Forecasting the Reference Evapotranspiration Using Time Series Model

Reference evapotranspiration is one of the most important factors in irrigation timing and field management. Moreover, reference evapotranspiration forecasting can play a vital role in future developments. Therefore in this study, the seasonal autoregressive integrated moving average (ARIMA) model was used to forecast the reference evapotranspiration time series in the Esfahan, Semnan, Shiraz, Kerman, and Yazd synoptic stations.
In the present study in all stations (characteristics of the synoptic stations are given in Table 1), the meteorological data, including mean, maximum and minimum air temperature, relative humidity, dry-and wet-bulb temperature, dew-point temperature, wind speed, precipitation, air vapor pressure and sunshine hours were collected from the Islamic Republic of Iran Meteorological Organization (IRIMO) for the 41 years from 1965 to 2005. The FAO Penman-Monteith equation was used to calculate the monthly reference evapotranspiration in the five synoptic stations and the evapotranspiration time series were formed. The unit root test was used to identify whether the time series was stationary, then using the Box-Jenkins method, seasonal ARIMA models were applied to the sample data.
Table 1. The geographical location and climate conditions of the synoptic stations
Station Geographical location Altitude (m) Mean air temperature (°C) Mean precipitation (mm) Climate, according to the De Martonne index classification
Longitude (E) Latitude (N) Annual Min. and Max.
Esfahan 51° 40' 32° 37' 1550.4 16.36 9.4-23.3 122 Arid
Semnan 53° 33' 35° 35' 1130.8 18.0 12.4-23.8 140 Arid
Shiraz 52° 36' 29° 32' 1484 18.0 10.2-25.9 324 Semi-arid
Kerman 56° 58' 30° 15' 1753.8 15.6 6.7-24.6 142 Arid
Yazd 54° 17' 31° 54' 1237.2 19.2 11.8-26.0 61 Arid
The monthly meteorological data were used as input for the Ref-ET software and monthly reference evapotranspiration were obtained. The mean values of evapotranspiration in the study period were 4.42, 3.93, 5.05, 5.49, and 5.60 mm day−1 in Esfahan, Semnan, Shiraz, Kerman, and Yazd, respectively. The Augmented Dickey-Fuller (ADF) test was performed to the time series. The results showed that in all stations except Shiraz, time series had unit root and were non-stationary. The non-stationary time series became stationary at 1st difference. Using the EViews 7 software, the seasonal ARIMA models were applied to the evapotranspiration time series and R2 coefficient of determination, Durbin–Watson statistic (DW), Hannan-Quinn (HQ), Schwarz (SC) and Akaike information criteria (AIC) were used to determine, the best models for the stations were selected. The selected models were listed in Table 2. Moreover, information criteria (AIC, SC, and HQ) were used to assess model parsimony. The independence assumption of the model residuals was confirmed by a sensitive diagnostic check. Furthermore, the homoscedasticity and normality assumptions were tested using other diagnostics tests.
Table 2- The selected time series models for the stations
Station Seasonal ARIMA model Information criteria R2 DW
SC HQ AIC
Esfahan ARIMA(1, 1, 1)×(1, 0, 1)12 1.2571 1.2840 1.2396 0.8800 1.9987
Semnan ARIMA(5, 1, 2)×(1, 0, 1)12 1.5665 1.5122 1.4770 0.8543 1.9911
Shiraz ARIMA(2, 0, 3)×(1, 0, 1)12 1.3312 1.2881 1.2601 0.9665 1.9873
Kerman ARIMA(5, 1, 1)×(1, 0, 1)12 1.8097 1.7608 1.8097 0.8557 2.0042
Yazd ARIMA(2, 1, 3)×(1, 1, 1)12 1.7472 1.7032 1.6746 0.5264 1.9943
The seasonal ARIMA models presented in Table 2, were used at the 12 months (2004-2005) forecasting horizon. The results showed that the models produce good out-of-sample forecasts, which in all the stations the lowest correlation coefficient and the highest root mean square error were obtained 0.988 and 0.515 mm day−1, respectively.
In the presented paper, reference evapotranspiration in the five synoptic stations, including Esfahan, Semnan, Shiraz, Kerman, and Yazd, were calculated using the FAO Penman-Monteith method for the 41 years, and the time series were formed. The selected models gave good out-of-sample forecasts of the monthly evapotranspiration for all the stations. The models can be used in the short-term prediction of monthly reference evapotranspiration. Note that, the use of models in long-term forecasting was not recommended. The time series model can be used in lost data. Even though more methods are available for model building, the use of time series models in water resources are advocated in modeling and forecasting. Time series can be used as a tool to find lost data.