Description of the Study area The study has been conducted in the remnant the natural forest and existence plantation Moulavi Bazar and Kaptai range under Sylhet South and Rangamati South Forest Division, Bangladesh. The study area contains tropical evergreen and semi-evergreen tree species with natural and plantation stands. The study area Lawachara in Moulavi Bazar range geographically, lies between 24°30'–24°32' N and 91°37'–91°47' E. The soil texture of this area is yellowish brown to reddish brown in colour, loam to clay loam and sandy clay loam of Pliocene origin (Hossain et al. 2019). Soil is acidic, organic matter and fertility levels are generally low. The moist tropical climate of this study area is generally warm and humid, turning cool in the winter. Moderately cool and lovely, and dry conditions exist from mid-November to the end of February while June to September is the time of the highest precipitation. Average maximum and minimum temperatures are 35 °C and 15 °C, respectively (Khatun et al. 2016). The average annual rainfall is 3800 mm, and humidity ranges from 70% to 85% in most parts of the year (Khatun et al. 2016). The study area Kaptai range under Rangamati South Forest Division approximately at the intersection 92°13' E and 22°32' N. The soil of this area is mainly yellowish-brown to reddish-brown loams which grade into broken shale or sandstone at a variable depth. The valley soil is mainly acid loams and clays subject to seasonal flooding (Hossain et al. 2014). Soil is acidic and organic matter content of the topsoil varies from 0.15 to 3.32%. While, total nitrogen concentration varies from 0.03 to 0.24% (Hossain et al. 2014). The average temperature of the study area ranges from 19.9 °C to 28.3 °C, while the average annual rainfall is about 2900 mm and average annual relative humidity is about 78 %. The climate of this region is characterized by mainly three distinct seasons: a hot, humid summer from March to June; a cool, rainy monsoon season from June to October, and a cool, dry winter from October to March (Khatun et al. 2016).
A sampling of trees A sufficient number of sample tree were selected on the basis of girth at breast height (GBH) classes and height classes at random. All sample trees were selected to avoid specimens with broken top, hollow trunk, damage caused by natural calamities or animals, and evidence of suppression or disease. We collected 316 individuals in different GBH class to derive the mathematical volume models and tables of jarul. Sample trees were selected purposively and covered existence wide GBH range and height range. Another 30 sample trees at all girth classes were recollected for model validation.
Measurement of trees Standing sampled jarul trees in representing different girth classes were selected at random for preparation of mathematical volume functions and tables. Trees girth at breast height in cm and total height in meter were measured with diameter tape and Haga-altimeter respectively and the nondestructive method was used to estimate the volume. The girth and bark thickness at one meter intervals in the stem and branches girth were measured by climbing the trees with a ladder. The bark thicknesses of the samples were measured with a bark gauge. To calculate branch volume for bigger branches whose mid-girth ≥30 cm in particular length also taken and smaller branches not taken. The collected fitting data set were categorized on the basis of GBH and height of the trees.
Compilation of data Volumes of all sections except top and bottom section were determined by using the mean cross-sectional areas of the two ends of each section following Smalian’s formula cubic volume = [(B+b)/2]L, where B= the cross-sectional area at the large end of the log, b= the cross-sectional area at the small end of the log, and L= log length. In determining the volume of bottom sections, the formulae used for calculating the volume of a cylinder was considered. Assuming the top section as cone the volume was computed to one third of the cylindrical volume of the portion. We considered the top end diameter measurement for each tree as the base diameter of the cone. In computing the under bark volume of the tree the volume of top section i.e. cone was ignored. The total volume of the tree is the sum of the volume of all sections and branches volume found in a tree. The individual tree total volumes (V), GBH (G) and total height (H) were variable in regression techniques using various functions and transformations as required in the models.
Computation of volume function Multiple regression analysis techniques were used to select the best suited model equations. The following 21 models (Clutter et al. 1983; Bi Hamilton 1998; Latif and Islam 2014 and Islam et al. 2017) were tested to select the equation of best fit with different variables.
Where: V = total volume over bark in cubic meters, G = girth at breast height in centimeters, H = is total height in meters, a is the regression constant and b, c and d are regression coefficients. The logarithmic functions are to the base e. Following original and transformed variables were used to select the best suited regression models: Dependent variables: V, Log (V), Independent variables: G, G2, G-1, G-2, H, H-1, GH, G2H, Log (G), Log(H) The dependent variables mentioned above were regressed with the independent variables. The equations of the best fit based on the highest multiple coefficients of determination; F-ratio and lowest residual mean square and AIC value statistic were chosen. Models for estimation of the total volume over bark were selected and the conversion factors to estimate under bark volume and volume to top end girths of approximately 30, 35, 40 and 45 cm were also estimated.
Model validation The best suited models were tested with a set of data recollected from 30 trees of different diameter class and compiled in the same procedure as earlier. The actual volumes of these trees were collectively compared with the corresponding volume predicted by the selected models. The independent tests for validation were chi-square test of goodness of fit, paired t-test and Percent absolute deviation (%AD). This was also compared with 45 degree line test by plotting the observed values and the predicated value in the graph.
Data Analysis Data collected were organized and screened (removing the outliers) for analysis. Descriptive statistical analysis was further carried out in order to summarize the data. All analysis carried out were conducted using MS Excel 2013, SPSS 17 Inc and EViews (Quantitative Micro Software, LLC) statistical package version 9.