TY - JOUR
AU - Goudie, James W.
AB - Mixed-effects models described sapwood area distribution along stems of 60 lodgepole pine and 124 western hemlock trees. The Gallant-Fuller (GF) segmented polynomial model for stem taper equations comprising three joined polynomial segments and a functional components (FC) model derived from pipe model theory containing three additive components were compared. A fixed-effects model was constructed, and then random tree effects were added. Random-effect variability was highest at the tree base for both models and species. Random variation was regressed against tree-level measurements to identify additional fixed-effects crown or height variables. All models fit well, but root mean square error was lower for pine than for hemlock and lower for the FC than the GF final fixed-effects model. Sapwood area conformed to pipe model theory assuming variable stem permeability that was associated with tree crown and total height. Pine validation data from four Alberta and two British Columbia sites showed that the FC model underpredicted (9%) and the GF model overpredicted (7%) sapwood area, mainly in the lower stem. Reasonable predictions of sapwood area could be derived from ground-based or remote sensing methods, allowing classification of sites, trees, and logs based on this wood quality characteristic. sapwood area, tree crown and stem, pipe model theory, Pinus contorta, Tsuga heterophylla Sapwood tissue conducts water and nutrients from the roots to the foliage and provides storage for both water and metabolites. It is distinguished from heartwood by a higher moisture content, a lower chemical extractive content, and a cell microstructure that favors the flow of liquid water. The boundary between sapwood and heartwood gradually advances into the sapwood with time, but the process by which sapwood is transformed into heartwood is not well understood (Bamber and Fukazawa 1985). The transformation involves the death of ray parenchyma cells, possibly caused by a rapid increase in the content of toxic extractives (Taylor et al. 2002, Bergström 2003). The differing characteristics of sapwood and heartwood affect the processing and value of forest products (Panshin and de Zeeuw 1970, p. 474, 478). Moisture content must be taken into account in drying schedules for solid wood to avoid warping. Solid wood products consisting entirely of sapwood or heartwood are more homogeneous than products that include both types of wood. The superior hydraulic conductivity of sapwood allows better uptake of liquid-borne wood treatments. On the other hand, the lower chemical extractive content of sapwood makes it less resistant to decay organisms, resulting in reduced durability of wood products (Taylor et al. 2002). In pulping, the higher extractive content of heartwood can necessitate additional treatment to achieve desired whiteness. In some species, color differences between heartwood and sapwood lead to differentiation of wood products based on appearance. In accordance with pipe model theory (Shinozaki et al. 1964), the amount and distribution of sapwood in tree stems is closely related to the amount and distribution of foliage in the crown (Waring et al. 1982). The general pattern of the distribution of sapwood area along the tree stem is an increase from the apex of the tree downward (Long et al. 1981, Mörling and Valinger 1999, DeBell and Lachenbruch 2009). In the uppermost part of the tree stem, the xylem is entirely sapwood; heartwood typically begins to appear well above the base of the live crown. Sapwood area increases rapidly near the stem base and is larger at wide tree spacings, presumably because these trees have larger crowns (Mörling and Valinger 1999, DeBell and Lachenbruch 2009). Correspondingly, sapwood area at breast height increases with tree diameter, age and radial growth rate (Sellin 1994). Knowledge of sapwood distribution in stems is valuable not only for understanding tree physiological processes but also for optimal use of sapwood and heartwood in forest products. Optimization along the entire forest product value chain from forest to market is possible for sapwood or heartwood inventories. An attempt to estimate heartwood percentage in Pinus sylvestris stems from site, stand, and tree variables available in forest inventory was unsuccessful (Björklund 1999), possibly because the crown variables were limited to crown length or the model form was unsuitable. The extensive evidence for the pipe model theory implies that tree hydraulic function could provide a sound basis for models of sapwood area distribution along the stem. Dean and Long (1986) explained most of the variation in Pinus contorta sapwood area using leaf area and distance from the crown midpoint as independent variables. A functional approach may not be necessary to model sapwood area relations. Maguire and Batista (1996) successfully described stem sapwood area using models originally developed for stem taper. Their sapwood area models require dbh, total height, and height to crown base. In a related approach, Ojansuu and Maltamo (1995) estimated sapwood area as the difference between simultaneous stem and heartwood taper models. The overall aim of this study was to develop models using variables obtainable from remote sensing, such as tree height and crown dimensions, to describe the longitudinal sapwood distribution in lodgepole pine (Pinus contorta [Doug. Ex. Loud.]) and western hemlock (Tsuga heterophylla [Raf.] Sarg.). These two species are abundant in western Canada and have contrasting ecological requirements (Farrar 1995). Specific objectives were to assess whether functional models have greater descriptive ability than purely empirical models and to assess the predictability of models by applying models to a separate region. Materials and Methods Data for Model Development Data from sample trees represented a wide range of tree (Table 1) and plot (Table 2) characteristics. Sapwood area and tree attributes were obtained from data previously collected by the British Columbia Ministry of Forests, Lands and Natural Resource Operations (BCMFLNR) (Figure 1). Lodgepole pine trees (Pinus contorta [Doug. Ex. Loud.]) were selected from two geographic locations near Kamloops, British Columbia (50.44°N, 120.55°W) and another two locations near Quesnel, British Columbia (53.00°N, 121.99°W). Western hemlock trees (Tsuga heterophylla [Raf.] Sarg.) were also selected in British Columbia locations near Adams Lake (50.40°N, 126.20°W), Cowichan Lake (48.78°N, 124.13°W), Port McNeill (50.57°N, 127.17°W), Mission (49.20°N, 122.30°W), Port Renfrew (48.59°N, 124.29°W), and the Malcolm Knapp Research Forest maintained by the University of British Columbia (49.28°N, 122.57°W). These species are referred to as pine and hemlock hereafter. Trees encompassed a range of diameters and crown classes in each stand (i.e., dominant, codominant, intermediate, and overtopped). Open grown trees from the same stand were included to expand the range of crown conditions. Trees were restricted to those free of defects or damage from pests, diseases, or abiotic causes and disks to 0.3 m and above. Further description is found in Mansfield et al. (2007) and Nemec et al. (2012). Summary of sample tree characteristics. Table 1. Summary of sample tree characteristics. Pl, lodgepole pine; Hw, western hemlock. View Large Table 1. Summary of sample tree characteristics. Pl, lodgepole pine; Hw, western hemlock. View Large Summary of sample characteristics for trees in sample plots. Table 2. Summary of sample characteristics for trees in sample plots. Pl, lodgepole pine; Hw, western hemlock. View Large Table 2. Summary of sample characteristics for trees in sample plots. Pl, lodgepole pine; Hw, western hemlock. View Large Figure 1. View largeDownload slide Map of study site locations in British Columbia and Alberta: KM, Kamloops; QS, Quesnel; AD, Adams Lake; CW, Cowichan Lake; PN, Port McNeill; MI, Mission; PR, Port Renfrew; UB, Malcolm Knapp Research Forest; maintained by the University of British Columbia; PS, Parsons; CB, Cranbrook; MK, MacKay; MC, McCardell; TF, Teepee Pole Flat; TN, Teepee Pole North. ■, hemlock sites; ▴, pine sites. Figure 1. View largeDownload slide Map of study site locations in British Columbia and Alberta: KM, Kamloops; QS, Quesnel; AD, Adams Lake; CW, Cowichan Lake; PN, Port McNeill; MI, Mission; PR, Port Renfrew; UB, Malcolm Knapp Research Forest; maintained by the University of British Columbia; PS, Parsons; CB, Cranbrook; MK, MacKay; MC, McCardell; TF, Teepee Pole Flat; TN, Teepee Pole North. ■, hemlock sites; ▴, pine sites. Before felling, a fixed-area plot including about 30 neighbors was established around each sample tree to describe the neighborhood. The crown projection area was mapped by measuring the radial distance from the tree stem to the perimeter of the crown at several points around the tree. Cross-sectional disks were cut at heights of 0.3, 0.7, and 1.3 m and at 6–10 additional locations equally spaced between 1.3 m and the tree top. The transition between heartwood and sapwood was visually determined based on wood color differences (Panshin and de Zeeuw 1970) and measured on two radii per disk. Color difference is distinct in pine and, although less so in hemlock, was still estimable after wetting the surface with water and observing the difference in adhesion of a water-soluble pencil across the sapwood to heartwood boundary. Radius positions were determined by methods adapted from Chapman and Meyer (1949, p. 345–346). Average radius inside bark was estimated by measuring the outside bark diameter with a diameter tape and subtracting the average bark thickness from half of this diameter. Two radii of this length were located and marked on the disk. The sapwood area is the difference between the inside bark area and heartwood area, with areas determined as the average area from each radius using the formula for the area of a circle. Pine Validation Data Sets Validation data were obtained from four sites in Alberta (MacKay [53.55°N, 115.53°W, 862 m], McCardell [53.2°N, 115.18°W, 1,408 m], Teepee Pole Flat [51.9°N, 115.18°W, 1,472 m], and Teepee Pole North [53.87°N, 115.17°W, 1,409m]) and two sites in British Columbia (Parsons [51°N, 116.7°W, 1,152 m] and Cranbrook [49.4°N, 115.6°W, 1,339 m]) (Figure 1). The Alberta sites are long-term pine juvenile thinning trials with various initial basal areas ranging from 2 to 26 m2/ha, and up to 41 m2/ha in the unthinned control plots, all established between 42 and 55 years before measurement. The British Columbia sites are long-term commercial thinning trials, with basal area ranging from 11 to 21 m2/ha in thinned plots and up to 46 m2/ha in control plots at establishment 16 years before measurement. For all sites, stem dbh (1.3 m) and tree height were measured for every tree. Live crown height was measured from the ground level to the base of the continuous live crown defined by a whorl with more than two branches with green leaves. Crown radius was the distance from the center of the tree bole to the edge of the crown margins (drip zone) averaged over four cardinal directions. Stem disks were cut at 30 cm (stump height), at 130 cm (breast height), at one-third and two-thirds of the distance between breast height and the merchantable top of 7.5 cm diameter, at the internode below the lowest live branch, at the base of the live crown, at the most vigorous part of the live crown, and at the merchantable top. Sapwood area was determined by locating the maximum and minimum radius on a disk, measuring the total length and sapwood width along those radii, and computing areas of circles from the corresponding radii. Development of Fixed-Effects, Mixed-Effects, and Tree Covariate Models For each species, we fitted the two model forms representing the empirical and functional approaches described in the Introduction: a segmented polynomial model as described by Gallant and Fuller (1973) (GF model) and a functional components model (FC model) comprising three additive components. For each model form we first derived and fitted a fixed-effects (FX) base model and then introduced tree-level random effects in several combinations of one, two, or three parameters to produce a mixed-effects (MX) model, evaluating alternatives based on Akaike's information criterion (AIC) and −2 log likelihood values. From the final MX model, the random effects for each tree (BLUPS) were estimated using best linear unbiased prediction. Next we examined additional tree-level covariates as predictors of the BLUPS from the MX model and introduced those prediction equations into a mixed-effects TX model and refitted all the parameters. Use of the TX model to predict the sapwood profile for a new tree (i.e., one not used in the fitting) requires a complex calibration to estimate the empirical BLUPs of the random effects (Nigh 2012). We refitted the TX model without the random effects to construct the tree covariate fixed-effects (TF) model, which does not require the calibration. Decisions related to the inclusion of the BLUP prediction equations were made from the TX model for which the random effects and covariance structure accounted for within-tree clustering of the data. We checked that the parameter estimates for the terms common to both TF and TX models were at least similar without and with covariance structure, respectively. All models were fit using the SAS procedure NLMixed (SAS version 9.2) with model variables defined in Table 3. Pearson residuals were plotted against the actual and predicted values, D, relative D, H, CL, CA, CRT, and CR for all models. Box plots of Pearson residuals were also plotted for the categorical variables site location, plots within sites, and four crown classes (open grown to suppressed), but because these did not suggest any major influence, they were not considered further. Model terms were included only if they had a significant effect according to a likelihood ratio test. The root mean square error (RMSE) also provided a measure of model comparison. Variable definitions and abbreviations. Table 3. Variable definitions and abbreviations. View Large Table 3. Variable definitions and abbreviations. View Large GF Model for Pine and Hemlock GF-FX and GF-MX Models The original GF segmented polynomial model (Gallant and Fuller 1973) comprises intercept, linear, and quadratic terms. We dropped the intercept term to constrain sapwood to zero at the top of the tree. Early in the analysis we encountered cases in which the upper stem polynomial would produce negative estimates of sapwood area and consequently we also removed the linear term. The resulting model was   where Sij is the sapwood area (cm2), i refers to tree disk level and j to tree level, D is the distance (m) from the apex, CL is the crown length (m), H is the tree height (m), β2, β3, β4, α1, and α2 are fixed-effects parameters to be estimated, b2, b3, b4 are the random effects; and ε is the residual error. The upper and lower join points were implemented by the I1 and I2 operators, with the upper join point a proportion of crown length CL and the lower join point a proportion of tree height (H) where   The GF-FX model version is Equation 1 with the random effects omitted, and the GF-MX model version is Equation 1 with the random effects included (see the source code in the online Supplemental Data). For the GF-MX model, when estimates of the off-diagonal elements were very close to zero, we refitted the model setting those elements to zero and used a likelihood ratio test to evaluate whether the data supported the use of a more parsimonious covariance matrix. We addressed heteroscedasticity by modeling the residual variance as a power function of the main covariate, D (Pinheiro and Bates 2000, p. 206)   Identification of Additional Tree Covariates for Inclusion in the GF-TX and GF-TF Models The GF tree covariate models were developed through linear and nonlinear regression analysis using additional tree crown covariates to predict the BLUPs from the GF-MX model. Component regressions of each random parameter on tree crown covariates were developed, and the substitution of these component regressions resulted in the GF-TX and GF-TF model versions. Pine For pine, the correlation between β2 + b2 and β3 + b3 was ρ = −0.98, so β3 + b3 was predicted from β2 + b2. None of the tree-level covariates helped to explain the variation in β4 + b4 so it was left unchanged for pine. The TX and TF models for pine were   where β4, α1, and α2 are fixed-effects parameters and b2, b3, b4 are random effects as in Equation 1; r0–r3 are fixed-effects parameters arising from the substitution of the BLUP equations. The GF-TX model version is Equation 3 refitted with the random effects included, and the GF-TF model version is Equation 3 with the random effects omitted and refitted. Hemlock The tree covariate models for hemlock included component regressions for β2 + b2, β3 + b3, and β4 + b4 from Equation 1, and substituting these component regressions into Equation 1 resulted in   where α1 and α2 are fixed-effects parameters and b2, b3, b4 are random effects as in Equation 1; r4–r9 are fixed-effects parameters arising from the substitution of the BLUP equations. The GF-TX model version is Equation 3 refitted with the random effects included, and the GF-TF model version is Equation 3 with the random effects omitted and refitted. FC Model for Pine and Hemlock Derivation of FC-FX and FC-MX Models The FC model comprises three overlapping components central to pipe model theory: the area of sapwood at a given height is proportional to the amount of foliage above it. The first component deals with the region within the crown where the amounts of foliage and sapwood area increase with distance from the apex to the crown base (DIC); this component is then constant below the crown base. The second component allows for a linear increase or decrease or no change in the sapwood area with distance below the crown base (DBC). Finally, the third component allows the possibility of nonlinear sapwood area butt flare, which is not an element of pipe model theory but which is widely observed. This component is represented as an exponential decay function of relative height (HR) along the stem, where HR ranges from 0 at the soil surface to 1 at the tree apex. The resulting three-component equation is   where   and where Sij is the sapwood area (cm2), i refers to the tree disk level and j to tree level, DIC is the distance (m) from the tree tip within the crown, DBC is the distance below the crown base (m), HR is the relative height, τ0–τ5 are fixed-effects parameters to be estimated, t0, t2, t3 are random tree-level effects, and ε is the residual error. The inclusion of the τ5 parameter defines an “effective crown base” that might be slightly above (τ5 <1) or below (τ5 >1) the observed crown base. The FC-FX model version is Equation 5 with the random effects omitted, and the FC-MX model version is Equation 5 with the random effects included, both refitted. We modeled the residual variance for all FC models as described in Equation 2. Identification of Tree Covariates for Inclusion in the FC-TX and FC-TF Models The FC tree covariate models were developed through linear and nonlinear regression analysis using additional tree crown covariates to predict the BLUPS from the FC-MX model. Component regressions of each random parameter on tree crown covariates were developed, and the substitution of these component regressions resulted in the FC-TX and FC-TF model versions. Pine After substituting the component regressions into Equation 5, the resulting FC-TX and FC-TF models for pine were   where τ1, τ4, and τ5 are fixed-effects parameters to be estimated and t0, t2, t3 are random tree-level effects as in Equation 5; d0−d5 are fixed-effects parameters arising from the substitution of the BLUP equations. The FC-TX model version is Equation 6 refitted with the random effects included, and the FC-TF model version is Equation 6 with the random effects omitted and refitted. Hemlock After substituting the component regressions into Equation 5, the resulting FC-TX and FC-TF model for hemlock was   where τ1, τ4, and τ5 are fixed-effects parameters to be estimated and t0, t2, t3 are random tree-level effects as in Equation 5; d6−d12 are fixed-effects parameters arising from the substitution of the BLUP equations. The FC-TX model version is Equation 7 refitted with the random effects included, and the FC-TF model version is Equation 7 with the random effects omitted and refitted. To examine the influence of tree height and crown ratio on sapwood area, we generated predictions for both species using the GF-TF and FC-TF models with tree height set at the middle of its range in our data and with crown ratio set at 0.2, 0.5, and 0.8 of its range. We computed crown length from tree height and crown ratio. The remaining TF covariates (crown radius and crown area) were derived from a regression that estimated crown radius from crown length. We also compared the sapwood area profiles between species for trees of the same size using a tree height of 25 m and a crown ratio of 0.5. These data were summarized in the figures for sapwood area against relative distance from apex for both tree species. Validation Data We predicted sapwood area profile for trees in the pine validation data set using the GF-TF and FC-TF models fitted to the BCMFLNR data. Absolute differences (actual minus predicted) were plotted against basic measures of site, tree (height, crown class, and social status) and silvicultural treatment to explore differences between the GF and FC models. Models were also assessed by comparing prediction bias at different points along the stem. Results FX Models The β2 values in the GF-FX model indicated that the within-crown sapwood area increased more rapidly with distance from the apex (D) in pine than in hemlock (Tables 4 and 5). In the GF-FX middle segments, β2 and β3 parameters were opposite in sign but almost equal in magnitude in pine; furthermore, because the coefficients of D are additive below the upper join point α1CL, the pine middle segment had a roughly linear increase toward the base (Table 4). The larger β4 parameter for the hemlock GF-FX model indicated greater butt flare than that for pine (below the lower join point α2H). The α1 values also indicated that the upper join point occurred much lower in the crown for hemlock than for pine, but the lower join points (α2 values) were in similar relative positions for the two species. Parameter estimates for the GF models fitted to the BCMFLNR pine data. Table 4. Parameter estimates for the GF models fitted to the BCMFLNR pine data. View Large Table 4. Parameter estimates for the GF models fitted to the BCMFLNR pine data. View Large Parameter estimates for the GF models fitted to the BCMFLNR hemlock data. Table 5. Parameter estimates for the GF models fitted to the BCMFLNR hemlock data. View Large Table 5. Parameter estimates for the GF models fitted to the BCMFLNR hemlock data. View Large The upper transition point for pine was much higher in the GF-FX than in the FC-FX model (α1 = 0.239 versus τ5 = 0.688) (Tables 4 and 6). For hemlock, the estimated transition point locations were slightly lower in the GF-FX model than in the FC-FX model (Tables 5 and 7, α1 versus τ5). The parameter values for both species for the middle component (τ2) were similar in the FC-FX models, but it was difficult to compare species because the top component is additive to the middle component (Tables 6 and 7). The FC-FX bottom sapwood component was a function of τ3, τ4, and tree height and displayed greater butt flare for hemlock than for pine. In summary, for both species, the FC-FX and GF-FX models shared similarities in that the sapwood area in the upper stem was a function of crown length and the butt flare of sapwood area was a function of tree height. Parameter estimates for the FC models fitted to the BCMFLNR pine data. Table 6. Parameter estimates for the FC models fitted to the BCMFLNR pine data. View Large Table 6. Parameter estimates for the FC models fitted to the BCMFLNR pine data. View Large Parameter estimates for the FC models fitted to the BCMFLNR hemlock data. Table 7. Parameter estimates for the FC models fitted to the BCMFLNR hemlock data. View Large Table 7. Parameter estimates for the FC models fitted to the BCMFLNR hemlock data. View Large MX Models For both pine and hemlock, the best-fitting mixed-effects models incorporated three random effects (Tables 428532853–7 variance components) and resulted in substantially improved model fit compared with the FX model. This variation indicates that individual tree differences were not adequately described in the FX models. The butt region had the largest amount of random tree-level variation for both the GF-MX and FC-MX models (b4 and t3 parameters, respectively). Tree-level random effects considerably reduced the standard error of the fixed-effects population parameters at the butt (β4, τ3) compared with that for the FX model, as expected (Tables 428532853–7, FX versus MX). Inclusion of tree-level random effects for middle and upper segments or components affected parameter estimates to a lesser degree than for the butt region. The GF-MX pine parameters for upper and middle sections (β2 and β3) still remained opposite in sign and of similar magnitude, predicting the nearly linear but increasing sapwood middle sections at the population level, but now allowing the two parameters to vary by tree through b2 and b3; consequently, this allowed the possibility of nonlinear increases in sapwood midsections for certain trees. The addition of the random effects did not shift the join or transition point locations in either the GF or FC model, except that the upper transition point was higher in the hemlock FC-MX model (Table 7, τ5). Another important aspect of the MX model versions was the variance-covariance structure. The covariances proved to be more important for hemlock, especially between the middle and bottom segments for the FC model compared to the pine (Tables 428532853–7, Cov). The pine MX models only required covariance between the top and middle segments (Cov b2, b3), indicating less dependence between the lower sections (Tables 4 and 6). TX Models In all models, the variances and standard errors of the random terms were reduced in the TX models compared to the MX models (Tables 428532853–7) which substantially improved fits compared to the FX model, especially the FC-FX model (Figures 2 and 3). Covariances were similarly reduced in all cases in the TX model compared to the MX model except the pine GF-TX (Table 4). Figure 2. View largeDownload slide Pine FC and GF model fits showing the sapwood profile along the stem from the apex for the trees of median RMSE in three height classes. Figure 2. View largeDownload slide Pine FC and GF model fits showing the sapwood profile along the stem from the apex for the trees of median RMSE in three height classes. Figure 3. View largeDownload slide Hemlock GF model fits showing the sapwood profile along the stem from the apex for the trees of median RMSE in three height classes. Figure 3. View largeDownload slide Hemlock GF model fits showing the sapwood profile along the stem from the apex for the trees of median RMSE in three height classes. TF Models Estimates of the fixed-effects parameters were similar between the TX and TF models except for the butt flare sapwood parameter in the GF pine model (Table 4, β4). In the GF-TF and FC-TF pine models, sapwood increased with larger crown but more slowly with DIC on shorter trees (trees with lower height to crown base or larger crown ratio). In the GF-TF and FC-TF models for hemlock, the sapwood area in the upper component decreased with a larger crown ratio. For both species, the fit of the top component was always improved by accounting for trees with wider crowns or larger crown areas, regardless of model form. Crown variables had varying effects on the middle segment or component of both models. In general for both species, crown variables had opposite effects in the midsection compared to the upper section; as a consequence, this allows less rapidly changing but still increasing sapwood area. Further, trees with smaller crown dimensions had less sapwood area in the middle sections, but this was increased when the crown extended lower proportionately on the tree. In the butt region, the other components were additive to those controlling this section, but these had a small effect because of the size and configuration of the functions controlling the butt area. The lower component of the FC-TF model declined to 10% of its ground-level sapwood value at a relative height of 0.05 for pine and a relative height of 0.01 for hemlock [from ln(0.1)/τ4]; consequently, the influence of the lower component was confined to the lowermost portion of the FC tree model. The sapwood area in the pine butt segment of the GF-TF model was associated with greater sapwood area in taller trees. In addition, the pine FC-TF model showed increased butt sapwood area with crown area and length (Table 6, d5). For hemlock, in the GF-TF model, butt sapwood area increased with tree height (Table 5, r9) and in the FC-TF model with crown area (Table 7, d12). That is, larger crowns on taller trees were associated with greater sapwood area at the butt. The join or transition points in all models were largely unchanged by the additional tree-level fixed-effects covariates in the TF models compared with those in the MX models. Comparison of Model Fit and Shape As expected, the goodness of fit in every case was best in the TX model, followed by the MX, TF, and FX models, respectively (Table 8). The goodness of fit between the MX and TX models was close, and the standard errors of parameters in common to all models were nearly always lowest in the MX or TX model (Tables 428532853–7). The reduction in RMSE obtained by adding tree-level covariates (TF versus FX) was greater for the FC than for the GF models (Table 8). Over all trees, the model goodness-of-fit statistics for both species indicated better fit in the FC-TF than in the GF-TF model (Table 8), with the difference mainly resulting from overprediction by the GF model of a few of the larger observations (Figure 4). Actual versus predicted values of all trees for pine and hemlock indicated adequate fit (Figure 4), and residual plots as a function of relative distance from the apex showed similar results (Figure 5). Goodness-of-fit statistics for the FX, MX, and tree-level TF models for the FC and GF models fitted to the BCMFLNR pine and hemlock data. Table 8. Goodness-of-fit statistics for the FX, MX, and tree-level TF models for the FC and GF models fitted to the BCMFLNR pine and hemlock data. View Large Table 8. Goodness-of-fit statistics for the FX, MX, and tree-level TF models for the FC and GF models fitted to the BCMFLNR pine and hemlock data. View Large Figure 4. View largeDownload slide FC model (left) and GF model observed data versus predicted tree-level covariates model (TF) fitted to the BCMFLNR pine and hemlock data. Figure 4. View largeDownload slide FC model (left) and GF model observed data versus predicted tree-level covariates model (TF) fitted to the BCMFLNR pine and hemlock data. Figure 5. View largeDownload slide Pearson residuals versus relative tree height (0 = top) for fixed-effects models with tree-level covariates model (TF) fitted to the BCMFLNR pine (top) and hemlock data (bottom). The FC model is on the left and the GF model on the right. The horizontal solid black line represents the model prediction, and the gray horizontal line represents a LOESS curve fitted to the residuals. Figure 5. View largeDownload slide Pearson residuals versus relative tree height (0 = top) for fixed-effects models with tree-level covariates model (TF) fitted to the BCMFLNR pine (top) and hemlock data (bottom). The FC model is on the left and the GF model on the right. The horizontal solid black line represents the model prediction, and the gray horizontal line represents a LOESS curve fitted to the residuals. Particularly for pine and to a lesser degree for hemlock the incorporation of tree-level covariates in the FC-TF model resulted in altered sapwood magnitude and slope compared with those in the FC-FX model (Figures 2 and 3, TF versus FX). For the pine and hemlock GF-TF models, the additional tree-level covariates mainly allowed greater nonlinearity in the sapwood area along the stem than in the GF-FX model (Figures 2 and 3). For both pine and hemlock, the FC models had sharper transitions between the top and middle components than for the GF models for all tree sizes (Figures 2 and 3), which arises from the model specification of its upper transition point. For trees of the same height, pine models tended to predict a slightly more convex shape with moderate and larger crown ratios than in hemlock (Figure 6). For pine, the GF-TF and FC-TF models predicted similar shaped sapwood areas across three classes of crown ratios (Figure 6, top). For hemlock, the GF-TF model predicted that the sapwood area was more nonlinear in the middle to lower sections of trees with median or greater crown ratios (Figure 6, middle). Comparisons between species indicated that the GF and FC models had substantial differences for trees with crown ratios of 0.5; pine displayed greater sapwood area than hemlock above the midpoint of the stem regardless of the model form (Figure 6, bottom). Figure 6. View largeDownload slide Graphs of predicted sapwood area versus relative distance from tree apex (0 = top) for pine (top) and hemlock (middle) fit with the tree-level covariates model (TF). The GF and FC model versions are shown in each panel for each species. The top two panels show the effect of crown ratio at 20th, 50th, and 80th percentiles of the range for trees at the 50th height percentile. The bottom panel compares the sapwood distributions for trees with height 25 m tall and a crown ratio of 0.5 for both species and model type. Figure 6. View largeDownload slide Graphs of predicted sapwood area versus relative distance from tree apex (0 = top) for pine (top) and hemlock (middle) fit with the tree-level covariates model (TF). The GF and FC model versions are shown in each panel for each species. The top two panels show the effect of crown ratio at 20th, 50th, and 80th percentiles of the range for trees at the 50th height percentile. The bottom panel compares the sapwood distributions for trees with height 25 m tall and a crown ratio of 0.5 for both species and model type. Model Validation for Pine We tested the TF predicted values from the models developed for pine against the actual values using a separate validation data set in different locations of British Columbia and Alberta. Overall, the FC model tended to overpredict at large values of sapwood to a greater degree than the GF model (Figure 7, top). However, the largest differences occurred nearer the tree base, especially for the FC model (Figure 7, bottom; Table 9) by up to 67% bias. Over all disks, the FC model overpredicted by 9% (observed − predicted) and the GF model underpredicted by 7% for all disks combined, with the largest bias in the butt flare region (Table 9). Parallel results were obtained for the RMSE over all disks, which showed the greatest differences at the butt with the GF model having superior prediction (Table 9). Some residual differences were evident in site MacKay (Alberta), where a few trees in heavily thinned plots had less sapwood area near the tree base than predicted compared with that for the unthinned or the lightly thinned plots on that site. Site MacKay was also one of the least productive of the sites. Site location Parsons (British Columbia) also had some trees with negative residuals at the tree base, but these occurred in both the thinned and unthinned plots. These residuals were related to trees with smaller sapwood area for their crown length or crown width. However, most of the residuals were clustered around the zero value for all sites and not considered important enough for model inclusion. Figure 7. View largeDownload slide The top panel depicts the FC model (left) and the GF model for the validation pine observed versus predicted (TF model) sapwood area. The bottom panel shows differences in sapwood area for the FC model (left) and GF model between the predicted (TF model) and the observed validation pine data versus relative height (0 = apex). Gray lines are LOESS curve fits. Figure 7. View largeDownload slide The top panel depicts the FC model (left) and the GF model for the validation pine observed versus predicted (TF model) sapwood area. The bottom panel shows differences in sapwood area for the FC model (left) and GF model between the predicted (TF model) and the observed validation pine data versus relative height (0 = apex). Gray lines are LOESS curve fits. Pine validation data set bias (actual − predicted) versus relative height (0 = base) and RMSE for the FC-TF and GF-TF models. Table 9. Pine validation data set bias (actual − predicted) versus relative height (0 = base) and RMSE for the FC-TF and GF-TF models. View Large Table 9. Pine validation data set bias (actual − predicted) versus relative height (0 = base) and RMSE for the FC-TF and GF-TF models. View Large Discussion Model Fitting Our base models fitted to sapwood area along the bole comprised a segmented polynomial model (GF model) previously used to describe stem taper and an FC model derived from pipe model theory. The base FX models were greatly improved by introducing tree-level random effects (MX models). Some of the tree-level random variation was related to crown variables and tree height, especially near the tree base. Some of the tree-level variation may arise through use of simple crown measures as abstractions to the complex three-dimensional distribution of foliage, but we expect this contribution to be small compared to the disk error for the lower stem. Disk-level variation can be increased by sampling error when the sapwood area is estimated from disk radii if the sampling strategy does not provide an unbiased estimate when disk shape is not exactly circular. Disks from near the base of the tree are especially prone to sampling error because cross-sections tend to be irregular shapes (Cruickshank 2002). GF versus FC Model Model Building Data The FC models placed the upper transition point relatively close to the crown base, but the GF models placed this transition higher up. This probably occurs because the polynomial structure of the GF middle component allows this section to describe the increasing sapwood area taper beginning near the base of the crown and into the crown some distance. This frees the upper transition point to float to a higher position near the top where there is a change in sapwood curvature direction. Unlike the FC model, the midsection component of the GF model can accommodate nonlinear taper in the crown section below this point and below the crown. In the FC model, the middle component is essentially linear so the location of the upper transition point must correspond with an abrupt change in modeled taper that occurs near the crown base while also adjusting for middle and lower sections. Both the GF and FC models predicted substantial flare in sapwood area at the base of the trees, which increased with tree height (GF model) or also with crown area (FC model). The FC model predicted greater curvature in the sapwood area at the butt than the GF model, particularly in smaller trees. Model Validation Data for Pine Model validation with independent pine data indicated that the GF model predictions had lower bias and RMSE, mostly, but not completely, due to the differences in the fit in the butt flare region. The GF model polynomial function appears to be more suitable than the FC model exponential decay function for prediction of sapwood area at the base of the validation trees. The validation sites are located further east of the sites used for model building and possibly have regional adaptation in hydraulic function. Both model forms adequately predicted the middle and upper stem sections despite having different bias signs. Model validation revealed positive prediction bias in the GF model, but negative and larger prediction bias in the FC model, either of which could be removed from the predicted values for forestry application. However, the GF model provides lower actual and percentage bias for prediction of sapwood in independent sites and therefore would be preferred for application. There were also model similarities such as the inclusion of tree height and crown variables, some of which are known to relate to the amount of sapwood, but these are discussed more fully below. Except at the butt flare, we obtained reasonably good prediction estimates of pine sapwood area over large differences in geographic locations using simple measures of height and crown. Biological Interpretation of Models The simple crown and tree variables in the TF and TX models probably act as surrogates for leaf area. Stem sapwood area has been shown to be strongly related to leaf area of the crown in lodgepole pine (Dean and Long 1986) and in eastern hemlock (Kenefic and Seymour 1999). We found that both model types showed a nonlinear increase of sapwood within the crown consistent with leaf area increasing with DIC. Pine and hemlock maximum leaf area first increased and then decreased nonlinearly with relative distance from the apex, with a maximum at 0.33 in pine (Garber and Maguire 2005) and at 0.65 in western hemlock trees of similar size (Kershaw and Maguire 1995). This result is consistent with hemlock's higher shade tolerance and greater needle longevity. We found that the sapwood area inflection or transition point in the GF model occurred at 0.25 and 0.63 relative distance into crown for pine and hemlock, respectively, consistent with the vertical distribution of leaf area of these species described above. The same inflections were predicted to occur nearer the crown base for the FC model, less consistent with changes in vertical distribution of leaf area and pipe model theory, for model configuration relating to the upper join point discussed in the previous section. Further, tree-level parameters such as crown length and width were positively associated with crown sapwood, consistent with pipe model theory and with expected linkages between vertical foliage distribution and crown size (Kershaw and Maguire 1995). The greater leaf area in the upper pine crown might explain the greater amount of pine sapwood area compared with that from the same location in hemlock of similar size. Nevertheless, other factors might also contribute; for example, greater sapwood area in upper stems might provide water storage needed for growth in drier and warmer habitats in which pine grows or there may be differences in hydraulic conductivity. We found increasing sapwood area below the crown toward the base of the tree in both model forms and tree species, suggesting that tracheid geometry or conductivity of the sapwood does not remain constant along the stem below the crown. Tracheids become longer and wider with increasing age or stem diameter, from branch to root (top to bottom), and within annual rings from latewood to earlywood (Zimmermann 1983, p. 89). Wider rings are found within the crown region of conifers than in lower stem positions, except in dominant trees (Farrar 1961); however, we did not find any relationship between model residuals and social status, probably because this was partly reflected in model parameters for tree and crown size. In conifers, the percentage of latewood also decreases with the wider rings (Smith 1980) that are characteristic of the upper stem. Latewood forms first at the base of the tree and then progresses upward (Young 1952), probably related to the diminishing auxin levels (Larson 1962), resulting in more latewood closer to the tree base because of the time lag. In Douglas-fir, latewood proportion increases from apex to base (Gartner et al. 2004) and latewood has lower conductivity (Domec and Gartner 2002). Increasing latewood proportion toward the base probably increases stem mechanical strength, but results in lower sapwood conductivity from apex to tree base. This would be a reasonable explanation for the increase in sapwood area toward the lower stem. It is not clear why the sapwood area greatly increases in the bottom 10% or less of the sample trees (study tree disks start at 0.3 m), but it may satisfy a requirement to maintain water flow locally through this region. We found that the sapwood in this area was most consistently affected by tree height and to a lesser extent by crown variables in all models. Increases in both variables would be associated with increased bending stress at the butt section. Douglas-fir seedlings artificially bent and held in one direction had impaired permeability of the stem in the bent area (Spicer and Gartner 2002). Unlike for the seedlings, bending tension and compression in the butt sapwood and extending into the buttressed roots of tall trees takes place in all directions, especially in dominant trees. Trees that were artificially prevented from sway showed higher growth in the area of sway prevention and lower growth below in the butt flare region and buttressed roots, suggesting that stem growth is altered by local mechanical bending stress (Fayle 1968). Spruce roots showed decreasing root hydraulic conductivity with a larger root system size (Rüdinger et al. 1994), which might be related to structural requirements to accommodate bending stress in the wood needed for taller trees or more frequent tracheid damage in this area. Sapwood tracheids may be affected by mechanical stress or increased cavitation risk, especially in older sapwood at the tree base, causing low or nonexistent functionality with age (Spicer and Gartner 2001). It is possible that a tradeoff between conductivity and mechanical requirement in the butt flare and buttressed roots is compensated for by an increase in sapwood area. Supporting this notion is the finding that sapwood area is related to leaf area in the stem, is not related in the butt flare region or in the buttressed roots, but is again related at more distal root positions (Coutts 1987) that are under lower bending stress. Proximal root sections have cells with lower radial and tangential diameter and length and have thicker cell walls than those for more distal sections (Fayle 1968), which combined would restrict sapwood permeability. The same cell attributes probably extend upward into the stem in addition to the changes in latewood proportion noted above. Application of Sapwood/Heartwood Estimates to Forest Management Attributes such as tree height, species composition, and stocking can be used to estimate volume by species chiefly by aerial photography and field measurements. Recently, application of terrestrial laser imaging detection and ranging (LiDAR) has provided a wide range of structural metrics related to fiber attributes and wood quality (Blanchette et al. 2015) that add to volume and species information. Airborne LiDAR systems have potential for estimating individual tree attributes over broader areas including tree height and crown height and diameter (Wulder et al. 2008), which are surrogates for estimating the fiber properties of biological and economic value. Wood quality is directly related to wood formation, which is largely driven by physiological processes originating in the foliage (Larson 1969). There is also a new generation of cube satellites that will be able to offer data at fine resolution and high revisitation rates (Butler 2014) that could provide sophisticated low-cost enhanced inventory tools for forest management. Sapwood estimates from remote inventory would allow a complete sapwood inventory of all trees spatially within stands at the planning stage of forest management. Sapwood and heartwood area based on crown or height dimensions of individual trees has practical application for processing raw material in the wood-based industry. When the target is to maximize product sapwood content, the trees can be selected by crown and height through spatial referencing, and bucking and transport operations could take into account this within-piece variation along the bole. At the mill level, processing of wood veneer is affected by the difference between sapwood and heartwood moisture content (Huang et al. 2012). Segregating logs based on sapwood could increase the quality and lower the cost of the raw materials especially if they can be allocated to product needs and forecasting and scheduling at the harvesting stage. Sapwood and heartwood also have chemical and physical properties that can guide segregation of raw materials. Heartwood differs from sapwood in chemical composition (Bertaud and Holmbom 2004), which can affect pulping (Esteves et al. 2005), drying rate (Pang 2000), checking (Lee et al. 2004), natural durability (Taylor et al. 2002), and uptake of preservatives (Hansmann et al. 2002). Our models also suggest that sapwood might be manipulated by silvicultural treatments. Lowering stand density increased the amount of sapwood and heartwood in black spruce (Yang and Hazenberg 1992) and increased the area of both sapwood and heartwood as well as the proportion of heartwood in western redcedar (DeBell and Lachenbruch 2009). In contrast, thinning Scots pine increased sapwood area marginally, with little effect on the heartwood area (Mörling and Valinger 1999). Fertilization increased the amount of leaf area but not sapwood area in Sitka spruce (Whitehead et al. 1984). Clearly more research is needed to sort out the effect of stand manipulation on sapwood and heartwood formation along the stem and to relate this to tree height, crown dimensions, and foliage biomass and distribution. Supplementary data are available with this article at http://dx.doi.org/10.5849/forsci.14-206. 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TI - Models of the Vertical Distribution of Sapwood Area for Lodgepole Pine and Western Hemlock in Western Canada
JO - Forest Science
DO - 10.5849/forsci.14-206
DA - 2015-12-01
UR - https://www.deepdyve.com/lp/springer-journals/models-of-the-vertical-distribution-of-sapwood-area-for-lodgepole-pine-9cucbZ90Ig
SP - 973
EP - 987
VL - 61
IS - 6
DP - DeepDyve
ER -