Volume tables for Abies borisii regis

Forest Ecology and Management, 25 (1988) 73-77 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

73

Short C o m m u n i c a t i o n

Volume Tables for Abies borisii regis KONSTADINOS G. MATIS

Dept. of Forestry and Natural Environment, P.O. Box, 237, University of Thessaloniki, GR-540 05 Thessaloniki (Greece) (Accepted 11 November 1987)

ABSTRACT Matis, K.G., 1988. Volume tables for Abies borisii regis. For. Ecol. Manage., 25: 73-77. Thirteen models expressing the relationship between volume and diameter at breast height, total height, and upper stem diameter at 5.3 m from ground level, were tested using a sample of 3011 trees. The results indicated that, in all cases, models using the log-log transformation and weighted models with weights of d 2 and d2h were much better than the untransformed and unweighted models. Untransformed and unweighted models with two or three independent variables gave values for Furnivars index of fit which were larger or almost equal to these obtained for transformed and weighted models with only one independent variable. The equation in (v) = - 0.912 + 0.9611n (d2h) was used to construct a double-entry table.

INTRODUCTION T h e U n i v e r s i t y f o r e s t a t P e r t o u l i , Greece, is a n a l m o s t p u r e fir (Abies borisii regis M a t f . ) forest. F o r t h a t forest, t h r e e s i n g l e - e n t r y v o l u m e t a b l e s ( O e c o n o m o p o u l o s , 1964; A n o n y m o u s , 1978; A s t e r i s a n d M a t i s , 1979 ) a n d o n e doublee n t r y v o l u m e t a b l e ( M a t i s a n d N e o p h y t o u , 1978) exist. All h a v e b e e n develo p e d w i t h o u t t a k i n g i n t o a c c o u n t t h e n o n - h o m o g e n e i t y of t h e v a r i a n c e of t h e variable 'tree volume'. B e c a u s e of t h e i m p o r t a n c e of v o l u m e t a b l e s to i n v e n t o r y i n g t h e f o r e s t for m a n a g e m e n t p u r p o s e s , a n d in e s t i m a t i n g t h e t o t a l v o l u m e of s t a n d i n g t r e e s for h a r v e s t i n g p u r p o s e s , it w a s d e c i d e d t h a t t o t a l - t r e e v o l u m e e q u a t i o n s s h o u l d be developed. METHODS T h e f o r e s t was s a m p l e d u s i n g s y s t e m a t i c s a m p l i n g . P a r a l l e l lines were d r a w n 200 m a p a r t on a m a p of t h e f o r e s t so t h a t all p o s s i b l e site c o n d i t i o n s w o u l d be covered. I n t h e field, o n e t r e e w a s m e a s u r e d e v e r y 50 m a l o n g t h e s e lines. T h e

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74 TABLE 1 N u m b e r of sample trees in each 4 - c m - d i a m e t e r a n d 2 - m - h e i g h t class D i a m e t e r T o t a l - h e i g h t class (h, m ) class (d,cm) 6 8 10 12 14 16 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94

2 37 17

Total

56

72 53 76 108 23 80 7 39 1 12 1 7 2

17 56 117 77 48 13 3 1 -1

3 28 73 98 58 27 17 6 1 2 1

Total 18

4 38 7 69 42 76 78 67 75 34 55 21 46 14 18 4 15 3 4 3 4 -1 1 --

20

8 29 68 67 66 39 23 17 7 4 1 3

1

22

5 22 45 50 36 29 30 21 9 7 3 3 -1 -2

24

26

1 9 19 28 42 43 36 18 9 5 4 6 2 1 2

28

30 32 34 36

1 1 6 15 - 2 20 4 -29 8 4 1 25 17 6 2 19 17 8 - - - 10 15 10 3 1 6 13 5 5 1 12 14 5 5 -1 4 3 2 2 1 4 1 4 4 1 1 1 -3 2 -1

1 180 301

333

314 335

345

332

263

226

148

100 51

21

5

1

2 182 289 338 340 309 290 248 235 174 159 141 98 62 44 46 22 7 13 8 3 1

1 3011

following measurements were taken on each tree: (1) breast height diameter (d) to the nearest cm, using calipers; (2) total height (h) to the nearest dm, using a Blume-Leiss hypsometer; and (3) upper stem diameters at 4-m height intervals above breast height (i.e. at 5.3 m, 9.3 m etc. ), using a Speigel relascope. The total tree volume over bark (v; m 3 ) was calculated from these measurements by computer. The 3011 sample trees were grouped into 22 4-cm diameter classes and 16 2m height classes; the numbers of trees in each diameter and height class, and the totals, are given in Table 1. S E L E C T I O N OF M A T H E M A T I C A L M O D E L S

In the forestry literature there are a large number of mathematical models which have been used by various authors in volume-tables construction (Spurr, 1952; Loetsch et al., 1973), while recent developments in this area include the

75 work of Sadiq and Smith (1983), Schreuder and Anderson (1984), Green and Strawderman (1985) and Meng and Tsai (1986). In most cases, those models have been found suitable for use in specific forests. In this study 13 models were chosen and compared. The 13 equations were classified into three groups: Group 1 includes five equations having only one independent variable, the breast-height diameter over bark (d). The 1st model is Kopezky-Gehrhard's equation, the 2nd is Hohenadl-Krenn's equation, and the 3rd is the local volume-table equation (Loetsch et al., 1973). Models 4 and 5 are forms of the Models 1 and 2, respectively, weighted by 1/d 2. In Group 2 there are also five equations, each with two independent variables, d and the total height (h). Models 6 and 8 are the combined variable and the logarithmic combined variable equations, respectively (Spurr, 1952). Model 7 was chosen after the initial use of another model having as independent variables d, h, dh, d 2, h 2 and d2h. Using this model, the stepwise regression search method showed that the independent variable d2 and h 2 contributed very little to R 2. Consequently, only the other variables were retained. Models 9 and 10 are Models 6 and 7 given a weighting of 1/d2h. In Group 3 are three equations each having three independent variables, d, h and the upper stem diameter at 5.3 m from ground level (ds.3). Models 11 and 13 were selected using the same techniques as for Model 7; in this case a stricter attitude was taken in the rejection of variables because of the higher coefficients of determination for the first variables added in the models. Model 12 is Ogaya's equation (Loetsch et al., 1973) with the difference that the diameter at half the tree height was substituted by d5.3. RESULTS AND DISCUSSION The 13 regression equations were fitted to the data of the 3011 sample trees. Analysis of variance on all regressions gave very large F values corresponding to probabilities of 0 to 4 decimal places. The test of the null hypothesis (H0): Ho :bi =0, i=O, 1, ..., 4 against the alternatives (He):

He :bi 50, i=0, 1, ..., 4 resulted in the rejection of the null hypothesis in all cases. The probabilities of the computed F values were equal to 0 to 4 decimal places in all cases but one. The test of the regression coefficient bl of Model 2 yielded an F value equal to 9.938, corresponding to a P = 0.002 for the appropriate degrees of freedom. The coefficient of determination cannot be used to compare the entire set of the 13 equations, but can be used to compare equations with the same response variable. The model:

76 In ( v ) = 0.566 + 0.5891n (dhds.3) + 0.9851n (d) R2=0.9912;

I=0.0698;

I%=6.04

gave the minimum Furnival's index of fit I (Furnival, 1961). This model requires the knowledge of three tree characteristics d, h and d5.3, the last one obtainable only with difficulty and imprecision and not, in most cases, known. Furthermore, the improvement on the estimate of tree volume, compared with the best models of Group 2, was very small (less than 1% ). In Group 1 the log-log transformation model: ln(v)=2.596+2.6481n(d)

R2=0.9662;

I=0.1368;

I%=11.84

(1)

had the minimum I, but the two weighted models of this group gave a value of I which was not greatly different. The most important fact in this case was the reduction in the value of I from the ordinary least-squares models. The untransformed and unweighted equations had about twice as large a value for I as the transformed and the two weighted models. In Group 2 the minimum I was associated with the model: v / d 2h = 0.300 + 0.060 ( 1/d) - 0.009 ( 1/d 2) - O. 162 ( 1/dh ) + 0.049 ( 1/d 2h)

R2=0.1628;

I=0.0775;

I%=6.71

(2)

The two next-best models had values of I very close to the I of the best model. Again, the untransformed and unweighted models in this group had I values about 2.5 × those of the transformed and weighted models. In Group 3, the inclusion in the best model of a 3rd independent variable, d5.3, resulted in obtaining an index of fit equal to 0.0698 (6.04%). The improvement in the value of I from the best model in Group 2 with only two independent variables was, however, negligible 0.0077 (0.67%). Considering the difficulty in measuring this variable, there is no justification for using models of this group in the forest under study. The values of Furnival's index for the other two models in this group are almost the same as the values observed for the transformed and weighted models in Group 1 with only one independent variable. From the equations with only one variable (d) the best, Equation (1), was chosen as the basis for a single-entry volume table for fir at the University Forest at Pertouli, Greece. This can be used when it is impossible to obtain an estimate of total tree height, and is only for rough volume estimates, because the standard error of estimate is 11.84%. The double-entry volume table, constructed by using the second-best equation in Group 2: ln(v)=-O.912+O.9611n(d2h)

R2=0.9888;

I=0.0787;

I%=6.82

must be used for more accurate estimates. This equation has been used because

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the best, Equation (2), gave inconsistent results for estimating the volume of small trees.

REFERENCES Anonymous, 1978. Management Plan of the University Forest at Pertouli 1979-1988. University Forests Administration, Pertouli, Greece, 209 pp. (in Greek). Asteris, C.J. and Matis, K.G., 1979. Derivation of three single-entry volume tables from a doubleentry one for fir of the University Forest at Pertouli, Thessaloniki, Greece. University Forests Administration, 24 pp. (in Greek). Furnival, G., 1961. An index for comparing equations used in construction of volume tables. For. Sci., 7: 337-341. Green, E.J. and Strawderman, W.E., 1985. The use of Bayes/empirical Bayes estimation in individual tree volume equation development. For. Sci., 31: 975-990. Loetsch, F., ZShrer, F. and Haller, K., 1973. Forest Inventory, vol. 2. BLV Verlagsgesellschaft, Munich, 469 pp. Matis, K.G. and Neophytou, C.N., 1978. Double-entry volume table for fir of the University Forest at Pertouli, Thessaloniki, Greece. University Forests Administration, 31 pp. (in Greek). Meng, C.H. and Tsai, W.Y., 1986. Selection of weights for a weighted regression of tree volume. Can. J. For. Res., 16: 671-673. Oeconomopoulos, A., 1964. The Forestry at Pertouli, Greece, Thessaloniki, Greece, 352 pp. (in Greek). Sadiq, R.A. and Smith, V.G., 1983. Estimation of individual tree volumes with age and diameter. Can. J. For. Res., 13: 32-39. Schreuder, H.T. and Anderson, J., 1984. Variance estimation for volume when d2h is the covariate in regression. Can. J. For. Res., 14: 818-821. Spurr, S., 1952. Forest Inventory. Ronald Press, New York, 476 pp.