Modelling stem selection in northern hardwood stands: assessing the effects of tree vigour and spatial correlations using a copula approach

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1 Forestry Advance Access published September 11, 2014 Forestry An International Journal of Forest Research Forestry 2014; 0, 1 11, doi: /forestry/cpu034 Modelling stem selection in northern hardwood stands: assessing the effects of tree vigour and spatial correlations using a copula approach Simon Delisle-Boulianne 1, Mathieu Fortin 2,3, Alexis Achim 1,4 and David Pothier 1 * 1 Centre d étude de la forêt, Département des sciences du bois et de la forêt, Faculté de foresterie, de géographie et de géomatique, Université Laval, 2405, rue de la Terrasse, Québec, QC, Canada G1V 0A6 2 AgroParisTech, Centre de Nancy, Laboratoire d Etude des Ressources Forêt-Bois (LERFoB, UMR 1092), 14 rue Girardet, Nancy, France 3 INRA, Centre de Nancy Lorraine, Laboratoire d Etude des Ressources Forêt-Bois (LERFoB, UMR 1092), 1 rue de l Arboretum, Champenoux, France 4 Centre de recherche sur les matériaux renouvelables, Université Laval, 2425, rue de la Terrasse, Québec, QC, Canada G1V 0A6 *Corresponding author. david.pothier@sbf.ulaval.ca Received 14 April 2014 Growth and yield simulators are often used to assess the long-term sustainability of selection cuttings in northern hardwood stands. Because residual tree growth is affected by tree removal, growth simulators must be coupled with a harvest model to adequately predict future stand development. Harvest models may consider variables such as species, diameter at breast height (DBH), spatial coordinates of the trees and indicators of stem quality and vigour. Through the development of a harvest model applicable to selection cuttings in northern hardwood stands, we aimed to quantify the loss of accuracy caused by the omission of either tree spatial coordinates or tree quality/vigour descriptors from the predictor variables. Spatial correlations were handled using both generalized linear mixed-effect models and a copula approach. Tree vigour and quality were assessed using three different classification systems used in practice. The inclusion of tree vigour and spatial correlations through the copula approach led to a 12 per cent improvement in maximum log-likelihood. However, the inclusion of the spatial coordinates of each tree only accounted for one quarter of this 12 per cent. In cases where a growth simulator uses tree vigour class as input, we recommend the inclusion of this variable in the associated harvest sub-model. Introduction Northern hardwood forests cover large areas in southern Canada and the northern US. They extend from the Great Lakes region to the Atlantic Ocean and make a transition with the oak-hickory forest to the South and with the boreal forest to the North (Bailey, 1983). These forests are located around densely populated areas and are composed of valuable species used by the appearance wood products industries (furniture, flooring, cabinets, interior finishing, etc.), giving them considerable socio-economic importance. Sugar maple (Acersaccharum Marsh.) often dominates these forests together with yellow birch (Betula alleghaniensis Britt.), American beech (Fagus grandifolia Ehrh.) and several other hardwood and conifer species in varying proportions. Their shade tolerance allows these species to regenerate after small-scale mortality events, which are relatively frequent (return cycle of years) in northern hardwood forests compared with large-scale disturbances (return cycle of years; Lorimer and Frelich, 1994; Seymouret al., 2002).Such spatio-temporal gap dynamics generally produces multi-cohort stands with an uneven-aged structure, which are mainly harvested using the selection cutting system according to the principles of ecosystem-based management (Nyland, 1998; Seymour et al., 2002). From an ecological perspective, selection cutting may thus be viewed as a sustainable silvicultural system for northern hardwood stands. However, this brings important challenges from an industrial point of view since at each cutting cycle the harvested trees must be of sufficient quality to supply the appearance wood products industry. At the same time, the treatment must also leave enough vigorous residual trees to ensure the sustainability of the wood supply in future cutting cycles. Leaving a large proportion of low-vigour trees can affect future stand yield because of the likely increase in mortality (Fortin et al., 2008; Guillemette et al., 2008). In addition, a systematic bias towards the harvesting of large, high-value trees can affect the diameter distribution of the residual stand and, in turn, future growth. To avoid the depletion of the stand, it is therefore recommended that trees are evenly harvested across all diameter classes and that a balance is maintained between the quality of harvested stems and the vigour of residual trees (Nyland, 1998; Pothier et al., 2013). Unlike in regions with a long history of sustainable management (Favre and Oberson, 2002; Ciancio et al., 2006), there is limited empirical knowledge about the effects of silvicultural practices on the growth of northern hardwood forests. Foresters often have to rely on growth and yield simulators to assess the long-term sustainability of different silvicultural practices. The growth and yield # Institute of Chartered Foresters, All rights reserved. For Permissions, please journals.permissions@oup.com. 1of11

2 Forestry simulators used for northern hardwood stands are typically designed to make tree-level predictions and are composed of submodels that dynamically and interactively account for tree growth and mortality, as well as recruitment (e.g. Kiernan et al., 2008; Fortin and Langevin, 2012). Because each of these components is affected by tree removal, future partial harvesting must also be considered in long-term simulations (Fortin, 2014). Trees to be harvested in simulated silvicultural treatments can be identified according to their current characteristics through a statistical harvest sub-model (e.g. Thurnher et al., 2011), which can then be coupled with the growth simulator. To assess whether the potentially competing objectives of promoting future growth and meeting current wood supply objectives have been adequately balanced, harvest models may additionally consider indicators of stem quality and vigour as well as species and diameter at breast height (DBH). Since harvesting is also likely to follow a spatial pattern (e.g. Arii et al., 2008), further information about the spatial coordinates of each tree could help refine descriptions of the growing conditions of residual trees. However, tree coordinates or tree-level assessments of vigour and quality are rarely available from conventional forest inventories. This paper presents a harvest model developed from detailed stand inventory data collected immediately before and after the application of a selection cut in northern hardwood stands. The modelling steps were designed to test the hypothesis that ignoring variables used to inform tree marking decisions, such as the spatial coordinates of neighbouring trees or descriptors of the quality and/ or vigour of subject trees, would result in a significant loss of model accuracy. This was achieved by successively fitting harvest models with different levels of detail and comparing the maximum likelihood of the resulting models. We then discuss the silvicultural implications of the main patterns highlighted by this modelbuilding process. Material and methods Study area The study area is located northwest of Montréal in the province of Québec, Canada (46829 N, W), in a region characterized by rounded hills with low-to-moderate slopes and a mean elevation of 350 m (Robitaille and Saucier, 1998). The main surface deposit is thin glacial till with occasional rockyoutcrops. The climate is continental, with a mean annual temperature of 2.58C and mean annual precipitation of 930 mm, of which 320 mm falls as snow. The growing season is relatively short but warm, with an average duration of 150 days and a total of 2400 degree days. The study area is part of the western sugar maple yellow birch bioclimatic subdomain (Robitaille and Saucier, 1998). Forest stands are mainly uneven-aged and dominated by sugar maple, American beech, yellow birch and balsam fir (Abies balsamea [L.] Mill.). Since the mid-twentieth century, these publicly owned forests have mainly been managed using partial cuts of varying intensities carried out by private companies under the control of government authorities. Provincial forest maps indicated that stands of the study area had not been affected by natural or anthropogenic disturbances since the end of the 1970s. Selection cut During the fall of 2009, a fully mechanized selection cut was conducted as part of normal forest operations. Following standard practice for this silvicultural treatment (MRNF, 2004), the objective of the selection cut was to reduce stand basal area to a minimum of 16 m 2 ha 21, while increasing stand vigour. Trees in the study area were hence operationally marked for harvesting, using a classification system that prioritized the removal of defective trees (Boulet, 2007 see the full description in a further section). However, because of a significant market downturn for some species at that time, special measures were put in place to increase the profitability of the treatment. Some trees marked for harvesting were hence left standing. These included the following: (1) cull trees with absolutely no sawing potential, (2) 50 per cent of American beech and red maple (Acer rubrum L.) trees, irrespective of stem quality and (3) all eastern white cedar (Thuja occidentalis L.), eastern hemlock (Tsuga canadensis (L.) Carrière) and basswood (Tilia americana L.) trees. Plot measurements After the trees were marked, and prior to harvesting, a total of 55 randomly distributed sample plots were established over the study area (Figure 1). Within each circular 900-m 2 plot, all living trees with a DBH (1.3 m above ground) of at least 9.1 cm were identified and measured. The number of trees in each plot ranged from 26 to 62, with an average of 46. The position of each individual tree was recorded in the polar coordinate system. Distance and azimuth angle were measured from the plot centre using a hypsometer and a theodolite. Besides recording the species and DBH, each tree was assessed according to three separate classification systems described in the following section. Because some species were scarce in the dataset, they were grouped with others based on their silvicultural and ecological characteristics. This grouping resulted in nine species groups, which had the added advantage of simplifying the statistical analyses (Table 1). The mean merchantable basal area prior to harvest was 27 m 2 ha 21. Despite the special measures allowing for some marked trees to be left standing, removal still reached an average of 7.3 m 2 ha 21, or 27 per cent of the initial basal area. In total, 493 out of the 2507 sampled trees were harvested, which represents an average of nine trees per plot, with a minimum of 1 and a maximum of 20. Tree classifications The first classification system, which will be referred to as the V-system, focuses only on tree vigour. Its four vigour classes are assumed to be closely related to the probability of mortality during the upcoming cutting cycle, i.e. a 25-year period (Boulet, 2007). The distinction among classes is based on the presence and severity of certain types of defects. These are grouped into eight categories: (1) conks and stromata, (2) cambium necrosis, (3) stem deformations and injuries, (4) stem base and root defects, (5) stem and bark cracks, (6) woodworms and sap wells, (7) crown decline and (8) forks and pruning defects. Dichotomous keys basedon these various defects are used to assign any tree to one of the four following classes: M: trees with major defects that will likely die during the next cutting cycle. S: trees with major defects that do not compromise their likelihood of survival in the medium term. C: trees with some minor defects and that are still growing. R: trees almost free from anydefects that should be preserved for the future. Growth-driven survival probabilities tended to confirm that trees from the lowest and highest vigour classes had different survival probabilities (Hartmann et al., 2008). However, in the absence of long-term monitoring results, a complete validation of this classification system has not yet been achieved. Nevertheless, the V-system is currently being used to guide tree marking decisions on public land in the province of Québec. The second classification system, referred to as the Q-system, focuses on the assessment of stem quality for trees with a DBH of atleast 23.1 cm. It comprises four categories, which aim to predict log grade and sawing volume recovery (Monger, 1991). The best 3.7-m section within the lower 5 m of each stem is visually assessed. It is then separated into four virtual faces, which are ranked according to the presence and extent of visible 2of11

3 Modelling stem selection in northern hardwood stands Figure 1 Location of the sample plots within the study area. Table 1 Species groups used for the harvest model Species group Abbreviation Grouped species Sugar maple SM Sugar maple (Acer saccharum Marsh.) Yellow birch YB Yellow birch (Betula alleghaniensis Britt.) American beech AB American beech (Fagus grandifolia Ehrh.) Red maple RM Red maple (Acer rubrum L.) Ironwood IW Ironwood (Ostrya virginiana (Mill.) K. Koch) Other valuable hardwoods OH Northern red oak (Quercus rubra L.), black ash (Fraxinus nigra Marshall), white pine a (Pinus strobus L.) and basswood (Tilia americana L.) Short-lived hardwoods SH White birch (Betula papyrifera Marshall), pin cherry (Prunus pensylvanica L.f.) and bigtooth aspen (Populus grandidentata Michaux) Balsam fir BF Balsam fir (Abies balsamea [L.] Mill.) Long-lived conifers LC White spruce (Picea glauca (Moench) Voss), eastern hemlock (Tsuga canadensis (L.) Carrière) and eastern white cedar (Thuja occidentalis L.) a White pine was grouped with the other valuable hardwoods because it is used in the appearance product industries and is subjected to the same silvicultural prescription. defects. A quality class is finallyattributed to each tree, according to its DBH, the length of the clear (defect-free) bole on the third-best face and deductions for stem sinuosity, curvature and an estimated proportion of decay. The four classes, namely, A, B, C and D in decreasing order of quality, are used to estimate the wood volume allocations for different types of wood processing industries (i.e. veneer, lumber or pulp and paper). There is a strong correlation between the four quality classes and the presence and volume of some log grades in standing trees (Fortin et al., 2009). The third classification system, referred to as the VQ-system, also assigns each tree to one of four categories. Unlike the previous two classifications, this system takes into account both tree vigour and stem quality, each being defined as a binary variable (Majcen et al., 1990). The vigour assessment determines whether or not a tree is associated with a high risk of mortality over the next 25-year period using simple criteria, such as the visual detection of decay, conks and stromata, cambium necrosis or significant crown dieback. A tree with a low risk of mortality belongs 3of11

4 Forestry Table 2 Number of trees before cutting for each species group within each classification system for all the 55 plots Species group s a SM AB YB IW RM BF OH SH LC DBH range [ ] [ ] [ ] [ ] [9.1 47] [ ] [ ] [ ] [ ] Total number of trees V-system R C S M Q-system A n.a n.a. B n.a n.a. C n.a n.a. D n.a. 2 1 n.a. N b n.a n.a. VQ-system I II III IV a Species indices used in statistical analyses. b N ¼ trees that were not classified, i.e. those with DBH smaller than 23.1 cm. to class I or II, whereas trees in classes III and IV have a high risk of mortality. Depending on tree vigour, odd-numbered classes (I or III) represent trees with sawlog potential, while classes II and IV represent trees with no sawlog potential. A particular tree is considered to have sawlog potential if it includes a 1.8-m log with a large-end diameter.23 cm and at least one defect-free face. Therefore, class Icorresponds to trees with high vigourand high quality (HV-HQ), class II to trees with high vigour and low quality (HV-LQ), class III to trees with low vigour and high quality (LV-HQ) and class IV to trees with low vigour and low quality (LV-LQ). The classification of sample trees within each system is summarized in Table 2. Statistical analyses Let us define i and j as the plot and the tree indices, respectively, such that i ¼ 1, 2,..., 55 and j ¼ 1, 2,..., p i, where p i is the number of trees in plot i. The post-harvest status of tree j in plot i follows a Bernoulli distribution y ij Bernoulli(p ij ) (1) where y ij takes the value of 1 if the tree has been harvested or 0 if it is still standing after harvesting and p ij is the probability of being harvested. This type of analysis is usually performed using a generalized linear model (cf., McCullagh and Nelder, 1989). The probability of being harvested is expressed through a link function, typicallya logit link function, as follows: ex ijb p ij = 1 + e x (2) ijb where x ij is a row vector of covariates for tree j in plot i and b is a column vector of unknown population parameters. The parameter vector b can be estimated using a maximum likelihood estimator. Modelling the probabilities of being harvested is challenging from a statistical standpoint, because the probabilities may exhibit a sinusoidal trend with respect to DBH. In this case, a segmented approach might yield a better fit than a traditional logistic model (Fortin, 2014). In the segmented approach, the idea is to have two quadratic segments that can be joined to fit the sinusoidal trend. The location of a join point remains subjective. However, it could be suspected that lumbermen would not behave the same depending on whether a particular stem was merchantable or not. Consequently, it seemed natural to distinguish non-merchantable stems from merchantable ones by setting the join point on the merchantable limit. In the province of Québec, the merchantable DBH limits are set to 23.1 and 9.1 cm for broadleaved and coniferous species, respectively. These values were selected as locations for the join points between segments. To achieve this, the covariate d ij was computed as d ij ¼ dbh ij and d ij ¼ dbh ij 2 9.1, depending on the species type. An additional binary variable, m ij, was also created to distinguish each segment. In summary, if d ij is positive, then m ij ¼ 1 and the tree is merchantable. Otherwise, d ij is negative and m ij ¼ 0. Using these two covariates, we designed a general model based on the model currently used in growth forecasts in the province of Québec (see Fortin, 2014). Our general model was defined as follows: eb 0+b 1,s+(b 2 +b 3 m ij)d ij+b 4 m ijd2 p ij = 1 + e b 0+b 1,s +(b 2 +b 3 m ij )d ij +b 4 m ij d 2 ij where s is the species group index as defined in Table 2. Model (3) converges at p ij = e b 0+b 1,s /1 + e b 0 +b 1,s when d ij ¼ 0, which corresponds to the join point of the two segments. Compared with the general model in Fortin (2014, Equation 2), the effect of d 2 ij is enabled only for merchantable stems. During preliminary trials, it was found that this effect was not significant for non-merchantable (3) 4of11

5 Modelling stem selection in northern hardwood stands stems and consequently, it was not kept in the model. Also, coniferous species are assumed to have a single segment since all coniferous stems with DBH 9.1 cm are merchantable by definition. This trend was confirmed by visual inspection. It can be reasonably expected that the probabilities of being harvested are spatially correlated due to the clustered nature of the data. To account for these possible correlations, we fitted two additional models: the first one using a random plot effect and the second one using a copula approach. The model with the plot random effect was specified as follows: eb 0+b 1,s+(b 2 +b 3 m ij)d ij+b 4 m ijd2 +b i p ij = 1 + e b 0+b 1,s +(b 2 +b 3 m ij )d ij +b 4 m ij d 2 ij +b i where b i is a random effect for plot i, which was assumed to be normally distributed with b i N(0, s 2 plot ). Model (4) belongs to the class of models known as generalized linear mixed models (GLMMs) and hereafter it will be referred to as such. It was fitted using a maximum likelihood estimator for both the parameter vector b and s 2 plot.the methodology behind the copula approach is described in detail in Fortin et al.(2013). Briefly, in such models, the likelihood function is slightly modified to include what is known as a copula. The copula accounts for the joint distribution of the responses within a given cluster, namely, a plot. The copula can also be modified to express this correlation asafunction of thedistance between the observations(bhat andsener, 2009). In the present case, the fixed part of the model was the same as for model (3). The copula used the following function to model the distancedependent correlations: u ijj = e dz ijj, where d is a parameter and z ijj is the Euclidian distance between trees j and j in plot i. For simplicity, the mathematical details related to the inclusion of the copula in the likelihood function are omitted here. Readers willing to learn more about this are invited to read Fortin et al.(2013)and Bhat and Sener (2009). In addition to these spatial considerations, we also tested the effect of the three classification systems described previously. The VQ, V and Q systems were alternately tested with these three modelling approaches, i.e. the general model, the GLMM model and the copula model, for a total of 12 combinations. Since all the fits relied on the same vector of response variables and maximum likelihood estimators, the 22 log-likelihood value, Akaike s information criterion (AIC) and the Bayesian information criterion (BIC) were used to discriminate between the models. Lower values of these statistics indicate a better fit (cf. Pinheiro and Bates, 2000, p. 84). Hosmer et al. (2013) also proposed a test to assess the goodness-of-fit, which is commonly referred to as the Hosmer Lemeshow test. This test consists of dividing the observations into 10 groups of even size. The differences between the expected and the observed events across the 10 groups are then used to calculate a x 2 statistic with 8 degrees of freedom. A significant probability for this statistic indicates a lack of fit. All GLMM models do not provide population-averaged predictions when the plot random effect is set to its expected value, i.e. zero. In fact, population-averaged predictions are obtained by integrating the prediction conditional on the random effect over the distribution of the random effect. In this study, we used a Gauss Hermite quadrature to approximate this integral as suggested in Fortin (2013). The expected numbers of events required in the Hosmer Lemeshow test were calculated using the corrected population-averaged predictions. Finally, the fit was also assessed visually. The observations were grouped into 2-cm diameter classes. Within each class, the predicted probabilities were averaged and compared with the observed proportion of harvested trees. Diameter classes with fewer than 20 observations were not included in those graphs because the observed proportions had large variances. Results For all modelling approaches (general, GLMM and copula), the inclusion of a standing-tree classification system improved the fit (4) Table 3 Fit statistics of the different harvest models Model 22 log-likelihood a AIC BIC Hosmer Lemeshow statistic (x 2 ) P. x 2 General V-system 1988 (9.0%) Q-system 2132 (2.4%) VQ-system 1990 (8.9%) GLMM 2148 (1.6%) V-system 1939 (11.2%) Q-system 2099 (3.9%) VQ-system 1955 (10.5%) Copula 2133 (2.3%) V-system 1922 (12.0%) Q-system 2085 (4.5%) VQ-system 1945 (10.9%) a Relative improvement with respect to the general model without tree classification appears in parentheses. statistics, with the best fits obtained using the V-system (Table 3). In all cases, the V-system improved the 22 log-likelihood value byat least 9 per cent when compared with the general model with no tree classification. The model including the VQ-system gave the second-best fits, with statistics that were very close to those of the models with the V-system. Finally, the Q-system yielded a moderate improvement of the model fit when compared with the general model. There was no evidence of lack of fit as indicated by the Hosmer Lemeshow test, except when the VQ-system was specified in any of the three models. The consideration of spatial correlations also improved the fit of the model. The assumption of a more-or-less constant spatial correlation using the GLMM approach resulted in an improvement of the 22 log-likelihood of 1.5 per cent. The copula approach, which assumed decreasing spatial correlations with an increasing distance between trees, yielded the best fits, with an improvement of more than 2 per cent compared with the general model. In the end, the best model included the V-system classification and the copula, which yielded a 12 per cent improvement in model fit. The average predicted probabilities and the observed proportions by 2-cm diameter classes are shown in Figure 2 for this model. The other copula models exhibited similar patterns. For the sake of simplicity and because they produced the best overall fits to the data, we will hereafter focus only on the copula models. The parameter estimates from the four fits are given in Table 4. Most parameter estimates were significantly different from zero according to a confidence interval a ¼ Except for parameters b 5, for which the specifications depend on the model, the parameter estimates were relatively constant among models. Of particular interest in the copula is the ˆd parameter, for which negative values indicate a decreasing correlation as the distance between the trees increases. All copula models indicated that the correlations followed this decreasing pattern. Using the parameter estimates given in Table 4, mean probabilities of being harvested were produced, as well as their 0.95 Waldbased confidence intervals (cf. Hosmer et al., 2013, p. 15). Some predictions of the copula model without any consideration of 5of11

6 Forestry tree classification are shown in Figure 3. The probability of a tree being harvested was affected by both the species and DBH, but the effect of these variables depended on whether or not the minimum merchantable DBH (23.1 cm for hardwoods) was Figure 2 Average predicted probabilities (dots) and observed proportions (triangles) of harvested trees by 2-cm DBH classes based on the copula model with the V-system. reached (Figure 3). The harvesting probabilities of the most abundant commercial hardwoods in this study (sugar maple, yellow birch, American Beech, red maple and ironwood) followed similar patterns as a function of DBH, i.e. a decreasing probability with an increase in DBH for pole-size trees (9.1 cm DBH, 23.1 cm) and an increase thereafter, so that a minimum removal probability was observed at the minimum merchantable DBH threshold (Figure 3). The harvesting probability of the other commercial hardwoods was low across all DBH values, particularly for pole-size trees. Balsam fir was the most harvested species, with harvesting probabilities approaching 90 per cent for the largest trees. Overall, these results indicate that there was a preference for harvesting large diameter trees in all species, but this trend was more apparent in sugar maple, American beech, yellow birch, red maple and balsam fir. The predictions of the copula model with the V-system are presented in Figure 4. The model predicted decreasing harvesting probabilities from class M to class R, i.e. from the least to the most vigorous trees. The confidence intervals around each class suggest that there were no statistical differences in harvesting probabilities between classes M and S, as well as between classes C and R. The inclusion of the Q-system in the copula model yielded the predictions shown in Figure 5. Trees in the lowest stem quality Table 4 Parameter estimates for harvesting models based on the copula approach (standard errors in parentheses) Parameter Covariate Copula Copula with V-system Q-system VQ-system ˆb 0 Intercept (0.551) (0.571) (0.575) (0.597) ˆb 1,1 Species SM (0.531) (0.559) (0.542) (0.561) ˆb 1,2 Species AB (0.540) (0.559) (0.554) (0.579) ˆb 1,3 Species YB (0.566) (0.576) (0.575) (0.617) ˆb 1,4 Species IW (0.574) (0.585) (0.584) (0.602) ˆb 1,5 Species RM (0.582) (0.610) (0.592) (0.622) ˆb 1,6 Species BF (0.570) (0.577) (0.597) (0.619) ˆb 1,7 Species OH ref. a ref. ref. ref. ˆb 1,8 Species SH (0.620) (0.629) (0.630) (0.654) ˆb 1,9 Species LC (0.686) (0.711) (0.702) (0.743) ˆb 2 DBH ij ( ) ( ) ( ) ( ) ˆb 3 m ij DBH ij ( ) ( ) ( ) ( ) ˆb 4 m ij DBH 2 ij ( ) ( ) ( ) ( ) ˆb 5,1 Class R, A or I n.a. b ref. ref. ref. ˆb 5,2 Class C, B or II n.a (0.271) (0.295) (1.070) ˆb 5,3 Class S, C or III n.a (0.184) (0.208) (0.141) ˆb 5,4 Class M, D or IV n.a (0.183) (0.230) (0.287) ˆd Dependence ( ) ( ) ( ) ( ) The parameter estimates ˆb 5,1, ˆb 5,2, ˆb 5,3 and ˆb 5,4 account for the classes of the different tree classifications when they apply. These parameters are added in a linear manner in the model. a ref. ¼ indicates the reference category. b n.a. ¼not applicable (the covariate is not taken into account in the model). 6of11

7 Modelling stem selection in northern hardwood stands classes (classes C and D) were more likely to be harvested than trees in the higher classes. Finally, the copula model with the VQ-system resulted in higher predictions of harvesting probabilities in classes III and IV compared with classes I and II, i.e. also from the least to the most vigorous trees (Figure 6). Figure 3 Predicted harvesting probabilities from the copula model (without tree classification) and their 0.95 confidence intervals as a function of DBH for species or species groups with at least 100 individuals (dash line: American beech; solid line: yellow birch). Discussion The fact that both the GLMM and the copula approaches gave an improved fit compared with the general models indicates that the responses were spatially correlated within the plots. This implies that the probability of a tree being harvested was not only dependent on its own characteristics, but was also influenced by the status of neighbouring trees. Whereas the GLMM approach assumes a more-or-less uniform correlation within the plots, the copula approach described a decreasing correlation with increasing distance between the trees. Since the GLMM models were outperformed by the copula models in all cases, our results support the assumption of decreasing spatial correlations within the plots rather than uniform correlations. This is in agreement with the results of Fortin et al. (2013) who compared empirical Spearman s correlation coefficients with predicted correlations and found out that spatial correlations decreased rapidly as the distances between two individuals increased. In fact, to account for such spatial relationships, Arii et al. (2008) designed their selection cutting algorithm for northern hardwood stands to simulate harvests as a contagious spatial process. From a practical perspective, such correlations can be explained by the fact that the harvesting process follows an intrinsically spatial pattern. Indeed, entire rows of trees have to be harvested to allow the passage of machinery and cutting trees, especially large ones, often results in damage to neighbouring trees, which must then be salvaged if they are of commercial dimensions. To our knowledge, this is the first harvest model in which the effects of spatial correlations have been tested in combination Figure 4 Predicted harvesting probabilities from the copula model with the V-system and their 0.95 confidence intervals as a function of DBH for sugar maple, yellow birch, American beech and balsam fir. 7of11

8 Forestry Figure 5 Predicted harvesting probabilities from the copula model with the Q-system and their 0.95 confidence intervals as a function of DBH for sugar maple, yellow birch, American beech and red maple. Figure 6 Predicted harvesting probabilities from the copula model with the VQ-system and their 0.95 confidence intervals as a function of DBH for sugar maple, yellow birch, American beech and balsam fir. 8of11

9 Modelling stem selection in northern hardwood stands with tree-level classifications of both vigour and quality. Among the three classification systems tested in this study, the V-system brought the largest improvement in the prediction ability of a model that already included tree species and DBH. This result was anticipated, since tree marking guidelines are also based on the V-system, which was implemented in the province of Québec to promote the development of vigorous post-cut stands in northern hardwood forests. However, some vigorous trees were still being harvested, either accidentally or deliberately. In any case, since vigorous trees tend to be of a higher quality (Pothier et al., 2013), this likely resulted in an increase of the financial return of the harvesting operation. The similar harvesting probabilities of the two lower vigourclasses (M and S), coupled with the presence of residual trees of the worst vigour class (M), imply that some trees of the S vigour class were likely marked instead of trees of class M. This would have resulted in an increase of the harvested volume of higher quality logs (Fortin et al., 2009). On theother hand, thesimilarities in harvesting probabilities between classes M and S, as well as between classes C and R, suggest that a vigour classification system taking into account only two classes could be as efficient as the current four class V-system, at least for the studied stands. This is supported by the predicting abilityof the model that included the VQ system, which is onlyslightly lower than that of the model with the V system, despite the fact that it uses only two vigour classes. In accordance with the theory of selection cutting (Nyland, 1998),the tree marking rules of Boulet (2007)do not include a preference for trees of a certain diameter class. However, our results revealed that for a given vigour or quality class, DBH was an important variable explaining the variability in harvesting probabilities. For trees larger than the minimum merchantable diameter, all candidate models indicated that harvesting probabilities increased with increasing DBH, regardless of tree species, vigour or stem quality. One advantage of harvesting large trees is the higher volume per stem, which is likely to decrease logging costs. There could also be some potential benefits to the sawmill, because larger logs could lead to a higher productivity (Steele, 1984; Lin et al., 2011). However, stem quality has been shown to decrease markedly beyond species-specific DBH thresholds (Pothier et al., 2013; Havreljuk et al., 2014), a fact that may result from increasing occurrences of injuries associated with decay and/or wood discolouration (Baral et al., 2013; Havreljuk et al., 2013). Therefore, the observed tendency to harvest large trees may not generate the expected benefits. Indeed, Pothier et al. (2013) suggested replacing the harvesting of a proportion of large trees of low quality by an equivalent volume composed of smaller trees of low vigour but of higher quality. They reasoned that low-vigour trees should have a high probability of imminent mortality, regardless of their quality. Therefore, such a strategy should not affect the achievement of the main silvicultural objective of selection cuttings, which is to harvest trees while improving the vigour of the residual stand, and thus also maximize the long-term yield in high-quality timber and other values (Nyland, 1998). Unlike merchantable trees, the harvesting probabilities of trees smaller than 23.1 cm increased with a decreasing DBH for all species, except balsam fir for which the minimum merchantable DBH was set at 9.1 cm. These trees were not marked prior to harvest and consequently, their removal can only be an indirect consequence of harvesting operations. Many were either damaged by the machinery or accidentally felled when a larger neighbour was harvested. Our post-harvest measurements indicated that these two causes accounted for 77 per cent of the harvested pole-sized trees. In preliminary trials, we tested two of the tree classification systems (V and VQ) for the non-merchantable stems and found that there was no significant effect on the harvest probabilities. Their removal therefore seems to be unplanned. Yet, future merchantable trees will be recruited from current pole-sized trees, so there may be important benefits for future yield if low-value species as well as low-vigour and lowquality individuals are marked for harvesting at this stage. Based on the observed decreasing removal probabilities as DBH approached 23 cm, a more deliberate selection of pole-size trees seems achievable with simple tree marking guidelines. Tree species was another important variable explaining the harvest probabilities. However, the inclusion of this variable in the models was not to differentiate the harvest probabilities between the main commercial hardwood species (sugar maple, yellow birch, American beech and red maple), but to distinguish the harvesting probabilities of the latter from those of balsam fir on the one hand, and from other companion species such as northern red oak, white spruce, white pine and basswood on the other. First, the similarity in harvesting probabilities between the main commercial hardwood species is not surprising, since the tree marking guidelines consider these species to be equally desirable. Despite this, the application of species-specific guidelines might have helped to improve the targetting of high-qualityand low-vigour trees, which differ in proportions among hardwood species (Pothier et al., 2013). Secondly, differences in harvesting probabilities between the main commercial hardwoods and balsam fir can be explained by the latter s lower minimum merchantable DBH (9.1 cm), as well as by its shorter lifespan. Indeed, balsam fir is known to suffer from the early development of stem decay on good quality sites (Whitney, 1989) and is also vulnerable to spruce budworm defoliation (Pothier et al., 2012). Consequently, it is the most intensively harvested species in these northern hardwood stands, especially among trees with a DBH larger than 15 cm, which is generally considered by softwood sawmills in the region to be the minimum profitable DBH. Thirdly, the higher harvesting probabilities of the main commercial hardwood species compared with companion species can be partly explained by the lower abundance of the latter in the sampling area, which also limits the demand. However, for valuable species such as northern red oak, eastern white pine and white spruce, there was a higher proportion of vigorous stems in the dataset, which implies that they were less likely to be marked for harvesting. In fact, the low harvesting probabilities of these companion species promote higher tree species diversity after selection cuts. In the context of ecosystem-based management, this can be viewed as a positive attribute of the selection cuts that were monitored in this study (Seymour et al., 2002). The segmented approach in our models was adapted from a general model that is currently in use to make growth forecasts in the province of Québec (Fortin, 2014). The location of the join point in the segments is an issue that cannot be overlooked. In our study, it matched the lowest probabilities, which is the reason why the effect of d 2 ij was significant for merchantable stems only. This match is not a requirement of this approach. In Fortin (2014), the join point was also set to 23 cm, but the lowest probabilities were observed 18 cm in DBH. A notable difference between our study and that of Fortin (2014) is the use of single segment for coniferous species. 9of11

10 Forestry Whereas Fortin (2014) set the join point to 23 cm for all species, we found out that coniferous species could be modelled using a single segment. Although a two-segment model can be used with coniferous species, it appears that the pattern of the harvest probabilities is not as complex as for broadleaved species. Setting the join point to the merchantable limit appears to be the less subjective decision to make as it can be assumed that merchantable and non-merchantable stems are not treated equally by lumbermen. This distinction can even facilitate the interpretation of the results. A fully objective approach would consist in letting the model make the decision as regards to the location of the join point. A new parameter to be estimated could be substituted for the merchantable limit. However, this re-parameterization yields a nonlinear expression on the logit transformed scale and fitting such parameterized models might prove more difficult. Future improvements through the parameterization of the join point remain to be investigated. Another option would have been to use an n th order polynomial of tree DBH instead of the segmented approach. We tried this polynomial approach with the general model during preliminary trials. However, even with a fourth order polynomial of tree DBH, the fit was not as good as it was with the segmented approach. Considering this lack of fit and the difficulty related to interpreting higher orders of tree DBH, we deemed that this approach did not deserve further investigation. Conclusions The addition of spatial correlations and tree vigour classes in our harvest model improved model accuracy by 12 per cent, as estimated from the difference in maximum log-likelihood. However, the inclusion of spatial correlations may not be practicable due to the likely high cost of data acquisition and the fact that their inclusion only accounted for one quarter of the improvement in the model fit. On the other hand, tree vigour was confirmed as an important factor for stem selection in partial cuts. In cases where a growth simulator can include a measure of tree vigour, we recommend the inclusion of this variable in the associated harvest submodel. This will give us a better representation of tree removal in simulated partial cuttings performed for northern hardwood stands. In this study, the harvesting of non-vigorous stems was prioritized, which is clearly beneficial for the future growth of the stand. However, there was also a strong bias towards larger stems that did not appear to comply with either the current tree marking guidelines or the underlying principles behind the use of selection cutting system. Indeed, these state that a silvicultural intervention should be applied to stems of all development stages within an uneven-aged stand. To improve future wood supplies, tree harvesting should not be biased against smaller stems of commercial dimensions and some deliberate culling of unpromising pole-sized stems should be applied. These measures may be made financially viable by targetting smaller stems that have a low-vigour status, but that have retained their quality. Acknowledgements We also wish to thank David Auty, two anonymous reviewers and the Associate Editor for their helpful suggestions for improving the text. Funding This work was part of the first author s master thesis and was funded by the Fonds de recherche du Québec Nature et technologies (FRQNT). We are grateful to the people involved in the field work: Jean-François Provencher, Annabelle Moisan-De Serres, Jean-François Belzile, David Bélanger and Pascal Gauthier, the latter three from the Coopérative forestière des Hautes-Laurentides. Conflict of interest statement None declared. 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