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http://journals.tubitak.gov.tr/agriculture/

Turkish Journal of Agriculture and Forestry Turk J Agric For(2015) 39: 851-863© TÜBİTAKdoi:10.3906/tar-1501-27

Compatible merchantable stem volume and taper equations for eucalyptusplantations in the Eastern Mediterranean Region of Turkey

Ramazan ÖZÇELİK1,*, Mehmet Fatih GÖÇERİ2

1Deparment of Forest Engineering, Faculty of Forestry, Süleyman Demirel University, East Campus, Isparta, Turkey 2Graduate School of Natural and Applied Sciences, Süleyman Demirel University, East Campus, Isparta, Turkey

* Correspondence: ramazanozcelik@sdu.edu.tr

1. IntroductionThe eucalyptus genus has been one of the forest resources most used industrially around the world. In Turkey, it is represented by two species: Eucalyptus camaldulensis Dehn (EC) and Eucalyptus grandis W.Hill ex Maiden (EG). The harvested wood is used for purposes such as domestic consumption, charcoal production, building construction, and cellulose pulp. The first eucalyptus plantations were established with EC in Tarsus–Karabucak in 1939 (Gürses, 1990). There have been eucalyptus plantations of greater than 10,000 ha in size in only the Eastern Mediterranean Region of Turkey (Özkurt, 2000). Although the eucalyptus plantations have high potential commercial value and can make an important contribution to the forest products industry of Turkey, there is little reference information regarding growth, yield, and management of eucalyptus plantations (Birler et al., 1995; Özkurt, 2000; Yıldızbakan et al., 2007). Estimating individual tree volume is one of several necessary components for forest growth and yield modeling. In this concept, standard volume equations have been published for EG (Özkurt, 2000) and EC

(Yıldızbakan et al., 2007) plantations in the Tarsus district. A site index model along with volume and dry matter weight yield tables has also been developed for EC coppices in the Tarsus district (Yıldızbakan et al., 2007). However, additional work is needed in this area to refine the volume equations.

Schröder et al. (2014) stated that the estimation of volume at harvest in planted forests is the main concern for forest managers as a way to determine economic yield and therefore species choice, silvicultural treatments, and rotation at any given plantation. Among the different ways to estimate tree volume, modeling methods may represent the most accurate and versatile approach. de-Miguel et al. (2012) indicated that when calculating assortment volumes foresters need to predict stem diameter at different heights along the stem. Taper models allow the prediction of stem diameter at any point along the stem, allowing one to calculate the accumulated volume at any height or diameter (Kozak, 2004). Since rotation in forest plantations for sawn wood production takes decades, any change in market demands related to log sizes during this

Abstract: Eucalyptus plantations are an important source of raw materials for the forest products industry in Turkey. Despite its economic and ecological value, there is little reference information regarding the growth and yield of eucalyptus plantations. Thus, a segmented compatible stem taper and merchantable tree volume equation system was developed for eucalyptus plantations in the Eastern Mediterranean Region of Turkey. The data used in this study come from 190 Eucalyptus grandis (EG) and 149 Eucalyptus camaldulensis (EC) destructively sampled from even-aged plantations. The equation systems were fitted by the simultaneous estimation of parameters by maximum likelihood with the full information estimation method in order to optimize fitting while simultaneously minimizing the errors in a combined way. The equation systems produced similar results for the two species analyzed in this study. Based on the overall fit statistics, the taper and volume equations showed consistent performance for different sections of stem using relative height classes, different tree height, and diameter classes in estimating stem diameter and merchantable volume. Differences in the taper equations among eucalyptus species were examined and tested using the F-test. The result of the F-test indicated differences in species-specific taper equations for the EG and EC species. A different taper equation should therefore be used for each eucalyptus species. The segmented taper and merchantable stem volume equations can help forest managers to estimate the stem diameter and volumes of standing trees of both eucalyptus species, which is important in practical forestry applications.

Key words: Stem diameter, simultaneous estimations, commercial volume, forest management, plantations

Received: 07.01.2015 Accepted/Published Online: 21.05.2015 Printed: 30.11.2015

Research Article

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period would require new estimates in standing volume, which could be met by flexible taper equations (Sharma and Zhang, 2004; Schröder et al., 2014), in contrast to the need for developing new commercial volume tables.

Two major forms of taper equations have been successfully used in the forestry literature. The first approach includes variable–exponent or variable form taper equations, which describe the profile of the tree stem with varying exponents from the baseline to the top to account for neiloid, paraboloid, and conic forms (Burkhart and Tome, 2012). This model form has the drawbacks that it cannot be analytically integrated to compute stem and log volume and iterative methods must be used to estimate merchantable heights at specified stem diameters. The second major form includes segmented polynomial taper equations that describe the taper of different tree sections using different equation forms. This approach can be directly integrated to calculate volume and can be rearranged algebraically to directly estimate heights for specified stem diameters (Kalıpsız, 1984; Kozak and Smith, 1993). As summarized by many researchers, a volume model developed by integrating a taper model often has the advantage of estimating total stem volume or volume at a specific height or diameter versus the limited estimating potential of a regressed volume model (Sharma and Oderwald, 2001). When integration is possible, the taper and volume models are preferred in compatible models. The most important benefit of a compatible system is to obtain consistent results from both the volume and the taper equations. Comprehensive reviews of taper-volume estimating systems have been reported by Sharma and Oderwald (2001), Corral-Rivas et al. (2007), Barrio-Anta et al. (2007), and Pompa-Garcia et al. (2009).

Compatible taper and volume equation systems do not have widespread use in Turkey. Some studies have been carried out to evaluate the suitability and applicability of previously published equations for simulating the stem taper of some tree species in Turkey (Yavuz and Saraçoğlu, 1997; Brooks et al., 2008; Sakıcı et al., 2008; Özçelik et al., 2011; Özçelik and Alkan, 2012; Özçelik and Brooks, 2012; Özçelik and Bal, 2013). To our knowledge, there are no detailed publications on compatible stem taper and merchantable volume equations for the EG and EC plantations in the Eastern Mediterranean Region of Turkey.

The purpose of this study was to develop an equation of systems for estimating stem diameters and merchantable stem volume for the EG and EC plantations found in the Eastern Mediterranean Region of Turkey. For this aim, a second alternative was selected to develop a compatible merchantable stem volume and taper equation with the segmented-stem form factors presented by Fang et al. (2000). The segmented taper models have been shown by several studies (Jiang et al., 2005; Diéguez-Aranda et

al., 2006; Crecente-Campo et al., 2009; Li and Weiskittel, 2010) to provide reliable and accurate predictions for both diameters and tree volume (merchantable and total tree volume).

2. Materials and methods 2.1. Study sitesThis study was carried out in Tarsus–Karabucak in the Eastern Mediterranean Region of Turkey, which represents a region with a high density of eucalyptus plantations. The summer growing season is hot and dry and winters are cool and wet. The area is 5–10 m above sea level and has a mean annual temperature of 18 °C. The mean annual rainfall is 609.5 mm and the area’s main soils are deep alluvial with relatively rich organic materials. The main factor that limits distribution of eucalyptus species in Turkey is low temperature (Gürses et al., 1995). We sampled parcels from 10 to 16 years of age belonging to the Forest Service of Turkey. 2.2. DataThe EG and the EC data consist of measurements on 190 and 149 individual trees, respectively, collected in the industrial plantations of the Eastern Mediterranean Region,. A previous inventory in the selected stands provided an overview of the diameter distribution, and the selected trees were then sampled to ensure a representative distribution by diameter and height class. Before felling the trees, diameter at breast height (dbh) at 1.30 m was measured using a digital caliper to the nearest 0.01 cm on each tree. After felling, total tree height was measured to the nearest 0.1 cm and then diameters along the upper tree boles were measured and recorded approximately every 1 m above breast height. The actual cubic meter volume for each stem section was calculated using Smalian’s formula. The top section volume was estimated using the volume formula for a cone. The individual tree volume (above the stump) was then calculated from summing all of the sectional volumes.

The scatter plot of relative diameter (d/D) against relative height (h/H) was visually examined for each eucalyptus species to detect possible anomalies in the data. This systematic approach for detecting abnormal data points was adopted to increase efficiency (Bi, 2000). A locally quadratic fitting with a smoothing parameter of 0.25 was used for both eucalyptus species after some iterative fitting and visual examination of the smoothed taper curves and the data (Figure 1). The number of these extreme data points accounted for 0.02% and 0.05% of the data for EG and EC, respectively. The paired data points of relative heights against relative diameters used for this study, together with the loess regression line, are shown in Figure 1. Table 1 shows descriptive statistics of the data for both tree species.

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2.3. Stem taper and volume equationMax and Burkhart (1976) developed the first segmented model, for which the tree stem was divided into three sections represented by separate subfunctions, and then the segments were joined mathematically (Burkhart and Tome, 2012). A number of additional studies involving segmented taper equations have been published (Clark et al., 1991; Petersson, 1999; Fang et al., 2000; Sharma and

Burkhart, 2003; Jiang et al., 2005; Jordan et al., 2005). The equation system of Fang et al. (2000) was used in this study because this model was superior to the other taper equation forms for simulating the stem taper and estimating stem volume in many studies in different countries (Diéguez-Aranda et al., 2006; Corral-Rivas et al., 2007; Crecente-Campo et al., 2009; Pompa-Garcia et al., 2009; Sevillano-Marco et al., 2009; Li and Weiskittel, 2010).

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Figure 1. Plot of the relative height versus relative diameter outside bark for Eucalyptus grandis (a) and Eucalyptus camaldulensis (b).

Table 1. Summary statistics for the measured characteristics of eucalyptus trees used in this study.

Species Mean SD Minimum Maximum

Eucalyptus grandis (EG) (n = 190)

Variable

DBH (D, cm) 33.70 7.60 15.60 51.40

Total height (H, m) 27.70 5.20 16.40 38.00

Disk dob (d, cm) 21.97 10.21 1.80 55.00

Disk height (h, m) 13.33 8.32 0.30 36.30

Volume (m3) 1.15 0.61 0.16 2.66

Eucalyptus camaldulensis (EC) (n = 149)

Variable

DBH (D, cm) 26.30 6.00 12.00 51.70

Total height (H, m) 22.10 4.80 13.60 32.10

Disk dob (d, cm) 17.14 8.31 1.30 61.50

Disk height (h, m) 10.93 6.95 0.30 29.30

Volume (m3) 0.57 0.32 0.09 2.19

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The form of the Fang et al. (2000) taper model is

d = c1 H k−b1( ) b1 1−q( ) k−β( ) β α1I1+I2α2

I2

(1)

where k = π/40,000 q = h/H

I1 =1 if p1 ≤ q ≤ p2 ; 0 otherwiseI2 =1 if p2 < q ≤1; 0 otherwise⎧⎨⎩

p1 = h1/H and p2 = h2/H (h1 and h2 are the heights from ground level where the two inflection points assumed in the model occur)

β =b11− I1+I2( )b2

I1b3I2

, α1 = 1− p1( ) b2−b1( )k b1b2

,

α2 = 1− p2( ) b3−b2( )k b2b3

, r0 = 1−hst( ) H( )k b1

,

r1 = 1− p1( )k b1

, r2 = 1− p2( )k b2

,

c1 =a0D

a1Ha2−k b1

b1 r0 −r1( )+b2 r1 −α1r2( )+b3α1r2 ;

ai, bi, and pi are the parameters to be estimated.

The expression of merchantable volume model is

v = c12Hk b1 b1r0 + I1 + I2( ) b2 −b1( )r1 +(I2 b3 −b2( )α1r2 −β 1−q( )k β α1

I1+I2α2I2 )

(2)

In order to test the possibility of combining both eucalyptus species and using a single taper equation and merchantable volume equation, an F-test (Bates and Watts, 1988; Judge et al. 1988) was used. The F-test requires the fitting of both reduced and full models. The full model incorporates different sets of parameters for the two species, whereas in the reduced model the same sets of parameters are used for both species (Xu, 2012).

The test statistics for comparing the reduced and full models is an F-test:

F =SSER −SSEF( ) / dfR −dfF( )

SSEF /dfF (3)

where the error sum of squares for the full model is denoted as SSEF, its degrees of freedom is written as dfF, the error sum of squares for the reduced model is denoted as SSER, and its degrees of freedom is written as dfR. Generally, the

F-test is significant if the P-value for the test is less than α = 0.05. 2.4. Model performance criteriaTo test model performance, the following evaluation statistics were used: the model’s average bias (E), percent bias (%E), standard error of the estimate (SEE), percent of the standard error of the estimate (%SEE), and coefficient of determination (R2). R2 indicates how much variation in the dependent variable can be explained by the independent variables. Although there are several shortcomings associated with the use of R2 in nonlinear regression, the general usefulness of some global measure of model adequacy appears to override some of those limitations (Ryan, 1997); nevertheless, it must not be used as the only criterion for selecting the best model (Myers, 1990). These measures have been computed by the following equations:

E =yi − yi( )

i=1

i=n∑

n(4)

E%=yi − yi( )

i=1

i=n∑ /n

y×100 (5)

SEE =yi − yi( )2

i=1

n

∑n – p

(6)

SEE%=yi − yi( )2

i=1

n

∑n – p

y

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

×100 (7)

R2 =1−yi − yi( )2

i=1

n

∑

yi − y( )2

i=1

n

∑

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

(8)

where yi are the observed values, ŷi are the values estimated by the model, y– is the mean of the observed values, n is the total number of data used for fitting the model, and p is the number of parameters which have to be estimated.

The MODEL procedure in SAS (SAS Institute, 2013) was used for estimation of the nonlinear simultaneous taper and merchantable volume equation systems in which different parameter estimation methods are available. These parameter estimation methods are compatible systems that can be fitted by ordinary least squares estimation (OLS), generalized least squares (GLS) methods, seemingly unrelated regression (SUR), the generalized method of moments (GMM), and the maximum likelihood with full

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information estimation method (FIML). When equation errors have a multivariate normal distribution, FIML can be used to obtain efficient parameter estimates (Fang et al., 2000). Since with our data the errors are almost normally distributed and the correlation between the error terms of the equations is not so high, the FIML method was selected for the estimation of parameters (e.g., Cov (εd, εVm) = 0.076 and 0.045 for EG and EC, respectively).

Assessment of the model performance with an independent data set would be the most desirable (Kozak and Kozak, 2003). In case of difficulty acquiring an independent data set, different model validation techniques (cross-validation or double cross-validation) can be used. However, the results reported by Kozak and Kozak (2003) showed that these techniques provide little additional information in the process of evaluating a regression model over the fit of the model with the whole date set. Therefore, a model validation process was not applied for evaluation of model performance.

3. Results Eqs. (1) and (2) were fitted to the data using MODEL in SAS. To obtain consistent parameter estimation based on the FIML method, the taper and merchantable volume equations were fitted simultaneously. All parameters were shared by both the taper and volume equations and all parameters were found to be significant at the 0.0001 level (Table 2). The overall fit statistics (bias, bias%, SEE, SEE%, and R2) were calculated and are presented for the entire merchantable stem in Table 3. The average biases were negative for the taper model, which indicates that diameter was overestimated for both eucalyptus species. About 97% of the total variation in predicting upper stem diameters was explained by Fang et al. (2000) for both species. The estimated SEE is about 1.7 cm in predicting stem diameter for both species. Statistics of fit (bias, %bias, SEE, %SEE, and R2) are presented in Table 3 for volume over bark. Eq. (2) explained more than 97% of the merchantable stem volume variability. The model had an average bias less than

Table 2. Parameter estimates and SE (standard errors) for the compatible taper and merchantable volume sys-tems for Eucalyptus grandis (EG) and Eucalyptus camaldulensis (EC) plantations.

ParametersEG EC

P-valueEstimates SE Estimates SE

a0 0.000058 1.426E–6 0.000105 1.515E–6 <0.0001

a1 2.027231 0.00563 1.828332 0.00419 <0.0001

a2 0.803353 0.00646 0.824602 0.00568 <0.0001

b1 0.000011 1.461E–6 0.000012 6.598E–7 <0.0001

b2 0.000031 4.4E–7 0.000032 1.191E–7 <0.0001

b3 0.000032 9.417E–8 0.000031 3.497E–7 <0.0001

p1 0.03626 0.00726 0.064301 0.00463 <0.0001

p2 0.455905 0.0921 0.709985 0.03710 <0.0001

Table 3. The fit statistics for compatible stem taper and merchantable tree volume systems for Eucalyptus grandis (EG) and Eucalyptus camaldulensis (EC) plantations.

Species E %E SEE %SEE R2 CN*

EG

Taper (cm) –0.0231 –0.1091 1.8113 8.5635 0.972170

Volume (m3) 0.0006 0.0750 0.0852 9.9346 0.9800

EC

Taper (cm) –0.0511 –0.3124 1.5424 9.4243 0.969749

Volume (m3) 0.0003 0.0634 0.0509 12.1116 0.9740

E = average bias, %E = percent bias, SEE = standard error of the estimate, %SEE = percent of the standard error of the estimate, R2 = coefficient of determination, and CN = condition number.

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0.07 m3 for volume over bark. The model underestimated volume for both eucalyptus species. Moreover, the diameters and merchantable tree volumes of EG may be slightly better predicted than those of EC using Eqs. (1) and (2).

The inflection points for the Fang et al. (2000) model occur at 3.6% and 6.4% of the height of tree near the dbh, and at 45.6% and 71.0% of relative height of the bole for EG and EC, respectively (Table 2). These results suggest that a segmented model with two inflection points is more appropriate to describe the stems of eucalyptus species.

To evaluate the performance of the equation of systems, values for average bias, %bias, SEE, and %SEE were evaluated by dbh class (Table 4), total height class (Table 5), and relative height class (Table 6) for the prediction

of stem diameters and merchantable volume estimations. As is known, larger trees possess more volume and value. However, it is possible to determine the predictive abilities for diameter and volume by overall fit statistics for different tree sizes. Therefore, to evaluate the models performance for different tree sizes, the models were further evaluated by dbh classes. Ten dbh classes were used for model evaluation. Average bias and SEE were calculated for diameter and merchantable stem volume prediction by dbh class (Table 4). The model showed similar performances for all dbh classes in terms of fit statistics for both species (Table 4). Based on the goodness of fit statistics values, the Fang et al. (2000) model was equally good in predicting stem diameter and merchantable volumes for different dbh classes and total height class for both species (Table

Table 4. Bias, percent bias (%bias), standard error of estimates (SEE), and percent standard error of estimates (%SEE) by dbh classes for outside bark diameter and merchantable volume of EG and EC plantations.

Species Dbh class (cm) n

Upper stem diameter Merchantable volume

Bias (cm) % Bias SEE (cm) %SEE Bias (m3) % Bias SEE (m3) %SEE

EG 16 35 –0.2711 –2.6528 0.5616 5.4955 –0.0037 –3.4021 0.0060 5.5052

EG 20 218 –0.1378 –1.0968 0.9450 7.5203 –0.0062 –3.3728 0.0136 7.4136

EG 24 328 0.3752 2.5206 1.9106 12.8346 0.0054 1.8332 0.0498 16.9178

EG 28 504 0.3738 2.1523 1.4784 8.5126 0.0065 1.4033 0.0434 9.4121

EG 32 915 0.0884 0.4610 1.6789 8.7526 –0.0026 –0.4417 0.0654 10.9413

EG 36 1019 –0.1866 –0.8867 1.7594 8.3623 –0.0109 –1.4232 0.0644 8.4183

EG 40 893 –0.1595 –0.6843 2.0561 8.8195 0.0046 0.4280 0.1123 10.5546

EG 44 753 –0.0025 –0.0096 1.9042 7.3892 0.0133 0.9869 0.0976 7.2634

EG 48 389 –0.1577 –0.5724 2.1291 7.7279 0.0098 0.5992 0.1408 8.5671

EG 52 34 –2.1837 –7.7599 3.0256 10.7516 –0.1362 –7.1962 0.1878 9.9224

All 5088 –0.0231 –0.1091 1.8113 8.5635 0.0006 0.0750 0.0852 9.9346

EC 12 15 –0.6023 –7.5918 1.0622 13.3888 –0.0045 –7.6321 0.0058 9.8016

EC 16 115 –0.1001 –1.0806 0.8530 9.2043 –0.0043 –4.0416 0.0100 9.4454

EC 20 405 –0.2465 –2.0607 1.2396 10.3644 –0.0082 –4.5725 0.0210 11.6746

EC 24 678 –0.1806 –1.2646 1.3823 9.6782 –0.0084 –3.0850 0.0299 10.9785

EC 28 814 0.2670 1.6000 1.2939 7.7539 0.0071 1.7586 0.0350 8.6293

EC 32 874 –0.3142 –1.7410 1.7697 9.8056 –0.0071 –1.3602 0.0585 11.1928

EC 36 295 0.5839 2.7721 1.7237 8.1833 0.0357 4.6059 0.0787 10.1673

EC 40 69 0.0658 0.2844 2.8126 12.1544 0.0264 3.2295 0.1086 13.2914

EC 44 25 1.1628 4.5466 1.6172 6.3230 0.0950 9.2992 0.1155 11.3039

EC 52 31 –2.4370 –8.8867 3.7496 13.6735 –0.1239 –7.7968 0.1911 12.0243

All 3321 –0.0511 –0.3124 1.5424 9.4242 0.0003 0.0634 0.0509 12.1116

*The totals for bias, %bias, SEE, and %SEE are calculated as overall means.

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5). Average bias, %bias, SEE, and %SEE were calculated for each pair of equation systems by relative height class and were used to evaluate stem diameter and volume estimation for each species (Table 6). The equation systems of Fang et al. (2000) behaved well and did not show large prediction errors in any sections of the stem. There is no clear tendency from the baseline to the top, which indicates that the segmented-taper equation reasonably reflects the butt part as well as the top part of the tree stems for relative height classes. SEE ranged from 1 to 2 cm and 0.005 to 0.13 m3 for the merchantable volume for both species (Table 6).

Another way to evaluate and compare the equations is to look at the graphics of the residuals (Heidarsson and Pukkala, 2011). The box plots of diameter and merchantable volume residuals against the relative height classes of the Fang et al. (2000) model did not show any clear systematic bias for eucalyptus species. As a result of the plots, the error distribution along the stem is equal for the eucalyptus species (Figure 2).

The Fang et al. (2000) model showed relatively big problems estimating diameter and merchantable volumes for the larger diameter classes than for the smaller trees (Figure 3). This trend found in some other studies as well (Diéguez-Aranda et al., 2006; Barrio-Anta et al. 2007; Schröder et al. 2014), a fact that could be considered

worrisome as larger trees are more valuable (Schröder et al. 2014). However, when considering prediction errors for diameter and merchantable volume percentage (%SEE), deviations tend to be more evenly distributed among diameter classes (Table 4; Figure 3). Similar results can be drawn from Figure 4, which shows the bias formulated as diameter and merchantable volume residuals against height classes. While the error distribution along the stem is equal for the eucalyptus species for diameter estimations, the error distribution is not equal for volume predictions. The Fang et al. (2000) model had bigger problems estimating volumes for higher height class for the EG plantations than for the EC ones (Figure 4).

Figure 5 shows the predicted taper equations using fitted taper equations on a typical tree with average dbh and total height values for EG and EC. We found that the EG stands tended to have greater stem diameter estimates in the lower and middle sections of the tree stem compared with the EC stands. However, this difference was minimal.

The results of the F-test for differences between eucalyptus species are shown in Table 7 for the Fang et al. (2000) model. This result indicated that there were statistically significant differences among the taper equations. Therefore, separate volume and taper equations are needed for each species. The comparison in Figure

Table 5. Bias, percent bias (%bias), standard error of estimates (SEE), and percent standard error of estimates (% SEE) by height classes for outside bark diameter and merchantable volume for EG and EC plantations.

Species Height class (m) n

Upper stem diameter Merchantable volume

Bias (cm) % Bias SEE (cm) %SEE Bias (m3) % Bias SEE (m3) %SEE

EG 16 17 1.1417 9.4536 2.2856 18.9259 0.0045 3.3755 0.0122 9.1978

EG 20 330 –0.2581 –1.5167 1.5357 9.0253 –0.0099 –2.5713 0.0371 9.6803

EG 24 825 –0.4737 –2.6550 1.7891 10.0267 –0.0083 –1.7448 0.0505 10.5558

EG 28 1514 0.2665 1.2994 1.8490 9.0162 0.0116 1.5966 0.0743 10.1969

EG 32 1243 0.3376 1.5301 1.8629 8.4429 0.0142 1.5120 0.0942 10.0570

EG 36 870 –0.5110 –2.0966 1.7607 7.2244 –0.0272 –2.0795 0.1027 7.8446

EG 40 289 –0.1322 –0.5161 1.9364 7.5610 0.0063 0.4211 0.1443 9.5667

All 5088 –0.0231 –0.1091 1.8113 8.5635 0.0006 0.0750 0.0852 9.9346

EC 16 355 –0.1326 –0.9896 1.1748 8.7651 –0.0041 –2.1482 0.0183 9.6473

EC 20 896 –0.0006 –0.0038 1.5519 9.9991 0.0000 –0.0077 0.0424 13.6473

EC 24 906 –0.2127 –1.3040 1.8314 11.2299 –0.0042 –1.0446 0.0601 14.9790

EC 28 532 –0.0112 –0.0678 1.4055 8.4969 0.0013 0.2776 0.0489 10.2183

EC 32 632 0.1210 0.6303 1.3768 7.1733 0.0086 1.1984 0.0613 8.5333

All 3321 –0.0511 –0.3123 1.5424 9.4242 0.0003 0.0634 0.0509 12.1116

*The totals for bias, %bias, SEE, and %SEE are calculated as overall means.

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5 reveals the importance of developing separate taper equations for different species.

4. DiscussionIn this study, systems of equations were developed for the economically important industrial eucalyptus plantations taking into account both practical and statistical considerations. The exponentially segmented compatible taper equation of Fang et al. (2000) was selected in this study. Several studies have shown that this equation can provide reliable and accurate estimations for both stem diameters and merchantable tree volume in different countries (Corral-Rivas et al., 2007; Crecente-Campo et al.,

2009; Pompa-Garcia et al., 2009; Li and Weiskittel, 2010). Moreover, the Fang et al. (2000) model has the merits of being a flexible and analytically integrable system that should provide very accurate estimates of stem volume for any stem segment, which is important in practical terms.

To ensure numeric consistency, a simultaneous fitting procedure based on the FIML estimation method was used and all equation parameters were shaped by both taper and volume equations. The adjustment by FIML homogenizes and minimizes the standard error of the parameters and allows total compatibility of the taper and merchantable volume system (Borders, 1989). The system proposed by Fang et al. (2000) exhibited good performance in predicting

Table 6. Bias, percent bias (%bias), standard error of estimates (SEE), and percent standard error of estimates (% SEE) by relative height (RH) classes for outside bark diameter and merchantable volume for EG and EC plantations.

Species RH nUpper stem diameter Merchantable volume

Bias (cm) % Bias SEE (cm) %SEE Bias (m3) % Bias SEE (m3) %SEE

EG 0.0–0.1 556 –0.1452 –0.4102 1.1930 3.3699 0.0003 0.2563 0.0061 5.2435

EG 0.1–0.2 524 –0.0997 –0.3169 1.5522 4.9317 0.0000 –0.0046 0.0243 6.1779

EG 0.2–0.3 521 –0.1280 –0.4486 1.6568 5.8068 –0.0011 –0.1829 0.0442 7.2406

EG 0.3–0.4 516 –0.1179 –0.4582 1.8223 7.0801 –0.0012 –0.1451 0.0628 7.8876

EG 0.4–0.5 520 –0.0971 –0.4252 1.9378 8.4875 –0.0001 –0.0082 0.0806 8.5590

EG 0.5–0.6 520 –0.0567 –0.2877 2.0131 10.2075 –0.0041 –0.3965 0.0958 9.1729

EG 0.6–0.7 529 0.1371 0.8232 2.1034 12.6318 0.0008 0.0743 0.1044 9.1841

EG 0.7–0.8 513 0.2034 1.5468 2.0721 15.7572 –0.0008 –0.0669 0.1150 9.6325

EG 0.8–0.9 492 0.0237 0.2614 2.0971 23.1704 0.0049 0.3921 0.1191 9.5938

EG 0.9–1.0 397 0.0883 3.0383 1.4686 50.5519 0.0103 0.8051 0.1254 9.7840

All 5088 –0.0231 –0.1091 1.8113 8.5635 0.0006 0.0750 0.0852 9.9346

EC 0.0–0.1 350 –0.1988 –0.7054 1.2778 4.5348 0.0002 0.3068 0.0048 9.6818

EC 0.1–0.2 325 –0.1555 –0.6210 1.2877 5.1418 0.0010 0.4906 0.0120 6.1589

EC 0.2–0.3 338 –0.1498 –0.6637 1.4836 6.5715 0.0002 0.0774 0.0206 6.8814

EC 0.3–0.4 330 –0.2123 –1.0478 1.5960 7.8750 –0.0007 –0.1723 0.0316 8.1136

EC 0.4–0.5 325 –0.1602 –0.8812 1.6798 9.2395 –0.0012 –0.2547 0.0443 9.4140

EC 0.5–0.6 332 0.1045 0.6565 1.9231 12.0834 –0.0011 –0.2053 0.0527 10.1666

EC 0.6–0.7 328 0.1174 0.8703 1.9538 14.4830 –0.0003 –0.0468 0.0648 11.1883

EC 0.7–0.8 336 0.0599 0.5789 1.6920 16.3638 0.0018 0.3121 0.0702 11.8493

EC 0.8–0.9 327 0.0310 0.4494 1.4549 21.0842 0.0013 0.2222 0.0765 12.6323

EC 0.9–1.0 330 0.0595 2.8035 0.8558 40.2920 0.0013 0.2229 0.0710 12.0421

All 3321 –0.0511 –0.3124 1.5424 9.4242 0.0003 0.0634 0.0509 12.1116

*The totals for bias, %bias, SEE, and %SEE are calculated as overall means.

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Eucalyptus grandis Eucalyptus camaldulensis

–15.0

–10.0

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ls (c

m)

–0.50

–0.25

0.00

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0.75

5 15 25 35 45 55 65 75 85 95

v re

sidua

ls (m

3 )

Relative height class (RH)5 15 25 35 45 55 65 75 85 95

Relative height class (RH)

Figure 2. Residuals box plot of estimated diameters and merchantable tree volumes over bark by relative height (RH) classes of the Fang et al. (2000) systems. The diamond signs represent the means of prediction errors for the corresponding relative height classes. The boxes represent the interquartile ranges. The maximum and minimum diameter over bark and merchantable volume prediction errors are represented respectively by the upper and lower small horizontal lines crossing the vertical lines.

Eucalyptus grandis Eucalyptus camaldulensis

–15.0

–10.0

–5.0

0.0

5.0

10.0

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d re

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ls (c

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0.75

16 20 24 28 32 36 40 44 48 52

v re

sidua

ls (m

3 )

Diameter classes (cm)12 16 20 24 28 32 36 40 44 48

Diameter classes (cm)

Figure 3. Residuals box plot of estimated diameters and merchantable tree volumes over bark by dbh classes of the Fang et al. (2000) systems. The diamond signs represent the means of prediction errors for the corresponding diameter classes. The boxes represent the interquartile ranges. The maximum and minimum diameter over bark and merchantable volume prediction errors are represented respectively by the upper and lower small horizontal lines crossing the vertical lines.

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–15.0

–10.0

–5.0

0.0

5.0

10.0

15.0

d re

sidua

ls (c

m)

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–0.25

0.00

0.25

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16 20 24 28 32 36 40

v re

sidua

ls (m

3 )

Height classes (m)16 20 24 28 32

Height classes (m)Figure 4. Residuals box plot of estimated diameters and merchantable tree volumes over bark by height classes of the Fang et al. (2000) systems. The diamond signs represent the means of prediction errors for the corresponding height classes. The boxes represent the interquartile ranges. The maximum and minimum diameter over bark and merchantable volume prediction errors are represented respectively by the upper and lower small horizontal lines crossing the vertical lines.

0

0.2

0.4

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1

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0 0.2 0.4 0.6 0.8 1

Pred

icte

d re

lativ

e dia

met

er

Relative height

Eucalyptus grandisEucalyptus camaldulensis

Figure 5. Predicted relative over bark diameter over relative height for typical tree with average total tree height and average dbh for two species.

Table 7. Results of the F-test for the taper models based on the model of Fang et al. (2000) for the different eucalyptus species analyzed in the study.

Model pair nFull model Reduced model

F value P valuedfF SSEF dfR SSER

EG–EC 8071 8051 24374.7 8061 24721.7 11.46 <0.0001

The F values were calculated according to Eq. (11). n is the number of observations, SSEF, dfF, SSER, and dfR are the sums of squared errors and the degrees of freedom associated with the full and reduced models, respectively.

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stem diameters and merchantable volume along the bole for overall fit and also different dbh classes, height classes, and relative height classes. Moreover, stem diameters and merchantable volumes of EG may be estimated slightly better than those of EC. When considering the prediction bias of diameter, residual variance was clearly more homogeneous along the stem among the relative height classes for EG (results not shown). Similar results were reported by Özkurt (2000).

The inflection points for EG occur at 3.6% of the tree height near the dbh base and below the midpoint of the total tree height (45.6%), so that much of the volume is concentrated in one section of the conical structure, whereas there is only 42% in a cylindrical structure. The inflection points for EC based on Fang et al. (2000) are located at 6.4% and 70.0% of the tree height. Thus, of the three sections segmented by the two inflection points, the first is very short as a percentage of total height. These results are similar to those of previous studies. For example, Crecente-Campo et al. (2009) determined that the first inflection points for Pinus sylvestris L. in different regions of Spain ranged from 8% to 12% of the tree height and the second inflection points ranged from 20% to 71% of the total tree height. When considering the position of first and second inflection points, we can say that EG has a more cylindrical stem form than EC. On the other hand, when considering position of first inflection points for both eucalyptus species, we can say that the butt sections of EG have a slightly smaller variation than those of EC. The trends of error in estimating stem diameter and volume along the stem are more important in the lower half of the stem and especially in the butt section, where most of the volume and value is concentrated for saw timber sections of the bole. When evaluated in this respect, the lower half of the stem of EG has a more cylindrical stem form than that of EC.

The breast high form factors for the three segments as found by dividing the predicted form factors βi by k are 0.140, 0.394, and 0.407 for EG and 0.153, 0.407, and 0.395 for EC. The form factors for EG are smallest at the bottom, moderate in the middle, and largest at the top, while the form factors for EC are smallest at the bottom, largest in the middle, and moderate at the top. As indicated by Fang et al. (2000), these form factors also compare almost remarkably to 0.250, 0.333, and 0.500 for a neiloid, cone,

and paraboloid, respectively, which is consistent with the results obtained by Fang et al. (2000) and Pompa-Garcia et al. (2009).

Based on the results of this study, it is recommended that the Fang et al. (2000) equation systems be considered operationally for practical forestry of eucalyptus tree species to estimate stem diameter and merchantable volume to a given height or diameter. In addition to providing better total volume predictions, the suggested model can also be utilized to estimate product volumes to any desired top diameter limit and can permit multiproduct volume prediction for the same tree, which is not possible with the existing total volume tables (Burkhart and Tome, 2012).

Multicollinearity is one of problems when using regression analysis in empirical forest modeling. The presence of correlations among variables in the model has been examined using a condition number (CN). According to Belsey (1991), in regression analysis, if the CN is in range of 30–100, then there are problems associated with collinearity, but this value is tolerated when models contain polynomial and cross-product terms. The equation systems of Fang et al. (2000) showed weak multicollinearity (Table 3) for both eucalyptus species. Kozak (1997) stated that multicollinearity does not seriously affect the predictive ability of the model.

There appears to be no statistical justification for combining the EG and EC species to form a single merchantable volume or taper equation. The results of the F-test revealed that EG and EC are different in shape and taper, which warrants the use of separate merchantable taper and stem volume equations.

With the equations systems used in this study, an option is given for plantation forest managers of the region to use a more efficient tool to calculate the distribution of timber and thus to determine the value and proper use of the raw material from forests in the forest products industries (Perez et al., 2013).

AcknowledgmentsThis study was supported by the Scientific and Technological Research Council of Turkey under project 113 O 834. We appreciate the valuable comments, constructive suggestions, and language revision from 3 reviewers and the editor that improved the content and quality of the manuscript.

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