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Variable-exponent taper equations for jack pine, black spruce, and balsam fir in eastern Canada
Forest Ecology and Management 198 (2004) 39–53
Variable-exponent taper equations for jack pine, black spruce, and balsam fir in eastern Canada Mahadev Sharma*, S.Y. Zhang Resource Assessment and Utilization Group, Forintek Canada Corp., 319 rue Franquet, Sainte-Foy, Que., Canada G1P 4R4 Received 3 December 2003; received in revised form 2 March 2004; accepted 9 March 2004
Abstract A variable-exponent taper equation was developed for jack pine (Pinus banksiana Lamb.), black spruce (Picea mariana (Mill.) B.S.P.) and balsam fir (Abies balsamea (L.) (Mill.)) trees grown in eastern Canada. The equation was derived from the dimensionally compatible taper equation. Stem analysis data from several of Forintek’s research projects on these tree species grown across eastern Canada were used to fit the taper equation, and the equation was then evaluated with independent data collected by Forintek and other organizations in eastern Canada. The taper equation presented in this paper was superior to the segmented polynomial, variable-exponent, and variable-form taper equations developed for different tree species in estimating diameters along the bole. Moreover, stand density or thinning effect on the taper of these tree species was examined, and the taper equation was modified in order to incorporate this effect. # 2004 Elsevier B.V. All rights reserved. Keywords: Tree taper and form; Stand density; Thinning; Nonlinear regression
1. Introduction Stem profile equations, commonly known as taper equations, are very useful in estimating the stem volume of trees and reconstructing the three-dimensional (3D) stem form for the simulation and optimization of stem bucking and log sawing. These equations can provide estimates of the diameter at any height along the bole of a tree and hence have long been the subject of research of many forest scientists. Although tree boles cannot be completely described in mathematical terms, it is common and convenient to
* Corresponding author. Tel.: þ1-418-659-2647; fax: þ1-418-659-2922. E-mail addresses: [email protected] (M. Sharma), [email protected] (S.Y. Zhang).
assume that segments of a tree stem approximate various geometric solids. The lower bole portion is generally assumed to be a neiloid frustum, the middle portion a paraboloid frustum, and the upper portion a cone (Avery and Burkhart, 2002). Valentine and Gregoire (2001), however, reported that the segments of slash pine (Pinus elliottii Engelm.), ponderosa pine (Pinus ponderosa Doug. Ex P. & C. Laws.), and yellow poplar (Liriodendron tulipifera L.) did not conform to these shapes. They further reported that the top and middle segments of sweet gum (Liquidambar styraciflua L.) trees resembled cones and paraboloid, respectively. It is obvious that a simple regression model may not be adequate to describe the tree form. Numerous models of varying complexity have been advanced in attempts to describe tree taper. Taper models presented by Kozak et al. (1969), Ormerod
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.03.035
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
(1973), Amidon (1984), Reed and Byrne (1985), Sharma and Oderwald (2001), and Sharma et al. (2002) are examples of simple taper functions. On the other hand, more complex taper equations have been developed that account for the three segments of a tree bole. The most common approaches applied to produce complex taper models are: (i) assuming a tree is made up of two or more segments with form being constant within a segment and different between segments, and (ii) allowing the tree form to vary from one point to another along the bole and expressing the variable form by a single continuous function. Segmentedpolynomial taper models developed by Max and Burkhart (1976), Demaerschalk and Kozak (1977), Cao et al. (1980), and Fang et al. (2000) are examples of the first approach where each segment of a tree is described by a polynomial. Examples of the second approach are the variable-exponent taper model elaborated by Kozak (1988) and the variable-form taper model presented by Newnham (1992). In these models, a single continuous function with an exponent changing from stump to top describes the neiloid, paraboloid, and several intermediate forms (Kozak, 1997). Flewelling and Raynes (1993) also developed a variable-form taper model based on a system of three equations. Unfortunately, most of these taper equations are species-specific i.e. the model accuracy in estimating diameters depends on tree species. Moreover, fitting a segmented-polynomial taper equation requires all diameter–height pair measurements for all segments from the stump to the tip of a tree. This type of data, however, is not always available. Diameter–height pair data collected by wood industries and some research organizations often only cover up to a merchantable top (e.g. 10 cm (inside bark) upper diameter class). Therefore, these organizations either use taper equations produced for other species or rely on inaccurate ones for volume estimations. Furthermore, foresters have long realized that trees grown at different stand densities do not necessarily have the same tree height or tree form. Stand density can be regulated either by planting the trees at different initial spacings or by thinning the stands to different densities. However, the trees of a particular species grown in a plantation and in a natural stand thinned to the same density may not have the same
form, especially if the thinning is carried out at a later age. Tree taper is affected by thinning (Sharma et al., 2002; Tasissa et al., 1997; Zhang et al., 1998; Koga et al., 2002). Although other stand variables have been added to the taper equation studied in the past, little attempt has been made to quantify the stand density or thinning effect on tree taper or other growth characteristics that may affect product recovery (Burkhart and Bredenkamp, 1989; Zhang et al., in press). Muhairwe et al. (1994) studied the effects of adding crown class, crown ratio, site class, breast height and age to Kozak’s variable-exponent taper equation for Douglas fir (Pseudotsuga menziesii (Mirb.) Franco), western red cedar (Thuja plicata Donn) and aspen (Populus tremuloides Michx.) trees. Similarly, Burkhart and Walton (1985) examined the effect of incorporating crown ratio into a segmented-polynomial taper equation for loblolly pine (Pinus taeda) trees. Both studies found that adding these stand variables into the taper equations did not significantly improve their goodness of fit. The objective of this study was to investigate a taper equation that is appropriate for jack pine (Pinus banksiana Lamb.), black spruce (Picea mariana (Mill.) B.S.P.), and balsam fir (Abies balsamea (L.) (Mill.)), the most important commercial tree species in eastern Canada. This study was also intended to investigate and quantify the effect of stand density and/or thinning on the stem taper of these species.
2. Data Data used to estimate the parameters and to evaluate the taper equations in this study were obtained from many different sources. Parameters of the taper equations for jack pine were estimated using the data collected from a jack pine initial spacing (IS) and pre-commercial thinning (PCT) study. These studies, funded by the Living Legacy Trust and Tembec, are part of a jack pine task force initiated by Forintek Canada Corp (Zhang, 1999). The plots used for the jack pine IS study were established in 1941 near Wellston in Manistee County, Michigan. One-yearold seedlings were planted at spacings of 0:46 m 0:46 m, 0:91 m 0:91 m, 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m, with each spacing
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
replicated five times on square 0.162 ha plots arranged in a Latin Square Design. One 1:52 m 1:52 m plot was thinned to a 2:13 m 2:13 m spacing at age 13. In 2001, 151 trees (3–8 trees in each diameter class were sampled for stem analysis from spacings 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m). For more information on this trial, refer to Guilkey and Westing (1956). The plots used for the jack pine PCT study were established by the New Brunswick Department of Natural Resources and Energy at Eel River Bridge on the Lower Miramichi in 1966. They were originated from fire in 1941. At the time of the establishment, some stands were thinned to densities of 7215, 4278, 2212, and 1413 trees/ha resulting in approximately 1:22 m 1:22 m, 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m spacings, respectively. Unthinned stands contained approximately 72,000 trees/ha. In 2000, 154 trees (6 trees per diameter class per spacing excluding 2:74 m 2:74 m) were sampled for stem analysis and product recovery studies. For more information about the trials, refer to a report by the New Brunswick Department of Natural Resources and Energy (Nbdnre, 1987). The parameters of the taper equations for black spruce trees were estimated using the data from a black spruce initial spacing study. The trial was established in 1950 by the Ontario Department of Lands and Forests, now the Ministry of Natural Resources, at two locations near Thunder Bay. Initially, trees were planted at three spacings: 1:83 m 1:83 m (3086 trees/ha), 2:74 m 2:74 m (1372 trees/ha), and 3:66 m 3:66 m (780 trees/ha). Three replications per spacing regime were initially included in the trial. Some plots resulted in 2066 and 2500 trees/ ha because the seedlings that died after planting were not replaced. For more information about these trials, refer to a report by Zhang and Chauret (2001). In 1998, approximately six trees per merchantable diameter class were randomly selected from stands with densities of 1372, 2066, 2500, and 3086 trees/ha. In total, 139 sample trees with no major defects were selected from the four densities. No tree was sampled from the stand with a density of 780 trees/ha as the stand density was obviously too low for black spruce. The parameters of the taper equations for balsam fir trees were estimated using the data from a PCT study (Zhang et al., 1998). The trial is located in the Bas
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Saint-Laurent Region of Quebec. The balsam fir stand for the trial was regenerated from a clearcutting in 1948. A 4 ha stand was divided into 100 plots of 0.04 ha each, and was manually thinned to densities ranging from 500 to 6270 trees/ha in 1960. In total, 150 trees (6 trees per merchantable diameter class per density) were sampled from three stands with densities of 1500, 3000, and 12,100 trees/ha in 1995. These densities correspond to heavily thinned, moderately thinned, and unthinned (control) stands, respectively. Jack pine tree data for the evaluation of taper equation were obtained from three even-age natural stands of 50-, 70-, and 90-year-old studied by the jack pine task force. These stands were located at the Romeo Malette Forest Management Unit in the district of Timmins, Northeastern Ontario and were regenerated from fire. The stands were composed of approximately 73% jack pine, 26% black spruce and the remainder birch. In total, 147 trees (47, 51, and 49 trees from stands aged 50, 70, and 90 years, respectively) were collected for the evaluation. The evaluation data for black spruce came from Northern Ontario and were provided by the Ontario Forest Research Institute. Similarly, balsam fir tree data for the evaluation were collected by Laval University from a naturally regenerated pre-commercial thinning trial located in the Bas Saint-Laurent Region of Quebec (Koga et al., 2002). The regeneration of the trail was taken place in 1955 and the sample trees were collected in 1993. In each study described above, the sample trees were selected and felled for stem analysis. In addition to total tree height and outside bark diameter at breast height (dbh), inside bark diameters were measured at the stump and every meter along the bole to a minimum merchantable diameter (e.g. 10, 8 or 4 cm top). Data used to evaluate the taper equations for black spruce and balsam fir, however, consisted of diameter– height observations measured every meter up to the tip of the tree. Summary statistics for total height and dbh of all trees used in fitting and evaluating the taper equations are presented in Table 1.
3. Taper equations The taper equations widely used in the United States and Canada are the segmented polynomial of
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Table 1 Summary statistics for total height and dbh of jack pine, balsam fir, and black spruce trees used in this study Species
Number of trees
Mean height (m)
S.D.
Mean dbh (cm)
S.D.
Fit data Jack pine IS Jack pine PCT Balsam fir Black spruce
140 154 150 135
17.68 15.44 13.88 15.08
1.96 1.71 2.15 2.27
18.77 16.36 17.65 16.85
4.57 3.88 5.11 3.73
Validation data Jack pine (age 50) Jack pine (age 73) Jack pine (age 90) Balsam fir Black spruce
47 50 49 59 131
19.39 19.74 22.16 11.26 14.48
2.03 2.39 2.02 1.77 5.56
18.07 19.72 21.97 13.48 13.99
4.59 4.84 5.26 2.95 4.30
Max and Burkhart (1976) and the variable exponent of Kozak (1988). These equations are written as follows. Segmented polynomial: 2 d ¼ b1 ðz 1Þ þ b2 ðz2 1Þ þ b3 ða1 zÞ2 I1 D þ b4 ða2 zÞ2 I2
(1)
where d is the diameter inside bark at any given height h; D the diameter at breast height outside bark; H the total tree height from ground to tip; z the h/H; a1 and a2 the upper and lower join points, respectively, of three segments; and b1–b4 the parameters. Similarly 1 if a1 z 0 I1 ¼ and 0 otherwise 1 if a2 z 0 I2 ¼ 0 otherwise Variable exponent: 2 þg
d ¼ g0 Dg1 g2 D X g3 z
4
lnðzþ0:001Þþg5 z0:5 þg6 ez þg7 ðD=HÞ
(2)
where X ¼ ð1 ðzÞ1=2 Þ=ð1 p1=2 Þ, p the lower join point (inflection point), g0–g7 the parameters, and other variables are as defined above. Zakrzewski (1999) developed a variable-form taper equation for jack pine grown in Ontario as 2
caz ¼ K
3
z þ y 1 z þ y2 z zs
z0 ¼ 11:3=H, C the inside bark cross-sectional area at the breast height (1.3 m), y1 and y2 the parameters, and other variables are as defined previously. Sharma and Oderwald (2001) developed a dimensionally compatible taper equation based on dimensional analysis as 2d h Hh 2 dout ¼ D2 (4) hD H hD where dout is the outside bark diameter at height h, hD the breast height and d a parameter known as the form parameter. Even though Eq. (4) has the simplest form with only one parameter, it was superior to the segmented-polynomial taper equation in predicting inside bark diameters of loblolly pine trees up to 60% of total height. Above 60% of the total height, however, Eq. (4) resulted in slightly larger bias than Eq. (1) (Sharma and Oderwald, 2001). In addition, Eq. (4) cannot be used to estimate inside bark diameter using the dbh measured outside bark. However, the equation can be modified in order to estimate the inside bark diameters using outside bark dbh by including a constant a as follows: 2d h Hh 2 2 d ¼ aD hD H hD The constant, a can be calculated as
4
(3)
where K ¼ Cðz0 sÞ=ðz20 þ y1 z30 þ y2 z40 Þ, caz the inside bark cross-sectional area, s ¼ 1 þ H=D,
a¼
D2i D2
where Di is the inside bark dbh. Eq. (4) can be further modified to result in a variable-exponent taper
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
equation where the tree form changes continuously along the bole. Since the form at a particular point depends on its relative height from the ground, the form parameter can be expressed in terms of the relative height (z) to result in a variable-exponent taper equation, i.e. 2f ðzÞ h Hh 2 2 d ¼ aD hD H hD Several functions of z (linear, quadratic, exponential, and their combinations) were examined. The taper equation with a quadratic function as a variable exponent, f ðzÞ ¼ d1 þ d2 z þ d3 z2 , where, d1, d2, and d3 are parameters, well described the tree profiles for the tree species used in this study. Rearranging the variables this taper equation results in: 2 2ðd1 þd2 zþd3 z2 Þ d h Hh ¼ d0 (5) D hD H hD Since the constant a is unknown, it has been replaced by another parameter d0. Similarly, Eq. (5) can be easily modified to accommodate stand density effect. Since the exponent of the height solely determines the tree form, the stand density effect can be incorporated by adding a stand density function to the exponent, i.e. 2 2ðd1 þd2 zþd3 z2 þd4 gðsdÞÞ d h Hh ¼ d0 D hD H hD (6) where g(sd) is a function of stand density and d4 a parameter. The best function that described the stand density effect in this case was gðsdÞ ¼
1 sd
where sd is the stand density (trees/ha) at the time of plot establishment. Thus, Eq. (6) with the added stand density function becomes 2 2ðd1 þd2 zþd3 z2 þd4 =sdÞ d h Hh ¼ d0 (7) D hD H hD Eq. (5) was compared with the segmented-polynomial (Eq. (1)), variable-exponent (Eq. (2)), and variableform (Eq. (3)) taper equations. Eq. (7) was evaluated
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by comparing with and without the density effect in the model. As we see Eqs. (1) and (5) contain the same dependent variable (d2/D2). Dependent variables in the other two equations, however, differ from d2/D2. Therefore, the model coefficient of determinations (R2) and model mean square errors (MSE) cannot be compared directly for evaluation. As a result, comparison was performed by examining the estimated coefficient of determination (I2) of the original (untransformed) independent variable (d) and the overall standard error of estimate (S.E.E.). These parameters can be calculated (Kozak and Smith, 1993) as follows: I2 ¼
SSd SSres SSd
(8)
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ 2 i¼1 ðdi d i Þ S:E:E: ¼ nk
(9)
where SSd ¼
n X
2; ðdi dÞ
SSres ¼
i¼1
n X ðdi d^i Þ2 i¼1
di is the actual observation (untransformed) of the dependent variable (diameter inside bark (cm) in this case), d^i the predicted value of the actual observation, d the average of the actual observations, n the number of observations, k the number of estimated parameters. These equations were also compared by calculating bias in estimating diameters at different points along the bole of the trees set aside for model evaluation. Since no independent data including density information were available, Eq. (7) was evaluated by comparing bias in estimating diameters along the bole of the fit data set with and without the stand density function in the model.
4. Results and discussion Eqs. (1)–(3) and (5) were fitted to the estimation data set (the data designated for parameter estimation) using the NLIN procedure in SAS (1984). Since most of the data contained height–diameter pairs measured up to a 10 cm (inside bark) top, the upper inflection point (join point) in the case of a
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Table 2 Parameter estimates (standard errors in parentheses) for Eqs. (1)–(3) and (5) fit to parameter estimation data using nonlinear regression Parameter
Jack pine (IS)
Eq. (1)
b1 b2 b3 b4 S.E.E. I2
2.1151 (0.2860) 0.7342 (0.1724) 0.8273 (0.2080) 8.4616 (0.7804) 0.86675 0.9785
2.7838 (0.1974) 1.1285 (0.1183) 1.4219 (0.1458) 39.0441 (0.7510) 0.69936 0.9823
4.6363 (0.3074) 2.3322 (0.1851) 2.3278 (0.2256) 26.3322 (0.9936) 0.93129 0.9797
4.1998 (0.3769) 2.0288 (0.2269) 2.0953 (0.2764) 29.9728 (1.1620) 0.94234 0.9710
Eq. (2)
g0 g1 g2 g3 g4 g5 g6 g7 S.E.E. I2
0.2568 (0.1078) 1.0943 (0.0547) 0.0061 (0.0030) 0.1180 (0.1551) 0.00086 (0.0368) 0.2194 (0.3587) 0.3210 (0.1931) 0.0164 (0.0144) 0.86046 0.9788
0.1496 (0.0908) 1.0600 (0.0525) 0.0074 (0.0034) 2.1828 (0.1472) 0.6062 (0.0331) 4.6381 (0.2984) 2.3417 (0.1324) 0.0187 (0.0133) 0.73817 0.9804
0.0193 (0.1095) 0.9879 (0.0591) 0.0012 (0.0033) 2.3849 (0.2535) 0.5452 (0.0561) 4.6684 (0.5092) 2.4280 (0.2831) 0.1313 (0.0122) 0.89277 0.9813
0.5334 (0.1607) 1.3522 (0.0901) 0.0309 (0.0054) 2.8929 (0.2548) 0.7779 (0.0562) 6.8986 (0.5245) 3.4106 (0.2920) 0.0813 (0.0171) 0.91725 0.9726
Eq. (3)
y1 y2 S.E.E. I2
1.5324 (0.0145) 0.6771 (0.0132) 1.01093 0.9707
1.8833 (0.0069) 1.0135 (0.0063) 0.81141 0.9762
1.7324 (0.0127) 0.8822 (0.0115) 0.92619 0.9799
1.8311 (0.0122) 0.9709 (0.0111) 0.99204 0.9678
Eq. (5)
d0 d1 d2 d3 S.E.E. I2
0.8646 (0.0026) 2.0399 (0.0029) 0.1749 (0.0298) 0.2429 (0.0434) 0.85877 0.9789
0.8501 (0.0024) 2.1577 (0.0025) 0.6530 (0.0272) 0.7022 (0.0385) 0.68464 0.9831
0.8960 (0.0043) 2.1152 (0.0034) 0.5918 (0.0482) 1.0853 (0.0759) 0.89187 0.9813
0.9256 (0.0049) 2.1177 (0.0038) 0.5137 (0.0491) 0.8377 (0.0745) 0.89658 0.9738
segmented-polynomial taper equation could not be estimated using the data available in this study. However, Sharma and Burkhart (2003) found that the fit statistics were insensitive to lower and upper join points in the range of 6–15 and 60–85% of total tree height, respectively, for loblolly pine trees. Their analysis also showed that in estimating tree diameters, a four-parameter model with fixed inflection points at 11 and 75% of the total tree height was slightly superior to a six-parameter model with estimated inflection points. Therefore, the lower and upper join points (inflection points) in Eq. (1) were assumed to be 11 and 75%, respectively, for the tree species used in this study. Similarly, a study by Demaerschalk and Kozak (1977) on Douglas-fir and broadleaf maple (Acer macrophyllum Pursh) species showed that the lower inflection point occurred at a constant relative height regardless of the size class and tree species. Therefore,
Jack pine (PCT)
Balsam fir
Black spruce
the lower joining point for Eq. (2) was assumed to be 11% of the total tree height for all three species. Estimated values of parameters for Eqs. (1)–(3) and (5) for different tree species, along with other fit statistics, are presented in Table 2. Though the difference in the estimated standard error of estimate (S.E.E.) resulted from Eqs. (2) and (5) for balsam fir trees is not appreciable, the S.E.E. is smaller for Eq. (5) than for all other equations for all species. The estimated coefficient of determination (I2), however, is less sensitive to the model form. Parameter estimates for the trees from the jack pine PCT plots were significantly different from their counterparts for the trees from the jack pine IS plots for all models. These estimates for both jack pine PCT and IS trees were also significantly different from their counterparts for balsam fir and black spruce trees. Estimates of all four parameters were significantly different between balsam fir and black spruce for
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Eqs. (2) and (3) only. In the case of Eqs. (1) and (5), however, only the estimates for b4 and d1, respectively, were significantly different between these two species. Parameters, g0 and g2 in the case of balsam fir and g3, g4, and g5 in the case of jack pine for the IS plots were not significantly different from zero for Eq. (2). These comparisons were made based on the confidence limits of the parameters obtained by linear and nonlinear regressions. The level of significance a ¼ 0:05 was chosen as the threshold for the tests throughout this paper. Taper Eqs. (1)–(3) and (5) were further analyzed based on the validation data set, the data set withheld to be used as independent data. To evaluate the predictive ability of the models over the entire length of the stem, the independent variable, relative height, was divided into ten sections. Within each section, the diameters available from the withheld data sets were compared to the diameters predicted from the taper equations. Since the data for the evaluation of the taper equations for jack pine trees were available from three different age classes, models for jack pine were evaluated at each age class. This provided additional information regarding the model dependency on stand age. Excepting Eqs. (1) and (2) at age 50, overall diameter prediction bias (observed predicted) along the bole of jack pine trees for Eqs. (1), (2) and (5) was smaller using the models with parameters estimated using the IS data than using the PCT data (Table 3). For Eq. (3), however, the overall diameter prediction bias for jack pine trees was larger using the IS data than using the PCT data. Although the overall bias increases with stand age for all equations, the increment is smaller for Eq. (5) than for the three other equations. Table 3 shows that Eqs. (2) and (3) are inferior to Eqs. (1) and (5) in terms of bias in estimating diameters along the bole of jack pine trees for all three age groups. Eq. (1) is identical to Eq. (5) in terms of bias at age 50. At ages 70 and 90, however, Eq. (5) is superior to Eq. (1) in estimating diameters. Moreover, Eq. (5) is more robust to stand age and stand origin than Eq. (1) in estimating diameters. The overall increment in bias from one stand age to another for Eq. (5) is smaller than for Eq. (1). The largest difference between the biases, resulted from models with parameters estimated using PCT and IS
45
data, is also smaller for Eq. (5) than for Eq. (1). Similarly, the absolute values of the maximum average bias in estimating the bole diameters across all three tree species were 1.161, 3.568, 2.259, and 0.869 cm for Eqs. (1)–(3) and (5), respectively. In the case of balsam fir, Eq. (5) is better than other equations in estimating diameters along the bole (Table 4). For black spruce trees, Eq. (5) is superior to Eqs. (2) and (3) but equally competitive to Eq. (1) in terms of bias. Eq. (2), however, is inferior to all other equations in terms of overall diameter prediction bias along the bole. The data used for parameter estimation for Jack pine trees were sampled from the stands with ages 59 and 60 for PCT and IS, respectively. Bias smaller at age 50 than at 70 and 90 indicates that the best results are obtained if the predictions are made close to the stand age of the fit data set for all models. The older are the trees the larger is the bias in predicting the diameters along the boles for jack pine trees. Black spruce and balsam fir trees were sampled from the stands with ages 48 and 47, respectively. In both cases, the trees used for model evaluation were younger than for parameter estimation. Other trees from older stands were not available for the evaluation. Therefore, the models for these tree species could not be evaluated against the stand age (Table 5). It is well known that the observed inside bark diameter at a given height could be different among trees of the same dbh and total height. Nevertheless, the estimated diameter at any given height for trees of the same dbh and total tree height would be the same. As a result, the positive and negative biases from different trees could average to zero or close to zero. Therefore, absolute value of the bias in estimating diameters along the bole was calculated for each tree using each equation. These absolute values were averaged for each section of the trees for each species (Tables 4 and 6). Overall, average absolute biases were smaller for Eq. (5) than for the other equations, and this case held true in all three species. This also proves that Eq. (5) is better than other equations in estimating diameters along the bole of the tree species studied here. To further analyze the characteristics of Eq. (5), tree profiles were generated using different parameter sets for a tree with a dbh of 17 cm and a total tree height of 15 m (average values). Although the profiles are
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Table 3 Bias (cm) (observed predicted) in predicting diameters along the bole of jack pine trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
IS
PCT
IS
PCT
IS
PCT
IS
PCT
Age ¼ 50 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
96 94 84 84 80 78 61 59 17 46
0.546 0.136 0.192 0.186 0.161 0.075 0.010 0.217 0.047 0.013
0.614 0.007 0.091 0.107 0.097 0.016 0.075 0.122 0.178 0.016
0.546 0.077 0.116 0.156 0.151 0.039 0.124 0.006 0.419 1.346
0.865 0.145 0.019 0.057 0.071 0.317 0.514 0.230 0.293 0.468
0.273 0.607 0.715 0.761 0.633 0.221 0.363 0.795 1.744 0.072
0.253 0.213 0.363 0.025 0.278 0.315 0.039 0.244 1.199 0.063
0.407 0.017 0.086 0.125 0.109 0.005 0.102 0.160 0.052 0.012
0.347 0.459 0.308 0.076 0.046 0.022 0.207 0.109 0.493 0.024
Age ¼ 70 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
112 103 100 100 97 84 80 75 50 52
0.257 0.456 0.542 0.450 0.186 0.158 0.148 0.315 0.394 0.112
0.984 0.314 0.432 0.364 0.116 0.094 0.220 0.417 0.531 0.117
0.315 0.337 0.409 0.357 0.122 0.064 0.322 0.590 0.826 3.568
1.043 0.002 0.228 0.163 0.185 0.373 0.776 0.842 0.630 2.550
0.048 0.910 1.021 0.950 0.560 0.181 0.654 1.492 2.259 0.199
0.559 0.024 0.165 0.062 0.169 0.254 0.240 0.919 1.721 0.189
0.256 0.307 0.415 0.365 0.113 0.064 0.260 0.387 0.390 0.110
0.623 0.232 0.022 0.142 0.048 0.037 0.373 0.681 0.869 0.127
Age ¼ 90 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
131 105 112 108 106 107 99 97 62 52
0.062 0.681 0.908 0.636 0.304 0.122 0.230 0.280 0.013 0.042
1.161 0.529 0.786 0.541 0.227 0.054 0.307 0.392 0.143 0.031
0.207 0.474 0.681 0.465 0.171 0.024 0.465 0.628 0.567 0.200
1.012 0.016 0.360 0.149 0.251 0.549 1.017 0.956 0.347 0.711
0.200 1.093 1.355 1.109 0.654 0.140 0.800 1.601 2.240 0.138
0.528 0.061 0.699 0.051 0.143 0.151 0.417 1.031 1.682 0.112
0.279 0.436 0.587 0.479 0.166 0.028 0.408 0.413 0.046 0.040
0.548 0.206 0.178 0.232 0.099 0.054 0.541 0.743 0.606 0.000
similar, the diameters are smaller for balsam fir than for black spruce (Fig. 1). The difference in diameters between balsam fir and black spruce trees above 60% of the total tree height is greater than below. Butt diameter is smaller for jack pine grown at the IS plots than for black spruce, balsam fir, and jack pine grown at the PCT plots. Upper stem diameters for jack pine grown at the PCT stands are smaller than for the other species up to 33% of the total tree height. Above this point, however, diameters for jack pine grown at the PCT and IS study plots are identical. Taper profiles were also generated for balsam fir trees with different dbh values and the same total tree
height (Fig. 2) and with the same dbh value and different total tree heights (Fig. 3). As expected, trees with larger dbh values also demonstrated larger butt diameters and more taper than did the trees with smaller dbh values for the same total tree height (Fig. 2). Similarly, taller trees contained thicker bark and smaller inside bark diameters than did shorter trees at the same relative height for the same dbh (Fig. 3). Kozak and Smith (1993) compared Eqs. (1) and (2) for coastal Douglas-fir trees and found Eq. (2) to be superior in estimating inside bark diameter. This study, however, ranked the Equations in the order
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
47
Table 4 Absolute bias (cm) (observed predicted) in predicting diameters along the bole of jack pine trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
IS
PCT
IS
PCT
IS
PCT
IS
PCT
Age ¼ 50 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
96 94 84 84 80 78 61 59 17 46
0.723 0.397 0.529 0.497 0.599 0.654 0.734 0.835 0.702 0.015
0.751 0.415 0.497 0.480 0.599 0.663 0.736 0.807 0.781 0.024
0.711 0.363 0.498 0.469 0.567 0.606 0.688 0.779 0.785 1.346
1.184 0.341 0.432 0.401 0.499 0.575 0.762 0.781 0.714 0.468
0.794 0.636 0.763 0.804 0.782 0.596 0.675 0.980 1.744 0.072
0.568 0.376 0.484 0.397 0.546 0.633 0.647 0.749 1.264 0.063
0.723 0.361 0.491 0.463 0.567 0.617 0.700 0.816 0.674 0.012
0.680 0.512 0.512 0.456 0.557 0.612 0.695 0.788 0.768 0.025
Age ¼ 70 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
112 103 100 100 97 84 80 75 50 52
0.744 0.537 0.654 0.616 0.578 0.632 0.628 0.767 1.004 0.116
1.089 0.417 0.551 0.572 0.576 0.639 0.627 0.812 1.103 0.128
0.743 0.478 0.568 0.558 0.567 0.601 0.632 0.848 1.125 3.568
1.452 0.399 0.479 0.483 0.586 0.622 0.859 0.998 1.038 2.550
0.891 0.928 1.034 0.971 0.724 0.637 0.825 1.550 2.265 0.199
0.833 0.420 0.475 0.478 0.615 0.649 0.635 1.037 1.767 0.189
0.728 0.478 0.581 0.563 0.566 0.605 0.625 0.777 0.971 0.110
0.930 0.429 0.451 0.480 0.559 0.601 0.651 0.901 0.938 0.127
Age ¼ 90 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
131 105 112 108 106 107 99 97 62 52
0.814 0.736 0.989 0.741 0.616 0.529 0.522 0.700 0.667 0.035
1.255 0.546 0.836 0.671 0.613 0.534 0.523 0.766 0.710 0.022
0.772 0.659 0.846 0.599 0.530 0.512 0.650 0.845 0.753 0.378
1.464 0.607 0.693 0.494 0.545 0.739 1.092 1.079 0.699 0.711
1.048 1.127 1.383 1.121 0.764 0.572 0.940 1.645 2.241 0.138
0.888 0.532 0.560 0.504 0.520 0.560 0.703 1.145 1.699 0.112
0.744 0.613 0.819 0.623 0.529 0.517 0.610 0.637 0.673 0.040
0.912 0.541 0.592 0.513 0.545 0.521 0.696 0.922 0.791 0.017
(5), (1), (2), and lastly (3) in estimating inside bark diameters for the three species grown in eastern Canada. 4.1. Stand density effect In order to examine stand density or thinning effect on tree taper, Eq. (5) was fitted to the fit data set for each stand density for each species separately. Trees per hectare at the time of plot establishment was used as the stand density for all species. In black spruce, all
the parameters at one stand density were significantly different from their counterparts at the other densities. In the case of balsam fir, only the parameters for heavily thinned stands were significantly different from their counterparts for unthinned and moderately thinned stands. Parameters between unthinned and moderately thinned balsam fir stands, however, were not significantly different from each other. In the case of jack pine trees, parameters for one stand density were not significantly different from their counterparts for the other densities for both IS
48
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Table 5 Bias (cm) (observed predicted) in predicting diameters along the bole of black spruce and balsam fir trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
Black spruce 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
207 135 141 142 146 136 136 133 136 257
0.124 0.321 0.239 0.079 0.094 0.061 0.053 0.032 0.182 0.051
0.432 0.381 0.426 0.136 0.059 0.312 0.357 0.175 0.333 1.029
0.294 0.035 0.079 0.016 0.315 0.427 0.467 0.369 0.189 0.018
0.027 0.378 0.331 0.212 0.215 0.078 0.000 0.037 0.093 0.176
Balsam fir 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
120 91 65 64 71 63 61 65 65 110
0.175 0.189 0.138 0.098 0.064 0.207 0.150 0.101 0.103 0.101
0.243 0.291 0.398 0.364 0.205 0.189 0.056 0.198 0.451 0.303
0.390 0.470 0.221 0.320 0.438 0.691 0.665 0.686 0.543 0.100
0.016 0.096 0.025 0.104 0.087 0.114 0.061 0.029 0.145 0.174
20 18
Inside bark diameter (cm)
16 14 12 10 8 6 4 2 0 0 BS
1
2 BF
3 JP PCT
4
5 JP IS
6
7
8
9
10
11
12
13
14
15
Height from ground (m)
Fig. 1. Tree profiles generated from different parameter sets estimated using Eq. (5) and black spruce (BS), balsam fir (BF), jack pine PCT (JP PCT), and jack pine IS (JP IS) tree data with the same values of DBH (17.0 cm) and total height (15.0 m).
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
49
Table 6 Absolute bias (cm) (observed predicted) in predicting diameters along the bole of black spruce and balsam fir trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
Black spruce 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
207 135 141 142 146 136 136 133 136 257
0.758 0.421 0.449 0.399 0.420 0.499 0.537 0.570 0.512 0.226
0.971 0.568 0.575 0.430 0.422 0.527 0.577 0.596 0.603 1.050
0.852 0.349 0.346 0.361 0.445 0.539 0.591 0.578 0.458 0.214
0.742 0.413 0.410 0.373 0.425 0.499 0.528 0.578 0.549 0.224
Balsam fir 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
120 91 65 64 71 63 61 65 65 110
0.598 0.252 0.285 0.313 0.366 0.461 0.525 0.531 0.430 0.170
0.516 0.311 0.405 0.398 0.342 0.379 0.431 0.496 0.560 0.325
0.861 0.480 0.286 0.372 0.494 0.727 0.752 0.789 0.621 0.161
0.540 0.225 0.248 0.298 0.341 0.414 0.459 0.472 0.434 0.216
28
Inside bark diameter (cm)
24 20 16 12 8 4 0 0 12
1
2 16
3 20
4
5 24
6
7
8
9
10
11
12
13
14
15
Height from ground (m)
Fig. 2. Tree profiles generated from Eq. (5) using the same value of total height (15.0 m) and different values of dbh (12.0, 16.0, 20.0, and 24.0 cm) for balsam fir.
50
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53 18 16
Inside bark diameter (cm)
14 12 10 8 6 4 2 0 0
0.1
0.2
0.3
12
14
16
18
0.4
0.5
0.6
0.7
0.8
0.9
1
Relative height
Fig. 3. Tree profiles generated from Eq. (5) using the same value of dbh (17.0 cm) and different values of total height (12.0, 14.0, 16.0, and 18.0 m) for balsam fir.
and the PCT stands. In fact, the model fit to the data so well that the largest bias in estimating diameters along the bole averaged over every 10% of total tree height was less than 0.1 cm. Therefore, Eq. (7) was fitted to the black spruce estimation data set using the NLIN procedure in SAS (1984). Estimated values of parameters d0, d1, d2, d3, and d4 were 0.9253, 2.0139, 0.5235, 0.8453, and 217.3 with approximate standard errors of 0.0045, 0.0069, 0.0450, 0.0683, and 12.1052, respectively. The mean square error (MSE) decreased by 16% when the stand density variable was included in the model. Table 7 displays the bias in estimating diameters along the boles of the trees used for parameter estimation with and without incorporation of the stand density variable. It is clear that including the stand density variable improved the model efficacy significantly for black spruce trees. Taper profiles generated for a tree with a dbh of 17 cm and a total tree height of 15 m using Eq. (7) at different stand densities (Fig. 4) show that trees have larger butt diameters and more taper at a lower stand
density than they do at higher stand density. For the same difference between stand densities, however, the difference in bole diameter between lower and higher stand densities diminishes as stand density increases.
Table 7 Bias (cm) (observed predicted) in predicting diameters along the bole of black spruce trees for Eqs. (5) and (7) (with and without the density variable in the model) using fit data sets Black spruce relative height 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
h/H 0.1 < h/H 0.2 < h/H 0.3 < h/H 0.4 < h/H 0.5 < h/H 0.6 < h/H 0.7 < h/H 0.8 < h/H 0.9 < h/H 1.0
Number of trees
Eq. (5)
Eq. (7)
259 201 201 176 193 183 145 106 44 136
0.044 0.009 0.020 0.097 0.125 0.080 0.129 0.066 0.276 0.032
0.073 0.038 0.001 0.079 0.097 0.059 0.087 0.048 0.179 0.032
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
51
24
Inside bark diameter (cm)
20
16
12
8
4
0 0 1000
1
2 2000
3
4 3000
5
6
4000
7
8
9
10
11
12
13
14
15
Height from ground (m )
Fig. 4. Tree profiles generated from Eq. (7) using dbh ¼ 17:0 cm and total height ¼ 15:0 m at different stand densities (1000, 2000, 3000, and 4000 trees/ha) for black spruce.
5. Conclusions A variable-exponent taper equation was developed for jack pine, black spruce and balsam fir, the three most important commercial tree species in eastern Canada. The equation was derived from the dimensionally compatible taper equation of Sharma and Oderwald (2001) and was compared with the segmented polynomial, variable-exponent, and variable-form taper equations by Max and Burkhart (1976), Kozak (1988) and Zakrzewski (1999), respectively. The taper of black spruce trees was affected by stand density or thinning. Therefore, the equation was modified in order to incorporate this stand density effect. Without the consideration of the stand density effect on tree taper, all four taper equations were very competitive in terms of estimated coefficient of determination (I2). The dimensionally compatible variableexponent taper equation presented in this study was superior to all other equations in terms of the estimated standard error of estimate (S.E.E.) and bias in estimating diameters along the bole of the trees. Moreover, the dimensionally compatible variable-exponent taper equation was less sensitive to the stand age compared
to the other three taper equations in estimating diameters. Since the segmented polynomial (with known inflection points) and dimensionally compatible variable-exponent taper equations have the same number of parameters and the same dependent variable, the mean square errors (MSE) for these two equations were also compared. MSE for the dimensionally compatible variable-exponent taper equation was smaller than for the segmented-polynomial taper equation in all cases. The MSE for the other two taper equations, however, could not be directly compared since they did not have the same dependent variable. Kozak variable-exponent and Zakrzewski variableform taper equations are inferior to the segmented polynomial and dimensionally compatible variableexponent taper equations for all species in terms of bias in estimating diameters. The dimensionally compatible variable-exponent taper equation is superior to the segmented-polynomial taper equation for jack pine and balsam fir. For black spruce, the dimensionally compatible variable-exponent taper equation with density term in the model is superior to the one without the density and segmented-polynomial taper
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
equation in terms of fit statistics and bias in estimating diameters for the fit data set. An independent data set was not available to evaluate the dimensionally compatible variable-exponent taper equation with the density term in the model. However, the fit statistics and average biases in estimating diameters for the fit data set indicate that the equation can be applied to accurately estimate diameters of black spruce trees grown at varying densities. This equation can be applied to other tree species by fitting the model to data specific for those species.
Acknowledgements This study was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Living Legacy Trust and Tembec’s Forestry Research Partnership. Our thanks go to Dr. W.T. Zakrzewski, Ontario Forest Research Institute in Sault Ste. Marie, Ontario, Dr. Jean Be´ gin, Laval University in Sainte-Foy, Quebec, and Dr. Isabelle Duchesne, Forintek Canada Corporation in SainteFoy, Quebec for providing stem analysis data for validating the taper equation. We would also like to thank Mr. Yvon Grenier, University of Quebec in Abitibi-Te´ minscaming, Quebec, Dr. C.H. Ung, Canadian Forest Service in Sainte-Foy, Quebec, Mr. Edwin Swift, Canadian Forest Service in Fredericton, New Brunswick, and Mr. Jon Vivian and Mr. Will Smith, British Columbia Ministry of Sustainable Resource Management in Victoria, British Columbia for their assistance in this study and are grateful to two anonymous reviewers for their constructive comments.
References Amidon, E.L., 1984. A general taper functional form to predict bole volume for five mixed-conifer species in California. For. Sci. 30, 166–171. Avery, T.E., Burkhart, H.E., 2002. Forest Measurements, 5th ed. McGraw-Hill, New York, p. 456. Burkhart, H.E., Bredenkamp, B.V., 1989. Product-class proportions for thinned and unthinned loblolly pine plantations. S. J. Appl. For. 13, 192–195. Burkhart, H.E., Walton, S.B., 1985. Incorporating crown ratio into taper equations for loblolly pine trees. For. Sci. 31, 478–484.
Cao, Q.V., Burkhart, H.E., Max, T.A., 1980. Evaluation of two methods for cubic volume prediction of loblolly pine to any merchantable limit. For. Sci. 26, 71–80. Demaerschalk, J.P., Kozak, A., 1977. The whole bole system: a conditioned dual equation system for precise prediction of tree profiles. Can. J. For. Res. 7, 488–497. Fang, Z., Borders, B.E., Bailey, R.L., 2000. Compatible volumetaper models for loblolly and slash pine based on system with segmented-stem form factors. For. Sci. 46, 1–12. Flewelling, J.W., Raynes, L.M., 1993. Variable-shape stem-profile predictions for western hemlock. Part I. Predictions from DBH and total height. Can. J. For. Res. 23, 520–536. Guilkey, P., Westing, A.H., 1956. Effects of initial spacing on the development of young jack pine in northern Lower Michigan. Mich. Acad. Sci. Arts, Lett. Pap. XLI, 45–50. Koga, S., Zhang, S.Y., Be´ gin, J., 2002. Impact of precommercial thinning on radial diameter growth and wood density in balsam fir. Wood Fiber Sci. 34 (4), 625–642. Kozak, A., 1988. A variable-exponent taper equation. Can. J. For. Res. 18 (11), 1363–1368. Kozak, A., 1997. Effects of multicollinearity and autocorrelation on the variable-exponent taper functions. Can. J. For. Res. 27, 619–629. Kozak, A., Munro, D.D., Smith, J.H.G., 1969. Taper functions and their application in forest inventory. For. Chron. 45, 278–283. Kozak, A., Smith, J.H.G., 1993. Standards for evaluating taper estimating systems. For. Chron. 69, 438–444. Max, T.A., Burkhart, H.E., 1976. Segmented polynomial regression applied to taper equations. For. Sci. 22, 283–289. Muhairwe, C.K., LeMay, V.M., Kozak, A., 1994. Effects of adding tree, and site variables to Kozak’s variable-exponent taper equation. Can. J. For. Res. 24, 252–259. NBDNRE, 1987. 1966 jack pine spacing trials at Eel River Bridge. Infor. Rep. No. 87-05. New Brunswick Department of Natural Resources and Energy, Fredericton, NB. Newnham, R.M., 1992. Variable-form taper functions for four Alberta tree species. Can. J. For. Res. 22, 210–223. Ormerod, D.W., 1973. A simple bole model. For. Chron. 49, 136– 138. Reed, D.D., Byrne, J.C., 1985. A simple, variable form volume estimation system. For. Chron. 61, 87–90. SAS, 1984. SAS Institute Inc., Cary, NC, USA. Sharma, M., Burkhart, H.E., 2003. Selecting a level of conditioning for the segmented polynomial taper equation. For. Sci. 49, 324– 330. Sharma, M., Oderwald, R.G., 2001. Dimensionally compatible volume and taper equations. Can. J. For. Res. 31, 797–803. Sharma, M., Oderwald, R.G., Amateis, R.L., 2002. A consistent system of equations for tree and stand volume. For. Ecol. Manage. 165, 179–187. Tasissa, G., Burkhart, H.E., Amateis, R.L., 1997. Volume and taper equations for thinned and unthinned loblolly pine trees in cotover, site-prepared plantations. S. J. Appl. For. 21, 146–152. Valentine, H.T., Gregoire, T.G., 2001. A switching model of bole taper. Can. J. For. Res. 31, 1400–1409. Zakrzewski, W.T., 1999. A mathematically tractable stem profile model for jack pine in Ontario. N. J. Appl. For. 16, 138–143.
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Variable-exponent taper equations for jack pine, black spruce, and balsam fir in eastern Canada Mahadev Sharma*, S.Y. Zhang Resource Assessment and Utilization Group, Forintek Canada Corp., 319 rue Franquet, Sainte-Foy, Que., Canada G1P 4R4 Received 3 December 2003; received in revised form 2 March 2004; accepted 9 March 2004
Abstract A variable-exponent taper equation was developed for jack pine (Pinus banksiana Lamb.), black spruce (Picea mariana (Mill.) B.S.P.) and balsam fir (Abies balsamea (L.) (Mill.)) trees grown in eastern Canada. The equation was derived from the dimensionally compatible taper equation. Stem analysis data from several of Forintek’s research projects on these tree species grown across eastern Canada were used to fit the taper equation, and the equation was then evaluated with independent data collected by Forintek and other organizations in eastern Canada. The taper equation presented in this paper was superior to the segmented polynomial, variable-exponent, and variable-form taper equations developed for different tree species in estimating diameters along the bole. Moreover, stand density or thinning effect on the taper of these tree species was examined, and the taper equation was modified in order to incorporate this effect. # 2004 Elsevier B.V. All rights reserved. Keywords: Tree taper and form; Stand density; Thinning; Nonlinear regression
1. Introduction Stem profile equations, commonly known as taper equations, are very useful in estimating the stem volume of trees and reconstructing the three-dimensional (3D) stem form for the simulation and optimization of stem bucking and log sawing. These equations can provide estimates of the diameter at any height along the bole of a tree and hence have long been the subject of research of many forest scientists. Although tree boles cannot be completely described in mathematical terms, it is common and convenient to
* Corresponding author. Tel.: þ1-418-659-2647; fax: þ1-418-659-2922. E-mail addresses: [email protected] (M. Sharma), [email protected] (S.Y. Zhang).
assume that segments of a tree stem approximate various geometric solids. The lower bole portion is generally assumed to be a neiloid frustum, the middle portion a paraboloid frustum, and the upper portion a cone (Avery and Burkhart, 2002). Valentine and Gregoire (2001), however, reported that the segments of slash pine (Pinus elliottii Engelm.), ponderosa pine (Pinus ponderosa Doug. Ex P. & C. Laws.), and yellow poplar (Liriodendron tulipifera L.) did not conform to these shapes. They further reported that the top and middle segments of sweet gum (Liquidambar styraciflua L.) trees resembled cones and paraboloid, respectively. It is obvious that a simple regression model may not be adequate to describe the tree form. Numerous models of varying complexity have been advanced in attempts to describe tree taper. Taper models presented by Kozak et al. (1969), Ormerod
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.03.035
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
(1973), Amidon (1984), Reed and Byrne (1985), Sharma and Oderwald (2001), and Sharma et al. (2002) are examples of simple taper functions. On the other hand, more complex taper equations have been developed that account for the three segments of a tree bole. The most common approaches applied to produce complex taper models are: (i) assuming a tree is made up of two or more segments with form being constant within a segment and different between segments, and (ii) allowing the tree form to vary from one point to another along the bole and expressing the variable form by a single continuous function. Segmentedpolynomial taper models developed by Max and Burkhart (1976), Demaerschalk and Kozak (1977), Cao et al. (1980), and Fang et al. (2000) are examples of the first approach where each segment of a tree is described by a polynomial. Examples of the second approach are the variable-exponent taper model elaborated by Kozak (1988) and the variable-form taper model presented by Newnham (1992). In these models, a single continuous function with an exponent changing from stump to top describes the neiloid, paraboloid, and several intermediate forms (Kozak, 1997). Flewelling and Raynes (1993) also developed a variable-form taper model based on a system of three equations. Unfortunately, most of these taper equations are species-specific i.e. the model accuracy in estimating diameters depends on tree species. Moreover, fitting a segmented-polynomial taper equation requires all diameter–height pair measurements for all segments from the stump to the tip of a tree. This type of data, however, is not always available. Diameter–height pair data collected by wood industries and some research organizations often only cover up to a merchantable top (e.g. 10 cm (inside bark) upper diameter class). Therefore, these organizations either use taper equations produced for other species or rely on inaccurate ones for volume estimations. Furthermore, foresters have long realized that trees grown at different stand densities do not necessarily have the same tree height or tree form. Stand density can be regulated either by planting the trees at different initial spacings or by thinning the stands to different densities. However, the trees of a particular species grown in a plantation and in a natural stand thinned to the same density may not have the same
form, especially if the thinning is carried out at a later age. Tree taper is affected by thinning (Sharma et al., 2002; Tasissa et al., 1997; Zhang et al., 1998; Koga et al., 2002). Although other stand variables have been added to the taper equation studied in the past, little attempt has been made to quantify the stand density or thinning effect on tree taper or other growth characteristics that may affect product recovery (Burkhart and Bredenkamp, 1989; Zhang et al., in press). Muhairwe et al. (1994) studied the effects of adding crown class, crown ratio, site class, breast height and age to Kozak’s variable-exponent taper equation for Douglas fir (Pseudotsuga menziesii (Mirb.) Franco), western red cedar (Thuja plicata Donn) and aspen (Populus tremuloides Michx.) trees. Similarly, Burkhart and Walton (1985) examined the effect of incorporating crown ratio into a segmented-polynomial taper equation for loblolly pine (Pinus taeda) trees. Both studies found that adding these stand variables into the taper equations did not significantly improve their goodness of fit. The objective of this study was to investigate a taper equation that is appropriate for jack pine (Pinus banksiana Lamb.), black spruce (Picea mariana (Mill.) B.S.P.), and balsam fir (Abies balsamea (L.) (Mill.)), the most important commercial tree species in eastern Canada. This study was also intended to investigate and quantify the effect of stand density and/or thinning on the stem taper of these species.
2. Data Data used to estimate the parameters and to evaluate the taper equations in this study were obtained from many different sources. Parameters of the taper equations for jack pine were estimated using the data collected from a jack pine initial spacing (IS) and pre-commercial thinning (PCT) study. These studies, funded by the Living Legacy Trust and Tembec, are part of a jack pine task force initiated by Forintek Canada Corp (Zhang, 1999). The plots used for the jack pine IS study were established in 1941 near Wellston in Manistee County, Michigan. One-yearold seedlings were planted at spacings of 0:46 m 0:46 m, 0:91 m 0:91 m, 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m, with each spacing
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
replicated five times on square 0.162 ha plots arranged in a Latin Square Design. One 1:52 m 1:52 m plot was thinned to a 2:13 m 2:13 m spacing at age 13. In 2001, 151 trees (3–8 trees in each diameter class were sampled for stem analysis from spacings 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m). For more information on this trial, refer to Guilkey and Westing (1956). The plots used for the jack pine PCT study were established by the New Brunswick Department of Natural Resources and Energy at Eel River Bridge on the Lower Miramichi in 1966. They were originated from fire in 1941. At the time of the establishment, some stands were thinned to densities of 7215, 4278, 2212, and 1413 trees/ha resulting in approximately 1:22 m 1:22 m, 1:52 m 1:52 m, 2:13 m 2:13 m, and 2:74 m 2:74 m spacings, respectively. Unthinned stands contained approximately 72,000 trees/ha. In 2000, 154 trees (6 trees per diameter class per spacing excluding 2:74 m 2:74 m) were sampled for stem analysis and product recovery studies. For more information about the trials, refer to a report by the New Brunswick Department of Natural Resources and Energy (Nbdnre, 1987). The parameters of the taper equations for black spruce trees were estimated using the data from a black spruce initial spacing study. The trial was established in 1950 by the Ontario Department of Lands and Forests, now the Ministry of Natural Resources, at two locations near Thunder Bay. Initially, trees were planted at three spacings: 1:83 m 1:83 m (3086 trees/ha), 2:74 m 2:74 m (1372 trees/ha), and 3:66 m 3:66 m (780 trees/ha). Three replications per spacing regime were initially included in the trial. Some plots resulted in 2066 and 2500 trees/ ha because the seedlings that died after planting were not replaced. For more information about these trials, refer to a report by Zhang and Chauret (2001). In 1998, approximately six trees per merchantable diameter class were randomly selected from stands with densities of 1372, 2066, 2500, and 3086 trees/ha. In total, 139 sample trees with no major defects were selected from the four densities. No tree was sampled from the stand with a density of 780 trees/ha as the stand density was obviously too low for black spruce. The parameters of the taper equations for balsam fir trees were estimated using the data from a PCT study (Zhang et al., 1998). The trial is located in the Bas
41
Saint-Laurent Region of Quebec. The balsam fir stand for the trial was regenerated from a clearcutting in 1948. A 4 ha stand was divided into 100 plots of 0.04 ha each, and was manually thinned to densities ranging from 500 to 6270 trees/ha in 1960. In total, 150 trees (6 trees per merchantable diameter class per density) were sampled from three stands with densities of 1500, 3000, and 12,100 trees/ha in 1995. These densities correspond to heavily thinned, moderately thinned, and unthinned (control) stands, respectively. Jack pine tree data for the evaluation of taper equation were obtained from three even-age natural stands of 50-, 70-, and 90-year-old studied by the jack pine task force. These stands were located at the Romeo Malette Forest Management Unit in the district of Timmins, Northeastern Ontario and were regenerated from fire. The stands were composed of approximately 73% jack pine, 26% black spruce and the remainder birch. In total, 147 trees (47, 51, and 49 trees from stands aged 50, 70, and 90 years, respectively) were collected for the evaluation. The evaluation data for black spruce came from Northern Ontario and were provided by the Ontario Forest Research Institute. Similarly, balsam fir tree data for the evaluation were collected by Laval University from a naturally regenerated pre-commercial thinning trial located in the Bas Saint-Laurent Region of Quebec (Koga et al., 2002). The regeneration of the trail was taken place in 1955 and the sample trees were collected in 1993. In each study described above, the sample trees were selected and felled for stem analysis. In addition to total tree height and outside bark diameter at breast height (dbh), inside bark diameters were measured at the stump and every meter along the bole to a minimum merchantable diameter (e.g. 10, 8 or 4 cm top). Data used to evaluate the taper equations for black spruce and balsam fir, however, consisted of diameter– height observations measured every meter up to the tip of the tree. Summary statistics for total height and dbh of all trees used in fitting and evaluating the taper equations are presented in Table 1.
3. Taper equations The taper equations widely used in the United States and Canada are the segmented polynomial of
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Table 1 Summary statistics for total height and dbh of jack pine, balsam fir, and black spruce trees used in this study Species
Number of trees
Mean height (m)
S.D.
Mean dbh (cm)
S.D.
Fit data Jack pine IS Jack pine PCT Balsam fir Black spruce
140 154 150 135
17.68 15.44 13.88 15.08
1.96 1.71 2.15 2.27
18.77 16.36 17.65 16.85
4.57 3.88 5.11 3.73
Validation data Jack pine (age 50) Jack pine (age 73) Jack pine (age 90) Balsam fir Black spruce
47 50 49 59 131
19.39 19.74 22.16 11.26 14.48
2.03 2.39 2.02 1.77 5.56
18.07 19.72 21.97 13.48 13.99
4.59 4.84 5.26 2.95 4.30
Max and Burkhart (1976) and the variable exponent of Kozak (1988). These equations are written as follows. Segmented polynomial: 2 d ¼ b1 ðz 1Þ þ b2 ðz2 1Þ þ b3 ða1 zÞ2 I1 D þ b4 ða2 zÞ2 I2
(1)
where d is the diameter inside bark at any given height h; D the diameter at breast height outside bark; H the total tree height from ground to tip; z the h/H; a1 and a2 the upper and lower join points, respectively, of three segments; and b1–b4 the parameters. Similarly 1 if a1 z 0 I1 ¼ and 0 otherwise 1 if a2 z 0 I2 ¼ 0 otherwise Variable exponent: 2 þg
d ¼ g0 Dg1 g2 D X g3 z
4
lnðzþ0:001Þþg5 z0:5 þg6 ez þg7 ðD=HÞ
(2)
where X ¼ ð1 ðzÞ1=2 Þ=ð1 p1=2 Þ, p the lower join point (inflection point), g0–g7 the parameters, and other variables are as defined above. Zakrzewski (1999) developed a variable-form taper equation for jack pine grown in Ontario as 2
caz ¼ K
3
z þ y 1 z þ y2 z zs
z0 ¼ 11:3=H, C the inside bark cross-sectional area at the breast height (1.3 m), y1 and y2 the parameters, and other variables are as defined previously. Sharma and Oderwald (2001) developed a dimensionally compatible taper equation based on dimensional analysis as 2d h Hh 2 dout ¼ D2 (4) hD H hD where dout is the outside bark diameter at height h, hD the breast height and d a parameter known as the form parameter. Even though Eq. (4) has the simplest form with only one parameter, it was superior to the segmented-polynomial taper equation in predicting inside bark diameters of loblolly pine trees up to 60% of total height. Above 60% of the total height, however, Eq. (4) resulted in slightly larger bias than Eq. (1) (Sharma and Oderwald, 2001). In addition, Eq. (4) cannot be used to estimate inside bark diameter using the dbh measured outside bark. However, the equation can be modified in order to estimate the inside bark diameters using outside bark dbh by including a constant a as follows: 2d h Hh 2 2 d ¼ aD hD H hD The constant, a can be calculated as
4
(3)
where K ¼ Cðz0 sÞ=ðz20 þ y1 z30 þ y2 z40 Þ, caz the inside bark cross-sectional area, s ¼ 1 þ H=D,
a¼
D2i D2
where Di is the inside bark dbh. Eq. (4) can be further modified to result in a variable-exponent taper
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
equation where the tree form changes continuously along the bole. Since the form at a particular point depends on its relative height from the ground, the form parameter can be expressed in terms of the relative height (z) to result in a variable-exponent taper equation, i.e. 2f ðzÞ h Hh 2 2 d ¼ aD hD H hD Several functions of z (linear, quadratic, exponential, and their combinations) were examined. The taper equation with a quadratic function as a variable exponent, f ðzÞ ¼ d1 þ d2 z þ d3 z2 , where, d1, d2, and d3 are parameters, well described the tree profiles for the tree species used in this study. Rearranging the variables this taper equation results in: 2 2ðd1 þd2 zþd3 z2 Þ d h Hh ¼ d0 (5) D hD H hD Since the constant a is unknown, it has been replaced by another parameter d0. Similarly, Eq. (5) can be easily modified to accommodate stand density effect. Since the exponent of the height solely determines the tree form, the stand density effect can be incorporated by adding a stand density function to the exponent, i.e. 2 2ðd1 þd2 zþd3 z2 þd4 gðsdÞÞ d h Hh ¼ d0 D hD H hD (6) where g(sd) is a function of stand density and d4 a parameter. The best function that described the stand density effect in this case was gðsdÞ ¼
1 sd
where sd is the stand density (trees/ha) at the time of plot establishment. Thus, Eq. (6) with the added stand density function becomes 2 2ðd1 þd2 zþd3 z2 þd4 =sdÞ d h Hh ¼ d0 (7) D hD H hD Eq. (5) was compared with the segmented-polynomial (Eq. (1)), variable-exponent (Eq. (2)), and variableform (Eq. (3)) taper equations. Eq. (7) was evaluated
43
by comparing with and without the density effect in the model. As we see Eqs. (1) and (5) contain the same dependent variable (d2/D2). Dependent variables in the other two equations, however, differ from d2/D2. Therefore, the model coefficient of determinations (R2) and model mean square errors (MSE) cannot be compared directly for evaluation. As a result, comparison was performed by examining the estimated coefficient of determination (I2) of the original (untransformed) independent variable (d) and the overall standard error of estimate (S.E.E.). These parameters can be calculated (Kozak and Smith, 1993) as follows: I2 ¼
SSd SSres SSd
(8)
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ 2 i¼1 ðdi d i Þ S:E:E: ¼ nk
(9)
where SSd ¼
n X
2; ðdi dÞ
SSres ¼
i¼1
n X ðdi d^i Þ2 i¼1
di is the actual observation (untransformed) of the dependent variable (diameter inside bark (cm) in this case), d^i the predicted value of the actual observation, d the average of the actual observations, n the number of observations, k the number of estimated parameters. These equations were also compared by calculating bias in estimating diameters at different points along the bole of the trees set aside for model evaluation. Since no independent data including density information were available, Eq. (7) was evaluated by comparing bias in estimating diameters along the bole of the fit data set with and without the stand density function in the model.
4. Results and discussion Eqs. (1)–(3) and (5) were fitted to the estimation data set (the data designated for parameter estimation) using the NLIN procedure in SAS (1984). Since most of the data contained height–diameter pairs measured up to a 10 cm (inside bark) top, the upper inflection point (join point) in the case of a
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Table 2 Parameter estimates (standard errors in parentheses) for Eqs. (1)–(3) and (5) fit to parameter estimation data using nonlinear regression Parameter
Jack pine (IS)
Eq. (1)
b1 b2 b3 b4 S.E.E. I2
2.1151 (0.2860) 0.7342 (0.1724) 0.8273 (0.2080) 8.4616 (0.7804) 0.86675 0.9785
2.7838 (0.1974) 1.1285 (0.1183) 1.4219 (0.1458) 39.0441 (0.7510) 0.69936 0.9823
4.6363 (0.3074) 2.3322 (0.1851) 2.3278 (0.2256) 26.3322 (0.9936) 0.93129 0.9797
4.1998 (0.3769) 2.0288 (0.2269) 2.0953 (0.2764) 29.9728 (1.1620) 0.94234 0.9710
Eq. (2)
g0 g1 g2 g3 g4 g5 g6 g7 S.E.E. I2
0.2568 (0.1078) 1.0943 (0.0547) 0.0061 (0.0030) 0.1180 (0.1551) 0.00086 (0.0368) 0.2194 (0.3587) 0.3210 (0.1931) 0.0164 (0.0144) 0.86046 0.9788
0.1496 (0.0908) 1.0600 (0.0525) 0.0074 (0.0034) 2.1828 (0.1472) 0.6062 (0.0331) 4.6381 (0.2984) 2.3417 (0.1324) 0.0187 (0.0133) 0.73817 0.9804
0.0193 (0.1095) 0.9879 (0.0591) 0.0012 (0.0033) 2.3849 (0.2535) 0.5452 (0.0561) 4.6684 (0.5092) 2.4280 (0.2831) 0.1313 (0.0122) 0.89277 0.9813
0.5334 (0.1607) 1.3522 (0.0901) 0.0309 (0.0054) 2.8929 (0.2548) 0.7779 (0.0562) 6.8986 (0.5245) 3.4106 (0.2920) 0.0813 (0.0171) 0.91725 0.9726
Eq. (3)
y1 y2 S.E.E. I2
1.5324 (0.0145) 0.6771 (0.0132) 1.01093 0.9707
1.8833 (0.0069) 1.0135 (0.0063) 0.81141 0.9762
1.7324 (0.0127) 0.8822 (0.0115) 0.92619 0.9799
1.8311 (0.0122) 0.9709 (0.0111) 0.99204 0.9678
Eq. (5)
d0 d1 d2 d3 S.E.E. I2
0.8646 (0.0026) 2.0399 (0.0029) 0.1749 (0.0298) 0.2429 (0.0434) 0.85877 0.9789
0.8501 (0.0024) 2.1577 (0.0025) 0.6530 (0.0272) 0.7022 (0.0385) 0.68464 0.9831
0.8960 (0.0043) 2.1152 (0.0034) 0.5918 (0.0482) 1.0853 (0.0759) 0.89187 0.9813
0.9256 (0.0049) 2.1177 (0.0038) 0.5137 (0.0491) 0.8377 (0.0745) 0.89658 0.9738
segmented-polynomial taper equation could not be estimated using the data available in this study. However, Sharma and Burkhart (2003) found that the fit statistics were insensitive to lower and upper join points in the range of 6–15 and 60–85% of total tree height, respectively, for loblolly pine trees. Their analysis also showed that in estimating tree diameters, a four-parameter model with fixed inflection points at 11 and 75% of the total tree height was slightly superior to a six-parameter model with estimated inflection points. Therefore, the lower and upper join points (inflection points) in Eq. (1) were assumed to be 11 and 75%, respectively, for the tree species used in this study. Similarly, a study by Demaerschalk and Kozak (1977) on Douglas-fir and broadleaf maple (Acer macrophyllum Pursh) species showed that the lower inflection point occurred at a constant relative height regardless of the size class and tree species. Therefore,
Jack pine (PCT)
Balsam fir
Black spruce
the lower joining point for Eq. (2) was assumed to be 11% of the total tree height for all three species. Estimated values of parameters for Eqs. (1)–(3) and (5) for different tree species, along with other fit statistics, are presented in Table 2. Though the difference in the estimated standard error of estimate (S.E.E.) resulted from Eqs. (2) and (5) for balsam fir trees is not appreciable, the S.E.E. is smaller for Eq. (5) than for all other equations for all species. The estimated coefficient of determination (I2), however, is less sensitive to the model form. Parameter estimates for the trees from the jack pine PCT plots were significantly different from their counterparts for the trees from the jack pine IS plots for all models. These estimates for both jack pine PCT and IS trees were also significantly different from their counterparts for balsam fir and black spruce trees. Estimates of all four parameters were significantly different between balsam fir and black spruce for
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Eqs. (2) and (3) only. In the case of Eqs. (1) and (5), however, only the estimates for b4 and d1, respectively, were significantly different between these two species. Parameters, g0 and g2 in the case of balsam fir and g3, g4, and g5 in the case of jack pine for the IS plots were not significantly different from zero for Eq. (2). These comparisons were made based on the confidence limits of the parameters obtained by linear and nonlinear regressions. The level of significance a ¼ 0:05 was chosen as the threshold for the tests throughout this paper. Taper Eqs. (1)–(3) and (5) were further analyzed based on the validation data set, the data set withheld to be used as independent data. To evaluate the predictive ability of the models over the entire length of the stem, the independent variable, relative height, was divided into ten sections. Within each section, the diameters available from the withheld data sets were compared to the diameters predicted from the taper equations. Since the data for the evaluation of the taper equations for jack pine trees were available from three different age classes, models for jack pine were evaluated at each age class. This provided additional information regarding the model dependency on stand age. Excepting Eqs. (1) and (2) at age 50, overall diameter prediction bias (observed predicted) along the bole of jack pine trees for Eqs. (1), (2) and (5) was smaller using the models with parameters estimated using the IS data than using the PCT data (Table 3). For Eq. (3), however, the overall diameter prediction bias for jack pine trees was larger using the IS data than using the PCT data. Although the overall bias increases with stand age for all equations, the increment is smaller for Eq. (5) than for the three other equations. Table 3 shows that Eqs. (2) and (3) are inferior to Eqs. (1) and (5) in terms of bias in estimating diameters along the bole of jack pine trees for all three age groups. Eq. (1) is identical to Eq. (5) in terms of bias at age 50. At ages 70 and 90, however, Eq. (5) is superior to Eq. (1) in estimating diameters. Moreover, Eq. (5) is more robust to stand age and stand origin than Eq. (1) in estimating diameters. The overall increment in bias from one stand age to another for Eq. (5) is smaller than for Eq. (1). The largest difference between the biases, resulted from models with parameters estimated using PCT and IS
45
data, is also smaller for Eq. (5) than for Eq. (1). Similarly, the absolute values of the maximum average bias in estimating the bole diameters across all three tree species were 1.161, 3.568, 2.259, and 0.869 cm for Eqs. (1)–(3) and (5), respectively. In the case of balsam fir, Eq. (5) is better than other equations in estimating diameters along the bole (Table 4). For black spruce trees, Eq. (5) is superior to Eqs. (2) and (3) but equally competitive to Eq. (1) in terms of bias. Eq. (2), however, is inferior to all other equations in terms of overall diameter prediction bias along the bole. The data used for parameter estimation for Jack pine trees were sampled from the stands with ages 59 and 60 for PCT and IS, respectively. Bias smaller at age 50 than at 70 and 90 indicates that the best results are obtained if the predictions are made close to the stand age of the fit data set for all models. The older are the trees the larger is the bias in predicting the diameters along the boles for jack pine trees. Black spruce and balsam fir trees were sampled from the stands with ages 48 and 47, respectively. In both cases, the trees used for model evaluation were younger than for parameter estimation. Other trees from older stands were not available for the evaluation. Therefore, the models for these tree species could not be evaluated against the stand age (Table 5). It is well known that the observed inside bark diameter at a given height could be different among trees of the same dbh and total height. Nevertheless, the estimated diameter at any given height for trees of the same dbh and total tree height would be the same. As a result, the positive and negative biases from different trees could average to zero or close to zero. Therefore, absolute value of the bias in estimating diameters along the bole was calculated for each tree using each equation. These absolute values were averaged for each section of the trees for each species (Tables 4 and 6). Overall, average absolute biases were smaller for Eq. (5) than for the other equations, and this case held true in all three species. This also proves that Eq. (5) is better than other equations in estimating diameters along the bole of the tree species studied here. To further analyze the characteristics of Eq. (5), tree profiles were generated using different parameter sets for a tree with a dbh of 17 cm and a total tree height of 15 m (average values). Although the profiles are
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
Table 3 Bias (cm) (observed predicted) in predicting diameters along the bole of jack pine trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
IS
PCT
IS
PCT
IS
PCT
IS
PCT
Age ¼ 50 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
96 94 84 84 80 78 61 59 17 46
0.546 0.136 0.192 0.186 0.161 0.075 0.010 0.217 0.047 0.013
0.614 0.007 0.091 0.107 0.097 0.016 0.075 0.122 0.178 0.016
0.546 0.077 0.116 0.156 0.151 0.039 0.124 0.006 0.419 1.346
0.865 0.145 0.019 0.057 0.071 0.317 0.514 0.230 0.293 0.468
0.273 0.607 0.715 0.761 0.633 0.221 0.363 0.795 1.744 0.072
0.253 0.213 0.363 0.025 0.278 0.315 0.039 0.244 1.199 0.063
0.407 0.017 0.086 0.125 0.109 0.005 0.102 0.160 0.052 0.012
0.347 0.459 0.308 0.076 0.046 0.022 0.207 0.109 0.493 0.024
Age ¼ 70 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
112 103 100 100 97 84 80 75 50 52
0.257 0.456 0.542 0.450 0.186 0.158 0.148 0.315 0.394 0.112
0.984 0.314 0.432 0.364 0.116 0.094 0.220 0.417 0.531 0.117
0.315 0.337 0.409 0.357 0.122 0.064 0.322 0.590 0.826 3.568
1.043 0.002 0.228 0.163 0.185 0.373 0.776 0.842 0.630 2.550
0.048 0.910 1.021 0.950 0.560 0.181 0.654 1.492 2.259 0.199
0.559 0.024 0.165 0.062 0.169 0.254 0.240 0.919 1.721 0.189
0.256 0.307 0.415 0.365 0.113 0.064 0.260 0.387 0.390 0.110
0.623 0.232 0.022 0.142 0.048 0.037 0.373 0.681 0.869 0.127
Age ¼ 90 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
131 105 112 108 106 107 99 97 62 52
0.062 0.681 0.908 0.636 0.304 0.122 0.230 0.280 0.013 0.042
1.161 0.529 0.786 0.541 0.227 0.054 0.307 0.392 0.143 0.031
0.207 0.474 0.681 0.465 0.171 0.024 0.465 0.628 0.567 0.200
1.012 0.016 0.360 0.149 0.251 0.549 1.017 0.956 0.347 0.711
0.200 1.093 1.355 1.109 0.654 0.140 0.800 1.601 2.240 0.138
0.528 0.061 0.699 0.051 0.143 0.151 0.417 1.031 1.682 0.112
0.279 0.436 0.587 0.479 0.166 0.028 0.408 0.413 0.046 0.040
0.548 0.206 0.178 0.232 0.099 0.054 0.541 0.743 0.606 0.000
similar, the diameters are smaller for balsam fir than for black spruce (Fig. 1). The difference in diameters between balsam fir and black spruce trees above 60% of the total tree height is greater than below. Butt diameter is smaller for jack pine grown at the IS plots than for black spruce, balsam fir, and jack pine grown at the PCT plots. Upper stem diameters for jack pine grown at the PCT stands are smaller than for the other species up to 33% of the total tree height. Above this point, however, diameters for jack pine grown at the PCT and IS study plots are identical. Taper profiles were also generated for balsam fir trees with different dbh values and the same total tree
height (Fig. 2) and with the same dbh value and different total tree heights (Fig. 3). As expected, trees with larger dbh values also demonstrated larger butt diameters and more taper than did the trees with smaller dbh values for the same total tree height (Fig. 2). Similarly, taller trees contained thicker bark and smaller inside bark diameters than did shorter trees at the same relative height for the same dbh (Fig. 3). Kozak and Smith (1993) compared Eqs. (1) and (2) for coastal Douglas-fir trees and found Eq. (2) to be superior in estimating inside bark diameter. This study, however, ranked the Equations in the order
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
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Table 4 Absolute bias (cm) (observed predicted) in predicting diameters along the bole of jack pine trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
IS
PCT
IS
PCT
IS
PCT
IS
PCT
Age ¼ 50 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
96 94 84 84 80 78 61 59 17 46
0.723 0.397 0.529 0.497 0.599 0.654 0.734 0.835 0.702 0.015
0.751 0.415 0.497 0.480 0.599 0.663 0.736 0.807 0.781 0.024
0.711 0.363 0.498 0.469 0.567 0.606 0.688 0.779 0.785 1.346
1.184 0.341 0.432 0.401 0.499 0.575 0.762 0.781 0.714 0.468
0.794 0.636 0.763 0.804 0.782 0.596 0.675 0.980 1.744 0.072
0.568 0.376 0.484 0.397 0.546 0.633 0.647 0.749 1.264 0.063
0.723 0.361 0.491 0.463 0.567 0.617 0.700 0.816 0.674 0.012
0.680 0.512 0.512 0.456 0.557 0.612 0.695 0.788 0.768 0.025
Age ¼ 70 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
112 103 100 100 97 84 80 75 50 52
0.744 0.537 0.654 0.616 0.578 0.632 0.628 0.767 1.004 0.116
1.089 0.417 0.551 0.572 0.576 0.639 0.627 0.812 1.103 0.128
0.743 0.478 0.568 0.558 0.567 0.601 0.632 0.848 1.125 3.568
1.452 0.399 0.479 0.483 0.586 0.622 0.859 0.998 1.038 2.550
0.891 0.928 1.034 0.971 0.724 0.637 0.825 1.550 2.265 0.199
0.833 0.420 0.475 0.478 0.615 0.649 0.635 1.037 1.767 0.189
0.728 0.478 0.581 0.563 0.566 0.605 0.625 0.777 0.971 0.110
0.930 0.429 0.451 0.480 0.559 0.601 0.651 0.901 0.938 0.127
Age ¼ 90 years 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
131 105 112 108 106 107 99 97 62 52
0.814 0.736 0.989 0.741 0.616 0.529 0.522 0.700 0.667 0.035
1.255 0.546 0.836 0.671 0.613 0.534 0.523 0.766 0.710 0.022
0.772 0.659 0.846 0.599 0.530 0.512 0.650 0.845 0.753 0.378
1.464 0.607 0.693 0.494 0.545 0.739 1.092 1.079 0.699 0.711
1.048 1.127 1.383 1.121 0.764 0.572 0.940 1.645 2.241 0.138
0.888 0.532 0.560 0.504 0.520 0.560 0.703 1.145 1.699 0.112
0.744 0.613 0.819 0.623 0.529 0.517 0.610 0.637 0.673 0.040
0.912 0.541 0.592 0.513 0.545 0.521 0.696 0.922 0.791 0.017
(5), (1), (2), and lastly (3) in estimating inside bark diameters for the three species grown in eastern Canada. 4.1. Stand density effect In order to examine stand density or thinning effect on tree taper, Eq. (5) was fitted to the fit data set for each stand density for each species separately. Trees per hectare at the time of plot establishment was used as the stand density for all species. In black spruce, all
the parameters at one stand density were significantly different from their counterparts at the other densities. In the case of balsam fir, only the parameters for heavily thinned stands were significantly different from their counterparts for unthinned and moderately thinned stands. Parameters between unthinned and moderately thinned balsam fir stands, however, were not significantly different from each other. In the case of jack pine trees, parameters for one stand density were not significantly different from their counterparts for the other densities for both IS
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Table 5 Bias (cm) (observed predicted) in predicting diameters along the bole of black spruce and balsam fir trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
Black spruce 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
207 135 141 142 146 136 136 133 136 257
0.124 0.321 0.239 0.079 0.094 0.061 0.053 0.032 0.182 0.051
0.432 0.381 0.426 0.136 0.059 0.312 0.357 0.175 0.333 1.029
0.294 0.035 0.079 0.016 0.315 0.427 0.467 0.369 0.189 0.018
0.027 0.378 0.331 0.212 0.215 0.078 0.000 0.037 0.093 0.176
Balsam fir 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
120 91 65 64 71 63 61 65 65 110
0.175 0.189 0.138 0.098 0.064 0.207 0.150 0.101 0.103 0.101
0.243 0.291 0.398 0.364 0.205 0.189 0.056 0.198 0.451 0.303
0.390 0.470 0.221 0.320 0.438 0.691 0.665 0.686 0.543 0.100
0.016 0.096 0.025 0.104 0.087 0.114 0.061 0.029 0.145 0.174
20 18
Inside bark diameter (cm)
16 14 12 10 8 6 4 2 0 0 BS
1
2 BF
3 JP PCT
4
5 JP IS
6
7
8
9
10
11
12
13
14
15
Height from ground (m)
Fig. 1. Tree profiles generated from different parameter sets estimated using Eq. (5) and black spruce (BS), balsam fir (BF), jack pine PCT (JP PCT), and jack pine IS (JP IS) tree data with the same values of DBH (17.0 cm) and total height (15.0 m).
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Table 6 Absolute bias (cm) (observed predicted) in predicting diameters along the bole of black spruce and balsam fir trees for Eqs. (1)–(3) and (5) using validation data sets Relative height
Number of trees
Eq. (1)
Eq. (2)
Eq. (3)
Eq. (5)
Black spruce 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
207 135 141 142 146 136 136 133 136 257
0.758 0.421 0.449 0.399 0.420 0.499 0.537 0.570 0.512 0.226
0.971 0.568 0.575 0.430 0.422 0.527 0.577 0.596 0.603 1.050
0.852 0.349 0.346 0.361 0.445 0.539 0.591 0.578 0.458 0.214
0.742 0.413 0.410 0.373 0.425 0.499 0.528 0.578 0.549 0.224
Balsam fir 0.0 h/H 0.1 0.1 < h/H 0.2 0.2 < h/H 0.3 0.3 < h/H 0.4 0.4 < h/H 0.5 0.5 < h/H 0.6 0.6 < h/H 0.7 0.7 < h/H 0.8 0.8 < h/H 0.9 0.9 < h/H 1.0
120 91 65 64 71 63 61 65 65 110
0.598 0.252 0.285 0.313 0.366 0.461 0.525 0.531 0.430 0.170
0.516 0.311 0.405 0.398 0.342 0.379 0.431 0.496 0.560 0.325
0.861 0.480 0.286 0.372 0.494 0.727 0.752 0.789 0.621 0.161
0.540 0.225 0.248 0.298 0.341 0.414 0.459 0.472 0.434 0.216
28
Inside bark diameter (cm)
24 20 16 12 8 4 0 0 12
1
2 16
3 20
4
5 24
6
7
8
9
10
11
12
13
14
15
Height from ground (m)
Fig. 2. Tree profiles generated from Eq. (5) using the same value of total height (15.0 m) and different values of dbh (12.0, 16.0, 20.0, and 24.0 cm) for balsam fir.
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53 18 16
Inside bark diameter (cm)
14 12 10 8 6 4 2 0 0
0.1
0.2
0.3
12
14
16
18
0.4
0.5
0.6
0.7
0.8
0.9
1
Relative height
Fig. 3. Tree profiles generated from Eq. (5) using the same value of dbh (17.0 cm) and different values of total height (12.0, 14.0, 16.0, and 18.0 m) for balsam fir.
and the PCT stands. In fact, the model fit to the data so well that the largest bias in estimating diameters along the bole averaged over every 10% of total tree height was less than 0.1 cm. Therefore, Eq. (7) was fitted to the black spruce estimation data set using the NLIN procedure in SAS (1984). Estimated values of parameters d0, d1, d2, d3, and d4 were 0.9253, 2.0139, 0.5235, 0.8453, and 217.3 with approximate standard errors of 0.0045, 0.0069, 0.0450, 0.0683, and 12.1052, respectively. The mean square error (MSE) decreased by 16% when the stand density variable was included in the model. Table 7 displays the bias in estimating diameters along the boles of the trees used for parameter estimation with and without incorporation of the stand density variable. It is clear that including the stand density variable improved the model efficacy significantly for black spruce trees. Taper profiles generated for a tree with a dbh of 17 cm and a total tree height of 15 m using Eq. (7) at different stand densities (Fig. 4) show that trees have larger butt diameters and more taper at a lower stand
density than they do at higher stand density. For the same difference between stand densities, however, the difference in bole diameter between lower and higher stand densities diminishes as stand density increases.
Table 7 Bias (cm) (observed predicted) in predicting diameters along the bole of black spruce trees for Eqs. (5) and (7) (with and without the density variable in the model) using fit data sets Black spruce relative height 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
h/H 0.1 < h/H 0.2 < h/H 0.3 < h/H 0.4 < h/H 0.5 < h/H 0.6 < h/H 0.7 < h/H 0.8 < h/H 0.9 < h/H 1.0
Number of trees
Eq. (5)
Eq. (7)
259 201 201 176 193 183 145 106 44 136
0.044 0.009 0.020 0.097 0.125 0.080 0.129 0.066 0.276 0.032
0.073 0.038 0.001 0.079 0.097 0.059 0.087 0.048 0.179 0.032
M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
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24
Inside bark diameter (cm)
20
16
12
8
4
0 0 1000
1
2 2000
3
4 3000
5
6
4000
7
8
9
10
11
12
13
14
15
Height from ground (m )
Fig. 4. Tree profiles generated from Eq. (7) using dbh ¼ 17:0 cm and total height ¼ 15:0 m at different stand densities (1000, 2000, 3000, and 4000 trees/ha) for black spruce.
5. Conclusions A variable-exponent taper equation was developed for jack pine, black spruce and balsam fir, the three most important commercial tree species in eastern Canada. The equation was derived from the dimensionally compatible taper equation of Sharma and Oderwald (2001) and was compared with the segmented polynomial, variable-exponent, and variable-form taper equations by Max and Burkhart (1976), Kozak (1988) and Zakrzewski (1999), respectively. The taper of black spruce trees was affected by stand density or thinning. Therefore, the equation was modified in order to incorporate this stand density effect. Without the consideration of the stand density effect on tree taper, all four taper equations were very competitive in terms of estimated coefficient of determination (I2). The dimensionally compatible variableexponent taper equation presented in this study was superior to all other equations in terms of the estimated standard error of estimate (S.E.E.) and bias in estimating diameters along the bole of the trees. Moreover, the dimensionally compatible variable-exponent taper equation was less sensitive to the stand age compared
to the other three taper equations in estimating diameters. Since the segmented polynomial (with known inflection points) and dimensionally compatible variable-exponent taper equations have the same number of parameters and the same dependent variable, the mean square errors (MSE) for these two equations were also compared. MSE for the dimensionally compatible variable-exponent taper equation was smaller than for the segmented-polynomial taper equation in all cases. The MSE for the other two taper equations, however, could not be directly compared since they did not have the same dependent variable. Kozak variable-exponent and Zakrzewski variableform taper equations are inferior to the segmented polynomial and dimensionally compatible variableexponent taper equations for all species in terms of bias in estimating diameters. The dimensionally compatible variable-exponent taper equation is superior to the segmented-polynomial taper equation for jack pine and balsam fir. For black spruce, the dimensionally compatible variable-exponent taper equation with density term in the model is superior to the one without the density and segmented-polynomial taper
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M. Sharma, S.Y. Zhang / Forest Ecology and Management 198 (2004) 39–53
equation in terms of fit statistics and bias in estimating diameters for the fit data set. An independent data set was not available to evaluate the dimensionally compatible variable-exponent taper equation with the density term in the model. However, the fit statistics and average biases in estimating diameters for the fit data set indicate that the equation can be applied to accurately estimate diameters of black spruce trees grown at varying densities. This equation can be applied to other tree species by fitting the model to data specific for those species.
Acknowledgements This study was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Living Legacy Trust and Tembec’s Forestry Research Partnership. Our thanks go to Dr. W.T. Zakrzewski, Ontario Forest Research Institute in Sault Ste. Marie, Ontario, Dr. Jean Be´ gin, Laval University in Sainte-Foy, Quebec, and Dr. Isabelle Duchesne, Forintek Canada Corporation in SainteFoy, Quebec for providing stem analysis data for validating the taper equation. We would also like to thank Mr. Yvon Grenier, University of Quebec in Abitibi-Te´ minscaming, Quebec, Dr. C.H. Ung, Canadian Forest Service in Sainte-Foy, Quebec, Mr. Edwin Swift, Canadian Forest Service in Fredericton, New Brunswick, and Mr. Jon Vivian and Mr. Will Smith, British Columbia Ministry of Sustainable Resource Management in Victoria, British Columbia for their assistance in this study and are grateful to two anonymous reviewers for their constructive comments.
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