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Compatible stem volume and taper equations for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey
Forest Ecology and Management 256 (2008) 147–151
Contents lists available at ScienceDirect
Forest Ecology and Management journal homepage: www.elsevier.com/locate/foreco
Compatible stem volume and taper equations for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey John R. Brooks a,*, Lichun Jiang b, Ramazan Ozc¸elik c a
Division of Forestry, West Virginia University, 322 Percival Hall, P.O. Box 6125, Morgantown, WV 26506-6125, United States College of Forestry, Northeast Forestry University, 26 Hexing Rd., Donglin District, Harbin, Heilongjiang 150040, China c Faculty of Forestry, Su¨leyman Demirel University, East Campus, 32260 Isparta, Turkey b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 12 December 2007 Received in revised form 7 April 2008 Accepted 9 April 2008
Compatible segmented taper and volume functions were developed for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey. The proposed models generally performed better for the whole tree, especially for Cilicica fir. Average diameter prediction error was less than 2.2 cm and average volume error was less than 0.009 m3. The proposed models provide needed merchantable stem volume and diameter estimates to any point in the bole based on the 10 relative height classes examined for the three species. Model estimates compared well to existing volume tables currently employed for these three important commercial species. ß 2008 Elsevier B.V. All rights reserved.
Keywords: Segmented models Taper and volume Pinus brutia Cedrus libani Abies cilicica
1. Introduction Brutian pine (Pinus brutia Ten.), Cedar of Lebanon (Cedrus libani A. Rich.), and Cilicica fir (Abies cilicica Carr.) are major commercial tree species in Turkey. Brutian pine forests cover an area of about 5.4 million ha with a current standing volume of approximately 270 million m3, Cedar of Lebanon forests cover an area of about 417188.5 ha with a current standing volume of approximately 27.4 million m3, and Cilicica fir forests cover an area of about 279020.6 ha with a current standing volume of approximately 41.7 million m3 (Anonymous, 2006). Individual total tree volume estimation is currently based on existing volume tables developed in this country. With ever changing market conditions, there is a need to accurately estimate tree volumes utilizing multiple upper stem merchantability limits. This is not currently possible with the existing total stem volume tables for these three species. One of the most accurate approaches to estimating upper stem diameter and volume to any merchantability limit is through the use of compatible volume and taper equations (Jiang et al., 2005). However, taper systems do not currently exist in Turkey for these three commercial species. The form and taper of tree stems have been studied for more than 100 years. Taper is the rate of change in diameter in relation to the increase in height along the tree stem (Gray,
* Corresponding author. Tel.: +1 304 293 2941; fax: +1 304 293 2441. E-mail address: [email protected] (J.R. Brooks). 0378-1127/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2008.04.018
1956). Taper functions are the mathematical expression of the change in stem diameter as a function of stem height based on tree species, stand age, density, and the many factors that affect site quality. In most cases, taper functions utilize the measurements of total height, diameter at breast height (dbh) and height above the ground as independent variables, since these variables are commonly measured during forest inventory activities. Numerous taper functions of various forms have been developed over the past 100 years from simple taper functions (Behre, 1923; Kozak et al., 1969; Demaerschalk, 1972; Ormerod, 1973; Hilt, 1980; Biging, 1984; McTague and Bailey, 1987; Thomas and Parresol, 1991; Sharma and Oderwald, 2001; Zakrzewski and MacFarlane, 2006) to more complex forms (Max and Burkhart, 1976; Demaerschalk and Kozak, 1977; Cao et al., 1980; Clark et al., 1991; Kozak, 1988, 2004; Bi, 2000; Fang et al., 2000; Jiang et al., 2005; Jordan et al., 2005). Relatively simple taper functions can effectively describe the general taper of trees; however, they lack the ability to describe the entire stem accurately. They may provide reasonable estimates in the mid-portion of the bole, but usually are less accurate in estimating the profile in the butt or upper stem segments (Max and Burkhart, 1976; Demaerschalk and Kozak, 1977; Kozak, 1988; Newnham, 1992). The purpose of this research was to develop volume estimating equations that can accurately predict individual tree volume to any upper stem merchantability limit. In addition, total tree volume estimates, based on the proposed equations, were compared to existing total stem volume tables in Turkey.
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151
148
Table 1 Model fitting data summary statistics for three commercial tree species in Turkey Species
Mean
S.D.
Minimum
Maximum
Brutian pine (n = 127 trees) dbh (cm) 28.96 Total height (m) 13.71 Disk dob (cm) 18.41 Disk height (m) 6.48
10.45 2.97 11.09 4.20
10.5 6.80 1.00 0.30
54.00 21.50 62.50 20.30
Cedar of Lebanon (n = 78 dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
trees) 31.91 16.50 19.97 8.05
10.47 4.78 11.70 5.48
12.00 8.00 0.50 0.30
58.00 27.60 66.00 26.30
Cilicica fir (n = 82 trees) dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
40.23 18.87 24.25 9.10
14.81 5.08 14.22 6.14
16.00 9.00 2.00 0.30
73.00 28.85 76.00 28.30
Sample trees were selected for Brutian pine, Cedar of Lebanon, and Cilicica fir from the Mut, Elmalı, and Mut Forest Enterprise in Turkey on lands owned by the Forest Service, respectively. Trees were felled and total height was measured to the nearest 0.03 m. Diameter outside bark (dob) at breast height (1.3 m) was measured and recorded to the nearest 0.25 cm. Diameter outside bark was measured at 0.3, 1.3, 2.3 m and then at intervals of 1 m along the remainder of the stem. For each species, 20% of data were randomly selected as a validation data set, while the rest were used for model fitting. Descriptive statistics for the both data sets are shown in Tables 1 and 2. Actual volume for each bolt and tree was calculated using the overlapping bolts method as described by Bailey (1995). 2.1. Taper and volume equation The Max and Burkhart (1976) segmented polynomial taper equation was selected for this study as this equation has been shown to provide accurate results for many species. It consists of three equations that describe the neiloid of the lower section, the paraboloid frustum of the middle section, and the conical shape of the upper bole section. The three equations are combined in a single equation using two join points. This equation is of the form: d
2
¼ b1 ðZ 1Þ þ b2 ðZ 2 1Þ þ b3 ða1 ZÞ2 I1 þ b4 ða2 ZÞ2 I2
(1)
Table 2 Model validation data summary statistics for three commercial tree species in Turkey Species
Mean
S.D.
Minimum
Maximum
Brutian pine (n = 32 trees) dbh (cm) 26.97 Total height (m) 12.96 Disk dob (cm) 17.14 Disk height (m) 6.21
8.78 2.06 10.09 3.89
8.00 6.00 1.00 0.30
46.50 16.90 50.00 16.30
Cedar of Lebanon (n = 19 dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
trees) 30.35 15.80 18.69 7.67
11.56 3.84 11.91 5.01
13.00 7.80 1.00 0.30
55.00 21.20 61.00 19.30
Cilicica fir (n = 21 trees) dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
35.04 17.93 21.80 8.57
6.23 2.69 10.77 5.30
21.00 12.00 2.00 0.30
46.00 22.10 49.00 21.30
Z ai Z > ai
i ¼ 1; 2
Z = h/H, h is the height above the ground to the measurement point (m), H the total tree height (m), D the diameter outside bark at breast height (cm), d the diameter outside bark (cm) to measurement point at height h, ai are the join points to be estimated from the sample data (i = 1, 2), and bi are the regression coefficients (i = 1,. . ., 4). The volume equation, derived through integration of the Max and Burkhart taper equation, is of the form: 9 8 b2 3 b1 2 > > 3 2 > > > ðZu Zl Þ þ ðZu Zl Þ ðb1 þ b2 ÞðZ u Z l Þ > > > > > 2 = <3 b3 2 3 3 (2) V ¼ KD H ½ða1 Z u Þ J1 ða1 Z l Þ K 1 > > 3 > > > > > > b > > 4 3 3 ; : ½ða2 Z u Þ J ða2 Z l Þ K 2 2 3
2. Data
D2
where 1; Ii ¼ 0;
where K = 0.0000785, Zl = hl/H, Zu = hu/H, hl the lower height of interest (m), hu the upper height of interest (m), 1; Z u ai Ji ¼ i ¼ 1; 2 0; Z u > ai Ki ¼
1; 0;
Z l ai Z l > ai
i ¼ 1; 2
All other variables as previously defined. 2.2. Model evaluation To evaluate model performance, Bias, the standard error of the estimate (SEE) and a fit index (FI), as described by Schlaegel (1981), were employed. These evaluation statistics are defined as Pn ðY Yˆi Þ Bias ¼ i¼1 i n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 ˆ i¼1 ðY i Yi Þ SEE ¼ nk "P 2# n ðY Yˆi Þ FI ¼ 1 Pi¼1 i n ¯ 2 i¼1 ðY i YÞ where Yi is the observed value for the ith observation, Yˆi the predicted value for the ith observation, Y¯ the mean of the Yi, k the number of estimated parameters, n the number of observations in the dataset, SEE the standard error of the estimate, and FI is the fit index. To concurrently minimize taper and volume errors, both equations were fitted simultaneously using SAS PROC MODEL (SAS Institute Inc., 2002). All parameters were shared by both the taper and volume equations. After an initial evaluation to determine whether individual species equations were justified, the models were independently fitted to the data for each of the three commercial species. Correlated error structure in the data was not taken into account in SAS MODEL procedure. Prediction accuracy is little affected by the correlated error structure, even when the correlated errors structure is accounted for in the equation fitting process (Williams and Reich, 1997; Kozak, 1997). 3. Results and discussion 3.1. Comparisons of taper functions among species To test whether species-specific parameters for the Max and Burkhart equation was justified, a nonlinear extra sum of squares
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151 Table 3 Total stem fit statistics for the compatible taper and volume equation system for three species in Turkey Species
Bias
SEE
FI
Brutian pine Taper (cm) Volume (m3)
0.3005 0.0006
1.9793 0.0062
0.9682 0.9736
Cedar of Lebanon Taper (cm) Volume (m3)
0.4075 0.0012
2.1969 0.0088
0.9648 0.9562
Cilicica fir Taper (cm) Volume (m3)
0.7813 0.0021
2.1403 0.0083
0.9774 0.9844
procedure was employed (Neter et al., 1996). The significance of the full and reduced model comparisons are based on an F-test of the form: F¼
ðSSER SSEF Þ=ðd:f:R d:f:F Þ SSEF =d:f:F
where SSER is the error sum of squares of the reduced model with the same set of parameters for all species, SSEF the error sum of squares of the full model with a different set of parameters for each species using dummy variables, d.f.R the degrees of freedom for the reduced model, and d.f.F is the degrees of freedom for the full model. Generally, the F-test was considered significant if the P-value for the test is less than 0.05. The results of the F-test indicated that there were differences among species (P < 0.0001) for the combined data. Since these differences may be caused by only two species, full and reduced model F-tests were also conducted for each possible pair of species so that the source of the difference could be identified. For the proposed taper function, there were significant differences between the three species (P < 0.0001), so separate parameter estimates by species were developed. 3.2. Taper equation Parameter estimates were obtained by simultaneously fitting the taper and volume equations for each species. All parameters for each equation were found to be significant at the 0.0001 level. The overall statistics of fit (Bias, SEE and FI) for the entire merchantable stem were calculated and presented in Table 3 for the species tested. The results indicated that the Max and Burkhart equation explained more than 96% of the variation for predicting upper stem diameter. The estimated SEE are approximately 2 cm for each species.
149
The Max and Burkhart equation was further evaluated by relative height (h/H) classes in order to evaluate its performance at different positions throughout the merchantable stem. The data for each species were divided into ten relative height classes. Bias and SEE were calculated for each species by relative height class and used to evaluate taper and volume estimates (Tables 4 and 5). SEE in predicting diameter was relatively small near the bottom of the bole for both Cedar of Lebanon and Cilicica fir (Table 4). In contrast, Bias and SEE values for Brutian pine were higher at the base of the tree, although this error is not considered large. The proposed equations showed consistent sectional performance for each species in the validation data set. Overall, biases were larger for the validation samples of the Brutian pine and Cilicica fir and lower for the Cedar of Lebanon. However, average errors (SEE) were slightly lower for the validation samples for the three species. 3.3. Volume prediction Statistics of fit (Bias, SEE, and FI) for total stem volume for each species are presented in Table 3. The proposed equation explained approximately 95%, 97%, and 98% of the variation for predicting volume for Cedar of Lebanon, Brutian pine, and Cilicica fir, respectively. The estimated SEE are less than 0.009 m3 for each species. Volume prediction by relative height class was also evaluated (Table 5). Average error (SEE) ranged from 0.0003 to 0.0079 m3 for Brutian pine, between 0.0005 and 0.0127 m3 for Cedar of Lebanon, and between 0.0016 and 0.0121 m3 for Cilicica fir. For the validation data set, the proposed equation showed better outside volume prediction for most sections with lower biases and SEE. In order to compare the proposed equation to the existing total volume tables for Brutian pine (Alemdag˘, 1962), Cedar of Lebanon (Evcimen, 1963), and Cilicica fir (Bozkus¸ and Carus, 1997) total volume estimates were compared to actual bole volumes based on the validation dataset. A plot of total volume residuals for the proposed Max and Burkhart model and the existing total stem volume tables indicated that differences were minimal for Brutian pine and Cedar of Lebanon. Volume table prediction errors for Cilicica fir were found to be larger than those for the proposed volume equation (Fig. 1). Prediction errors were generally larger in larger diameter trees. For all trees in the validation data set, the proposed Max and Burkhart model exhibited lower Bias and SEE than the existing volume tables. These differences were small for Brutian pine and Cedar of Lebanon. For Cilicica fir, the SEE for the proposed model was one-quarter the size of that for the existing volume table (Table 6). In addition to providing as good or better total volume estimates, the proposed models can also be utilized to predict product volumes to any desired top diameter limit and
Table 4 Diameter Bias and standard error of the estimate by relative height (RH) class for three Turkish tree species RH
0.0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0 All
Brutian pine
Cedar of Lebanon
Cilicica fir
n
Bias (cm)
SEE (cm)
n
Bias (cm)
SEE (cm)
n
Bias (cm)
SEE (cm)
185 165 167 167 159 165 161 167 152 103 1591
0.9653 0.4257 0.1368 0.0309 0.3795 0.5224 0.2783 0.1694 0.0622 0.1734 0.3005
2.6857 1.3043 1.6538 1.8093 1.9959 2.0785 2.3010 2.2664 1.9059 1.0308 1.9793
131 122 117 114 113 124 117 118 121 100 1177
0.4113 0.3116 0.3701 0.5802 0.5443 0.5422 0.4144 0.4085 0.2588 0.2156 0.4075
1.9839 1.6363 2.0860 2.1204 2.4920 2.6602 2.8353 2.6663 1.9329 1.1770 2.1969
157 138 143 146 140 142 141 141 140 100 1388
0.8819 1.1444 1.2251 1.1923 0.9022 0.8772 0.6597 0.2976 0.1640 0.3002 0.7813
1.7187 1.8338 2.1484 2.2475 2.2649 2.3295 2.2634 2.1422 2.3030 2.3666 2.1403
Note: The all class for Bias and SEE are calculated as overall mean.
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151
150
Table 5 Volume Bias and standard error of the estimate by relative height (RH) class for three Turkish tree species RH
0.0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0 All
Brutian pine
Cedar of Lebanon 3
3
Cilicica fir 3
3
n
Bias (m )
SEE (m )
n
Bias (m )
SEE (m )
n
Bias (m3)
SEE (m3)
185 165 167 167 159 165 161 167 152 103 1591
0.0026 0.0002 0.0005 0.0004 0.0015 0.0013 0.0002 0.0002 0.0001 0.0000 0.0006
0.0079 0.0064 0.0079 0.0076 0.0076 0.0061 0.0054 0.0038 0.0018 0.0003 0.0062
131 122 117 114 113 124 117 118 121 100 1177
0.0014 0.0009 0.0015 0.0021 0.0015 0.0015 0.0007 0.0006 0.0014 0.0001 0.0012
0.0068 0.0095 0.0104 0.0095 0.0107 0.0093 0.0076 0.0050 0.0127 0.0005 0.0088
157 138 143 146 140 142 141 141 140 100 1388
0.0035 0.0052 0.0041 0.0033 0.0016 0.0014 0.0008 0.0001 0.0002 0.0001 0.0021
0.0080 0.0102 0.0121 0.0104 0.0098 0.0084 0.0068 0.0052 0.0039 0.0016 0.0083
Note: The all class for Bias and SEE are calculated as overall mean.
permit multi-product volume estimation for the same tree, a feature not supported in the existing total stem volume tables. Finally, the proposed taper and volume models were refit to the entire data set (model fitting and validation data sets). The resultant parameter estimates by species are provided in Table 7.
Table 6 Comparison of total volume Bias and SEE for the proposed Max and Burkhart model and existing total stem volume table estimates for three commercial species in Turkey Species
Bias (m3)
SEE (m3)
Brutian pine Max and Burkhart Existing volume table
0.0087 0.0127
0.0410 0.0470
Cedar of Lebanon Max and Burkhart Existing volume table
0.0303 0.0627
0.0985 0.1230
Cilicica fir Max and Burkhart Existing volume table
0.0419 0.0964
0.0039 0.0159
Table 7 Parameter estimates for the compatible taper and volume equations for three commercial Turkish species based on all sample data Parameter
Brutian pine
Cedar of Lebanon
Cilicica fir
b3 b2 b3 b4 a1 a2
3.0832 1.4860 0.9304 17.9703 0.7313 0.1307
3.6549 1.7947 1.3658 25.9476 0.7593 0.1116
2.9364 1.3965 0.7093 5.4083 0.8490 0.1710
4. Conclusions In this study, compatible taper and volume equations were developed for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey. To ensure numeric consistency, a simultaneous fitting procedure was used for each compatible taper and volume system of equations. Parameter estimates were obtained that simultaneously minimized taper and volume error. The Max and Burkhart model showed consistent performance in terms of overall fit statistics, sectional performance, Bias and SEE in estimating diameter and volume, respectively. The proposed equation was superior in predicting total volume than currently employed volume tables for all species using the validation data. In addition, the proposed equation can also be used to estimate bole volume to any top diameter or height limit, and can be utilized to estimate multi-product volumes in the same tree.
References Fig. 1. Total volume residuals (validation data) for the proposed Max and Burkhart model and the existing total stem volume tables for (a) Brutian pine, (b) Cedar of Lebanon, and (c) Cilicica fir, respectively.
Alemdag˘, S¸., 1962. Development, Yield and Management Rules of Brutian Pine (Pinus brutia Ten.) Forests in Turkey. Forestry Research Publication, Technical Bulletin No. 11, Ankara, 160 pp.
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151 Anonymous, 2006. Forest Resources. The General Directorate of Forests, Ankara, 159 pp. Bailey, R.L., 1995. Upper stem volumes from stem analysis data: an overlapping bolts method. Can. J. For. Res. 26, 170–173. Behre, C.E., 1923. Preliminary notes on studies of tree form. J. For. 21, 507–511. Bi, H., 2000. Trigonometric variable-form taper equations for Australian eucalyptus. For. Sci. 46, 397–409. Biging, G.S., 1984. Taper equations for second-growth mixed conifers of northern California. For. Sci. 30, 1103–1117. Bozkus¸, H.F., Carus, S., 1997. Double accessed stem volume tables of Cilicica fir (Abies cilicica Carr.) and Cedar (Cedrus libani Link.) and the comparison of them with the ones existing. Rev. Faculty For., Univ. Istanbul Ser. A 47 (1), 51–70. Cao, Q.V., Burkhart, H.E., Max, T.A., 1980. Evaluation of two methods for cubicvolume prediction of loblolly pine to any merchantable limit. For. Sci. 26, 71–80. Clark III, A., Souter, R.A., Schlaegel, B.E., 1991. Stem profile equations for southern tree species. USDA For. Serv. Res. Pap. SE-282. Demaerschalk, J.P., Kozak, A., 1977. The whole-bole system: a conditioned dualequation system for precise prediction of tree profiles. Can. J. For. Res. 7, 488–497. Demaerschalk, J.P., 1972. Converting volume equations to compatible taper equations. For. Sci. 18, 241–245. Evcimen, B.S., 1963. The Economical Value, Yield, and Management Basis of Cedar (Cedrus libani A. Rich.) Forests in Turkey. General Directorate of Forest Publications, Publication No. 355/16, 199 pp. Fang, Z., Borders, B.E., Bailey, R.L., 2000. Compatible volume taper models for loblolly and slash pine based on system with segmented-stem form factors. For. Sci. 46, 1–12. Gray, H.R., 1956. The form and taper of forest-tree stems. Oxford Univ., Imp. For. Inst. Pap. 32, 1–74. Hilt, D.E., 1980. Taper-based system for estimating stem volume of upland oaks. USDA For. Serv. Res. Pap. NE-458 12 p. Jiang, L., Brooks, J.R., Wang, J., 2005. Compatible taper and volume equations for yellow-poplar in West Virginia. For. Ecol. Manage. 213, 399–409.
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Jordan, L., Berenhaut, K., Souter, R., Daniels, R.F., 2005. Parsimonious and completely compatible taper, total, and merchantable volume models. For. Sci. 51, 578– 584. Kozak, A., Munro, D.O., Smith, J.H.G., 1969. Taper functions and their application in forest inventory. For. Chron. 45, 278–283. Kozak, A., 1988. A variable-exponent taper equation. Can. J. For. Res. 18, 1363–1368. Kozak, A., 1997. Effects of multicollinearity and autocorrelation on the variableexponent taper functions. Can. J. For. Res. 27, 619–629. Kozak, A., 2004. My last words on taper equations. For. Chron. 80, 507–515. Max, T.A., Burkhart, H.E., 1976. Segmented polynomial regression applied to taper equations. For. Sci. 22, 283–289. McTague, J.P., Bailey, R.L., 1987. Simultaneous total and merchantable volume equations and a compatible taper function for loblolly pine. Can. J. For. Res. 17, 87–92. Neter, J., Kutner, M.H., Nachtsheim, C.J., Wasserman, W., 1996. Applied Linear Statistical Models. Times Mirror Higher Education Group Inc., 1048 p. 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. SAS Institute Inc., 2002. SAS/ETS User’s Guide, Version 9.0. SAS Institute Inc., Cary, NC. Schlaegel, B.E., 1981. Testing, reporting, and using biomass estimation models. In: Proceedings of the Southern Forest Biomass Workshop, pp. 95–112. Sharma, M., Oderwald, R.G., 2001. Dimensionally compatible volume and taper equations. Can. J. For. Res. 31, 797–803. Thomas, C.E., Parresol, B.R., 1991. Simple, flexible trigonometric taper equations. Can. J. For. Res. 21, 1132–1137. Williams, M.S., Reich, R.M., 1997. Exploring the error structure of taper equations. For. Sci. 43, 378–386. Zakrzewski, W.T., MacFarlane, D.W., 2006. Regional stem profile model fro crossborder comparisons of harvested red pine (Pinus resinosa Ait.) in Ontario and Michigan. For. Sci. 52, 468–475.
Contents lists available at ScienceDirect
Forest Ecology and Management journal homepage: www.elsevier.com/locate/foreco
Compatible stem volume and taper equations for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey John R. Brooks a,*, Lichun Jiang b, Ramazan Ozc¸elik c a
Division of Forestry, West Virginia University, 322 Percival Hall, P.O. Box 6125, Morgantown, WV 26506-6125, United States College of Forestry, Northeast Forestry University, 26 Hexing Rd., Donglin District, Harbin, Heilongjiang 150040, China c Faculty of Forestry, Su¨leyman Demirel University, East Campus, 32260 Isparta, Turkey b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 12 December 2007 Received in revised form 7 April 2008 Accepted 9 April 2008
Compatible segmented taper and volume functions were developed for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey. The proposed models generally performed better for the whole tree, especially for Cilicica fir. Average diameter prediction error was less than 2.2 cm and average volume error was less than 0.009 m3. The proposed models provide needed merchantable stem volume and diameter estimates to any point in the bole based on the 10 relative height classes examined for the three species. Model estimates compared well to existing volume tables currently employed for these three important commercial species. ß 2008 Elsevier B.V. All rights reserved.
Keywords: Segmented models Taper and volume Pinus brutia Cedrus libani Abies cilicica
1. Introduction Brutian pine (Pinus brutia Ten.), Cedar of Lebanon (Cedrus libani A. Rich.), and Cilicica fir (Abies cilicica Carr.) are major commercial tree species in Turkey. Brutian pine forests cover an area of about 5.4 million ha with a current standing volume of approximately 270 million m3, Cedar of Lebanon forests cover an area of about 417188.5 ha with a current standing volume of approximately 27.4 million m3, and Cilicica fir forests cover an area of about 279020.6 ha with a current standing volume of approximately 41.7 million m3 (Anonymous, 2006). Individual total tree volume estimation is currently based on existing volume tables developed in this country. With ever changing market conditions, there is a need to accurately estimate tree volumes utilizing multiple upper stem merchantability limits. This is not currently possible with the existing total stem volume tables for these three species. One of the most accurate approaches to estimating upper stem diameter and volume to any merchantability limit is through the use of compatible volume and taper equations (Jiang et al., 2005). However, taper systems do not currently exist in Turkey for these three commercial species. The form and taper of tree stems have been studied for more than 100 years. Taper is the rate of change in diameter in relation to the increase in height along the tree stem (Gray,
* Corresponding author. Tel.: +1 304 293 2941; fax: +1 304 293 2441. E-mail address: [email protected] (J.R. Brooks). 0378-1127/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2008.04.018
1956). Taper functions are the mathematical expression of the change in stem diameter as a function of stem height based on tree species, stand age, density, and the many factors that affect site quality. In most cases, taper functions utilize the measurements of total height, diameter at breast height (dbh) and height above the ground as independent variables, since these variables are commonly measured during forest inventory activities. Numerous taper functions of various forms have been developed over the past 100 years from simple taper functions (Behre, 1923; Kozak et al., 1969; Demaerschalk, 1972; Ormerod, 1973; Hilt, 1980; Biging, 1984; McTague and Bailey, 1987; Thomas and Parresol, 1991; Sharma and Oderwald, 2001; Zakrzewski and MacFarlane, 2006) to more complex forms (Max and Burkhart, 1976; Demaerschalk and Kozak, 1977; Cao et al., 1980; Clark et al., 1991; Kozak, 1988, 2004; Bi, 2000; Fang et al., 2000; Jiang et al., 2005; Jordan et al., 2005). Relatively simple taper functions can effectively describe the general taper of trees; however, they lack the ability to describe the entire stem accurately. They may provide reasonable estimates in the mid-portion of the bole, but usually are less accurate in estimating the profile in the butt or upper stem segments (Max and Burkhart, 1976; Demaerschalk and Kozak, 1977; Kozak, 1988; Newnham, 1992). The purpose of this research was to develop volume estimating equations that can accurately predict individual tree volume to any upper stem merchantability limit. In addition, total tree volume estimates, based on the proposed equations, were compared to existing total stem volume tables in Turkey.
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151
148
Table 1 Model fitting data summary statistics for three commercial tree species in Turkey Species
Mean
S.D.
Minimum
Maximum
Brutian pine (n = 127 trees) dbh (cm) 28.96 Total height (m) 13.71 Disk dob (cm) 18.41 Disk height (m) 6.48
10.45 2.97 11.09 4.20
10.5 6.80 1.00 0.30
54.00 21.50 62.50 20.30
Cedar of Lebanon (n = 78 dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
trees) 31.91 16.50 19.97 8.05
10.47 4.78 11.70 5.48
12.00 8.00 0.50 0.30
58.00 27.60 66.00 26.30
Cilicica fir (n = 82 trees) dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
40.23 18.87 24.25 9.10
14.81 5.08 14.22 6.14
16.00 9.00 2.00 0.30
73.00 28.85 76.00 28.30
Sample trees were selected for Brutian pine, Cedar of Lebanon, and Cilicica fir from the Mut, Elmalı, and Mut Forest Enterprise in Turkey on lands owned by the Forest Service, respectively. Trees were felled and total height was measured to the nearest 0.03 m. Diameter outside bark (dob) at breast height (1.3 m) was measured and recorded to the nearest 0.25 cm. Diameter outside bark was measured at 0.3, 1.3, 2.3 m and then at intervals of 1 m along the remainder of the stem. For each species, 20% of data were randomly selected as a validation data set, while the rest were used for model fitting. Descriptive statistics for the both data sets are shown in Tables 1 and 2. Actual volume for each bolt and tree was calculated using the overlapping bolts method as described by Bailey (1995). 2.1. Taper and volume equation The Max and Burkhart (1976) segmented polynomial taper equation was selected for this study as this equation has been shown to provide accurate results for many species. It consists of three equations that describe the neiloid of the lower section, the paraboloid frustum of the middle section, and the conical shape of the upper bole section. The three equations are combined in a single equation using two join points. This equation is of the form: d
2
¼ b1 ðZ 1Þ þ b2 ðZ 2 1Þ þ b3 ða1 ZÞ2 I1 þ b4 ða2 ZÞ2 I2
(1)
Table 2 Model validation data summary statistics for three commercial tree species in Turkey Species
Mean
S.D.
Minimum
Maximum
Brutian pine (n = 32 trees) dbh (cm) 26.97 Total height (m) 12.96 Disk dob (cm) 17.14 Disk height (m) 6.21
8.78 2.06 10.09 3.89
8.00 6.00 1.00 0.30
46.50 16.90 50.00 16.30
Cedar of Lebanon (n = 19 dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
trees) 30.35 15.80 18.69 7.67
11.56 3.84 11.91 5.01
13.00 7.80 1.00 0.30
55.00 21.20 61.00 19.30
Cilicica fir (n = 21 trees) dbh (cm) Total height (m) Disk dob (cm) Disk height (m)
35.04 17.93 21.80 8.57
6.23 2.69 10.77 5.30
21.00 12.00 2.00 0.30
46.00 22.10 49.00 21.30
Z ai Z > ai
i ¼ 1; 2
Z = h/H, h is the height above the ground to the measurement point (m), H the total tree height (m), D the diameter outside bark at breast height (cm), d the diameter outside bark (cm) to measurement point at height h, ai are the join points to be estimated from the sample data (i = 1, 2), and bi are the regression coefficients (i = 1,. . ., 4). The volume equation, derived through integration of the Max and Burkhart taper equation, is of the form: 9 8 b2 3 b1 2 > > 3 2 > > > ðZu Zl Þ þ ðZu Zl Þ ðb1 þ b2 ÞðZ u Z l Þ > > > > > 2 = <3 b3 2 3 3 (2) V ¼ KD H ½ða1 Z u Þ J1 ða1 Z l Þ K 1 > > 3 > > > > > > b > > 4 3 3 ; : ½ða2 Z u Þ J ða2 Z l Þ K 2 2 3
2. Data
D2
where 1; Ii ¼ 0;
where K = 0.0000785, Zl = hl/H, Zu = hu/H, hl the lower height of interest (m), hu the upper height of interest (m), 1; Z u ai Ji ¼ i ¼ 1; 2 0; Z u > ai Ki ¼
1; 0;
Z l ai Z l > ai
i ¼ 1; 2
All other variables as previously defined. 2.2. Model evaluation To evaluate model performance, Bias, the standard error of the estimate (SEE) and a fit index (FI), as described by Schlaegel (1981), were employed. These evaluation statistics are defined as Pn ðY Yˆi Þ Bias ¼ i¼1 i n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 ˆ i¼1 ðY i Yi Þ SEE ¼ nk "P 2# n ðY Yˆi Þ FI ¼ 1 Pi¼1 i n ¯ 2 i¼1 ðY i YÞ where Yi is the observed value for the ith observation, Yˆi the predicted value for the ith observation, Y¯ the mean of the Yi, k the number of estimated parameters, n the number of observations in the dataset, SEE the standard error of the estimate, and FI is the fit index. To concurrently minimize taper and volume errors, both equations were fitted simultaneously using SAS PROC MODEL (SAS Institute Inc., 2002). All parameters were shared by both the taper and volume equations. After an initial evaluation to determine whether individual species equations were justified, the models were independently fitted to the data for each of the three commercial species. Correlated error structure in the data was not taken into account in SAS MODEL procedure. Prediction accuracy is little affected by the correlated error structure, even when the correlated errors structure is accounted for in the equation fitting process (Williams and Reich, 1997; Kozak, 1997). 3. Results and discussion 3.1. Comparisons of taper functions among species To test whether species-specific parameters for the Max and Burkhart equation was justified, a nonlinear extra sum of squares
J.R. Brooks et al. / Forest Ecology and Management 256 (2008) 147–151 Table 3 Total stem fit statistics for the compatible taper and volume equation system for three species in Turkey Species
Bias
SEE
FI
Brutian pine Taper (cm) Volume (m3)
0.3005 0.0006
1.9793 0.0062
0.9682 0.9736
Cedar of Lebanon Taper (cm) Volume (m3)
0.4075 0.0012
2.1969 0.0088
0.9648 0.9562
Cilicica fir Taper (cm) Volume (m3)
0.7813 0.0021
2.1403 0.0083
0.9774 0.9844
procedure was employed (Neter et al., 1996). The significance of the full and reduced model comparisons are based on an F-test of the form: F¼
ðSSER SSEF Þ=ðd:f:R d:f:F Þ SSEF =d:f:F
where SSER is the error sum of squares of the reduced model with the same set of parameters for all species, SSEF the error sum of squares of the full model with a different set of parameters for each species using dummy variables, d.f.R the degrees of freedom for the reduced model, and d.f.F is the degrees of freedom for the full model. Generally, the F-test was considered significant if the P-value for the test is less than 0.05. The results of the F-test indicated that there were differences among species (P < 0.0001) for the combined data. Since these differences may be caused by only two species, full and reduced model F-tests were also conducted for each possible pair of species so that the source of the difference could be identified. For the proposed taper function, there were significant differences between the three species (P < 0.0001), so separate parameter estimates by species were developed. 3.2. Taper equation Parameter estimates were obtained by simultaneously fitting the taper and volume equations for each species. All parameters for each equation were found to be significant at the 0.0001 level. The overall statistics of fit (Bias, SEE and FI) for the entire merchantable stem were calculated and presented in Table 3 for the species tested. The results indicated that the Max and Burkhart equation explained more than 96% of the variation for predicting upper stem diameter. The estimated SEE are approximately 2 cm for each species.
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The Max and Burkhart equation was further evaluated by relative height (h/H) classes in order to evaluate its performance at different positions throughout the merchantable stem. The data for each species were divided into ten relative height classes. Bias and SEE were calculated for each species by relative height class and used to evaluate taper and volume estimates (Tables 4 and 5). SEE in predicting diameter was relatively small near the bottom of the bole for both Cedar of Lebanon and Cilicica fir (Table 4). In contrast, Bias and SEE values for Brutian pine were higher at the base of the tree, although this error is not considered large. The proposed equations showed consistent sectional performance for each species in the validation data set. Overall, biases were larger for the validation samples of the Brutian pine and Cilicica fir and lower for the Cedar of Lebanon. However, average errors (SEE) were slightly lower for the validation samples for the three species. 3.3. Volume prediction Statistics of fit (Bias, SEE, and FI) for total stem volume for each species are presented in Table 3. The proposed equation explained approximately 95%, 97%, and 98% of the variation for predicting volume for Cedar of Lebanon, Brutian pine, and Cilicica fir, respectively. The estimated SEE are less than 0.009 m3 for each species. Volume prediction by relative height class was also evaluated (Table 5). Average error (SEE) ranged from 0.0003 to 0.0079 m3 for Brutian pine, between 0.0005 and 0.0127 m3 for Cedar of Lebanon, and between 0.0016 and 0.0121 m3 for Cilicica fir. For the validation data set, the proposed equation showed better outside volume prediction for most sections with lower biases and SEE. In order to compare the proposed equation to the existing total volume tables for Brutian pine (Alemdag˘, 1962), Cedar of Lebanon (Evcimen, 1963), and Cilicica fir (Bozkus¸ and Carus, 1997) total volume estimates were compared to actual bole volumes based on the validation dataset. A plot of total volume residuals for the proposed Max and Burkhart model and the existing total stem volume tables indicated that differences were minimal for Brutian pine and Cedar of Lebanon. Volume table prediction errors for Cilicica fir were found to be larger than those for the proposed volume equation (Fig. 1). Prediction errors were generally larger in larger diameter trees. For all trees in the validation data set, the proposed Max and Burkhart model exhibited lower Bias and SEE than the existing volume tables. These differences were small for Brutian pine and Cedar of Lebanon. For Cilicica fir, the SEE for the proposed model was one-quarter the size of that for the existing volume table (Table 6). In addition to providing as good or better total volume estimates, the proposed models can also be utilized to predict product volumes to any desired top diameter limit and
Table 4 Diameter Bias and standard error of the estimate by relative height (RH) class for three Turkish tree species RH
0.0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0 All
Brutian pine
Cedar of Lebanon
Cilicica fir
n
Bias (cm)
SEE (cm)
n
Bias (cm)
SEE (cm)
n
Bias (cm)
SEE (cm)
185 165 167 167 159 165 161 167 152 103 1591
0.9653 0.4257 0.1368 0.0309 0.3795 0.5224 0.2783 0.1694 0.0622 0.1734 0.3005
2.6857 1.3043 1.6538 1.8093 1.9959 2.0785 2.3010 2.2664 1.9059 1.0308 1.9793
131 122 117 114 113 124 117 118 121 100 1177
0.4113 0.3116 0.3701 0.5802 0.5443 0.5422 0.4144 0.4085 0.2588 0.2156 0.4075
1.9839 1.6363 2.0860 2.1204 2.4920 2.6602 2.8353 2.6663 1.9329 1.1770 2.1969
157 138 143 146 140 142 141 141 140 100 1388
0.8819 1.1444 1.2251 1.1923 0.9022 0.8772 0.6597 0.2976 0.1640 0.3002 0.7813
1.7187 1.8338 2.1484 2.2475 2.2649 2.3295 2.2634 2.1422 2.3030 2.3666 2.1403
Note: The all class for Bias and SEE are calculated as overall mean.
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Table 5 Volume Bias and standard error of the estimate by relative height (RH) class for three Turkish tree species RH
0.0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0 All
Brutian pine
Cedar of Lebanon 3
3
Cilicica fir 3
3
n
Bias (m )
SEE (m )
n
Bias (m )
SEE (m )
n
Bias (m3)
SEE (m3)
185 165 167 167 159 165 161 167 152 103 1591
0.0026 0.0002 0.0005 0.0004 0.0015 0.0013 0.0002 0.0002 0.0001 0.0000 0.0006
0.0079 0.0064 0.0079 0.0076 0.0076 0.0061 0.0054 0.0038 0.0018 0.0003 0.0062
131 122 117 114 113 124 117 118 121 100 1177
0.0014 0.0009 0.0015 0.0021 0.0015 0.0015 0.0007 0.0006 0.0014 0.0001 0.0012
0.0068 0.0095 0.0104 0.0095 0.0107 0.0093 0.0076 0.0050 0.0127 0.0005 0.0088
157 138 143 146 140 142 141 141 140 100 1388
0.0035 0.0052 0.0041 0.0033 0.0016 0.0014 0.0008 0.0001 0.0002 0.0001 0.0021
0.0080 0.0102 0.0121 0.0104 0.0098 0.0084 0.0068 0.0052 0.0039 0.0016 0.0083
Note: The all class for Bias and SEE are calculated as overall mean.
permit multi-product volume estimation for the same tree, a feature not supported in the existing total stem volume tables. Finally, the proposed taper and volume models were refit to the entire data set (model fitting and validation data sets). The resultant parameter estimates by species are provided in Table 7.
Table 6 Comparison of total volume Bias and SEE for the proposed Max and Burkhart model and existing total stem volume table estimates for three commercial species in Turkey Species
Bias (m3)
SEE (m3)
Brutian pine Max and Burkhart Existing volume table
0.0087 0.0127
0.0410 0.0470
Cedar of Lebanon Max and Burkhart Existing volume table
0.0303 0.0627
0.0985 0.1230
Cilicica fir Max and Burkhart Existing volume table
0.0419 0.0964
0.0039 0.0159
Table 7 Parameter estimates for the compatible taper and volume equations for three commercial Turkish species based on all sample data Parameter
Brutian pine
Cedar of Lebanon
Cilicica fir
b3 b2 b3 b4 a1 a2
3.0832 1.4860 0.9304 17.9703 0.7313 0.1307
3.6549 1.7947 1.3658 25.9476 0.7593 0.1116
2.9364 1.3965 0.7093 5.4083 0.8490 0.1710
4. Conclusions In this study, compatible taper and volume equations were developed for Brutian pine, Cedar of Lebanon, and Cilicica fir in Turkey. To ensure numeric consistency, a simultaneous fitting procedure was used for each compatible taper and volume system of equations. Parameter estimates were obtained that simultaneously minimized taper and volume error. The Max and Burkhart model showed consistent performance in terms of overall fit statistics, sectional performance, Bias and SEE in estimating diameter and volume, respectively. The proposed equation was superior in predicting total volume than currently employed volume tables for all species using the validation data. In addition, the proposed equation can also be used to estimate bole volume to any top diameter or height limit, and can be utilized to estimate multi-product volumes in the same tree.
References Fig. 1. Total volume residuals (validation data) for the proposed Max and Burkhart model and the existing total stem volume tables for (a) Brutian pine, (b) Cedar of Lebanon, and (c) Cilicica fir, respectively.
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