Open Access

Spatial prediction of basal area and volume in Eucalyptus stands using Landsat TM data: an assessment of prediction methods

  • Aliny Aparecida dos Reis1Email author,
  • Mônica Canaan Carvalho1,
  • José Marcio de Mello1,
  • Lucas Rezende Gomide1,
  • Antônio Carlos Ferraz Filho2 and
  • Fausto Weimar Acerbi Junior1
New Zealand Journal of Forestry Science201848:1

https://doi.org/10.1186/s40490-017-0108-0

Received: 3 February 2017

Accepted: 27 December 2017

Published: 16 January 2018

Abstract

Background

In fast-growing forests such as Eucalyptus plantations, the correct determination of stand productivity is essential to aid decision making processes and ensure the efficiency of the wood supply chain. In the past decade, advances in remote sensing and computational methods have yielded new tools, techniques, and technologies that have led to improvements in forest management and forest productivity assessments. Our aim was to estimate and map the basal area and volume of Eucalyptus stands through the integration of forest inventory, remote sensing, parametric, and nonparametric methods of spatial prediction.

Methods

This study was conducted in 20 5-year-old clonal stands (362 ha) of Eucalyptus urophylla S.T.Blake x Eucalyptus camaldulensis Dehnh. The stands are located in the northwest region of Minas Gerais state, Brazil. Basal area and volume data were obtained from forest inventory operations carried out in the field. Spectral data were collected from a Landsat 5 TM satellite image, composed of spectral bands and vegetation indices. Multiple linear regression (MLR), random forest (RF), support vector machine (SVM), and artificial neural network (ANN) methods were used for basal area and volume estimation. Using ordinary kriging, we spatialised the residuals generated by the spatial prediction methods for the correction of trends in the estimates and more detailing of the spatial behaviour of basal area and volume.

Results

The ND54 index was the spectral variable that had the best correlation values with basal area (r = − 0.91) and volume (r = − 0.52) and was also the variable that most contributed to basal area and volume estimates by the MLR and RF methods. The RF algorithm presented smaller basal area and volume errors when compared to other machine learning algorithms and MLR. The addition of residual kriging in spatial prediction methods did not necessarily result in relative improvements in the estimations of these methods.

Conclusions

Random forest was the best method of spatial prediction and mapping of basal area and volume in the study area. The combination of spatial prediction methods with residual kriging did not result in relative improvement of spatial prediction accuracy of basal area and volume in all methods assessed in this study, and there is not always a spatial dependency structure in the residuals of a spatial prediction method. The approaches used in this study provide a framework for integrating field and multispectral data, highlighting methods that greatly improve spatial prediction of basal area and volume estimation in Eucalyptus stands. This has potential to support fast growth plantation monitoring, offering options for a robust analysis of high-dimensional data.

Keywords

Forest inventoryMachine learning algorithmsMultiple linear regressionRandom forestSupport vector machineArtificial neural networks

Background

The Brazilian forestry sector represents an important share of the products, taxes, jobs, and income generation of the country and accounts for 3.5% of the national GDP (IBÁ 2015). This is in large part due to the successful establishment of fast-grown plantations of Eucalyptus species, which currently occupy around 5.6 million hectares (71.9% of the total planted forest area in Brazil) and represent 17% of the harvested wood in the world (IBÁ 2014, 2015).

The Eucalyptus genus has more than 500 species, and a subset of these are used in fast-growing plantations (Barrios et al. 2015), commonly located in tropical and sub-tropical regions, and more recently in temperate regions. Spain (González-García et al. 2015), Portugal (Lopes et al. 2009), Uruguay (Barrios et al. 2015), Chile (Watt et al. 2014), South Africa (Dye et al. 2004), Australia (Verma et al. 2014), and the USA (Wear et al. 2015) are some examples of productive Eucalyptus plantations in temperate regions that have cutting cycles ranging from 8 to 12 years. In tropical regions such as Brazil, the cutting cycles of Eucalyptus plantations range from 5 to 7 years (Guedes et al. 2015, Scolforo et al. 2016).

Timber production is the main ecosystem service of planted forests and the main management objective for these plantations (Gao et al. 2016). In the case of fast-growing plantations, the correct determination of stand productivity is essential to support forest management planning strategies (González-García et al. 2015, Retslaff et al. 2015). Traditionally, productivity assessments of a plantation are carried out based on field measurements of the diameter at breast height (DBH) and tree height via forest inventory. However, in fast-growing plantations, field-based inventory programmes may not be sufficient to capture productivity differences across the entire area, such as those arising from losses due to pest and disease attacks (Coops et al. 2006), or from climatic anomalies (González-García et al. 2015, Scolforo et al. 2016).

In the past decade, advances in geographical information systems (GIS), global positioning systems (GPS), and remote sensing have provided new tools, techniques, and technologies to support forest management. Thus, low-cost and accurate forest productivity assessment can be made, as well as allowing the collection of information in areas not sampled by forest inventory (Morgenroth and Visser 2013). The analysis of remote sensing information combined with field data has been used by several authors to fill the information gap left by data collected only in the field (Watt et al. 2016, Boisvenue et al. 2016, Moreno et al. 2016, Fayad et al. 2016, Vicharnakorn et al. 2014). Ponzoni et al. (2015) used data collected from Landsat 5 thematic mapper (TM) images for spectral-temporal characterisation of Eucalyptus canopies. Berra et al. (2012) estimated the volume of a Eucalyptus plantation in the southern region of Brazil from Landsat 5 TM images. Canavesi et al. (2010) used hyperspectral data from the Hyperion EO-1 sensor for the volume estimation of Eucalyptus plantations under different relief conditions. The results found by these authors corroborate the potential use of data collected by remote sensing to estimate the productivity of Eucalyptus plantations.

In parallel to the advances in remote sensing, computational techniques, such as machine learning algorithms (MLA), have been increasingly used to model spectral and biological data. These techniques overcome the difficulties of classical statistical methods such as spatial correlation, non-linearity of data, and overfitting (Were et al. 2015). In addition, these algorithms allow the use of categorical data, with statistical noise and incomplete data, and therefore are able to address needs under different dataset scenarios (Breiman 2001).

Several studies have shown the superiority of machine learning algorithms in relation to classical statistics in several areas, such as in forest management. For instance, Ahmed et al. (2015) modelled a Landsat time-series data structure in conjunction with LiDAR data and found that the random forest algorithm achieved better results than multiple regression for all forest classes. In another study, García-Gutiérrez et al. (2015) found that machine learning algorithms (mainly support vector machine) were superior for modelling a range of forest variables (viz., aboveground biomass, basal area, dominant height, mean height, and volume) compared with multiple linear regression. Machine learning algorithms have also been shown to provide an economical and accurate way to estimate aboveground biomass in forests from Landsat satellite images (Wu et al. 2016). These studies highlight the benefits of applying more robust techniques in solving problems previously resolved by traditional statistical modelling.

In this context, the aims of this study were: (i) to estimate and map basal area and volume of a Eucalyptus plantation through the integration of forest inventory, remote sensing, and parametric and nonparametric methods of spatial prediction; (ii) to compare the performance of machine learning algorithms (random forest, support vector machine, and artificial neural networks) with the linear regression model; and (iii) to assess the improvement in basal area and volume estimation with the addition of residual kriging in spatial prediction methods.

Methods

Study area

The study area is located in Minas Gerais state, the fourth largest state in Brazil, with an area of 586,521 km2. Minas Gerais state has the largest area occupied by plantations of the Eucalyptus genus in the country (1,400,232 ha), corresponding to 25.2% of Brazilian Eucalyptus plantations. The wood from these plantations is mainly used for the production of charcoal, as well as pulp, lumber, and panels (IBÁ 2015).

The Eucalyptus clonal stands under study are located in Lagoa Grande municipality, in the northwest of Minas Gerais state (lat. 17° 43′ 00″ S–17° 44′ 00″ S, long. 46° 32′ 00″ W–46° 33′ 00″ W, elevation 560 m a.s.l.) (Fig. 1). According to the Köppen climatic classification system, the climate in this region is Aw, classified as a tropical savanna climate, with drier months during the winter, high annual precipitation in the summer and average temperature of all months greater than 18 °C (Alvares et al. 2013). The average annual rainfall and the average monthly rainfall of the dry and wet seasons are 1430, 8, and 257 mm, respectively.
Fig. 1

Geographic location of the Eucalyptus stands and sampling grid

Field data description and sampling

This study was undertaken in a set of 20 clonal stands of Eucalyptus urophylla S.T.Blake x Eucalyptus camaldulensis Dehnh, totalling an area of 362.2 ha. These stands were planted in April and May 2004, with initial spacing of either 3 × 2 m or 3 × 3 m. The forest inventory was carried out in June and July 2009 on a set of 35 georeferenced square plots of 400 m2. The plots were georeferenced in the field with GPS (Garmin 60CSx, Garmin Ltd., Olathe, Kansas, USA). The sampling procedure adopted was systematic, allocating approximately one plot per 10 ha of forest. In each plot, the diameter at breast height (DBH) of all stems was measured, as well as the total height of the first 15 trees with normal stems (without bifurcation or any other defect) and height of dominant trees (the 100 largest diameter trees per hectare). Descriptive statistics of the variables collected in the field are shown in Table 1. Estimates of basal area (m2 ha−1), and total stem volume (m3 ha−1) were obtained from the information collected in the plots.
Table 1

Descriptive statistics of the variables collected in the field

Statistic

DBH

H

Hd

Minimum

11.98

16.98

19.40

Maximum

15.45

24.63

26.38

Mean

14.02

21.18

22.98

Standard deviation

0.85

2.33

1.92

DBH diameter at breast height (cm), H total height (m), Hd dominant height (m)

Remote sensing data and processing

Spectral data were obtained from a Landsat 5 TM satellite image, with spatial resolution of 30 m, on the date of June 25, 2009, corresponding with field data collection, in orbit 220, point 072, in bands TM1 (0.45–0.52 μm), TM2 (0.52–0.60 μm), TM3 (0.63–0.69 μm), TM4 (0.76–0.90 μm), TM5 (1.55–1.75 μm), and TM7 (2.18–2.35 μm). The Landsat 5 TM Surface Reflectance Climate Data Record (CDR) was used, which is a Landsat Level-2A product generated by the Landsat Ecosystem Disturbance Adaptive Processing System (LEDAPS) (Masek et al. 2006) obtained from the USGS (United States Geological Survey) database (USGS 2017). These images already contain radiometric calibration, and geometric and atmospheric corrections.

In addition, vegetation indices using the red, near infrared and short wave infrared spectral bands of Landsat 5 TM (Table 2) were calculated, as described by Lu et al. (2004) and Ponzoni et al. (2012). The normalised difference vegetation index (NDVI) is the most widely used vegetation index for retrieval of forest biophysical parameters (Rouse et al. 1973, Lu et al. 2004). The soil-adjusted vegetation index (SAVI) and modified soil-adjusted vegetation index (MSAVI) are soil adjusted vegetation indices used to reduce the effect of soil background reflectance (Qi et al. 1994). The enhanced vegetation index (EVI) was developed to optimise the vegetation signal, correcting reflected light distortions caused by particulate matter suspended in the air, as well as by influence of background data under the vegetation canopy (Justice et al. 1998). The global environment monitoring index (GEMI) minimises atmospheric effects, similar to the EVI and minimises observational angular effects in the observed vegetation index signal (Pinty and Verstraete 1992).
Table 2

Vegetation indices used in the spectral characterisation of the Eucalyptus stands

Vegetation indices

Formulation

Reference

NDVI

(TM4 − TM3)/(TM4 + TM3)

Rouse et al. (1973)

ND53

(TM5 − TM3)/(TM5 + TM3)

Huete et al. (2002)

ND54

(TM5 − TM4)/(TM5 + TM4)

Huete et al. (2002)

ND57

(TM5 − TM7)/(TM5 + TM7)

Huete et al. (2002)

SAVI

[(TM4 − TM3)/(TM4 + TM3 + 0.5)].(1.5)

Huete (1988)

MSAVI

\( \left[\left(2\mathrm{TM}4+1\right)-\sqrt{{\left(2\mathrm{TM}4+1\right)}^2-8\left(\mathrm{TM}4-2\mathrm{TM}3\right)\ }\right]/2 \)

Qi et al. (1994)

EVI

2.5 × [(TM4 − TM3)/(TM4 + 6TM3 − 7.5TM1 + 1)]

Justice et al. (1998)

GEMI

n(1 − 0.25n). [(TM3 − 0.125)/(1 − TM3)]

\( n=\frac{2\left(\mathrm{TM}{4}^2-\mathrm{TM}{3}^2\right)+1.5\mathrm{TM}4+0.5\mathrm{TM}3}{\mathrm{TM}4+\mathrm{TM}3+0.5} \)

Pinty and Verstraete (1992)

TM thematic mapper, ND normalised difference, NDVI normalised difference vegetation index, SAVI soil-adjusted vegetation index, MSAVI modified soil-adjusted vegetation index, EVI enhanced vegetation index, GEMI global environment monitoring index

Dataset integration

The choice of an appropriate pixel size is one of the issues to be considered when using remote sensing data to estimate dendrometric characteristics. Due to easy accessibility and affordability, a number of studies have employed Landsat images and found statistically significant correlations between remotely sensed data and dendrometric characteristics using ground plots ranging from 315 to 2500 m2 (Dube and Mutanga 2015, López-Sánchez et al. 2014, Zhang et al. 2014, López-Serrano et al. 2016).

Although the size of a single plot (20 × 20 m) in this study does not cover a Landsat pixel, we considered that a plot represents an area larger than its size. As the sampling design was one plot per hectare, we ensured that each plot matched with the reference pixel in order to extract reliable data.

Spatial modelling and prediction methods

Exploratory data analysis

Spectral response was extracted from the Landsat TM bands and vegetation indices from the geographical coordinates of the forest inventory plots. Thus, the plot database was composed of basal area (m2 ha−1), volume (m3 ha−1), spectral band values, and vegetation index values. The total database (35 plots) was systematically divided into two datasets: prediction or fitting set (70% of the database) and validation set (30% of the database). Therefore, 25 plots were used for basal area and volume predictions, and 10 plots were used for validation of the different approaches to estimate basal area and volume in the Eucalyptus stands under study.

Pearson correlation analysis was carried out among basal area, volume, values of spectral bands, and vegetation indices. From these correlations, the relationship between the dendrometric characteristics of Eucalyptus stands and its spectral response in Landsat images was explored.

Multiple linear regression (MLR) analysis

Basal area and volume estimation were accomplished through MLR analysis. A stepwise variable elimination method was used in conjunction with the Akaike information criterion (AIC) to select only those spectral variables that “best” explained basal area and volume variation. The residuals from regression models were analysed to assess the existence of trends in the errors. The variance inflation factor (VIF) was used to detect possible correlations between explanatory variables (multicollinearity). The adopted VIF cutoff value was 10.

Random forest (RF)

The RF algorithm, initially proposed by Breiman (2001), is an ensemble method that generates a set of individually trained decision trees and combines their results. The greatest advantage of these decision trees as regression methods is that they are able to accurately describe complex relationships among multiple variables, and by aggregating these decision trees, more accurate solutions are generated (Gleason and Im 2012). In addition to these characteristics, RF is an easy parameterisation method (Immitzer et al. 2012). This method has shown great potential in regression studies with integration of spectral data, in some cases generating better results than conventional techniques (Stojanova et al. 2010, Dube et al. 2014, García-Gutiérrez et al. 2015, Görgens et al. 2015, Wu et al. 2016). The RF algorithm fitted in this study is implemented in the open-source software WEKA 3.8 (Frank et al. 2016). Tests were carried out with the exchange of tree numbers and attribute numbers to be drawn. Then, 20 trees with 10 attributes to be drawn by node for basal area and 80 trees and 11 attributes for volume were fixed.

Support vector machine (SVM)

SVMs operate by assuming that each set of inputs will have a unique relation to the response variable and that the grouping and the relation of these predictors to one another is sufficient to identify rules that can be used to predict the response variable from new input sets. For this, SVMs project the input space data into a feature space with a much larger dimension, enabling linearly non-separable data to become separable in the feature space. This method has been successfully used in forestry classification problems (Huang et al. 2008, Shao and Lunetta 2012) and more recently in regression problems with the use of spectral data (García-Gutiérrez et al. 2015, Wu et al. 2016). The Kernel function used in the present study was the Gaussian or radial basis function (RBF). The algorithm used is implemented in WEKA 3.8 software under the sequential minimal optimization (SMO) function. Values of parameters C and σ (bandwidth or influence range of each training point in the RBF) were tested within the interval (10i)i =  − 3, − 2, − 1, 0, 1, 2, 3, where the least squared mean error configuration was chosen for application. For basal area and volume, selected C and σ values were 10 and 0.1, and 100 and 0.01, respectively.

Artificial neural networks (ANNs)

ANNs are a parallel-distributed information processing system that simulates the working of neurons in the human brain, being able to learn from examples. Artificial neural networks are widely used to model complex and non-linear relations between inputs and outputs or to determine patterns in data (Diamantopoulou 2012). The use of this technique in conjunction with remote sensing data is consolidated in several studies (Cluter et al. 2012, García-Gutiérrez et al. 2015, Rodriguez-Galiano et al. 2015, Were et al. 2015). We used the ANN obtained by running the Multilayer Perceptron function (of the multilayer perceptron type) provided by WEKA 3.8 software. The training of neural networks occurred through the back-propagation algorithm, which fit the weights of all the layers of the network from the backpropagation of the error, obtained in the output layer. The weights updating was carried out according to the error, learning rate, and momentum terms (Delta rule). The sigmoidal activation function was employed in all neurons. Determined by previous tests, ANNs were structured with 14 neurons in the input layer (number of variables), 1 neuron in the hidden layer, and 1 neuron in the output layer, corresponding to estimated basal area or volume. The learning rate, the momentum term, and iteration numbers were fixed at 0.3, 0.5, and 500 for basal area, and 0.2, 0.7, and 500 for volume, respectively.

Relative importance evaluation

The variable importance was assessed for each model with a removal-based approach in order to avoid the limited interpretability of the MLA and to verify how each independent variable contribute to the performance of machine learning algorithms (RF, SVM, and ANN). All algorithms were adjusted n times, with n being the number of available variables. At each time, one variable was removed from the training set and then the root mean square error (RMSE) of the algorithm was quantified. At the end, the obtained errors were normalised by the ratio of the largest RMSE so that they were between 0 and 1 and multiplied by 100 (Were et al. 2015). The variable that results in the highest RMSE when removed from the database is the variable with the highest relative importance within the model. This methodology was chosen because it can be consistently applied to all algorithms, allowing comparisons of variable contribution between the methods.

Geostatistical modelling of prediction methods errors

Spatial prediction methods capture the average behaviour of the main variable, allowing the identification of its general spatial behaviour, without detailing more specific areas or regions. For details of specific regions, estimates obtained exclusively from the auxiliary variables need to be corrected. Thus, residuals generated by spatial prediction methods (MLR, RF, SVM, and ANN) were used for the correction of trends in the estimates and for detailing the spatial behaviour of the main variables (basal area and volume) using ordinary kriging. The interpolated values of the residuals were then added to the estimates of the spatial prediction methods (MLR, RF, SVM, and ANN). Thus, we obtained the basal area and volume estimates corrected by the ordinary kriging of the residuals for each spatial prediction method.

For the application of ordinary kriging to the spatial prediction method residuals, we considered the stationarity presupposition of the intrinsic hypothesis (Journel and Huijbregts 1978), through fitting of theoretical functions to experimental semivariogram models. Spherical, exponential, and Gaussian models were fitted to the semivariogram of the residuals from each spatial prediction method using weighted least squares. The semivariogram parameters (nugget (τ2), sill (σ2), and range (ϕ)) were calculated from the best fitted models, which provided information about the spatial structure as well as input parameters for the kriging interpolation. The nugget represents the minimum semivariance among different sampling intervals. Nugget values greater than zero represent a combination of experimental error and of unresolved spatial variability occurring at scales smaller than inter-sampling lag distance. Sill is the plateau reached by the values of semivariance and indicates the amount of variation than can be explained by the spatial structure of the data. Range is the distance at which the semivariogram reaches the plateau, indicating the distance which values are spatially correlated. The evaluation of the performance of each semivariogram model and the selection of the best models were based on cross-validation, which estimates the reduced average error (RAE) and the standard deviation of the reduced average error (SRE) (Yamamoto and Landim 2013).

Validation and assessment of the prediction methods

The different approaches to basal area and volume estimation of Eucalyptus stands were evaluated by comparing the basic statistics of the predicted maps (mean and standard deviation) with the estimates obtained from the forest inventory, and through the discrepancies between observed and predicted values in the fitting and validation datasets. These discrepancies were evaluated using the mean error (ME), the mean absolute error (MAE), and the root mean square error (RMSE), as described in Eqs. 1–3.
$$ \mathrm{ME}=\frac{1}{N}{\sum}_{i=1}^N\left({X}_i-{\widehat{X}}_i\right) $$
(1)
$$ \mathrm{MAE}=\frac{1}{N}{\sum}_{i=1}^N\left|{X}_i-{\widehat{X}}_i\right| $$
(2)
$$ \mathrm{RMSE}=\sqrt{\frac{1}{N}\ {\sum}_{i=1}^N{\left({X}_i-{\widehat{X}}_i\right)}^2\ } $$
(3)
where N is the number of values in the dataset; \( {\widehat{X}}_i \) is the estimated value of the main variable; X i is the observed value in the prediction and validation sets.
The relative improvement (RI) achieved by residual kriging for a particular spatial prediction method was calculated by comparing the change in RMSE when the residual kriging was applied using Eq. 4.
$$ \mathrm{RI}=\frac{{\mathrm{RMSE}}_{\mathrm{spm}}-{\mathrm{RMSE}}_{\mathrm{spm}\hbox{-} \mathrm{RK}}}{{\mathrm{RMSE}}_{\mathrm{spm}}}\times 100\% $$
(4)

where RMSEspm is the root mean square error of a spatial prediction method, RMSEspm ‐ RK is the root mean square error of the spatial prediction method when residual kriging is added to this method.

Data analysis for this study was performed using the following software: R (R Core Team 2016) with the geoR package (Ribeiro Júnior and Diggle 2001), WEKA 3.8 (Frank et al. 2016), and ArcGis version 10.1 (Esri 2010) with Geostatistical Analyst extension (Esri 2010).

Results

Descriptive statistic of the measured basal area and volume

Basal area ranged from 10.07 to 21.63 m2 ha−1, with average of 16.86 m2 ha−1 and standard deviation of 2.4 m2 ha−1 (Table 3). The average volume was 169.34 m3 ha−1 with a standard deviation of 29.66 m3 ha−1 and range from 95.80 up to 213.85 m3 ha−1. Basal area had a lower coefficient of variation (CV = 14.26%) compared to volume (CV = 17.51%), demonstrating a considerable homogeneity of this dendrometric characteristic in the evaluated Eucalyptus stands.
Table 3

Descriptive statistics obtained from forest inventory processing using the estimators of simple random sampling (SRS)

Estimators

Basal area

Volume

(m2)a

(m2 ha−1)

(m3)a

(m3 ha−1)

Minimum

0.91

10.07

8.62

95.80

Maximum

1.95

21.63

19.25

213.85

Mean

1.52

16.86

15.24

169.34

Standard deviation

0.22

2.4

2.67

29.66

Coefficient of variation (%)

14.26

17.51

Sampling error (%)

4.89

6.00

Total confidence interval

5807.9–6405.0

57,652.7–65,018.7

aEstimates obtained for an area of 900 m2 (corresponding to the area of each pixel of the Landsat images)

Correlation among basal area, volume, spectral bands, and vegetation indices

The correlation between plot basal area and the different spectral bands and their ratios (Table 4) ranged from − 0.91 (ND54) to 0.15 (TM2). The SAVI, MSAVI, GEMI, and EVI were also highly correlated with basal area (r > 0.85). The correlation between plot volume and the spectral bands and ratios ranged from − 0.52 (ND54) to − 0.02 (TM2). The NDVI (r = 0.49) and SAVI (r = 0.47) also had high correlations with volume, but these were lower in magnitude when compared with those for basal area. Many of the spectral bands and ratios were also highly correlated with each other (r > 0.90), which can be considered a drawback due possible to multicollinearity problems in linear regression models.
Table 4

Pearson’s correlation coefficient (r) among basal area, volume, and spectral data for the Eucalyptus stands

Variables

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1. G

1.00

               

2. V

0.70*

1.00

              

3. TM1

− 0.24ns

0.10ns

1.00

             

4. TM2

0.15ns

− 0.02ns

0.59*

1.00

            

5. TM3

− 0.20ns

− 0.10ns

0.80*

0.72*

1.00

           

6. TM4

0.82*

0.41*

− 0.05ns

0.43*

0.12ns

1.00

          

7. TM5

− 0.66*

− 0.36ns

0.53*

0.31ns

0.68*

− 0.40*

1.00

         

8. TM7

− 0.68*

− 0.40*

0.56*

0.32ns

0.66*

− 0.42*

0.90*

1.00

        

9. NDVI

0.83*

0.49*

− 0.53*

− 0.13ns

− 0.55*

0.75*

− 0.78*

− 0.82*

1.00

       

10. ND53

− 0.60*

− 0.32ns

− 0.29ns

− 0.50*

− 0.37ns

− 0.66*

0.43*

0.31ns

− 0.31ns

1.00

      

11. ND54

− 0.91*

− 0.52*

0.31ns

− 0.09ns

0.30ns

− 0.86*

0.80*

0.78*

− 0.93*

0.65*

1.00

     

12. ND57

0.45*

0.27ns

− 0.49*

− 0.28ns

− 0.49*

0.27ns

− 0.50*

− 0.82*

0.60*

0.00ns

− 0.48*

1.00

    

13. SAVI

0.88*

0.47*

− 0.23ns

0.25ns

− 0.12ns

0.97*

− 0.57*

− 0.60*

0.89*

− 0.57*

− 0.94*

0.41*

1.00

   

14. MSAVI

0.88*

0.45*

− 0.36ns

0.13ns

− 0.27ns

0.92

− 0.65*

− 0.67*

0.94*

− 0.50*

− 0.95*

0.46*

0.99*

1.00

  

15. GEMI

0.86*

0.45*

− 0.14ns

0.34ns

0.00ns

0.99*

− 0.49*

− 0.52*

0.83*

− 0.62*

− 0.91*

0.35ns

0.99*

0.96*

1.00

 

16. EVI

0.87*

0.42*

− 0.41*

0.12ns

− 0.28ns

0.92*

− 0.64*

− 0.67*

0.94*

− 0.48*

− 0.94*

0.47*

0.98*

1.00*

0.96*

1.00

V volume (m3 ha−1), G basal area (m2 ha−1), TM thematic mapper, ND normalised difference, NDVI normalised difference vegetation index, SAVI soil-adjusted vegetation index, MSAVI modified soil-adjusted vegetation index, GEMI global environment monitoring index, EVI enhanced vegetation index

ns Not significant at 5%; *significant at 5%

Spatial prediction of basal area and volume by MLR, RF, SVM, and ANN

The spectral data examined had several significant correlations with the basal area and volume data (Table 4). However, they contributed in a reduced form to the regression models due to multicollinearity problems, which resulted in final regression models with few significant explanatory variables (Table 5). The basal area model only included the ND54 vegetation index (Table 5), while the volume model included the TM1 band and NDVI. The coefficient of determination was high for the basal area model (R2 = 0.81), but was much lower for the volume model (R2 = 0.37).
Table 5

Regression model fitted for basal area and volume estimation for the Eucalyptus stands

Model

β 0

β 1

β 2

R 2 aj

S xy

S xy (%)

G = β0 + β1ND54

0.78***

− 1.80***

0.81

0.09

5.76

V = β0 + β1NDVI + β2TM1

− 24.11*

42.69***

241.61*

0.37

2.01

13.08

G basal area (m2), V volume (m3), β0, β1, and β2 coefficients, R2aj adjusted coefficient of determination, S xy residual standard error, TM thematic mapper, ND normalised difference, NDVI normalised difference vegetation index

***Significant at 1%; *significant at 10%

In the case of basal area and volume predictions using machine learning algorithms, the increases in RMSEs when the predictors were excluded one by one from the SVM, ANN, and RF models are shown in Fig. 2. The variable ranking by relative importance differed for each algorithm. The ND54 index, chosen for basal area model by the MLR, also had the greatest effect on the accuracy of the RF model, both for basal area and volume. The TM2 band had the highest relative importance for the ANN and SVM models of both basal area and volume. The TM1 band, selected by the MLR for volume estimation, also had high importance in the ANN and SVM models of volume.
Fig. 2

Relative importance of the variables within each machine learning algorithm: RF, SVM, and ANN for basal area and volume

Comparisons of measured values and estimated values of basal area (Fig. 3) showed that basal area was underestimated by the ANN model (Fig. 3d). The model fitted using the RF algorithm produced values of basal area that were in closer agreement with measured values (Fig. 3b). Similar results were seen for the volume models, but with a slight overestimation for the plots with small volumes and an underestimation of the plots with high volumes. The model fitted using ANN algorithm did not produce estimates of volume that were consistent with measured values (Fig. 3h). The models fitted using the MLR and SVM (Fig. 3e, g) algorithms produced predicted values that were more closely related to the measured values than those from the ANN algorithm.
Fig. 3

Scatter plots of measured values versus estimated values by: MLR (a) and (e); RF (b) and (f); SVM (c) and (g); and ANN (d) and (h) for basal area and volume, respectively. A 1:1 line (black, dashed) is provided for reference

Prediction and validation sets of basal area and volume were compared by means of Student’s t test, in order to check if they provided unbiased subsets of the original data (Viana et al. 2012). Average basal area (17.03 m2 ha−1) and volume (171.10 m3 ha−1) obtained from the prediction set did not statistically differ from average basal area (16.45 m2 ha−1) and volume (164.92 m3 ha−1) obtained from the validation set, considering two-tailed Student’s t test (Basal area: t = 0.629ns, df = 33, p value = 0.533; volume: t = 0.550ns, df = 33, p value = 0.585).

The evaluation of spatial prediction methods, based on prediction and validation sets, was done by comparing the statistics presented in Eqs. 1 through 4 (Table 6). The mean error (ME) should ideally be close to zero if the prediction method is unbiased, and the values of this parameter suggested that all predictions generated impartial estimates when evaluated from both prediction and validation sets. Both the MAE and RMSE showed that basal area estimates were more accurate than volume estimates for all spatial prediction methods. The MAE and RMSE results obtained from the validation set demonstrated that there were no significant differences among the MLR, RF, SVM, and ANN for basal area estimates. For the volume estimates, the models fitted by SVM had the best performance and MLR the poorest performance.
Table 6

Prediction methods evaluation using the prediction and validation sets for the Eucalyptus stands

Method

Statistic

Basal area error (m2)

Volume error (m3)

Prediction set

Validation set

Prediction set

Validation set

MLR

ME

0.00

− 0.05

0.00

− 0.74

MAE

0.07

0.09

1.56

2.08

RMSE

0.08

0.14

1.89

2.48

RMSE (%)

5.50

9.42

12.27

16.72

RF

ME

0.01

− 0.03

0.08

− 0.90

MAE

0.03

0.09

0.62

1.63

RMSE

0.04

0.14

0.73

2.21

RMSE (%)

2.48

9.54

4.77

14.91

SVM

ME

− 0.01

− 0.05

0.00

− 0.66

MAE

0.04

0.09

1.19

1.59

RMSE

0.06

0.14

1.60

2.02

RMSE (%)

4.14

9.39

10.41

13.58

ANN

ME

0.09

0.03

0.94

0.45

MAE

0.10

0.09

1.70

1.68

RMSE

0.14

0.13

1.98

2.05

RMSE (%)

8.87

8.52

12.88

13.82

MLR multiple linear regression, RF random forest, SVM support vector machine, ANN artificial neural networks, ME mean error, MAE mean absolute error, RMSE root mean square error

Geostatistical modelling of prediction method errors

The semivariogram models were selected based on RAE and SRE values close to 0 and 1, respectively (Yamamoto and Landim 2013). The experimental semivariograms constructed from the residuals of the basal area and volume prediction methods had a spatial dependence structure defined in six of the eight analysed situations (Fig. 4 and Table 7). The volume residuals from MLR and ANN methods had a pure nugget effect, i.e. no spatial dependence structure. This result indicated a random spatial distribution of the residuals in these two situations.
Fig. 4

Experimental semivariograms of residuals from: MLR (a) and (e); RF (b) and (f); SVM (c) and (g); and ANN (d) and (h) for basal area and volume, respectively

Table 7

Nugget (τ2), sill (σ2), and range (ϕ) parameters for the selected semivariance function models for each of the variables in study

Variables

Residual

Selected model

τ 2

σ 2

ϕ (m)

RAE

SRE

Basal area

MLR

Exponential

0.0016

0.0067

1350

− 0.0092

1.0818

RF

Spherical

0.0004

0.0009

737

− 0.0079

1.0586

SVM

Gaussian

0.0017

0.0037

1577

0.0089

0.9610

ANN

Exponential

0.0000

0.0119

1430

− 0.0303

1.1393

Volume

MLR

Exponential

PNE

PNE

PNE

PNE

PNE

RF

Spherical

0.3316

0.2505

773

− 0.0051

1.0258

SVM

Exponential

0.0000

2.5582

858

− 0.0039

0.9958

ANN

Exponential

PNE

PNE

PNE

PNE

PNE

MLR multiple linear regression, RF random forest, SVM support vector machine, ANN artificial neural networks, RAE reduced average error, SRE standard deviation of the reduced average error, PNE pure nugget effect

The residuals of the spatial prediction methods that had defined spatial dependence structures (Fig. 4) were interpolated using ordinary kriging, and their estimates were added to basal area and volume estimates of the respective spatial prediction methods. The relative improvement (RI) of the addition of basal area residual kriging by the ANN method was 25%, i.e. there was a reduction from 8.52 to 6.37% in the RMSE (Table 8). For the RF method, the RMSE increased from 9.54 to 10.08%, which corresponds to a 5.7% increase in the error of the basal area estimates by kriging of the residuals. For the volume, the addition of residual kriging improved the precision of SVM estimates and reduced the precision of the RF estimates.
Table 8

Prediction methods with addition of the residual estimation by ordinary kriging using the prediction and validation sets for the Eucalyptus stands

Method

Statistic

Basal area error (m2)

Volume error (m3)

Prediction set

Validation set

Prediction set

Validation set

MLR-RK

ME

0.00

− 0.03

MAE

0.03

0.09

RMSE

0.04

0.14

RMSE (%)

2.80

9.30

RI

49.09

1.27

RF-RK

ME

0.01

− 0.05

0.00

− 1.03

MAE

0.04

0.10

0.63

1.70

RMSE

0.05

0.15

0.77

2.26

RMSE (%)

3.08

10.08

5.02

15.25

RI

− 24.19

− 5.66

− 5.24

− 2.28

SVM-RK

ME

0.01

− 0.03

− 0.32

− 0.57

MAE

0.05

0.10

0.80

1.22

RMSE

0.06

0.15

1.11

1.74

RMSE (%)

4.09

9.83

7.19

11.72

RI

1.21

− 4.69

30.93

13.70

ANN-RK

ME

0.02

− 0.06

MAE

0.04

0.06

RMSE

0.09

0.09

RMSE (%)

5.79

6.37

RI

34.72

25.23

MLR multiple linear regression, RF random forest, SVM support vector machine, ANN artificial neural networks, RK residual estimation by ordinary kriging, ME mean error, MAE mean absolute error, RMSE root mean square error, RI relative improvement

Mapping of basal area and volume for Eucalyptus stands

Basal area and volume estimates obtained by different spatial prediction methods (Table 9) had average values very close to each other, and were in agreement with the forest inventory estimates (Table 3). Only the ANN method generated underestimated values for both basal area and volume, so that the total values of basal area and volume were not within the confidence interval generated by the forest inventory.
Table 9

Statistics of basal area and volume maps estimated by spatial predictions methods MLR, RF, SVM, and ANN

Method

Basal area (m2)

Volume (m3)

Min

Max

Mean

Standard deviation

Total estimate

Min

Max

Mean

Standard deviation

Total estimate

MLR

0.62

1.83

1.52

0.20

6151.9

4.51

19.99

15.30

2.30

61,739.5

MLR-RK

0.65

1.93

1.52

0.21

6141.0

RF

0.96

1,89

1.51

0.17

6101.5

9.26

18.08

15.27

1.81

61,600.1

RF-RK

0.93

1.93

1.53

0.17

6166.6

9.02

18.37

15.36

1.91

61,965.7

SVM

0.88

2.12

1.57

0.18

6326.2

1.36

19.64

15.31

2.57

61,760.7

SVM-RK

0.76

2.10

1.56

0.19

6284.2

1.10

21.78

15.29

2.92

61,683.8

ANN

0.97

1.65

1.42

0.22

5715.3

8.32

15.68

13.93

2.70

56,223.8

ANN-RK

0.90

1.94

1.50

0.23

6070.3

Min minimum value, Max maximum value, MLR multiple linear regression, RF random forest, SVM support vector machine, ANN artificial neural networks, RK residual estimation by ordinary kriging

Maps showing the spatial distribution of basal area and volume identified the same areas with high and low productivity, regardless of the spatial prediction method (Figs. 5 and 6). The maps obtained by ANN had a smaller difference between maximum and minimum estimated values for basal area and volume, while the mapping obtained from the SVM models had a greater difference between these values. MLR and RF methods provided similar estimates in the basal area and volume mapping.
Fig. 5

Spatial distribution of the basal area in Eucalyptus stands, estimated by: MLR (a), RF (b), SVM (c), and ANN (d)

Fig. 6

Spatial distribution of the volume in Eucalyptus stands, estimated by: MLR (a); RF (b); SVM (c); and ANN (d)

The addition of residual kriging in the basal area and volume mapping (Fig. 7) resulted in a greater difference between maximum and minimum estimated values in all spatial prediction methods. For ANN, residual kriging resulted in estimates that were more in agreement with the field observations, correcting the basal area underestimation behaviour for the Eucalyptus stands under study. However, the addition of residual kriging to the models fitted by RF and SVM methods did not result in significant differences in basal area and volume mapping, and also led to increases in estimation errors in non-sampled areas in the field (Table 8).
Fig. 7

Spatial distribution of the basal area in Eucalyptus stands estimated by: MLR (a): RF (b); SVM (c); and ANN (d) with addition of the residual estimation by ordinary kriging; and for volume estimated by RF (e) and SVM (f) with addition of the residual estimation by ordinary kriging

Discussion

Remote detection of forest canopies is complex due to the size, shape, and dielectric properties of its scatter elements (leaves, branches, and stems) (Galeana-Pizaña et al. 2014). The spatial diversity of forest canopies makes the relationship between forest parameters and remote sensing data a major challenge, although several studies have already demonstrated correlation between spectral data and forest characteristics of interest (Stojanova et al. 2010, Viana et al. 2012, Castillo-Santiago et al. 2013, Fayad et al. 2016, Gao et al. 2016). For instance, plantations comprised of different Eucalyptus species may have very similar values of basal area and volume, but have different spectral characteristics due to differences in spectral behaviour of the species that form the canopies. Also, according to Ponzoni et al. (2015), the canopy reflectance of older Eucalyptus plantations (between 4 and 6 years) tend to contain a greater contribution from green leaves and a lower contribution from shadows, the background, and from dry branches inside the canopies than the canopy reflectance of young Eucalyptus plantations (< 4 years). Thus, the canopy reflectance of older Eucalyptus plantations generated highest correlations with bands of the infrared region of the electromagnetic spectrum and, therefore, with vegetation indices that include these bands in their compositions (Ponzoni et al. 2015). These results are consistent with the best correlations found in this study among the infrared bands, vegetation indices derived from these bands, basal area, and volume. This same behaviour was observed in the studies of Gebreslasie et al. (2008), Canavesi et al. (2010), Berra et al. (2012), and Pacheco et al. (2012).

Basal area was more strongly correlated with the spectral data because this variable is derived from only the diameter of the trees, which is directly related to size of the tree canopies, and determines the canopy reflectance (Ponzoni et al. 2012). On the other hand, volume is derived from the diameter, form factor, and height of the trees. Height estimates are obtained from empirical equations that add errors during the volume estimation process. This acts to reduce the strength of relationships between volume and variables obtained from remotely sensed images. The ND54 index was the spectral variable that had the strongest correlation with basal area (r = − 0.91) and volume (r = − 0.52). However, it was also significantly correlated with the other spectral variables. During multiple linear regression analysis, the fact that two or more explanatory variables are highly correlated may generate multicollinearity problems in the fitted models, since one of the regression assumptions is that no linear relationship may exist between any independent variables or linear combinations of these (Montgomery et al. 2006).

For the MLR method, the best volume estimation model was obtained from the TM1 band and the NDVI (Table 5), yet was only able to explain approximately 37% of the variation in this stand attribute. Conversely, the best model for basal area estimation used the ND54 index as the predictor variable and was able to explain more than 80% of the variation in this attribute, confirming the explanatory power of spectral data for basal area estimation in Eucalyptus stands. Gebreslasie et al. (2010) assessed the suitability of both visible and shortwave infrared ASTER data and vegetation indices for estimating forest structural attributes of Eucalyptus species in southern KwaZulu Natal, South Africa. These authors applied a MLR using MSAVI and band 3 as predictor variables and were able to explain slightly more of the variation in basal area (R2 = 0.67) than volume (R2 = 0.65). Although the MLR model for volume does not have a high coefficient of determination, the spectral data can efficiently explain the volumetric variations in non-sampled areas in the field. In a similar study for Eucalyptus stands located in the southern region of Brazil, Berra et al. (2012) concluded that spectral data obtained from Landsat images were efficient in mapping the volume in the study area, even when the regression models did not present high coefficients of determination (R2 < 0.70).

Divergence among variables that were deemed important between the different methods was observed with the machine learning algorithms. For basal area modelling, the ND54 index and NDVI had a higher importance value for RF. Statistically, these indices had high correlation values with the variable of interest (r = − 0.91 and 0.83, respectively) and high multicollinearity (r = − 0.93). The ND54 index also was the variable that most contributed to the volume estimate by the RF method. The fact that the explanatory variables are correlated does not affect the performance of these algorithms. These methods do not rely on underlying assumptions about the data, which allows them to work with all available explanatory variables, without loss of information in the process of variable selection and reduction (Görgens et al. 2015). For the models fitted using ANN and SVM algorithms, the TM2 band was the most important predictor variable for basal area and volume. The linear correlation between this variable and basal area and volume is low to non-existent (r = 0.15 and − 0.02, respectively). However, this band is usually applied in vegetation vigour assessment (Meng et al. 2009), a characteristic that is indirectly related to volume and basal area, and which may explain the greater contribution of the TM2 band in the ANN and SVM algorithms, since trees that are more vigorous tend to have higher values of basal area and volume.

The models of basal area and volume developed by the RF algorithm had smaller errors compared with those developed by other machine learning algorithms and MLR. The performance of this algorithm has been proven in many modelling and remote sensing studies (Lafiti et al. 2010, Rodriguez-Galiano et al. 2015, Wu et al. 2016). In the study by Shataee et al. (2012), volume prediction models developed by RF performed better than those developed using k-nearest neighbour (k-NN) and SVM. Employing ASTER satellite data, the relative RMSE obtained for all three volume models was higher than for the models developed in our study: 28.54% for k-NN, 25.86% for SVM, and 26.86% for RF, and only the RF algorithm produced unbiased volume estimations. For basal area, RF produced models with lower RMSE (18.39%) when compared with SVM (RMSE = 19.35%) and k-NN (RMSE = 20.20%); however, only k-NN was able to generate unbiased estimation compared with the other two algorithms used.

One of the positive features of RF is that it achieves satisfactory performance even with a limited number of samples and with many independent variables (attributes), as in the case of this current study. It is an ensemble method, which combines several regression trees to generate an average estimate, in which different attributes are used in each tree, making the results take into account the information of all available attributes. Stojanova et al. (2010) also concluded that ensemble methods (RF) were significantly better in height and canopy cover modelling using remote sensing data than single- and multi-target regression trees. The ANN and SVM algorithms also have proven good performance and robustness in several studies (e.g. Shao and Lunetta 2012, Were et al. 2015). However, the parameterisation of these methods is laborious, and they are very sensitive to the variation of input parameters, with ANN being more sensitive than other methods (Rodriguez-Galiano et al. 2015). This same behaviour was observed in this study, where the use of a restricted dataset by ANN resulted in estimates that were not compatible with the forest inventory estimates (Tables 3 and 9).

The addition of residual kriging in spatial prediction methods did not necessarily result in relative improvements in the estimation of these methods. In the case of MLR and ANN methods, residual kriging contributed to better accuracy of the basal area estimates. These results are consistent with the results of Dai et al. (2014), who reported that the combination of the residual kriging with artificial neural networks provides an improvement in the estimate accuracy of the variables of interest. The combination of MLR with residual kriging also provided improvements in estimates in the studies of Viana et al. (2012), Castillo-Santiago et al. (2013), and Galeana-Pizaña et al. (2014). For basal area and volume estimation, the addition of residual kriging in the RF and SVM methods resulted in a lower precision of the estimates. Hybrid methods are advantageous in the ability to use spatial information (ordinary kriging of residuals) and non-spatial information (multiple linear regression analysis and machine learning algorithms). However, in some situations, hybrid methods provide less-accurate estimates in regions where the data collected in the field are sparse (Palmer et al. 2009).

The high growth rate of Eucalyptus stands in Brazil reinforces the importance of robust methods that consider auxiliary information in the process of estimating variables of interest, such as basal area and volume. The methodologies presented here are powerful tools for estimating basal area and volume from spectral data obtained from Landsat 5 TM or from other multispectral optical sensors. According to Görgens et al. (2015), machine learning algorithms can continuously learn from new data and keep all the accumulated knowledge of previous datasets. This fact allows the implementation of these algorithms in other situations where only limited amounts of data are available. The use of all auxiliary variables in the estimation process is another advantage over traditional regression methods, since machine learning algorithms are not restricted by correlation between input variables, thus avoiding the loss of important information in the estimation process of the variable of interest. Nevertheless, these methods have as disadvantage the transparency of the resulting models, so an alternative to overcome this obstacle is the evaluation of the relative importance of the explanatory variables. Furthermore, the causal relation between inputs and outputs of the estimation process is not clear, which implies a limited biological interpretation (Aertsen et al. 2010, Özçelik et al. 2013).

The results from the current study do need to be interpreted cautiously, as they are limited to a homogenous and relatively small study area. While this work uses a small number of plots, it represents the sampling intensity adopted by most Brazilian forestry companies, i.e. one plot (usually 200–500 m2 in size) for each 10 ha of Eucalyptus plantation (Raimundo et al. 2017, Scolforo et al. 2016) and the results from this research showcase the importance of using remotely sensed data and robust prediction methods for basal area and volume estimation. The data used here were also from a relatively old sensor, Landsat 5 TM, and a study by Fassnacht et al. (2014) concluded that predictor data (sensor) type is the most important factor for the accuracy of biomass estimates and that the prediction method had a substantial effect on accuracy and was generally more important than the sample size. Fassnacht et al. (2014) also suggested that choosing the appropriate statistical method may be more effective than obtaining additional field data for obtaining good biomass estimates.

Considering the cost of improving accuracy of timber production estimates by field measurements in Eucalyptus stands, it seems sensible to invest in further studies that focus on more test sites and a wider range of sensor systems (particularly RADAR and LIDAR). This would further increase our understanding of the role of the statistical model set-up in remote sensing-based estimates of forest variables in Eucalyptus stands. Further studies could also investigate whether other prediction methods, such as nonlinear regression or partial least squares regression (PLSR) approaches, alter our findings. The integration of additional predictors (e.g. topographic information or climate variables) would be a further possible extension of our work.

Conclusions

Machine learning algorithms, particularly the random forest (RF) and support vector machine (SVM) algorithms, were able to develop models that estimate basal area and volume in Eucalyptus stands using spectral data collected from Landsat 5 TM images. The artificial neural network (ANN) method did not perform well in this context, due in part to the limited data availability.

Random forest was the best method of spatial prediction and mapping of basal area and volume in Eucalyptus stands in Minas Gerais state. However, due to the close performance to the support vector machine and multiple linear regression methods, we propose that both methods should be tested and then the best result applied for spatial prediction of basal area and volume in other regions with Eucalyptus stands. The approaches used in this study provide a framework for integrating field and multispectral data, highlighting methods that greatly improve spatial prediction of basal area and volume estimation in Eucalyptus stands. Although the sensor TM of Landsat satellites is no longer operational, the concepts presented in this study are expected to be consistent regardless of the sensor. Thus, the approach used in this study can be more broadly applied to basal area and volume estimation in Eucalyptus stands using the new optical sensors such as Landsat 8 OLI and Sentinel-2.

The combination of spatial prediction methods with residual kriging should be used with caution, since the relative improvement of spatial prediction accuracy of basal area and volume did not occur in all methods, and there is not always a spatial dependency structure in the residuals of a spatial prediction method.

Abbreviations

AIC: 

Akaike information criterion

ANN: 

Artificial neural networks

EVI: 

Enhanced vegetation index

G: 

Basal area

GDP: 

Gross domestic product

GEMI: 

Global environment monitoring index

GIS: 

Geographical information systems

GPS: 

Global positioning systems

MAE: 

Mean absolute error

ME: 

Mean error

MLA: 

Machine learning algorithms

MLR: 

Multiple linear regression

MSAVI: 

Modified soil-adjusted vegetation index

ND: 

Normalised difference

NDVI: 

Normalised difference vegetation index

PNE: 

Pure nugget effect

R 2 aj

Adjusted coefficient of determination

RAE: 

Reduced average error

RBF: 

Radial basis function

RF: 

Random forest

RI: 

Relative improvement

RK: 

Residual estimation by ordinary kriging

RMSE: 

Root mean square error

SAVI: 

Soil-adjusted vegetation index

SMO: 

Sequential minimal optimization

SRE: 

Standard deviation of the reduced average error

SVM: 

Support vector machine

S xy

Residual standard error

TM: 

Thematic mapper

USGS: 

United States Geological Survey

V: 

Volume

VIF: 

Variance inflation factor

Declarations

Acknowledgements

We thank CAPES - Coordenadoria de Aperfeiçoamento do Pessoal do Ensino Superior (Brazilian Federal Agency for Support and Evaluation of Graduate Education) for the scholarships provided to AAR and MCC.

Funding

Not applicable

Availability of data and materials

Not applicable

Authors’ contributions

All authors contributed substantially to the work reported here. AAR, MCC, LRG, and ACFF analysed and interpreted the data. ARR and MCC wrote the manuscript. JMM, ACFF, LRG, and FWAJ reviewed and edited the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Publisher’s Note

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Forest Science, Federal University of Lavras – UFLA, Lavras, Brazil
(2)
CPCE, Federal University of Piauí, Bom Jesus, Brazil

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