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Copyright Ó 2013, Paper 17-003; 51581 words, 7 Figures, 0 Animations, 5 Tables. http://EarthInteractions.org
Neural Network and Multiple Linear Regression for Estimating Surface Albedo from ASTER Visible and Near-Infrared Spectral Bands Mohammad H. Mokhtari* Department of Remote Sensing, Faculty of Geo-Information and Real Estate, University Technology of Malaysia, Johor Bahru, Malaysia, and Faculty of Natural Resources, Yazd University, Yazd, Iran
Ibrahim Busu Department of Remote Sensing, Faculty of Geo-Information and Real Estate, University Technology of Malaysia, Johor Bahru, Malaysia
Hossein Mokhtari Faculty of Science, Yazd University, Yazd, Iran
Gholamreza Zahedi Process System Engineering Center (PROSPECT), University Technology of Malaysia, Johor Bahru, Malaysia
Leila Sheikhattar and Mohammad A. Movahed Faculty of Chemical Engineering, University Technology of Malaysia, Johor Bahru, Malaysia Received 26 September 2011; accepted 25 January 2013 * Corresponding author address: Mohammad H. Mokhtari, Department of Remote Sensing, Faculty of Geo-Information and Real Estate, Universiti Teknologi Malaysia, Skudai, Johor 81310, Malaysia. E-mail address:
[email protected] DOI: 10.1175/2011EI000424.1
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ABSTRACT: The current Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER)-based broadband albedo model requires shortwave infrared bands 5 (2.145–2.185 nm), 6 (2.185–2.225 nm), 8 (2.295–2.365 nm), and 9 (2.360–2.430 nm) and visible/near-infrared bands 1 (0.52–0.60 nm) and 3 (0.78–0.86 nm). However, because of sensor irregularities at high temperatures, shortwave infrared wavelengths are not recorded in the ASTER data acquired after April 2008. Therefore, this study seeks to evaluate the performance of artificial neural networks (ANN) in estimating surface albedo using visible/nearinfrared bands available in the data obtained after April 2008. It also compares the outcomes with the results of multiple linear regression (MLR) modeling. First, the most influential spectral bands used in the current model as well as band 2 (0.63–0.69 nm) (which is also available after April 2008 in the visible/nearinfrared part) were determined by a primary analysis of the data acquired before April 2008. Then, multiple linear regression and ANN models were developed by using bands with a relatively high level of contribution. The results showed that bands 1 and 3 were the most important spectral ones for estimating albedo where land cover consisted of soil and vegetation. These two bands were used as the study input, and the albedo (estimated through a model that utilized bands 1, 3, 5, 6, 8, and 9) served as a target to remodel albedo. Because of its high collinearity with band 1, band 2 was identified less effectively by MLR as well as ANN. The study confirmed that a combination of bands 1 and 3, which are available in the current ASTER data, could be modeled through ANN and MLR to estimate surface albedo. However, because of its higher accuracy, ANN method was superior to MLR in developing objective functions. KEYWORDS: Artificial neural network; ASTER; Albedo; Multiple linear regression; Visible/near-infrared
1. Introduction Surface albedo is a key physical parameter for estimating shortwave radiation balance and, as a result, for modeling land surface energy balance (Bhattacharya et al. 2009). Albedo is used to estimate the amount of energy reflected back to the space in the energy balance equation. The energy absorbed by Earth’s surface, considered as residual, is determined when the incoming solar energy is subtracted from the outgoing solar energy in the energy balance equation. Areas with higher albedo values absorb less radiative energy and, as a result, lower convective overturning causes a reduction in precipitation and evaporation (Liang 2001). Therefore, it affects the amount of the energy absorbed by Earth’s surface and defines Earth’s features temperature, which is important in climatic and environmental studies (Dickinson 1983; Liang et al. 1999). It has been proved that land cover, surface roughness, and surface water change (through human manipulation of agricultural infrastructures and other anthropogenic factors) have an important role in continent-scale albedo trends and, as a result, in climatic changes (Doughty et al. 2012; Loarie et al. 2010). Currently, land surface albedo is estimated from bidirectional reflectance measurements using satellite sensors. The detected reflectance data are influenced by sensors’ field of view, viewing geometry meteorological conditions, and atmospheric optical parameters (Pinker and Stowe 1990). Therefore, the knowledge of atmospheric conditions and land surface attributes is a requisite for determining surface albedo by satellite data (Liang et al. 2002). Since satellite sensors measure narrow bands at the top of the atmosphere, reflectance data are
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Table 1. ASTER bands and radiometric characteristic (Thome et al. 1998). Spectral range Visible/near-infrared
Shortwave infrared
Thermal infrared
Band No. 1 2 3N 3B 4 5 6 7 8 9 10 11 12 13 14
Spectral range (mm)
Spatial resolution
Used for albedo estimation
0.52–0.60 0.63–0.69 0.78–0.86 0.78–0.86 1.600–1.700 2.145–2.185 2.185–2.225 2.235–2.285 2.295–2.365 2.360–2.430 8.125–8.475 8.475–8.825 8.925–9.275 10.25–10.95 10.95–11.65
15 m
* *
30 m * * * * 90 m
converted to surface broadband albedo by extensive radiative transfer simulation and then by linking simulated top-of-atmosphere reflectance with surface broadband albedo using a nonparametric regression algorithm (Liang et al. 2002) or following the method presented by the National Aeronautics and Space Administration (NASA) Moderate Resolution Imaging Spectroradiometer (MODIS) science team (Schaaf et al. 2002). The Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) moderate-resolution satellite data were provided by the Japanese Ministry of International Trade and Industry for NASA. It was launched on 18 December 1999 as a part of NASA’s Earth Observing System (EOS) AM-1 spacecraft (Allen et al. 1998). It has been widely used for the field-scale modeling of surface energy balance and estimation of evapotranspiration (Van Der Kwast et al. 2009; Sarwar and Bill 2007; Gowda et al. 2007). ASTER instrument characteristics are shown in Table 1. Because of the irregular temperature of sensors, ASTER shortwave infrared (SWIR) detectors stopped functioning. So, the ASTER data acquired after April 2008 are useless for estimating the shortwave balance and, in turn, for surface energy balance modeling (ERSDAC 2010). Moreover, the current available ASTER-based total shortwave albedo model, which requires shortwave infrared bands of 5, 6, 8, and 9 in addition to bands 1 and 3 of the visible/near-infrared (VNIR) part, is not applicable to the data acquired after April 2008. Over the past decades, broad attention has been paid to certain flexible computational techniques such as multivariate analysis and data mining in order to develop complex biophysical modeling processes in studies of natural resources such as water resources and hydrology (French et al. 1992; Minns and Hall 1996; Jain et al. 2004; Strobl and Forte 2007), evapotranspiration (Kisi 2006; Kumar et al. 2010; Rahimi Khoob 2008; Kisi and Guven 2010; Landeras et al. 2009; Kisi 2008; Martı´ et al. 2011), drainage prediction (Yang et al. 2000), streamflow prediction (Besaw et al. 2010; Jeong and Kim 2005), estimation of solar radiation (Senkal and Kuleli 2009), spatiotemporal interpolation of climatic variables (Antonic et al. 2001), and estimation of spatially varying albedo and optical thickness in radiation
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slab (Bokar 1999). The applicability of these methods in selecting the best variables and predicting dependent variables has been proved where a limited number of input variables are available (Konno and Takaya 2008; Kisi 2007; Jain et al. 2008; Rahimi Khoob 2009). Likewise, a neural network has been identified as an efficient technique for estimating broadband surface inherent albedos from satellite data without atmospheric information (Liang et al. 1999). Linear regression analysis is the most commonly used method to convert satellite-derived narrowband albedo into broadband albedo (Key 1996; Stroeve et al. 1997; Valiente et al. 1995). Several models with different band combinations for various satellite data have been developed via a regression analysis of the surface spectra reflectance and reflective spectral region of satellite data. However, the estimation of surface broadband albedo from multispectral data—including ASTER, Advanced Very High Resolution Radiometer (AVHRR), Goddard Earth Observing System (GEOS), Landsat thematic mapper (TM)/Enhanced Thematic Mapper Plus (ETM1), Multiangle Imaging Spectroradiometer (MISR), MODIS, Polarization and Directionality of Earth Reflectances (POLDER), and VEGETATION (VGT)—has been considered broadly and modeled accurately (Liang et al. 1999; Liang 2004; Liang 2001). For instance, based on extensive radiative transfer simulations, certain linear conversion formulas have been presented for total shortwave albedo with six bands, total visible albedo with eight bands, and total near-infrared with four bands for ASTER data (Liang et al. 2002). This study aims at evaluating the capability of the available ASTER spectral bands (VNIR) that are currently recorded by sensors to predict surface albedo using computerized mathematical methods. The estimation of albedo was initially decided to be based on the ASTER data acquired before April 2008. It was done with all the necessary spectral bands at hand using weighing factors for spectrally integrated narrowband reflectance and following the model presented in the literature (Liang et al. 2002). This model has been simulated and developed in various atmospheric conditions, on a variety of surface types such as dark water and snow, from different solar zenith angles, and with various visibility values using extensive surface reflectance spectra (256) collected from the U.S. Geological Survey (USGS) digital library (Clark et al. 1993), Jet Propulsion Laboratory (JPL) spectral library, measurements by Dr. Salisbury at Johns Hopkins University (Salisbury et al. 1991) (available at http://speclab.cr.usgs.gov), and reflectance spectra of different surface cover types extracted from Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) imagery. The ground-based validation (including diverse soils, crops, and natural vegetation) results show that this model has met the required accuracy for the estimation of surface broadband albedo from ASTER data (R2 5 0.839). The residual standard errors of fitting and the ground-based validation were reported to be 0.0135 and 0.019, respectively, for this model (Liang et al. 2002; Liang 2004; Liang 2001). What this model offered as an outcome has been utilized as a truth in this study. In this study, the relationship between ASTER spectral bands and surface albedo was examined, and the most influential independent variables were identified. Then, the performance of the artificial neural network (ANN) and multiple linear regression models was inspected statistically for the estimation of surface albedo from the selected bands. Finally, via the ANN and multiple linear regression, an examination was made of the potential of ASTER visible/near-infrared spectral bands in producing surface broadband
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Figure 1. The location of study area: ASTER image with color composites of 3, 2, 1 obtained on 4 Jul 2000.
albedo equivalent to the existing model. As far as the available literature is concerned, there is no evaluation ever reported of the potential of ASTER VNIR spectral bands in estimating broadband albedo from soil and vegetation surface, nor is there any model ever presented for these classes, which are common in agricultural studies. The presented methodology is expected to solve the limitations of using the ASTER data acquired after April 2008 to estimate surface broadband albedo and, as a result, to model surface energy balance.
2. Methodology 2.1. Study area and dataset The study site is located in an agricultural area in the southern part of the Yazd province, Iran. It expands between 31.288 and 31.488N and between 54.808 and 55.008E (Figure 1). Pistachio gardens and bare agricultural soil are the only land cover in the selected subset area. The red color produced by the ASTER band combination of 4, 3, and 2 in Figure 1 indicates vegetated areas (croplands). Non-croplands (desert pavement coverage) outside the agricultural area are indicated by white to dark gray colors. The areas under topographic consideration are almost flat. The average elevation in the subset area is about 1600 m MSL. According to DeMarton’s classification method (Hipps et al. 1985), this area is regarded as an arid region based on data from the Islamic Republic of Iran Meteorological Organization (IRIMO).
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2.2. ASTER data and preprocessing ASTER cloud-free level 1B satellite data taken on 4 July 2000 were obtained from NASA Land Processes Distributed Active Archive Center (LP DAAC). Level 1B data, the recorded radiance at the sensor (W m22 sr21 mm21 ), are produced from level 1A by applying a radiometric coefficient. The data taken before April 2008 allowed the estimation of surface albedo from spectral bands both with and without shortwaves. In this study, atmospheric correction was carried out to remove the effects of changes in sun–satellite geometry and aerosol scattering. Also, the atmospheric correction of solar reflective (VNIR and SWIR) bands was done to provide radiation reflected at the surface (Kimes and Sellers 1985; Allen et al. 1998). The atmospheric correction of ASTER data required outside sources since the system was not designed to repossess the atmospheric information. In this study, Fast Line-of-Sight Atmospheric Analysis of Spectral Hypercubes (FLAASH) was performed on ASTER VNIR and SWIR bands using climatic parameters and atmospheric modeling. The output was then converted to the reflectance using radiative transfer codes (atmospheric modeling). The at-sensor radiance was converted to the at-surface reflectance through this process (Allen et al. 2001). The image was geo-registered to the Universal Transverse Mercator (UTM) World Geodetic System 1984 (WGS84) zone 40 (the zone of the study site) coordinate system by using 25 well-distributed ground control points (GCPs). The result was evaluated by calculating the root-mean-square error (RMSE; RMSE , 0.32). First-order polynomial transformation with the nearest neighbor resampling method was applied to the images to fit the image coordinates to the coordinates of the ground control points. As a final preprocessing stage, the subset of the agricultural area was extracted from the image scene. The following model, which is presented by Liang (Liang 2004) and utilizes six independent variables with their weighing coefficients, was used to convert the narrowband reflectance of NIR and SWIR spectral bands to broadband albedo. The model was generated by extensive radiative transfer simulation using atmospheric radiative transfer (SBDRT) codes (Ricchiazzi et al. 1998) aASTER 5 0:484a1 1 0:335a3 2 0:324a5 1 0:551a6 1 0:305a8 2 0:367a9 2 0:0015 , (1) where aASTER is the ASTER image based on surface albedo and a1 through a9 are the ASTER spectral bands in reflectance. 2.3. Multiple linear regression analysis To examine the strength and direction of the relationship between ASTER spectral bands and the estimated albedo through Equation (1) as well as to explore the amount of albedo variance accounted for by each spectral band, a bivariate correlation analysis was performed based on Pearson product-moment correlation coefficient. Then, a multiple regression analysis was performed to evaluate the amount of albedo variances explained by seven ASTER spectral bands (bands 1, 2, 3, 5, 6, 8, and 9) for predictive calculations of surface albedo. It should be noted that band 2 was not used in Equation (1) but, due to its availability on the data acquired after April 2008, its potential too was evaluated in predicting surface
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broadband albedo. The general form of the linear regression model is presented as follows: y 5 a0 1 a1 x1 1 a2 x2 1 1 ak xk 1 e,
(2)
where y is the dependent variable (DV) (albedo); a0 , a1 , x2 , . . . , ak are parameters that are estimated from the observed data of yt , x1 , x2 1 xkt ; and t 51, 2, . . . , T. Therefore, the equation is solved to find xi1 , xi2 1 xis that best fits the observed data: e 5 yt 2 (a0 1 a1 x1t 1 a2 x2t 1 1 ak xkt ),
t 5 1, 2, . . . , T .
(3)
Considering Equation (3), the aforementioned spectral bands were entered in the linear regression model, and the significance of the interrelationship between the spectral bands and albedo estimated via Equation (1) was evaluated based on the yielded prediction of each band with regard to the predictability of the other spectral bands. Then, the fundamental ASTER bands that present the patterns of correlations within a set of ASTER data bands were determined. The total variability of the dependent variable accounted for by the multiple linear regression was measured through the coefficient of determination (Tabachnick and Fidell 2007), f (a0 , a1 , x2 , . . . , ak ) 5
T P
[yt 2 (a0 1
P aj xjt )]2 ,
(4)
t51
and the sum of the absolute values of errors as g(a0 , a1 , x2 , . . . , ak ) 5
T P
jyt 2 (a0 1
P
aj xjt )j .
(5)
t51
The model presented in Equation (1) was used to estimate surface albedo from the preprocessed ASTER data acquired on 4 July 2000. Then, albedo as a dependent variable and the raster sets of bands 1, 2, 3, 5, 6, 8, and 9 in reflectance viewed as predictors were analyzed, in which each layer consisted of 513 608 pixels. In addition, to check against any infringement of the assumptions, the relationship between the independent variables (multicollinearity) and the extreme scores of the variables (outliers) was evaluated. Also, examination was made of various aspect of the data distribution such as normality, linearity, homoscedasticity, and independence of the residuals of variables required in the multiple linear regression analysis. The graphical and numerical results were evaluated by visual and statistical inspection, respectively. Subsequently, a forward regression analysis was conducted to evaluate the potential of particular ASTER bands in predicting albedo when the effects of the other bands were controlled. The selection was based on the variance explained by each variable, and the most informative bands were selected successively in order. All the predictive models were generated via a stepwise forward regression analysis of the selected bands; estimates of albedo were then derived, which were plotted against the albedo computed from Equation (1). The MLR-based models were evaluated by calculating the correlation coefficients, mean-square error
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(MSE), and mean absolute error (MAE) between the albedo values of Equation (1) and those of new models. 2.4. Artificial neural network concept Although ANN models do not give an explanation on the estimation of parameters as much as MLR does, they are flexible computerized models that have the ability to accommodate both linear and nonlinear relationships among variables without the assumptions required in MLR. Therefore, in addition to MLR, the potential of ANN in retaining correct surface albedo from ASTER VNIR bands was examined in this study. Artificial networks consist of input, output, and computational units (nodes), namely, artificial neurons based on whose distribution various types of networks are determined (Bokar 1999). These interlinked units that are arranged in one or more layers are liable to calculate mathematical functions. In an ANN, each node is defined by three parameters, including the weights associated with each neuron input, activation function, and bias (Krose and Van der Smagt 1996). In this study, the optimal weights were determined when training the network was completed. In regard to the estimation of surface albedo, ASTER bands 1 and 3 were the inputs and the albedo estimated from six ASTER spectral bands [Equation (1)] was postulated as the target. Basically, a neuron, as a fundamental processing element, computes a weighed sum of its input signals for numbers of hidden layers and then applies an activation function to produce output signals. However, in a network, the number of layers and hidden neurons needs to be determined properly (Zhang et al. 1998). The determination of these parameters is based on several factors, such as the number of input and output units, the amount of available training patterns, the complexity of the problem, and the training algorithm. Therefore, overfitting and underfitting can be controlled and the ability of ANN generalization can be increased by selecting an appropriate architecture (Cerden˜a et al. 2007). Moreover, the function of training is increased by an increase in the network complexity. Nevertheless, this results in poor generalization particularly when the applicants deal with a small, noisy, and partially inaccurate training dataset (Liang 2003). In this study, the widely used MLP topology was selected to retrieve albedo from ASTER VNIR spectral bands. The back-propagation algorithm is the most common learning rule for multilayer perceptron. Back propagation consists of two major phases: namely, feed forward and backward. In the feed-forward phase, the external input information at the input nodes is propagated forward to compute the output information signal at the output unit (Cigizoglu 2005). Consequently, in the backward phase, the weights as connection strengths are determined based on the differences between the computed and the observed information signals at the output units. These processes are continued progressively and the estimated value is compared with the known value up to the end of the network training. The mathematical form of a neuron input–output can be expressed as yj 5
p P
wji yi ,
(6)
j50
where p is the number of inputs of a neuron, wji is the weight, yi is the weighed sum of input signals, and i represents the number of hidden layers. For the detailed
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Figure 2. A scheme of MLP-based ANN with two inputs, one hidden layer (six nodes), and one output. (Gray and blue lines indicate synaptic weight > 0 and synaptic weight < 0, respectively.)
information of neural network structures, the reader is referred to Haykin (Haykin 1999). A scheme of the network generated in this study is shown in Figure 2. In this figure, the synaptic weight bars indicate the connection weights. The gray bars tend toward positivity and suggest values above zero, while the blue ones show values in the negative domain. The bar thickness implies how big or small the weight is; thicker gray bars refer to more positive values and bigger weights, while thicker blue ones suggest more negative values and smaller weights. The training process in ANN minimizes possible errors by adjusting the weights and the biases. For the present research, the most efficient training algorithm was selected based on the primary data analysis and the evaluation of the MSE of the training process. The Levenberg–Marquardt (LM) training algorithm (Marquardt 1963), which uses Newton’s method of minimizing error function, was selected to set up a purposeful relationship between inputs (bands 1 and 3 of ASTER data) and albedo in the equation as a target variable, E5
n 1P 2 (yi9 2 yi ) . n i51
(7)
The widely used log-sigmoid and linear (purelin) transfer functions were applied to the hidden layers and the output layer, respectively. The number of the units in the
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Table 2. Correlation matrix for ASTER spectral bands and albedo. Albedo B1 B2 B3 B5 B6 B8 B9
Albedo
B1
B2
B3
B5
B6
B8
B9
1
0.960* 1
0.957* 0.972* 1
0.785* 0.626* 0.659* 1
0.699* 0.751* 0.710* 0.338* 1
0.704* 0.754* 0.708* 0.325* 0.994* 1
0.789* 0.805* 0.808* 0.340* 0.694* 0.700* 1
0.724* 0.791* 0.749* 0.286* 0.957* 0.964* 0.828* 1
* Pearson correlation is significant at the P value of 0.01.
hidden layers was determined by trial and error as well as evaluating the corresponding errors. The standard error of training was set at 1 3 1025 in the networks. The data of the study were divided into three sets. First, 70% and 30% of the total data were assigned to training and testing, respectively. To avoid network overtraining and to control overfitting, the training set was divided into two sets: training and cross validation. The use of a small percentage (about 10%) of the training data as a crossvalidation set has been suggested by many researchers (Kucuk 2008; Kohavi 1995). Therefore, in this study, 10% of the training dataset (7% of total) was randomly taken and used for cross validation. As much as 63%, 30%, and 7% of the total data were thus used for training, testing the performance of the model, and cross validation, respectively. The output of the network was evaluated using MSE and MAE, MSE 5
n 1P 2 (yi9 2 yi ) n i51
MAE 5
n 1P jyi9 2 yi j. n i51
and
(8)
(9)
3. Results and discussion 3.1. Primary evaluation of the input variables As shown in Table 2, the data analysis proved a significant positive linear correlation between the spectral bands and albedo at the significance level (p value) of 0.01. The interrelationship of the spectral bands also suggested a high correlation between bands 1 and 2 of the visible/near-infrared as well as among bands 5, 6, and 9 of shortwave infrared region (presence of collinearity). Correlations between band 1 of the visible part and the shortwave bands (B5, B6, B8, and B9) were also remarkable. To verify the assumptions of data normality and presence of outliers, a normal probability plot and a residual scatterplot were created via multiple regression practice. Regarding the normal probability plot (Figure 3a), all the points were located along a strait diagonal line. This certified the absence of any major deviation of the variables from normality. Nonlinearity assumption was rejected since the scatterplot showed a point distribution in the form of a rectangle (versus a curved shape in the case of nonlinearity). Moreover, each predicted score value and normally distributed
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Figure 3. (a) Normal probability plot and (b) residual scatterplot (dependent variable 5 albedo).
residuals formed a well-centralized rectangle, with the residual points lying symmetrically from the center as shown in Figure 3b. Thus, in regard to this figure, any high violation of the assumption was rejected, although a small number of points were observed as outliers (presented cut of point value of 63.3 by Tabachnick and Fidell 2007). Heteroscedasticity would occur when the band enclosing width of residuals became large at a higher predicted value. With reference to the same scatterplot, this assumption was discarded as well. Based on Mahalanobis distance test and chi-square (x2) critical value of 24.32 at P value , 0.001 (Tabachnick and Fidell 2007), 3325 points out of 513 608 (0.64%) were identified as outliers. It should be noted that, in remote sensing and energy balance application, the decision about removing outliers is difficult since the prediction of a parameter for a feature is sometimes based on another parameter, especially when they are associated with the highest and lowest values in the selected subset area. Thus, no action was taken until the most predictive spectral bands were determined. These points were located in the range of 0.20–0.32 of estimated albedo, including vegetation and soil land covers. In this study, variance inflation factor (VIF) and tolerance indices (1 2 R2) were used to investigate the presence of collinearity in the independent variables. In regard to the presented critical value of VIF, variance inflation factor (.10), and tolerance index (,0.1) for occurrence of multicollinearity (Pallant 2007), band 3 with the tolerance of 0.402 had less collinearity with the other independent variables. Next to band 3, there were bands 8, 1, and 2, with the tolerance values of 0.11, 0.041, and 0.042, respectively. This could be roughly predicted from the correlation matrix of the variables as presented in Table 2. 3.2. Regression results The result of variance analysis showed that, based on the predictive potential of ASTER spectral band, six models can be presented at the significance level of 0.01. The zero value of the standard error of regression for all the models indicated that
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Table 3. Statistics of information of the first three albedo models presented by MLR. Unstandardized coefficients Model
B
1a Constant 0.077 Band 1 0.868 2b Constant 0.008 Band 1 0.696 Band 3 0.299 3c Constant 20.028 Band 1 0.507 Band 3 0.356 Band 8 0.177 a b c
Standardized coefficients Standard error Beta 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
— 0.960 — 0.770 0.302 — 0.560 0.360 0.216
Adjusted Standard error R2 of the estimate
t
Significance
R2
972.564 2443.142 108.337 2801.967 1100.138 2450.739 1922.070 1958.116 891.858
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.921
0.921
0.007 258
0.976
0.976
0.003 961
0.991
0.991
0.002 481
Predictors: (constant), band 1. Predictors: (constant), band 1, and band 3. Predictors: (constant), band 1, band 3, and band 8.
parameter estimates do not vary from sample to sample. The first model consisted of band 1 and a constant value. Therefore, if one ASTER spectral band tends to contribute in the model, band 1 of this dataset is the choice that explains 92% of the variability of albedo. Using two variables, preferably bands 1 and 3 offered by multiple linear regression analysis, 0.976% of the variability of albedo is accounted for. If three independent variables are preferred to be included in the model, band 8 is the next choice. In this case, these three bands account for 99% of the variability of albedo. The coefficients of the first three proposed models are illustrated in Table 3. Band 6, as the next independent variable, does not impart a considerable variance of dependent variables if bands 1, 3, and 8 are selected beforehand in the model. However, in case of unavailability of bands 5, 6, 8, and 9 (data acquired after April 2008), model 2 (constant, bands 1 and 3) is the option. It should be noted that band 2, due to its collinearity with band 1 (0.972), was not included in the models by this method. If other bands are preferred to be included, bands 9, 8, 6, and 5 are proposed, respectively. The statistical analysis of the models showed that collinearity occurs when bands 6, 9, and 5 are added to model 3 by turns (tolerance . 0.1 and VIF , 10). In terms of collinearity of spectral bands, band 3 was identified to be the least collinear with the other bands. Since this study sought to evaluate the potential of MLP and ANN in estimating albedo and compare their performance, the same dataset of training selected for ANN modeling was used to develop a regression model. Subsequently, the model generated by MLR was applied to estimate albedo from the test dataset. The result of the two-band (bands 1 and 3) regression model is shown in Equation (10). The outputs of Equation (10) were then compared with the albedo estimated by Equation (1). The coefficients presented for model 2 in Table 3 are slightly varied from those presented in Equation (10) since the former are generated from the regression of all the cases (training, cross validation, and testing datasets), aASTER
VNIR
5 0:697a1 1 0:298a3 1 0:008 .
(10)
The goodness of fit of the regression model was evaluated by plotting the albedo of Equation (1) against those calculated by Equation (10). The plots of the train and
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Figure 4. The scatterplots of MLR-predicted albedo vs albedo calculated from Equation (1) for (a) train and (b) test datasets.
test data versus Equation (1) are shown in Figures 4a and 4b, respectively. The regression equation had an R2 and MSE of 0.976 and 1.6 3 1026, respectively. An MSE of 7.557 3 1026was observed when the independent variables of the test data were used to estimate albedo in the regression model. 3.3. ANN results The importance of analyzing ASTER spectral bands to estimate albedo via ANN is shown in Figure 5. This figure indicates that bands 1 and 3 are the most important variables and they are followed by bands 6, 9, 8, 5, and 2. Therefore, these bands, as the most influential ones, were used as inputs for modeling albedo via an artificial neural network method. In this study, different neural network (NN) architectures were employed. In each structure, hidden neurons, activation function, and different training parameters such as learning rate were examined. The best ANN architecture was chosen based on the minimum MSE computed for the training data. After applying all the structures in an MATLAB environment, it was found that NN with a back-propagation algorithm and Levenberg–Mardquart (trainlm) training algorithm leads to the best performance (i.e., higher performance of the algorithm as shown in Table 4). The model consisted of two inputs, one hidden layer with six units, and one output. To evaluate the model performance, the test data were laid out against the ANN estimates. The plots of the ANN model-predicted albedo versus the albedo calculated from Equation (1) are shown in Figures 6a and 6b for training and testing sets, respectively. The line 1:1 indicates a good agreement between the ANN output and those calculated by Equation (1) for both the training and testing sets. The training was converged at MSE of 1.4 3 1026 and R2 of 0. 979. The MSE of 6.454 3 1026 with the same R2 of training process was observed when the ANN model performance was evaluated by the test data.
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Figure 5. Importance and normalized importance of each ASTER spectral band in determining the ANN model.
3.4. Comparison of MLR and ANN results Bands 1 and 3 of the ASTER data were identified as the most informative ones for estimating albedo by both MLR and ANN. However, a comparison of the ranks of bands by MLR and ANN showed that the order was changed from the third variable on. This is possibly due to the presence of a slight nonlinearity relationship between these bands and albedo. Band 2 was identified as the least influential parameter by both ANN and MLR methods when the other bands were included. A t test was carried out to compare the output means of ANN and MLR models. The result suggested a significant difference (P value of 0.05) between ANN and MLR Table 4. Performance of training algorithms. Function
Algorithm
MSE
Traingdm Traingdx Traincgf Traincgp Traincgb Trainscg Trainlm Trainrp Trainoss Trainbfg
Gradient descent with momentum back propagation Gradient descent with momentum and adaptive learning rate back propagation Fletcher–Powell conjugate gradient back propagation Polak–Ribiere conjugate gradient back propagation Powell–Beale conjugate gradient back propagation Scaled conjugate gradient back propagation Levenberg–Marquardt back propagation Resilient back propagation (Rprop) One step secant back propagation Broyden–Fletcher–Goldfarb–Shanno quasi-Newton back propagation
0.004 41 0.004 87 0.003 71 0.004 97 0.001 49 0.003 85 0.000 98 0.004 50 0.003 39 0.004 05
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Figure 6. The scatterplots of ANN-predicted albedo vs albedo calculated from Equation (1) for (a) training and (b) testing datasets.
estimates. The MSE of estimates in ANN was observed to be less than that in MLR in both the test and train datasets. Moreover, the ANN model produced a relatively improved R2 value (0.979) as compared to the MLR method (0.976). To evaluate the errors in detail, absolute error percentiles (5, 10, 25, 50, 75, 90, 95, and 99) were calculated and shown in Table 5. The table indicates that ANN has a higher predictive performance as compared with MLR. Nevertheless, as shown in Figure 7, albedo computed by MLR was compared with those showing a higher performance (ANN), and a high correspondence was found between the two statistical methods. In this figure, line 1:1 indicates a high level of correlation between MLR and ANN models although a slight deviation is observed from the albedo values higher than 0.35. As a result, the good performance of the mathematical methods presented in this study is hereby confirmed. Similarly, a good agreement was observed where projection pursuit regression albedo results were compared with those presented by ANN over the highly variable land surface conditions in the study by Liang (Liang 2003).
4. Conclusions This study was conducted with a focus on the potential of ANN and MLR methods in estimation of surface albedo from ASTER visible/near-infrared Table 5. Absolute error percentiles. Absolute error percentiles 5 Train data MLR ANN Test data MLR ANN
0.000 0.000 0.000 0.000
10 246 220 249 216
0.000 0.000 0.000 0.000
25 498 439 493 430
0.001 0.001 0.001 0.001
50 258 116 259 105
0.002 0.002 0.002 0.002
75 651 376 640 367
0.004 0.004 0.004 0.004
90 478 105 470 104
0.006 0.006 0.006 0.006
95 375 030 366 010
0.007 0.007 0.007 0.007
99 678 397 619 362
0.010 0.010 0.010 0.010
605 613 624 580
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Figure 7. Comparison of MLR and ANN model outputs.
spectral bands. According to the results, bands 1 and 3 of the ASTER data are considered sufficient to estimate albedo where land cover consists of soil and vegetation. A good agreement (R2 . 0.97) is found between the models output in this study and the output of model published in the literature [Equation (1)]. After applying the corresponding coefficients, the sum of the four omitted band ranges (bands 5, 6, 8, and 9) is found to be between 0.025 and 0.055 [Equation (1)]; it is far less than the sum of the ranges of bands 3 and 1 in this case study. In other words, ASTER shortwave infrared bands are less correlated with albedo than are visible/ near-infrared spectral bands in this area. Although bands 1 and 2 of the ASTER satellite data have been already used to model visible albedo (Liang 2001), these bands are found highly correlated in this case study. Therefore, band 2 causes collinearity and is not recommended to be used in combination with band 1 in a model where land cover consists of soil and vegetation. This band has been identified as the most ineffective variable within the ASTER bands 1, 2, 3, 5, 6, 8, and 9 for estimating surface broadband albedo from MLR and ANN methods. The presence of collinearity among bands 1, 5, 6, 8, and 9 in a regression model suggests that these bands do not provide much independent information in the model, and thus the coefficients cannot be estimated with a lot of certainty if they are incorporated in an albedo model where land cover consists of soil and vegetation. In the case of collinearity, it is difficult to separate the impact of variables on the model output. Therefore, it is difficult to make sure whether the model estimates have been representative of the population. As a result, a larger number of observations are required. An albedo model produced from the most responsive predictor(s) (spectral bands) for a limited number of land-cover classes (e.g., soil and vegetation, which are common in agricultural studies) can be hence more efficient and reliable than a generalized model developed from collinear variables
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for various land-cover types. The regression model of this study with a maximum error of 13% can be used to calculate surface broadband albedo from reflective bands 1 and 3 of the ASTER data where land cover consists of soil and vegetation (albedo ranging from 0.15 to 0.45). The statistical results confirm a significant difference between MLR and ANN outputs at P value , 0.05. It is also proved that ANN with lower MSE and MAE can achieve a relatively higher accuracy rate of prediction than the equation established by the regression method. Unlike MLR, ANN does not require any assumptions about underlying population distributions. As a result, reasonable models can be developed on small datasets. Because of accurate pattern recognition and generalization, the ANN model can be more transferable than the regression function. ANN has also shown to be efficient in predicting surface total shortwaves and near-infrared inherent albedos from TOA narrowband albedos without atmospheric information (Liang et al. 1999; Liang 2003). Because of its higher accuracy and flexibility, the ANN method is superior to the MLR method in developing certain objective functions. However, it somehow lacks the explanation of parameters. So, developing an optimal network takes some time. In this case, statistical information derived through the MLR process can be useful in interpreting the relationship between variables even if modelers use ANN to estimate albedo. It does not lie within the scope of this study to determine whether bands 1 and 3 are capable of estimating surface albedo via MLR and ANN on lands with other cover types. More verification is needed for locations where albedo values are out of the range as considered in this study (i.e., albedo beyond 0.15–0.45 and snow or water bodies with different levels of turbidity). Acknowledgments. The authors are indebted to Dr. Shunlin Liang and his co-researchers for the valuable information and research outputs, which were essential for the outcome of this study. Also, we would like to acknowledge NASA Land Processes Distributed Active Archive Center (LPDAAC) for providing the ASTER data of this research.
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