the leveling off is a reasonable conjecture. Further, the average height model would be incorporated into an existing growth and yield model as later illustrated in this paper. The volume per ha would be reduced over time due to plantation mortality (as would biomass, etc.) in other submodels. 29 Random coefficients (aka, parameter prediction) modelling (Clutter et al. 1983, pp. 54-56, Schabenberger and Pierce 2002, Chapter 7, Littell et al. 2006 Chapter 8) was used to modify the base models for planting sites and provenances for each meta-dataset, following the approaches of McLane et al. (2011) and Leities et al. (2012). McLane et al. (2011) replaced the coefficients of a nonlinear basal area growth model with functions of source (i.e., genetic population) and location (i.e., planting site) climate variables. Leites et al. (2012) used a linear mixed-effects model fitted using plot-level information from different studies to get the predicted height growth response of Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) populations. Their model used three levels of blocking, namely test locations, planting site within test location, and block within planting site, as random effects in the model. Then, a number of variables that represented climate transfer distances were used to predict height growth response of each provenance and site. In this study, the Chapman-Richard growth model (Eq. 2.1) was modified to predict average height of each provenance at each site by replacing the parameters with functions of climate transfer distances and other variables: [2.2] H𝑖𝑖𝑖𝑖𝑖𝑖 = 𝜃𝜃1𝑖𝑖𝑖𝑖(1 − exp (1 − 𝜃𝜃2𝑖𝑖𝑖𝑖age𝑖𝑖𝑖𝑖𝑖𝑖))( 11−𝜃𝜃3𝑖𝑖𝑖𝑖) + 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 [2.2a] 𝜃𝜃1𝑖𝑖𝑖𝑖 = f�climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿1𝑖𝑖𝑖𝑖 [2.2b] 𝜃𝜃2𝑖𝑖𝑖𝑖 = f�climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿2𝑖𝑖𝑖𝑖 [2.2c] 𝜃𝜃3𝑖𝑖𝑖𝑖 = f�climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿3𝑖𝑖𝑖𝑖 where H𝑖𝑖𝑖𝑖𝑖𝑖, age𝑖𝑖𝑖𝑖𝑖𝑖, and 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 were previously defined; and 𝛿𝛿1𝑖𝑖𝑖𝑖, 𝛿𝛿2𝑖𝑖𝑖𝑖 and 𝛿𝛿3𝑖𝑖𝑖𝑖 are error terms. 30 Several random coefficients models were fitted using combinations of predictor variables in sub-models 2.2a to 2.2c. Possible predictor variables were: climate transfer distance variables (DMAT, DMAP, DDdif), planting density, site elevation, and provenance elevation. Transformations of some variables were included to represent expected nonlinear trends with climatic variables. Also, the climate transfer model used by Rehfeldt et al. (1999) was used as a guide for predicting the asymptote parameter [2.2a]. PROC MODEL of SAS software version 9.3 was used to fit the models (SAS Institute Inc. 2014). To ensure a global optimum for each model solution, several starting parameter sets were included and each model was carefully compared to measured values. All fitted models were then compared and one model was selected for each meta-dataset. 2.2.3.2 Model Selection To evaluate the accuracy of each model and to select among alternative models, the goodness of fit statistics used were: 1. Pseudo R2 calculated as: Pseudo R2 = 1 − 𝑆𝑆𝑆𝑆𝑟𝑟𝑟𝑟𝑟𝑟𝑖𝑖𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟 (𝑐𝑐𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑡𝑡𝑟𝑟𝑟𝑟) = 1 − ∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑡𝑡−H�𝑖𝑖𝑖𝑖𝑡𝑡�2𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑡𝑡−H��2𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1 where 𝑖𝑖 = 1, … ,𝑛𝑛 planting sites; 𝑗𝑗 = 1, … ,𝑚𝑚𝑖𝑖 provenances within planting sites; 𝑡𝑡 = 1, … , 𝑙𝑙𝑖𝑖𝑖𝑖 measurement times for a provenance with a planting site; and H�𝑖𝑖𝑖𝑖𝑖𝑖 is the predicted height. 2. Root mean squared error (RMSE, m) calculated as: RMSE = �∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑡𝑡−H�𝑖𝑖𝑖𝑖𝑡𝑡�2𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1𝑀𝑀−𝑝𝑝 31 where 𝑀𝑀 is the total number of observations; and 𝑝𝑝 is the number of fixed-effects parameters in the model. 3. Akaike’s information criterion (AIC) calculated as: 𝐴𝐴𝐴𝐴𝐴𝐴 = −2𝑙𝑙𝑙𝑙𝑙𝑙L + 2𝑝𝑝 where L is the likelihood evaluated at its maximum. As well as these fit statistics, graphs of predictions superimposed on the observed data by provenance and planting site were used to check for any model lack of fit. 2.2.3.3 Model validation Model validation was applied only to the model selected for each meta-dataset. First, these two selected models were evaluated using the “leave-one-out” model validation approach. For this purpose, one planting site along with all provenances at that site was excluded from model fitting, and then the model was applied to the excluded planting site. This was repeated for each planting site and results were summarized into the root mean square predicted error (RMSPE) and mean prediction error (MPE): RMSPE = �∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑖𝑖 − H�𝑖𝑖𝑖𝑖𝑖𝑖�2𝑙𝑙𝑖𝑖𝑖𝑖𝑖𝑖=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1𝑀𝑀 MPE = ∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑖𝑖 − H�𝑖𝑖𝑖𝑖𝑖𝑖�𝑙𝑙𝑖𝑖𝑖𝑖𝑖𝑖=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1𝑀𝑀 Further validation of each of the two selected models was done using the “0.632 bootstrap” validation approach (Efron and Tibshirani 1997; McNelis 2005). For this, the model was fitted using 30 planting sites selected using simple random sampling with replacement; the fitted 32 model was applied to the non-sampled sites (i.e., test set). Results were then summarized to obtain RMSPEc and error variance (𝜎𝜎𝑐𝑐2): RMSPE𝑐𝑐 = �∑ ∑ ∑ �𝐻𝐻𝑖𝑖𝑖𝑖𝑖𝑖 − 𝐻𝐻�𝑖𝑖𝑖𝑖𝑖𝑖�2𝑙𝑙𝑖𝑖𝑖𝑖𝑖𝑖=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑟𝑟𝑖𝑖=1 𝑘𝑘 𝜎𝜎𝑐𝑐2 = ∑ ∑ ∑ 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖2𝑙𝑙𝑖𝑖𝑖𝑖𝑖𝑖=1 − �∑ ∑ ∑ 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖𝑙𝑙𝑖𝑖𝑖𝑖𝑖𝑖=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑟𝑟𝑖𝑖=1 �2 𝑘𝑘�𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑟𝑟𝑖𝑖=1𝑘𝑘 − 1 where k is the number of observations in the test set and 𝑛𝑛𝑠𝑠 is the number of non-sampled planting sites. After repeating this process 1000 times, the average RMSPEs (RMSPE���������𝑐𝑐) and average error variances were calculated (𝜎𝜎�𝑐𝑐2). Next, the model was fitted using all meta-data and applied to the same meta-data; these results were also summarized into a root mean squared error (RMSPEM) and error variance (𝜎𝜎𝑀𝑀2 ). Then, the weighted averages of these (i.e., using the test data-sets and using the full meta-dataset) were used to calculate the 0.632 bootstrap root mean square predicted error (RMSPE0.632) and error variance (Err0.632) as follows: Err0.632 = 0.368 𝜎𝜎𝑀𝑀2 + 0.632 𝜎𝜎�𝑐𝑐2 RMSPE0.632 = 0.368 RMSPE𝑀𝑀 + 0.632 RMSPE���������𝑐𝑐 2.2.3.4 Using the Average Height Trajectory Model to Forecast Yields To illustrate the use of the selected average height trajectory model, the Petawawa National Forestry Institute (PNFI) site in Ontario was chosen from the meta-data since this was the oldest plantation (last measured at 44 years since planting) with the largest number of provenances (25 provenances, Figure 2-3 and Table 2-7). For each provenance, the predicted versus measured heights were visually compared. 33 Figure 2-3. Geographic locations of the selected planting site at the Petawawa National Forestry Institute (PNFI) in Ontario along with the 25 provenances that were tested at the site. 34 Table 2-7. Provenances at the Petawawa National Forestry Institute site (46° latitude, -77.5° longitude, 170 m elevation, 847 mm MAP, 5°C MAT and 98 days DD) in Ontario. aID Provenance (Prov.) bProv. MAP (mm) Prov. MAT (°C) Prov. DD. (days) DMAT (°C) DMAP (mm) DDdif (days) Prov. Elevation (m) 1 Ashley Mines 861 2.0 72.6 2.72 -13.2 25.5 340 2 Bissett Creek South 877 5.1 83.9 -0.33 -29.1 14.3 260 3 Chalk River 846 4.8 86.4 -0.05 1.0 11.8 160 4 Chequamegon National Forest 803 5.3 171.4 -0.60 44.9 -73.2 198 5 Cobourg 908 7.3 74.9 -2.59 -60.1 23.3 260 6 Cook County 678 3.8 171.0 0.89 169.0 -72.9 198 7 Cushing 1,075 6.0 71.8 -1.27 -227.3 26.3 76 8 Edmundston 1,020 3.6 70.5 1.17 -172.3 27.7 198 9 Grand Rapids 685 4.2 170.9 0.52 162.2 -72.7 411 10 Grandes-Piles 1,084 3.9 60.6 0.87 -236.8 37.5 370 11 Huron Nat. For. 789 6.3 190.5 -1.61 58.4 -92.4 210 12 Kakabeka 739 2.8 67.8 1.95 108.0 30.4 274 13 Lac Baskatong 1,035 3.5 56.6 1.25 -188.0 41.5 240 14 Lac McNally 928 4.2 62.2 0.55 -80.3 35.9 240 15 Lac Mitchinamicus 1,039 3.5 56.8 1.20 -191.3 41.3 400 16 Luce County 864 4.5 178.9 0.23 -16.6 -80.8 210 17 Manitoulin Island 883 5.1 88.9 -0.42 -35. 7 9.2 210 18 Marquette County 749 5.2 187.2 -0.51 98.1 -89.1 270 19 Miller Lake 1,132 6.9 76.3 -2.16 -284.9 21.9 180 20 Notre-Dame-du-Laus 1,088 4.2 63.2 0.57 -240.4 34.9 320 21 Pagwachuan Lake 822 1.4 67.5 3.36 25.9 30.6 300 22 Price 1,099 2.3 55.3 2.43 -252.0 42.8 300 23 Shipshaw River 983 2.4 50.6 2.28 -135.4 47.5 296 24 Swastika 871 2.0 87.0 2.74 -23.6 11. 2 300 25 Valcartier 1,273 4.1 80.8 0.64 -425.4 17.4 300 a See Figure 2-3 for provenance locations. bMAT (°C) is the mean daily temperature; MAP (mm) is the mean annual precipitation; DD (days) is the mean degree days greater than 5°C; and DMAT (°C) is mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the provenance. To examine the sensitivity of the three meta-model parameters (θ1ij, θ2ij, and θ3ij, Eq. 2 to 2c) to climatic transfer distances, average height trajectories to plantation age 80 years were obtained for eight simulated provenances from a wide spatial range (i.e., from northern and southern 35 locations across the boreal and hemiboreal forests of Canada, Table 2-8) planted at the PNFI site. These were compared to the local provenance (i.e., Chalk River) with small climatic transfer distances. Two planting densities were also simulated (500 and 2,500 stems per ha). Since the motivation for developing this meta-model was to forecast yields for a broad range of provenances and planting sites over a large spatial extent, the selected Sw-Se model was then incorporated into an existing white spruce plantation growth and yield model by Prégent et al. (2010). The developed average height trajectory meta-model replaced the average height model as shown in the Figure 2-4, but no other model components were changed (implemented in R, version 3.0.1). Since this model is “height-driven”, this average height model replacement might be expected to provide accurate yield predictions for volume per ha and other variables as later discussed. However, the precision of these predictions was not explicitly tested since this information is not commonly published on provenance trials and was therefore not available in the meta-datasets compiled for this research. 36 Table 2-8. Geographic locations and climatic variables of provenances in the boreal and hemiboreal Forests of Canada used to illustrate height trajectories. Provenance (Prov.) aProv. Lat. Prov. Long. Prov. Elev. (m) Prov. MAP (mm) Prov. MAT (°C) Prov. DD. (days) DMAT (°C) DMAP (mm) DDdif (days) Alberta North 60.0 -117.0 279 371 -2 42 7 477 56 Alberta South 55.0 -114.0 596 470 2 75 3 377 23 BC North 60.0 -124.0 244 456 -1 62 5 391 36 BC South 55.0 -121.0 1,100 520 3 66 2 327 32 Ontario North 52.0 -91.0 378 688 0 72 5 159 26 Ontario South 46.0 -82.0 219 933 4 78 0 -86 20 Quebec North 48.0 -75.0 477 998 3 56 1 -151 42 Yukon North 63.0 -138.0 723 331 -3 42 8 517 56 Chalk River 45.9 -77.4 160 846 5 86 0 1 12 a Lat. is the latitude; Long. is the longitude; Elev. (m) is the elevation; MAT (°C) is the mean daily temperature; MAP (mm) is the mean annual precipitation; DD (days) is the mean degree days greater than 5°C; and DMAT (°C) is mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the provenance. 37 Figure 2-4. Stand-level model by Prégent et al. (2010) modified for yields of different provenances and planting sites. Start t=𝑡𝑡 + 1 age𝑖𝑖𝑖𝑖𝑖𝑖+1 = age𝑖𝑖𝑖𝑖𝑖𝑖 + 1 year Outputs at t=1 to end of the forecast • Average height (m) • Basal area (m2/ha) • Quadratic mean DBH (cm) • Volume (m3/ha) • SPH (stems/ha) • Mean annual increment (MAI) and current annual increment (CAI) of Basal area, Quadratic mean DBH, and Volume Spacing, Sp𝑖𝑖 = 𝑓𝑓(density𝑖𝑖) Plantation yields of plantation age at time t Basal area, G�𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑓𝑓 (Sp𝑖𝑖, H�𝑖𝑖𝑖𝑖𝑖𝑖 , age𝑖𝑖𝑖𝑖𝑖𝑖 ) Volume, V�𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑓𝑓 (Sp𝑖𝑖, H�𝑖𝑖𝑖𝑖𝑖𝑖 , G�𝑖𝑖𝑖𝑖𝑖𝑖) Quadratic mean DBH, 𝑞𝑞DBH� 𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑓𝑓 (Sp𝑖𝑖, H�𝑖𝑖𝑖𝑖𝑖𝑖 , age𝑖𝑖𝑖𝑖𝑖𝑖 ) Input planting site and provenance characteristics • Planting density (stems/ha) • Planting site and provenance elevations (m) • Climate distances: DMAT(°C), DMAP (mm), DDdif (days) • Plantation age at time t 𝜃𝜃�3𝑖𝑖𝑖𝑖 = f�DMAP𝑖𝑖𝑖𝑖� Predict average height for planting site (i) and provenance (j) H�𝑖𝑖𝑖𝑖𝑖𝑖 = 𝜃𝜃�1𝑖𝑖𝑖𝑖(1 − 𝑒𝑒𝑒𝑒𝑝𝑝 (1 − 𝜃𝜃�2𝑖𝑖𝑖𝑖age𝑖𝑖𝑖𝑖𝑖𝑖))� 11−𝜃𝜃�3𝑖𝑖𝑖𝑖� 𝜃𝜃�1𝑖𝑖𝑖𝑖 = f�site elevation𝑖𝑖, DMAT𝑖𝑖𝑖𝑖, DMAT𝑖𝑖𝑖𝑖2,provenance elevation𝑖𝑖 � 𝜃𝜃�2𝑖𝑖𝑖𝑖 = f�density𝑖𝑖, DDdif𝑖𝑖𝑖𝑖� Initialize model 38 2.3 Results 2.3.1 Average Height Trajectory Models Using the Sw meta-dataset, several combinations of climate transfer variables (DMAT, DMAP, DDdif), density, site elevation, and provenance elevation were used to model the parameters of the Chapman-Richard’s model (Eq. 2.2 with 2.2a, 2.2b, and 2.2c). Among the many models fitted, a selection of best models is listed along with fit statistics in Table 2-9. Of these models, Model 6 had highest Pseudo R2 (0.8826), smallest RMSE (1.0372 m) and smallest AIC (4,558). Also, the predicted average height trajectories superimposed on the observed data showed no lack of fit. For this selected model, all parameters associated with predictor variables in submodels 2.2a to 2.2c were statistically significantly different from 0 (α = 0.05) (Table 2-10). The RMSPE was 1.1 m and the MPE was 0.15 m using the “leave-one-out” model validation procedure. Using the “0.632 bootstrap” model validation approach the RMSPE0.632 (i.e., weighted RMSPEs) was 1.3 m and Err0.632 (i.e., weighted variances) was 1.6 m2. 39 Table 2-9. List of fitted models and fit statistics using the Sw meta-dataset. Model No. Parameters Fit Statistics aAsymptote (θ�1𝑖𝑖𝑖𝑖) Shape 1 (θ�2𝑖𝑖𝑖𝑖) Shape 2 (θ�3𝑖𝑖𝑖𝑖) Pseudo R2 RMSE AIC 1 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 𝑏𝑏20 + 𝑏𝑏21Density +𝑏𝑏22DDdif 𝑏𝑏30 + 𝑏𝑏31DMAP 0.8805 1.0463 4,586 2 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2 𝑏𝑏20 + 𝑏𝑏21DMAT +𝑏𝑏22DDdif 𝑏𝑏30 + 𝑏𝑏31DMAP 0.8749 1.0706 4,658 3 𝑏𝑏10 + 𝑏𝑏11site elevation+ 𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2+ 𝑏𝑏14log (density) 𝑏𝑏20 + 𝑏𝑏21DMAP +𝑏𝑏22DDdif +𝑏𝑏23density 𝑏𝑏30 + 𝑏𝑏31DMAP+ 𝑏𝑏32density 0.8406 1.2086 5,040 4 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2 +𝑏𝑏14 log(Density) + 𝑏𝑏15DDdif 𝑏𝑏20 + 𝑏𝑏21DMAT +𝑏𝑏22density 𝑏𝑏30 + 𝑏𝑏31DMAP+ 𝑏𝑏32density 0.8399 1.2112 5,046 5 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2 +𝑏𝑏14log (density) 𝑏𝑏20 + 𝑏𝑏21DMAT +𝑏𝑏22DDdif +𝑏𝑏23density 𝑏𝑏30 + 𝑏𝑏31DMAP+ 𝑏𝑏32DDdif+ 𝑏𝑏33density 0.8405 1.2088 5,042 6 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAT 𝑏𝑏20 + 𝑏𝑏21density +𝑏𝑏22DDdif 𝑏𝑏30 + 𝑏𝑏31DMAP 0.8826 1.0372 4,558 aDMAT (°C) is mean daily temperature distance, DDdif (days) is degree days distance, and DMAP (mm) is mean annual precipitation distance between the site and the provenance; density (stems ha-1) is the planting density; and site elevation is in metres. 40 Table 2-10. Parameter estimates (standard errors of estimates) and 95% confidence intervals for the selected model using Sw meta-dataset. Parameter aVariable Parameter Estimate (Standard Error) 95% Confidence Intervals Asymptote (θ1𝑖𝑖𝑖𝑖) Intercept 30 (2.7245) 24.6601 30.0000 Site elevation (m) -0.01825 (0.0018) -0.0218 -0.0147 DMAT2 (°C) -0.07599 (0.0276) -0.0760 -0.1301 DMAT (°C) -0.5719 (0.0966) -0.7612 -0.3825 Shape 1 (θ2𝑖𝑖𝑖𝑖) Intercept 0.029583 (0.00344) 0.0228 0.0363 Density (stems ha-1) 4.018E-7 (4.711E-8) 3.095E-7 4.941E-7 DDdif (days) 0.000024 (4.012E-6) 0.000016 0.000032 Shape 2 (θ3𝑖𝑖𝑖𝑖) Intercept 0.574084 (0.0256) 0.5238 0.6243 DMAP (mm) 0.00004 (9.874E-6) 0.000021 0.000059 aDMAT (°C) is mean daily temperature distance, DDdif (days) is degree days distance, and DMAP (mm) is mean annual precipitation distance between the site and the provenance; density (stems ha-1) is the planting density; and site provenance elevations are in metres. Using the Sw-Se meta-dataset, again many models were fitted; a selection is listed along with fit statistics in Table 2-11. Model 6 had highest Pseudo R2 (0.9234), smallest RMSE (0.9866 m) and smallest AIC (5,894), and no lack-of-fit was noted on the graph of predicted average height trajectories superimposed on the observed data. For this selected meta-model, all parameters associated with predictor variables in sub-models 2a to 2c were statistically significantly different from 0 (α = 0.05) (Table 2-12). 41 Table 2-11. List of fitted models and fit statistics using the Sw-Se meta-dataset. Model No. Parameters Fit Statistics aAsymptote (θ�1𝑖𝑖𝑖𝑖) Shape 1 (θ�2𝑖𝑖𝑖𝑖) Shape 2 (θ�3𝑖𝑖𝑖𝑖) Pseudo R2 RMSE AIC 1 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 𝑏𝑏20 + 𝑏𝑏21density +𝑏𝑏22DDdif + 𝑏𝑏23 provenance elevation 𝑏𝑏30 +𝑏𝑏31DMAP 0.9224 0.9933 5,922 2 𝑏𝑏10 + 𝑏𝑏11site elevation+ 𝑏𝑏12DMAT2+ 𝑏𝑏13 log(density)+ 𝑏𝑏14provenance elevation 𝑏𝑏20 + 𝑏𝑏21DDdif 𝑏𝑏30 +𝑏𝑏31DMAP 0.9211 1.0017 5,956 3 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2 +𝑏𝑏14 DDdif 𝑏𝑏20 + 𝑏𝑏21DMAT +𝑏𝑏22 density 𝑏𝑏30+ 𝑏𝑏31DMAP+ 𝑏𝑏32density 0.7873 1.6442 8,032 4 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 𝑏𝑏20 + 𝑏𝑏21 DDdif+ 𝑏𝑏22DMAP+ 𝑏𝑏23 provenance elevation 𝑏𝑏30+ 𝑏𝑏31DMAP+ 𝑏𝑏32density 0.6280 2.1744 9,202 5 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12DMAT2 + 𝑏𝑏13DMAP2 +𝑏𝑏14 log(density) +𝑏𝑏15provenance elevation 𝑏𝑏20 + 𝑏𝑏21DMAT +𝑏𝑏22DDdif + 𝑏𝑏23density 𝑏𝑏30+ 𝑏𝑏31DMAP+ 𝑏𝑏32DDdif+ 𝑏𝑏33density 0.8972 1.1432 6,488 6 𝑏𝑏10 + 𝑏𝑏11site elevation +𝑏𝑏12 DMAT2 + 𝑏𝑏13DMAT +𝑏𝑏14 provenance elevation 𝑏𝑏20 + 𝑏𝑏21density +𝑏𝑏22 DDdif 𝑏𝑏30+ 𝑏𝑏31DMAP 0.9234 0.9866 5,894 aDMAT (°C) is mean daily temperature distance, DDdif (days) is degree days distance, and DMAP (mm) is mean annual precipitation distance between the site and the provenance; density (stems ha-1) is the planting density; and site provenance elevations are in metres. 42 Table 2-12. Parameter estimates (standard errors of estimates) and 95% confidence intervals for the selected model using Sw-Se meta-dataset. Parameter aVariable Parameter Estimate (Standard Error) 95% Confidence Intervals Asymptote (θ1𝑖𝑖𝑖𝑖) Intercept 30 (2.6612) 24.7841 30.0000 Site elevation (m) -0.00513 (0.000819) -0.00674 -0.00353 DMAT (°C) -0.8768 (0.1163) -1.1048 -0.6488 DMAT2 (°C) -0.1230 (0.0297) -0.1813 -0.0647 Provenance elevation (m) 0.00796 (0.000908) 0.00618 0.00974 Shape 1 (θ2𝑖𝑖𝑖𝑖) Intercept 0.0260 (0.00268) 0.0207 0.0312 Density (stems ha-1) 3.601E-7 (3.835E-8) 2.849E-7 4.352E-7 DDdif (days) 0.000020 (3.066E-6) 0.000014 0.000026 Shape 2 (θ3𝑖𝑖𝑖𝑖) Intercept 0.5674 (0.0204) 0.5274 0.6074 DMAP (mm) 0.000039 (7.475E-6) 0.000025 0.000054 aDMAT (°C) is mean daily temperature distance, DDdif (days) is degree days distance, and DMAP (mm) is mean annual precipitation distance between the site and the provenance; and density (stems ha-1) is the planting density. Likely because of the wider range elevations in the Sw-Se meta-dataset relative to the Sw meta-dataset, both site and provenance elevations were included in the function for the asymptote (Eq. 2.2a). The RMSPE was 1.1 m and the MPE was 0.10 m using the “leave-one-out” model validation procedure. Using the “0.632 bootstrap” model validation approach, RMSPE0.632 was 1.2 m and Err0.632 was 1.3 m2. 43 For both of the selected Sw and Sw-Se models, the asymptote changed in a nonlinear trend with DMAT; as a result, both DMAT and DMAT2 were included. To illustrate the impacts of DMAT, using the Sw-Se model and provenance and site elevations of 500m, the asymptote peaks at a DMAT of around -5° C. Since the range of DMATs within the Sw-Se meta-dataset was from -6.83 to 8.66 ° C (Table 2-6), the asymptote mostly declines with an increase in DMAT meaning that the asymptote is lower when a provenance from a cold location is planted in a warmer location. In terms of the two shape parameters, it is interesting that planting density was included in the parameter prediction model for 𝜃𝜃2𝑖𝑖𝑖𝑖 indicating that average height is affected by planting density. Although dominant height is commonly not affected across a broad range by planting density, average height may be affected as later discussed. Interpreting the two shape parameters is more difficult since they work in concert. As a result, model illustrations were used to examine other impacts on the average height trajectory meta-model as shown in the next section. 2.3.2 Using the Average Height Trajectory Model to Forecast Yields The selected model for the Sw-Se meta-dataset was used to obtain predicted height to plantation age 44 for each of the 25 provenances planted at the PNFI planting site (Figure 2-5). Most of the predicted average height trajectories were very close to the measured average heights except for five provenances (i.e., 1, 3, 4, 9 and 24) where average heights were underestimated. At age 15, all predicted averages were close to the measured average heights. 44 Figure 2-5. Predicted average height (dashed line) and measured average height (solid line and box shape to show actual measurement time) for the 25 provenances (provenance IDs given in Table 2.7) planted at the Petawawa National Forestry Institute planting site. 45 Morgenstern et al. (2006) mentioned that in the PNFI planting site at age 15, eight provenances were top performers in height (i.e., 2, 3, 5, 10, 13, 17, 20 and 24); however, this list of top performing provenances changed somewhat at age 44 (i.e., 1, 3, 4, 7, 10, 14, 20 and 24). In their paper, Leites et al. (2012) found that the Pearson correlation coefficient between population-averaged predictions and measured heights of Douglas-fir provenances was 0.59. However, this correlation improved when subject-specific predictions were used. The predicted height trajectories for the eight simulated provenances over a wide spatial range (Table 2-8) showed a large variation with a range of about 12 to 27 m at age 80 for both densities (Figure 2-6). This was largely due to mean annual temperature climatic transfer distances corresponding with latitude. Provenances from northern areas of Alberta, BC, Ontario, and Yukon had lower predicted height trajectories (shown in grey on Figure 2-6) since DMAT was large for these simulated provenances (5 to 8 °C, Table 2-8). Southern provenances (shown in black on Figure 2-6) had the same or higher predicted height trajectories relative to the Chalk River provenance. Similar results were obtained for both densities indicating that density has only a small effect on θ2ij. 46 Figure 2-6. Predicted average height trajectories for simulated white spruce provenances using the Petawawa National Forestry Institute planting site. 47 The stand-level growth and yield model developed by Prégent et al. (2010) based on over 40 years of data for white spruce plantations in Québec was modified to predict yields of different provenances and planting sites (Figure 2-4). Specifically, the average height trajectory model was replaced by the selected model using the Sw-Se meta-dataset (Table 2-12). To run the model, planting density, site and provenance elevations, DMAT, DMAP, and DDdif are input into the model and used to obtain the predicted average height for the plantation age at a particular time, t. The predicted average height is then used as an input to models for basal area per ha, volume per ha, and quadratic mean diameter at breast height (1.3 m above ground; quadratic mean DBH). This is then repeated via incrementing the plantation age by one year to obtain the yield trajectories. Once the yield trajectories are calculated, mean annual and current annual increments (MAI and CAI) can be calculated. Using this modified model and the same set of planting densities, provenances, and planting site used for the average height trajectories shown in Figure 2-6, the quadratic mean DBH and volume per ha trajectories were predicted. Although the planting density had little impact on the average height trajectories (Figure 2-6), these other yield variables are responsive to planting density as would be expected (Figure 2-7). Most important, the predicted volume yields for the different provenances ranged from around 200 to 600 m3/ha at 80 years for the 2,500 stems/ha planting density compared to the Chalk River local provenance with about 425 m3/ha, indicating the wide range of possible productivity. 48 Figure 2-7. Predicted quadratic mean DBH (cm) and volume (𝑚𝑚3/ha) over plantation age (years) for simulated white spruce/hybrid spruce provenances using the Petawawa National Forestry Institute planting site. 49 2.4 Discussion Tree improvement programs can provide options for ameliorating fiber shortages; however, forecasting possible changes in yields is very challenging. In this study, a meta-modelling approach was used as a mechanism for providing yield forecasts by combining meta-data from white and hybrid spruce provenance trials across the wide expanse of the Canadian boreal forest. Height trajectories were modelled using the Chapman-Richards model. Although other models have been used to forecast height, the Chapman-Richards model has been widely used (e.g., Huang et al. 1992, Temesgen et al. 2007, Weiskittel et al. 2011). Also, the derivation of the model was based on growth of organisms (Von Bertalanffy 1957; Rathbun et al. 2011). Further, the complex biological yield of organisms has been translated into a simplified mathematical form with only three parameters (Pienaar and Turnbull 1973) representing a wide range of curve shapes. As a result, the three parameters of the Chapman-Richards model can be replaced via random coefficients (aka, parameter prediction) modelling to model the changes in the asymptote and shape parameters with relative ease. Since climatic transfer distances between the provenance and planting site locations can represent growth potential (Matyas 1994, Rehfeldt et al. 1999), the base average height trajectory using the Chapman-Richards model was modified using random coefficients models based on climate variables for the provenance and site locations. Rehfeldt et al. (1999) predicted mean height growth of different provenances and planting sites using a simple quadratic surface and a single climate transfer variable. They tested a large number of climate transfer variables, namely: mean annual temperature, mean temperatures in the warmest and coldest months, difference between warmest and coldest months mean temperatures, the mean annual precipitation, summer 50 mean precipitation, average duration of the frost-free days, degree days greater than 5°C, degree days less than 0°C, summer moisture index, and annual moisture index. Of these variables, mean annual temperature was the best variable for predicting average height growth of Larix sukaczewii Dylis, Larix sibirica Ledeb., and Larix gmelinii (Rupr.) Rupr. provenances ; other effective variables were mean temperature in the coldest month, degree days less than 0°C, annual moisture index, and difference between warmest and coldest months mean temperatures. They found that best height growth were obtained when climate transfer variables were zero (planting site and provenances had similar climates) which was the overall findings of their study. Rehfeldt et al. (1999) also fitted a model to predict height by excluding one planting site (Brooks) to avoid confounding due to irrigation performed only in one site. The result from their fitted model showed that the optimal height growth was obtained when provenances moved from warmer to colder temperatures when planting sites were similar in terms of silvicultural treatments (Rehfeldt et al. 1999, p. 1665). For lodgepole pine, McLane et al. (2011) used summer and winter temperatures and precipitation in a random coefficients model for basal area growth. Basal area growth was larger when provenances came from a warm temperature and planted at a warmer and wetter site. Leites et al. (2012) selected the mean temperature of the coldest month transfer distance for modelling three-year height of Douglas-fir provenances in USA. Their final model indicated that provenances coming from colder climates had improved growth if planted at locations with warmer winter temperatures. These support my results in that I found that climate transfer distances impacted the three parameters of the average height trajectory model, in particular, the asymptote. 51 For the asymptote, a nonlinear trend with DMAT was found. Planting density and elevations of the planting site and provenance were also included, but these variables had less impact on the asymptote than DMAT. Similar nonlinear trends with climate transfer distances were found by Rehfeldt et al. (1999) and McLean et al. (2011). Unlike Leites et al. (2012), but consistent with Rehfeldt et al. (1999), the average height asymptote for white spruce/spruce hybrids was generally higher for provenances that were planted in a colder location (i.e., a negative DMAT), although this asymptote declined for very large negative DMATs. Other studies showed inconsistent results with climatic transfer distances. For example, Carter (1996) found that the height growth of several species common in eastern North America decreased when provenances from colder areas were transferred to warmer sites for some species. However, the reverse of this trend was reported in other studies. For example, Andalo et al. (2005) found that height growth decreased substantially when a provenance from a warmer temperature (40C more) was planted at a colder site. Even though planting density was also included in one of the two shape parameters submodels, it had only a minimal effect on predicted average heights. These results are not unexpected since many authors have found that planting density has little effect on dominant height (Clutter et al. 1983, Pienaar and Shiver 1984, Smith et al. 1997), and average and dominant heights are quite similar in plantations with similar genetic stocks. As further support, Krieger (1998) found no statistically significant effects of planting density on white spruce average height using Nelder plots. However, even though planting density did not have a large impact on average height trajectories, there would be impacts on volume per ha and on quadratic mean DBH as reflected in the other sub-models of a growth and yield model. 52 Commonly, yields of selected provenances are compared with a baseline representing “regular” plantation stocks to assess impacts of tree improvements (e.g., McInnis and Tosh 2004, Petrinovic et al. 2009). In this study, I applied the average height trajectory model for the PNFI planting site and included a wider range of provenances than those actually planted at that site. I found a maximum increase of 5 m at 80 years relative to average heights for the Chalk River (i.e., local) provenance. However, there are two important cautions. First, this forecast was almost 40 years beyond the range of the meta-datasets. Also, the model does not reflect any improvements that have or could be made through successive generations of breeding (White et al. 2007). Finally, to provide the desired yield estimates for differing provenances, the height model in the Prégent et al. (2010) simulator was modified using the average height trajectory meta-model. Since models for basal area per ha, volume per ha, and stems per ha were linked with the height model, the trajectories of these other variables were altered in consequence. An alternative approach often used in other studies is to use a single percentage (or multiplier) at all ages to modify the growth and yield of regular stocks for changes in genetic stock. For example, Petrinovic et al. (2009) modified the Bolghari and Bertrand (1984) yield tables by increasing the dominant average height trajectory by 20%. McInnis and Tosh (2004) projected height, volume and diameter using the STAMAN stand growth model (Roussell et al.1993) and obtained the percent gain at age 40 by comparing improved and unimproved stocks. However, using a simple multiplier assumes that any changes in yield are anamorphic, whereas I allowed the shape parameters to also change allowing for polymorphic changes over time. I also modified only the 53 average height trajectory model, which affected other yields in turn. Some authors modified several simulator component models. For example, Carson et al. (1999) altered mean top height and basal area in order to predict growth and yield of improved radiata pine (Pinus radiata D. Don.). Since I used a meta-modelling approach to increase the temporal and spatial extents of the data used in modelling, information on variables other than average height was not commonly available. Further, since models of stand-level variables are linked in growth and yield simulators to reflect biological reality (Weiskittel et al. 2011), my change in the average height trajectory might be expected to produce realistic forecasts of the other yield variables. The main contribution in this study is the use of a meta-modelling approach to expand the spatial and temporal extents of the meta-data resulting in a larger range of climate transfer distances and plantation ages. However, the meta-data still only extended to 44 years of age, whereas the harvest ages of white spruce plantations are more commonly about 80 to 100 years (Delong 1997, Xie and Yanchuk 2003, Alden 2006). Since the developed average height trajectory meta-model used the Chapman Richards model as a base, logical forecasts beyond the age range of the data would be expected given the biological basis for this model and the asymptotic nature which should reflect the trend in average height as noted earlier. Further, the average height trajectory meta-model would be expected to underestimate successive generation yields (White et al. 2007). A final limitation in this study was that the climate transfer distance variables used for predicting height were not measured at that exact planting location, but rather were spatially interpolated using the closest weather stations. 54 2.5 Conclusions Using meta-modelling, average height trajectory models for white and hybrid spruces were developed that can be used over a broad spatial and temporal range of planting sites and provenances of the boreal and hemiboreal forests of Canada. The fitted models closely followed the observed patterns of average height over time. Further, the trajectories are polymorphic, allowing for the possibility that a provenance that performs well at an early stage may not, ultimately, be the best provenance at harvest. The average height yield trajectory model was incorporated in a growth and yield simulator to obtain the yield trajectories of other variables, notably volume per ha, as a means of estimating the potential of tree improvement programs for mitigating fiber shortages. However, caution should be used since harvests commonly used for white and hybrid spruces are beyond the 44 years represented in the meta-data. 55 3. Gain of White Spruce and Hybrid Spruce Provenances in the Canadian Boreal Forest 3.1 Introduction Tree improvement programs in the boreal and hemiboreal forests of Canada provide improved stock to promote plantation success. The first step in these programs is to select trees from particular locations (i.e., provenances) and perform provenance trials to determine the ‘gain’ (also referred to as selection differential or genetic gain) based on the best performing provenances relative to a baseline (Fins et al. 1992). From these provenance or family trials, further gains can be realized through successive breeding of best performers (Zobel and Talbert 1984; White et al. 2007). Using this process, substantial improvements in growth and/or yield have been noted for boreal forest and other temperate forest species (Newton 2003; McInnis and Tosh 2004; Petrinovic et al. 2009). However, the specific definition of gain varies among studies. A specific seedlot may be included at all test sites as a common reference or, alternatively, the average of all provenances (or some percentile) may be used as the baseline. The specific criteria for selecting top-performing provenances can vary and the age at which this evaluation takes place is also not consistent among studies. For example, Nance and Wells (1981) used heights of local seed sources as a baseline and compared these to heights of non-local seed sources for loblolly pine (Pinus taeda L.) in USA. They calculated the time needed for non-local seed sources to achieve the same heights as local seed sources and then translated the time gain (or loss) into a gain (or loss) in site index. Hamilton and Rehfeldt (1994) used heights of ponderosa pine (Pinus 56 ponderosa Dougl.) stock from natural stands as a baseline and compared these to heights of improved stock using: [3.1] 𝐺𝐺𝐺𝐺𝑖𝑖𝑛𝑛 = 𝑀𝑀∗𝑀𝑀 where 𝑀𝑀∗ is the multiplier for improved stock and 𝑀𝑀 is the multiplier for the baseline. 𝑀𝑀∗ was determined by modifying the Stand Prognosis Model (Stage 1973; Wykoff et al.1982) until the simulated height matched with the average height of improved stock at ages 8, 14 and 19 years. A similar approach was followed to obtain M using stock from natural stands. They found that gain ranged from 1.07 (7%) to 1.21 (21%). Magnussen and Yanchuk (1994) showed a 2 to 5% gain for coastal Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco var. menziesii) aged 4 to 23 in British Columbia, Canada using the average height of best performing provenances, based on ranking breeding values, relative to the average height of all provenances. Newton (2003) conducted a meta-analysis using published literature for four conifers (black spruce (Picea mariana (Mill.) B.S.P.), white spruce (Picea glauca (Moench) Voss), jack pine (Pinus banksiana Lamb.) and red pine (Pinus resinosa Ait.)) tested at sites in central and eastern Canada. He estimated the relative height gain at the last measurement of each provenance trial using: [3.2] 𝐺𝐺𝐺𝐺𝑖𝑖𝑛𝑛 % = 𝐻𝐻𝐷𝐷𝐷𝐷−𝐻𝐻𝐷𝐷𝑛𝑛𝐻𝐻𝐷𝐷𝑛𝑛× 100 where 𝐻𝐻𝐷𝐷𝐷𝐷 is the average height of improved stock for the tallest quartile and 𝐻𝐻𝐷𝐷𝑛𝑛 is the average height of all provenances. McInnis and Tosh (2004) showed height gains of 7 to 12% and volume gains of 18 to 30% at age 20 using the average height of the top 25% of all families versus the average height of unimproved stock for black spruce and jack pine plantations in New Brunswick, Canada. Petrinovic et al. (2009) estimated an average 10% height gain based on seven white spruce progeny trials in Quebec, Canada. They used the top height, defined as the mean height of the 200 tallest stems per ha (𝐻𝐻�𝑆𝑆) from selected families relative to the top height 57 (𝐻𝐻�𝑈𝑈𝑆𝑆) of all other families. The reference ages were 14, 13 and 22 years depending on the progeny trial and height gains ranged from 7.06% to 12.40%. Ahtikoski et al. (2012) found a 3 to 15% height gain for improved versus unimproved stocks for Scots pine (Pinus sylvestris L.) in Finland, based on an assessment at harvest age. As well as differences in the calculation of gain, the approaches used to alter existing growth and yield models to forecast possible gains in plantation yields at harvest varies. An approach that has been widely used is to assume that a single gain metric can be applied to yields (or growth) at all ages (i.e., gain is invariant with age). A number of authors increased the dominant height trajectory using a single gain multiplier for all ages (e.g., Buford and Burkhart 1987; Carson et al. 1999; Adams et al. 2006; Gould et al. 2008; Lu and Charrette 2008; Gould and Marshall 2010; Ahtikoski et al. 2012; Ahtikoski et al. 2013). Alternatively, other authors altered the site index (e.g., Nance and Wells 1981; Newton 2003; McInnis and Tosh 2004; Petrinovic et al. 2009). For example, Petrinovic et al. (2009) increased site indices by 10% to evaluate the economic benefits of using improved white spruce trees in Quebec, Canada. A further variation is to alter a number of yield variables, but still assume that gain is invariant over time. For example, Hamilton and Rehfeldt (1994) estimated the gain for a number of variables for ponderosa pine (Pinus ponderosa Laws.) and modified the Stand Prognosis Model (Stage 1973; Wykoff et al.1982). Similarly, Carson et al. (1999) modified several growth variables for radiata pine (Pinus radiata D.Don). Other approaches have allowed for gain to change in time. For example, Lambeth (1980) developed a correlated-gain approach where the change in phenotypic correlation between two 58 ages for the trait of interest changed depending on the ratio of the two ages. For this purpose, he used a linear model: [3.3] 𝑟𝑟𝑃𝑃𝐽𝐽,𝑀𝑀 = 𝛽𝛽0 + 𝛽𝛽1 𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒 �𝑦𝑦𝑦𝑦𝑦𝑦𝑛𝑛𝐷𝐷𝑒𝑒𝑠𝑠𝑖𝑖 𝑎𝑎𝐷𝐷𝑒𝑒𝑦𝑦𝑙𝑙𝑜𝑜𝑒𝑒𝑠𝑠𝑖𝑖 𝑎𝑎𝐷𝐷𝑒𝑒 � + 𝜖𝜖 where 𝑟𝑟𝑃𝑃𝐽𝐽,𝑀𝑀 is the phenotypic correlation between age 𝐽𝐽 (juvenile or youngest age) and 𝑀𝑀 (oldest, mature or harvest age). He fitted this model using a meta-regression approach by using a database of results from previous studies. Lambeth argued his fitted model could be applied to any species at any location. However, Lambeth and Dill (2001) found that the parameters of Eq. [3.3] varied with site. Di Lucca (1999) and Xie and Yanchuk (2003) adapted Lambeth’s method and estimated genetic gain for use in the TIPSY (Table Interpolation Program for Stand Yields) model developed for British Columbia, Canada. Rather than using Lambeth’s approach (or some variation of this), McInnis and Tosh (2004) fitted a nonlinear power function to predict percent height gain in black spruce using age as a predictor variable; they used the resulting model to project gains to age 40. In a more restricted approach, Ahtikoski et al. (2012) allowed for a different gain metric at each of two stand development stages. For stands at the sapling stage (dominant height ≤ 7m), gain was calculated as a single value and used to increase the average height and average diameter. For older stands, the asymptotes of the average height and diameter versus average age trajectories were increased using an estimated gain. The goal in this study was to estimate the gain for improved stock of white spruce and hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) in the boreal and hemiboreal forests of Canada. A high level of genetic diversity exists in white spruce (Petrinovic et al. 2009) as it covers a very wide geographical range across the breadth of Northern America. At the western boundary, white spruce hybridizes with other spruce species, 59 particularly Engelmann spruce at mid-elevations. Because of this broad spatial extent and also because forest management is a provincial mandate in Canada, there are no common benchmarks for assessing and quantifying gains across white and hybrid spruce tree improvement programs. To achieve this goal, meta-data from provenance trials4 across the Canadian boreal and hemiboreal forests were compiled. These meta-data were used to address the specific research objectives, namely: (i) to assess the effects of the evaluation age (i.e., the age when top performers are selected) in concert with the definitions of the top performers and the baseline on the estimated gain; (ii) to assess whether gain changes over time and/or with planting site using a modelling approach and based on a selected definition of the evaluation age, the top performers, and the baseline; and (iii) to provide a final model that estimates the changes in gain for white spruce over a wide spatial and temporal range. 3.2 Methods 3.2.1 Meta-Data and Height Trajectories The meta-data for this study was previously used in Chapter 2 of this dissertation. Briefly, the meta-data for 18 studies representing 38 planting sites in Canada (Figure 3-1) with provenances from Canada and the United States were collected from published articles, internal reports, government reports, and other sources principally using the Web of Science® (Science Citation 4 Data for white spruce from ATISC (2011) were from progeny rather than provenance trials. Data for hybrid spruce were also from progeny rather than provenance trials. 60 Index expanded) search engine, supplemented by additional data provided by authors in some cases. Figure 3-1. Geographic locations of planting sites in the meta-data set. Following the approach by Magnussen and Yanchuk (1994), average heights were predicted from 1 to 45 years for each provenance at each planting site in the meta-data using the average height model developed in Chapter 2: [3.4] H�𝑖𝑖𝑖𝑖𝑖𝑖 = 𝜃𝜃�1𝑖𝑖𝑖𝑖(1 − 𝑒𝑒𝑒𝑒𝑝𝑝 (1 − 𝜃𝜃�2𝑖𝑖𝑖𝑖age𝑖𝑖𝑖𝑖𝑖𝑖))( 11−𝜃𝜃�3𝑖𝑖𝑖𝑖) [3.4a] 𝜃𝜃�1𝑖𝑖𝑖𝑖 = 30 − 0.00513 site elevation𝑖𝑖 − 0.1230 × DMAT𝑖𝑖𝑖𝑖2 −0.8768 × DMAT𝑖𝑖𝑖𝑖 + 0.00796 × provenance elevation𝑖𝑖 [3.4b] 𝜃𝜃�2𝑖𝑖𝑖𝑖 = 0.0260 + 3.601E − 7 × density𝑖𝑖 + 0.000020 × DDdif𝑖𝑖𝑖𝑖 [3.4c] 𝜃𝜃�3𝑖𝑖𝑖𝑖 = 0.5674 + 0.000039 × DMAP𝑖𝑖𝑖𝑖 61 where H�𝑖𝑖𝑖𝑖𝑖𝑖 is the predicted average height (m) of planting site i and provenance j measured at time t; age (years) is the plantation age; 𝜃𝜃�1𝑖𝑖𝑖𝑖 is the estimated asymptote; 𝜃𝜃�2𝑖𝑖𝑖𝑖 and 𝜃𝜃�3𝑖𝑖𝑖𝑖 are shape parameters; site and provenance elevations are in metres; Density is the planting density (stems per ha; DMAT, DDdif, and DMAP are the distances for mean annual daily temperature (MAT, °C), degree days > 5°C (DD, days), and mean annual precipitation (MAP, mm) between the planting site and provenance, respectively. These climatic distances were calculated for each provenance at each planting site as described in Chapter 2 (Tables 2-5 and 2-6), using climate normal for 1981-2010 from the Environment Canada database (www.climate.weatheroffice.ec.gc.ca, accessed 10th September, 2013) and from the National Oceanic and Atmospheric Administration’s (NOAA) National Climatic Data Center (NCDC) (ftp://ftp.ncdc.noaa.gov/pub/data/normals/1981-2010, accessed 11th September, 2013). A positive DMAT indicates the planting site is warmer, a positive DMAP indicates that the planting site has higher precipitation, and a positive DDdif indicates that the planting site has more days with a mean temperature > 5°C than the provenance location. The forecast was limited to age 45, since the oldest planting age of all study sites was 44. Using this approach, average height trajectories were obtained for the 38 planting sites representing 337 provenances in the meta-data; some are illustrated in Figure 3-2. 62 Figure 3-2. Predicted average height trajectory of some white spruce provenances (Prov.) and planting sites (Site) of the meta-data. 3.2.2 Effects of Age and Definition of Top Performers Using the height trajectories from age 1 to 45 years, the relative height gain (RH) was calculated by planting site and age as: [3.5] Gain, RH = average height of top x%−average height of y%average height of y% where top x% is percentile used to define top performers selected at a particular evaluation age, 63 and y% is the percentile used to define the baseline population. To illustrate this definition, Figure 3-3 shows 15% as the percentile for top performers based on an evaluation of 15 years relative to the baseline using all provenances (100%). Figure 3-3. Hypothetical predicted average height trajectory of white spruce populations showing top performing provenances (top 15%) versus all provenances (baseline 100%) at an evaluation age of 15 years. Since there is no common definition of evaluation age, top performers, or baseline, a sensitivity analysis was conducted on the impacts of different definitions on gain. Changes in the evaluation 64 age affect gain because the top performing provenances selected at a particular juvenile age may not be same provenances selected as top performers at other evaluation ages. The effects of differing evaluation ages on gain will depend on the percentage of provenances that switch in or out of the top performing group at later ages. As a result, an evaluating age closer to the harvest age is preferred. However, early evaluation of stocks is desirable, and few provenance trials are maintained until harvest age, particularly for boreal and hemiboreal species with harvest ages of 80 years or more. Changing the percentile used to define the top performers will also affect gain. If the percentile is too low (e.g., top 5% or less), then the estimated gain would be unrealistically high and the number of provenances switching in or out with different evaluation ages may be high. Conversely, if the percentile for top performers is too high, the estimated gain would be unrealistically low, but the evaluation age would likely have little impact with few or no provenances switching in or out of the top performing group. Finally, the definition of the baseline also affects the gain. Two options have been commonly used: using 100% of all provenances in the trial, or using only the provenances excluded from the top performing group. Four evaluation ages (5, 15, 25 and 45 years) were included for the sensitivity analysis and the top performers definition was altered to reflect three different percentiles (5%, 15% and 25%). The baseline was 100% for all of these combinations of evaluation ages and top performers definitions, except that the baseline was also altered to be the bottom 75% percentile when coupled with the 25% top performers. These definitions generally cover the ranges found in the literature, except that the evaluation age was extended down to 5 years and up to 45 years. For 65 each combination in this sensitivity analysis, the trend in gain over time was examined along with how these trends change with the gain definition. Based on this sensitivity analysis, as well as comparison with other studies and experts’ opinions, one gain definition for the gain trajectory model was chosen. 3.2.3 Gain Trajectory Model Using the chosen definition of gain, the gain was calculated for ages 1 to 45 for each of the 38 sites. Then, using an approach similar to Petrinovic et al. (2009), a flexible nonlinear gain model (i.e., base model) was fitted: [3.6] RH𝑖𝑖𝑖𝑖 = 𝛽𝛽0age𝑖𝑖𝑖𝑖𝛽𝛽1𝛽𝛽2 age𝑖𝑖𝑡𝑡 + 𝜖𝜖𝑖𝑖𝑖𝑖 where RH𝑖𝑖𝑖𝑖 is the gain at planting site 𝑖𝑖 at measurement time 𝑡𝑡; age𝑖𝑖𝑖𝑖 is the plantation age (years); 𝛽𝛽0 is a scale parameter; 𝛽𝛽1 and 𝛽𝛽2 are shape parameters; and 𝜖𝜖𝑖𝑖𝑖𝑖 is the error term. This base model represents the average gain trajectory over all planting sites. Other models were considered but were not selected since the functional form of this model better represents the relatively higher gains for very young plantations and the lower and nearly constant gain for later ages. Also, since this base model inherently includes a zero-intercept, the gain is restricted to be equal to 0 for a plantation age of 0. An intercept only model (i.e., null model) was also fitted. Because the meta-data for most planting sites included repeated measures with irregular intervals in plantation ages, first and second order continuous autoregressive parameters (CAR(x)) were added to the models. The models were then fitted using PROC MODEL of SAS software version 9.3 (SAS Institute Inc. 2014), which allows for autoregressive parameters. Several sets of 66 starting parameters were used to ensure a global optimum solution for each model. Akaike’s information criterion (AIC) was calculated for the base and null models as follows: [3.7] AIC = −2𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒𝐿𝐿 + 2𝑝𝑝 where 𝐿𝐿 is the maximum likelihood of the equation; and p is the number of parameters in the model. A large change in AIC relative to the null model was used as evidence that the gain changed with plantation age. The base model was then altered using a random coefficients (aka, parameter prediction) modelling approach to examine whether gains also differed with planting site characteristics. Specifically, the parameters of Eq. [3.6] were allowed to vary by replacing them with functions of planting site climatic variables and other site characteristics: [3.8] RH𝑖𝑖𝑖𝑖 = 𝛽𝛽0𝑖𝑖age𝑖𝑖𝑖𝑖𝛽𝛽1𝑖𝑖𝛽𝛽2𝑖𝑖 age𝑖𝑖𝑡𝑡 + 𝜖𝜖𝑖𝑖𝑖𝑖 [3.8a] 𝛽𝛽0𝑖𝑖 = f(site climatic𝑖𝑖, other site characteristics𝑖𝑖) + 𝛿𝛿0𝑖𝑖 [3.8b] 𝛽𝛽1𝑖𝑖 = f(𝑠𝑠𝑖𝑖𝑡𝑡𝑒𝑒 climatic𝑖𝑖, other characteristics𝑖𝑖) + 𝛿𝛿1𝑖𝑖 [3.8c] 𝛽𝛽2𝑖𝑖 = f(𝑠𝑠𝑖𝑖𝑡𝑡𝑒𝑒 climatic𝑖𝑖 , other characteristics𝑖𝑖) + 𝛿𝛿2𝑖𝑖 where RH𝑖𝑖𝑖𝑖 , age𝑖𝑖𝑖𝑖, and 𝜖𝜖𝑖𝑖𝑖𝑖 are as previously defined; and 𝛿𝛿0𝑖𝑖, 𝛿𝛿1𝑖𝑖 and 𝛿𝛿2𝑖𝑖 are random effects at the site level. A number of submodels (Eq. 3.8a to 3.8c) were fitted using combinations of predictor variables, specifically: planting site climatic variables (MAT, MAP, DD), planting density, and planting site elevation. Again, all of these models were fitted using PROC MODEL of SAS software version 9.3 (SAS Institute Inc. 2014) allowing for continuous autoregressive parameters and several sets of starting parameters were used to try to obtain a global optimum solution for each model fitted. Changes in AIC for these models relative to the base model were used to indicate whether site characteristics affected gain, after accounting for any changes due 67 to plantation age. Changes in AIC were also used to evaluate alternative variable subsets. As a further indicator of model fit, the predicted gain trajectory for each possible model was superimposed on the measured gains for each planting site. Finally, one model was selected using the AIC values and the observed versus estimated gain graphs. For the selected model, validation statistics were calculated using the “leave-one-out” approach. For this purpose, one planting site was excluded from model fitting and the model was applied to the excluded planting site. This was repeated for each planting site and results were summarized into the root mean square predicted error (RMSPE): RMSPE = � ∑ ∑ �RH𝑖𝑖𝑡𝑡−RH� 𝑖𝑖𝑡𝑡�2𝑟𝑟𝑡𝑡=1𝑛𝑛𝑖𝑖=1𝑇𝑇 where 𝑖𝑖 = 1, … ,𝑛𝑛 planting sites, 𝑡𝑡 = 1, … , 𝑙𝑙 measurement times within planting sites, RH� 𝑖𝑖𝑖𝑖 is the predicted gain, and 𝑇𝑇 is the total number of observations. 3.3 Results 3.3.1 Gain Definition Sensitivity Analysis The general trend of gain over time was nonlinear regardless of evaluation age as illustrated for four planting sites using 15% for the top performers and 100% for the baseline (Figure 3-4). The gain trends were similar in shape among different planting sites, but the levels changed (e.g., Petawawa, Ontario vs. Calling Lake, Alberta). 68 Figure 3-4. Gain over time for four planting sites where the top performers (15%) were selected at various evaluation ages (baseline is 100% of provenances). Gain based on selection of top performers at each age (“changing performers”) is included for comparison. When the selection of top performers was allowed to change by selecting the top 15% at each age (“changing performers”), the gains were similar to those using a fixed evaluation age of 5 years up to plantation age 15, but then were more similar to evaluation ages 15 to 45 years. This 69 indicates that the selected top performers at a particular evaluation age were not top performers at other ages. Evaluation age 5 years was the most different from the “changing performers” trend as might be expected. Since the top performing provenances selected at evaluation age 15 years were quite similar to the “changing performers” and to evaluation ages 25 and 45 and, also, this evaluation age is easier to apply in practice, 15 years was selected as the evaluation age to be used in the gain calculations used for the gain trajectory model. Using a selected evaluation age of 15 years and the baseline as the average of all provenances (i.e., 100%), both the levels and the shapes of the gain trajectories changed with the definition of top performers as illustrated for four planting sites in Figure 3-5. The levels were similar using the top 15% and 25%, but quite different using top the 5%. As shown by these four planting sites (Figure 3-5), the gain trends vary by planting site. Calling Lake had lower gains using top the 15% and top 25% in particular. The shape generally became less curved when going from top the 25% to the top 5%, except for Aleza, BC where the shape became concave using the top 5%. Using the top 25% but varying the baseline from 100% to 75% (i.e., using only the excluded provenances) resulted in a perception of higher gains compared to using a baseline of 100% for that top percentage. 70 Figure 3-5. Gain over time at evaluation age 15 for different definitions of top performers (i.e., top 5%, 15% or 25%) and baseline (i.e., 75% or 100%). Overall, this sensitivity analysis indicates the importance of the definition used in calculating the gain. Using the top 15% is perhaps a more realistic indicator of gain than either 5% or 25%. In terms of the baseline, the use of 100% as the baseline is perhaps more common in literature. Further, if the gain will be used to adjust an existing plantation growth and yield model, it can be 71 argued that the existing model could represent the average of all provenances planted and the 100% baseline would be more suitable. 3.3.2 Gain Trajectory Model Using the selected gain definition (i.e., evaluation age 15 years, top 15%, and 100% baseline), the relative height gain was calculated by planting site for ages 1 to 45 years as: [3.9] Gain, RH𝑖𝑖𝑖𝑖 = average height of top 15%−average height of 100%average height of 100% for which the top 15% were selected at 15 years. The resulting trajectories for all planting sites are shown in Figure 3-6. Figure 3-6. Gain over time by planting site. The baseline model was then fitted (Eq. 3.6), along with the null model. Then, several combinations of planting site climatic variables (MAT, MAP, DD) and other variables (e.g. planting density and site elevation) were used to model the scale and shape parameters (Eq. 3.8, 72 including 3.8a to 3.8c). A selection of these models based on improvements in AIC are listed in Table 3-1. As noted, first and second order continuous autoregressive parameters were added to the models, but the second order continuous autoregressive parameter was not statistically significant (p-value > 0.05) for any model and was dropped. Table 3-1. Fit statistics for a subset of height gain trajectory models. Model Parameter AIC aScale (β�0) Shape 1 (β�1) Shape 2 (β�2) Null Intercept only model -8124 Base (I) β�0 β�1 β�2 -8206 II 𝑏𝑏00 + 𝑏𝑏01MAP2 + 𝑏𝑏02 DD 𝑏𝑏10+ 𝑏𝑏11 ln (density) 𝑏𝑏20 + 𝑏𝑏21MAT+ 𝑏𝑏22MAP -15298 III 𝑏𝑏00 + 𝑏𝑏01MAT2 +𝑏𝑏02 site elevation 𝑏𝑏10+ 𝑏𝑏11 ln (density) 𝑏𝑏20 + 𝑏𝑏21MAT+ 𝑏𝑏22MAP -18684 IV 𝑏𝑏00 + 𝑏𝑏01MAT2+ 𝑏𝑏02 site elevation 𝑏𝑏10+ 𝑏𝑏11 ln (density) 𝑏𝑏20 + 𝑏𝑏21MAT -18666 V 𝑏𝑏10 + 𝑏𝑏11MAP2 + 𝑏𝑏12 DD 𝑏𝑏10+ 𝑏𝑏11 ln (density) 𝑏𝑏20+ 𝑏𝑏21MAP -15244 VI 𝑏𝑏00 + 𝑏𝑏01MAT2+ 𝑏𝑏02 site elevation 𝑏𝑏10+ 𝑏𝑏11 ln(density) 𝑏𝑏20 + 𝑏𝑏21DD -18665 VII 𝑏𝑏00 + 𝑏𝑏01MAT2 + 𝑏𝑏02MAP2+ 𝑏𝑏03 DD + 𝑏𝑏04 site elevation 𝑏𝑏10+ 𝑏𝑏11 ln(density) 𝑏𝑏20 + 𝑏𝑏21MAT+ 𝑏𝑏22MAP -19136 aMAT (°C) is mean annual daily temperature; DD (days) is degree days greater than 5°C ; MAP (mm) is mean annual precipitation ; density is the planting density (stems ha-1); and site elevation is in metres. The change in AIC for the null versus the base model (Model I) indicated that the height gain does changed with plantation age (Table 3-1). To illustrate the differences in using a single multiplier versus allowing for changes with plantation age, the actual gains versus estimated gains using the null model (intercept = average gain = 0.157182) and using the base model (?̂?𝛽0 = 0.173562, ?̂?𝛽1 = −0.03347, and ?̂?𝛽2 = 0.999845) were graphed for two planting sites, 73 “Owen Sound 194E” and “Gander”, both with a planting density of 3,086 stems per ha (i.e., 1.8 m spacing) (Figure 3-7). For both sites, the base model reflected a drop in gain with plantation age unlike the commonly used multiplier that does not change with age represented by the null model. However, for Gander, the base model also mimics the gain levels by age, whereas these are underestimated by about 0.15 for Owen Sound 19E. These two sites illustrate the improvements in allowing for changes in gain with plantation age, but they also illustrate that other site variables are needed to represent the actual gains by plantation age at the site level. Figure 3-7. Predicted height gains (RH) over age using the null model and using the base model that changes with plantation age versus the actual gains for two planting sites. 74 Further improvements in predicting height gain were obtained by allowing the parameters of the base model to change with the planting site characteristics (Models II to VII) as noted by lower AIC values for relative to the base model (Model I) (Table 3-1). The scale parameters estimated for each site (Eq. 3.8) showed a nonlinear trend with MAT and MAP; as a result, MAT2 and MAP2 were included as possible predictor variables for submodel 8a. One or both of these variables were included in all models shown in Table 3-1. A nonlinear trend with planting density was also noted for the first shape parameter (Eq. 3.8b). A number of transformations were evaluated and the natural logarithm of planting density was the only predictor variable selected for this submodel. Only climate variables were included for the second shape parameter submodel (Eq. 3.8c). Model VII, which had the largest number of predictor variables had smallest AIC (-19136), substantially lower than the base model (-8206). Further, Model VII had smaller AIC values than the other models (about 4000 less than Models II and V and about 400 less than Models III, IV, and VI). As well, the predicted height trajectories superimposed on the observed data showed the best fit for Model VII also. As a result, Model VII was selected (Table 3-2).The RMSPE for this model was 0.0042 using the “leave-one-out” model validation procedure. 75 Table 3-2. Parameter estimates (and standard errors) for the selected gain trajectory model. Eq. 3.8 Parameter Variablea Parameter Estimate (standard error) β0 (scale) Intercept 0.12455 (0.000271) MAT2(°C) 0.0019668 (5.395E-6) MAP2 (mm) 1.6164E-8 (1.94E-10) DD (days) -0.000011170 (2.293E-6) Site elevation (m) 9.6782E-6 (2.129E-7) β1𝑖𝑖 (shape 1) Intercept -0.051836 (0.00463) ln(density) (stems ha-1) 0.0010935 (0.000586) β2𝑖𝑖 (shape 2) Intercept 1.0053153 (0.000947) MAT (°C) 0.00025649 (0.000209) MAP (mm) -6.9969E-6 (1.35E-6) aMAT is mean annual daily temperature (°C); DD (days) is degree days greater than 5°C; MAP (mm) is mean annual precipitation (mm); density (stems ha-1) is the planting density; and site elevation is in metres. To illustrate the improvements from using the predicted height gain trajectories allowing for changes due to planation age and site characteristics (Model VII) as opposed to for changes due to plantation ages only (base model, Model I), the gain trajectories are again shown for the “Owen Sound 194E” and “Gander” planting sites (Figure 3-8). Generally, the shapes were very similar for Model VII versus the base model, whereas the scale differed, indicating the importance of allowing submodel 3.8a to vary with site characteristics and site climate. Unlike the base model, Model VII also represents the levels of gain for both sites and for all other sites (not shown) due to including site level variables as predictor variables. 76 Figure 3-8. Predicted height gain (RH) over age for two planting sites using: (i) Model VII; (ii) base model; and (iii) actual gain. The predictor variables of Model VII interact with each other (e.g., elevation, MAP, MAT) making it difficult to examine the effects of each predictor variable on the gain trajectory. However, combinations of these variables within the range of meta-data were selected and the resulting gain trajectories were graphed. For this purpose, the planting site elevation and planting density were set to 400 m and 2,500 stems per ha, respectively, based on the means of these variables in the meta-data. Then, all combinations of two MAPs (683 and 1,100 mm), two MATs (-0.5 and 3.5°C), and two DDs (46 and 80 days) were input in to Model VII along with ages from 0 to 50 years (Figure 3-9). Although these values represent the ranges within the meta-data, they may not strictly reflect the multivariate relationships within the predictor variables. 77 Figure 3-9. Changes in the predicted height gain (RH) over age for combinations of values for site degree days (DD), mean annual daily temperature (MAT), and mean annual precipitation (MAP), given a planting site elevation of 400 m and a planting density of 2,500 stems per ha. Changes in DD from 46 to 80 days had little impact (difference in gain less than 0.005) on the height gain trajectory (Figure 3-9). Increasing MAT from -0.5 to 3.5°C resulted in an increase in gain of about 0.03 across the plantation ages. Noticeable changes in the shape occurred when MAP changed from 683 to 1,100 mm with a steeper and more curved shape for the higher 78 precipitation level. Other graphs allowing the planting density and site elevation to change by fixing climate variables (not shown) indicated that an increase in planting density from 1,000 stems per ha to 2,500 stems per ha resulted in a very small increase in the estimated gain (gain was less than 0.005), but only for older plantations. An increase in the site elevation from 279 m to 600 m resulted in an increase in gain of 0.003 for most plantation ages. As noted earlier, the scale parameter was more affected by site climate and other characteristics than by the two shape parameters. Although it is not possible to completely interpret the scale parameter separately from the other two parameters, simulations were used to examine the effects of planting site climate on this parameter. For this purpose, the planting site elevation and planting density were kept constant at 400 m and 2,500 stems per ha, respectively. Then, the DD, MAT, and MAP were allowed to vary within the range of values in the meta-data (Figure 3-10). Slightly larger values were obtained for a DD of 46 days. The scale parameter increased with increases in MAT and MAP, indicating that sites with better growing conditions have higher height gain trajectories. 79 Figure 3-10. Changes in the estimated scale parameter (?̂?𝛽0𝑖𝑖) with changes in site degree days (DD), mean annual daily temperature (MAT), and mean annual precipitation (MAP) given a planting site elevation of 400 m and a planting density of 2,500 stems per ha. 3.4 Discussion Relative height gain has been used as a basis for estimating gains in yields resulting from superior provenances in Canada and elsewhere. In many research papers, a simple relative height gain multiplier was estimated using provenance trials data, thereby implicitly or explicitly assuming that this gain multiplier does not vary over the plantation age and site characteristics (e.g., Buford and Burkhart 1987; Carson et al. 1999; Adams et al. 2006; Gould et al. 2008; Lu and Charrette 2008; Petrinovic et al. 2009; Gould and Marshall 2010; Ahtikoski et al. 2012; 80 Ahtikoski et al. 2013). These assumptions may not hold, particularly over the large spatial extents of spruce plantations in Canada. Meta-data were used to examine these assumptions by including plantation age and site characteristics as possible predictor variables in a model to estimate height gain. For this purpose, height trajectories for each provenance at all planting sites in the meta-data were created using climatic information of the provenance and planting site locations and the model developed in Chapter 2. Once these height trajectories were obtained, models of the actual height gain over the plantation age and for different planting site characteristics were developed. I needed to address two issues before these gains could be calculated: (i) the age at which to identify top performers; and (ii) the percentiles to use to define the top performers and the baseline. To address these issues, sensitivity analyses were used to examine the impacts of various evaluation ages and top performers/baselines definitions. With regards to the first issue, the sensitivity analyses showed that the choice of evaluation age resulted in different gain trajectories, but mostly when comparing an evaluation age of 5 years (plantation age) to older ages. Given that there was little change between using 15 and 45 years as evaluation ages, and also that earlier identification of improved stock is useful in practice, the evaluation age of 15 years was selected. Previous work supports this and other values. Johnson (2003) suggested using an evaluation age at ¼ of the harvest age for Douglas-fir for predicting gain. Applying this to white spruce and given a harvest age of approximately 60-80 years for plantations, the recommended evaluation age would be 15 to 20 years, similar to that selected in this study. McInnis and Tosh (2004) used age 20 as the evaluation age for black spruce and jack pine to estimate gains in height, volume and diameter and found that height gain declined sharply until 81 age 20. Conversely, Newton (2003) used 50 years as the evaluation age to estimate relative height gain for black spruce, jack pine, white spruce and red pine. Newton’s choice of 50 years would be closer to the biological rotation ages for these species, but this may not be practical in application and may not be necessary based on the sensitivity analysis presented in this paper. Petrinovic et al. (2009) used data from seven white spruce progeny trials covering plantation ages of 13 to 22 years, depending upon the trial, to estimate gain in dominant height. With respect to the second issue, the definitions of top performers and the baseline affected both the gain trajectory level and the shape. The average of the top 15% of the provenances was selected to define the top performers and the average of all provenances (i.e., 100%) was selected as the baseline to estimate gain for each planting site and plantation age. The values used for identifying top performers and baselines vary across other studies, often without specific justification for these values used. Zobel and Talbert (1984) used 100% for the baseline as did Magnussen and Yanchuk (1994). Newton (2003) used the top 25% relative to the remaining stock (i.e., 75%). McInnis and Tosh (2004) also used the top 25% of improved stock, but used 100% of unimproved stock as the baseline. In some studies, top performers were based on “plus-trees” selected based on larger heights and diameters outside bark at breast height (i.e., DBHs measured at 1.3 m above ground) rather than on a particular percentiles (e.g., Xie et al. 1998; Xie 2003). Using the selected definition, I found the estimated height gain was 0.12 to 0.25 (i.e., 12 to 25%) for young plantations and 0.12 to 0.22 for 45-year old plantations depending on the planting site. These values are similar to those found in other studies, although, as noted, the gain definition 82 differs among studies. Newton (2003) estimated the height gain as 0.1240 (i.e., 12.40%) at age 50 years. McInnis and Tosh (2004) reported that the first generation black spruce gain in height was 0.12 at age 20 and then projected the gain to be 0.07 to 0.08 by age 40 years. Changes in height gain were found with plantation age, indicating that this is not invariant over age for white spruce. Carson et al. (1999) also found changes in height gain over plantation age for radiata pine; the gain decreased from age 15 to age 45 years. Similarly, McInnis and Tosh (2004) noted a downward trend with age for black spruce. A number of other studies found differences in gain with plantation age, but simple gain multipliers were still used for simplicity (Petrinovic et al. 2009; Ahtikoski et al. 2012; Ahtikoski et al. 2013). Conversely, Newton (2003) found that gain was relatively constant over time in his meta-analysis for white spruce. However, he used the top 25% versus 75% as noted earlier, which does not appear to vary greatly with age based on the sensitivity analysis in this study. The height gain for white spruce also varied with planting site characteristics. This does not agree with the meta-analysis for white spruce reported by Newton (2003), which found no differences in gain among planting sites. However, a more extensive meta-data set was used than that used by Newton (2003), covering a wider range of planting sites across a broad spatial extent, including sites from Alberta and hemiboreal forests of British Columbia. In particular, the height gain trajectories varied with site climatic variables, specifically MAT, MAP and DD, along with site elevation and planting density. Generally, height gains were larger for warmer sites (i.e., higher MAT) with more precipitation (i.e., higher MAP). This result is supported by evidence in the genetics literature on environment interactions (e.g., Morgenstern 1996; ATISC 83 2004; Rweyongeza et al. 2007; ATISC 2007; Rweyongeza et al. 2011). Also, Gould and Marshall (2010) found that gains in volume, diameter and height increased for higher productivity sites in Douglas-fir stands. Similarly, others (e.g., Buford and Burkhart 1987 for loblolly pine) have found that genetic gains increased with site index. Although planting density was included in the final model, the effects on gain were small relative to other site variables. Dominant height growth in even-aged stands is relatively unaffected by density (Pienaar and Shiver 1984). Gould and Marshall (2010) in their study of Douglas-fir plantations also found that the predicted height growth for improved stock was generally unaffected by initial planting density. Although this research has contributed to the understanding of the impacts of the gain definition on the reported gains and trends in gain with time, and also examined the changes over plantation age and over planting densities, a number of difficulties in conducting this work should be noted. First, the so-called actual gain trajectories for each provenance on every site were based on height trajectories produced via a previously fitted model using these data (Chapter 2) as used in other studies (e.g., Magnussen and Yanchuk 1994). This allowed for both the extensive spatial range represented in the meta-data and the longer temporal range that was represented in only a portion of the meta-data. However, using the model-based height to age 45 trajectories could have introduced biases in the so-called actual gains. Further, gains were based on percentiles of provenances included at each provenance trial. As noted by Wright (1976) and Namkoong (1979), the selection of which provenances to include in each provenance trial affects the calculation of gain. Finally, although a very broad spatial coverage was included in the meta-data, there were still spatial gaps. However, these missing locations may not be suitable for white 84 spruce plantations, because of site characteristics (e.g., poor soils, etc.) or because they may be too far from mills and other facilities that might make use of these plantations. 3.5 Conclusions Based on the comprehensive sensitivity analysis, along with comparisons to other studies and experts’ opinions, one definition of potential gain was chosen. In particular, the top 15% of all provenances relative to all provenances evaluated at 15 years since planting was recommended. This could be applied across the boreal and hemiboreal forests of Canada thereby facilitating comparisons and aggregation of information. Using this definition, height gains of white and hybrid spruces varied with plantation age and planting site characteristics. Consequently, the commonly used approach of a single gain multiplier for all ages and planting site characteristics may not reflect the gain trajectories. As an alternative, a height gain trajectory model was developed using a random coefficients (aka, parameter prediction) modelling approach to model the changes in over time under varying site characteristics. The developed gain trajectory model provided accurate gain estimates for each planting site using site climatic, planting density, and elevation as predictor variables. This was largely affected by the mean annual daily temperature of planting sites, but was also affected by mean annual precipitation and degree days > 5°C and, to a lesser degree, by planting density. The developed height gain model could be incorporated into an existing growth and yield model to forecast potential gains in crop yields over time. Overall, this model shows the potential for improving yields based on tree improvement program in the boreal and hemiboreal forests of Canada. 85 4. Alternative Methods to Forecast Yields of Hybrid Spruce Progenies 4.1 Introduction In tree improvement programs, provenances or progenies are planted at several sites and the growth and yield of trees are measured over time. These provenance or progeny trials provide the opportunity to compare alternative planting stocks and also provide information needed to further improve stocks in tree breeding. However, in Canada, tree breeding programs are relatively recent, with major activities starting in 1950 (Fowler and Morgenstern 1990). In the 1970s, the New Brunswick Tree Improvement Council (NBTIC) established tree improvement trials on black spruce (Picea mariana (Mill.) B.S.P), white spruce (Picea glauca (Moench) Voss), tamarack (Larix laricina (Du Roil) K. Koch) and jack pine (Pinus banksiana Lamb.)(Simpson and Tosh 1997). In Alberta, tree improvement activities started in 1975; the Alberta Tree Improvement and Seed Centre performs provenance and progeny trials on major Alberta conifer species (i.e., lodgepole pine (Pinus contorta var. latifolia Douglas ex Louden) and white spruce) (Rweyongeza et al. 2010; Rweyongeza et al. 2011). Tree improvement activities started in British Columbia in the 1960s focusing on major commercial species (lodgepole pine, coastal Douglas-fir (Pseudotsuga menziesii var. menziesii) interior Douglas-fir (Pseudotsuga menziesii var. glauca (Mirb.)), interior spruce (Picea glauca and Picea engelmannii), yellow cedar (Chamaecyparis nootkatensis), western white pine (Pinus monticola), western red cedar (Thuja plicata), western larch (Larix occidentalis), and western hemlock (Tsuga heterophylla)) (Xie and Yanchuk 2003). Consequently, most Canadian provenance and progeny trials are relatively young making it difficult to anticipate the impacts of using improved stock on yield at harvest, typically 80 years or more. 86 In spite of these difficulties, provenance and progeny trials have been used in some studies to evaluate yield performances (Beaulieu 1996; Morgenstern et al. 2006; Rweyongeza et al. 2010; Rweyongeza et al. 2011). For example, Beaulieu (1996) analyzed yields of white spruce provenances and associated breeding performances based on trials in Quebec at 25 years since planting. Morgenstern et al. (2006) evaluated yields of provenance trials at the Petawawa Research Forest in Ontario at 44 years after planting. However, methods that provide accurate yield forecasts at expected harvest ages would further help to assess provenance performance. Mixed-effects models could be particularly useful for forecasting yields of provenances or progenies at particular planting sites, since they can: (i) account for autocorrelation among repeated measures of each provenance or progeny within a site, thereby improving fixed-effects parameter estimates; (ii) provide subject-specific (i.e., provenance or progeny within site) random effects to localize estimates; and (iii) provide an autocorrelation model that can be used to forecast repeated measures. For this purpose, some authors have used linear mixed-effects models (e.g., Magnussen and Yanchuk 1994). However, nonlinear mixed-effects models (NLMM) are more commonly used to project tree height (and other yields) over time since they better reflect growth and yield trends over time (e.g., Wang et al. 2007; Meng and Huang 2009 & 2010; Huang et al. 2011). The fitted model of Chapter 2 can be used to obtain predicted yields over time for any planting site and provenance using population-averaged predictions as was shown. A similar NLMM model could be developed and used to obtain population-averaged progeny yields. However, subject-specific estimates may provide improved forecasts of provenances or progenies that are within the data used to fit the model and also for others with 87 repeated measures that were not part of the model-building dataset, as was shown for permanent sample plots by Meng and Huang (2009). Alternatively, repeated measures for progenies could be used in the fitted autocorrelation model to improve forecasts (Judge et al. 1988) as demonstrated Meng and Huang (2010) for repeated measures of permanent sample plots. Although alternative methods to forecast yields of permanent sample plots have been demonstrated and these methods could be used to forecast yields of provenances or progenies in trials, the accuracy of these methods depends upon the accuracy of the population-averaged model. Only minor (or negligible) improvements might be expected to result from using a subject-specific or autocorrelation model to forecast yields if the population-averaged model already includes predictor variables that account for differences among provenances or progenies within a site based on research aimed at forecasting repeated measures of permanent sample plots (Meng and Huang 2009; Yang and Huang 2011; Meng et al. 2012). In this research, a NLMM model for the height yields of hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) progenies using trials in British Columbia was developed. Then, alternative methods to forecast yields were compared, specifically using: (i) population-averaged forecasts based on the fixed-effects part of the NLMM model only; (ii) subject-specific (i.e., a progenies within sites) forecasts using the random effects developed using available repeated measures; and (iii) forecasts using the population-averaged model along with prior repeated measures in the autocorrelation model. Validation methods were then used to compare forecast accuracies. Since the accuracy of the fixed-effects part of the NLMM model 88 would be expected to affect the results, two models were included in the comparison, with one having a more accurate fixed-effects portion based on Akaike’s information criterion (AIC). 4.2 Methods 4.2.1 Progeny Repeated-Measures Data Data for progeny trials initiated under the hybrid spruce tree breeding program in British Columbia (BC)5 were provided by the BC Ministry of Forests, Lands and Natural Resources Operations. The data consisted of repeated measures for a number of hybrid spruce progenies at three planting sites in BC. At each planting site, 25 open-pollinated families were tested in progeny trials and these were selected from British Columbia. The initial planting density was 1600 stems per ha (i.e., spacing of 2.5m). For each planting site, the following information was obtained: (i) site latitude, longitude, and elevation; (ii) measurement dates; (iii) average height of each progeny at each measurement time; and (iv) latitude, longitude, and elevation of each progeny. For each progeny at every planting site, climate information, specifically, climate normals were added to the data. Weather station data for the 1981-2010 period were extracted from the Environment Canada climate normals database (www.climate.weatheroffice.ec.gc.ca, accessed 10th September, 2013). These data were used to interpolate climate normals for each progeny location and planting site using inverse distance weighting (IDW). The climate variables retained 5 Dr. Alvin Yanchuk and Barry Jaquish provided data for these trials in May, 2013. 89 for this study were: (i) mean daily temperature (MAT); (ii) mean annual precipitation (MAP); and (iii) mean annual number of degree days greater than 5°C (DD). Using these data, climate transfer distances for each progeny at a planting site were calculated as: 1. Mean daily temperature distance (DMAT) = site MAT – progeny MAT, where a positive DMAT occurs for a progeny from a colder area that is planted at a warmer planting site, and a negative DMAT occurs for a progeny from a warmer area that is planted at a colder planting site. 2. Mean annual precipitation difference (DMAP) = site MAP – progeny MAP, where a positive DMAP occurs when precipitation is higher at the planting site than at the progeny location, and a negative DMAP occurs when precipitation is lower at the planting site than at the progeny location. 3. Degree days difference (DDdif) = site DD – progeny DD, where a positive DDdif occurs when DD is higher at the planting site than at the progeny location, and a negative DDdif occurs when DD is lower at the planting site than at the progeny location. Summary statistics for the climate data are shown in Table 4-1. 90 Table 4-1. Summary statistics for the three hybrid spruce progeny trials in BC. aVariable Minimum Mean Maximum Plantation age (i.e., years since planting) 2.00 17.00 42.00 Planting sites Latitude 52.99 53.58 54.05 Longitude -122.72 -122.36 -122.11 Elevation (m) 610.00 743.75 915 MAT (°C) 4.32 4.58 4.73 MAP (mm) 584.69 605.92 628.61 Degree days 52.89 60.83 75.22 Progenies Latitude 52.82 53.81 54.83 Longitude -122.52 -122.20 -121.5 Elevation (m) 610.00 844.68 1,220.00 MAT (°C) 2.94 4.35 5.12 MAP (mm) 535.04 618.66 935.74 Degree days 53.67 60.40 85.21 Differences DMAT -0.80 0.25 1.79 DMAP -351.05 -12.67 93.57 DDdif -32.21 0.75 21.56 aMAT (°C) is the mean daily temperature; MAP (mm) is the mean annual precipitation; Degree days is the number of days with mean temperature greater than 5°C; DMAT (°C) is the mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the progeny. For all planting sites, repeated measures were obtained from 1 to 42 years since planting; however, the number of repeated measures differed. At the Aleza planting site there were six repeated measures (2, 5, 8, 12, 27 and 42 years since planting), whereas at the PGTIS and 91 Quesnel planting sites, there were seven repeated measures (2, 5, 8, 12, 17, 32 and 42 years since planting). The average height trajectories of all progenies at all sites are shown in Figure 4-1. Figure 4-1. Observed average height trajectories for of hybrid spruce progenies in BC. 4.2.2 Height Trajectory Model Using the same approach applied in Chapter 2, an NLMM model was fitted using the hybrid spruce data for the three planting sites. The Chapman-Richard’s growth model (Pienaar and Turnbull 1973) was used as the base model, and modified to predict average height of each progeny at each site by replacing the parameters with functions of climate transfer distances and other variables: [4.1] H𝑖𝑖𝑖𝑖𝑖𝑖 = 𝜃𝜃1𝑖𝑖𝑖𝑖(1 − exp (1 − 𝜃𝜃2𝑖𝑖𝑖𝑖age𝑖𝑖𝑖𝑖𝑖𝑖))( 11−𝜃𝜃3𝑖𝑖𝑖𝑖) + 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 ~𝑁𝑁(0,𝑹𝑹𝑖𝑖𝑖𝑖) 92 [4.1a] 𝜃𝜃1𝑖𝑖𝑖𝑖 = 𝛽𝛽10 + f� climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿1𝑖𝑖𝑖𝑖 [4.1b] 𝜃𝜃2𝑖𝑖𝑖𝑖 = 𝛽𝛽20 + f�climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿2𝑖𝑖𝑖𝑖 [4.1c] 𝜃𝜃3𝑖𝑖𝑖𝑖 = 𝛽𝛽30 + f�climate transfer distance𝑖𝑖𝑖𝑖, other variables𝑖𝑖𝑖𝑖� + 𝛿𝛿3𝑖𝑖𝑖𝑖 where H𝑖𝑖𝑖𝑖𝑖𝑖 is the average height of planting site i and progeny j at measurement time t; 𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 is the error term; age𝑖𝑖𝑖𝑖𝑖𝑖 is the plantation age; and 𝛿𝛿1𝑖𝑖𝑖𝑖, 𝛿𝛿2𝑖𝑖𝑖𝑖 and 𝛿𝛿3𝑖𝑖𝑖𝑖 are error terms. This model has three parameters: the asymptote (𝜃𝜃1) specifies the maximum average height and the other two are shape parameters (𝜃𝜃2 and 𝜃𝜃3). The error covariance matrix was defined for all progenies at every site as: [4.2] 𝑹𝑹𝑖𝑖𝑖𝑖 = 𝜎𝜎2𝜳𝜳𝑖𝑖𝑖𝑖 = 𝜎𝜎2⎣⎢⎢⎢⎡ 1 𝜌𝜌𝑜𝑜12𝜌𝜌𝑜𝑜21 1 ⋯ 𝜌𝜌𝑜𝑜1𝑟𝑟𝑖𝑖𝑖𝑖⋯ 𝜌𝜌𝑜𝑜2𝑟𝑟𝑖𝑖𝑖𝑖⋮ ⋮𝜌𝜌𝑜𝑜𝑟𝑟𝑖𝑖𝑖𝑖1 𝜌𝜌𝑜𝑜𝑟𝑟𝑖𝑖𝑖𝑖2⋱ ⋮ ⋯ 1 ⎦⎥⎥⎥⎤ where 𝜎𝜎2 is the constant error variance; 𝜌𝜌 is the autocorrelation parameter; 𝑑𝑑 represents distance between two measurements; 𝜳𝜳𝑖𝑖𝑖𝑖 is the error correlation structure; and 𝑙𝑙𝑖𝑖𝑖𝑖 is the number of repeated measures for planting site i and progeny j (varies by progeny within planting site). For sub-models 4.1a to 4.1c, possible predictor variables were climate transfer distance variables (DMAT, DMAP, DDdif), site elevation, and progeny elevation. Several models were fitted using combinations of predictor variables as in Chapter 2. Transformations of some variables were included to represent expected nonlinear trends with climatic variables. 93 The NLMM can be modified to include subject-specific random effects on any or all fixed-effects parameters. In the repeated measures data in this chapter, progenies were nested within planting sites and all repeated measures that share the same levels of prov(site) (i.e., progenies are nested within planting sites) were represented as a single subject. In this study, random effects to represent each progeny within each planting of the sample data were added to each sub-model of the model; specifically, 𝑏𝑏10𝑖𝑖𝑖𝑖, 𝑏𝑏20𝑖𝑖𝑖𝑖 or 𝑏𝑏30𝑖𝑖𝑖𝑖 were added to the intercepts of Eq. 4.1a to 4.1c, respectively. These random effects parameters were assumed to follow a multivariate normal distribution with an estimated variance-covariance matrix 𝑫𝑫� defined as: [4.3] 𝑫𝑫� = � 𝜎𝜎�𝑏𝑏10𝑖𝑖𝑖𝑖2 𝜎𝜎�𝑏𝑏10𝑖𝑖𝑖𝑖𝑏𝑏20𝑖𝑖𝑖𝑖 𝜎𝜎�𝑏𝑏10𝑖𝑖𝑖𝑖𝑏𝑏30𝑖𝑖𝑖𝑖𝜎𝜎�𝑏𝑏20𝑖𝑖𝑖𝑖𝑏𝑏10𝑖𝑖𝑖𝑖 𝜎𝜎�𝑏𝑏20𝑖𝑖𝑖𝑖2 𝜎𝜎�𝑏𝑏20𝑖𝑖𝑖𝑖𝑏𝑏30𝑖𝑖𝑖𝑖𝜎𝜎�𝑏𝑏30𝑖𝑖𝑖𝑖𝑏𝑏10𝑖𝑖𝑖𝑖 𝜎𝜎�𝑏𝑏30𝑖𝑖𝑖𝑖𝑏𝑏20𝑖𝑖𝑖𝑖 𝜎𝜎�𝑏𝑏30𝑖𝑖𝑖𝑖2� For all models, the %NLINMIX macro of SAS software version 9.4 (SAS Institute Inc. 2014) was used to estimate parameters, including the subject-specific random effects (expand=EBLUP option), and the spatial power function (i.e., SP(POW)) to account for correlations among irregular repeated measures for each progeny within a planting site. Residuals were also checked for heteroscedasticity and normality. Several starting values were used to ensure the global maximum. To evaluate the accuracy of each model and to select among alternative models, Akaike’s information criterion (𝐴𝐴𝐴𝐴𝐴𝐴 = −2𝑙𝑙𝑙𝑙𝑙𝑙𝐿𝐿 + 2𝑝𝑝, where 𝐿𝐿 = maximum likelihood and 𝑝𝑝 = the number of fixed- and random-effects parameters; Littell et al. 2006, page 574-585) was used. Also, graphs of predictions superimposed on the observed data by progeny and planting site were used 94 to check for any model lack-of-fit. One NLMM was then selected for use in forecasts. However, a simpler, less accurate NLMM was also included in the forecast methods comparisons, since the relative accuracy of alternative forecast methods would likely depend on the ability of the fixed-effects part of the NLMM model to represent differences among progenies across sites. 4.2.3 Forecasting Methods 4.2.3.1 Population-Averaged Forecasts Once one of the NLMM models had been selected, forecasts to future times were made by imputing the fixed-effects variables for the progeny at a planting site, along with the future plantation age. This was repeated for the simplified, less accurate model. 4.2.3.2 Subject-Specific Forecasts For forecasts of heights to future ages using the selected NLMM, the subject-specific parameters for progenies and sites within the sample data (i.e., including the EBLUP estimates of 𝑏𝑏10𝑖𝑖𝑖𝑖, 𝑏𝑏20𝑖𝑖𝑖𝑖 or 𝑏𝑏30𝑖𝑖𝑖𝑖 in sub-models 4.1a, 4.1b, and 4.1c) can be used instead of using the population-averaged forecasts. For progenies and sites not included in the data used to fit the model (i.e., new progeny r at a new planting site q), estimates of these random-effects parameters could also be obtained if 95 repeated measures data become available (Lindstrom and Bates 1990; Vonesh and Chinchilli 1997). Using the revised NLMM and the new repeated measures data, the EBLUP expansion can be solved for 𝒃𝒃𝑞𝑞𝑞𝑞 (subject-specific estimate of a new progeny) using a first order Taylor series expansion and an iterative procedure as described in Meng and Huang (2009): [4.4] 𝒃𝒃�𝑞𝑞𝑞𝑞 = 𝑫𝑫�𝒁𝒁𝑞𝑞𝑞𝑞𝑇𝑇 �𝒁𝒁𝑞𝑞𝑞𝑞𝑫𝑫�𝒁𝒁𝑞𝑞𝑞𝑞𝑇𝑇 + 𝑹𝑹�𝑞𝑞𝑞𝑞�−1(𝐇𝐇𝑞𝑞𝑞𝑞 − 𝑓𝑓�𝜽𝜽�𝒒𝒒𝒒𝒒,𝒃𝒃�𝑞𝑞𝑞𝑞 , 𝐚𝐚𝐚𝐚𝐚𝐚𝑞𝑞𝑞𝑞� + 𝒁𝒁𝑞𝑞𝑞𝑞𝒃𝒃�𝑞𝑞𝑞𝑞) where 𝑫𝑫� is based on the model-fitting data (defined in Eq. 4.3); 𝒃𝒃�𝑞𝑞𝑞𝑞 is a unknown vector of the estimated random effects for the planting site q and progeny r; 𝐇𝐇𝑞𝑞𝑞𝑞 is the vector of k repeated height measures for the new progeny at plantation ages given in 𝐚𝐚𝐚𝐚𝐚𝐚𝑞𝑞𝑞𝑞; 𝜽𝜽�𝒒𝒒𝒒𝒒 is the vector of fixed-effects using the vector of covariates for the new progeny ( 𝐱𝐱𝑞𝑞𝑞𝑞) in Eq. 4.1a to 4.1.c; 𝑹𝑹�𝑞𝑞𝑞𝑞 is the error covariance matrix for the k repeated measures of the new progeny based on the structure given in Eq. 4.2 and using estimates of 𝜎𝜎2 and 𝜌𝜌 from fitting the subject-specific NLMM; 𝑓𝑓�𝜽𝜽�𝒒𝒒𝒒𝒒,𝒃𝒃�𝑞𝑞𝑞𝑞 , 𝐚𝐚𝐚𝐚𝐚𝐚𝑞𝑞𝑞𝑞� is a vector of subject-specific height estimates for the new progeny; and 𝒁𝒁𝑞𝑞𝑞𝑞 is defined as: [4.5] 𝒁𝒁𝑞𝑞𝑞𝑞 =⎣⎢⎢⎢⎡𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟1)𝜕𝜕𝑏𝑏10𝑞𝑞𝑟𝑟𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟1)𝜕𝜕𝑏𝑏20𝑞𝑞𝑟𝑟𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟1)𝜕𝜕𝑏𝑏30𝑞𝑞𝑟𝑟⋮ ⋮ ⋮𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟𝑞𝑞)𝜕𝜕𝑏𝑏10𝑞𝑞𝑟𝑟𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟𝑞𝑞)𝜕𝜕𝑏𝑏20𝑞𝑞𝑟𝑟𝜕𝜕𝜕𝜕(𝜽𝜽�𝑞𝑞𝑟𝑟,𝒃𝒃�𝑞𝑞𝑟𝑟,age𝑞𝑞𝑟𝑟𝑞𝑞)𝜕𝜕𝑏𝑏30𝑞𝑞𝑟𝑟 ⎦⎥⎥⎥⎤ A three-step iterative algorithm was used to solve for 𝒃𝒃�𝑞𝑞𝑞𝑞, using the Newton-Raphson search algorithm programmed using SAS PROC IML (SAS Institute Inc. 2014) and following the description given in Meng and Huang (2009, page 240). Once these random-effects were estimated for the new progeny, average heights could be forecasted using the subject-specific height forecasts to future ages. 96 4.2.3.3 Forecasting using the autocorrelation portion of the model Another option for forecasts is to use the population-averaged forecasts adjusted using repeated measures and the autocorrelation portion of the model (Judge et al. 1988). First, the fixed-effects portion of the NLMM model (i.e., not using subject-specific random effects) is used to obtain a population-averaged prediction. Then, repeated measures of average heights for a specific progeny within a planting site are input to the autocorrelation part of the model as in Meng and Huang (2010): [4.6] H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑝𝑝(𝜽𝜽�𝑖𝑖𝑖𝑖 , age𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖) + 𝑽𝑽𝑇𝑇𝜳𝜳𝑖𝑖𝑖𝑖−1(𝐇𝐇𝑖𝑖𝑖𝑖 − 𝑓𝑓(𝜽𝜽�𝑖𝑖𝑖𝑖 ,𝐚𝐚𝐚𝐚𝐚𝐚𝑖𝑖𝑖𝑖)) where H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖 is the forecasted average height for progeny j, planting site i, and future time t; age𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖 is the future plantation age; 𝜽𝜽�𝑖𝑖𝑖𝑖 are the estimated fixed-effects for the NLMM model (Eq.4.1 to 4.1c); 𝑓𝑓𝑝𝑝(𝜽𝜽�𝑖𝑖𝑖𝑖, age𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖) is the population-averaged forecast of average height at future time t; 𝐇𝐇�𝑖𝑖𝑖𝑖 is a vector of prior average height measures with associated plantation 𝐚𝐚𝐚𝐚𝐚𝐚𝑖𝑖𝑖𝑖; 𝑓𝑓�𝜽𝜽�𝑖𝑖𝑖𝑖 ,𝐚𝐚𝐚𝐚𝐚𝐚𝑖𝑖𝑖𝑖� is a vector of prior average height population-averaged predictions; 𝑽𝑽 is a correlation vector between (H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖 − 𝑓𝑓𝑝𝑝(𝜽𝜽�𝑖𝑖𝑖𝑖 , age𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖)) and (𝐇𝐇�𝑖𝑖𝑖𝑖 − 𝑓𝑓(𝜽𝜽�𝑖𝑖𝑖𝑖 ,𝐚𝐚𝐚𝐚𝐚𝐚𝑖𝑖𝑖𝑖)); and 𝜳𝜳𝑖𝑖𝑖𝑖 is the correlation matrix as defined in Eq. 4.2. The calculations were programmed using SAS PROC IML (SAS Institute Inc. 2014) to solve for H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖. These forecasts can be obtained for any progeny within a site for which repeated measures data are available, regardless of whether the data were used to originally fit the model. 97 4.2.4 Evaluating Forecast Accuracies To compare alternative forecast methods, “leave-one-out” validation approaches were used. First, to evaluate forecast accuracies for a progeny with repeated measures data that were not used in fitting the model (termed “new progeny”), one progeny was selected and excluded from all three sites. Then, the selected NLMM was fit using the remaining data. Forecasts for the omitted progeny were then obtained for plantation ages 27 and 42 for the Aleza site, and for plantation ages 17, 32 and 42 for the PGTIS and Quesnel planting sites. The other measures (i.e., plantation ages 2, 5, 8 and 12 years) were: (i) not used for the population-averaged model forecasts; (ii) used to obtain the subject-specific estimates; or (iii) used as inputs to the autocorrelation part of the NLMM model. Once this process was repeated for each progeny, differences between the measured (H𝑖𝑖𝑖𝑖𝑖𝑖) and forecasted heights (H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖) were calculated and summarized by forecast method into root mean square predicted error (RMSPE) and mean absolute prediction error (MAPE) statistics defined as: [4.7] RMSPE = �∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑡𝑡−H�𝑝𝑝𝑖𝑖𝑖𝑖𝑡𝑡�2𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1𝑀𝑀 [4.8] MAPE = ∑ ∑ ∑ �H𝑖𝑖𝑖𝑖𝑡𝑡−H�𝑝𝑝𝑖𝑖𝑖𝑖𝑡𝑡�𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡=1𝑚𝑚𝑖𝑖𝑖𝑖=1𝑛𝑛𝑖𝑖=1𝑀𝑀 where 𝑖𝑖 = 1, … ,𝑛𝑛 planting sites; 𝑗𝑗 = 1, … ,𝑚𝑚𝑖𝑖 progenies within planting sites; 𝑡𝑡 = 1, 2, or 3 (i.e., excluded measurement times only); and 𝑀𝑀 is the number of excluded observations. Second, to evaluate forecasts for progenies within sites that were included in model fitting, again one progeny was selected and excluded from all three sites. However, the omitted progeny data were split into repeated measures used in model fitting (i.e., plantation ages 2, 5, 8 and 12 years) 98 versus repeated measures reserved for testing forecasts. Forecasts were then obtained for the reserved measures (i.e., plantation ages 27 and 42 for the Aleza site and 17, 32 and 42 years for the PGTIS and Quesnel planting sites). The other measures that were included in model fitting (i.e., plantation ages 2, 5, 8 and 12 years) were: (i) not used for the population-averaged model forecasts; (ii) used to obtain the subject-specific estimates; or (iii) used as inputs to the autocorrelation model. Once this process was repeated for each progeny, differences between the measured (H𝑖𝑖𝑖𝑖𝑖𝑖) and forecasted heights (H�𝑝𝑝𝑖𝑖𝑖𝑖𝑖𝑖) were calculated and summarized by forecast method into root mean square predicted error (RMSPE) and mean absolute prediction error (MAPE) statistics. Since the number of repeated measurements may affect the forecast accuracies, I repeated these two validation approaches using three repeated measures (plantation ages 2, 5 and 8 years) and then again using only two repeated measures (plantation ages 2 and 5 years). Regardless of the number of repeated measures available (i.e., four, three or two), forecasts were always obtained for plantation ages 27 and 42 years for the Aleza planting site and for plantation ages 17, 32 and 42 years for the other two planting sites to enable comparisons. As a final examination of the impacts of having more repeated measures, all available repeated measures except for the final measurement were used to forecast height for the final plantation age of 42 years using the selected NLMM model. This final validation approach was repeated using the simpler, less accurate NLMM model also. 99 4.3 Results 4.3.1 NLMM models NLMM height yield models were fit using several combinations of climate transfer variables (DMAT, DMAP, DDdif), along with site and progeny elevation in the fixed-effects part of the model (Eq. 4.1 to 4.1c). For the subject-specific random effects, several combinations of 𝑏𝑏10𝑖𝑖𝑖𝑖, 𝑏𝑏20𝑖𝑖𝑖𝑖 , and 𝑏𝑏30𝑖𝑖𝑖𝑖 were evaluated. Based on fit criteria (not shown), only the 𝑏𝑏10𝑖𝑖𝑖𝑖 random effect that alters the asymptote was retained. A selection of best models is listed in Table 4-2 along with fit statistics. Of these models, Model I had smallest AIC (7.3) and the predicted average height trajectories superimposed on the observed trajectories showed no lack-of-fit. For this selected NLMM model, all variables were significant in sub-models 4.1a to 4.1c (𝑝𝑝 < 0.05) (Table 4-3). As noted in the methods, a simplified, less accurate NLMM model was also selected since the fixed-effects part is expected to affect the outcomes of alternative forecast methods. For this purpose, Model III with an AIC about three times as large as the selected NLMM model (AIC=22.9) was included (parameters estimates in Table 4-4). 100 Table 4-2. Selected fitted models with fit statistics. Model No. Parameter Prediction Models (4.1a to 4.1c) with random effects AIC Asymptote (𝜃𝜃�1𝑖𝑖𝑖𝑖) 𝜃𝜃�2𝑖𝑖𝑖𝑖 𝜃𝜃�3𝑖𝑖𝑖𝑖 I ?̂?𝛽10 + ?̂?𝛽11progeny elevation +?̂?𝛽12DMAP2 + ?̂?𝛽13 DMAT + 𝑏𝑏�10𝑖𝑖𝑖𝑖 ?̂?𝛽20 + ?̂?𝛽21DDdif ?̂?𝛽30 7.3 II ?̂?𝛽10 + ?̂?𝛽11progeny elevation +?̂?𝛽12DMAP2 + ?̂?𝛽13DMAT + 𝑏𝑏�10𝑖𝑖𝑖𝑖 ?̂?𝛽20 ?̂?𝛽30 13.1 III ?̂?𝛽10 + ?̂?𝛽12DMAP2 + ?̂?𝛽13DMAT + 𝑏𝑏�10𝑖𝑖𝑖𝑖 ?̂?𝛽20 ?̂?𝛽30 22.9 IV ?̂?𝛽10 + ?̂?𝛽11DMAT + 𝑏𝑏�10𝑖𝑖𝑖𝑖 ?̂?𝛽20 ?̂?𝛽30 23.9 aDMAT (°C) is the mean daily temperature distance, DDdif (days) is the degree days distance, and DMAP (mm) is the mean annual precipitation distance between the site and the progeny; and progeny elevation is in m. 101 Table 4-3. Parameter estimates (standard errors) for the selected NLMM model (Model I). Parameter Prediction Models aVariable Parameter Estimate (standard error) Asymptote (𝜃𝜃�1𝑖𝑖𝑖𝑖) Intercept 102.51 (11.51) Progeny elevation (m) -0.03629 (0.009512) DMAP2(mm) 0.000269 (0.000071) DMAT (°C) -10.0581 (2.035900) Shape 1 (𝜃𝜃�2𝑖𝑖𝑖𝑖) Intercept 0.01318 (0.000951) DDdif (days) -0.00002 (6.72E-6) Shape 2 (𝜃𝜃�3𝑖𝑖𝑖𝑖) Intercept 0.4824 (0.008521) 𝜎𝜎𝑏𝑏10𝑖𝑖𝑖𝑖2 39.0305 (7.572800) Temporal correlation 𝑆𝑆𝑝𝑝(𝑝𝑝𝑙𝑙𝑝𝑝),𝜌𝜌 0.9361 (0.009149) Residual variance (𝜎𝜎𝜖𝜖2) 0.06480 (0.007796) aDMAT is mean daily temperature distance, DDdif is degree days distance, and DMAP is mean annual precipitation distance between the site and the progeny. 102 Table 4-4. Parameter estimates (standard errors) for the less accurate NLMM model (Model III). Parameter Prediction Models aVariable Parameter Estimate (standard error) Asymptote (𝜃𝜃�1𝑖𝑖𝑖𝑖) Intercept 69.9894 (5.483400) DMAP2(mm) 0.000058 (0.000040) DMAT (°C) -4.1070 (1.211100) Shape 1 (𝜃𝜃�2𝑖𝑖𝑖𝑖) Intercept 0.01348 (0.000981) Shape 2 (𝜃𝜃�3𝑖𝑖𝑖𝑖) Intercept 0.4851 (0.008780) 𝜎𝜎𝑏𝑏10𝑖𝑖𝑖𝑖2 44.6917 (8.864500) Temporal correlation 𝑆𝑆𝑝𝑝(𝑝𝑝𝑙𝑙𝑝𝑝),𝜌𝜌 0.9377 (0.0089460) Residual variance (𝜎𝜎𝜖𝜖2) 0.06687 (0.008118) aDMAT is the mean daily temperature distance and DMAP is mean annual precipitation distance between the site and the progeny. 4.3.2 Forecast Accuracies Forecast accuracies for progenies for which no measures were used in model fitting (i.e., “new” progenies) and for progenies for which a subset of prior measures were used model-fitting were calculated using the selected NLMM model (Tables 4-5 and 4-6). For new progenies, the forecast accuracy was best using four prior measures to obtain subject-specific forecasts (RMSPE =0.9231 m, MAPE= 0.6933 m). However, given only two prior measures, the forecast accuracy for new progenies was better using the autocorrelation part of the selected NLMM model. In fact, using only two prior measures, forecasts using the subject-specific random effects 103 were worse than using the population-averaged forecasts. Population-averaged forecasts improved slightly also, when more repeated measures on a progeny were included in the model-fitting data (RMSPE=1.002 m for two prior measures versus RMSPE=0.9737 m for four prior measures (Table 4-5). Overall, given two prior measures for a progeny used in model fitting, or for a new progeny not used in model fitting, using the autocorrelation model was best, whereas given four prior measures, the subject-specific forecasts were best. Table 4-5. RMSPEs (m) for plantation ages 27 and 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) Four prior measures (2, 5, 8 and 12) 0.9811 0.9231 0.9553 Two prior measures (2 and 5) 0.9862 0.9793 Progenies where a subset of prior measures were used in model fitting Four prior measures (2, 5, 8 and 12) 0.9737 0.9341 0.9500 Two prior measures (2 and 5) 1.0002 1.0087 0.9980 104 Table 4-6. MAPEs (m) for plantation ages 27 and 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) Four prior measures (2, 5, 8 and 12) 0.7417 0.6933 0.6961 Two prior measures (2 and 5) 0.7434 0.7370 Progenies where a subset of prior measures were used in model fitting Four prior measures (2, 5, 8 and 12) 0.7224 0.6932 0.6763 Two prior measures (2 and 5) 0.7463 0.7500 0.7407 To further indicate improvements using more prior measures, validation statistics for forecasts to plantation age 42 years using all other prior measures were obtained using the selected NLMM model (Tables 4-7 and 4-8). For comparison, forecasts to plantation age 42 years only using four or two prior measures were also included. Using progenies where subsets of prior measures were used in model fitting, there was only a very small improvement in the population-averaged forecasts when more prior measures were included. Substantial gains in the forecast accuracies were obtained using all available prior measures and the subject-specific forecasts, with the RMSPEs reduced to about 0.5 m. This is similar to indirect height measurement accuracies using older devices (e.g., tape measure for distance along with a clinometer for angles; Hendee et al. 2012; West 2015). There was also a gain in accuracy using more prior measures and the 105 autocorrelation forecasts, but this was less than using subject-specific forecasts (RMSPE of about 0.9 m and 0.5 m, respectively, Table 4-7). Table 4-7. RMSPEs (m) for plantation 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.3112 0.5049 0.9612 Four prior measures (2, 5, 8 and 12) 1.2782 1.3026 Two prior measures (2 and 5) 1.3265 1.3117 Progenies where a subset of prior measures were used in model fitting All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.2121 0.5117 0.8990 Four prior measures (2, 5, 8 and 12) 1.3040 1.2975 1.2969 Two prior measures (2 and 5) 1.3400 1.3602 1.3403 106 Table 4-8. MAPEs (m) for plantation age 42 years only using population-averaged, subject-specific, or autocorrelation forecast methods and the selected NLMM model (Model I). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.0620 0.4195 0.7711 Four prior measures (2, 5, 8 and 12) 1.0524 1.0574 Two prior measures (2 and 5) 1.0708 1.0627 Progenies where a subset of prior measures were used in model fitting All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 0.9544 0.4234 0.7053 Four prior measures (2, 5, 8 and 12) 1.0305 1. 0296 1.0292 Two prior measures (2 and 5) 1.0688 1.0942 1.0695 Using the same validation approach, forecast accuracies were obtained for the less accurate NLMM model (Model III) (Tables 4-9 and 4-10). For a new progeny, the population-averaged forecasts using the less accurate NLMM showed about a 16% higher RMSPE and a 15% higher MAPE (Tables 4-9 and 4-10) relative to the selected NLMM model (Tables 4-7 and 4-8). As expected, using the autocorrelation forecasts in all cases resulted in greater improvements than using the population-averaged forecasts with this less accurate NLMM model than when the 107 more accurate selected NLMM model was used. The subject-specific forecasts showed the best RMSPE and MAPE and were much less affected when using simplified model (Model III) compared to other forecasting methods. Table 4-9. RMSPEs (m) for plantation age 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the less accurate NLMM model (Model III). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.5196 0.5720 1.1324 Progeniess where a subset of prior measures were used in model fitting All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.3302 0.5379 1.0128 108 Table 4-10. MAPEs (m) for plantation age 42 years using population-averaged, subject-specific, or autocorrelation forecast methods and the less accurate NLMM model (Model III). Available prior measures (plantation ages in years) Population-averaged forecasts Subject-specific forecasts Autocorrelation forecasts New progenies (i.e., prior measures not used in model fitting) All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.2179 0.4592 0.8824 Progenies where a subset of prior measures were used in model fitting All ages less than 42 (five prior measures for Aleza and six for other two planting sites) 1.0663 0.4477 0.8013 4.3.3 Demonstration of Alternative Forecast Methods As a demonstration of the forecast methods assessed in this study, the ‘Fraser-Fort George10’ progeny was selected as a new progeny not used in model fitting. All other progenies and prior measures were used to refit the selected NLMM model. Then, all seven prior measures including the measure at plantation age 42 years were used in the subject-specific and autocorrelation forecasts. Forecasts were then obtained for plantation ages 43 to 80 years (Figure 4-2). 109 Figure 4-2. Observed (≤ plantation age 42 years) and forecasted (plantation ages 43 to 80 years) average heights for the ‘Fraser-Fort George10’ progeny at the Quesnel planting site. The population-averaged forecast results in a “drop” at plantation age 43 years, since this represents the average for all progenies with the same set of climate distance and other variables using in Eq. 4.1 to 4.1c. As noted, the prior measures were not included in model-fitting; a similar drop was obtained using population-averaged forecasts when these measures were included in model fitting (not shown). Using the autocorrelation forecast prevents this drop at age 43 years; however, following this, the trend is much flatter and approaches the population-averaged forecasts. The subject-specific forecasts also start with values similar to those at plantation age 42, and then the forecasts appear to follow the trend represented by the prior 110 measures more closely. Although only a demonstration of forecast differences, these forecasts reflect the forecast accuracies obtained via validation. 4.4 Discussion The impacts of tree improvement programs on hybrid spruce yields is difficult to assess since the time from plantation establishment to harvest is commonly 80 or more years, whereas most provenance or progeny trials in Canada have been monitored for much shorter time periods. As a result, an accurate forecasting method is needed for more informed selection of provenances or progenies. In this study, three alternative forecasting methods were evaluated using data from hybrid spruce progenies of BC. Since the available number of repeated measures would likely affect forecast accuracies, these were virtually altered to examine effects on outcomes for alternative forecasting methods. Also, since forecast accuracies across these forecasting techniques are likely dependent on the accuracy of the fixed-effects part of the model, forecasts were compared to a less accurate NLMM model. In general, subject-specific and autocorrelation forecasting methods showed more accurate forecasts than using population-averaged forecasts. Specifically, the subject-specific forecasting method was the most accurate when more repeated measures were available. Using five prior measures, the accuracy of average height forecasts was similar to that obtained for heights measured with standard equipment. When there were a limited number of prior measures, the autocorrelation forecasts were more accurate than the subject-specific forecasts, but were less accurate than using common height-measuring devices. Also, as expected, the accuracy of the fixed-effects part of the model did affect forecast results. Greater improvements over the population-averaged forecasts were obtained for both NLMM 111 models; however, forecasts were less accurate overall when using a less accurate fixed-effects part of the model. In an earlier alternative to the model forecast approaches uses in this study, Lambeth (1980) proposed a “correlated gain” approach, where the correlation between the gain at one plantation age relative to another plantation age was used in a simple linear model. Initially, Lambeth suggested that this correlated gain should be independent of species and sites. However, in a later paper, Lambeth and Dill (2001) showed that the parameters of the correlated gain model change with planting sites. This correlated gain approach has been used in a number of subsequent papers to estimate gain and then incorporate this gain into growth and yield models to forecast yields of improved stock (e.g., Di Lucca 1999; Xie and Yanchuk 2002; Chen et al. 2003; Xie and Yanchuk 2003). The correlated gain approach has similarities with the autocorrelation forecasts used in this paper, with the main difference being that multiple prior measures in time can be used in the autocorrelation forecast. The use of autocorrelation forecasts has a long history, particularly in economics modelling (Judge et al. 1988). This approach has also been used to forecast permanent sample plots in forests and plantations (e.g., Gregoire 1987). In fact, with the advent of improved software for fitting mixed-effects models, this approach has become relatively common for forecasting permanent sample plot yields (e.g., Gregoire and Schabenberger 1996; Hall and Bailey 2001; Jansson et al. 2003; Meng and Huang 2009; Yang and Huang 2011; Meng et al. 2012). However, few authors have used this approach to forecast provenances or progenies in provenance or 112 progeny trials. Magnussen and Yanchuk (1994) used a linear mixed-effects model for predicting breeding values and accounted for the autocorrelation among repeated measures; their forecasts were based on population-averaged values only. No research was found on using autocorrelation forecasts for provenance or progeny trials as employed in this study Although the autocorrelation approach has a long history, the use of subject-specific forecasts can provide improved estimates when prior measures of the particular subject are available (Pinheiro and Bates 2000). Improvements in permanent sample plot forecasts have been shown when prior measures were used in model fitting (e.g., Fang and Bailey 2001; Wang et al. 2007; Meng et al. 2009; Meng and Huang 2010). Subject-specific estimates can also be obtained for a new permanent sample plot with repeated measures not used in model fitting. Fang and Bailey (2001), Calama and Montero (2004) and Wang et al. (2007) used the approach by Vonesh and Chinchilli (1997) where estimated random effects are obtained by setting 𝒃𝒃𝑞𝑞𝑞𝑞 = 0 in the right hand side of the Eq. 4.4 to simplify the estimation procedure, and only the fixed-effects part of the model are included in the derivative matrix 𝒁𝒁𝑞𝑞𝑞𝑞. Meng and Huang (2009) noted that resulting predictions can be poor although this procedure can provide good initial estimates of random effects. Instead, they used a first order Taylor series expansion at 𝒃𝒃𝑞𝑞𝑞𝑞 = 𝒃𝒃𝑞𝑞𝑞𝑞∗ (EBLUP expansion) followed by an iterative procedure as described in Lindstrom and Bates (1990). This later approach has been also used by a number of authors including Temesgen et al. (2008) for subject-specific Douglas-fir tree heights, Mehtätalo et al. (2015) for subject-specific tree heights for a wide range of species, and Sirkiä et al. (2015) for subject-specific dominant heights. As with the autocorrelation approach for forecasting, no prior research papers were found on using subject-specific forecasts for provenance or progeny trials. In this study, subject-specific 113 forecasts showed improvements over the population-averaged forecasts. The more accurate procedure recommended by Meng and Huang (2009) was used to obtain the subject-specific estimates for a progeny not included in the model-fitting data. A larger number of prior measures is essential for getting accurate subject-specific predictions (Pinheiro and Bates 2000) and has also been shown to affect forecasts using an autocorrelation approach. Meng et al. (2009) found that using one prior measure of lodgepole pine provided much less accurate height forecasts than using all available prior measures for subject-specific forecasts. As in their study, Meng and Huang (2010) compared subject-specific and autocorrelation approaches using differing numbers of prior measures. As expected, subject-specific effects were better estimated using more prior measures. Using the autocorrelation approach, they found that the choice of autocorrelation models affected the results. Specifically, since their data included repeated measures in time that were almost regularly spaced, they compared the use of the SP(POW) error structure to the TOEP(X) error structure and found greater improvements using the TOEP(X) error structure. However, their results are not applicable to circumstances where measures are irregular in time as is often the case for permanent sample plots and for provenance or progeny trials. In their study forecasting the average heights of permanent sample plots, Meng et al. (2009) used six alternative nonlinear fixed-effects components. They found that the accuracy of the NLMM model depended on the fixed-effects part of the model. The results in our study supported their finding in that the accuracy of the fixed-effects part of the NLMM model affected the accuracies 114 of progeny forecasts. Greater improvements were obtained using the subject-specific or autocorrelation forecasts over the population-averaged forecasts with the less accurate model; however, forecasts were less accurate than using the more accurate model. The research presented in this study compares alternative approaches to forecast progeny yields, which, to my knowledge, is novel. Although these approaches were tested using hybrid spruce progenies in a limited number of locations, similar results would be expected for other locations and species based on prior literature using permanent sample plots and based on the long-standing research behind autocorrelation and subject-specific forecasts. Further, these approaches could be applied to other phenotypic characteristics, including volume and biomass per unit area. These forecasts can fill a knowledge gap critical to evaluating alternative provenances or progenies for planting selection. 4.5 Conclusions Alternative approaches for forecasting average heights of progenies on a particular planting site were evaluated in this research. Using available prior measures resulted in improved forecasts both for progenies where these data were used in fitting the model and for progenies with repeated measures that were not used. Forecasts were more accurate using subject-specific forecasting, unless only a few prior measures were available. In that case, autocorrelation forecasts performed better. Forecast accuracies were negatively impacted by having a less accurate fixed-effects part of the NLMM model, indicating the importance of careful selection and testing of the fixed-effects part of the model. However, the subject-specific forecasts were 115 comparatively much less affected when a less accurate fixed-effects part of the model was used, since this approach may partly compensate for progeny and site variables not included in the fixed-effects part of the NLMM model. Overall, these forecast methods can provide accurate forecasts for provenances or progenies at a particular site leading to better information when selecting amongst provenances or progenies for plantations. 116 5. Conclusions 5.1 Contributions to Knowledge 5.1.1 Summary In this research, knowledge gaps concerning the impacts of tree improvement programs on timber supply were narrowed by modelling and forecasting yields of white spruce and hybrid spruce in the boreal and hemiboreal forests in Canada using a comprehensive compilation of available meta-data from provenance and progeny trials. The major contribution of this research is providing a mechanism for estimating the yields at harvest of improved spruce stocks over a large spatial and temporal extent. Since improved stocks could be used to ameliorate possible timber supply shortages in the boreal and hemiboreal forests of Canada, this mechanism could be used to evaluate timber management options. The particular contributions towards narrowing the knowledge gap were: i. A meta-dataset of white and hybrid spruce provenance and progeny trials across the boreal and hemiboreal forests of Canada was developed (described in Chapter 2). ii. A mixed-effects nonlinear model (NLMM) to predict height yields of provenances in planting sites in the boreal and hemiboreal forests in Canada was developed (Chapter 2). iii. Using a random coefficients modelling approach, the effects of climatic variables, provenances and site characteristics on the height trajectory were examined (Chapter 2). iv. An existing growth and yield model (Prégent et al. 2010) was modified to predict provenance yields for long temporal and large spatial extents (Chapter 2). 117 v. A definition of gain was selected via sensitivity analyses; this definition could be applied to any planting site in the boreal and hemiboreal forests in Canada thereby facilitating comparisons (Chapter 3). vi. A meta-model of height gain was developed using the gain definition; this could be used to evaluate the yield potentials of improved white and hybrid spruce stocks for a planting site or larger spatial extent (Chapter 3). vii. Forecast methods to forecast the repeated average height measures of a particular provenance at a location were compared (i.e., population-averaged, subject-specific, and autocorrelation) and the subject-specific approach was recommended (Chapter 4). viii. Additional evidence was obtained concerning the negative impacts on forecast accuracies given a less accurate fixed-effects part of the model; however, these impacts are largely mitigated if a larger number of repeated measures is available and these are used to obtain subject-specific forecasts (Chapter 4). 5.1.2 Chapter 2: Research Conclusions and Implications A meta-modelling approach was used in the first study of this dissertation to develop meta-models of average height trajectories of white and hybrid spruces across the boreal and hemiboreal forests of Canada. The process used was: (i) a meta-dataset was developed using published summaries of tree improvement trials; and (ii) models were then developed using these summary data. The Chapman-Richard growth model (Pienaar and Turnbull 1973) was used as a base to model the average trajectory. The effects of provenance climate, planting site climate, site elevation, provenance elevation and planting site density on height trajectories were then 118 examined and incorporated into the meta-model using a random coefficients modeling approach (aka, parameter prediction approach; Clutter et al. 1983, Schabenberger and Pierce 2002, Littell et al. 2006) as used by McLane et al. (2011) and Leites et al. (2012). Separate meta-models were fitted for the boreal forest, and then for the boreal and hemiboreal forests of Canada combined. Both meta-models were strongly affected by planting site and provenance climates, and by planting density, site elevation and provenance elevation. The average height trajectory meta-model for the boreal and hemiboreal forests was incorporated into an existing growth and yield model (Prégent et al. 2010). This modified model can be used to predict the yield of any provenance on any planting site in the boreal and hemiboreal forests of Canada to harvest age. These study results are being applied by affiliate members of GE3LS team in the SMarTForests project to evaluate possible economic benefits of the white and hybrid spruces tree improvement programs in Canada. 5.1.3 Chapter 3: Research Conclusions and Implications The definition of gain varies in reports on tree improvement program trials. In the second study, the definition of gain was assessed using sensitivity analysis altering the evaluation age (i.e., the age when top performers are selected), along with the definitions of baseline and top performers. From this analysis, the average of the top 15% of the provenances determined at 15 years was selected to define the top performers relative to the average of all provenances (i.e., 100%). Using the meta-model developed in the first study and this gain definition, a meta-dataset of the gain for plantation ages 1 to 45 years by planting site was developed. Using this gain meta-dataset, a gain trajectory model was fitted for white spruce and hybrid spruce provenance trials across the Canadian boreal and hemiboreal forests. For this gain trajectory model, the planting 119 site mean annual daily temperature, mean annual precipitation, and the number of degree days > 5°C had large impacts. This gain trajectory model can be used to predict gain to harvest age for white and hybrid spruce on any planting site in the boreal and hemiboreal forests of Canada. Further, these gain trajectories could be averaged over a region as an indication of yield potential of the white spruce and hybrid spruce tree improvement programs. 5.1.4 Chapter 4: Research Conclusions and Implications In the third study, approaches to forecast the average height trajectory beyond the available repeated measures for a particular provenance at a planting site were compared. Specifically, the best approach was selected for forecasting hybrid spruce progenies using data for three sites in BC. Generally, subject-specific forecasts provided more accurate forecasts than using the autocorrelation method, which, in turn, provided more accurate forecasts than the population-averaged approach. The subject-specific approach showed noticeable improvements over other approaches when the number of available prior repeated measures was increased. All forecasting methods were negatively affected when a less accurate NLMM model was used, but these effects were considerably less when the subject-specific forecasting method was used. Based on these results, I recommend using the subject-specific approach which gave accurate forecasts even for the less accurate population-averaged model and only a few prior measures. 120 5.2 Limitations As noted earlier, a major limitation in accessing the success of tree breeding programs in Canada is the lack of information on provenance (and progeny) to harvest age and also information is only available for a limited spatial coverage. Although the studies in this dissertation reduced this knowledge gap via meta-modelling and forecasting provenances (or progeny), the meta-dataset used in the first study collected from all available sources extended to plantation age 44 years only. Similarly, the spatial distribution of planting sites was limited to the southern portions of several Canadian provinces (i.e., British Columbia, Quebec, Alberta, Ontario and Newfoundland). Another limiting factor was that summary reports for provenance trials were given for a single plantation age only, whereas repeated measures are needed to reflect real height trajectories. Finally, the climate transfer distance variables were calculated using spatial interpolation from the closest weather stations for provenances and planting sites, and these stations are often quite distant given the vast forest area of Canada. However, the meta-data developed in the first study did cover a larger spatial and temporal scale than that used in the very limited number of prior studies on this topic. Since the height trajectory model developed in the first study was used to obtain the gain trajectory meta-dataset for the second study, the limitations and advantages of the first study also impact on the second study. For the third study on forecasting methods, only three planting sites from the hemiboreal forest in Canada were used. Although a broader spatial range would have been advantageous for that study, results might be expected to be similar since they were supported by fundamental theories on forecasting and by prior research studies on forecasting permanent sample plots. 121 5.3 Future Research Other than white and hybrid spruces, tree improvement programs on various species are ongoing in Canada. The major species involved include: black spruce (Picea mariana (Mill.) B.S.P.), jack pine (Pinus banksiana Lamb.), Norway spruce (Picea abies (L.) Karst.), and Engelmann spruce (Picea engelmannii Parry) (Carlisle 1970; Newton 2003; McInnis and Tosh 2004). To analyze the growth and yield potential of these species at a larger scale, a meta-database could be developed for each species using the published summaries of tree improvement trials similar to the meta-modelling approach followed in this research. In this dissertation research, the impacts of improved stocks on only one major component of growth and yield models (i.e., height) were directly modelled. Although this could be used in an existing growth and yield model to indirectly predict yields of improved stocks as shown in the first study, this assumes that altering the heights alone is sufficient for accurately predicting yields of other components (i.e., volume per ha, average diameter, basal area per ha). A meta-dataset that includes information on these other components would be needed to test this assumption. Since these data are not often reported, obtaining this information would require contact with each researcher to obtain either summary data or the original data from which summaries can be made. Another area for further research is examining changes in survival and risk of failure in using improved compared to regular stocks. The risk of failures using stocks from tree improvement programs can be higher than using natural stocks because of lower genetic heterogeneity (White 122 et al. 2007). Conversely, in the selection process of tree breeding programs, parent trees can be selected because of their tolerance to frost and drought, and/or resistance to insects and diseases resulting in lower risks of plantation failures (Zobel and Talbert 1984; White et al. 2007). 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Impacts of tree improvement programs on yields of white spruce and hybrid spruce in the Canadian boreal… Ahmed, Suborna Shekhor 2016
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Title | Impacts of tree improvement programs on yields of white spruce and hybrid spruce in the Canadian boreal forest |
Creator |
Ahmed, Suborna Shekhor |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Local and regional timber shortages may be ameliorated via planting improved stocks with higher yields. In this dissertation, I addressed an important knowledge gap on the impacts of tree improvement programs on yields of white spruce (Picea glauca (Moench) Voss) and hybrid spruce (Picea engelmannii Parry ex Engelmann x Picea glauca (Moench) Voss) plantations across the boreal and hemiboreal forests of Canada using meta-modelling approaches. In particular, I used meta-data for white and hybrid spruce provenance trials extracted from the literature to: (i) forecast provenance yields over time for broad spatial and temporal extents; (ii) model yields of provenances relative to standard stocks (termed “gain”) over time; and (iii) test alternatives for forecasting each provenance at a location using available repeated-measures data. In the first study, provenance height over time trajectories were modelled by incorporating the effects of climatic variables, provenances and site characteristics into mixed-effects nonlinear models via a random coefficients modelling approach. Height trajectories were strongly affected by planting site and provenance climates, along with planting site characteristics. The height trajectory meta-model was incorporated into an existing growth and yield model, which can be used to predict provenance yields for long temporal and large spatial extents. In the second study, the impacts of the particular gain definition (i.e., selection age, proportion of top performers) were examined using the model from the first study, and one definition was selected. A meta-model of gain as a function of plantation age, planting density, and planting site climate was developed. Planting site climate strongly affected these gain trajectories. The gain definition and trajectory model can be used to evaluate potential gains of using improved white and hybrid spruce stocks. Forecasts are needed to evaluate provenance (or progeny) performance at harvest, often 80 or more years from planting. In the third study, three alternative procedures (population-averaged, subject-specific, and autocorrelation) to forecast repeated measures for a particular progeny at a location were compared and evaluated by virtually removing some repeated measures. The subject-specific forecasts were best with accuracies similar to the measurement precision using standard height measurement devices given five or more prior measurements. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-10-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0319209 |
URI | http://hdl.handle.net/2429/59494 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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