Spatial Estimate and mapping of reference evapotranspiration in Khuzestan Province

1- Introduction Evapotranspiration (ETo) is one of the main factors determining water balance in every area in which spatial and temporal distribution plays an important role in water resource management, estimation of water requirement and agricultural planning. Commonly, evapotranspiration is calculated locally at each station; however, it can now be estimated throughout a broad area using interpolation methods. The objective of the present study is to determine the most adequate interpolation method for spatial estimation and mapping of ETo in Khuzestan province. 2- Theoretical framework 2-1-Ordinary Kriging Kriging is a estimation based on moving average weighted. The ordinary kriging estimator consists of two steps. In the first step, understanding and modeling the spatial structure of variable is considered using of the semi-variogram analysis. The second step; being dependent on the first, is the estimation of intended variable. As a condition for being used in Kriging methods, the variable that is detectable by semi-variogram should be static (Hasani Pak, 1998: 181) 2-2-Cokriging In cases where there are few samples of the primary variable, yet a greater number of auxiliary variables, Cokriging is a useful method. This method can be used where there is a strong and significant correlation between primary and auxiliary variables (Majani,B.S, 2007:33). 2-3- Regression-Kriging Regression-kriging is an interpolation method that is a synthesis of the regression of the variable being dependent on auxiliary variables (such as longitude, latitude and altitude) and the kriging of the residuals (Boer & et al., 2001:150). 2-4- Kriging with an External Drift This method is an extension of kriging with trend. If the auxiliary variables are available at all predictor grid-nodes as well as the position of target variable, kriging with external drift can be used. The auxiliary variables should also be highly correlated with the target variable (Hengel. & et al., 2003:3). 2-5- Inverse distance weighting (IDW) method This method is based on an observed value closer to the prediction location, which has more influence on the predicted value than other observed values located farther apart. In IDW method, the weight is represent based on the inverse distance of power (p) and the power that has the least error (root mean square error) is selected as the most optimal (Ha. & et al., 2011: 2797). 2-6- Spline Splines are Non-parametric functions with high elasticity. Spline interpolation methods can be called piecewise polynomial functions, i.e. complex functions that include fragments of a polynomial function with different degrees of smooth connections between two points in space (Mahdizadeh, 2002:22). 2-7- 3D linear gradient In this method, it is assumed that a multiple linear relationship can be fitted into the predicted variables or independent variables (typically topological variables such as longitude, latitude and altitude) and a known depend variable (reference evapotranspiration). This method can be used to estimate the dependent primary variable in places where it has not been estimated. (Majani, B.S, 2007:33). 3- Methodology For estimating ETo values, the meteorological data derived from 42 meteorology stations over a 28-year study period (1982-2009) were used and FAO Penman-Monteith method was applied. Seven interpolation Inverse Distance Weighted (IDW), Spline, 3D linear gradient, Ordinary Kriging, Cokriging, Kriging with External Drift and Kriging-Regression were assessed for estimating spatial ETo. For variographic analysis in Kriging, five variogram models i.e. Spherical, Exponential, linear, linear to sill and Gaussian were fitted into ETo data. Based on the lower Residual Sums of Squares errors and higher correlation coefficiencies, the best model was selected. The most adequate interpolation method was determined based on the calculation of Root Mean Square Error, Mean Bias Error and Mean Absolute Error indices from different methods. 4- Discussion The Anderson-Darling test results obtained from different months indicated that evapotranspiration data was not significantly different from normal distribution at 95% confidence level during most months. Only March, April, May, June and December did not follow the normal distribution. Probability distributions for these months were transformed by alogarithmic transformation. Variographic analysis results of ordinary Kriging method indicated that the role of structured to unstructured component is much more effective in more than 95% of the months. Therefore, the analysis represents an appropriate spatial structure of ETo data in the region. Furthermore, it is concluded that the optimal theoretical variogram model in all case is the spherical model. The results of elevation structured analysis in CoKriging revealed that the proportion of [C/(C+C0)] is 1, which indicates a lack of a nugget effect and represents a strong spatial structure of elevation data in the region. In addition, the optimal theoretical cross-variogram model for refence evapotranspiration-eleveation data in monthly and annual scales is assumed to be the spherical model. In inverse distance weighting method, power 3 was selected as the optimal power. The data also indicated that in this method (IDW), value of 6 was considered to be the least points for error. The results of three-dimensional linear gradient method showed that in all months, the coefficiency of the independent variables Longitude, latitude and elevation were negative. In other words, reference evapotranspiration decreased in the region from west to east and from south to north. The results of Root Mean Square Error in all months showed Ordinary Kriging, Cokriging and Inverse Distance Weighted methods to have the least amount of error for estimating ETo than any other method. The outcome of Mean Absolute Error represent revealed that Cokriging method had the least amount of error in most months. The investigation of Mean Bias Error indicates Ordinary Kriging, Cokriging and 3D linear gradient to have the lowest Bias Error. 5- Conclusion and Suggestions Error investigation of indices showed Cokriging method with Gaussian semivariogram model to have the lowest error and to be introduced as the best method for analysing monthly and annual ETo data in the province. In the Gaussian semivariogram model of Cokriging, the ratio of structured to unstructured component is 1 in most months, which indicates a strong spatial structure in the simultaneous changes of ETo variables and altitude. However, it should be considered that Cokriging method has an underestimated property in most months.