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Sal (Shorea robusta Gaertn.) is one of the most important commercial tree species 
in Nepal and far-western Terai is renowned for its forest. This study was carried out 
in far-western Terai to develop volume models of Sal at tree level using destructive 
sampling. Out of 99 sample trees, 81 data were used to develop the models and 18 
data for validation of the selected models. Over bark stem diameters were measured 
at an interval of 0.5 m in lowermost three sections, at an interval of 1 m for one section 
and at an interval of 2 m in upper part of the trunk from the ground level. Smalian’s 
formula was used to compute tree volume. Seven regression models were tested 
using DBH as a predictor variable. Cross validation of the independent data set was 
used to validate the selected models. The graphical analysis and fit statistics of the 
models were evaluated to select the best fit model. The selected model for total over 
bark stem volume is ln V = - 8.04674 + 2.26641 ln DBH with R2 of 92 % and standard 
error of 0.18. Similarly, the selected models for over bark volume up to 10 and 20 cm 
top diameter have R2 of 82.41% and 79.97% and standard errors of 0.35 and 0.42, 
respectively. The prediction error of the selected model was found to be less than 6%. 
Forest managers can use the recommended model in estimation of timber volume of 
Sal in a particular forest area of this region for effective forest management.

Key words: Destructive sampling, far-western Terai, fit statistics, regression equation

Volume models for Sal (Shorea robusta Gaertn.) in 
far-western Terai of Nepal

T. Subedi1

Tree volume provides vital information in forest 
management for estimating current and future 

stock of forest. However, direct measurement of 
volume is a tedious and impractical in the field. 
Thus, models or mathematical functions are 
necessary to estimate the volume using some 
measurable variables such as height, diameter 
and form of the tree. Further, volume models have 
been used as one of the best means to estimate 
trees and stand volume and have played vital role 
in forest inventory, management and silvicultural 
research (Ozçelik et al., 2010).

In principle, height and diameter are measured in 
the routine forest measurement. However, height 
measurement is not always practical due to more 
time consuming and cost and possibility of being 
less accurate (Wagle and Sharma, 2011; Sharma 
and Pukkala, 1990b; Chaturvedi and Khanna, 
1982). On one hand, the possibility of error 
increases more in the dense forest measuring tree 

height, on the other, DBH can be measured more 
easily and accurately with less time and cost. 
Further, volume table produced using a model 
with predictor variable DBH only is particularly 
useful for quick timber inventory. It can be tallied 
with species and only DBH (Sharma and Pukkala, 
1990a; Chaturvedi and Khanna, 1982; Özçelik, 
2008).

Department of Forest Research and Survey 
(DFRS) have been involved in producing various 
volume and biomass models (Sharma and 
Pukkala, 1990a; Laamanen et al, 1995, Tamrakar, 
2000; Acharya et al., 2003) of tree species 
required for forest management for long time. The 
general volume tables of 21 tree species and two 
species groups were developed in 1990 using data 
collected in 1960s (Sharma and Pukkala, 1990a). 
During 1990s there were few studies on general 
volume and growth models of Sal, especially in 
central Bhabar forests of Nepal (Laamanen et al., 

1 Department of Forest Research and Survey, Babarmahal, Kathmandu, Nepal
E-mail: ecothakur@yahoo.co.in



Banko Janakari, Vol 27 No. 2, 2017

4

1995). In addition to this, DFRS has developed 
biomass and volume models of some trees and 
bamboo species and some forest types. However, 
the diameter used in developing the models was ≤ 
35 cm and the data were collected from thinning 
operation applied in trial plots may not represent 
the natural forest and again, the data were from 
only central part of the country (Acharya and 
Acharya, 2004; Acharya et al., 2003; Tamrakar, 
2000; Pukkala et al, nd.).

S. robusta Gaertn., only one species found in 
Nepal of tropical family Dipterocarpaceae, is a 
multipurpose tree species. Sal is a valuable and 
important timber species for construction, and 
fuel wood. Sal seeds are used as raw material in 
industries and leaf used for making plates and 
as fodder for livestock (Jackson,1994). It is still 
most predominant species in the terai of Nepal 
(DFRS, 2014). It is found from Terai region to 
1500 m but common up to 1000 m. It occurs 
mostly in Terai, Siwalik and low land of hilly 
areas. In most areas, almost pure Sal forest can 
be found or in association with Terminalia alata. 
In some places, it grows along with broadleaved 
species. The Sal forests in the Terai (plain area) 
are mostly large and differ from hill Sal forest. In 
higher rainfall and moist areas, it is replaced by 
mixed forest. Dobremez (1976) listed nine types 
of Shorea forests, but Champion and Seth (1968) 
listed more than that, most of them are expected 
to be found in Nepal (cited by Jackson,1994).

The climate of far-western terai is drier than other 
parts of the country. Out of the total forest area of 
Kailali district, 32.16% is covered by Sal forest 
and 31.39% by Terai Mixed Hardwood (TMH) 
with Sal (DDC Kailali, 2015). The Sal forest in 
far-western Terai is similar with minor variations 
due to topographic and climatic similarity. Thus, 
DFRS, 2014 described as far-western forest 
clusters, which is different from those in other 
parts of the country because its climate is drier 
than other parts of the country. Therefore, the 
volume models based on one variable viz. DBH 
alone will be applicable.

In recent years, efforts have been made towards 
the scientific forest management in Nepal. 
However, there is lack of appropriate technical 
tool for volume and biomass estimation of timber. 
There is need for precise and site specific volume 
estimation equations using easily and accurately 

measurable independent variables of the trees. 
Realizing the situation, Department of Forests 
has called for preparing district wise local volume 
tables to estimate the timber quantity (DoF, 2014). 
Therefore, preparation of local volume equations 
of this species for the natural forest of far-western 
Terai is essential. The purpose of the study was 
to prepare local volume models of S. robusta for 
specific heights of tree trunk in far-western Terai.

Materials and methods

Study area

The study site is Balchaur area located at the 
eastern part of Kailali district and other sites such 
as Bani, Banka, Motipur, Gwalabari, Krishnapur, 
Basahan, Singapurur, etc. located at eastern and 
southern part of Kanchanpur district (Fig. 1).

Fig 1: Map of the study area

These sites are in an altitudinal range of 109 m to 
200 m above the mean sea level. Kailali district 
has an area of 3,235 sq. km, in which 40 per cent 
is covered by Terai (flat land) and 60 per cent by 
Chure hills. The total area of Kanchanpur district 
is 1610 sq. km in which 11.7 percent covered by 
Chure region, 55 percent is covered by forest and 

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Banko Janakari, Vol 27 No. 2, 2017

5

streams and 548.5 sq. km. of wildlife reserves and 
its buffer zone area. The far-western Terai forest 
extends from the Karnali River in the east to the 
western border of Nepal. It covers 97,622 ha 
forest outside the protected area (DFRS, 2014). 
Sal forest, Terai Mixed Hardwood forest, Sal with 
Terai Mixed Hardwood forest and Khair-Sissoo 
forest are the dominant types of forest in this area 
(DFRS, 2014).

The southern parts of the districts consist of plain 
area with deep fine sandy loamy soil. The climate 
is generally sub-tropical. The precipitation and 
temperature from Dhangadhi and Mahendranagar 
stations represent eastern and western study 
sites, respectively. The data of precipitation and 
temperature in Dhangadhi are average figures of 
25-years and 6-years, respectively. The data of 
precipitation and temperature in Mahendranagar 
are average figures of 10-years and 4-years, 
respectively. Average annual precipitation ranges 
from 1547 mm in Mahendranagar to 1725 mm 
in Dhangadhi. Average monthly temperature 
ranges from 23.7o C in Dhangadhi to 23.6o C in 
Mahendranagar. Absolute maximum temperature 
is 43.5o C whereas absolute minimum temperature 
is 2.0o C in Dhangadhi. Absolute maximum 
temperature is 43.0o C whereas absolute minimum 
temperature is 2.6o C in Mahendranagar. 
Dhangadhi annual autumn temperature is 
maximum 43o C to minimum 24o C and winter 
temperature is maximum 19o C to minimum 2o C. 
In this way, average temperature is found as 30.5o 
C. These climatic data taken from Department 
of Hydrology and Meteorology/Department of 
Irrigation, Hydrology and Meteorology are stated 
by Jackson (1994).

Data collection

According to the Forest Act (1993), plants above 
30 cm diameter are regarded as tree and selected 
as samples for the study. The data of selected 
sample trees were collected above 30 cm DBH 
to develop models particularly for estimation 
of Sal timber. The data were collected from the 
forests of far-western Terai. The trees above 30 
cm diameter were divided into 10 cm diameter 
class up to 90 cm and one class above this. 
For each diameter class, at least 10 trees were 
selected for developing a model. The data were 
collected from different forest types, quality 
class, crown class and density to represent all the 

possible local minor variations of natural forest. 
The forest type, quality class, crown class and 
density were measured as in FRA (2010). The 
representative sample trees with respect to size 
(diameter) were chosen randomly among the 
available trees from all parts of the selected area. 
Twenty-one sample trees were selected from pure 
Sal forest, 23 from Sal-Asna mixed forest and 37 
from Terai Mixed Hardwood forest to develop 
the model . The crown cover of these forests 
ranged from 30 to 85% having median of 60% 
crown cover. Similarly, the selected sample trees 
to develop the model were 67 from predominant, 
12 from co-dominant and two from suppressed 
trees. In this way, 81 (32 from Kailali and 49 
from Kanchanpur) sample trees were selected 
to develop the model. Similarly, 18 sample trees 
were selected from representative study area for 
validation of the models.

The basic characteristics of the site and sampled 
trees were recorded before felling the sample 
trees. After measuring DBH, trees were felled and 
over bark diameters were measured at an interval 
of 0.5 m in lowermost three sections, at an 
interval of 1 m for one section and at an interval 
of 2 m in upper part of the trunk (Sharma and 
Pukkala, 1990a; Eerikainen, 2001). Further the 
height of the sampled tree up to 10 cm and 20 cm 
over bark top diameters were recorded. The over 
bark diameters were measured by a diameter tape 
with an accuracy of 0.1 cm. Sectional volume 
was calculated using Smalian’s formula and 
then total volume and volume up to top 10 and 
20 cm diameters were obtained by summing up 
sectional volumes (Laamanen et al.,1995, Segura 
and Kanninen, 2005 Özçelik, 2008, Ozçelik et 
al., 2010).

Data structure and model development

The average DBH and height of the sample trees 
were approximately 59 cm and 30 m (Table 1). 
The detailed descriptive statistics of 81 sample 
trees is given in table 1.

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Table 1: Descriptive statistics of data of sample trees

Variables Number of 
sample trees Minimum Median Mean Maximum Standard 

deviation

Total height (m) 81 19.00 30.05 29.91 41.20 5.05

DBH (cm) 81 30.10 57.50 59.21 108.50 16.86

Height diameter ratio 81 0.38 0.50 0.53 0.76 0.10

Crown height (m) 72 3.40 12.20 12.39 23.30 5.28

Total volume (m3) 81 0.58 2.90 3.69 11.20 2.35

Volume up to 20 cm (m3) 81 0.38 2.80 3.59 11.14 2.36

Volume up to 10 cm (m3) 81 0.56 2.89 3.67 11.18 2.35

The following different models were tested using 
R statistical software (R core Team, 2012).

V = a + b *D ...............................(i)
ln V = a + b *ln D........................(ii)
V = a + b *ln D............................(iii)
ln V = a + b *D............................(iv)
V = a + b *D2...............................(v)
V = a + b*D+c*D2.......................(vi)
V = aDb........................................(vii)

Moreover, the following models were tested 
for predicting the volume of the proportion of 
sample tree in top 10 cm and top 20 cm over bark 
diameter (Sharma and Pukkala, 1990a; Laamanen 
et al., 1995).

lnV1/ V = a + b* ln D ................(viii)
lnV2/Vt = a + b*ln D .................(ix)

where,
a, b and c are parameters to be estimated,
V is total volume of tree,
V1 is volume beyond 10 cm top diameter,
V2 is volume between top 10 and 20 cm diameter, 
Vt is volume up to 10 cm top diameter,
DBH is diameter at breast height and ln is the 
natural logarithm.

Model selection and validation

T-test and F-test were used for testing the 
significance of the parameters and whole 
equation, respectively. The best fit model was 
selected based on residual analysis (whether the 
model fulfilled regression assumption or not), and 
fit statistics (standard error, bias and coefficient 
of determination). The back transformation was 
done with bias correction by adding exp (SE2/2) 

to the intercept (Sprugel, 1983).

The method of cross validation technique was used 
(Hawkins, 1987; Kozak and Kozak, 2003). The 
models were evaluated by testing cross validation 
of separate data sets of 18 trees. The prediction 
statistics was estimated using following equation 
and percentage error were plotted against the 
explained variable (Hawkins,1987; Acharya et 
al., 2003; Ducey and Williams, 2011).

Prediction error = (∑ actual volume − ∑ predicted 
volume)/ ∑ actual volume ∗ 100)………(xii)

Results and discussion

Relationship between total volume and tree 
variables

The relation of volume with both the height and 
DBH was found to be strong and positive (Fig. 
2). The Pearson Correlation between total volume 
and DBH was found to be 0.9897. Similarly, the 
correlation between volume and height was found 
to be 0.8610 whereas it was

Fig 2: Relationship among different variables

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7

0.8011 between DBH and total height. Since, the 
total height has strong and positive relation with 
DBH, the height adds very little effect on volume 
than that of the DBH alone.

Total stem volume model

Above mentioned models were fitted and checked 
both by graphically and numerically in order to 
identify the best predicted model. The fitted 
models were overlaid on the observed data (Fig. 
3). In second graph, the observations are better 
distributed around the model throughout the DBH 
range. However, in all other figure, the models 
better capture the observation only in some part 
of the DBH range.

The seven different regression models as given 
above were fitted and checked both by graphically 
and numerically to test the best predicted model. 
The fitted models were overlaid on the observed 
data (Fig. 3).

Except the fifth and sixth models, the parameters 
of other five models are significant at 5% level of 
significance or even less. In general, all models 

fitted to the data well, as their good statistical fits in 
terms of R2 explained greater than 80% of variability 
(Table 2). There were variations in standard error of 
estimate (SEE) in different models and equation 2 
has the lowest SEE which is less than 0.2.

V = aDb

Fig. 3: Visualization of seven models

Table 2: Values of regression constants, std.error, R2 with t and p-values of tested models

Mo
del no.

Para
meter Estimate Standard

Error t value Pr(>|t|)
Sd. Error

of 
Estimate

R2 p-value

1 a -4.3588 0.3291 -13.2500 <2e-16*** 0.8053 0.8915 <2.2e-16
b 0.1360 0.0053 25.6600 <2e-16 ***

2 a -8.0640 0.2930 -27.5300 <2e-16 *** 0.1859 0.9249 <2.2e-16
b 2.2664 0.0722 31.4100 <2e-16 ***

3 a -27.2900 1.6610 -16.4300 <2e-16 *** 1.054 0.8142 <2.2e-16
b 7.6680 0.4090 -18.7500 <2e-16 ***

4 a -1.1329 0.0937 -12.1000 <2e-16 *** 0.2292 0.8873 <2.2e-16
b 0.0376 0.0015 24.9400 <2e-16 ***

5 a -0.3144 0.1626 -1.9330 0.568 0.724 0.9135 <2.2e-16
b 0.0011 0.0000 28.8680 <2e-16 ***

6 a -0.7848 0.8744 -0.8970 0.372 0.7272 0.9137 <2.2e-16
b 0.0154 0.0281 0.5480 0.586
c 0.0009 0.0002 4.3460 4.14e-05***

7 a 0.0006 0.0002 2.9250 0.00449** 0.7306 0.8928 6.11e-06
b 2.1144 0.0786 26.8970 <2e-16*** (converge

nce 
tolerance)

Note: ** shows significant at 95% and *** shows significant at 99% confidence level

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In the past, similar models were tested for different 
species (Hawkins, 1987; DFRS, 2006). Laamanen 
et al. (1995) developed a general volume model 
with R2 of 97.1% and standard error of estimate 
of 0.13 for Adabhar, Bara district. The model 
developed for Sal by Sharma and Pukkala (1990) 
for producing general volume table has R2 of 
98.3% and standard error of estimate of 0.13 and 
it is widely used. In both cases, DBH and height 
were used as predictor variables but in this study 
only DBH was used as a predictor variable, which 
alone explained 92.5% variation of the observed 
tree volume. Lower R2 value in this study may be 
due to the use of single predictor variable, DBH. 
SEE is greater in one explained variable than 
two variables (Pukkala et al. n.d.). Other similar 
models were also developed in the past but their 
statistics of fit was not mentioned (Acharya et al., 
2003).

The residuals of all models were analyzed. Due 
to the brevity of space, only graphs of the most 
suitable model (M2) were presented in figure 4. 
It is important to note that only model 2 showed 
homoscadascity and normality of residual 
distribution. The curve of the residuals was not 
seen sharply decrease or increase in the selected 
model. Similar trend was found for standardized 
residuals. Three outliers were found in which two 
outliers had underestimated and one outlier had 
overestimated values. The outlier samples were 
identified and analyzed. Though, removal of 
outliers improves the model reliability but they 
were not removed so as to represent the data from 
all parts of the study area.

Fig. 4: Histogram of residuals and residuals 
versus Ln (DBH)

Over bark stem volume up to top 10 cm and 
top 20 cm top diameter

The distribution of ratios (M8 and M9) against its 
predictor variable clearly indicates that the data 
were distributed negative exponentially (Fig. 5). 
Hence the models with both side logarithms were 
used as in the past in similar cases (Sharma and 
Pukkala, 1990a; Laamanen et al., 1995).

In this case, only logarithmic model was tested as 
used in similar past studies (Sharma and Pukkala, 
1990a ; Laamanen et al., 1995). All the parameters 
of both models were significant (Table 3).

Volume between 10 cm 
to 20 cm top diameter/

Volume up to 10 cm top 
diameter vs DBH

Vol. beyond 10 cm top 
diameter/Total volume 

vs DBH 

Fig. 5: Volume ratio of top 10 and 20 cm 
diameter vs. DBH

In comparison to total stem volume equation, the 
R2 values of both equations were lower, while 
standard error of estimates was higher. It may 
be due to error accumulation from total volume 
model. Sharma and Pukkala (1990a) reported 
78.9 and 74.1 R2 values for the ratio of top 10 and

20 cm over bark diameter, respectively. Similarly 
their standard error of estimates was 0.51 for both 
the equations. These two models were better than 
Sharma and Pukkala (1990a) in terms of R2 and 
SEE (Table 3). However, the fit statistics reported 
by Laamanen et al. (1995) was better than that of 
the study.

Table 3: Values of regression constants, R2 and standard error of the models

Model no. Model a b R2 SEE n
8 ln(V1/V) = a + b*ln D 5.0445 -2.6094 0.8241 0.3469 81

9 ln(V2/Vt) = a + b*ln D 8.221 -2.954 0.7997 0.4253 81

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The residuals of both models (M8 and M9) were 
found to be satisfactory and they were distributed 
evenly without any trend, so these models can be 
recommended. The residuals against DBH are 
shown in figure 6.

Fig. 6: Residuals vs. DBH in two models

Model validation

The definition and method of validation among 
the researchers were not found uniform (Kozak 
and Kozak, 2003; Bellocchi et al., 2010). Most of 
them considered components of fit statistics and 
graphical inspection for validation (Kozak and 
Kozak, 2003; Bellocchi et al., 2010). In addition 
to the analysis of fit statistics, Meehl et al. (2005) 
compared different models, which are used for 
similar purposes. Vanclay (1994) calculated 
the prediction statistics of independent data 
sets for model validation. Iles (2003) strongly 
recommended for checking few independent 
trees to measure accuracy of the volume table 
(cited by Ducey and Williams, 2011). Therefore, 
for validation of these equations, in addition to 
compare different equations, fit statistics and 
graphical inspection, the prediction statistics of 
18 trees were analyzed.

The bias, RMSE and prediction error of models 
for total volume, and volume up to 10 and 20 
cm diameter are almost similar (Table 4). The 
equation for total volume over bark was biased 
to 0.1 8 m3 with RMSE of nearly 0.4 m3, which 
is acceptable since validation data sets were 
fewer than modeled data, resulting losses of the 
information (Kozak and Kozak, 2003). Moreover, 
Hawkins (1987) recommended that the overall 
prediction error should be within 10 to 15 % 
of the actual value. In this study, the prediction 

errors of selected model was found to be within 
6% and were lower than that of Acharya et al. 
( 2003). But some statisticians argue that due to 
fewer validations data sets rather than modeled 
data validation losses the information (Kozak and 
Kozak, 2003).

The predicted values of all suggested models 
were plotted against the actual values of test data 
sets (Fig. 7). There is slightly underestimation of 
volume mainly in large-sized trees (Fig. 7).

Fig. 7: Predicted versus actual volume

The individual error of test data set was evaluated 
by plotting against the diameter (Fig. 8). The 
overall prediction errors of all models were 
within limit but the error percentage of individual 
trees was a bit high in some cases. The models 
consistently underestimated volume of trees over 
the range of DBH (Fig. 8).

Fig. 8: Percent error vs. DBH of individual tested tree

Table 4: Prediction statistics of the models

Model RMSE Bias Prediction error (%)
Total volume over bark 0.3958158 0.1836834 5.812
Volume up to 10 cm over bark 0.395469 0.1827535 5.809
Volume up to 20 cm over bark 0.4009819 0.1759837 5.769

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Conclusion

Among the tested seven models, the recommended 
logarithmic model has the smallest RMSE and 
bias, and higher R2. The model for total stem 
volume is ln V = 8.04674 + 2.26641 ln DBH. 
The models for ratio of volume beyond 10 cm 
top diameter to total stem volume is ln (V1/V) 
= 5.0618 - 2.6094*ln DBH and ratio of volume 
between 10 cm and 20 cm diameter to volume 
up to 10 cm top diameter is ln (V2/Vt) = 8.31144 
- 2.954*ln (DBH). This study recommends for 
application of the equations within the range of 
sample data. Since, the samples are site specific, 
the models should be used cautiously in other 
places of Nepal after validating the models.

Acknowledgements

I highly acknowledge to Dr. Binaya Pasakhala for 
linguistic improvement and valuable comments. I 
also want to thank Mr. Kiran Kumar Pokharel for 
inspiring me to write this article.

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