Path Analysis in Identification of Dominant Effective Meteorological Parameters on ET0 in East Azarbaijan

Introduction
Reference potential evapotranspiration (ET₀) is one of the main elements of hydrologic cycle which can be estimated from weather data. This element can be used in calculating crop water requirements, scheduling irrigation systems, preparing input data to hydrological water-balance models, regional water resources assessment, and planning and management of water in a region and/or basin. The use of ET₀ permits a physically realistic characterization of the effect of the microclimate in a field on the evaporative transfer of water from the soil-plant system to the atmosphere. It provides a measure of the integrated effects of radiation, wind speed, temperature and humidity on evapotranspiration. The long-term mean ET₀ value in a certain timescale (month, season or year) can be changed during the recent decades in a given station. By decreasing ET₀, crop water demand decreases. Conversely, by increasing ET₀ the crop water requirements can also increase accordingly. Therefore, it can be suggested that change in the rates of ET₀ due to climate change would have great importance for agriculturalists and water decision makers. On the other hand, accurate estimation of ET₀ is crucial in improving the irrigation efficiency in a region. Many climatic parameters impacted the ET₀ value in a single site. On the other hand, these parameters are correlated to each other.  
Materials and methods
The climatic data from the synoptic stations with at least 20 years of continuous records in East Azarbaijan province were gathered from the Islamic Republic of Iran Meteorological Organization (IRIMO). The obtained data are including maximum air temperature (Tmax), minimum air temperature (Tmin), wind speed in 10 m height (U), maximum relative humidity (RHmax), minimum relative humidity (RHmin), and duration of sunshine hours (n). The well-known FAO-PM56 method was used to calculate the ET₀. There are many methods for ET0 estimation. The Penman–Monteith (PM) method is recommended as the standard by the United Nations Food and Agriculture Organization (UNFAO) and has gained worldwide acceptance and received much research interests. The PM equation has been widely used in ET₀ estimation. However, this method needs more meteorological data which is not available in many regions. This led scientists to use other methods which do not need more parameters. Among the empirical methods which estimate ET₀ using less climatic parameters are Hargreaves, Tornth-Wait, Belaney-Criddle and Priestley–Taylor. Unfortunately, outputs of these models are not accurate in all the sites. Therefore, for using these simple empirical models the calibration process should be done as well. Therefore, the following issues need urgent study: (1) selection of as few dominant meteorological variables as possible in meteorological parameters affecting ET₀, and (2) universal application of an established model in more regions.
The alternative method namely Multiple Linear Regression (MLR) can be used to estimate the ET₀. In order to evaluate the performance of the MLR method, some measures were calculated by comparing the results of MLR with FAO56-PM method. These measures are the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Nash–Sutcliffe Efficiency (NSE), and the coefficient of determination (R²). Then, the correlation coefficients (r_yxi) calculated for the ET₀ time series (y) and each of the meteorological parameters (x_i). Then, the partial correlation coefficients (r_ij) are calculated for the explanatory variables (x_i and x_j) as well. Both of the direct and indirect effects of each climatic parameter on ET₀ were evaluated by path analysis. These effects are denoted by P and R_dc, respectively. By solving the Eq. 6, the elements of P (direct effect of x_i on y or ET₀) are obtained. By multiplying the obtained P vector (direct effects) on  , we calculated the indirect effect of x_i through the x_j on ET₀. This process is repeated for all the selected sites.
Path analysis was first proposed in 1921 as a mathematical and statistical method by the geneticist Sewell Wright. Nowadays, the method is broadly used in agriculture and energy demands to reveal direct or indirect relationships between some morphological characters. However, information is available on the use of this technique to evaluate the factors affecting ET₀. Given the fact that all the meteorological variables are strongly correlated and ultimately lead to multi-collinearity, traditional trend and correlation analyses cannot quantify the interactions among the meteorological factors when filtering the suitable parameters.
Path analysis is a type of multivariate statistical analysis for studying relationships among variables. It can reveal the strength of the effects of independent variables on a dependent variable. Path analysis can determine direct and indirect effects of independent variables on the dependent variable; multi-collinear independent variables resulted from their own strong correlations, and optimal regression equations without unnecessary independent variables. The path coefficient is a type of standard partial regression coefficient (without units) that expresses causalities among related variables. It is also a directional correlation coefficient between independent and dependent variables. This analysis was conducted for each of the selected stations in East Azarbaijan province, Iran. To do this, we initially calculated the correlation coefficients between each of the climatic parameters and ET₀ time series. Similarly, correlation coefficients matrix between the climatic parameters affecting ET₀ was obtained for each of the stations.  
Results and discussion
The results of this research showed that the values of MAPE obtained for the stations were between 0.433 and 0.874. However, the R² values were between 0.972 and 0.9953. Similarly, the RMSE were between 0.042 (mm/day) and 0.982 (mm/day), and the obtained MAE values were between 0.033 and 0.057. It was also found that the wind speed at the stations namely Tabriz, Jolfa, Sarab, Sahand, Maragheh and Mianeh had significant correlation (at the 0.01% level) with ET₀. The strongest correlation was related to the station Ahar, which was between ET₀ and the wind speed (at the 0.01% level). The results of path analysis showed that the maximum value of P (direct effects of meteorological parameters on ET₀ belonged to the wind speed). The P values of wind speed in the stations Tabriz, Julfa, Sahand, Sarab, Maragheh, and Mianeh were equal to 0.637, 0.787, 0.877, 0.578, 0.850, and 0.780, respectively. In the station Ahar, the highest value of the P observed belong to the T_max (equal to 0.398).  
Conclusion 
Accurate estimation of ET₀ is very important from the view of optimal water management in any region. Wind speed was found to be the dominant direct climatic parameter due to having the largest value of the P. In general, it can be concluded that the causal analysis method can be considered as an effective way to investigate the direct and indirect effects of meteorological parameters on ET₀. Overall, it is more reasonable to apply path analysis method to evaluate dominant meteorological parameters of the ET₀ in direct and indirect ways. The further research can be oriented in analysis on why dominant factors vary with meteorological stations. Development of other soft computing techniques calculated ET₀ using the climatic methods (such as firefly algorithm, artificial neural networks, support vector regression, and genetic expression programming) and comparing their accuracy with that of the MLR are recommended for further studies.