Development of Hybrid Method for the Modeling of Evaporation and Evapotranspiration

contains 23 related to the modeling and simulation of evapotranspiration (ET) and remote sensing-based energy balance determination of ET. These areas are at the forefront of technologies that quantify the highly spatial ET from the Earth's surface. The topics describe mechanics of ET simulation from partially vegetated surfaces and stomatal conductance behavior of natural and agricultural ecosystems. Estimation methods that use weather based methods, soil water balance, the Complementary Relationship, the Hargreaves and other temperature-radiation based methods, and Fuzzy-Probabilistic calculations are described. A critical review describes methods used in hydrological models. Applications describe ET patterns in alpine catchments, under water shortage, for irrigated systems, under climate change, and for grasslands and pastures. Remote sensing based approaches include Landsat and MODIS satellite-based energy balance, and the common process models SEBAL, METRIC and S-SEBS. Recommended guidelines for applying operational satellite-based energy balance models and for overcoming common challenges are made.

compared the results of the lysimeters with those of 20 different empirical equations and methodologies for ET o measurements. It was found that PM ET o model showed the optimal results over all the climatic zones. If the observed/measured data for ET o does not exist, therefore, PM ET o model can be considered as a standard methodology to calculate ET o . In Gwangju and Haenam stations which were selected for this study, there are no observed data for ET o using a lysimeter. The data calculated using PM ET o model can be assumed as the observed ET o , whose reliability was verified by many previous studies. All calculation procedures as used in PM ET o model are based on the FAO guidelines as laid down in the publication No. 56 of the Irrigation and Drainage Series of FAO ″Crop Evapotranspiration-Guidelines for Computing Crop Water Requirements″ (1998 Allen et al. (1998) and can be shown as the following equation (1 (1) where FAO-56 PM ET o = the grass reference evapotranspiration (mm/day); R n = the net radiation at the crop surface (MJ/m 2 day); G = the soil heat flux density (MJ/m 2 day); T = the mean air temperature at 2m height (•C); u 2 = the wind speed at 2m height (m/sec); e s = the saturation vapor pressure (kPa); e a = the actual vapor pressure (kPa); e s -e a = the saturation vapor pressure deficit (kPa); Δ = the slope vapor pressure curve (kPa/•C); and γ = the psychometric constant (kPa/•C). FAO CROPWAT 8.0 computer program has been used to calculate FAO-56 PM ET o and extraterrestrial radiation (R a ). FAO CROPWAT 8.0 computer program allows the user to enter the climatic data available including maximum temperature (T max ), minimum temperature (T min ), mean relative humidity (RH mean ), mean wind speed (WS mean ), and sunshine duration (SD) for calculating FAO-56 PM ET o . On the base of climatic data available, FAO CROPWAT 8.0 computer program calculates the solar radiation reaching soil surface. Fig. 1

Stochastic model
3.1 Univariate seasonal periodic autoregressive moving average model Stationary ARMA models have been widely applied in stochastic hydrology for modeling of annual time series where the mean, variance, and the correlation structure do not depend on time. For seasonal hydrologic time series, such as monthly series, seasonal statistics including the mean and standard deviation may be reproduced by a stationary ARMA model by means of standardizing the underlying seasonal series. Hydrologic time series such as monthly streamflows, PE, and FAO-56 PM ET o are usually characterized by different dependence structure depending on the season (Salas, 1993). One may extend Univariate Seasonal periodic autoregressive (PAR) model to include periodic moving average (MA) parameters. Such a model is Univariate Seasonal periodic autoregressive moving average (PARMA) model and is expressed as Univariate Seasonal PARMA(p,q) model. The stochastic models are generally simple to use. When the errors, however, happen in model identification and parameter estimation, the generated data using the stochastic models cannot reconstruct the statistical properties of the observed data exactly. Furthermore, the high-order PARMA(p,q) models have generally many parameters, and the calculation process is much complex (Salas et al., 1980). In this study, the author determined in advance 4 kinds of Univariate Seasonal PARMA(p,q) models including PARMA(1,1), PARMA(1,2), PARMA (2,1), and PARMA(2,2), which are the low-order models and contain the seasonal properties. In general, the low-order Univariate Seasonal PARMA(p,q) models are useful for the periodic hydrologic time series modeling (Salas et al., 1982). Furthermore, the author generated 100 years data in advance using each Univariate Seasonal PARMA(p,q) model for the climatic variables of the neural networks models, respectively. As a result, the author selected Univariate Seasonal PARMA(1,1) model, which shows the best statistical properties and is simple in parameter estimation. Univariate Seasonal PARMA(1,1) model has been applied to monthly streamflow time series from the previous studies (Tao and Delleur, 1976;Hirsch, 1979), and is shown as the following equation (2).

Construction of Univariate Seasonal PARMA(p,q) model
The author used Univariate Seasonal PARMA(1,1) model to generate the sufficient training dataset, and obtained two generated samples. They included the input nodes/variables including mean temperature (T mean ), maximum temperature (T max ), minimum temperature (T min ), mean dew point temperature (DP mean ), minimum relative humidity (RH min ), mean relative humidity (RH mean ), mean wind speed (WS mean ), maximum wind speed (WS max ), and sunshine duration (SD) in mean values and the output nodes/variables including PE and FAO-56 PM ET o in total values, respectively. Furthermore, they were generated for 100 years (Short-term), 500 years (Mid-term), and 1000 years (Long-term), respectively. The author selected the second generated sample, and the first 50 years of the 100, 500, and 1000 years was abandoned to eliminate the biases, respectively. The parameters of Univariate Seasonal PARMA(1,1) model were determined using the method of approximate least square, respectively.

Support Vector Machine Neural Networks Model (SVM-NNM)
SVM-NNM has found wide application in several areas including pattern recognition, regression, multimedia, bio-informatics and artificial intelligence. Very recently, SVM-NNM is gaining recognition in hydrology (Dibike et al., 2001;Khadam & Kaluarachchi, 2004). SVM-NNM implements the structural risk minimization principle which attempts to minimize an upper bound on the generalization error by striking a right balance between the training performance error and the capacity of machine. The solution of traditional neural networks models such as MLP-NNM may tend to fall into a local optimal solution, whereas global optimum solution is guaranteed for SVM-NNM (Haykin, 2009). SVM-NNM is a new kind of classifier that is motivated by two concepts. First, transforming data into a high-dimensional space can transform complex problems into simpler problems that can use linear discriminant functions. Second, SVM-NNM is motivated by the concept of training and using only those inputs that are near the decision surface since they provide the most information about the classification. The first step in SVM-NNM is transforming the data into a high-dimensional space. This is done using radial basis function (RBF) that places a Gaussian at each sample data. Thus, the feature space becomes as large as the number of sample data. RBF uses backpropagation to train a linear combination of the gaussians to produce the final result. SVM-NNM, however, uses the idea of large margin classifiers for the training performance. This decouples the capacity of the classifier from the input space and at the same time provides good generalization. This is an ideal combination for classification (Vapnik, 1992(Vapnik, , 2000Principe et al., 2000;Tripathi et al., 2006). In this study, the basic ideas of SVM-NNM are reviewed. Consider the finite training pattern   ii x, y . where n i x   = a sample value of the input vector x considering of N training patterns; and n i y   = the corresponding value of the desired model output. A nonlinear transformation function ()   is defined to map the input space to a higher dimension feature space, h n  . According to Cover's theorem (Cover, 1965), a linear function, f( )  , could be formulated in the high dimensional feature space to look for a nonlinear relationship between inputs and outputs in the original input space. It can be written as the following equation (3).
where y = the actual model output. The coefficient w and b are adjustable model parameters. In SVM-NNM, we aim at minimizing the empirical risk. It can be written as the following equation ( Following regularization theory (Haykin, 2009), the parameters w and b are calculated by minimizing the cost function. It can be written as the following equation (5).
subject to the constraints: 1) ii i yyεξ   i = 1, 2, ... , N, 2) ii i yyεξ     i = 1, 2, ... , N, and 3) ii ξ ,ξ 0   i = 1, 2, ... , N. where ε ψ (w,ξ,ξ )  = the cost function; ii ξ ,ξ  = positive slack variables; and C = the cost constant. The first term of the cost function, which represents weight decay, is used to regularize weight sizes and to penalize large weights. This helps in improving generalization performance (Hush and Horne, 1993). The second term of the cost function, which represents penalty function, penalizes deviations of y from y larger than  using Vapnik's ε-insensitive loss function. The cost constant C determines the amount up to which deviations from ε are tolerated. Deviations above ε are denoted by i ξ , whereas deviations below -ε are denoted by i ξ  . The constrained quadratic optimization problem can be solved using the method of Lagrangian multipliers (Haykin, 2009). From this solution, the coefficient w can be written as the following equation (6).
where ii α ,α  = the Lagrange multipliers, which are positive real constants. The data points corresponding to non-zero values for ii (αα )   are called support vectors. In SVM-NNM to calculate PE and FAO-56 PM ET o , there are several possibilities for the choice of kernel function, including linear, polynomial, sigmoid, splines and RBF. In this study, RBF is used to map the input data into higher dimensional feature space. RBF can be written as the following equation (7).
where i, j = the input layer and the hidden layer; , and has a constant value; and σ = the width/spread of RBF, which can be adjusted to control the expressivity of RBF. The function for the single node of the output layer which receives the calculated results of RBF can be written as the following equation (8).
where k = the output layer; k G = the calculated value of the single output node; and B = the bias in the output layer. Equation (8), finally, takes the form of equation (9) and (10) where 2 Φ ()  = the linear sigmoid transfer function; 1 W = the specific weights connected to the output variable of PE; and 2 W = the specific weights connected to the output variable of FAO-56 PM ET o . A number of SVM-NNM computer programs are now available for the modeling of PE and FAO-56 PM ET o . NeuroSolutions 5.0 computer program was used to develop SVM-NNM structure. Fig. 2 shows the developed structure of SVM-NNM. From the Fig. 2, the input nodes/variables of climatic data are mean temperature (T mean ), maximum temperature (T max ), minimum temperature (T min ), mean dew point temperature (DP mean ), minimum relative humidity (RH min ), mean relative humidity (RH mean ), mean wind speed (WS mean ), maximum wind speed (WS max ), and sunshine duration (SD) in mean values (01/1985-12/1990). The output nodes/variables of climatic data are PE and FAO-56 PM ET o in total values (01/1985-12/1990).
The developed structure of SVM-NNM

Study scope and data
In this study, Gwangju and Haenam stations from the Yeongsan River catchment are selected among the 71 weather stations including Jeju-do under the control of the Korea meteorological administration (KMA). They have possessed long-term climatic data dating back over at least 30 years. The Yeongsan River catchment covers an area of 3455 km 2 , and lies between latitudes 34.4°N and 35.2°N, and between longitudes 126.2°E and 127.0°E. Fig.  3 shows the Yeongsan River catchment including Gwangju and Haenam stations. The climatic data, which was necessary for the modeling of PE and FAO-56 PM ET o , were collected from the Internet homepage of water management information system (www.wamis.go.kr) and the Korea meteorological administration (www.kma.go.kr). The performance of SVM-NNM to account for calculating the monthly PE and FAO-56 PM ET o was evaluated using a wide variety of standard statistics index. A total of 3 different standard statistics indexes were employed; the coefficient of correlation (CC), root mean square error (RMSE), and Nash-Sutcliffe coefficient (R 2 ) (Nash & Sutcliffe, 1970;ASCE, 1993). Table 1 shows summary of the statistics index in this study. where i y (x) = the calculated PE and FAO-56 PM ET o (mm/month); i y (x) = the observed PE and FAO-56 PM ET o (mm/month); y u = mean of the calculated PE and FAO-56 PM ET o (mm/month); y u = mean of the observed PE and FAO-56 PM ET o (mm/month); and n = total number of the monthly PE and FAO-56 PM ET o considered. A model which is effective in the modeling of PE and FAO-56 PM ET o accurately, and efficient in capturing the complex relationship among the various inputs and output variables involved in a particular problem, is considered the best. CC, RMSE, and R 2 statistics quantify the efficiency of SVM-NNM in capturing the extremely complex, dynamic, and nonlinear relationships (Kim, 2011). Table 1. Summary of statistics indexes

Data normalization
The climatic data used in this study including mean temperature (T mean ), maximum temperature (T max ), minimum temperature (T min ), mean dew point temperature (DP mean ), minimum relative humidity (RH min ), mean relative humidity (RH mean ), mean wind speed (WS mean ), maximum wind speed (WS max ), and sunshine duration (SD) were normalized for preventing and overcoming problem associated with the extreme values. An important reason for the normalization of input nodes is that each of input nodes represents an observed value in a different unit. Such input nodes are normalized, and the input nodes in dimensionless unit are relocated. The similarity effect of input nodes is thus eliminated (Kim et al., 2009). According to Zanetti et al. (2007), by grouping the daily values into averages, ET o may be calculated due to their highest stabilization. For data normalization, the data of input and output nodes were scaled in the range of [0 1] using the following equation (11).
where norm Y = the normalized dimensionless data of the specific input node/variable; i Y = the observed data of the specific input node/variable; min Y = the minimum data of the specific input node/variable; and max Y = the maximum data of the specific input node/variable.

Training performance
The method for calculating parameters is generally called the training performance in the neural networks model category. The training performance of neural networks model is iterated until the training error is reached to the training tolerance. Iteration means one completely pass through a set of inputs and target patterns or data. In general, it is assumed that the neural networks model does not have any prior knowledge about the example problem before it is trained (Kim, 2004). A difficult task with the neural networks model is to choose the number of hidden nodes. The network geometry is problem dependent. This study adopted one hidden layer for the construction of SVM-NNM since it is well known that one hidden layer is enough to represent PE and FAO-56 PM ET o nonlinear complex relationship (Kumar et al., 2002;Zanetti et al., 2007). The testing performance in the modeling of PE and FAO-56 PM ET o , therefore, is carried out using the optimal parameters, which are calculated during the training performance. The hybrid method, which was developed in this study, consisted of the following training patterns. First, the stochastic model was selected. As explained previously, Univariate Seasonal PARMA(1,1) model, which consisted of 1 pattern only, was used to generate the training dataset. Second, the data, which were generated by Univariate Seasonal PARMA(1,1) model, consisted of 3 patterns including 100 years (Short-term), 500 years (Mid-term), and 1000 years (Long-term), respectively. Finally, the neural networks model, which consisted of 1 pattern only including SVM-NNM, was used for the training and testing performances, respectively. Therefore, the hybrid method consisted of 3 training patterns including 100/PARMA(1,1)/SVM-NNM, 500/PARMA(1,1)/SVM-NNM, and 1000 /PARMA(1,1)/SVM-NNM, respectively. For Gwangju and Haenam stations, the training dataset including the climatic, PE, and FAO-56 PM ET o data were generated by Univariate Seasonal PARMA(1,1) model using observed data (01/1985-12/1990) for 100 years (Shortterm), 500 years (Mid-term), and 1000 years (Long-term), respectively. After the first 50 years of the generated data for 100, 500, and 1000 years was abandoned to eliminate the biases, the training performance should be carried out using SVM-NNM. Therefore, the total amount of data used for the training performance consisted of 600, 5400, and 11400, respectively. For the training performance of SVM-NNM, NeuroSolutions 5.0 computer program was used to carry out the training performance. Fig. 4 Tokar and Johnson (1999) suggested that the data length has less effect on the neural networks model performance than the data quality. Sivakumar et al. (2002) suggested that it is imperative to select a good training dataset from the available data series. They indicated that the best way to achieve a good training performance seems to be to include most of the extreme events such as very high and very low values in the training dataset. Furthermore, Kim (2011)

Testing performance
The neural networks model is tested by determining whether the model meets the objectives of modeling within some preestablished criteria or not. Of course, the optimal parameters, which are calculated during the training performance, are applied for the testing performance of the neural networks model (Kim, 2004). For the testing performance, the monthly climatic data (01/1985-12/1990) in Gwangju and Haenam stations were used. The total amount of data used for the testing performance consisted of 72 data for the monthly time series. The testing performance applied the cross-validation method in order to overcome the over-fitting problem of SVM-NNM. The cross-validation method is not to train all the training data until SVM-NNM reaches the minimum RMSE, but is to cross-validate with the testing data at the end of each training performance. If the over-fitting problem occurs, the convergence process over the mean square error of the testing data will not decrease but will increase as the training data are still trained (Bishop, 1994;Haykin, 2009).   Kim (2011) suggested the similar results for the modeling of PE and ET r using the neural networks models. He suggested that the statistics results of ET r were better than those of PE for the modeling of PE and ET r using GRNNM-BP and GRNNM-GA, South Korea. Fig. 5(a)

The One-way ANOVA
The One-way ANOVA is a class of statistical analysis that is widely used because it encourages systematic decision making for the underlying problems that involve considerable uncertainty. It enables inferences to be made in such a way that sample data can be combined with statistical theory. It supposedly removes the effects of the biases of the individual, which leads to more rational and accurate decision making. The One-way ANOVA is the formal procedure for using statistical concepts and measures in performing decision making. The following six steps can be used to make statistical analysis of the Oneway ANOVA on the means and variances: 1) Formulation of hypotheses 2) Define the test statistic and its distribution 3) Specify the level of significance 4) Collect data and compute www.intechopen.com test statistic 5) Determine the critical value of the test statistic 6) Make a decision (McCuen, 1993;Salas et al., 2001;Ayyub and McCuen, 2003). The One-way ANOVA on the means was performed and computed t statistic using twosample t test between the observed PE/FAO-56 PM ET o and the calculated PE/FAO-56 PM ET o , respectively. The critical value of t statistic was computed for the level of significance 5 percent (5%) and 1 percent (1%). If the computed value of t statistic is greater than the critical value of t statistic, the null hypothesis, which is the means are equal, should be rejected and the alternative hypothesis should be accepted. The One-way ANOVA on the variances was performed and computed F statistic using two-sample F test between the observed PE/FAO-56 PM ET o and the calculated PE/FAO-56 PM ET o , respectively. The critical value of F statistic was computed for the level of significance 5 percent (5%) and 1 percent (1%). If the computed value of F statistic is greater than the critical value of F statistic, the null hypothesis, which is the population variances are equal, should be rejected and the alternative hypothesis should be accepted. Table 6 shows the results of the One-way ANOVA on the means of PE. The critical value of t statistic was computed as t 0.05 =1.981 and t 0.01 =2.620 for the level of significance 5 percent (5%) and 1 percent (1%) Table 6. Results of the One-way ANOVA on the means of PE

The Mann-Whitney U test
The Mann-Whitney U test is a nonparametric alternative to the two-sample t test for two independent samples and can be used to test whether two independent samples have been taken from the same population. It is the most powerful alternative to the two-sample t test. Therefore, when the assumptions of the two-sample t test are violated or are difficult to evaluate such as with small samples, the Mann-Whitney U test should be applied. The Mann-Whitney U test is to be used in the case of two independent samples, and the Kruskal-Wallis test is an extension of the Mann-Whitney U test for the case of more than two independent samples (McCuen, 1993;Salas et al., 2001;Ayyub and McCuen, 2003). The Mann-Whitney U test was performed and computed z statistic between the observed PE/FAO-56 PM ET o and the calculated PE/FAO-56 PM ET o , respectively. The critical value of z statistic was computed for the level of significance 5 percent (5%) and 1 percent (1%). If the computed value of z statistic is greater than the critical value of z statistic, the null hypothesis, which is the two independent samples are from the same population, should be rejected and the alternative hypothesis should be accepted in this study.     (12). The overall deviations are nearly zero, and this tendency always occurs for the BLRAM. The standard error ratios of the regression coefficient (b 1 ) and the intercept (b 0 ) are 0.035 and 0.345, which indicates that the regression coefficient is relatively more accurate than the intercept. If a large and negative intercept exists, it can create some problems for forecasting or modeling (McCuen, 1993). Fig. 7

Conclusions
The hybrid method was developed for the modeling of the monthly PE and FAO-56 PM ET o simultaneously. The author determined in advance 4 kinds of Univariate Seasonal PARMA(p,q) models including PARMA(1,1), PARMA(1,2), PARMA (2,1), and PARMA(2,2), which are the low-order models and contain the seasonal properties. As a result, the author selected Univariate Seasonal PARMA(1,1) model, which show the best statistical properties and is simple in parameter estimation. The data which were generated by Univariate Seasonal PARMA(1,1) model consisted of 100 years (Short-term), 500 years (Mid-term), and 1000 years (Long-term), respectively. The following conclusions can be drawn from this study.
[1] The statistics results of the testing performance for 100/PARMA(1,1)/SVM-NNM training pattern were better than those of the testing performances for 500/PARMA(1,1) /SVM-NNM and 1000/PARMA(1,1)/SVM-NNM training patterns for PE of Gwangju station. And, the statistics results of the testing performance for 500/PARMA(1,1)/SVM-NNM training pattern were better than those of the testing performances for 100 /PARMA(1,1)/SVM-NNM and 1000/ PARMA(1,1)/SVM-NNM training patterns for PE of Haenam station, respectively [2] The statistics results of the testing performance for 100/PARMA(1,1)/SVM-NNM training pattern were better than those of the testing performances for 500/PARMA(1,1) /SVM-NNM and 1000/PARMA(1,1)/SVM-NNM training patterns for FAO-56 PM ET o of Gwangju station. And, the statistics results of the testing performance for 1000 /PARMA(1,1)/SVM-NNM training pattern were better than those of the testing performances for 100/PARMA(1,1)/SVM-NNM and 500/PARMA(1,1)/SVM-NNM training patterns for FAO-56 PM ET o of Haenam station, respectively [3] Homogeneity evaluation consisted of the One-way ANOVA and the Mann-Whitney U test. The null hypothesis, which is the means are equal, was accepted using the One-way ANOVA on the means for PE and FAO-56 PM ET o of Gwangju and Haenam stations, respectively. And, the null hypothesis, which is the variances are equal, was accepted using the One-way ANOVA on the variances for PE and FAO-56 PM ET o of Gwangju and Haenam stations, respectively. The null hypothesis, which is the two independent samples are from the same population, was accepted using the Mann-Whitney U test for PE and FAO-56 PM ET o of Gwangju and Haenam stations, respectively.
[4] The BLRAM was adopted to calculate FAO-56 PM ET o simply using the observed PE and compare the observed PE and the calculated FAO-56 PM ET o of Gwangju and Haenam stations, respectively. A very good relationship was found with the BLRAM, which could calculate FAO-56 PM ET o . As PE and FAO-56 PM ET o are relatively important for the design of irrigation facilities and agricultural reservoirs, the spread of an automatic measuring system for PE and FAO-56 PM ET o is important and urgent to ensure the reliable and accurate data from the measurements of PE and FAO-56 PM ET o . Furthermore, the continuous research will be needed to establish the neural networks models available for the optimal training patterns and modeling of PE and FAO-56 PM ET o .