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A ratio method for predicting stem merchantable volume and associated taper equations for Cupressus lusitanica, Ethiopia
Forest Ecology and Management 204 (2005) 171–179 www.elsevier.com/locate/foreco
A ratio method for predicting stem merchantable volume and associated taper equations for Cupressus lusitanica, Ethiopia Tesfaye Teshome* Wondo Genet College of Forestry, Debub University, P.O. Box 5, Awassa, Ethiopia Received 9 September 2003; received in revised form 9 June 2004; accepted 21 July 2004
Abstract Volume ratio method is one of the methods used for estimating merchantable volume of a stem between a pre-specified stump height and top diameter. Volume ratios of merchantable volume to total volume were expressed as a function of the ratio of merchantable top diameter to breast height diameter (dm/D) and as a function of merchantable height to total tree height (hm/H). A number of mathematical equations were developed using dm/D and hm/H as independent variables for its estimation for Cupressus lusitanica. Fit index and standard error of estimate were used to evaluate and chose the best equations for each explanatory variable. The predictive abilities of the selected equations for estimating merchantable volume ratios were then examined using a test data. The results showed that the selected equations had an overall mean bias of 0.0111 and 0.0052, respectively. Development of merchantable volume ratio equations led to stem taper equations that predict diameter on a stem at any height and height at any diameter, based on given D and H. The performances of the derived taper equations were also tested. The overall absolute biases were 0.2276 cm and 0.2438 m for predicting merchantable top diameter and height at various relative heights and varying merchantable top diameters, respectively. # 2004 Elsevier B.V. All rights reserved. Keywords: Volume ratio method; Merchantable volume ratio; Taper equations
1. Introduction Total tree-stem volume is often estimated using volume equations. When it is expressed on unit area basis, it is a good indicator of the existing amount of timber a forest stand produces. It also provides information on the timber production capacity of the * Tel.: +251 620 4627; fax: +251 620 5421. E-mail address: [email protected].
forestland when related to the age of stands. This measure of timber productivity, however, does not give quantitative information on the amount of wood for a specific use standard, i.e., the amount of saw timber, pulp or fuel wood. Cupressus lusitanica plantations of Munessa Shashemene Forest, Ethiopia, comprise an important source of raw material for the forest products industry. Although C. lusitanica is one of the major exotic plantation species in Ethiopia, no effort has been
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.07.064
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devoted so far to develop merchantable volume and taper estimating equations. There are two methods to estimate merchantable volume of a stem: direct and indirect methods. Direct methods focus on measuring diameter over bark at different locations and calculating the volume by sections between specified points of interest. Merchantable volume estimates are also obtained indirectly utilizing one of the following procedures (Alemdag, 1988): (1) development of volume equation to a fixed top diameter or merchantable height, (2) development of a taper equation which can be integrated to estimate merchantable volume and (3) development of merchantable volume equation that predicts the ratio of merchantable volume to total stem volume for a variable top diameter or merchantable height. The direct method is easy to understand while the indirect method requires developing a mathematical equation using appropriate data. The development of separate equations using the first indirect method for each set of merchantability standards not only requires considerable effort, but also the equations do not always behave logically when considered together. To overcome such a problem, Bailey and Clutter (1970) suggested using constrained volume equations for merchantability limits that differ by a fixed amount. However, this procedure is cumbersome for new sets of merchantability standards. The second indirect approach that has been used to solve this problem was to develop taper equations that predict the upper stem diameter at any point on the tree bole. Integration of taper equations gives partial volume within the specified section of the bole (Kozak et al., 1969; Demaerschalk, 1972, 1973; Clutter, 1980; Biging, 1984; James and Kozak, 1984; Walters and Hann, 1986a; Tesfaye, 1996). The third indirect method estimates the ratio of merchantable volume to total stem volume. This indirect method employs merchantable top diameter or merchantable height as an independent variable together with breast height diameter, total height and in some instances stump height. The merchantable volume is calculated as a product of the merchantable volume conversion ratio and total stem volume, which is obtained by direct measurement or estimated from another volume equation or table. Merchantable volume to total stem volume ratio approach for a
variable top diameter or merchantable height was studied in detail by Bennett et al. (1959), Romancier (1961), Gingrich (1962), Burkhart (1977), Cao and Burkhart (1980), Monserud (1981), Van Deusen et al. (1981, 1982), Reed and Green (1984), Green and Reed (1985), Clutter (1980), Lynch (1986), Walters and Hann (1986b) and Alemdag (1988). The main objective of this study is thus to develop a system of merchantability prediction equations for C. lusitanica, a coniferous species commonly planted in Ethiopia, focusing on ratio method. The second objective is to derive taper equations from the ratio method that predict the diameter for any height and height for any diameter of the tree, and use the system developed for some other species in Ethiopia.
2. Data A total of 48 circular plots with radius of 11.28 m were laid out in representative sites of the three branches of the project. The site index equation developed by Tesfaye and Petty (2000) for C. lusitanica stands in Munessa Forest was instrumental in sample plots distribution in the three branches of the project. Four trees nearest to the center of the sample plots were selected for stem analysis in order to collect the taper and merchantable volume data. The diameter at breast height (D, cm) was measured on each of the 192 sample trees, and these sample trees were then felled and cut into five sections (10, 30, 50, 70 and 90%) of equal length of the total height of individual trees between a 30-cm stump height and the tip of the tree. For each section, the butt and top diameter over bark was measured by caliper at two opposite directions perpendicular to the longitudinal axis of the tree’s bole and the length of the section was measured with a tape. The section lengths were then summed to obtain total height (m) from stump to tip (H). The total volume over bark (without stump volume) of the first five sections was calculated using Smalian’s formula. The volume of the top bolt was calculated using a cone formula. The sum of the six sections provided the total volume over bark of each tree (Vt, m3). The volume of each section was accumulated in succession from the lowest section
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Table 1 Summary of model construction and testing data sets N
D (cm) Mean
dm/D
H (m) Range
Data for model development 135 17.9 5.7–35.2 Data for model testing 57 20.7 9–38.6
hm/H
R
Mean
Range
Mean
Range
Mean
Range
Mean
Range
15.9
5.3–29.5
0.6
0.02–1.15
0.5
0.09–0.91
0.806
0.284–1.00
18.5
8.0–26.0
0.6
0.03–1.08
0.5
0.09–0.90
0.793
0.326–1.00
N is the number of trees; there are 675 and 285 number of observations for the model development and testing data, respectively.
upwards in the tree for each merchantable top diameter (dm) or merchantable height (hm). Moreover, the ratios of these volumes to total stem volume were established. The values of merchantable volume conversion factor (R) are the ratios of merchantable volume (Vm) to total stem volume (Vt), and range between 0.284 and 1. Likewise the ratios of dm/D and hm/H were calculated. The data collected from 135 trees were used for equation development. One-third of the collected data (57 trees) were selected randomly and put aside as test data to examine the performance of the equations that best fit the model construction data (Table 1).
3. Method Merchantability limit can be expressed either by a diameter limit or height limit. In Canada, for instance, it is the merchantable top diameter that is generally used (Bonnor, 1978; Honer et al., 1983) to predict merchantable volume. This does not necessarily mean that merchantable height has no role in the estimation of merchantable volume, although to a lesser degree. Two basic components of a stem are identified, in this study, in relation to its total stem volume: merchantable volume up to the merchantable top diameter and top volume above the merchantable top diameter up to the tip of the tree. These two components are added together to make up total stem volume. Consequently, in ratio values, the sum of the ratios (Vm/Vt) of merchantable volume to merchantable top diameter (Vm) to total stem volume (Vt) and top volume (Vtop) above the merchantable top diameter up to the tip of the tree to the total stem volume (Vtop/Vt) is equal to 1. In light of these facts,
hence, the main objective of this study has been, first, to develop a prediction equation for the ratio of merchantable volume, and second, to establish a method of estimating the ratio of the top volume. Eventually, stem taper equations were derived from the merchantable volume equations. Eight equations (Alemdag, 1988; Burkhart, 1977 and Burkhart modified Cao et al., 1980) were selected for evaluation, as they are simple and provide merchantable volume conversion factor, R, and stem taper equations simultaneously (Table 2). The above equations predict merchantable volume ratios to various merchantable top diameters and heights. The first four involve dm/D and the second four hm/H. The input variables are also easily obtainable by direct measurement including the total tree volume, which can be either measured or estimated from existing volume tables. If the upper and lower diameters on the bole of the stem are given, the volume of a segment of a tree stem can also be easily calculated. Employing the equations that best fit the data, taper equations will be derived and presented which would be used to predict the diameter for any height and height for any diameter of the tree, which is an additional advantage for users.
4. Fitting the equations to the data The equations are conditioned so that R = 1 when hm = H and dm = 0. Hence, in terms of ratio values, the merchantable volume and the top volume when summed up yield 1. Similarly R will not exceed 1. As can be easily noticed from Table 2, none of the above equations is linear. For this reason, the equations were transformed to estimate initial values.
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Table 2 Models fitted for ratio of stump-to-limit volume and total volume Reference
Model form
Models using dm/D as an independent variable Alemdag (1988)
Burkhart (1977) Models using hm/H as an independent variable Alemdag (1988)
Burkhart modified Cao et al. (1980)
Model no.
b2
dm þei D b2 R ¼ 1 b1 ln dDm þ 1 þei b dm 2 þ ei R ¼ exp b1 D b 2 R ¼ 1 b1 Ddmb3 þ ei
R ¼ 1 b1
& 2 R ¼ 1 & 1 1 hHm þei &2 R ¼ 1 & 1 0:693147 ln hHm þ 1 þei &2 R ¼ exp &1 1 hHm þ ei &2 mÞ þ ei R ¼ 1 & 1 ðHh H &3
(1) (2) (3) (4)
(5) (6) (7) (8)
R: merchantable volume ratio; b1–b5, z1–z5: regression coefficients; e1–e5: error terms; dm: merchantable diameter; hm: merchantable height; D: diameter at breast height; H: total height.
To illustrate this, Eq. (1) is transformed logarithmically: dm lnðR 1Þ ¼ lnðb1 Þ þ b2 ln (9) D The same procedure was followed to transform the rest of the equations. Using these initial values, the parameters of the equations were then estimated using nonlinear regression (SAS, 1994). After fitting the equations and estimating the regression coefficients, merchantable volume (Vm) and top volume (Vtop) volumes are estimated from total volume (Vt) as Vm ¼ RVt
(10)
Vtop ¼ ð1 RÞVt
(11)
The equations that fit best the data are evaluated using Alemdag (1986) fit index (FI) (which expresses the goodness-of-fit) and standard error of the estimates (SEE) that were calculated as ! Pn ˆ 2 i¼1 ðYi Yi Þ FI ¼ 1:0 Pn (12) ¯ 2 i¼1 ðYi Yi Þ
SEE ¼
P 1=2 ðYi Yˆ i Þ2 =ðn kÞ Y¯
(13)
where Yi is the observed stump-to-limit volume ratio, Yˆ i the predicted stump-to-limit ratio, Y¯ i the mean of the observed stump–limit-volume ratio, n the number of observation and k the number of parameters estimated.
5. Model testing The selected equations were evaluated for their predictive abilities, accuracy and precision using withheld test data. Three criteria were used to evaluate the equations in terms of R (Cao et al., 1980): (a) bias (the mean of the differences between ¯ (b) mean the observed and predicted volumes, B), absolutes difference (the mean of the absolute ¯ and (c) standard deviation of the differences, jBj) differences, which is commonly called the standard error of estimate, SEE. The same criteria were applied to compare the observed dm to the predicted dm and the observed hm to predicted hm from the tree taper equations. Furthermore, average and percent bias, mean absolute differences, and average and percent standard errors of estimate were calculated and compared for the total tree height and separately for the five
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Table 3 Regression coefficients of the fitted equations Equation no.
Regression coefficients b1
b2
FI
SEE%
b3
dm/D as variable (1) (2) (3) (4)
0.524705 2.277772 0.786223 0.170985
(0.007639)a (0.123892) (0.017152) (0.015869)
2.970851 3.996374 4.033729 3.324962
(0.083702) (0.127014) (0.012690) (0.095954)
– – – 2.908758 (0.089993)
0.8062 0.8109 0.8260 0.8939
12.11 11.96 11.47 8.96
hm/H as variable (5) (6) (7) (8)
0.794743 1.953267 1.276185 0.499703
(0.007081) (0.039860) (0.016877) (0.019755)
2.977106 2.352786 3.733230 2.964786
0.042124) (0.033143) (0.052098) (0.037866)
– – – 2.794968 (0.040395)
0.9631 0.9625 0.9650 0.9704
5.22 5.33 5.14 4.72
a
Values in parentheses are standard error values for the estimated parameters.
relative height classes. Smaller values demonstrate improved performance compared to the other equations.
6. Results and discussion After fitting the equations and estimating the coefficients, the fit index (FI) and the mean-squared errors were calculated to assess the accuracy and performance of the equations in predicting R (Table 3). All equations fitted the data well, as indicated by high FI values (above 0.8062, Table 3). Equations that used hm/H as independent variable provided better results than those that used dm/D. These results are consistent with the findings of Van Deusen et al. (1982), Reed and Green (1984) and Alemdag (1988) who found that the ratio of hm/H was superior to the ratio of dm/D in predicting R. Eq. (4) as a function of dm/D and Eq. (8) as a function of hm/H gave the most accurate results. The FI values for the two equations were 0.8939 and 0.9704, respectively. Eqs. (4) and (8) were examined against D, dm, H and hm to check the behavior of the two equations. As it is depicted in Fig. 1, these two equations show reasonable trends for a given value of any of these variables, indicating the appropriateness of the equations. Furthermore, these graphical expressions show the stem volume distribution for any given breast height diameter, total tree height, diameter at various locations on the stem and height to any point on the bole.
Eqs. (4) and (8) were further tested using the test data (Table 1). The summary of how these two selected equations behaved for the test data is presented in Table 4. The overall mean bias was found to be only 0.0111 and 0.0052 for Eqs. (4) and (8), respectively. Only the mean biases by classes showed an overestimation and underestimation as high as 0.1256 and 0.0419, respectively; these were in the highest dm/D and smallest hm/H classes, respectively. The standard error percentages of all components for Eqs. (4) and (8) are 12.6 and 4.1%, respectively. The standard error values at five relative heights for Eq. (4) vary between 45.3 and 0.11% while these values for Eq. (8) vary between 10.8 and 0.4%. The percent bias and standard error of estimate are based on the average measured values of the class. The error in the estimation of merchantable volume ratio for the 10% height class is high because of underestimation of diameter near ground level due to butt swell. In general, the error percent decreases substantially to 0.11% with increasing relative heights. By and large these testing results depicted that the two equations are adequately accurate and precise for future use.
7. Taper and merchantable height equations Once the two equations that best fit the data are developed for the estimates of merchantable volume ratio, one by diameter (dm/D, Eq. (4)) and one by
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Fig. 1. Characteristics of selected merchantable volume ratio equations: (a1 and a2) for Eq. (4) and (b1 and b2) for Eq. (8).
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Table 4 Values of average and percent bias, mean absolute difference, and average and percent standard error of estimate of Eqs. (4) and (8) for the test data ¯ jBj
SEE
SEE%
Merchantable volume ratios to various merchantable diameters (Eq. (4)) 10% 0.1256 31.7 30% 0.0211 3.1 50% 0.0560 6.3 70% 0.0356 3.6 90% 0.0004 0.04 All components 0.0111 1.40
0.1397 0.0729 0.0690 0.0382 0.0007 0.0641
0.1793 0.0994 0.0899 0.0470 0.0011 0.0999
45.3 14.5 10.1 4.8 0.11 12.6
Merchantable volume ratios to various merchantable heights (Eq. (8)) 10% 0.0060 30% 0.0419 50% 0.0032 70% 0.0109 90% 0.0008 All components 0.0052
0.0339 0.0025 0.0228 0.0139 0.0016 0.0226
0.0427 0.0511 0.0293 0.0161 0.0035 0.0323
10.8 7.4 3.3 1.6 0.4 4.1
Height from stump to the tip of the tree
B¯
¯ B%
height (hm/H, Eq. (8)), they can be easily transformed into taper equation. Following the argument of Alemdag (1988), in two trees of the same shape, which are assumed to be proportionally similar in all dimensions, the ratio of merchantable volume to total volume (R) by (dm/D) will be equal to (R) by (hm/H). Thus we can write b2 d c1 ðH hm Þc2 Rðdm =DÞ ¼ 1b1 mb ; Rðhm =HÞ ¼ 1 H c3 D3
where b1, b2 and b3 and c1, c2 and c3 are estimated coefficients from the dm/D and hm/H as equations (Table 3). After rearranging this equation, the following taper equation is obtained for predicting merchantable diameter to any merchantable top height: c2 c3 1=b2 b3 c1 ðH hm Þ =H dm ¼ D (15) b1 The same procedure is followed to derive an equation that predicts merchantable height to any merchantable top diameter from Eq. (14):
which can be rewritten as 1 b1
d b2 Db3
¼1
1.5 6.1 0.4 1.1 0.1 0.7
c1 ðH hÞc2 H c3
hm ¼ H
(14)
b2 =Db3 ÞÞ H c3 ðb1 ðdm c1
1=c2 (16)
Table 5 Values of average and percent bias, mean absolute differences, and average and percent standard error of estimate of Eq. (15) (cm) and Eq. (16) (m) for the test data Height from stump to the tip of the tree
Equation
B¯
¯ B%
¯ jBj
SEE
SEE%
10%
(15) (16) (15) (16) (15) (16) (15) (16) (15) (16) (15) (16)
1.623 1.146 0.289 0.115 1.349 1.293 1.613 1.399 0.193 0.149 0.228 0.244
8.6 76.5 1.8 2.1 9.9 13.9 17.3 1.4 10.8 0.9 1.9 2.6
1.679 1.467 1.312 1.076 1.858 1.653 1.849 2.916 1.564 0.655 1.539 1.297
3.072 2.366 2.341 1.739 2.346 2.091 2.468 2.107 1.432 1.005 2.341 1.870
16.2 127.5 14.5 31.9 17.3 22.6 26.4 16.3 45.3 6.1 19.2 20.2
30% 50% 70% 90% All
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Table 6 An illustration of the estimated volume ratios of different components using Eqs. (4) and (8) dm/D
hm/H
Merchantable volume ratio
Top volume ratio
Total volume
0.7000
0.9514 0.9769
0.0486 0.0231
1.0 1.0
0.4666
As reported by Alemdag (1988), some anomalous prediction of dm and hm may be expected, as the equations were not developed by regression analysis through fitting to independent C. lusitanica taper data. The taper-predictive abilities of Eqs. (15) and (16) were thus examined using test data to check for their performance, accuracy and precision. The results are presented in Table 5. The over all mean biases are only 0.228 cm and 0.244 m for predicting merchantable top diameter and merchantable height at various relative heights and varying merchantable diameters, respectively. Only the mean biases by classes showed an overestimation of merchantable top diameter and merchantable height as high as 1.623 cm and 1.146 m, and these were in the highest dm/D and lowest hm/H classes, respectively. Overall these testing results indicate that the taper equations are adequately accurate and precise for future use.
8. Application The application of the two selected equations for estimating merchantable and top volume ratios and taper is illustrated below. The merchantable volume and top volume ratios are calculated for dm/ D = 0.4666 and hm/H = 0.7000 using Eqs. (4) and (8) and presented in Table 6 as an example. For taper equations, a stem with the total height and diameter at breast height of 20.4 m and 19.5 cm, Eq. (15) results in 11.67 cm upper diameter if merchantable top diameter is looked for at a merchantable height of 10.2 m while Eq. (16) provides 8.89 m merchantable height if height is searched at merchantable top diameter of 13 cm.
9. Summary The volume ratio method is one of the methods used to estimate tree-stem merchantable volume. In
this study, this ratio (R) is expressed as merchantable volume to total stem volume (Vm/Vt), with both volumes being outside bark. This ratio is strongly correlated with the ratios of dm/D and hm/H. Eq. (4) was best for estimating R as a function of dm/D as judged by statistical measures of FI and SEE% while Eq. (8) outperformed the rest of the equations when hm/H was used as independent variable. When evaluated using a set of test data, the two equations performed well. The overall biases were 0.0111 and 0.0052, respectively. The merchantable volume is calculated as a product of R or the merchantable volume conversion factor and the total stem volume of any unit of measurement. Eqs. (4) and (8) can also give estimates of volume distribution within the stem for various merchantable top diameters and merchantable heights. Developing merchantable volume ratio equations led to derivation of stem taper equations that predict merchantable top diameter on a stem at any merchantable height and merchantable height at any merchantable top diameter, based on given D and H. The performances of the derived taper equations (15) and (16) were also examined. The overall mean biases were 0.228 cm and 0.244 m for predicting merchantable diameter and merchantable height at various merchantable heights and varying merchantable top diameters, respectively. Only the mean biases by classes showed an overestimation as high as 1.623 cm and 1.146 m; these were in the highest d/D and h/H classes, respectively.
Acknowledgements I gratefully acknowledge the support provided by the Council for International Exchange Scholars (CIES), USA, as Fulbright scholar and Oregon State University, College of Forestry, Department of Forest Resources, USA, who have provided me with all necessary facilities. Special thanks go to Dr. D.W. Hann who has given me office facilities and overall
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professional support during my stay at OSU. My thanks go to Dr. Temesgen Hailemariam from the University of British Columbia for his valuable comments and suggestions. References Alemdag, I.S., 1988. A ratio method for calculating stem volume to variable merchantable limits, and associated taper equations. Forest. Chron. 64, 18–26. Alemdag, I.S., 1986. Estimating oven dry mass of trembling aspen and white birch using measurements from aerial photographs. Can. J. For. Res. 16, 163–165. Bailey, R.L., Clutter, J.L., 1970. Volume tables for old-field loblolly pine in the Georgia Piedmont. Ga Forest Research Council Report 22, Ser. 2, 4 pp. Bennett, F.A., MacGee, C.E., Swidel, J.L., 1959. Form class taper tables and volume tables and their applications. J. Agric. Res. 35, 673–744. Biging, G.S., 1984. Taper equation for second-growth mixed conifers of northern California. For. Sci. 30, 1103–1117. Bonnor, G.M. (Ed.), 1978. A guide to Canadian forest inventory terminology and usage. For. Serv., For. Manage. Inst. Burkhart, H.E., 1977. Cubic foot volume of loblolly pine to any merchantable top limit. Southern J. Appl. Forest. 1, 7–9. Cao, Q.V., Burkhart, H.E., 1980. Cubic-foot volume of loblolly pine to any height limit. Southern J. Appl. Forest. 4, 166–168. 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 (1), 71–80. Clutter, J.L., 1980. Development of taper functions from variabletop merchantable volume equations. For. Sci. 6, 117–120. Demaerschalk, J.P., 1972. Converting volume equations to compatible taper equations. For. Sci. 18, 241–245. Demaerschalk, J.P., 1973. Integrated systems for the estimation of tree taper and volume. Can. J. For. Res. 3, 90–94. Gingrich, S.F., 1962. Adjusting shortleaf pine volume tables for different limits of top utilization. Tech. Paper. 190. Cent. States For. Exp. Sta., USDA Forest Service. Green, E.J., Reed, D.D., 1985. Compatible tree volume and taper functions for pitch pine. North. J. Appl. For. 2 (1), 14–16.
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Honer, T.G., Ker, M.F., Alemdag, I.S., 1983. Metric timber tables for the commercial tree species of central and eastern Canada. Maritimes For. Res. Centre Inf. Rep. M-X-140. Can. For. Serv., Environment Canada. James, C.A., Kozak, A., 1984. Fitting taper equations from standing trees. Forest. Chron. 60, 157–161. Kozak, A., Munro, D.D., Smith, J.H.G., 1969. Taper functions and their application in forest inventory. Forest. Chron. 65, 278–283. Lynch, T.B., 1986. Total cubic volume equations implied by diameter-based merchantable to total volume ratio models. For. Sci. 32 (1), 262–267. Monserud, R.A., 1981. Estimating truncated tree volumes with the Behre hyperbolic and existing total volume equations. For. Sci. 27, 253–265. Reed, D.D., Green, E.J., 1984. Compatible stem taper and volume ratio equations. For. Sci. 30 (4), 977–990. Romancier, R.M., 1961. Weight and volume of plantation-grown loblolly pine. SE For. Exp. Sta. REs. Note 161. USDA Forest Service. SAS, 1994. SAS/STAT User’s Guide, Version 6, vol. 2, 4th ed. SAS Institute, Inc., p. 1673. Tesfaye, T., 1996. A growth and yield model for C. lusitanica in Munessa Shashemene State Forest Project, Ethiopia. Ph.D. Thesis. University of Aberdeen, Scotland, UK, 242 pp. Tesfaye, T., Petty, J.A., 2000. Site index equation for Cupressus lusitanica stands in Munessa Forest Ethiopia. For. Ecol. Manage. 126, 339–347. Van Deusen, P.C., Sulivan, A.D., Matney, T.G., 1981. A prediction system for cubic foot volume of loblolly pine applicable through much of its range. Southern J. Appl. For. 5, 186–189. Van Deusen, P.C., Matney, T.G., Sulivan, A.D., 1982. A compatible system for predicting the volume and diameter of sweetgum trees to any height. Southern J. For. 6, 159–163. Walters, D.K., Hann, D.W., 1986a. Taper equations for six conifer species in Southwest Oregon. Corvallis Research Bulletin 56. Forest Research Laboratory, Oregon State University, 41 pp. Walters, D.K., Hann, D.W., 1986b. Predicting merchantable volume in cubic feet to a variable top and in Scribner board feet to a 6inch top for six major conifers of Southwest Oregon. Research Bulletin 52. Forest Research Laboratory, Oregon State University, Corvallis, 107 pp.
A ratio method for predicting stem merchantable volume and associated taper equations for Cupressus lusitanica, Ethiopia Tesfaye Teshome* Wondo Genet College of Forestry, Debub University, P.O. Box 5, Awassa, Ethiopia Received 9 September 2003; received in revised form 9 June 2004; accepted 21 July 2004
Abstract Volume ratio method is one of the methods used for estimating merchantable volume of a stem between a pre-specified stump height and top diameter. Volume ratios of merchantable volume to total volume were expressed as a function of the ratio of merchantable top diameter to breast height diameter (dm/D) and as a function of merchantable height to total tree height (hm/H). A number of mathematical equations were developed using dm/D and hm/H as independent variables for its estimation for Cupressus lusitanica. Fit index and standard error of estimate were used to evaluate and chose the best equations for each explanatory variable. The predictive abilities of the selected equations for estimating merchantable volume ratios were then examined using a test data. The results showed that the selected equations had an overall mean bias of 0.0111 and 0.0052, respectively. Development of merchantable volume ratio equations led to stem taper equations that predict diameter on a stem at any height and height at any diameter, based on given D and H. The performances of the derived taper equations were also tested. The overall absolute biases were 0.2276 cm and 0.2438 m for predicting merchantable top diameter and height at various relative heights and varying merchantable top diameters, respectively. # 2004 Elsevier B.V. All rights reserved. Keywords: Volume ratio method; Merchantable volume ratio; Taper equations
1. Introduction Total tree-stem volume is often estimated using volume equations. When it is expressed on unit area basis, it is a good indicator of the existing amount of timber a forest stand produces. It also provides information on the timber production capacity of the * Tel.: +251 620 4627; fax: +251 620 5421. E-mail address: [email protected].
forestland when related to the age of stands. This measure of timber productivity, however, does not give quantitative information on the amount of wood for a specific use standard, i.e., the amount of saw timber, pulp or fuel wood. Cupressus lusitanica plantations of Munessa Shashemene Forest, Ethiopia, comprise an important source of raw material for the forest products industry. Although C. lusitanica is one of the major exotic plantation species in Ethiopia, no effort has been
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.07.064
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devoted so far to develop merchantable volume and taper estimating equations. There are two methods to estimate merchantable volume of a stem: direct and indirect methods. Direct methods focus on measuring diameter over bark at different locations and calculating the volume by sections between specified points of interest. Merchantable volume estimates are also obtained indirectly utilizing one of the following procedures (Alemdag, 1988): (1) development of volume equation to a fixed top diameter or merchantable height, (2) development of a taper equation which can be integrated to estimate merchantable volume and (3) development of merchantable volume equation that predicts the ratio of merchantable volume to total stem volume for a variable top diameter or merchantable height. The direct method is easy to understand while the indirect method requires developing a mathematical equation using appropriate data. The development of separate equations using the first indirect method for each set of merchantability standards not only requires considerable effort, but also the equations do not always behave logically when considered together. To overcome such a problem, Bailey and Clutter (1970) suggested using constrained volume equations for merchantability limits that differ by a fixed amount. However, this procedure is cumbersome for new sets of merchantability standards. The second indirect approach that has been used to solve this problem was to develop taper equations that predict the upper stem diameter at any point on the tree bole. Integration of taper equations gives partial volume within the specified section of the bole (Kozak et al., 1969; Demaerschalk, 1972, 1973; Clutter, 1980; Biging, 1984; James and Kozak, 1984; Walters and Hann, 1986a; Tesfaye, 1996). The third indirect method estimates the ratio of merchantable volume to total stem volume. This indirect method employs merchantable top diameter or merchantable height as an independent variable together with breast height diameter, total height and in some instances stump height. The merchantable volume is calculated as a product of the merchantable volume conversion ratio and total stem volume, which is obtained by direct measurement or estimated from another volume equation or table. Merchantable volume to total stem volume ratio approach for a
variable top diameter or merchantable height was studied in detail by Bennett et al. (1959), Romancier (1961), Gingrich (1962), Burkhart (1977), Cao and Burkhart (1980), Monserud (1981), Van Deusen et al. (1981, 1982), Reed and Green (1984), Green and Reed (1985), Clutter (1980), Lynch (1986), Walters and Hann (1986b) and Alemdag (1988). The main objective of this study is thus to develop a system of merchantability prediction equations for C. lusitanica, a coniferous species commonly planted in Ethiopia, focusing on ratio method. The second objective is to derive taper equations from the ratio method that predict the diameter for any height and height for any diameter of the tree, and use the system developed for some other species in Ethiopia.
2. Data A total of 48 circular plots with radius of 11.28 m were laid out in representative sites of the three branches of the project. The site index equation developed by Tesfaye and Petty (2000) for C. lusitanica stands in Munessa Forest was instrumental in sample plots distribution in the three branches of the project. Four trees nearest to the center of the sample plots were selected for stem analysis in order to collect the taper and merchantable volume data. The diameter at breast height (D, cm) was measured on each of the 192 sample trees, and these sample trees were then felled and cut into five sections (10, 30, 50, 70 and 90%) of equal length of the total height of individual trees between a 30-cm stump height and the tip of the tree. For each section, the butt and top diameter over bark was measured by caliper at two opposite directions perpendicular to the longitudinal axis of the tree’s bole and the length of the section was measured with a tape. The section lengths were then summed to obtain total height (m) from stump to tip (H). The total volume over bark (without stump volume) of the first five sections was calculated using Smalian’s formula. The volume of the top bolt was calculated using a cone formula. The sum of the six sections provided the total volume over bark of each tree (Vt, m3). The volume of each section was accumulated in succession from the lowest section
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Table 1 Summary of model construction and testing data sets N
D (cm) Mean
dm/D
H (m) Range
Data for model development 135 17.9 5.7–35.2 Data for model testing 57 20.7 9–38.6
hm/H
R
Mean
Range
Mean
Range
Mean
Range
Mean
Range
15.9
5.3–29.5
0.6
0.02–1.15
0.5
0.09–0.91
0.806
0.284–1.00
18.5
8.0–26.0
0.6
0.03–1.08
0.5
0.09–0.90
0.793
0.326–1.00
N is the number of trees; there are 675 and 285 number of observations for the model development and testing data, respectively.
upwards in the tree for each merchantable top diameter (dm) or merchantable height (hm). Moreover, the ratios of these volumes to total stem volume were established. The values of merchantable volume conversion factor (R) are the ratios of merchantable volume (Vm) to total stem volume (Vt), and range between 0.284 and 1. Likewise the ratios of dm/D and hm/H were calculated. The data collected from 135 trees were used for equation development. One-third of the collected data (57 trees) were selected randomly and put aside as test data to examine the performance of the equations that best fit the model construction data (Table 1).
3. Method Merchantability limit can be expressed either by a diameter limit or height limit. In Canada, for instance, it is the merchantable top diameter that is generally used (Bonnor, 1978; Honer et al., 1983) to predict merchantable volume. This does not necessarily mean that merchantable height has no role in the estimation of merchantable volume, although to a lesser degree. Two basic components of a stem are identified, in this study, in relation to its total stem volume: merchantable volume up to the merchantable top diameter and top volume above the merchantable top diameter up to the tip of the tree. These two components are added together to make up total stem volume. Consequently, in ratio values, the sum of the ratios (Vm/Vt) of merchantable volume to merchantable top diameter (Vm) to total stem volume (Vt) and top volume (Vtop) above the merchantable top diameter up to the tip of the tree to the total stem volume (Vtop/Vt) is equal to 1. In light of these facts,
hence, the main objective of this study has been, first, to develop a prediction equation for the ratio of merchantable volume, and second, to establish a method of estimating the ratio of the top volume. Eventually, stem taper equations were derived from the merchantable volume equations. Eight equations (Alemdag, 1988; Burkhart, 1977 and Burkhart modified Cao et al., 1980) were selected for evaluation, as they are simple and provide merchantable volume conversion factor, R, and stem taper equations simultaneously (Table 2). The above equations predict merchantable volume ratios to various merchantable top diameters and heights. The first four involve dm/D and the second four hm/H. The input variables are also easily obtainable by direct measurement including the total tree volume, which can be either measured or estimated from existing volume tables. If the upper and lower diameters on the bole of the stem are given, the volume of a segment of a tree stem can also be easily calculated. Employing the equations that best fit the data, taper equations will be derived and presented which would be used to predict the diameter for any height and height for any diameter of the tree, which is an additional advantage for users.
4. Fitting the equations to the data The equations are conditioned so that R = 1 when hm = H and dm = 0. Hence, in terms of ratio values, the merchantable volume and the top volume when summed up yield 1. Similarly R will not exceed 1. As can be easily noticed from Table 2, none of the above equations is linear. For this reason, the equations were transformed to estimate initial values.
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Table 2 Models fitted for ratio of stump-to-limit volume and total volume Reference
Model form
Models using dm/D as an independent variable Alemdag (1988)
Burkhart (1977) Models using hm/H as an independent variable Alemdag (1988)
Burkhart modified Cao et al. (1980)
Model no.
b2
dm þei D b2 R ¼ 1 b1 ln dDm þ 1 þei b dm 2 þ ei R ¼ exp b1 D b 2 R ¼ 1 b1 Ddmb3 þ ei
R ¼ 1 b1
& 2 R ¼ 1 & 1 1 hHm þei &2 R ¼ 1 & 1 0:693147 ln hHm þ 1 þei &2 R ¼ exp &1 1 hHm þ ei &2 mÞ þ ei R ¼ 1 & 1 ðHh H &3
(1) (2) (3) (4)
(5) (6) (7) (8)
R: merchantable volume ratio; b1–b5, z1–z5: regression coefficients; e1–e5: error terms; dm: merchantable diameter; hm: merchantable height; D: diameter at breast height; H: total height.
To illustrate this, Eq. (1) is transformed logarithmically: dm lnðR 1Þ ¼ lnðb1 Þ þ b2 ln (9) D The same procedure was followed to transform the rest of the equations. Using these initial values, the parameters of the equations were then estimated using nonlinear regression (SAS, 1994). After fitting the equations and estimating the regression coefficients, merchantable volume (Vm) and top volume (Vtop) volumes are estimated from total volume (Vt) as Vm ¼ RVt
(10)
Vtop ¼ ð1 RÞVt
(11)
The equations that fit best the data are evaluated using Alemdag (1986) fit index (FI) (which expresses the goodness-of-fit) and standard error of the estimates (SEE) that were calculated as ! Pn ˆ 2 i¼1 ðYi Yi Þ FI ¼ 1:0 Pn (12) ¯ 2 i¼1 ðYi Yi Þ
SEE ¼
P 1=2 ðYi Yˆ i Þ2 =ðn kÞ Y¯
(13)
where Yi is the observed stump-to-limit volume ratio, Yˆ i the predicted stump-to-limit ratio, Y¯ i the mean of the observed stump–limit-volume ratio, n the number of observation and k the number of parameters estimated.
5. Model testing The selected equations were evaluated for their predictive abilities, accuracy and precision using withheld test data. Three criteria were used to evaluate the equations in terms of R (Cao et al., 1980): (a) bias (the mean of the differences between ¯ (b) mean the observed and predicted volumes, B), absolutes difference (the mean of the absolute ¯ and (c) standard deviation of the differences, jBj) differences, which is commonly called the standard error of estimate, SEE. The same criteria were applied to compare the observed dm to the predicted dm and the observed hm to predicted hm from the tree taper equations. Furthermore, average and percent bias, mean absolute differences, and average and percent standard errors of estimate were calculated and compared for the total tree height and separately for the five
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Table 3 Regression coefficients of the fitted equations Equation no.
Regression coefficients b1
b2
FI
SEE%
b3
dm/D as variable (1) (2) (3) (4)
0.524705 2.277772 0.786223 0.170985
(0.007639)a (0.123892) (0.017152) (0.015869)
2.970851 3.996374 4.033729 3.324962
(0.083702) (0.127014) (0.012690) (0.095954)
– – – 2.908758 (0.089993)
0.8062 0.8109 0.8260 0.8939
12.11 11.96 11.47 8.96
hm/H as variable (5) (6) (7) (8)
0.794743 1.953267 1.276185 0.499703
(0.007081) (0.039860) (0.016877) (0.019755)
2.977106 2.352786 3.733230 2.964786
0.042124) (0.033143) (0.052098) (0.037866)
– – – 2.794968 (0.040395)
0.9631 0.9625 0.9650 0.9704
5.22 5.33 5.14 4.72
a
Values in parentheses are standard error values for the estimated parameters.
relative height classes. Smaller values demonstrate improved performance compared to the other equations.
6. Results and discussion After fitting the equations and estimating the coefficients, the fit index (FI) and the mean-squared errors were calculated to assess the accuracy and performance of the equations in predicting R (Table 3). All equations fitted the data well, as indicated by high FI values (above 0.8062, Table 3). Equations that used hm/H as independent variable provided better results than those that used dm/D. These results are consistent with the findings of Van Deusen et al. (1982), Reed and Green (1984) and Alemdag (1988) who found that the ratio of hm/H was superior to the ratio of dm/D in predicting R. Eq. (4) as a function of dm/D and Eq. (8) as a function of hm/H gave the most accurate results. The FI values for the two equations were 0.8939 and 0.9704, respectively. Eqs. (4) and (8) were examined against D, dm, H and hm to check the behavior of the two equations. As it is depicted in Fig. 1, these two equations show reasonable trends for a given value of any of these variables, indicating the appropriateness of the equations. Furthermore, these graphical expressions show the stem volume distribution for any given breast height diameter, total tree height, diameter at various locations on the stem and height to any point on the bole.
Eqs. (4) and (8) were further tested using the test data (Table 1). The summary of how these two selected equations behaved for the test data is presented in Table 4. The overall mean bias was found to be only 0.0111 and 0.0052 for Eqs. (4) and (8), respectively. Only the mean biases by classes showed an overestimation and underestimation as high as 0.1256 and 0.0419, respectively; these were in the highest dm/D and smallest hm/H classes, respectively. The standard error percentages of all components for Eqs. (4) and (8) are 12.6 and 4.1%, respectively. The standard error values at five relative heights for Eq. (4) vary between 45.3 and 0.11% while these values for Eq. (8) vary between 10.8 and 0.4%. The percent bias and standard error of estimate are based on the average measured values of the class. The error in the estimation of merchantable volume ratio for the 10% height class is high because of underestimation of diameter near ground level due to butt swell. In general, the error percent decreases substantially to 0.11% with increasing relative heights. By and large these testing results depicted that the two equations are adequately accurate and precise for future use.
7. Taper and merchantable height equations Once the two equations that best fit the data are developed for the estimates of merchantable volume ratio, one by diameter (dm/D, Eq. (4)) and one by
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Fig. 1. Characteristics of selected merchantable volume ratio equations: (a1 and a2) for Eq. (4) and (b1 and b2) for Eq. (8).
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177
Table 4 Values of average and percent bias, mean absolute difference, and average and percent standard error of estimate of Eqs. (4) and (8) for the test data ¯ jBj
SEE
SEE%
Merchantable volume ratios to various merchantable diameters (Eq. (4)) 10% 0.1256 31.7 30% 0.0211 3.1 50% 0.0560 6.3 70% 0.0356 3.6 90% 0.0004 0.04 All components 0.0111 1.40
0.1397 0.0729 0.0690 0.0382 0.0007 0.0641
0.1793 0.0994 0.0899 0.0470 0.0011 0.0999
45.3 14.5 10.1 4.8 0.11 12.6
Merchantable volume ratios to various merchantable heights (Eq. (8)) 10% 0.0060 30% 0.0419 50% 0.0032 70% 0.0109 90% 0.0008 All components 0.0052
0.0339 0.0025 0.0228 0.0139 0.0016 0.0226
0.0427 0.0511 0.0293 0.0161 0.0035 0.0323
10.8 7.4 3.3 1.6 0.4 4.1
Height from stump to the tip of the tree
B¯
¯ B%
height (hm/H, Eq. (8)), they can be easily transformed into taper equation. Following the argument of Alemdag (1988), in two trees of the same shape, which are assumed to be proportionally similar in all dimensions, the ratio of merchantable volume to total volume (R) by (dm/D) will be equal to (R) by (hm/H). Thus we can write b2 d c1 ðH hm Þc2 Rðdm =DÞ ¼ 1b1 mb ; Rðhm =HÞ ¼ 1 H c3 D3
where b1, b2 and b3 and c1, c2 and c3 are estimated coefficients from the dm/D and hm/H as equations (Table 3). After rearranging this equation, the following taper equation is obtained for predicting merchantable diameter to any merchantable top height: c2 c3 1=b2 b3 c1 ðH hm Þ =H dm ¼ D (15) b1 The same procedure is followed to derive an equation that predicts merchantable height to any merchantable top diameter from Eq. (14):
which can be rewritten as 1 b1
d b2 Db3
¼1
1.5 6.1 0.4 1.1 0.1 0.7
c1 ðH hÞc2 H c3
hm ¼ H
(14)
b2 =Db3 ÞÞ H c3 ðb1 ðdm c1
1=c2 (16)
Table 5 Values of average and percent bias, mean absolute differences, and average and percent standard error of estimate of Eq. (15) (cm) and Eq. (16) (m) for the test data Height from stump to the tip of the tree
Equation
B¯
¯ B%
¯ jBj
SEE
SEE%
10%
(15) (16) (15) (16) (15) (16) (15) (16) (15) (16) (15) (16)
1.623 1.146 0.289 0.115 1.349 1.293 1.613 1.399 0.193 0.149 0.228 0.244
8.6 76.5 1.8 2.1 9.9 13.9 17.3 1.4 10.8 0.9 1.9 2.6
1.679 1.467 1.312 1.076 1.858 1.653 1.849 2.916 1.564 0.655 1.539 1.297
3.072 2.366 2.341 1.739 2.346 2.091 2.468 2.107 1.432 1.005 2.341 1.870
16.2 127.5 14.5 31.9 17.3 22.6 26.4 16.3 45.3 6.1 19.2 20.2
30% 50% 70% 90% All
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Table 6 An illustration of the estimated volume ratios of different components using Eqs. (4) and (8) dm/D
hm/H
Merchantable volume ratio
Top volume ratio
Total volume
0.7000
0.9514 0.9769
0.0486 0.0231
1.0 1.0
0.4666
As reported by Alemdag (1988), some anomalous prediction of dm and hm may be expected, as the equations were not developed by regression analysis through fitting to independent C. lusitanica taper data. The taper-predictive abilities of Eqs. (15) and (16) were thus examined using test data to check for their performance, accuracy and precision. The results are presented in Table 5. The over all mean biases are only 0.228 cm and 0.244 m for predicting merchantable top diameter and merchantable height at various relative heights and varying merchantable diameters, respectively. Only the mean biases by classes showed an overestimation of merchantable top diameter and merchantable height as high as 1.623 cm and 1.146 m, and these were in the highest dm/D and lowest hm/H classes, respectively. Overall these testing results indicate that the taper equations are adequately accurate and precise for future use.
8. Application The application of the two selected equations for estimating merchantable and top volume ratios and taper is illustrated below. The merchantable volume and top volume ratios are calculated for dm/ D = 0.4666 and hm/H = 0.7000 using Eqs. (4) and (8) and presented in Table 6 as an example. For taper equations, a stem with the total height and diameter at breast height of 20.4 m and 19.5 cm, Eq. (15) results in 11.67 cm upper diameter if merchantable top diameter is looked for at a merchantable height of 10.2 m while Eq. (16) provides 8.89 m merchantable height if height is searched at merchantable top diameter of 13 cm.
9. Summary The volume ratio method is one of the methods used to estimate tree-stem merchantable volume. In
this study, this ratio (R) is expressed as merchantable volume to total stem volume (Vm/Vt), with both volumes being outside bark. This ratio is strongly correlated with the ratios of dm/D and hm/H. Eq. (4) was best for estimating R as a function of dm/D as judged by statistical measures of FI and SEE% while Eq. (8) outperformed the rest of the equations when hm/H was used as independent variable. When evaluated using a set of test data, the two equations performed well. The overall biases were 0.0111 and 0.0052, respectively. The merchantable volume is calculated as a product of R or the merchantable volume conversion factor and the total stem volume of any unit of measurement. Eqs. (4) and (8) can also give estimates of volume distribution within the stem for various merchantable top diameters and merchantable heights. Developing merchantable volume ratio equations led to derivation of stem taper equations that predict merchantable top diameter on a stem at any merchantable height and merchantable height at any merchantable top diameter, based on given D and H. The performances of the derived taper equations (15) and (16) were also examined. The overall mean biases were 0.228 cm and 0.244 m for predicting merchantable diameter and merchantable height at various merchantable heights and varying merchantable top diameters, respectively. Only the mean biases by classes showed an overestimation as high as 1.623 cm and 1.146 m; these were in the highest d/D and h/H classes, respectively.
Acknowledgements I gratefully acknowledge the support provided by the Council for International Exchange Scholars (CIES), USA, as Fulbright scholar and Oregon State University, College of Forestry, Department of Forest Resources, USA, who have provided me with all necessary facilities. Special thanks go to Dr. D.W. Hann who has given me office facilities and overall
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professional support during my stay at OSU. My thanks go to Dr. Temesgen Hailemariam from the University of British Columbia for his valuable comments and suggestions. References Alemdag, I.S., 1988. A ratio method for calculating stem volume to variable merchantable limits, and associated taper equations. Forest. Chron. 64, 18–26. Alemdag, I.S., 1986. Estimating oven dry mass of trembling aspen and white birch using measurements from aerial photographs. Can. J. For. Res. 16, 163–165. Bailey, R.L., Clutter, J.L., 1970. Volume tables for old-field loblolly pine in the Georgia Piedmont. Ga Forest Research Council Report 22, Ser. 2, 4 pp. Bennett, F.A., MacGee, C.E., Swidel, J.L., 1959. Form class taper tables and volume tables and their applications. J. Agric. Res. 35, 673–744. Biging, G.S., 1984. Taper equation for second-growth mixed conifers of northern California. For. Sci. 30, 1103–1117. Bonnor, G.M. (Ed.), 1978. A guide to Canadian forest inventory terminology and usage. For. Serv., For. Manage. Inst. Burkhart, H.E., 1977. Cubic foot volume of loblolly pine to any merchantable top limit. Southern J. Appl. Forest. 1, 7–9. Cao, Q.V., Burkhart, H.E., 1980. Cubic-foot volume of loblolly pine to any height limit. Southern J. Appl. Forest. 4, 166–168. 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 (1), 71–80. Clutter, J.L., 1980. Development of taper functions from variabletop merchantable volume equations. For. Sci. 6, 117–120. Demaerschalk, J.P., 1972. Converting volume equations to compatible taper equations. For. Sci. 18, 241–245. Demaerschalk, J.P., 1973. Integrated systems for the estimation of tree taper and volume. Can. J. For. Res. 3, 90–94. Gingrich, S.F., 1962. Adjusting shortleaf pine volume tables for different limits of top utilization. Tech. Paper. 190. Cent. States For. Exp. Sta., USDA Forest Service. Green, E.J., Reed, D.D., 1985. Compatible tree volume and taper functions for pitch pine. North. J. Appl. For. 2 (1), 14–16.
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