Scaling Approach for Estimating Stand Sapwood Area from Leaf Area Index in Five Boreal species

: This paper presents a scaling approach for estimating sapwood area at the stand level using knowledge obtained for individual trees of ﬁve boreal species: Populus tremuloides (Michx.), Pinus contorta (Doug. ex Loud. var. latifolia Engelm.), Pinus banksiana (Lamb.), Picea mariana (Mill.) BSP, and Picea glauca (Moench) Voss. Previously developed allometric models for sapwood depth and diameter at breast height for individual tree species were used to build stand level sapwood area estimates as well as stand level leaf area estimates, in pure and mixed vascular vegetation stands. A stand’s vegetation heterogeneity is considered in the scaling approach by proposing regression models for each species. The new combined scaling approach drew strong linear correlations at the stand scale between sapwood area and leaf area using observations taken in mixed stands of Southern Alberta, Canada. This last outcome suggests a good linear relationship between stand sapwood area and stand leaf area. The accuracy of the results was tested by observing each regression model’s adequacy and by estimating the error propagated through the whole scaling process. a correlation between LA plot and SA plot but there should be no correlation between SA plot and LAI in the deciduous data set. These results were in reasonable agreement with the results obtained with the Pearson’s correlation hypothesis test and the regression analysis.

Researchers studying the same species under different site conditions have reported different LA sp :SA sp regression models (linear and non-linear). The discrepancy between results have helped to species; and (3) to determine the absolute error propagated while scaling sapwood depth from an individual tree up to the stand level.

Model Approach and Sampling Design
For scaling purposes (from tree-to-stand level), a vegetated stand with vascular species has been conceptualized as an area of forested land (a stand) with a single tree with a sapwood area SA plot equal to the summation of all the individual trees' sapwood cross-sectional area (SA i ) inside the stand.
where SA i is species specific for each tree i in the stand containing n trees of different species. Similarly, the single tree's leaf area (LA plot ) will be the summation of all trees' leaf area inside the stand.
Despite the simplicity of the concept, it considers vegetation heterogeneity by using species specific models to scale up biophysical characteristics. At the tree-level, linear regression models were developed for each species sapwood depth sd versus outer bark diameter at breast height (D OB ) data so that Equation (3) below could be used to estimate SA i : where sd i is each i th tree's average sapwood depth and D OB i is the i th tree's diameter at breast height. Equation (3) calculates SA i as the region lying between two concentric circles within a tree's cross-section. The outermost circle borders the bark and vascular cambium, while the innermost one bounds the heartwood. These circles are naturally irregular but tree trunks are treated as having a cylindrical shape. The models obtained were used to estimate every tree's sapwood area SA i inside the delimited stand. Equation (1) is then used to estimate the stand level sapwood area (SA plot ) of that single tree representing the whole stand. The correlation between species specific leaf area and sapwood area (LA sp :SA sp ) and a mathematical scaling approach detailed in Section 3 were used to estimate LA plot :SA plot .
The leaf area for the stand LA plot , was obtained by measuring the Leaf Area Index (LAI) by light transmission, and the surface ground area of interest. Stands of four different sizes are used to validate the LA sp :SA sp relationship. Four stands were 60 × 60 m, and nine were 10 × 10 m. These stands were located in the Sibbald Areas of Kananaskis Country, Alberta, Canada. The species composition of these stands was either dominated by deciduous trees (mainly Populus tremulolides), or coniferous trees (Pinus contorta and Picea glauca). Figures 1 and 2 show the location and delimitation of these stands. Two more stands were delimited at Whitecourt, in northern Alberta: One was dominated by Pinus banksiana individuals (20 × 20 m) and one was dominated by Picea mariana (15 × 15 m) individuals. The stands whose species composition was dominated by a conifer tree were labelled as "coniferous", and for those whose species composition was dominated by a deciduous tree were labeled as "deciduous". The 10 × 10 m stands were distinguished from the larger ones (≥ 300 m 2 ) by adding a prime symbol in front of their assigned number. Field data collected at each stand included: (a) Number of trees per stand, (b) species identification, and outer bark diameter at breast height for every tree inside the stand, and (c) Leaf Area Index for the stand. LAI was measured in the 60 × 60 m stands using the Tracing Radiation and Architecture Canopies (TRAC, 3rd Wave Engineering Co.; Nepean, Ontario, Canada) device. The Canopy Analyzer LAI-2000 (LI-COR Incorporated; Lincoln, Nebraska, US) was used to measure LAI in the stands of 20 × 20 m and 15 × 15 m, located in Whitecourt.
Since saplings generally lack heartwood and being mostly composed of sapwood [21,31,39], saplings correlations between DOB and , or between DOB and cross-sectional area per species (SAsp) will be different than for mature trees and thus, were treated separately. Furthermore, due to their size and composition, saplings were considered part of the understory, and this study focused on scaling allometric correlations of the overstory. Saplings were considered those trees with DOB ranging between 2.41 cm and 10.2 cm, and heights between 38.1 cm and 76.2 cm. Trees found inside of the stands with a DOB ≤ 10cm were considered saplings, and were excluded from all allometric correlations.   . Figure 2. Geographical location of deciduous plots. The plots are in the Sibbald areas, south-east of Barrier Lake (Kananaskis Country).

Statistical Analysis
For each species, we performed statistical analyses to determine the most adequate regression model. These statistics include the Pearson's correlation coefficient, detection of outliers, the  Kananaskis Valley is a Montane closed forest formation [33] within the Rocky Mountains [34,35]. The Montane forest is classified as an ecoregion within the Cordilleran eco-province with a particular mix of physiography and air masses leading to unique climatic conditions [35]. Within Alberta, the Montane forest maintains the warmest temperatures during the winter than any other forested ecosystem. This type of forest has ridged foothills and a marked rolling topography. The Whitecourt forest is within the mid boreal mixed-wood ecoregion [35] and considered a closed forest formation [36] in the Southern Alberta uplands [35]. Further details on the field sites can be found in [31,37].

Treatment of Saplings
Since saplings generally lack heartwood and being mostly composed of sapwood [20,30,38], saplings correlations between D OB and sd, or between D OB and cross-sectional area per species (SA sp ) will be different than for mature trees and thus, were treated separately. Furthermore, due to their size and composition, saplings were considered part of the understory, and this study focused on scaling allometric correlations of the overstory. Saplings were considered those trees with D OB ranging between 2.41 cm and 10.2 cm, and heights between 38.1 cm and 76.2 cm. Trees found inside of the stands with a D OB ≤ 10cm were considered saplings, and were excluded from all allometric correlations.

Statistical Analysis
For each species, we performed statistical analyses to determine the most adequate regression model. These statistics include the Pearson's correlation coefficient, detection of outliers, the regression analysis, ANOVA, list of unusual observations, and the lack-of-fit test. Finally, each linear model's residuals were examined to check the model adequacy. Model adequacy checking included a normal plot of residuals and a plot of residuals versus fitted values.
In addition to the correlation analysis, a pairwise comparison of the coefficient of variation (COV) confidence intervals (C.I.) was used as an indication of the relationship between the scaling parameters at the stand scale (i.e., between SA plot and LAI, and between SA plot and LA plot ). The pairwise comparison consisted of comparing the parameters' C.I.s to see if they overlapped or not. This was another way to test if there was correlation between the variables. Because they are different parameters and the units differ, it was not suitable to use mean values or standard deviations to test the similarity between the two sample populations [39][40][41].
Another modified approach to MacKay C.I. is the suggested by [42]:

Model Implementation and Data Results
For all five species, the tree-level sapwood depth was measured by microscopical analysis of wood anatomy to differentiate between sapwood and heartwood, as well as to differentiate sapwood from cambium and phloem. The sampling, sapwood depth, average sapwood depth, and D OB results obtained for each species sampled to create the tree-level allometric correlations reported in [31].

Tree-Level Allometric Correlations
Pinus banksiana and Pinus contorta average sapwood depth sd was not well correlated with their respective D OB [31]. This was also true of Picea mariana. Picea glauca and Populous tremuloides did however, show significant, positive linear correlations between sd and D OB . For Picea glauca, the relationship from [31] and used here was: For Populous tremuloides the relationship obtained in [31] was used and is as follows: Depending on the correlation between sd and D OB (that is, whether there is no relationship or a significant linear relationship like those shown in Equations (6) and (7)), the methods for scaling up to the stand level using D OB required a species-based approach. Thus, for Picea glauca and Populous tremuloides, Equations (6) and (7) were used to determine stand level sd estimates from stand level estimates of D OB , respectively. For those species without a strong tree-level linear correlation, a simple stand average, i.e., sd, was proposed.

Scaling Up Sapwood Area from the Tree to the Stand Level
Since it was feasible to fit a linear regression model for two of the five studied species, the aggregation of cross-sectional sapwood area to the stand level to obtain SA plot was performed by fusing two approaches. To estimate the sapwood area at every stand, if Picea glauca and/or Populus tremuloides trees were present, their respective linear model was applied-this was the first scaling approach. But under the presence of Pinus banksiana and/or Pinus contorta and/or Picea mariana, their respective average sapwood area was applied, which is the second scaling approach. Therefore, which is further expanded as (8b): where L is the number of species present with a linear model (i.e., L = 0, 1 or 2 depending on whether none, one, or both are present of Picea glauca or Populus tremuloides; thus, if L = 0, the summation with j beginning at 1 is null), N sp,j is the total number of trees inside the stand of species j; D OB i,j is the i th individual's D OB of species j; and sd i,j is the i th individual's estimated sapwood depth for species j, and estimated for trees of species Picea glauca with Equation (6), or Populus tremuloides with Equation (7). The second term on the right-hand side of Equation (8a) pertains to those species that did not produce a significantly linear model. Thus, M is the number of species k present in the stand with k being Pinus banksiana, Pinus contorta, or Picea mariana. M = 0, 1, 2 or 3; and N sp,k is the number of trees in the stand of each of these species. Based on previous results [31], it was assumed that the sapwood cross-sectional area for each of these three species remained relatively constant as the tree grew. Therefore, for each of these species k, the average sapwood area was calculated based on results from [31] with important details given in the next paragraph for SA k . The study of [31] determined what relationships existed for these five boreal species between sd and D OB . Microscopically differentiated average sapwood depth (averaged over the four cardinal directions) from wood cores obtained for the fives species was correlated with D OB for each tree in that study. Average sapwood area for each species that did not have a linear relationship between sd and D OB was computed using Equation (9): where, sd i,k is the average sapwood depth for each cored tree i, of species k, in that study, and accordingly, D OB i,j was the outer bark diameter at breast height for tree i of species k and t k was the number of cored trees in the sample sets of [31]. Values of SA k and t k for all five species are reported in Table 1 and [31] provides greater details not contained here. Table 1. Values of average tree-level sapwood area for each species computed with data from [31].
Only for the first three species on the list, SA k was used to estimate stand level (SA) in Equation (8b). The SA plot calculated for each 60 × 60 m and 10 × 10 m stand, including tree species, tree quantity, stand D OB statistics and the error associated with SA plot , are given in Tables 2 and 3, respectively. The conifer sites were those sites containing Picea glauca, Pinus contorta, Pinus banksiana, and Picea mariana. The deciduous sites were mainly composed of Populus tremuloides. If deciduous individuals were present in the coniferous stands, their numbers counted for less than 10% of the total tree quantity or were mainly saplings. Such was the case of the site Conifer-5, with a Populus tremuloides tree quantity of 114; however, 92 individuals were saplings. Thus, the count of deciduous trees inside the stand was a small portion of the total in terms of sapwood area contribution. Most of the deciduous sites were 100% pure stands, and if any other tree was present, it counted for less than 5% of the total stand's tree quantity (e.g., Deciduous-6). Most of the coniferous sites (60 × 60 m and 10 × 10 m) were composed of Picea glauca and Pinus contorta trees. The two pure coniferous sites were Conifer-11 and Conifer-12. Table 2. Descriptive statistics of the 60 × 60 m stands located in the Sibbald areas of Kananaskis Country, Alberta, and Whitecourt, Alberta. Tree quantity is either N sp,k or N sp,j depending on the species; SA sp is either SA j or SA k in Equation (8b), depending on the species, ∆SA plot is the absolute error associated with SA plot . D OB units are in cm, and all areas are in units of m 2 .

Pinus banksiana
where α l is the woody-to-total area ratio (i.e., tree's woody fraction), γ E is the needle-to-shoot area ratio (i.e., fraction of needles per shoot), Ω E is the clumping index. In order to derive the α l and γ E values from the typical values reported by [44], the age and productivity of the stand must be known. Thus, age and productivity characteristics were defined by comparing the LAI eff with typical values reported in [45]. The LAI eff of the Pinus banksiana stand was similar to an intermediate/medium productivity stand (LAI eff = 2.20) with α l of 0.05 and γ E of 1.35. The Picea mariana stand LAI eff value was similar to a mature medium/high productivity stand (LAI eff = 2.78), thus, α l is assumed as 0.14 and γ E as 1.35. Finally, Ω E was derived from typical values reported by [45] as well; thus, the Ω E values for Picea mariana and Pinus banksiana were respectively set as 0.65 and 0.75. The fifteen stands estimated LAI (LAI plot ) concur with the previous reported values by [43,45,46]. The total leaf area per stand, LA plot was estimated based on the definition of Leaf Area Index: "The total one-sided (or one half of the total all-sided) green leaf area per unit ground surface area": Table 4 displays the estimates of LAI plot . The coniferous 10 × 10 m stands' areas were adjusted and reported as 150 m 2 in order to take into account what is known as the "TRAC footprint". The TRAC footprint was created by trees with shadows large enough to fall into the delimited stand influencing the TRAC measurements. Thus, the LAI measured in the field belonged to a larger area than the delimited one. This effect was not evident in the 10 × 10 m deciduous stands. Furthermore, the TRAC footprint was influenced by tree height and solar zenith angle (θ) [44]. Therefore, the TRAC footprint was adjusted by assuming that in Kananaskis Country, the coniferous trees' average height was approximately 15 m, and the solar noon zenith angle was about 45.21 • on the date when LAI was measured (day 238 of the year). Therefore, the extent of the footprint (tree height × tanθ) was 15 m. This extent would occur in only one side of the stand giving an LAI for a stand area of 10 × 15 m.

Stand Level Leaf Area Sapwood Area Allometric Correlations
In the case of the Populus tremuloides sites, a close similarity was found between the two sets of stands of 10 × 10 m and 60 × 60 m in terms of LAI. In order to support this observation, the equality of variances with Bartlett's test and the F-test was conducted. Based on the two statistical test results, the LAI variances of the two sample sets were equal (C.I. = 95%) and the two sets were merged to observe the correlation of Populus tremuloides leaf area and sapwood area at the stand level. The explanation for such a similarity between Populus tremuloides' leaf area at different scales may be related to the canopy type. Unlike coniferous trees, Populus tremuloides canopy is horizontally wide, creating small gaps between the canopy of neighboring trees. This effect of Populus tremuloides canopies is due to their wide-circular leaves and their alternate (not clumped) distribution. At the same time, the leaves are predominantly at the top of the tree, creating a rounded crown with a large diameter. Moreover, in the studied sites, the Populus tremuloides tree heights were all similar. The combination of these characteristics gives wider tree shadows that practically cover the entire stand's floor at noon. Thus, inside the stand there is little room for observing the footprint of external trees in any direction meaning that the TRAC device mostly measures the LAI inside the stand with a negligible footprint. This was proven by examining the LAI measured in the east-west and north-south transects. In both east-west and north-south directions, LAI was observed to have practically the same value (a paired T-test proved that the difference of the population mean was equal to zero with an α = 0.05). Hence, it was concluded that it was possible to merge both data sets for use in a linear regression model. For all Populus tremuloides sites, there was a strong linear correlation between SA plot and LA plot (ρ ≈ 0.999), and the p-value that equaled zero gave sufficient evidence to conclude that the correlation was not zero (α = 0.05). Results of the regression analysis supported the decision of fitting a linear model to the data, showing that LA plot resulted in a significant predictor of SA plot with a p-value < 0.0001 (α = 0.05). The ANOVA results determined that LA plot contributes significantly to the model (α = 0.05). The first fitted model ( Figure 3) had a high coefficient of determination (R 2 = 99.8%) that supported the theory of SA plot being fairly well explained by a linear model. The R 2 adj (99.8%) was close in value to the R 2 . The R 2 pred (57.77%) was weak with a large difference from R 2 adj . Thus, there was a slight indication of one value inflating the prediction, and thus, this model might not be suitable for other stands. The list of unusual observations draws attention to the large influence that the Deciduous-6 site gives to the model, considering it was an unusual observation. Even if the observation was not considered an outlier, it was removed from the sample set, and the regression model was fitted with only the four observations. In the second regression attempt (Figure 4), all the coefficients of determination values were lower than the ones of the first model. The R 2 was high (99.2%), and the R 2 pred (79.5%) was in reasonable agreement with the R 2 adj (98.78%). Thus, the second model was proposed for practical use. model to the data, showing that LAplot resulted in a significant predictor of SAplot with a p-value < 0.0001 (α = 0.05). The ANOVA results determined that LAplot contributes significantly to the model (α = 0.05). The first fitted model ( Figure 3) had a high coefficient of determination (R 2 = 99.8%) that supported the theory of SAplot being fairly well explained by a linear model. The R 2 adj (99.8%) was close in value to the R 2 . The R 2 pred (57.77%) was weak with a large difference from R 2 adj. Thus, there was a slight indication of one value inflating the prediction, and thus, this model might not be suitable for other stands. The list of unusual observations draws attention to the large influence that the Deciduous-6 site gives to the model, considering it was an unusual observation. Even if the observation was not considered an outlier, it was removed from the sample set, and the regression model was fitted with only the four observations. In the second regression attempt (Figure 4), all the coefficients of determination values were lower than the ones of the first model. The R 2 was high (99.2%), and the R 2 pred (79.5%) was in reasonable agreement with the R 2 adj (98.78%). Thus, the second model was proposed for practical use.   The coniferous SAplot and LAplot estimates for the 10 × 10 m and 60 × 60 m stands are given in Table  5. For all coniferous sites, there was a strong linear correlation between the SAplot and the LAplot, and the P-value supports that the correlation is not zero. However, if a regression model was derived by using both sample sets (10 × 10 m and 60 × 60 m stands), the lack-of-fit test is significant at a p-value of 0.019. The lack-of-fit test suggested a possible curvature in the model and that some other type of model should be fitted. It was assumed that the mismatch between the two data sets was due to the overestimation of LAI due to the influence of the footprint at the 10 × 10 m scale. Therefore, the sapwood area for the 10 × 10 m stands were underestimated between 20% and 29.6%. Hence, in the case of the coniferous sites, the obtained values for the 10 × 10 m stands were not suitable for combination with the 60 × 60 m stands because of the footprint caused by the canopy type (which The coniferous SA plot and LA plot estimates for the 10 × 10 m and 60 × 60 m stands are given in Table 5. For all coniferous sites, there was a strong linear correlation between the SA plot and the LA plot , and the p-value supports that the correlation is not zero. However, if a regression model was derived by using both sample sets (10 × 10 m and 60 × 60 m stands), the lack-of-fit test is significant at a p-value of 0.019. The lack-of-fit test suggested a possible curvature in the model and that some other type of model should be fitted. It was assumed that the mismatch between the two data sets was due to the overestimation of LAI due to the influence of the footprint at the 10 × 10 m scale. Therefore, the sapwood area for the 10 × 10 m stands were underestimated between 20% and 29.6%. Hence, in the case of the coniferous sites, the obtained values for the 10 × 10 m stands were not suitable for combination with the 60 × 60 m stands because of the footprint caused by the canopy type (which was not randomly distributed, that is, it was clumped, and had large open areas that allow trees outside of the stand to reflect their shadows inside of it).  1 Correlation between 10 × 10 m stands, 2 correlation between 60 × 60 m stands, 3 correlation includes all stands.
The sample set was then divided into the 10 × 10 m and the 60 × 60 m stands to fit two separate regression models for the conifer stands. The first model corresponded to the 10 × 10 m stands ( Figure 5), and the latter model was fit for the 60 × 60 m stands ( Figure 6). Regression analysis and ANOVA results suggest that the LA plot is a significant predictor of the SA plot (α = 0.05). However, the model's R 2 pred for the 10 × 10 m stands denote inadequacy for future predictions, and the R 2 pred (38.37%) significantly differs from the R 2 adj . The regression model for the 60 × 60 m stands, however, show a better agreement between its coefficients of determination ( Figure 6). Still the difference between R 2 adj and the R 2 pred was large (68.60%), but the model adequacy check gave enough evidence to support the decision for fitting a linear model to the 60 × 60 m data set. Both models have similar slopes that differ by just about 0.37 cm 2 . The sample set was then divided into the 10 × 10 m and the 60 × 60 m stands to fit two separate regression models for the conifer stands. The first model corresponded to the 10 × 10 m stands ( Figure  5), and the latter model was fit for the 60 × 60 m stands ( Figure 6). Regression analysis and ANOVA results suggest that the LAplot is a significant predictor of the SAplot (α = 0.05). However, the model's R 2 pred for the 10 × 10 m stands denote inadequacy for future predictions, and the R 2 pred (38.37%) significantly differs from the R 2 adj. The regression model for the 60 × 60 m stands, however, show a better agreement between its coefficients of determination ( Figure 6). Still the difference between R 2 adj and the R 2 pred was large (68.60%), but the model adequacy check gave enough evidence to support the decision for fitting a linear model to the 60 × 60 m data set. Both models have similar slopes that differ by just about 0.37 cm 2 .

COV Confidence Intervals
The pairwise comparison of C.I. for COV was an extra analysis to support the applicability and reliability of the regression models, because the sample size for both data sets was not as large as desired when fitting a linear regression model. Therefore, this analysis reinforced the suggested relationship between the scaling factors, no matter the sample size. Since most of the COVs were larger than 0.33, Payton's equation (Equation (4)) was used to estimate the C.I. Table 6 displays the obtained COVs and C.I. of SA plot , LAI, and LA plot . Table 6. Coefficient of variation (COV) results for the coniferous and deciduous stands' SA plot , LA plot and LAI with their 95% confidence intervals (95% C.I.s).

Site Type
Variable COV 95% C.L. For the coniferous data set, the LA plot , SA plot , and LAI confidence intervals were not significantly different; therefore, this suggested that there may be correlations amongst these scaling factors. For the deciduous data set, the LA plot and SA plot confidence intervals were not significantly different, while the LAI confidence interval was significantly different from the LA plot and SA plot confidence intervals. Therefore, this may be an indication of a correlation between LA plot and SA plot but there should be no correlation between SA plot and LAI in the deciduous data set. These results were in reasonable agreement with the results obtained with the Pearson's correlation hypothesis test and the regression analysis.

Estimates of Error Propagation
The absolute error on SA plot estimates (∆SA plot ) was calculated based on the rules of error propagation that are derived from a Taylor series [47]. Equation (8a) describes SA plot as the summation of each tree's SA, thus, ∆SA plot will be given by the summation of each tree's error contribution to SA: where ∆D OB i and ∆sd i are the absolute errors on the i th tree's D OB and estimated sd i , respectively. The measurement of D OB were carefully verified by measuring the circumference of the trees at breast height on the same tree, 50 times. Two trees were measured in this exercise. The average error calculated was ± 0.0024 m.
For Picea glauca sd estimates (Equation (6)), ∆sd i will be given by: In the Populus tremuloides linear model (Equation (7)), ∆sd i is given by: Solving the partial derivatives in Equation (12) and substituting ∆sd i for Equation (13), the Picea glauca ∆SA plot is: and for Populus tremuloides: For the other three species studied here that have no linear relationship between average sapwood depth and diameter at breast height, ∆SA plot was estimated by the following equation: Due to the direct method used in this study to measure T-the total number of trees inside a particular plot, it was considered an exact number (if some other approaches are used to indirectly estimate tree quantity, there may be an error associated with T); therefore, ∆SA plot = T∆SA sp (18) where ∆SA sp is the absolute error on the SA average value and is given by: where n is a constant; thus, the third term is null. The equation then becomes: According to previous research, the measure of sapwood depth with the microscopical analysis of wood anatomy gives an accuracy of 98% [48]. The sapwood depth measurements supporting this research were obtained from [37] and were measured using the microscopical method. It was estimated that the error on sd was related to the accuracy of the ocular scale of the microscope (with divisions of 1µ) and the ruler (with divisions of 1 mm) used to measure each core's sapwood depth [37]. Thus, the instrument limit of error (ILE) was estimated as half of the smallest measuring increment of the instrument (ruler). Hence, it was estimated that ∆sd i = ILE = 1 2 (1 mm) = ± 0.0005 m. ∆SA sp for the Pinus contorta and Pinus banksiana sample set was 0.0002 m 2 ; ∆SA sp for the Picea mariana sample set was also 0.0002 m 2 . Tables 2 and 3 report ∆SA plot for the coniferous and deciduous sites, respectively.
Equation (21) is used to estimate the error on LA plot (∆LA plot ): ∆LAI was ±0.10 for the deciduous stands and ±0.13 for the coniferous stands, while ∆A plot was given by: Thus, on the whole we obtain:  Table 4. Notice that the ∆LA plot became larger as the stand size increased, being more notorious in the larger stands. In addition, the contribution of ∆LAI to ∆A plot was small, but still the size of the stand influenced the first term of the Equation (21); however, if LAI increased in large stands, then ∆A plot became large (e.g., Conifer-4, Deciduous-6). The prediction of SA plot for deciduous and coniferous stands using the obtained linear models, establishes that SA plot = f(LA plot ). Thus, the error propagation on the linear models follows: For the deciduous sites linear model, the ∆SA plot produced the following: For the coniferous linear model, the error on the SA plot was:

Discussion and Conclusions
Reliable estimates of tree and stand sapwood area are necessary in the calculation of other eco-hydrological parameters such as the actual transpiration from trees, for example. The error associated with the estimates of SA and SA plot will influence transpiration values if used to scale tree sap flow measurements. Since the accuracy of water balance components is crucial in this area of work, the authors believe that determining SA, SA plot , LA and LA plot propagated errors are fundamental to any such research and modelling. A number of studies have already highlighted the relevance of creating reliable allometric models, and have analyzed the error associated with their allometric correlations in order to support their reliability [26,[49][50][51].
Regarding the allometric correlations, previous research [31] demonstrated that it was feasible to estimate a single tree's sapwood area for Picea glauca and Populus tremuloides using estimates of sd from D OB . In this study, D OB was not an adequate sapwood area predictor of Pinus banksiana, Pinus contorta and Picea mariana species. As a result, scaling cross-sectional sapwood area to the stand-level was approached by combining the estimates of SA obtained with the regression models for Picea glauca and Populus tremuloides, and by using an average of the sapwood area of the remaining three species in a combined approach that scaled up sapwood area from a single tree to the stand scale. To the authors' knowledge, the use of average sapwood area has not been previously studied as an option for scaling key, characteristic dimensions of trees. This combined approach gives reliable SA plot estimates that were significantly correlated to LA plot estimates derived from measurements of LAI. The final relationship (for both groups conifers and deciduous) allowed the development of linear models for predicting SA plot as function of LA plot . We attribute the reliability of the average sapwood area values to the careful and detailed methods that were used to measure a single tree's sapwood depth, and that contributed to an error that was almost negligible [31]. This statement was supported by a more recent research study that reached similar conclusions regarding accurate measures of sapwood depth [52].
The errors associated with the SA plot estimates were mainly influenced by the size of the stand that in turn influenced the LA plot values-the larger the plot, the more the associated error with LA plot . This error (∆LA plot ) had an impact on the estimates of SA plot having LA plot as predictor, and prevented stronger regression models by combining the 10 × 10 m and the 60 × 60 m coniferous stands. On the contrary, in the case of the Populus tremuloides, the addition of stands of a size in between 10 × 10 m and 60 × 60 m would improve the model even further and may allow it to be applicable for use on other data sets; and perhaps even allow the addition of the stand Deciduous-6 to the model. Another important point is that the reduction of error on leaf area estimates depends on accurate delimitation of the stands, and as the stand's size increases, it is more complicated to retain accurate stand delimitation. Still, stands larger than 10 × 10 m are needed in order to limit discrepancies associated with the footprint on LAI. Even though small stands give the smallest errors, there are large discrepancies between estimates of LA plot and SA plot due to the footprint influence. Consequently, we consider the analysis of the regression model for the 10 × 10 m stands helps to support the reliability of the 60 × 60 m linear models (since both follow the same linear pattern and the slope is practically the same). For future modelling, it might be suitable to add the slope as another predictor of sapwood area to the stand level, as suggested by [53] who determined that it is the slope that determines the correlation between stand-level leaf area and sapwood area. The authors recommend that at least for conifer trees, it is preferred to use stands that are bigger than 100 m 2 , and by trying to reduce the footprint influence on LAI derived from optical measurements. Furthermore, attention to the accuracy in the relationship between D OB and sd cannot be understated. In a simple computation to determine SA in the plots when simply using an average sapwood area (Equation (9)) for all species as opposed to Equation (9) only for Pinus contorta, Pinus banksiana and Picea mariana and the appropriate linear form of Equation (8) for Populus and Picea glauca, significant overestimates of SA were found (results not shown here). This in turn created an overestimate of the slope in the LA:SA models (as in Figures 3-6) for each set of plots.
Recent studies have stressed the significant influence that scaled values of sapwood area can exert on final estimates of tree transpiration [54]. Thus, studies that have carefully analyzed the different methods to scale up tree morphological characteristics and their associated uncertainty, as in this paper, this should be a good reference for choosing reliable scaling methods in order to reduce the error that may be propagated into stand, ecosystem, or even watershed estimates. It is also recommended that the errors associated with estimates of tree transpiration for instance, should be determined and added into the overall error propagation equation. Some studies have, however, focused on taking into account the errors associated with only a section of their models [54,55]. For reliability purposes, the authors strongly recommend that one determine all the errors associated with every single parameter of a scaling model.
In conclusion, these results establish the uniqueness of each species sapwood area, D OB and LA allometries. Thus, any attempt to use the reported correlations in this paper for other estimates should do so for similar site conditions, topographic and climatic characteristics. In this study, it was possible to aggregate cross-sectional sapwood area from the tree to the stand level in a mixed forest by differentiating between deciduous and coniferous groups of trees, and by combining two different approaches, with an error that is considered negligible.
In terms of scaling, the results suggest that the SA plot and LA plot relationship is maintained at larger scales in this particular area and for the five studied species. There is still some variation that is not explained by the models. Thus, it is recommended that allometric characteristics of the trees, such as crown class, soil moisture or even vapor pressure deficit and maximum summer temperature as suggested by [56], could be introduced in the allometry by season, in order to observe what other factors influences SA plot .
Author Contributions: M.R.Q.-P. conducted the lab and field experiments, and together with C.V. conducted the data analysis and wrote the paper.