Development of an allometric model to estimate above-ground biomass of forests using MLPNN algorithm, case study: Hyrcanian forests of Iran

Document Type : Research Paper


University of Tehran


This research develops an allometric model for estimation of biomass based on the height and DBH of trees in the Hyrcanian forests of Iran. An accurate allometric model reduces the uncertainty of allometric equation in biomass estimation using radar images. In this study, 317 trees were selected randomly from the 4 different dominant tree species for the development of an allometric model covering the wide range of DBH and height classes. The selected trees were measured by fieldwork in different parts and then volumes of these parts were calculated separately. Total volume of tree is obtained by the summation of these volumes. Twelve commonly used allometric models, three generalized models and a proposed model were tested and the most suitable model was selected based on some of the commonly measured statistical parameters including coefficient of determination (R2), Root-Mean-Square Error (RMSE) and Mean Error (ME). We showed that the biomass estimation accuracy was improved in a multilayer perceptron neural network (MLPNN) when density of wood, DBH and height were used in combination compared to estimating the biomass by current allometric models. The RMSE value was decreased when the proposed method was used (RMSE =0.163 Mg and R2=0.986) in comparison with Chave model, as the best current method (RMSE =0.404 Mg and R2=0.957) in this paper.



Development of an allometric model to estimate above-ground biomass of forests using MLPNN algorithm, case study: Hyrcanian forests of Iran

A. Sharifi1, J.Amini*1, F. Pourshakouri2


1. Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

2. Department of Forestry and Forest Economics, Faculty of Natural Resources, University of Tehran, Karaj, Iran

* Corresponding author’s E-mail:

(Received: Feb. 02. 2016 Accepted: May. 10. 2016)


This research develops an allometric model for estimation of biomass based on the height and DBH of trees in the Hyrcanian forests of Iran. An accurate allometric model reduces the uncertainty of allometric equation in biomass estimation using radar images. In this study, 317 trees were selected randomly from the 4 different dominant tree species for the development of an allometric model covering the wide range of DBH and height classes. The selected trees were measured by fieldwork in different parts and then volumes of these parts were calculated separately. Total volume of tree is obtained by the summation of these volumes. Twelve commonly used allometric models, three generalized models and a proposed model were tested and the most suitable model was selected based on some of the commonly measured statistical parameters including coefficient of determination (R2), Root-Mean-Square Error (RMSE) and Mean Error (ME). We showed that the biomass estimation accuracy was improved in a multilayer perceptron neural network (MLPNN) when density of wood, DBH and height were used in combination compared to estimating the biomass by current allometric models. The RMSE value was decreased when the proposed method was used (RMSE =0.163 Mg and R2=0.986) in comparison with Chave model, as the best current method (RMSE =0.404 Mg and R2=0.957) in this paper.


Key words:Allometric model, Biomass, DBH, Height, Hyrcanian forests, MLPNN.


Measurement of Above-Ground Biomass (AGB) is necessary for quantifying carbon and biomass storages and also for comparing the result of remotely sensed methods in biomass estimation (Chave 2005). The methods of biomass estimation can be divided into two groups i.e. direct and indirect (Overman et al. 1994). The direct method involves the complete harvesting of sample plots and subsequent extrapolation to an area unit (Araujo et al. 1999). The indirect method aims to construct a functional relationship between tree biomass and other tree dimensions, such as stem diameter, height and wood density, by means of regression analysis (Brown et al. 1989). Since the direct method is very time consuming, costly and completely destructive, and biomass expansion factors (BEFs) are complex in nature, field observations of biomass are normally based on allometric models that approximate the biomass of the tree component or the total biomass of single trees according to easily measured variables, such as DBH (Diameter at Breast Height) or height (Brown et al. 2001). The term allometry means the relationship between part of an organism and its whole (West 2009).

Many studies have already developed allometric equations for different purposes, different regions, and different species, for example species-specific allometric models (Saint-Andre et al. 2004; Cole & Ewel 2006; Arevalo et al. 2007), generalized allometric models (Crow 1978; Overman et al. 1994; Araujo et al. 1999; Ares & Fownes 2000; Segura & Kanninen 2005; Chave et al. 2005; Henry et al. 2011), allometric models for tropical forests (Brown 1997; Brown et al. 1998; Murali et al. 2005; Djomo 2011), simplified allometric models (Montagu et al. 2005; Ebuy et al. 2011), allometric models for regional and global level biomass estimations (Fang et al. 1998; Fang & Wang 2001; Genet et al. 2011; Iranmanesh et al. 2012; Sohrabi & Shirvani 2012; Parsapour et al. 2013; Vahedi et al. 2013), and there have even been studies on the uncertainty of using allometric models (Brown 1997; van Breugel 2011). All of these models have been effective for specific purposes so far, and there is no single optimal model which can provide a good calibration function for the estimation of AGB for all tree species and for all climatic regions because the calibration coefficients of allometric models are reported to vary with tree species, stand age, site quality, climate, and the stocking of stands (Ketterings et al. 2001).

Some studies (Brown et al. 1989; Arevalo et al. 2007; Picard et al. 2012) found that the allometric equation could be generalized to make it useful for local to regional levels, but they also recommended that an allometric model should be species-specific or site specific (Arevalo et al. 2007) for its effective use, or calibration of coefficients be performed before its use in other places (Brown et al. 1989).

This paper has two objectives: 1) Development of an allometric model based on the dominant tree species in Hyrcanian forest of Iran in order to reduction of the uncertainties from generalized allometric equations, and 2) Improve the accuracy of estimated biomass using a multilayer perceptron neural network (MLPNN).



Study area

The study area is located at Hyrcanian forests of Iran around the Asalem forest (Fig. 1). Hyrcanian forests of Iran are high forest and are managed using selection system method.  The natural forest vegetation is temperate deciduous broadleaved forest that the main dominant trees of this forest are Fagus orientalis, Alnus serrulata, Carpinus betulus, and Ulmus glabra. Fig. 1 shows the coordinates of this area that is considered as one of the rainiest areas (the mean annual precipitation: 300 mm, the highest annual range of temperature: 1-38 oC) in Iran which is a suitable habitat for the broadleaf species. This research was conducted in three parcels with 171 hectares area. Study areas were extended in range of 600-950 m from sea level.



Fig. 1. Position of Asalem forest with broadleaf trees in North of Iran.


Data Collection

Samples were randomly selected because the exact volume of the trees should be calculated. In some diameter classes specially the lower one, there were not sufficient cut trees in order to be used for sampling from the diameter classes. Hence completely random method was used instead of diagonal and height classes for sampling. Regarding total cost of inventory, 317 trees were selected. Trees were retrieved in the nature and desired characteristics were measured (Height and DBH) for this study. Minimum error and inventory costs are two determinant factors in samples components numbers (West 2009). Field data collection was based on a stratified sampling methodology. Stratified sampling is a probability sampling technique wherein the researcher divides the entire population into different subgroups or strata, then randomly selects the final subjects proportionally from the different strata. It is important to note that the strata must be non-overlapping. Having overlapping subgroups will grant some individuals higher chances of being selected as subject. This completely negates the concept of stratified sampling as a type of probability sampling. Trees were measured for each tree species in order to get a desired precision level (in this case, an error level of 10% expressed as the 95% confidence interval of the mean). For determination of total volume calculation of trunk over 20 cm diameters, firewood and stump volumes are necessary. Total volume of tree is obtained from the summation of these volumes. Volume of Trunks and branches were calculated using Smalian formula in 2 m pieces (Eq. 1):




where Vs is trunk volume (m3), l is piece length (m), and dL and dU are diagonals diameter of trunk (m3) at the beginning and the end of 2m piece, respectively.

Firewood volume of branches was divided into 1m pieces and their diameter was measured in the middle, then the volume of each branch was calculated by Huber formula (Eq.2):


where Vf is firewood volume, l is piece length, and dM is diagonal middle diameter of branch.

Stump volume was calculated from another form of Smalian formula (Eq. 3) as follows:


where Vt is Stump volume, H is piece length, and dL is diagonal diameter of tree in cut location. Finally total volume is the sum of trunk, firewood and stump volumes (West 2009).

Relating Volume to AGB

In this paper forest volume data for calculation of biomass is used. Required data for this method is the volume for sample trees that was determined in section 2.3. AGB in megagram (Mg) per hectare (ha) is estimated by Eq. 4.



where AGB is above-ground biomass, V is volume, and WD is basic density of the wood.

If we couldn’t calculate volume, another way was estimation of volume by multiplying trunk volume (Vt) into BEFs (Brown & Lugo 1992).

Wood density is defined as the mass of dry wood per green wood volume unit. Its unit is Mg per m3. In 1992, an equation (Eq. 5) was developed by Reyes (1992) to convert wood density with 12%  moisture content into wood density based on dry mass per green volume (Bergès et al. 2008).



where WD is average density of the wood, and X is wood density in 12%  moisture.

Wood densities in 12% moisture for Fagus orientalis, Alnus serrulata, Carpinus betulus, and Ulmus glabra species are 0.633, 0.535, 0.755, and 0.55, respectively (Kiaei & Samariha 2011). Table 1 summarizes the ground measurements and resulting calculations.



The relationship between the physical parameters (DBH or/and height) and the AGB of all sampled trees needed to be established in order to estimate the AGB of non-harvested trees. Although there are several empirical methods available, this study established this relationship using allometric equations because an allometric model is a useful tool which can approximate the AGB of single trees according to easily measured variables, such as diameter at breast height (DBH) or height (H) (Brown  1997(.

The most common allometric model in biomass studies is the power function (Brown, 1997) as follows:



AGB = a  (DBH) b                                           


where AGB is the total above-ground biomass, DBH is the diameter at breast height, a and  b are the scaling coefficient and scaling exponent, respectively. In most cases, the variability of AGB is largely explained by the variability of DBH. However, the values of a and b are reported to vary with species, stand age, site quality, climate, and stocking of stands (Ketterings et al. 2001), and the most common problem with allometric equations is that the raw data are non-linear and tend to be heteroscedastic. As such, the equation 6 cannot satisfy the relationship between AGB and the DBH. Hence, the standard method for obtaining estimates for the coefficients a and b is by the least-squares regression for DBH and H measured from destructively sampled trees, and the form of the model will be as follows (7):



ln (AGB) = ln (a) + b ln (DBH)                    


This transformation is appropriate when the standard deviation of AGB at any DBH increases in proportion to the value of DBH in many cases, log-transformation of real data results in homoscedasticity of the dependent variable AGB, a prerequisite for regression methods. However, even though the linear relationship of equation 7 mathematically equivalent to equation 6, they are not identical in a statistical sense and this transformation introduces a systematic bias that is generally corrected using a correction factor estimated from the standard error, but it has become conventional practice in allometric studies (Niklas 2006).

Different types of regression models and combinations of parameters have been used including ordinary least squares on log-transformed data (Overman et al. 1994; Montagu et al. 2005; Arevalo et al. 2007), weighted least-squares regression on log-transformed variables (Arevalo et al. 2007), and non-linear regression (Saint-Andre et al. 2004; Murali et al. 2005; Arevalo et al. 2007).

However, apparently there is no single optimal regression model that can give a good calibration function for the estimation of AGB because the values of coefficients are varied based on many factors (Ketterings et al. 2001). Considering this situation, this paper tested different types of regression models for north of Iran including linear and non-linear, but most emphasis was placed on the methods of Brown et al. (1989), Brown et al. (1997) and Chave et al. (2005) as the work of these researchers used in recently remote sensing researches for estimation of biomass from SAR images (Amini & Sumantyo 2009; Englhart et al. 2011; Saatchi et al. 2011; Carreiras et al. 2012; Englhart et al. 2012). Finally, proposed method was done with an MLPNN. A multilayer neural network is made up of sets of neurons assembled in a logical way and constituting several layers.

Three distinct types of layers are present in the MLPNN.

The input layer is not itself a processing layer but is simply a set of neurons acting as source nodes which supply input feature vector components to the second layer.

Typically, the number of neurons in the input layer is equal to the dimensionality of the input feature vector.

Then, there is one or more hidden layers, each of these layers comprising a given number of neurons called hidden neurons.

Finally, the output layer provides the response of neural network to the pattern vector submitted in the input layer. The number of neurons in this layer corresponds to the number of classes that the neural network should differentiate (Haykin 1999).

The neural network that is used in this paper is arranged in layers as follows. The number of neurons in the output layer is taken to be equal to the estimated biomass. The input layer contains three neurons corresponding to the number of attributes in the input vectors. The input vector to the network for pixel i of the data sets is of the form νios = {νi1, νi2, νi3}, where vi1 belongs to the height, vi2 belongs to DBH, and vi3 belongs to wood density.

After the determination of the input layer, the number of hidden layers required, as well as the number of neurons in these layers, still needs to be decided upon. An important result, established by the Russian mathematician Kolmogorov in the 1950s, states that any discriminate function can be derived by a three-layer feed forward neural network (Haykin 1999).

Increasing the number of hidden layers can then improve the accuracy of the fitting model, picking up some special requirements of the recognition procedure during the training, or enabling a practical implementation of the network. However, a network with more than one hidden layer is more prone to be poorly trained than one with only one hidden layer.

Thus, a three-layer neural network with the structure 3-2-1 (three input neurons, two hidden neurons and one output neurons) is used to fit a model to the data sets.Training the neural network involves tuning all the synaptic weights so that the network learns to recognize the given patterns or classes of samples sharing similar properties. The learning stage is critical for effective modeling, and the success of an approach by neural networks depends mainly on this phase.



Table 1. Characteristics of field data.


No. of trees

Mean height


Mean DBH


Mean Volume (m3)

Mean AGB


Fagus orientalis






Alnus serrulata






Carpinus betulus






Ulmus glabra






† Diameter at breast height

‡ Above-ground biomass


Independent Validation of AGB Estimation Models

Testing the goodness of fit of each model is very important in order to find the most suitable model for AGB estimation. The statistics of accuracy assessment included the Root-Mean-Square Error (RMSE), and the relative errors to the mean value of AGB. The value of the RMSE is affected by large errors which give disproportionately large weights because of the squaring process. The ME is a signed measure of error which indicates whether the predicted AGB is biased.

The predicted AGB is underestimated (UE) with a negative ME and overestimated (OE) with a positive ME. Additionally, the coefficient of determination (R2) was calculated as the square of Pearson’s correlation coefficient.


The correlations coefficient (r) between DBH, and height with AGB were 0.93 and 0.86, respectively (Fig. 2). Thus using these parameters together for modeling may lead to better result. The results of models are shown in Table 2. The simple regression models (models 1, 2 & 3) were not found to be suitable. The power-function models (models 4, 5 & 6) displayed very good performances. The log-transformed models (models 7, 8 & 9) were found to be effective for AGB measurement because of the fact that log-transform has the potential to correct for the heterogeneous variance of AGB. The methods of Brown et al. (1989), Brown et al. (1997) and Chave et al. (2005) (models 10, 12 & 14, respectively) using DBH, height, and wood density achieved very good accuracies.

Although we achieved better result than these models when we used sample data of north of Iran for calibration of coefficients of these equations (models 11, 13 & 15, respectively). From the 15 models tested, model 15, or the calibrated Chave et al. (2005) model was found to give the best fit considering all of the statistic parameters among current methods. A fit of about 95.72% and a RMSE of 0.404 Mg were obtained using this model. This is very satisfactory in comparison with other allometric model but we developed a novel method based on neural network that had the best result among all current models.

The neural network is trained by using a back-propagation rule (Paola 1995). The numbers of training data are 222 samples (70% of all samples) with their wood density. The set of training patterns is presented repeatedly to the neural network until it has learned to recognize them. A training pattern is said to have been learned when the absolute difference between the output of each output neuron and its desired value is less than a given threshold. Indeed, it is pointless to train the network to reach the target outputs of zero or one since the sigmoid function never attains its minimum and maximum. The network is trained when all training patterns have been learned. Once the network is trained, the weights of the network are applied on the data sets to fitting model. The result of the neural network is shown in Table 2 in comparison with current models. For accuracy assessment and calibration, 95 samples (30% of samples) were selected as the test samples, randomly. The values 98.6% and 0.163 Mg are achieved for R2 and RMSE, respectively. It’s the best result among current methods for biomass estimation. In comparison between the MLPNN and current models, the advantages of MLPNN that is used in this paper are as follow: 1) It can accept all kinds of numerical inputs, whether these conform to statistical distribution or not. 2) It can recognize inputs that are similar to those which have been used to train them. 3) Because the network consists of a number of layers of neurons, it is tolerant to noise present in the training patterns. Table 2 shows that highest R2 and lowest RMSE are related to models 1, 2 & 3, respectively among simple regression models. Better accuracy of model 2 (R2 = 0.935, RMSE = 0.577 Mg) compared to model 1 is due to DBH and height parameters used for modeling whereas in model 1 (R2 = 0.873, RMSE = 0.696 Mg) only DBH was applied. Probably the use of summation between DBH and height parameters leads to accuracy reduction in model 3 (R2 = 0.896, RMSE = 0.631 Mg) than model 2. As shown in Fig. 3, the density of points is high around y=x line for model 2 where proximity of these points to this axis indicates low ME, OE and UE in this model compared to models 1 & 3. Values of this error for each model are shown in Table 2, separately. Model 12 in Table 2 represents the Brown et al. (1997) model which is a second-order polynomial according to DBH but in contrast to models 1, 2 & 3 has lower R2 (0.827) and higher RMSE (0.812 Mg). It shows how the use of a univariate generalized model with no calibrated coefficients can cause error in AGB estimation. The model uncertainty greatly increases when this relation is applied as a source. Model 12 coefficients calibration based on the local data leads to model 13 which increases R2 up to 0.909 and decreases RMSE to 0.588. As shown in Table 2, Errors of model 13 declined sharply compared to model 12.

Fig. 3 indicates that density of points around identity line (y=x line) in model 12 is low which increases in model 13 after coefficients calibration. Results reveal that coefficients calibration of Brown et al (1997) can increase accuracy of AGB estimation. However, sufficient ground data of different species should be existed for calibration. In general, the optimal model (model 13) is proposed for those forests that have not feasibility to measure trees height due to their age or high density. Thus, only by measuring trees DBH and applying model 13, desirable accuracy for AGB estimating can be obtained.

Among models based on the power function, the highest R2 and lowest RMSE are related to models 5, 4 & 6, respectively. Better accuracy of model 5 (R2=0.911, RMSE=0.579 Mg) compared to model 4 (R2=0.909, RMSE=0.590 Mg) is due to the lack of power factor for offset, while accuracy is reduced in model 6 by eliminating offset. The ME, OE and UE of each model is represented in Table 2. In general, regarding to the use of DBH in power functions singly, acceptable accuracy was obtained for these relationships. Model 10 in Table 2 is the Brown et al (1992) model which is a power function model based on DBH and height.

As shown in Table 2, although this model uses both DBH and height parameter but has the lowest R2 and highest RMSE (R2=0.868, RMSE=0.711 Mg) compared to the other power-function such as 4, 5 and 6 models which are all univariate.

In addition to R2 and RMSE, the comparison of errors indicates that applying the univariate model with integer coefficients may leads to better consequence in contrast to a multivariate generalized model with non-calibrated coefficients. With calibrating the coefficients of model 10 according to local data, model 11 is obtained which makes R2 increased (0.946) and RMSE (0.453 Mg) decreased.

Table 2 shows errors reduction in model 11 compared to model 10.

As Fig. 3 exhibits, density of points around identity line is low in model 10.

After calibration of coefficients and producing model 11 the density of points is highly increased. Results indicate that with the coefficients calibration of the Brown et al. (1992) model, the accuracy of AGB estimation can be greatly increased into desirable extent. Generally, optimized model of brown et al. (1992) (model 13) is proposed for those forests which have measurement feasibility of trees height with DBH.

Therefore, measurement of trees height, DBH and coefficients calibration suited to local species leads to highest accuracy for ground biomass estimation by Brown et al. (1992).



Fig. 2. Relationship between DBH (point), and height (circle) with AGB, respectively.


Among the models based on logarithmic transformation, the highest R2 and lowest RMSE are belong to models 9, 7 and 8, respectively. Better accuracy of model 9 (R2=0.906, RMSE=0.761 Mg) compared to the two other models is due to the use of both DBH and the height parameter for modeling, whereas in model 7 (R2=0.869, RMSE=0.809 Mg) only DBH and in model 8 (R2=0.799, RMSE=0.875) only the height was used for modeling. Nonetheless, DBH-based models are more accurate than height-based models. In the case of ME, OE and UE, this sequence can also be expressed for logarithmic models. Although model 9 has the best performance among logarithmic models, but is not superior to the other methods and compared with the model 11, it cannot be considered as an accurate model.

As Fig. 3 illustrates, the density of points along the identity line in model 9 is better than models 7 & 8.


Table 2. Results obtained from different models for the development of an allometric model (Coefficient of determination (R2), Root-mean-square error (RMSE), Mean error (ME)).

Model No.

Regression Model


Value of coefficient



















































































ln(AGB)=a+b.ln(DBH)+ c.ln(H)











AGB=a. (BDH2.H)b

























- 6.95















AGB=a. (WD.BDH2.H)b









AGB=a. (WD.BDH2.H)b
















Model 14 demonstrated in Table 2 is the Chave et al. (2005) model which is a power function model based on the wood density, DBH and height that has been applied in most recent remote sensing papers. The basic pattern is the Brown et al.  (1992) model with the difference that the density of trees species has been considered in allometric relation in order to estimate AGB. This model has R2=0.887 and RMSE=0.657 Mg compared to the other two models (10 and 12), which leads to higher accuracy. However, by calibration of the Chave et al. (2005) model using local data and improved optimized model for the north of Iran, the best performance was observed among all allometric models in biomass estimation. As shown in Table 2, model 15 with R2=0.957 and RMSE=0.404 Mg has a better result compared to the other methods investigated to date. Model 15 is distinct from models 1 through 14, as shown in Fig. 3. The optimized model of Chave et al. (2005) is proposed for those forests that besides DBH and height measurement feasibility, have available data related to the density of trees species.

By applying this model the uncertainty of allometric equation in biomass estimation by radar images can be greatly reduced, because the main reference of ground forest biomass estimation for remote sensing investigations is allometric relations. Finally, the considered model of this study was implemented using the MLPNN. Model 16 in Table 2 is developed based on neural networks. This model leads to more accurate result than current methods with highest R2 (0.986) and lowest RMSE (0.163 Mg) among all the models. In addition, there is significant difference in ME, OE and UE compared to the rest of methods. Fig. 3 shows that the model 16 has more brilliant performance among all the models. High density of points around identity line and along the axis represents the accuracy of this model. In general, after MLPNN model, power function, logarithmic and simple regression models have the best accuracy for biomass estimation, respectively.




Fig. 3. Performance of different allometric models against MLPNN model.


Overall, Hyrcanian forests of Iran are the temperate deciduous broadleaved forests which must be met through scientific research aimed at reducing carbon emissions through a better land use/land cover management. Therefore, an accurate and spatially explicit AGB of the forest cover of these forests is paramount if carbon stocks and respective changes over time are to be quantified and assessed. It is often difficult to transfer a developed model of a specific study area to another due to many factors, such as tree species, stand age, site quality, climate, and the stocking of stands, which could affect the success of model transferability. This study aimed at modeling a novel allometric model from field data. Many different modeling approaches were tested and a proposed model was selected for biomass estimation. We have shown that the biomass estimation accuracy was improved when MLPNN was used in comparison with estimating biomass using the generalized allometric models and no need calibration. The proposed methods were assessed and resulting a RMSE of 0.163 Mg and coefficient of determination between observed and predicted AGB values of 0.986. However, accuracy of model using the wide range of tree species for a regional context would be better in future research.

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