Water and Irrigation Management

Water and Irrigation Management

Forecasting of SPEI index using the stochastic time-series models (Case study: Ardabil province)

Document Type : Research Paper

Authors
Yagob Dinpashoh
Farzad Abkar
Department of Water Engineering, University of Tabriz, Tabriz, Iran.
10.22059/jwim.2026.408263.1275
Abstract
Introduction: Drought is a natural disaster that affects more people comparing others. It has known as creeping phenomenon that evolve gradually. It has adverse effects on the ecosystem, economic growth, agricultural production, political and social stability. It affects human survival and all other living things. Four types of droughts are 1) Meteorological, 2) Hydrological, 3) Agriculture and 4) Social and economic droughts. This study is on meteorological drought which can be studied using different drought indices such as SPEI, SPI, and many others. The SPEI index, is based on both precipitation and reference crop potential evapotranspiration (ET0). SPEI has been widely used in hydrology and climatology in different countries. This index is obtained by inclusion of ET0 on SPI index. Undoubtedly accurate prediction of drought is very important in optimal management of water resources in any region. Among the widely used linear stochastic models, that can forecast SPEI time series, it can be refer to the random stochastic models namely ARIMA and SARIMA. These two mentioned models have been widely used in various hydrological and climatic studies. The aim of this study is to predict the SPEI using the time series models in Ardabil province.
Materials and Methods: Data used in this study are the precipitation, maximum air temperature (Tmax), minimum air temperature (Tmin), maximum relative humidity (RHmax), minimum relative humidity (RHmin), and wind speed at ten meters height (U10), and actual sunshine hours (n) in daily time scale. A 30-year statistical time period (1992–2021) was used. Three synoptic stations (Ardabil, Khalkhal, and Parsabad) having complete data were selected across the Ardabil province. Standardized Precipitation-Evaporation Index (SPEI) calculated in five distinct time scales (1, 3, 6, 9, and 12 months. This index is calculated by precipitation minus the ET0 in a given time scale. In the first step, to calculate the SPEI index, the potential reference crop evapotranspiration (ET0) is calculated using the recommended from of the Penman-Montheis method. In the second step, the parameter denoted by D is obtained by subtraction of ET0 from precipitation. In the third step, Di values are fitted by a three-parameter log-logistic distribution. Its parameters were estimated using the maximum likelihood method. Then, having the cumulative distribution function values at hand, the SPEI values were estimated as well. In the following, two random time series models (ARIMA and SARIMA) were used to model the SPEI index. Finally, the derived model was used to forecast SPEI in the selected stations.
Results and Discussion: The most prominent results are as follows: 1) In a 6-month time window the longest duration of drought in Ardabil station had 33 months length. This length was about 16 months in Khalkhal station for a 12-month time window. Also, in Parsabad station, the longest period of drought for a 12-month time window has lasted about 44 months. 2) The most severe drought period has been experienced in Ardabil, Khalkhal, and Parsabad stations, in a 12-month time window, had drought severity (with SPEI) equal to -3.89, -2.46, and -3.06, respectively. 3) The most suitable linear time series random model in Ardabil station was found for the SPEI6 index. This model was the SARIMA(2,0,0)(0,1,4)12. In Khalkhal station, the most suitable model belonged to SPEI1. The type of the model for this site is SARIMA model (0,0,0) (0,1,1)12. In Parsabad station the most suitable model was found for SPEI6. SARIMA model (2,0,0) (1,1,1)12 was detected as the best one for Khalkhal. 4: Predictions for short-term time windows (1, 3 and 6 months) were more accurate than that of the long-term time windows (9 and 12 months).
Conclusions: The results of this study showed that the length of meteorological drought using the SPEI index in the three selected stations and in the time scales (1, 3, and 6 months) fluctuates between 3 and 5 months. As the time scale increases, the number of drought events decreases. In addition, by increasing the time window duration of drought increases. In the present study, ARIMA and SARIMA models were used to predict the SPEI index. Time series models were found to be a suitable tool to predict the SPEI index in the study area having the arid and semi-arid climate. In this study, the SARIMA model was used to predict the SPEI index due to the presence of seasonal trends in the 1, 3, 6, and 9 months, time windows. Results showed that this model had reasonable accuracy for all time windows except 12 months. This window was found to be lack of the seasonal trends. Therefore, the non-seasonal ARMA model was used, for 12-month window.
Keywords
Forecasting
Drought
ARIMA
SPEI
Subjects
  1. Abebe, A., & Foerch, G. (2008). Stochastic simulation of the severity of hydrological drought. Water and Environment Journal, 22(1), 2-10. DOI: http://dx.doi.org/10.1111/j.1747-6593.2007.00080.x
  2. Ahmadi, F., Dinpashoh, Y., Fakheri fard, A., khalili, K., & Darbandi, S. (2015). Comparing nonlinear time series models and genetic programming for daily river flow forecasting (Case study: Barandouz-Chai River). Journal of Water and Soil Conservation, 22(1), 151-169. (In Persian)
  3. Ashrafzadeh, A., Kişi, O., Aghelpour, P., Biazar, S. M., & Masouleh, M. A. (2020). Comparative study of time series models, support vector machines, and GMDH in forecasting long-term evapotranspiration rates in northern Iran. Journal of Irrigation and Drainage Engineering, 146(6), 04020010. DOI: https://doi.org/10.1061/(ASCE)IR.1943-4774.0001471
  4. Aghelpour, P., Bahrami-Pichaghchi, H., & Varshavian, V. (2021). Hydrological drought forecasting using multi-scalar streamflow drought index, stochastic models and machine learning approaches, in northern Iran. Stochastic Environmental Research and Risk Assessment, 35(8), 1615-1635. DOI: http://dx.doi.org/10.1007/s00477-020-01949-z
  5. Aghelpour, P., Mohammadi, B., & Biazar, S. M. (2019). Long-term monthly average temperature forecasting in some climate types of Iran, using the models SARIMA, SVR, and SVR-FA. Theoretical and Applied Climatology, 138(3-4), 1471-1480. DOI: https://doi.org/10.1007/s00704-019-02905-w
  6. Aghelpour, P., & Varshavian, V. (2020). Evaluation of stochastic and artificial intelligence models in modeling and predicting of river daily flow time series. Stochastic Environmental Research and Risk Assessment, 34(1), 33-50. DOI: https://doi.org/10.1007/s00477-019-01761-4
  7. Allen, R. G., Jensen, M. E., Wright, J. L., & Burman, R. D. (1989). Operational estimates of reference evapotranspiration. Agronomy Journal, 81(4), 650-662. DOI: http://dx.doi.org/10.2134/agronj1989.00021962008100040019x
  8. Allen, R. G., Pereira, L. S., Raes, D., & Smith, M. (1998). Crop evapotranspiration-Guidelines for computing crop water requirements-FAO Irrigation and drainage paper 56. FAO, Rome, 300(9), D05109.
  9. Al Sayah, M. J., Abdallah, C., Khouri, M., Nedjai, R., & Darwich, T. (2021). A framework for climate change assessment in Mediterranean data-sparse watersheds using remote sensing and ARIMA modeling. Theoretical and Applied Climatology, 143, 639-658. DOI: http://dx.doi.org/10.1007/s00704-020-03442-7
  10. Babamiri, O., Nowzari, H., & Maroofi, S. (2017). Potential evapotranspiration estimation using stochastic time series models (case study: Tabriz). Journal of Watershed Management Research, 8(15):137-146. DOI: http://dx.doi.org/10.29252/jwmr.8.15.137
  11. Babamiri, O., & Dinpazhoh, Y. (2016). Comparison and evaluation of twenty methods for estimating reference evapotranspiration based on three general categories: air temperature, solar radiation and mass Transfer in the basin of lake Urmia. JWSS-Isfahan University of Technology, 20(77), 145-161. DOI: http://dx.doi.org/10.18869/acadpub.jstnar.20.77.145 (In Persian).
  12. Bazrafshan, J., Nadi, M., & Ghorbani, K. (2015). Comparison of empirical copula-based joint deficit index (JDI) and multivariate standardized precipitation index (MSPI) for drought monitoring in Iran. Water Resources Management, 29, 2027-2044. DOI: http://dx.doi.org/10.1007/s11269-015-0926-x
  13. Box, G.E.P., Jenkins, G.M., & Reinsel, G.C. (1994). Time Series Analysis; Forecasting and Control. 3rd Edition, Prentice Hall, Englewood Cliff, New Jersey.
  14. Byun, H. R., & Wilhite, D. A. (1999). Objective quantification of drought severity and duration. Journal of Climate, 12(9), 2747-2756.‏ DOI: http://dx.doi.org/10.1175/1520-0442(1999)012<2747:OQODSA>2.0.CO;2
  15. Chourishi, D. K., Rajput, A., Gour, S., & Mishra, K. (2022). Study to forecast drought using SARIMA model. Mathematical Statistician and Engineering Applications, 71(4), 11157-11167. DOI: https://doi.org/10.17762/msea.v71i4.2197
  16. Daneshvar Vousoughi, F., Dinpashoh, Y., & Aalami, M. (2011). Effect of drought on groundwater level in the past two decades (Case study: Ardebil Plain). Water and Soil Science, 21(4), 165-179. (In Persian).
  17. Dinpashoh, Y., & Heidari, N. (2025). Trend analysis of ET0 and relative importance of climatic variables on ET0 (Case study: Stations Khorramdareh, Gheidar, and Abbar).  Journal of Water and Irrigation Management, 15(1), 75-91. DOI: https://doi.org/‎10.22059/jwim.2024.367513.1126 (In Persian).
  18. Dinpashoh, Y., Jahanbakhsh-Asl, S., Rasouli, A.A., Foroughi, M., & Singh, V.P. (2019). Impact of climate change on potential evapotranspiration (case study: west and NW of Iran)‏. Theoretical and Applied Climatology, 136, 185-201.
  19. Dinpashoh, Y., Jhajharia, D., Fakheri-Fard, A., Singh, V. P., & Kahya, E. (2011). Trends in reference crop evapotranspiration over Iran. Journal of Hydrology, 399(3-4), 422-433. 10.1016/j.jhydrol.2011.01.021
  20. Dinpashoh, Y., & Shafaei, S. (2018). Analysis of drought characteristics of Tabriz (1951-2015). Water and Soil Science, 28(3), 117-130. (In Persian).
  21. Dong, J., Xing, L., Cui, N., Zhao, L., Guo, L., & Gong, D. (2023). Standardized precipitation evapotranspiration index (SPEI) estimated using variant long short-term memory network at four climatic zones of China. Computers and Electronics in Agriculture, 213, 108253. DOI: https://doi.org/10.1016/j.compag.2023.108253
  22. Ekhtiari Khajeh, S., & Dinpashoh, Y. (2018). Application of effective drought index (EDI) in characterizing drought periods (Case study: Tabriz, Bandar Anzali, and Zahedan). Irrigation Sciences and Engineering, 41(1), 133-145. doi: 10.22055/jise.2018.13455. (In Persian)
  23. Ekhtiary Khajeh, S., Negahban, F., & Dinpashoh, Y. (2019). Comparing univariate and multivariate indices in drought monitoring. JWSS-Journal of Water and Soil Science, 23(2), 433-446. URL: http://jstnar.iut.ac.ir/article-1-3647-fa.htm (In Persian).
  24. Fathian, F., Fakheri Fard, A., Ouarda, T.B.M.J., Dinpashoh, Y., & Mousavi Nadoushani, S.S. (2019). Modeling streamflow time series using nonlinear SETAR-GARCH models. Journal of Hydrology, 573, 82-97. https://doi.org/10.1016/j.jhydrol.2019.03.072.
  25. Ghorbani Aghdam, M., Dinpazhoh, Y., Fakheri Fard, A., & Darbandi, S. (2012). Regionalization of Urmia Lake basin from the view of drought using factor analysis. Water and Soil, 26(5), 1268-1276. doi: 10.22067/jsw.v1391i0.17069 (In Persian).
  26. Hoseini, S. M., Dinpashoh, Y., & Babamiri, O. (2025). Prediction of hydrological drought using the GRI index and linear random time series models (Study area: Ardabil Plain aquifer). Hydrogeology, 9(2), 77-92. doi: 10.22034/hydro.2025.64314.1321. (In Persian)
  27. Hosseini, B., Dinpazhoh, Y. and Nikbakht, J. (2015). Analysis of droughts of northwest of Iran using the reconnaissance drought index. Water and Soil, 29(2), 295-310. doi: 10.22067/jsw.v0i0.28050 (In Persian).
  28. Huang, X., Yang, X., Wu, F., & Zhang, J. (2025). Future propagation characteristics of meteorological drought to hydrological drought in the Yellow River basin. Journal of Hydrology, 649, 132444. https://doi.org/10.1016/j.jhydrol.2024.132443
  29. Khalili, K., Ahmadi, F., Dinpashoh, Y., & Behmanesh, J. (2014). Linear and nonlinear behavior analysis of hydrological time series (Case study: Western rivers of Lake Urmia). Iran-Water Resources Research, 10(2), 12-20. (In Persian)
  30. Kim, D., & Rhee, J. (2016). A drought index based on actual evapotranspiration from the Bouchet hypothesis. Geophysical Research Letters, 43(19), 10-277. DOI: http://dx.doi.org/10.1002/2016GL070302
  31. Lai, C., Zhong, R., Wang, Z., Wu, X., Chen, X., Wang, P., & Lian, Y. (2019). Monitoring hydrological drought using long-term satellite-based precipitation data. Science of the Total Environment, 649, 1198-1208. DOI: http://dx.doi.org/10.1016/j.scitotenv.2018.08.245
  32. Liu, S., Kang, W., & Wang, T. (2016). Drought variability in Inner Mongolia of northern China during 1960–2013 based on standardized precipitation evapotranspiration index. Environmental Earth Sciences, 75, 1-14. DOI: http://dx.doi.org/10.1007/s12665-015-4996-0
  33. Lorenzo-Lacruz, J., Vicente-Serrano, S.M., González-Hidalgo, J.C., López Moreno, J.I., & Cortesi, N. (2013). Hydrological drought response to meteorological drought in the Iberian Peninsula. Clim Res, 58, 117-131 https://doi.org/10.3354/cr01177
  34. Machiwal, D., & Jha, M. K. (2008). Comparative evaluation of statistical tests for time series analysis: application to hydrological time series. Hydrological Sciences Journal, 53(2), 353-366. DOI: http://dx.doi.org/10.1623/hysj.53.2.353
  35. McKee, T. B., Doesken, N. J., & Kleist, J. (1993, January). The relationship of drought frequency and duration to time scales. In Proceedings of the 8th Conference on Applied Climatology, 17(22), 179-183.
  36. Mehdizadeh, S., Ahmadi, F., Mehr, A. D., & Safari, M. J. S. (2020). Drought modeling using classic time series and hybrid wavelet-gene expression programming models. Journal of Hydrology, 587, 125017. DOI: https://doi.org/10.1016/j.jhydrol.2020.125017
  37. Mehdizadeh, S., Fathian, F., Safari, M. J. S., & Adamowski, J. F. (2019). Comparative assessment of time series and artificial intelligence models to estimate monthly streamflow: a local and external data analysis approach. Journal of Hydrology, 579, 124225. DOI: https://doi.org/10.1016/j.jhydrol.2019.124225
  38. Mirabbasi Najafabadi, R., Fakheri-Fard, A., Dinpashoh, Y., & Eslamian, S. (2014). Longterm drought monitoring of Urmia Using Joint Deficit Index (JDI). Water and Soil Science, 23(4), 87-103. (In Persian).
  39. Mishra, A. K., Desai, V. R., & Singh, V. P. (2007). Drought forecasting using a hybrid stochastic and neural network model. Journal of Hydrologic Engineering, 12(6), 626-638. DOI: http://dx.doi.org/10.1623/hysj.53.2.35310.1061/(ASCE)1084-0699(2007)12:6(626)
  40. Mohammadi, B., Mehdizadeh, S., Ahmadi, F., Lien, N. T. T., Linh, N. T. T., & Pham, Q. B. (2021). Developing hybrid time series and artificial intelligence models for estimating air temperatures. Stochastic Environmental Research and Risk Assessment, 35, 1189-1204. DOI: https://doi.org/10.1007/s00477-020-01898-7
  41. Otkin, J. A., Zhong, Y., Ford, T. W., Anderson, M. C., Hain, C., Hoell, A., et al. (2024). Multivariate evaluation of flash drought across the United States. Water Resources Research, 60, e2024WR037333. https://doi.org/10.1029/2024WR037333
  42. Palmer, W. C. (1968). Keeping track of crop moisture conditions, nationwide: the new crop moisture index. Weatherwise, 21(4), 156-161. DOI: https://doi.org/10.1080/00431672.1968.9932814
  43. Salas, J.D., Delluer, J.W., Yevjevich, V., & Lane, W.L. (1980). Applied Modelling of Hydrologic Time Series. Water Resources Publications. 498 Pages.
  44. Sanikhani, H., Dinpashoh, Y., & Ghorbani, M. (2015). Baranduz-Chay river flow modeling using the K-nearest neighbor and intelligent methods. Water and Soil Science, 25(1), 219-233. (In Persian)
  45. Santoso, H., & Marganingrum, D. (2022). Assessing the trend in low flows and the impact of climate change using the Standardized Precipitation–Evapotranspiration Index and total runoff: A case study of Bintan Island, Indonesia. DOI: http://dx.doi.org/10.21203/rs.3.rs-1900599/v1
  46. Stagge, J. H., Kohn, I., Tallaksen, L. M., & Stahl, K. (2015). Modeling drought impact occurrence based on meteorological drought indices in Europe. Journal of Hydrology, 530, 37-50. DOI: http://dx.doi.org/10.1016/j.jhydrol.2015.09.039
  47. Tayfur, G. (2021). Discrepancy Rainfall Index for Monitoring Meteorological Drought. Journal of Hydrology, 597. DOI: http://dx.doi.org/10.1016/j.jhydrol.2021.126174
  48. Toth, E., Montanari, A., & Brath, A. (1999). Real-time flood forecasting via combined use of conceptual and stochastic models. Physics and Chemistry of the Earth, Part B: Hydrology, Oceans and Atmosphere, 24(7), 793-798. DOI: http://dx.doi.org/10.1016/S1464-1909(99)00082-9
  49. Van Vuren, S., de Vriend, H., & Barneveld, H. (2016). A stochastic model approach for optimisation of lowland river restoration works. Journal of Earth Science, 27, 55-67. DOI: https://doi.org/10.1007/s12583-016-0629-0
  50. Vicente-Serrano, S. M., Beguería, S., & López-Moreno, J. I. (2010). A multiscalar drought index sensitive to global warming: the standardized precipitation evapotranspiration index. Journal of Climate, 23(7), 1696-1718. DOI: http://dx.doi.org/10.1175/2009JCLI2909.1
  51. Wang, Z., Zhong, R., Lai, C., Zeng, Z., Lian, Y., & Bai, X. (2018). Climate change enhances the severity and variability of drought in the Pearl River Basin in South China in the 21st century. Agricultural and Forest Meteorology, 249, 149-162. DOI: http://dx.doi.org/10.1016/j.agrformet.2017.12.077
  52. Xu, D., Zhang, Q., Ding, Y., & Huang, H. (2020). Application of a hybrid ARIMA–SVR model based on the SPI for the forecast of drought—a case study in Henan Province, China. Journal of Applied Meteorology and Climatology, 59(7), 1239-1259. DOI: http://dx.doi.org/10.1175/JAMC-D-19-0270.1
  53. Zhang, H., Ding, J., Wang, Y., Zhou, D., & Zhu, Q. (2021). Investigation about the correlation and propagation among meteorological, agricultural and groundwater droughts over humid and arid/semi-arid basins in China. Journal of Hydrology, 603, 127007. DOI: http://dx.doi.org/10.1016/j.jhydrol.2021.127007
  54. Zhang, Y., Li, W., Chen, Q., Pu, X., & Xiang, L. (2017). Multi-models for SPI drought forecasting in the north of Haihe River Basin, China. Stochastic Environmental Research and Risk Assessment, 31, 2471-2481. DOI: http://dx.doi.org/10.1007/s00477-017-1437-5
Water and Irrigation Management
Volume 16, Issue 2
Summer 2026
Pages 349-370