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Construction of site index equations for even aged stands of Tectona grandis (teak) from permanent plot data in India
Forest Ecology and Management 212 (2005) 14–22 www.elsevier.com/locate/foreco
Construction of site index equations for even aged stands of Tectona grandis (teak) from permanent plot data in India Anoop Upadhyay, Tron Eid *, Prem L. Sankhayan Department of Ecology and Natural Resource Management, Agricultural University of Norway, P.O. Box 5003, NO-1432 Aas, Norway Received 7 November 2003; received in revised form 5 November 2004; accepted 21 February 2005
Abstract A quantitative measure of the timber production potential of a given site may be provided by site index equations, which predict site index at a reference age from height and age information of a forest stand. Data from permanent sample plots laid down in even aged stands of Tectona grandis in six states of India were used to develop site index equations. Eleven growth functions were estimated from height–age data using the guide curve method and difference equation method with anamorphic and polymorphic formulations. The applied methods and estimated functions were compared with respect to residual variation, i.e., root mean square error (RMSE) from original data. The difference equation method gave lower RMSE than the guide curve method. A special solution of the difference equation derived from one of the growth functions, where a hyperbolic relationship between site index and a parameter was assumed, gave the best fit to the data. Site index curves with dominant heights in the 11– 28 m range at a reference age of 25 years are presented. # 2005 Elsevier B.V. All rights reserved. Keywords: Difference equation method; Guide curve method; India; Site index; Tectona grandis
1. Introduction Tectona grandis (family Verbenaceae) or teak is a predominantly tropical or sub-tropical tree. The natural distribution of the genus Tectona is in South and SouthEast Asia. India is one of the major teak producing countries where the genus is known only by T. grandis and has great genetic variability with distribution over * Corresponding author. Tel.: +47 64 94 8901; fax: +47 64 94 8890. E-mail address: [email protected] (T. Eid).
8.9 million hectares. Its natural zone of distribution is confined mostly to the peninsular region below 248 latitude (Tewari, 1992). Teak is a strong light demander, frost sensitive and a vigorous coppicer. It is capable of thriving on a variety of soils and geological formations, but requires good sub-soil drainage. Teak is the most important and general-purpose timber in India, suitable for several end uses. It is used for construction and making of furniture, railway sleepers, doors, windows, electric poles, etc., and is the best Indian timber in dimensional stability with very low fibre saturation point and shrinkage. In order to classify and group
0378-1127/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2005.02.058
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timber from different species in India according to their functional properties, suitability indices are derived taking teak as the standard for comparison (Tewari, 1992). Teak is reputed worldwide as a quality timber on account of its appropriate physical and mechanical properties and is one of the commercially most important and costly timbers in India. The initial and almost indispensable step in the prediction of growth and yield is to quantify differences in site quality. In the context of timber management, site quality can be perceived as the timber production potential of a site for a particular species or forest type (e.g., Clutter et al., 1983). In even-aged stands of a single species, the most common measure is site index, i.e., the expected height at a nominated age. Site index models provide a convenient and effective tool for determining the forest management practices as they help in predicting timber productivity, wood volume, potential rate of growth of a forest and the time of first and subsequent thinning. Site index is an indispensable input parameter in growth– yield and biomass prediction models. It has been used in growth and yield studies for estimation of timber volume and basal area (e.g., Clutter, 1963; Johnstone, 1976; Singh, 1979; Sharma, 1979). Site index equations predict site index from age and height information. A variety of different techniques and equational forms have been used with variations in application. Most of these techniques can be viewed as special cases of three general equation-development methods, namely the guide curve method, the difference equation method and the parameter prediction method (e.g., Clutter et al., 1983). The classification of height–age curve families may be anamorphic, polymorphic-disjoint or polymorphicnon-disjoint (Bailey and Clutter, 1974). A number of studies have been conducted in the past to estimate site index equations for temperate species like Norway spruce (e.g., Tveite, 1977), Sitka spruce (e.g., Farr and Harris, 1979), Douglas fir (Doolittle, 1958) and a variety of pines (e.g., Wakely and Marrero, 1958; Myers and Van Deusen, 1960; Beck, 1971; Graney and Burkhart, 1973; Alban, 1976; Newberry and Pienaar, 1978; Amaetis and Burkhart, 1985; Elfving and Kiviste, 1998). In comparison, work in the field of tropical species is relatively scarce. Top height models and site index curves have been
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developed for some tropical species like T. grandis (Malende and Temu, 1990; Nunifu and Murchison, 1999; Bermejo et al., 2004), Cupressus lusitanica (Teshome and Petty, 2000) and Azadirachta indica (Tewari and Kishan Kumar, 2002). In India, top heights of permanent sample plots of some tropical species at different sites have been recorded at various ages. Based on it, site qualities have been classified covering a range of top heights at different age (Directorate of Forest Education, 1970). Site index equations for these species that provide a specific index value at a reference age for any site, and that can serve as input for estimation of timber production are, however, not previously developed in India. In spite of India’s long and unique history of maintaining sample plots in a wide range of forest types, the knowledge generated from these ‘‘permanent laboratories’’ in the field (Ghosh and Kaul, 1977) has mostly failed to enter the scientific literature through proper analysis (Rogers, 1991). An exception from this, however, is provided by Pandey (1996) who applied permanent sample plots from teak plantations in India in order to study the influence of climatic factors on the productivity of the species. The aim of the present study was to make use of data from permanent sample plots of even aged forest to construct different site index equations using dominant height and age data of T. grandis at different locations in India. Different growth functions were tested with different construction methods, i.e., the guide curve method as well as the difference equation method with anamorphic and polymorphic formulations, to find the best solution.
2. Material and methods 2.1. Material Data from 150 permanent sample plots of T. grandis laid out in the states of Uttar Pradesh, Madhya Pradesh, Maharashtra, Orissa, Karnataka and Kerala of India were used in this study. These plots, in average approximately 0.2 ha in size and of variable density and spacing, were laid out by the Forest Research Institute, Dehradun (India) and cover a considerable range of natural distribution of teak throughout India. Remeasurements were carried out in
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G1: all 747 observations on height and age from the 150 sample plots were used for the guide curve method. G2: a set of 448 height and age pairs, i.e., height and age observations from measurements and remeasurements on the 150 sample plots, was used for the difference equation method. G3: an independent set of 110 observations from seven permanent sample plots not included in G1 and G2, from where height–age series have been measured and remeasured over a period of 75 years at 5-yearintervals, was used for validation of results. G3 constitutes a part of the permanent sample plots applied in developing the yield tables published by the Forest Research Institute, Dehradun (Directorate of Forest Education, 1970).
Fig. 1. Observed top height over age in the Tectona grandis sample plots.
these sample plots at age intervals of 4–10 years; most of them at an interval of 5 years. Age at which measurements have been recorded varied from 4 to 93 years. All together 747 observations were available from the sample plots (Fig. 1, Table 1). The scatter diagram in Fig. 1 gives the range of distribution of the observed top heights of the plots at different ages. The top height of the stand has been defined in this study as the average height of the 100 largest diameter trees per hectare. The heights of sample trees in all diameter classes were recorded and a height–diameter curve was drawn for each plot. The mean diameter of the 10 dominant trees in a plot was calculated from their total basal area, and the height corresponding to this mean diameter on the height–diameter curve was taken as the top height of the plot. Stand age has been defined as the age of the stand from the year of plantation. The data from the remeasured permanent plots were arranged in the following three G1–G3 groups.
2.2. Methods Eleven widely used (Kiviste, 1988) and highly flexible growth functions were selected (Table 2). Some of these functions have been analysed by Zeidi (1993) and most of them have been tested for construction of site index equations for Pinus svlvestris in Sweden by Elfving and Kiviste (1998). All these functions are of the general form: H ¼ f ðAÞ
(1)
where A denotes stand age (years) and H top height (m). The guide curve method was applied, both directly on all the 747 observations as well as after grouping the data in 10-year age classes, and performing smoothing as un-weighted, non-linear regression for arithmetic means of top heights and age for each age class.
Table 1 Top height (m) statistics over ages classes (age class 5 = 1–10 years, etc.) Age class
Number of observations
Mean
Standard deviation
Minimum value
Maximum value
5 15 25 35 45 55 65 75 85 95
68 116 109 101 84 85 85 64 31 4
11.8 16.7 21.3 22.7 23.7 27.0 28.4 29.1 29.7 30.4
3.1 4.1 4.1 5.9 5.5 6.1 5.9 6.6 7.0 5.1
5.1 7.2 10.2 10.2 11.7 12.9 16.5 18.0 19.2 33.0
17.4 24.3 28.2 32.1 34.2 36.3 37.8 39.6 40.5 40.2
Total
747
A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22 Table 2 Growth functions studied No.
Growth function
Functions of two parameters E1 E2
H = b0 exp(b1/A) H = b0/(1+b1/A)3
Functions of three parameters: E3 H ¼ b0 f1 expðb1 =AÞgb2 E4 H ¼ b0 =f1 þ ðb1 =AÞ þ ðb2 =A2 Þg E5 H ¼ b0 =f1 þ b1 =Ab2 g E6 H ¼ b0 =ð1 þ b1 =A2 Þb2 E7 H ¼ b0 expðb1 =Ab2 Þ E8 H ¼ b0 f1 expðb1 Ab2 Þg E9 H ¼ b0 =f1 þ b1 expðb2 AÞg Functions of four parameters E10 E11
H ¼ b0 =f1 þ b1 =Ab2 gb3 H ¼ b0 =f1 þ ðb1 =AÞ þ ðb2 =Ab3 Þg
H = height, A = age, b0, b1, b2 and b3 are parameters.
The guide curve method, when applied directly on all observations, will naturally give low R2 values and high RMSE values as compared to when the mean heights in 10-year age classes are considered. This is due to the effect of repeat runs, i.e., different observed values of top heights at the same age. In feet, it is not possible to attain a high value of R2 in such cases, no matter how appropriate the model is, since any model can explain only the variations due to lack of fit and not the pure error variations resulting from repeated runs (Draper and Smith, 1981). If
Using the above method, described by Draper and Smith (1981), the data set G1 was checked for the extent of pure error and lack of fit. The difference equation method was applied using the general form of the difference function H2 = f(H1, A1, A2) directly on the remeasurement data set G2. Here H1 and H2 are the top-heights at ages A1 and A2. The anamorphic formulation of the equation was obtained by solving (1) for the constant term, to give: H2 ¼ H1
nj m X X ðY ju Y¯ j Þ2 ;
(2)
j¼1 u¼1
where Yju is the uth observation Pn j (u = 1, 2, . . ., nj) at Xj (j = l, 2, . . ., m) and Y¯ j ¼ u¼1 Y ju with degrees of freedom ne ¼
m X j¼1
n j m:
(3)
f ðA2 Þ f ðA1 Þ
(4)
Substitution of one of the variable coefficients in (1) will give the polymorphic formulation of the difference equation. A special solution of the difference equation method was derived by Cieszewski and Bella (1989). The function E5 was transformed into the site index model: b1 1 þ b2 A0 H ¼ SI (5) 1 þ Abb12 where SI is the site index, i.e., the top height at reference age A0. A hyperbolic relationship between SI and parameter b1 was assumed to exist (b1 = b/SI). Thus, model (5) could be transformed into the form: SI þ H¼
Y11, Yl2, . . ., Y1n1 are n1 repeat observations at point X1, Y21, Y22, . . ., Y2n2 are n2 repeat observations at point X2, . . ., Ym1, Ym2, . . ., Ymnm are nm repeat observations at point Xm , then the total pure error is
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b b A0 2
(6)
b
1 þ ASIb2 A polymorphic difference equation, which was used effectively by Elfving and Kiviste (1998), was derived in the following form: H2 ¼
H1 þ d þ r 2þ
4b b A22 ðH1 dþrÞ
where d ¼
b b A02
and r ¼
(7) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1 dÞ2 þ 4b Hb12 . A1
3. Results and discussion 3.1. Model estimates The parameter estimates and the RMSE for the different functions listed in Table 2 by using the guide
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Table 3a Residual variation (RMSE) and parameter estimates of different growth functions by guide curve method with repeat runs of top heights and mean of top heights No.
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11
Repeat runs of top heights RMSE
b1
5.322 5.253 5.137 5.145 5.129 5.140 5.164 5.135 5.195 5.137 5.136
8.9802 3.6790 0.0142 21.9304 11.8930 8405.7781 6.3596 0.1157 1.9490 1.2501 28.3071
Mean of top heights b2
0.4690 39.1088 0.6243 0.2118 0.0480 0.5592 0.0530 0.3499 11.6235
b3
RMSE
b1
b2
b3
3.7575 0.2999
2.430 2.102 0.669 0.707 0.688 0.656 0.697 0.684 0.897 0.733 0.706
6.6312 2.8849 0.0084 23.5612 11.8202 15451.1655 4.1444 0.1263 1.7855 83.0898 206.0361
0.3770 63.3054 0.4506 0.1788 0.1138 0.4349 0.0471 0.8228 205.4978
0.4770 0.9576
curve methods with repeat runs and by taking mean of top heights in different age classes, are presented in Table 3a. The estimates for the functions using the difference equation method with anamorphic and polymorphic formulation are given in Table 3b. For the special solution of the difference Eq. (7), the smallest RMSE was obtained for a reference age of 25 years with the parameter estimates b = 345.217, b2 = 0.77858, and RMSE = 0.361. Substituting the above parameter estimates into Eqs. (6) and (7) give relations (8) and (9).
H¼
H25 þ 28:16 345:217
:
(8)
H25 1 þ A0:77858
H2 ¼
H1 þ r þ 28:16 ; 2 þ A0:778581380:868 ðH þr28:16Þ
(9)
1
where in relation (9), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r¼
ðH1 28:16Þ2 þ
1380:868H1 A0:77858 1
and H25 is top
height at age of 25 years, i.e., the site index. 3.2. Model evaluation The models were evaluated on the basis of RMSE. For the guide curve method (Table 3a), function E5 gave the lowest residual variation when the functions were estimated using the repeat runs method. Analysis
Table 3b Residual variation (RMSE) and parameter estimates of different growth functions by using anamorphic and polymorphic difference equation methods No.
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11
Anamorphic difference method RMSE
b1
1.138 0.972 0.446 0.437 0.407 0.482 0.402 0.424 0.878 0.402 0.398
4.7658 2.2271 0.0216 22.2558 16.0281 3401.6964 3.8552 0.0859 3.0486 0.8391 4.1452
Polymorphic difference method b2
0.6133 22.5470 0.7561 0.2804 0.2578 0.6847 0.1110 0.3439 1.7052
b3
RMSE
b1
5.8179 0.0580
1.197 0.978 0.580 0.467 0.375 2.292 0.365 0.414 1.347 0.348 0.453
29.7306 32.2790 350.8175 42.0167 54.0811 23.7860 112.2517 85.9830 36.5415 183.9246 73.4278
b2
b3
0.0001 20.2553 0.7482 0.0005
0.1503
0.2635 0.2264 0.0384 0.6881 4.8609
A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22 Table 4 ANOVA table for estimation of function, E5 using data set G3 with repeat runs Source
d.f.
SS
MS
R2
F-ratio
0.535
428.10
Regression (corrected) Residual Lack of fit Pure error
2
22526.81
11263.41
744 130 614
19574.68 3639.23 15935.45
26.31 27.99 25.95
Total (corrected)
746
42101.49
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the best when fitted with the anamorphic difference method. For the polymorphic method, functions E7 and E10 gave the lowest RMSE for three and four parameter functions. 3.3. Model validation and selection
1.08
Validation of the models was done using the data set G3. The final selection of an appropriate model was done by comparing the residual variation from the validation data of the estimated functions with the lowest RMSEs (Table 5). For the guide curve method, on validation of function E5 with data set G3, a lower residual variation was observed for the equation, when estimation was done by using the data directly (repeat runs), than when it was done by using the mean heights. This was in spite of much higher RMSE in the former case due to repeat runs. In general, low RMSE’s were found for the functions E7 and E10 fitted by the polymorphic difference method, while the lowest RMSE was found for the special solution of function E5 by difference method. Fig. 2 presents site index curves for T. grandis in India based on the selected Eq. (9) at a reference age of 25 years.
of variance for this function is presented in Table 4 showing lack of fit and pure error. The break-up of the residual sum of squares shows that 81.4% of it is due to pure error as against 8.6% due to lack of fit As no model can pick up the pure error variations, the R2 values are quite low. The F-ratio for lack of fit in Table 4 is not significant at 5% level for 130 and 614 degrees of freedom (d.f.), hence the model given by function E5 is considered adequate. Also, as no significant lack of fit is exhibited, the residual SS can be used as an estimate of variance (Y) to carry out an F-test for overall model regression. This F-ratio is significant at 5% probability level for all degrees of freedom. Residual variation was much higher for the curves constructed with the guide curve method (Table 3a) than for those constructed with the anamorphic and polymorphic difference method (Table 3b). When the mean of top heights for different age classes were used for finding the estimates, function E6 gave the lowest residual variation (Table 3a). The RMSE, due to repeat runs, are much higher in the former case than in the latter, as discussed in Section 2 of the paper. For the difference equation method (Table 3b), the three-parameter function E7 and the four-parameter functions E10 and E11 were among
3.4. Model limitations Though the top height models have been developed using data covering a wide range in terms of sitequality and age, yet, in practice, deviations from the predicted height growth pattern may take place due to climatic fluctuations, changes in nitrogen deposition and CO2 contents in the air, elimination of dominant heights due to disease or thinning, different management practices and provenance. Site qualities of teak
Table 5 Residual variation (RMSE) of the best equations tested on the independent sample plot data (G3) No.
Guide curve method
Difference equation method
Repeat runs of top heights
Mean of top heights
Anamorphic difference
Polymorphic difference
Special solution
E05 E07 E10 E11
1.4904
1.6319
0.7451 0.7239 0.7226 0.7181
0.7466 0.6173 0.3707
0.3348
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A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22
Fig. 2. Site index curves for Tectona grandis based on Eq. (9). The curves are labelled with site index values at reference age of 25 years.
stands in Trinidad have been known to drop in site quality with advancement of age (Lamb, 1957). Site index curves constructed for teak plantations in northern Ghana indicate earlier culmination of top height growth than in the present study (Fig. 3). Also a comparison of site index curves for teak in Tanzania, Java, Nilambur in India, Central America and Nigeria showed different height growth patterns over age for the same site index (Malende and Temu, 1990). The comparisons of site index curves of the present study and those made by Malende and Temu (1990) indicate that the top height of a stand may not be expected to develop along the same predicted site index curve throughout the rotation period. However, since the method used for construction of site index curves obviously has an impact on the results, the form of the height growth pattern in different data sets would probably be more appropriately analysed if it was based directly on the data, and not indirectly on the equations as in the present study. Zeidi (1993) analysed the most widely used growth equations in terms of their time decline patterns, but due to non-availability of top height data for higher ages, it was not possible to perform a similar study and to determine the time decline pattern of the growth functions taken up in this paper. This kind of time decline pattern should, therefore, be more closely studied when appropriate data becomes available.
Fig. 3. Site index curves for teak in northern Ghana and India for different site indexes at base age of 25 years.
A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22
4. Conclusions The tested growth functions gave site index equations with different RMSEs in different order depending on smoothing methods. Thus, the choice of smoothing method was as important as the choice of growth function. The efficiency of the guide curve method was found to be lower than difference equation method. The best results were obtained with functions E10, E7 and E5 fitted by the polymorphic difference method and the special solution of function E5 by the difference equation method. On comparison with the validation data from independent permanent sample plots, the special solution was found to give the lowest residual variation. Hence, Eq. (9) is recommended for determination of site index and height growth prediction of even aged stands of T. grandis in the age interval 5–85 years. Field studies of height growth in old T. grandis plantations are recommended in order to determine a possible time decline pattern of height for the species.
Acknowledgements This research was a part of INCO-DEV Bursary (ICFP5004PRO2) for young researchers from developing countries under the EU funded research project ‘‘An interdisciplinary approach to analyse the dynamics of forest and soil degradation and to develop sustainable agro-ecological strategies for fragile Himalayan watersheds’’, jointly being undertaken with Forest Research Institute, Dehradun, India. Financial assistance by the EU is gratefully acknowledged.
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Bermejo, I., Canellas, I., Miguel, A.S., 2004. Growth and yield models for teak plantations in Costa Rica. Forest Ecol. Manage. 189, 97–110. Cieszewski, C.J., Bella, I.E., 1989. Polymorphic height and site index curves for lodgepole pine in Alberta. Can. J. Res. 19, 1151–1160. Clutter, J.L., Fortson, J.C., Piennar, L.V., Brister, G.H., Bailey, R.L., 1983. Timber Management: A Quantitative Approach. John Wiley & Sons, New York, pp. 30–58. Clutter, J.L., 1963. Compatible growth and yield models for loblolly pine. Forest Sci. 9, 354–371. Directorate of Forest Education, 1970. Growth and Yield Statistics of Common Indian Timber Species, vol. II. Forest Research Institute Press, Dehradun, pp. 260–275. Doolittle, W.T., 1958. Site index comparisons for several forest species in the Southern Appalachians. Proc. Soil Sci. Soc. Am. 22, 455–458. Draper, N.R., Smith, H., 1981. Applied Regression Analysis. John Wiley & Sons, pp. 8–42. Elfving, B., Kiviste, A., 1998. Construction of site index equations for Pinus sylvestris L. using permanent plot data in Sweden. Forest Ecol. Manage. 98, 124–134. Farr, W.A., Harris, A.S., 1979. Site index of Sitka spruce along the Pacific coast related to latitude and temperatures. Forest Sci. 25, 145–153. Ghosh, R.C., Kaul, O.N., 1977. Nature reserves in India. Indian Forester 103, 497–511. Graney, D.L., Burkhart, H.E., 1973. Polymorphic site index curves for short leaf pine in the Ouachita Mountains. USDA Forest Serv. Res. Paper SO-85. Johnstone, W.D., 1976. Variable Density Yield Tables for Natural Stands of Lodgepole Pine in Alberta, Department of Fisheries and Environment, Canadian Forest Service, Forestry Technical Report 20, Ottawa. Kiviste, A., 1988. Forest Growth Functions. Estonian Agricultural Academy, Tartu, p. 108 and Appendix (in Russian). Lamb, A.F.A., 1957. Teak (Trinidad) in Tropical Silviculture, vol. II. FAO Forestry and Forest Products Studies No. 13. FAO Rome, pp. 179–186. Malende, Y.H., Temu, A.B., 1990. Site-index curves and volume growth of teak (Tectona grandis) at Mtibwa, Tanzania. Forest Ecol. Manage. 31, 91–99. Myers, C.A., Van Deusen, J.S., 1960. Site index of ponderosa pine in the Black Hills from soil and topography. J. Forest 58, 548–555. Newberry, J.D., Pienaar, L.V., 1978. Dominant height growth models and site index curves for site-prepared slash pine plantations in the lower coastal plain of Georgia and North Florida. University of GA. Plantation Mgt. Res. Coop. Res. Paper no. 4. Nunifu, T.K., Murchison, H.G., 1999. Provisional yield models of teak (Tectona grandis Linn F.) plantations in northern Ghana. Forest Ecol. Manage. 120, 171–178. Pandey, D., 1996. Estimating Productivity of Tropical Forest Plantations by Climatic Factors. Swedish University of Agricultural Sciences, Department of Forest Resource Management and Geomatics. Report 7. Rogers, W.A., 1991. Forest preservation plots in India—management status and value. Indian Forester 117, 425–433.
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Construction of site index equations for even aged stands of Tectona grandis (teak) from permanent plot data in India Anoop Upadhyay, Tron Eid *, Prem L. Sankhayan Department of Ecology and Natural Resource Management, Agricultural University of Norway, P.O. Box 5003, NO-1432 Aas, Norway Received 7 November 2003; received in revised form 5 November 2004; accepted 21 February 2005
Abstract A quantitative measure of the timber production potential of a given site may be provided by site index equations, which predict site index at a reference age from height and age information of a forest stand. Data from permanent sample plots laid down in even aged stands of Tectona grandis in six states of India were used to develop site index equations. Eleven growth functions were estimated from height–age data using the guide curve method and difference equation method with anamorphic and polymorphic formulations. The applied methods and estimated functions were compared with respect to residual variation, i.e., root mean square error (RMSE) from original data. The difference equation method gave lower RMSE than the guide curve method. A special solution of the difference equation derived from one of the growth functions, where a hyperbolic relationship between site index and a parameter was assumed, gave the best fit to the data. Site index curves with dominant heights in the 11– 28 m range at a reference age of 25 years are presented. # 2005 Elsevier B.V. All rights reserved. Keywords: Difference equation method; Guide curve method; India; Site index; Tectona grandis
1. Introduction Tectona grandis (family Verbenaceae) or teak is a predominantly tropical or sub-tropical tree. The natural distribution of the genus Tectona is in South and SouthEast Asia. India is one of the major teak producing countries where the genus is known only by T. grandis and has great genetic variability with distribution over * Corresponding author. Tel.: +47 64 94 8901; fax: +47 64 94 8890. E-mail address: [email protected] (T. Eid).
8.9 million hectares. Its natural zone of distribution is confined mostly to the peninsular region below 248 latitude (Tewari, 1992). Teak is a strong light demander, frost sensitive and a vigorous coppicer. It is capable of thriving on a variety of soils and geological formations, but requires good sub-soil drainage. Teak is the most important and general-purpose timber in India, suitable for several end uses. It is used for construction and making of furniture, railway sleepers, doors, windows, electric poles, etc., and is the best Indian timber in dimensional stability with very low fibre saturation point and shrinkage. In order to classify and group
0378-1127/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2005.02.058
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timber from different species in India according to their functional properties, suitability indices are derived taking teak as the standard for comparison (Tewari, 1992). Teak is reputed worldwide as a quality timber on account of its appropriate physical and mechanical properties and is one of the commercially most important and costly timbers in India. The initial and almost indispensable step in the prediction of growth and yield is to quantify differences in site quality. In the context of timber management, site quality can be perceived as the timber production potential of a site for a particular species or forest type (e.g., Clutter et al., 1983). In even-aged stands of a single species, the most common measure is site index, i.e., the expected height at a nominated age. Site index models provide a convenient and effective tool for determining the forest management practices as they help in predicting timber productivity, wood volume, potential rate of growth of a forest and the time of first and subsequent thinning. Site index is an indispensable input parameter in growth– yield and biomass prediction models. It has been used in growth and yield studies for estimation of timber volume and basal area (e.g., Clutter, 1963; Johnstone, 1976; Singh, 1979; Sharma, 1979). Site index equations predict site index from age and height information. A variety of different techniques and equational forms have been used with variations in application. Most of these techniques can be viewed as special cases of three general equation-development methods, namely the guide curve method, the difference equation method and the parameter prediction method (e.g., Clutter et al., 1983). The classification of height–age curve families may be anamorphic, polymorphic-disjoint or polymorphicnon-disjoint (Bailey and Clutter, 1974). A number of studies have been conducted in the past to estimate site index equations for temperate species like Norway spruce (e.g., Tveite, 1977), Sitka spruce (e.g., Farr and Harris, 1979), Douglas fir (Doolittle, 1958) and a variety of pines (e.g., Wakely and Marrero, 1958; Myers and Van Deusen, 1960; Beck, 1971; Graney and Burkhart, 1973; Alban, 1976; Newberry and Pienaar, 1978; Amaetis and Burkhart, 1985; Elfving and Kiviste, 1998). In comparison, work in the field of tropical species is relatively scarce. Top height models and site index curves have been
15
developed for some tropical species like T. grandis (Malende and Temu, 1990; Nunifu and Murchison, 1999; Bermejo et al., 2004), Cupressus lusitanica (Teshome and Petty, 2000) and Azadirachta indica (Tewari and Kishan Kumar, 2002). In India, top heights of permanent sample plots of some tropical species at different sites have been recorded at various ages. Based on it, site qualities have been classified covering a range of top heights at different age (Directorate of Forest Education, 1970). Site index equations for these species that provide a specific index value at a reference age for any site, and that can serve as input for estimation of timber production are, however, not previously developed in India. In spite of India’s long and unique history of maintaining sample plots in a wide range of forest types, the knowledge generated from these ‘‘permanent laboratories’’ in the field (Ghosh and Kaul, 1977) has mostly failed to enter the scientific literature through proper analysis (Rogers, 1991). An exception from this, however, is provided by Pandey (1996) who applied permanent sample plots from teak plantations in India in order to study the influence of climatic factors on the productivity of the species. The aim of the present study was to make use of data from permanent sample plots of even aged forest to construct different site index equations using dominant height and age data of T. grandis at different locations in India. Different growth functions were tested with different construction methods, i.e., the guide curve method as well as the difference equation method with anamorphic and polymorphic formulations, to find the best solution.
2. Material and methods 2.1. Material Data from 150 permanent sample plots of T. grandis laid out in the states of Uttar Pradesh, Madhya Pradesh, Maharashtra, Orissa, Karnataka and Kerala of India were used in this study. These plots, in average approximately 0.2 ha in size and of variable density and spacing, were laid out by the Forest Research Institute, Dehradun (India) and cover a considerable range of natural distribution of teak throughout India. Remeasurements were carried out in
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G1: all 747 observations on height and age from the 150 sample plots were used for the guide curve method. G2: a set of 448 height and age pairs, i.e., height and age observations from measurements and remeasurements on the 150 sample plots, was used for the difference equation method. G3: an independent set of 110 observations from seven permanent sample plots not included in G1 and G2, from where height–age series have been measured and remeasured over a period of 75 years at 5-yearintervals, was used for validation of results. G3 constitutes a part of the permanent sample plots applied in developing the yield tables published by the Forest Research Institute, Dehradun (Directorate of Forest Education, 1970).
Fig. 1. Observed top height over age in the Tectona grandis sample plots.
these sample plots at age intervals of 4–10 years; most of them at an interval of 5 years. Age at which measurements have been recorded varied from 4 to 93 years. All together 747 observations were available from the sample plots (Fig. 1, Table 1). The scatter diagram in Fig. 1 gives the range of distribution of the observed top heights of the plots at different ages. The top height of the stand has been defined in this study as the average height of the 100 largest diameter trees per hectare. The heights of sample trees in all diameter classes were recorded and a height–diameter curve was drawn for each plot. The mean diameter of the 10 dominant trees in a plot was calculated from their total basal area, and the height corresponding to this mean diameter on the height–diameter curve was taken as the top height of the plot. Stand age has been defined as the age of the stand from the year of plantation. The data from the remeasured permanent plots were arranged in the following three G1–G3 groups.
2.2. Methods Eleven widely used (Kiviste, 1988) and highly flexible growth functions were selected (Table 2). Some of these functions have been analysed by Zeidi (1993) and most of them have been tested for construction of site index equations for Pinus svlvestris in Sweden by Elfving and Kiviste (1998). All these functions are of the general form: H ¼ f ðAÞ
(1)
where A denotes stand age (years) and H top height (m). The guide curve method was applied, both directly on all the 747 observations as well as after grouping the data in 10-year age classes, and performing smoothing as un-weighted, non-linear regression for arithmetic means of top heights and age for each age class.
Table 1 Top height (m) statistics over ages classes (age class 5 = 1–10 years, etc.) Age class
Number of observations
Mean
Standard deviation
Minimum value
Maximum value
5 15 25 35 45 55 65 75 85 95
68 116 109 101 84 85 85 64 31 4
11.8 16.7 21.3 22.7 23.7 27.0 28.4 29.1 29.7 30.4
3.1 4.1 4.1 5.9 5.5 6.1 5.9 6.6 7.0 5.1
5.1 7.2 10.2 10.2 11.7 12.9 16.5 18.0 19.2 33.0
17.4 24.3 28.2 32.1 34.2 36.3 37.8 39.6 40.5 40.2
Total
747
A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22 Table 2 Growth functions studied No.
Growth function
Functions of two parameters E1 E2
H = b0 exp(b1/A) H = b0/(1+b1/A)3
Functions of three parameters: E3 H ¼ b0 f1 expðb1 =AÞgb2 E4 H ¼ b0 =f1 þ ðb1 =AÞ þ ðb2 =A2 Þg E5 H ¼ b0 =f1 þ b1 =Ab2 g E6 H ¼ b0 =ð1 þ b1 =A2 Þb2 E7 H ¼ b0 expðb1 =Ab2 Þ E8 H ¼ b0 f1 expðb1 Ab2 Þg E9 H ¼ b0 =f1 þ b1 expðb2 AÞg Functions of four parameters E10 E11
H ¼ b0 =f1 þ b1 =Ab2 gb3 H ¼ b0 =f1 þ ðb1 =AÞ þ ðb2 =Ab3 Þg
H = height, A = age, b0, b1, b2 and b3 are parameters.
The guide curve method, when applied directly on all observations, will naturally give low R2 values and high RMSE values as compared to when the mean heights in 10-year age classes are considered. This is due to the effect of repeat runs, i.e., different observed values of top heights at the same age. In feet, it is not possible to attain a high value of R2 in such cases, no matter how appropriate the model is, since any model can explain only the variations due to lack of fit and not the pure error variations resulting from repeated runs (Draper and Smith, 1981). If
Using the above method, described by Draper and Smith (1981), the data set G1 was checked for the extent of pure error and lack of fit. The difference equation method was applied using the general form of the difference function H2 = f(H1, A1, A2) directly on the remeasurement data set G2. Here H1 and H2 are the top-heights at ages A1 and A2. The anamorphic formulation of the equation was obtained by solving (1) for the constant term, to give: H2 ¼ H1
nj m X X ðY ju Y¯ j Þ2 ;
(2)
j¼1 u¼1
where Yju is the uth observation Pn j (u = 1, 2, . . ., nj) at Xj (j = l, 2, . . ., m) and Y¯ j ¼ u¼1 Y ju with degrees of freedom ne ¼
m X j¼1
n j m:
(3)
f ðA2 Þ f ðA1 Þ
(4)
Substitution of one of the variable coefficients in (1) will give the polymorphic formulation of the difference equation. A special solution of the difference equation method was derived by Cieszewski and Bella (1989). The function E5 was transformed into the site index model: b1 1 þ b2 A0 H ¼ SI (5) 1 þ Abb12 where SI is the site index, i.e., the top height at reference age A0. A hyperbolic relationship between SI and parameter b1 was assumed to exist (b1 = b/SI). Thus, model (5) could be transformed into the form: SI þ H¼
Y11, Yl2, . . ., Y1n1 are n1 repeat observations at point X1, Y21, Y22, . . ., Y2n2 are n2 repeat observations at point X2, . . ., Ym1, Ym2, . . ., Ymnm are nm repeat observations at point Xm , then the total pure error is
17
b b A0 2
(6)
b
1 þ ASIb2 A polymorphic difference equation, which was used effectively by Elfving and Kiviste (1998), was derived in the following form: H2 ¼
H1 þ d þ r 2þ
4b b A22 ðH1 dþrÞ
where d ¼
b b A02
and r ¼
(7) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1 dÞ2 þ 4b Hb12 . A1
3. Results and discussion 3.1. Model estimates The parameter estimates and the RMSE for the different functions listed in Table 2 by using the guide
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Table 3a Residual variation (RMSE) and parameter estimates of different growth functions by guide curve method with repeat runs of top heights and mean of top heights No.
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11
Repeat runs of top heights RMSE
b1
5.322 5.253 5.137 5.145 5.129 5.140 5.164 5.135 5.195 5.137 5.136
8.9802 3.6790 0.0142 21.9304 11.8930 8405.7781 6.3596 0.1157 1.9490 1.2501 28.3071
Mean of top heights b2
0.4690 39.1088 0.6243 0.2118 0.0480 0.5592 0.0530 0.3499 11.6235
b3
RMSE
b1
b2
b3
3.7575 0.2999
2.430 2.102 0.669 0.707 0.688 0.656 0.697 0.684 0.897 0.733 0.706
6.6312 2.8849 0.0084 23.5612 11.8202 15451.1655 4.1444 0.1263 1.7855 83.0898 206.0361
0.3770 63.3054 0.4506 0.1788 0.1138 0.4349 0.0471 0.8228 205.4978
0.4770 0.9576
curve methods with repeat runs and by taking mean of top heights in different age classes, are presented in Table 3a. The estimates for the functions using the difference equation method with anamorphic and polymorphic formulation are given in Table 3b. For the special solution of the difference Eq. (7), the smallest RMSE was obtained for a reference age of 25 years with the parameter estimates b = 345.217, b2 = 0.77858, and RMSE = 0.361. Substituting the above parameter estimates into Eqs. (6) and (7) give relations (8) and (9).
H¼
H25 þ 28:16 345:217
:
(8)
H25 1 þ A0:77858
H2 ¼
H1 þ r þ 28:16 ; 2 þ A0:778581380:868 ðH þr28:16Þ
(9)
1
where in relation (9), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r¼
ðH1 28:16Þ2 þ
1380:868H1 A0:77858 1
and H25 is top
height at age of 25 years, i.e., the site index. 3.2. Model evaluation The models were evaluated on the basis of RMSE. For the guide curve method (Table 3a), function E5 gave the lowest residual variation when the functions were estimated using the repeat runs method. Analysis
Table 3b Residual variation (RMSE) and parameter estimates of different growth functions by using anamorphic and polymorphic difference equation methods No.
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11
Anamorphic difference method RMSE
b1
1.138 0.972 0.446 0.437 0.407 0.482 0.402 0.424 0.878 0.402 0.398
4.7658 2.2271 0.0216 22.2558 16.0281 3401.6964 3.8552 0.0859 3.0486 0.8391 4.1452
Polymorphic difference method b2
0.6133 22.5470 0.7561 0.2804 0.2578 0.6847 0.1110 0.3439 1.7052
b3
RMSE
b1
5.8179 0.0580
1.197 0.978 0.580 0.467 0.375 2.292 0.365 0.414 1.347 0.348 0.453
29.7306 32.2790 350.8175 42.0167 54.0811 23.7860 112.2517 85.9830 36.5415 183.9246 73.4278
b2
b3
0.0001 20.2553 0.7482 0.0005
0.1503
0.2635 0.2264 0.0384 0.6881 4.8609
A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22 Table 4 ANOVA table for estimation of function, E5 using data set G3 with repeat runs Source
d.f.
SS
MS
R2
F-ratio
0.535
428.10
Regression (corrected) Residual Lack of fit Pure error
2
22526.81
11263.41
744 130 614
19574.68 3639.23 15935.45
26.31 27.99 25.95
Total (corrected)
746
42101.49
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the best when fitted with the anamorphic difference method. For the polymorphic method, functions E7 and E10 gave the lowest RMSE for three and four parameter functions. 3.3. Model validation and selection
1.08
Validation of the models was done using the data set G3. The final selection of an appropriate model was done by comparing the residual variation from the validation data of the estimated functions with the lowest RMSEs (Table 5). For the guide curve method, on validation of function E5 with data set G3, a lower residual variation was observed for the equation, when estimation was done by using the data directly (repeat runs), than when it was done by using the mean heights. This was in spite of much higher RMSE in the former case due to repeat runs. In general, low RMSE’s were found for the functions E7 and E10 fitted by the polymorphic difference method, while the lowest RMSE was found for the special solution of function E5 by difference method. Fig. 2 presents site index curves for T. grandis in India based on the selected Eq. (9) at a reference age of 25 years.
of variance for this function is presented in Table 4 showing lack of fit and pure error. The break-up of the residual sum of squares shows that 81.4% of it is due to pure error as against 8.6% due to lack of fit As no model can pick up the pure error variations, the R2 values are quite low. The F-ratio for lack of fit in Table 4 is not significant at 5% level for 130 and 614 degrees of freedom (d.f.), hence the model given by function E5 is considered adequate. Also, as no significant lack of fit is exhibited, the residual SS can be used as an estimate of variance (Y) to carry out an F-test for overall model regression. This F-ratio is significant at 5% probability level for all degrees of freedom. Residual variation was much higher for the curves constructed with the guide curve method (Table 3a) than for those constructed with the anamorphic and polymorphic difference method (Table 3b). When the mean of top heights for different age classes were used for finding the estimates, function E6 gave the lowest residual variation (Table 3a). The RMSE, due to repeat runs, are much higher in the former case than in the latter, as discussed in Section 2 of the paper. For the difference equation method (Table 3b), the three-parameter function E7 and the four-parameter functions E10 and E11 were among
3.4. Model limitations Though the top height models have been developed using data covering a wide range in terms of sitequality and age, yet, in practice, deviations from the predicted height growth pattern may take place due to climatic fluctuations, changes in nitrogen deposition and CO2 contents in the air, elimination of dominant heights due to disease or thinning, different management practices and provenance. Site qualities of teak
Table 5 Residual variation (RMSE) of the best equations tested on the independent sample plot data (G3) No.
Guide curve method
Difference equation method
Repeat runs of top heights
Mean of top heights
Anamorphic difference
Polymorphic difference
Special solution
E05 E07 E10 E11
1.4904
1.6319
0.7451 0.7239 0.7226 0.7181
0.7466 0.6173 0.3707
0.3348
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A. Upadhyay et al. / Forest Ecology and Management 212 (2005) 14–22
Fig. 2. Site index curves for Tectona grandis based on Eq. (9). The curves are labelled with site index values at reference age of 25 years.
stands in Trinidad have been known to drop in site quality with advancement of age (Lamb, 1957). Site index curves constructed for teak plantations in northern Ghana indicate earlier culmination of top height growth than in the present study (Fig. 3). Also a comparison of site index curves for teak in Tanzania, Java, Nilambur in India, Central America and Nigeria showed different height growth patterns over age for the same site index (Malende and Temu, 1990). The comparisons of site index curves of the present study and those made by Malende and Temu (1990) indicate that the top height of a stand may not be expected to develop along the same predicted site index curve throughout the rotation period. However, since the method used for construction of site index curves obviously has an impact on the results, the form of the height growth pattern in different data sets would probably be more appropriately analysed if it was based directly on the data, and not indirectly on the equations as in the present study. Zeidi (1993) analysed the most widely used growth equations in terms of their time decline patterns, but due to non-availability of top height data for higher ages, it was not possible to perform a similar study and to determine the time decline pattern of the growth functions taken up in this paper. This kind of time decline pattern should, therefore, be more closely studied when appropriate data becomes available.
Fig. 3. Site index curves for teak in northern Ghana and India for different site indexes at base age of 25 years.
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4. Conclusions The tested growth functions gave site index equations with different RMSEs in different order depending on smoothing methods. Thus, the choice of smoothing method was as important as the choice of growth function. The efficiency of the guide curve method was found to be lower than difference equation method. The best results were obtained with functions E10, E7 and E5 fitted by the polymorphic difference method and the special solution of function E5 by the difference equation method. On comparison with the validation data from independent permanent sample plots, the special solution was found to give the lowest residual variation. Hence, Eq. (9) is recommended for determination of site index and height growth prediction of even aged stands of T. grandis in the age interval 5–85 years. Field studies of height growth in old T. grandis plantations are recommended in order to determine a possible time decline pattern of height for the species.
Acknowledgements This research was a part of INCO-DEV Bursary (ICFP5004PRO2) for young researchers from developing countries under the EU funded research project ‘‘An interdisciplinary approach to analyse the dynamics of forest and soil degradation and to develop sustainable agro-ecological strategies for fragile Himalayan watersheds’’, jointly being undertaken with Forest Research Institute, Dehradun, India. Financial assistance by the EU is gratefully acknowledged.
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