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ISSN : 1598-5504(Print)
ISSN : 2383-8272(Online)
Journal of Agriculture & Life Science Vol.51 No.4 pp.55-64

Comparative Analysis of Simple Volume Models for Japanese Larch(Larix kaempferi) Species in the Central Region of South Korea

Sunjoo Lee, Nova D. Doyog, Young Jin Lee*
Department of Forest Reosurces, Kongju National University, Yesan-eup, Yesan-gun, Chungnam-do, 32439, Korea
Corresponding author : Young Jin Lee
20170525 20170710 20170714


This study was conducted to develop a simple volume model for the Larix kaempferi species in the Central Region of South Korea. General volume model forms were used and validated using a subset of the data collected for the L. kaempferi species. A total of 550 trees were collected in the different L. kaempferi stands through destructive sampling. The dataset was randomly divided into two: 80% for initial model development(seven volume models) and 20% for model validation. The 100% dataset was used for the final model fitting. Evaluation statistics including bias(Ē), absolute mean deviation(AMD), root mean square error(RMSE), coefficient of determination(R2), and the Akaike Information Criterion values(AIC) and weight (AICw) were used to assess performance of the different volume models. Rank analysis was employed and the first five best volume models were chosen for the model validation and final model fitting. The evaluations showed that volume model 4(V=aDBHb*Hc) had the best performance, while volume model 3(V=aDBH+bDBH2), which has a single variable, had the poorest performance.


    Korea Forestry Promotion Institute


    Volume models are used to estimate the average contents for standing trees of various sizes and species. The principal variables ordinarily associated with standing tree volume are diameter at breast height and tree height. Tree volume models that are based on the single variable of diameter at breast height are commonly referred to as local volume models or single volume models while models that require the combination of diameter at breast height and the tree height are referred to as standard volume or double entry volume models. Assumptions regarding the inferiority of local volume models over standard volume models are not necessary, particularly when the local equation is derived from a standard volume equation(Avery & Burkhart, 2002). On the other hand, previous research has shown that volume models having two predictors performed better than volume models with only one predictor(Lumbres & Lee, 2013; Seo et al., 2015). General volume models could be in the form of a linear model, constant form factor, single variable, quadratic, logarithmic, general logarithmic, or transformed variable(Clutter et al., 1983; Husch et al., 2003).

    Stem volume estimations are very important for forest managers, particularly in forest inventories. Forest inventories have often been used as starting points for the estimation of biomass and carbon storage in the forest of South Korea and forest inventories are very important for the timely monitoring and reporting the forest resources. South Korea has been routinely monitoring forest resources, especially the growing stock, since the establishment of the National Forest Inventory in 1972.

    One of the most commonly used procedures in estimating growing stock is developing equations based on the relationships between volume, diameter at breast height and total height(Akindele & LeMay, 2006) through regression analysis(Avery & Burkhart, 2002) and by allometric equation development(Teshome, 2005; Akindele & LeMay, 2006; Gonzalez-Benecke et al., 2014). Volume equations are very important in estimating the forests’ aboveground biomass by transformation of volume to dry weight, using density and biomass expansion factors for the biomass determination of the whole tree(Fang & Wang, 2001; Lehtonen et al., 2004; Tobin & Nieuwenhuis, 2007).

    As Larix kaempferi is among the main coniferous species of South Korea(Kang et al., 2014; Lee et al., 2014; Lee et al., 2015), it is one of the major contributors to the national forest stock of South Korea. Thus volume monitoring and reporting of this species is of significance. Although the species has been the subject of previous research(Son et al., 2002; Kang et al., 2014; Lee et al., 2014; Lee et al., 2015) the stem volume equation, with the use of a simple volume model has not yet been developed. This study aims to develop the stem volume equation of the L. kaempferi species in the Central Region of South Korea and to validate the performance of the volume model with different forms.

    Materials and Methods

    1.Study site

    The study sites were located in the Larix kaempferi stands in the Central Regions of South Korea, particularly in Bouen, Buyeo, Cheongju, Danyang, and Yeongju. The central zone stretches to 40°N in the east, 39°N in the west, and 38.5°N in the interior areas(Fig. 1). The region is hot and humid during the summer, and have cold and dry climate during the winter, and has a temperate climate.

    2.Data collection

    A total of 550 trees were felled for research purposes. The trees were selected to represent the range of diameters at breast height classes of South Korea: 53 trees represented diameter class 1(<6 cm); 146 trees diameter class 2(6 to 16 cm); 227 trees diameter class 3(18 to 28 cm) and 124 trees diameter class 4(>30 cm).

    The sampled trees had diameters at breast height (DBH, cm) ranging from 0.60 to 47.90 cm, with an average of 23.79 cm. The tree height(H) ranged from 2.0 to 33.00 m, with a mean of 21.23 m. DBH was measured at 1.20 m above the ground, following the recommendation of the Korea Forest Research Institute(KFRI, 2010), using a standard diameter tape. H was measured directly using standard measuring tape, after cutting the tree to a stump height of 0.2 m. The diameter outside bark(d, cm) was also measured at specific heights from 0.2 m of the tree section, 0.7 m, 1.2 m and then at two meters intervals. A total of 6, 415 paired d and heights at specific diameter(h, m) were recorded.

    3.Data analysis

    Stem section volumes were calculated using the Smalian’s formula except for the top section, for which the cone formula was used(Avery & Burkhart, 2002). The total stem volume for a single tree was determined by summing each of the stem section volumes. The scatter plot of DBH against tree volume was plotted(Fig. 2) to visually examine each sample tree of the L. kaempferi species to detect possible anomalies in the data(Özçelik & Göçeri, 2015) and to increase efficiency(Bi, 2000).

    The data was divided into two sets through random sampling: 80% of the data were used for initial model fitting, while the remaining 20% were used for model validation. The total dataset(100%) was used for the final model fitting. Descriptive statistics of the dataset are presented in Table 1.

    Seven commonly used model forms for estimating individual stem volumes were selected as candidate models for the initial model fitting(Table 2). The volume models were selected from forestry literature (Clutter et al., 1983; van Laar & Akça, 1997; Avery & Burkhart, 2002; Husch et al., 2003) and have been used for several studies(Dela Cruz & Bruzon, 2004; FAO, 2005; Segura & Kanninen, 2005; Lumbres & Lee, 2013; Seo et al., 2015). The volume models are in the forms of a linear model, constant form factor, single variable, quadratic, logarithmic, general logarithmic, and transformed variable(Clutter et al., 1983; Husch, 2003).

    The Statistical Analysis System(SAS) NLIN procedure (SAS Institute, 2004) was used for the determination of the different values of the coefficients of each equation.

    The performance of each seven volume models was evaluated using various statistics of fit including bias(Ē), absolute mean deviation(AMD), root mean square error(RMSE), coefficient of determination(R2), and the Akaike Criterion Information(AIC) of Akaike (1974). The raw AIC values were weighted through the Akaike weight(AICw). The AICw is used to evaluate how much statistical importance is attached to a difference in the AIC values between the best model and the next best model(Wagenmakers & Farrell, 2004). Just like the AMD and RMSE, AIC choose the model with lowest values as the best while for the AICw, the model which has the highest value is determined to be the best model, the same with R2. The statistics of fit for each equation is presented in Table 3.

    The ranking of methods(Poudel & Cao, 2013) was employed in order to determine the best model. The difference of the ranking of methods used in this study with that of the traditional standard or ordinal ranks is that the ranking of method shows not just the order of the models but also the magnitude of difference between the models. The first five volume models with the lowest value were chosen for the model validation and final model fitting. The rank analysis is in the form of:

    R i = 1 + [ ( m 1 ) * ( S i S min ) S max S min ]

    where Ri is the relative rank of model i(i=1, 2,… m); Si is the goodness-of-fit statistic produced by model i; Smin is the minimum value of the good-of-fit statistic; and Smax is the maximum value of the goodness-of-fit statistic.

    Numbers 1 and m was regarded as the best and worst rank, respectively, for each statistical criterion. In this ranking system, the order and also the magnitude of difference of the models are taken into consideration. For example, relative ranks of 1.00, 1.05, 1.07, 1.11, 1.12, 4.49, and 7.00 in the case of seven models suggest that the models fall into groups that are separated by a large gap.

    Although there is no set of specific standards or tests that can be easily applied to determine the appropriateness of a model, a minimum validation procedure must be established to ensure reliability and reasonable performance of a new model(Huang et al., 2003). The remaining 20% of the dataset was used for model validation. For the final model fitting with the use of the validated volume models, the combined data or the 100% dataset were used and rank analysis was performed to determine the best volume model for Larix kaempferi species in the Central Region of South Korea.

    Aside from the fit statistics criteria used for evaluation of the models, a simple linear regression (Zar, 1999) was used to compare the observed and predicted stem volumes of the Larix kaempferi species in the Central Korea. The observed and predicted stem volumes were related according to the following linear model: Predicted stem volume=b0+b1* Observed stem volume. If the simple volume model correctly estimated the stem volume of a tree, then the intercept(b0) would not be significantly different from zero and the slope(b1) would not be significantly different from one. A simultaneous F-test was also conducted to evaluate the hypothesis: Ho: (β0, β1) = (0, 1), Ha: (β0, β1) ≠ (0, 1). Simple linear regression and the simultaneous F-test had already been used by previous researches to evaluate the performance of a model including Lee & Coble, 2002.

    Results and Discussion

    1.Initial model fitting and model validation

    The initial model fitting was carried out with the seven volume models using the 80% of the dataset. These were evaluated using fit statistics including Ē, AMD, RMSE, R2, AIC, and AICw. The results showed that the volume models with two variables, in the form of a linear model, constant form, quadratic, logarithmic, and Honer’s transformed variable, had better fit statistics values than the volume models with a single variable(Table 4). The best five volume models, based on rank analysis, were chosen for model evaluation using the 20% dataset and final model fitting using the 100% dataset. Codes models 1 to 5 were assigned to the five chosen models. Four of the models use two variables, DBH and H. The model validation showed that model 1 provided the least Ē and models 3, with a single variable, and 4 over-predicted the volume. Models 2 and 5 provided an under-predicted result. The AMD ranged from 0.023(model 4) to 0.065(model 3); the RMSE ranged from 0.035(model 4) to 0.090(model 3); the R2 ranged from 0.978(model 3) to 0.997(model 4), while the AIC ranged from -725.25(model 4) to -522.59(model 3). The AICw, on the other hand, had values that ranged from <0.001(model 3) to 0.997 (model 4). The overall rank analysis of the model evaluation showed that model 4, which is in logarithmic form, had the best value. The volume model with a single variable(model 3) had the poorest value of the five. The result of the relative ranks of the model using the validation dataset based on fit statistics is presented in Fig. 3. The model with the smallest area inside the box represents the best model. Model 4 had the smallest area followed by model 1, model 5, model 2, and model 3, respectively.

    The Ē of the five models was plotted against the DBH class of South Korea(Fig. 4) for further evaluation. Fig. 4 showed that the model 4 had a Ē value of almost zero within the <6 cm and 6 to 16 cm DBH classes and provided underestimated results within the 18 to 28 and >30 cm DBH classes. For the other models, model 1 had an overestimated result within the diameter classes <6 and 6 to 16 and underestimated result within the diameter classes 18 to 28 and >30. Model 2 on the other hand, gave underestimated results in all diameter classes while model 3 overestimated the diameter classes <6, 6 to 16 and >30, and underestimated diameter class 18 to 28. Model 5 underestimated diameter classes <6, 6 to 16, and 18 to 28 but overestimated the >30 diameter class.

    2.Final model fitting

    The five models were refitted to the combined or the 100% dataset for the final model fitting. The resultant parameter estimates are provided in Table 5. Using the same fit statistics as before, the performance of the volume models was evaluated as shown in Table 6. Model 4 remained the number 1 rank among the five models with a Ē of 0.002 m3, AMD of 0.103 m3, RMSE of 0.172 m3, R2 of 0.924, and an AIC value of -1906 with an AICw of 1.0. Model 4 had the second nearest zero Ē, the second lowest AMD value, the lowest RMSE, and the highest R2. The lowest E was model 1(0) while the third lowest Ē was model 3(-0.004 m3), followed by models 2 (0.027 m3) and 5(0.33 m3). Only model 3 overpredicted the volume. Model 2 had the lowest AMD (0.102 m3) and the third is model 5(0.104 m3) followed by models 1(0.110 m3) and 3(0.143 m3). For the RMSE, the best was model 4. Model 1 had a value of 0.185 m3 followed by models 5, 2 and 3 with values of 0.189, 0.190 and 0.218 m3, respectively. For R2, where the higher value the better the performance, model 4 again had the best performance (0.924). Model 5(0.908) was the second best, followed by model 2(0.907), model 3(0.877) and model 1(0.798), respectively. For the AIC value(AICw), the models had the following values arranged from lowest to highest: model 1: -1824(<0.001), model 5: -1806 (<0.001), model 2: -1802(<0.001), model 3: -1651(<0.001).

    The overall ranking showed that the best performance was given by model 4(1.053), followed by model 2(3.001), model 1(3.014), model 5(3.114), and model 3(4.265), respectively. Although model 3, which uses only one predictor(DBH), had the poorest performance, the use of this model is still very important in cases where only DBH is available. Having only the DBH in a field inventory can be unavoidable considering the physical structure of the forest of South Korea. In addition, measuring height in the field is always accompanied by errors, as tree height measurement instruments do not have 100% accuracy. High accuracy is only guaranteed if a direct, destructive measurement is taken. Measuring DBH in the field is easy to do directly, so high accuracy is possible.

    In line with other Larix species in South Korea, the logarithmic model(V=aDBH2*H) had been suggested to be the best model for predicting the stem volume of L. leptolepis stand in Jinan, Chonbuk, South Korea(Jeon et al., 2007).

    The relationships of the observed and the predicted stem volume were plotted using a simple linear regression(Fig. 5). The stem volumes were found to be related according to the following linear model: Predicted stem volume=0.0794+0.8631*Observed stem volume. Model 4 gave an underestimated results in volumes <0.45 m3 and an overestimated in volumes >0.55 m3. The overestimation became greater as the volume increased. The results show that most of the observations are <1.00 volume, which agrees with the average volume of 0.8190 m3 for a single Larix kaempferi tree in South Korea(Kang & Son, 2016). Additionally, for model 4, a simultaneous F-test was conducted to evaluate the hypothesis: Ho: (β0, β1) =(0, 1), Ha: (β0, β1)≠(0, 1). A p-value of 0.5131 was computed indicating that there is no significant difference between the observed and predicted volumes.

    All four stem volume models with two variables, DBH, and total height, showed better predictive capability than the single variable volume model when estimating the stem volume of the L. kaempferi in the Central Region of South Korea. Model 4 consistently performed best, from initial model fitting to model evaluation and finally the final model fitting. On the other hand, model 3, the single variable model, is still recommended in situations where total height is unavailable. Volume models with only a single variable should not be totally ignored considering that the information gathered in forest inventories includes directly measured tree DBH but tree heights that are only remotely measured. Tree height is difficult to measure remotely with accuracy. This inaccuracy can result in biased estimates when tree height is included as an independent variable in volume and biomass models. Considering the sources of error, it is necessary to consider volume models using single variables such as DBH which can be measured accurately in the field.


    This study was carried out with the support of R&D Program for Forest Science Technology(Project No. 2013069D10 1719 AA03, 2014068E10 1719 AA03) provided by Korea Forest Service(Korea Forestry Promotion Institute).



    Locations of the study sites in the central region of South Korea.


    Diameter at breast height against the stem volume of the L. kaempferi species.


    Rank analysis of the simple volume models using the validation dataset(The model with the smallest area inside the box represents the best model).


    Bias against the volume models against the diameter classes of South Korea using the validation dataset.


    Comparison of observed and predicted stem volumes of L. kaempferi using simple linear regression.


    Statistics for the L. kaempferi data used for the comparative analysis of volume models

    where: DBH=diameter at breast height; H=total height; V=stem volume; n=number of observations; Mean with standard deviation in parenthesis.

    Simple volume models used for the initial model fitting(80% dataset)

    Note: V=volume(m3); DBH=diameter at breast height(cm); H=total height(m); a, b, c and d are parameters to be estimated.

    Fit statistics used to evaluate the performance of the different stem volume models

    Note: where Yi= observed volume of the ith tree; Ŷi= predicted volume of the ith tree; Ȳ= mean of the Yi; n= number of observations in the dataset; k= number of estimated parameters; RSS= residual sum of squares; ln= natural logarithm; Δi= difference between the AIC corrected of the best fitting model and that of the i(model), R= relative likelihood of all the models; and Δr= normalized relative likelihoods of the candidate models.

    The results of the rank analysis using the model validation dataset(20%)

    Note: V= stem volume(m3), H= tree height(m), DBH= tree diameter at breast height(cm), and the a, b and c= parameters to be estimated.

    Parameters estimates of the equations used for the final model fitting using the 100% dataset

    Note: Parameters estimates with standard error in parenthesis.

    Fit statistics of the equations used for the final model fitting using the 100% dataset

    Note: Fit statistics with relative rank in parenthesis.


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