Bivariate distribution functions for predicting twig leaf area within ...

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Bivariate distribution functions for predicting twig leaf area within hybrid spruce crowns H. Temesgen, Valerie M. LeMay, and Ian R. Cameron

Abstract: The vertical and horizontal distribution of leaf area per centimetre of twig (APCM) for hybrid spruce (Picea engelmannii Parry ex Engelm. × Picea glauca (Moench) Voss × Picea sitchensis (Bong.) Carrière) tree crowns was modelled using bivariate Weibull and beta distribution functions. Horizontal position was represented by relative position on the first-order branch. Vertical position was based on branch position from the tree apex relative to tree height, since the base of the live crown is often difficult to locate. Sample APCM measures were obtained using systematic sampling of 12 tree crowns taken from stands at three developmental stages (20, 60, and 140 years of age). For comparison, univariate Weibull and beta distribution functions using only vertical distribution were also fitted. Generally, APCM decreased from the tree apex downward and from the branch tips toward the stem, although variation in the values was quite high. Trees from the middle stand age (60 years) had the highest average APCM values, followed by the smallest, youngest trees (stand age 20), and the lowest values were found for the largest trees in the oldest stand (140 years). As anticipated, the bivariate Weibull and beta distribution functions resulted in more precise representations of APCM within tree crowns than the univariate Weibull and beta distribution functions, although the improvements were minimal. Results were generally poorer for trees from the oldest stand. These functions could be used to evaluate other variables distributed over the tree crown, such as specific leaf area. The resulting models from this study were used to reconstruct the entire crown of every sample tree for conducting sampling simulations. Résumé : Les distributions verticale et horizontale de la surface foliaire par centimètre de rameau dans la cime de l’épinette hybride (Picea engelmannii Parry ex Engelm. × Picea glauca (Moench) Voss × Picea sitchensis (Bong.) Carrière) ont été modélisées à l’aide des fonctions de Weibull à deux variables et de distribution bêta. La position horizontale est représentée par la position relative sur la branche de premier ordre. La position verticale est basée sur la position de la branche à partir du sommet de l’arbre relativement à sa hauteur étant donné que la base de la cime vivante est souvent difficile à localizer. Des mesures échantillons de la surface foliaire par centimètre de rameau ont été obtenues à l’aide d’un échantillonnage systématique de la cime de 12 arbres choisis dans des peuplements rendus à trois stades de développement (20, 60 et 140 ans). Des fonctions de Weibull à une variable et de distribution bêta ont également été ajustées pour fin de comparaison en utilizant seulement la distribution verticale. En général, la surface foliaire par centimètre de rameau diminuait du sommet des arbres vers le bas et de l’extrémité des branches vers le tronc mais les valeurs étaient très variables. Les arbres dans le peuplement d’âge moyen (60 ans) avaient les plus fortes valeurs moyennes de surface foliaire par centimètre de rameau, suivis par les plus petits et les plus jeunes arbres (peuplement âgé de 20 ans). Les plus faibles valeurs ont été observées chez les plus gros arbres dans le peuplement le plus vieux (140 ans). Comme prévu, les fonctions de Weibull à deux variables et de distribution bêta fournissaient une représentation plus précise de la surface foliaire par centimètre de rameau à l’intérieur de la cime des arbres que les fonctions de Weibull à une variable et de distribution bêta bien que les améliorations soient minimales. Les résultats étaient généralement moins bons pour les arbres du plus vieux peuplement. Ces fonctions pouvaient être utilizées pour évaluer d’autres variables distribuées dans la cime des arbres comme la surface foliaire spécifique. Les modèles développés dans cette étude ont été utilizés pour reconstituer toute la cime de chaque arbe échantillonné afin d’effectuer des échantillonnage simulés. [Traduit par la Rédaction]

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Received 19 December 2002. Accepted 21 May 2003. Published on the NRC Research Press Web site at http://cjfr.nrc.ca on 16 October 2003. H. Temesgen and V.M. LeMay.1 Department of Forest Resources Management, The University of British Columbia, 20452424 Main Mall, Vancouver, BC V6T 1Z4, Canada. I.R. Cameron. J.S. Thrower and Associates, 103-1383 McGill Road, Kamloops, BC V2C 6K7, Canada. 1

Corresponding author (e-mail: [email protected]).

Can. J. For. Res. 33: 2044–2051 (2003)

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doi: 10.1139/X03-127

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Temesgen et al.

Introduction Measuring needle area distribution within tree crowns is important for studying and measuring photosynthesis (Woodman 1971; Richardson et al. 2000) and transpiration (Brooks et al. 1991), developing light extinction models (Kellokami et al. 1980), and determining specific needle area and, by extension, canopy leaf area (Wykoff 2002). Needle area is used as a surrogate for light interception, photosynthetic capacity, and transpiration rates of a tree (Oren et al. 1986; Wykoff 2002). However, measurements are very time consuming, particularly for trees with many small needles. “Projected leaf area” refers to the area of a horizontal projection of the needle on a flat surface (Wykoff 2002), representing nearly one half of the total surface area of leaves. However, leaves mainly receive light from one side. Leaf area measures have been reported at various levels, including needles (Strong and Zavitkovski 1978), shoots (Schulze et al. 1977; Kramer and Kozlowski 1979), and branches (Kershaw and Maguire 1996). Relative measures of leaf area have also been defined for different scales of measurement including needle area density (NAD) (needle surface area (square metres) per crown volume (cubic metres)) (e.g., Wang et al. 1990; Webb and Ungs 1993) and specific leaf area (SLA) (projected leaf area (square centimetres) per unit of dry leaf mass (grams)) (Shelton and Switzer 1984; Gregg 1992). Leaf areas within a tree crown vary over vertical and radial (horizontal) directions. Studies of vertical and (or) horizontal distributions of leaf area, leaf mass, or relative measures (e.g., SLA and NAD) using a variety of scales have been conducted for several species including paper birch (Betula papyrifera Marsh.) (Ashton et al. 1998), Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco var. menziesii) (Woodman 1971; Massman 1982; Jensen and Long 1983; Kershaw and Maguire 1996; Maguire and Bennett 1996), lodgepole pine (Pinus contorta Dougl. ex Loud.) (Gary 1978), Scots pine (Pinus sylvestris L.) (Whitehead 1978), radiata pine (Pinus radiata D. Don) (Whitehead et al. 1990), red pine (Pinus resinosa Ait.) (Stephen 1969), loblolly pine (Pinus taeda L.) (Shelton and Switzer 1984; Xu and Harrington 1998), eastern larch or tamarack (Larix laricina (Du Roi) K. Koch) (Strong and Zavitkovski 1978), western larch (Larix occidentalis Nutt.) (Bidlake and Black 1989), western hemlock (Tsuga heterophylla (Raf.) Sarg.) (Kershaw and Maguire 1996), grand fir (Abies grandis (Dougl. ex D. Don) Lindl.) (Kershaw and Maguire 1996), eastern redcedar (Juniperus virginiana L.) (Gregg 1992), black spruce (Picea mariana (Mill.) BSP) (Strong and Zavitkovski 1978), Norway spruce (Picea abies (L.) Karst.) (Schulze et al. 1977), and hybrid spruce (Engelmann spruce (Picea engelmannii Parry ex Engelm.) × white spruce (Picea glauca (Moench) Voss) × Sitka spruce (Picea sitchensis (Bong.) Carrière)) (Richardson et al. 2000). Generally, more leaves are produced in the upper one half to two thirds of the crown, with better-lit portions producing more foliage than shaded portions (Flemming et al. 1990). For SLA, Gregg (1992) found higher values at the bottom of the crown than at the top of the tree crown for three eastern redcedar crowns. Similar trends in SLA were observed for other coni-

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fers such as loblolly pine (Shelton and Switzer 1984), black spruce, and eastern larch (Strong and Zavitkovski 1978). Schulze et al. (1977) found significant differences in annual growth of needles and twigs of Norway spruce along radial positions of a branch. Wang et al. (1990) found that maximum NAD (up to four times larger than the average NAD) was close to the radial centre of radiata pine tree crowns. Several univariate distribution models have been used to describe the vertical or radial distribution of foliage within tree crowns, including the normal (Stephen 1969), Weibull (Vose 1988; Gillespie et al. 1994), beta (Kellokami et al. 1980; Wang et al. 1990), and Johnson’s SB (Kershaw and Maguire 1996) distributions. In a comparison of models for the radial distribution of foliage, Kershaw and Maguire (1996) found no significant differences in accuracy among Weibull, beta, normal, and Johnson’s SB univariate probability distributions. However, while univariate distribution functions may be easier to interpret, bivariate distribution functions provide flexibility in assessing the clumpy nature of foliage distributions in tree crowns and provide more information on the vertical and radial distribution of foliage (Wang et al. 1990). In this study, bivariate Weibull and beta distribution functions were used in predicting twig leaf area from variables indicating the vertical and horizontal positions for hybrid spruce (Engelmann spruce × white spruce × Sitka spruce) tree crowns sampled from northwestern British Columbia. Each crown was considered a population of twigs, following the approach of Wilson (1989). The resulting spatial models were compared with spatial models based on vertical position alone. Hybrid spruce was selected for study, as this species occupies 15.2% of the forested lands of British Columbia (B.C. Ministry of Forests 1994). Horizontal position was represented by relative position of the twig on the first-order branch. Vertical position was defined as the branch position from the tree apex relative to tree height, since irregularly spaced branches at the base of the live crown often make the base difficult to locate. Since variation in twig length strongly contributes to the differences in amount and spatial distribution of leaf area and light interception in tree crowns (Wilson 1989), leaf area (square centimetres) per centimetre of twig (APCM) was used to represent leaf area in various vertical and horizontal points of tree crowns. As a measure of leaf area per unit twig length (centimetres), APCM could be used as input into light models (Brunner 1998). Other variables, such as SLA, could be modelled in a similar fashion using bivariate distribution functions, as demonstrated in this study. The final fitted models were used to populate the entire tree crowns of sampled trees and to estimate individual tree leaf area and were used in simulations to study sampling alternatives (Temesgen 2003).

Methods Study area The study sites were located in the moist, cold subzone of the Interior Cedar–Hemlock biogeoclimatic zone (ICHmc2) near Hazelton in northwestern British Columbia (55°55′N, 126°10′W) (Clement and Banner 1992). In this area, the Interior Cedar–Hemlock forest type represents a transition be© 2003 NRC Canada

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Can. J. For. Res. Vol. 33, 2003

tween true coastal and interior forests (Meidinger and Pojar 1991). Three study sites were selected with stand ages of 20, 60, and 140 years to obtain a range of tree bole and crown sizes. The two older stands were located within the Date Creek research area managed by the B.C. Ministry of Forests, while the youngest stand was in an adjacent wildfire stand known locally as the Ken Fire. The stands experience similar climates and are considered to be similar forest types. All of these stands were established following catastrophic fires (Ashton et al. 1998; Richardson et al. 2000). The stands are mixed species dominated by very shadetolerant species such as western hemlock and western redcedar (Thuja plicata Donn. ex D. Don). Other species present in the study sites included subalpine fir (Abies lasiocarpa (Hook.) Nutt.), lodgepole pine, paper birch, and hybrid spruce (Sutton et al. 1994). Data description From each stand, four of the tallest hybrid spruce trees were sampled between July and August. The tallest trees were selected to ensure that the selected trees had relatively uniform crowns and straight boles and were mainly free from past crown damage. The diameter outside bark at 1.3 m above ground (DBH) was measured and the four cardinal directions were marked at breast height with different colours of paint. Each sample tree was then felled between 0.0 and 0.3 m above ground and the total tree height (HT) (metres), height to the base of live crown (BLC) (metres), and crown width (CW) (metres) were measured on the cut stem. A disk was then removed at 1.3 m above ground and annual rings were counted (breast height age (BHAGE)). On each tree, each first-order branch in the crown was classed into one of the four cardinal directions (i.e., branches that were within 45° of north were classed as “north”), and the total number of first-order branches in each cardinal direction was counted. Only whorled first-order branches were included. To ensure complete coverage of the tree crown, samples were selected systematically in both vertical and horizontal directions. Some of the first-order branches did not have foliage close to the trunk; therefore, sampling at a fixed interval (e.g., samples taken at a 10- or 20-cm interval) was not used, since this might not provide large enough sample sizes or ensure good spatial coverage of the population of twigs. From the base of the live crown, the first and then every fourth first-order branch were selected by cardinal direction. On each selected first-order branch, the branch length and internodal distances between second-order branches were measured. From each selected first-order branch, the first and then every fourth second-order branch (starting from the branch tip inward) were systematically sampled. On the selected second-order branches, the branch length and distance

[1]

from the tip of the first-order branch were measured. When a selected second-order branch had more than one thirdorder branch, the number of third-order branches was counted and one of them was randomly selected. Also, the length of the selected third-order branch and its distance from the tip of the second-order branch were measured. The same selection procedure was used when a selected thirdorder branch had more than one fourth-order branch. Assuming that leaves older than 5 years do not contribute substantially to the total leaf area of hybrid spruce crowns, a twig was defined as an axis of a sampling unit arising from any order of branch, comprising the last 5 years of foliage for this study. On each selected second-, third-, or fourthorder branch, all needles less than 6 years old were clipped and placed in a paper bag. The length of each sampled twig was measured to the nearest centimetre. Samples were taken each morning and transported to a laboratory. Projected leaf area was measured using a leaf area meter (model CI-202, manufactured by CID, Inc., Camas, Wash.). Leaf area (square centimetres) and twig length (centimetres) were measured on the same day as clipping occurred to avoid measurement errors arising from leaf shrinkage. Preliminary calculations The crown length (CL = HT – BLC) and crown ratio (CR = CL/HT) were calculated for each tree. For each branch, the relative height above ground was calculated as the position of a branch from the apex of the tree relative to the total tree height (ZI), varying from 0 (tree apex) to 1 (ground level). For each sampled twig, the APCM and relative radial length (HI) were calculated. Since HI was calculated as the ratio of the radial distance (metres) from the tree trunk to the sampling point relative to the first-order branch length (metres), 1 indicates the tip of the branch, whereas 0 indicates the connection of the branch with the tree trunk. Needle area distribution models Although cardinal direction was recorded for each sampled branch, preliminary analyses indicated that there was very little difference in leaf area measures over the cardinal directions. This could be the result of selecting the tallest trees with more uniform crowns. Therefore, cardinal direction was not included in the spatial distribution model for APCM. The bivariate Weibull (Lu and Bhattacharyya 1990) and beta distribution functions (A-Grivas and Asaoka 1982) were selected to predict leaf APCM using the relative heights (vertical position) and the relative radial lengths (horizontal position) because of the wide variety of shapes that can be modelled using these functions. The bivariate Weibull distribution function is given as

β × β2  z=  1 [(x / β 3)β 1 / (β 5 −1) (y / β 4)β 2 / (β 5 −1) ] {[(x / β 3)β 1 / β 5 + (y / β 4)β 2 / β 5 ](β 5 − 2)} β 3 × β 4  × {[(x / β 3)β 1 / β 5 + (y / β 4)β 2 / β 5 ]β 5 + (1/ β 5 − 1)} exp{−[(x / β 3)β 1 / β 5 + (y / β 4)β 2 / β 5 ]}β 5 + ε1 © 2003 NRC Canada

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2047 Table 1. Summary of tree size and leaf area per centimetre of twig (APCM) values by stand age. Stand age (years)

No. of sample twigs

Variable

20 (4 trees)

60 (4 trees)

140 (4 trees)

All (12 trees)

DBH (cm) Height (m) Crown width (m) Crown length (m) APCM (cm2/cm) DBH (cm) Height (m) Crown width (m) Crown length (m) APCM (cm2/cm) DBH (cm) Height (m) Crown width (m) Crown length (m) APCM (cm2/cm) DBH (cm) Height (m) Crown width (m) Crown length (m) APCM (cm2/cm)

471

988

1289

2748

where β1 to β5 are model parameters to be estimated, 0 < β 5 ≤ 1, β1 to β4 > 0, and ε1 is the prediction error. For the leaf area models, z is APCM, x is ZI, and y is HI. This model was constructed based on survival functions by Marshall and Olkin (1967) and later updated by Lu and Bhattacharyya (1990). A-Grivas and Asaoka (1982) used the bivariate beta distribution function to assess the joint distribution of cohesive and friction coefficients in slope stability analysis. The bivariate beta distribution function is given as [2]

SD

Minimum

Maximum

1.96 0.75 0.69 1.02 0.92 8.99 1.05 0.13 0.35 1.21 1.69 2.07 0.44 1.94 1.28 14.15 11.25 0.78 4.19 1.21

5.90 4.70 1.80 3.50 0.14 16.50 14.00 4.10 8.82 0.16 40.00 30.40 3.70 12.65 0.01 5.90 4.70 1.80 3.50 0.01

11.00 6.55 3.50 6.29 5.94 38.50 16.90 4.45 9.80 8.15 44.40 36.40 4.86 17.22 12.22 44.40 36.40 4.86 17.22 12.22

tions given as eqs. 1 and 2 are not restricted to the 0 to 1 range, as they are not in the cumulative form. However, the multiplier was introduced to allow for greater flexibility. For comparison, the univariate Weibull and beta distribution functions using only the vertical position were also fitted to assess any improvements obtained using both relative vertical and horizontal position in the bivariate models. The univariate Weibull distribution function is given as [3]

z = {(β11 / β10)[(x − β 9)/ β10 ]β 11 −1} × exp{−[(x − β 9)/ β10 ]β 11 } + ε 3

 Γ(β 6 + β 7 + β 8)  z=   Γ(β 6) + Γ(β 7) + Γ(β 8)  × x

Mean 8.54 5.49 2.54 4.92 2.92 22.72 15.29 4.26 9.32 3.06 42.33 33.42 4.27 15.22 2.67 29.49 22.12 3.97 11.33 2.85

and the univariate beta distribution function is given as

β 6 −1 β 7 −1

y

β 8 −1

(1 − x − y)

+ ε2

where 0 ≤ (x + y)/2 ≤ 1, x ≥ 0, y ≥ 0, and β6 to β8 > 0 are model parameters to be estimated, ε2 is the prediction error, and Γ(t) is the gamma distribution with parameter t. For the prediction of APCM, z is APCM, x is ZI/2, and y is HI/2. Bivariate distribution functions were fitted for each sample tree using a derivative-free (DUD), nonlinear least squares optimization technique in PROC NLIN (SAS Institute Inc. 2000). To find the global minimum, starting values for each parameter were varied and several runs were performed, and the analysis was repeated using the Gauss– Newton method. Also, for the bivariate beta distribution, an initial grid search was performed, since there are only three parameters. Each of these two functions was also modified, for more flexibility, by introducing an overall scaling coefficient (multiplier) and refitted, allowing this multiplier to vary. Yang et al. (1978) used a multiplier to alter the cumulative form of the univariate Weibull function, since the cumulative form represents cumulative probabilities and is, therefore, restricted to predicted values from 0 to 1. Predicted values from the bivariate Weibull and beta distribu-

[4]

 Γ(β + β13)  z = [(x)β 12 −1 (1 − x)β 13 −1 ]  12  + ε4  Γ(β12)Γ(β13) 

where β9 < x < ∞, β9, β10, and β11 are the location, scale, and shape parameters, respectively (Clutter et al. 1983), ε3 and ε4 are the prediction errors, and β12 and β13 are model parameters to be estimated. For the prediction of APCM, z is APCM and x is ZI. As with the bivariate distribution functions, a multiplier was introduced, and the functions were refitted, allowing this multiplier to vary along with the other parameters. The two bivariate and the two univariate models were compared using I2 and root mean square error (RMSE) values defined as (y i − y$i) 2 (y i − y ) 2

[5]

I2 = 1 −

[6]

RMSE =

(y i − y$i) 2 n © 2003 NRC Canada

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Fig. 1. Leaf area (square centimetres) per centimetre of twig (APCM) versus relative vertical and horizontal position by stand age. Relative vertical position (Rel. Vert.) goes from the apex of the tree (0) to the tree base (1). Relative horizontal position (Rel. Hor.) goes from the branch tip (1) to the tree bole (0).

where yi and y$i are measured and predicted APCM, respectively, and n is the number of sampled twigs.

Results and discussion Because of the difficulty in obtaining detailed leaf area measures over hybrid spruce crowns, only 12 sample trees were measured; however, this resulted in 2748 measures of leaf area (Table 1) and is comparable with other studies of leaf area distributions (e.g., Wang et al. 1990; Whitehead et al. 1990). Sample trees ranged from 4.70 to 36.4 m in height, from 5.9 to 44.4 cm in DBH, from 1.8 to 4.68 m in CW, and from 3.50 to 17.22 m in CL. Within the tree crowns, APCM was found to be highly variable (Fig. 1), as is typical for spruce leaf attributes (Sutton et al. 1994; Richardson et al. 2000). The maximum APCM was four times as large as the average APCM (Table 1), with higher values for the older trees. Wang et al. (1990) found wide variability of leaf area measures within radiata pine crowns, with a maximum NAD four times larger than the average NAD. Negative correlations between APCM and relative height indicate a decrease in APCM from the apex of the tree downward (Table 2), as shown in Fig. 1. The strongest negative correlations occurred for the youngest trees. Correlations with relative position on the branch varied over the 12 trees, indicating a less predictable trend, likely confounded by the changes in APCM due to relative height differences. Generally, an increase toward the tree apex and branch tips would be expected because of the higher density of younger leaves (higher leaf area) at the tip of the first-order branches and the apex of the tree (Schultze et al. 1977; Kershaw and Maguire 2000). For most sample hybrid spruce tree crowns in this study, both the bivariate Weibull and beta distribution functions provided moderately precise APCM estimates for the young

Table 2. Pearson correlations for leaf area (square centimetres) per centimetre of twig (APCM) with relative height above ground (ZI) (increases from the apex to the base of the tree) and relative position on the branch (HI) (increases from the tree bole to the branch tip). Stand age (years) 20

60

140

Tree No.

No. of sampled twigs

ZI

HI

1 2 3 4 5 6 7 8 9 10 11 12

109 120 101 141 237 269 237 245 428 378 217 266

–0.5432* –0.4849* –0.3495* –0.4023* –0.3513* –0.2452* –0.4025* –0.5006* –0.1719* –0.2471* 0.2846* 0.2270*

0.5173* 0.1528 0.2229* 0.1214 0.0253 0.0846 0.0288 0.2021 –0.0189 0.0347 0.1906* –0.0459

*Correlations significantly different from zero (p < 0.05).

and middle-aged trees, except for tree 6, which was more variable (Table 3). For tree 3, the inclusion of a multiplier to scale the bivariate beta distribution resulted in an improvement in the sum of squared errors, but this did not occur for other trees or for the fits of the Weibull bivariate distribution. For the oldest stand, RMSE values were similar to those of the younger trees, but there were some negative I2 values, indicating that using the average of the APCM values would be better than using the predictions from the model. This was consistent with the simple correlations, which were poorer for the older trees (Table 2). Larger variability in older trees resulted in poorer predictions. Results © 2003 NRC Canada

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Table 3. Root mean square error (RMSE) (square centimetres per centimetre) and I2 values for leaf area (square centimetres) per centimetre of twig (APCM) for the bivariate and univariate prediction models. Bivariate

Univariate

Weibull Stand age (years) 20

60

140

Beta

Weibull*

Beta*

Tree No.

No. of sampled twigs

RMSE

I2

RMSE

I2

RMSE

I2

RMSE

I2

1 2 3 4 5 6 7 8 9 10 11 12

109 120 101 141 237 269 237 245 428 378 217 266

0.72 0.59 0.75 0.64 0.89 1.29 1.18 0.84 0.78 1.02 1.06 1.61

0.33 0.29 0.14 0.27 0.24 0.07 0.27 0.27 0.06 0.07 –0.11 –0.01

0.75 0.64 0.75* 0.72 0.93 1.32 1.27 0.87 0.78 1.02 0.96 1.56

0.28 0.18 0.14* 0.07 0.18 0.02 0.16 0.22 0.06 0.07 0.09 0.05

0.71 0.60 0.76 0.67 0.96 1.31 1.25 0.86 0.79 1.02 1.06 1.54

0.36 0.28 0.13 0.21 0.12 0.05 0.19 0.24 0.04 0.07 –0.09 0.07

0.71 0.60 0.76 0.67 0.96 1.30 1.25 0.86 0.78 1.02 1.00 1.54

0.36 0.28 0.13 0.21 0.12 0.06 0.19 0.25 0.05 0.07 0.03 0.07

Note: Relative height above ground (ZI) increases from the apex to the base of the tree and relative position on the branch (HI) increases from the tree bole to the branch tip. *An extra parameter, a multiplier, was added to the equation to improve the fit.

Fig. 2. Predicted leaf area (square centimetres) per centimetre of twig (Pred. APCM) using the fitted bivariate Weibull distribution function for tree 3 (young), tree 6 (middle stand age), and tree 9 (oldest age). Relative vertical position (Rel. Vert.) goes from the apex of the tree (0) to the tree base (1). Relative horizontal position (Rel. Hor.) goes from the branch tip (1) to the tree bole (0).

were slightly better for the bivariate Weibull distribution model for most trees. Predictions of APCM using the bivariate Weibull and beta distribution functions were expected to be more precise (lower RMSE) than those using the univariate distribution functions, since the horizontal position was also included as a predictor (Table 3). However, by adding the multiplier to the univariate distribution functions, similar fit statistics were obtained for the univariate versus the bivariate distributions for the youngest and oldest trees. Most gains in using

the horizontal position were for the middle-aged stand using the bivariate Weibull distribution, with an increase in I2 of over 0.10 for tree 5. These results indicate that the variation in the horizontal direction is small relative to the variation in the vertical direction and that variables other than spatial position would be needed to improve the APCM prediction. Wang et al. (1990) also observed similar trends for NAD in radiata pine crowns. Because of the nature of the functional form, all predicted values for APCM were greater than zero (Fig. 2). Using the © 2003 NRC Canada

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Can. J. For. Res. Vol. 33, 2003 Table 4. Estimated parameters for the bivariate Weibull leaf area (square centimetres) per centimetre of twig (APCM) prediction models using branch position from the apex of the tree relative to total tree height (ZI) and relative position in the branch (HI) as predictor variables. Bivariate Weibull Stand age (years) 20

60

140

Tree No.

β1

β2

β3

β4

β5

1 2 3 4 5 6 7 8 9 10 11 12

0.019 990 0.000 124 0.053 000 0.016 800 0.313 000 0.264 100 0.022 100 0.128 800 0.126 500 0.124 800 0.132 500 0.000 014

0.000 000 009 9 0.000 000 599 0 0.000 029 000 0 0.000 025 000 0 0.001 340 000 0 0.000 012 000 0 0.000 054 000 0 0.000 031 000 0 0.000 000 732 4 0.000 002 484 0 0.000 001 386 0 0.000 000 190 3

1.1684 1.0450 0.8960 1.0175 0.6532 0.6736 0.5776 0.7339 0.5316 5.0000 1.0193 1.5673

0.000 000 012 0 0.000 000 519 0 0.000 076 000 0 0.000 087 000 0 0.009 540 000 0 0.002 660 000 0 0.000 152 000 0 0.000 075 000 0 0.000 480 000 0 0.000 035 000 0 0.000 026 000 0 0.000 000 032 4

0.003 940 000 0 0.000 026 000 0 0.005 250 000 0 0.000 871 000 0 0.012 400 000 0 0.000 517 000 0 0.002 610 000 0 0.023 800 000 0 0.000 148 000 0 0.000 793 000 0 0.006 670 000 0 0.000 005 153 0

fitted bivariate Weibull functions (Table 4), predicted APCM values showed an increase toward the apex (relative vertical position of 0) and a very slight increase toward the branch tips, as indicated for the three trees in Fig. 2. As indicated by the very slight differences between the bivariate and univariate models, the trend with relative horizontal position (branch position) is only slight, whereas the trend with relative vertical position is more pronounced.

Acknowledgments The financial support of the B.C. Ministry of Forests, Research Branch, and the Natural Sciences and Engineering Research Council of Canada and the data collection and analysis of this research by Forest Renewal BC are gratefully acknowledged.

References Conclusions Leaf area measures for hybrid spruce are highly variable within the tree crown, particularly for older, larger trees. For modelling APCM, the bivariate Weibull distribution using both relative position and relative height as predictors resulted in slightly better models than using the bivariate beta distribution function or univariate Weibull or beta functions. Because of the very great variability, many sets of starting parameters and more than one search method were used to try to obtain a global minimum. For some trees, including a scaling parameter (multiplier) for greater model flexibility improved the search results, particularly for the univariate distribution functions. In this study, the crown was considered to be a population of twigs. The final fitted models were used to populate the entire tree crowns of sampled trees and used in a simulations study on sampling alternatives. Other variables, such as SLA, could be modelled in a similar fashion using bivariate distribution functions, as demonstrated in this study. Although the intensity of sampling for each tree was quite high, only 12 trees were included because of very high sampling costs. The models developed are therefore reliable for the sampled trees and indicate difficulties in fitting leaf area spatial distribution models for older trees. Also, only the tallest trees were selected. If hybrid spruce trees from other positions in the canopy were selected, cardinal direction, as well as vertical and horizontal position, could be useful in predicting the APCM for the crown population of twigs. Finally, the use of mixed nonlinear models could be used to model the variability of parameters among trees.

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