Abstract

Drought as an exigent natural phenomenon, with high frequency in arid and semi-arid regions, leads to enormous damage to agriculture, economy, and environment. In this study, the seasonal Standardized Precipitation Index (SPI) drought index and time series models were employed to model and predict seasonal drought using climate data of 38 Iranian synoptic stations during 1967–2014. In order to model and predict seasonal drought ITSM (Interactive Time Series Modeling) statistical software was used. According to the calculated seasonal SPI, within the study area, drought severity classes 4 and 3 had the greatest occurrence frequency, while classes 6 and 7 had the least occurrence frequency. Results indicated that the best fitted models were Moving-Average or MA (5) Innovations and MA (5) Hannan-Rissenen, with 60.53 and 15.79 percentage, respectively. On the other hand, results of the prediction as well, indicated that drought class 4 with the highest percentages, was the most abundant class over the study area and drought class 7 was the least frequent class. According to results of trend analysis, without attention to significance of them, observed seasonal SPI data series (1967–2014), in 84.21% of synoptic stations had a negative trend, but this percentage changes to 86.84% when studying the combination of observed and predicted simultaneously (1967–2019).

INTRODUCTION

In recent decades, increasing regard has been assigned to drought events and their tremendous detrimental influences on agriculture, economy, and environment, in scheduling water management programs (Zarei et al. 2016; Hao et al. 2017). As Iran is located in an arid and semi-arid region, prediction and modeling drought in the future is vital. Prediction of drought will lead to plans and administrations that are significantly effective for preparation against drought (Salhvand & Montazeri 2013). Drought is a costly and frequent phenomenon that usually occurs after an extraordinary water deficit due to low rainfall over a large geographical area. The frequency of drought occurrence increases as the region becomes more and more arid (Zarei et al. 2016; Bahrami et al. 2017). There exists no precise definition for drought, but in general, almost all the definitions give the concept of deficit of precipitation resulting in water shortages in applications related to water usage (Wilhite & Glants 1985).

The most well-known indices are Palmer Drought Severity Index, Reconnaissance Drought Index (RDI), Standardized Precipitation Evapotranspiration Index, Multivariate Standardized Drought Index, China Z Index (CZI), and Standardized Precipitation Index (SPI) among many indices that have been studied and utilized all around the world by many scientists (Salhvand & Montazeri 2013; Azadi et al. 2015; Tajbakhsh et al. 2015; Zarei & Mahmoudi 2017; Zarei 2018). The SPI index (McKee et al. 1993) has been utilized, due to being modest, flexible, with a precise approach to calculating and the least number of meteorological parameters needed (Choubin et al. 2016). This index can be used in different time scales (3, 6, 12, 24, 48-month scale). Various studies have been done on drought analysis using SPI (Shakiba et al. 2010; Ansari et al. 2010; Tabari et al. 2011; Paulo et al. 2012; Hanafi et al. 2012; Hasanzade et al. 2013; Zehtabian et al. 2013; Moreira et al. 2014; Shah et al. 2015; Huang et al. 2016).

Zarei et al. (2017) used SPI index and Markov chain method to monitor and predict monthly drought in Iran. The results showed that, in the most synoptic stations, normal, moderately dry, and severe dry classes of drought have the highest frequency of occurrence. Roshan et al. (2018) evaluated a new application for TOPSIS: monitoring drought and wet periods in Iran using RDI index. Sadeghi & Hazabvi (2017) evaluated the spatiotemporal variation of watershed health propensity through a reliability–resilience–vulnerability based drought index (case study: Shazand watershed in Iran).

Time series datasets are a sequence taken set of data collected successively in a prolonged period, in various time scales. The natural order of time series datasets is the most striking feature of time series analysis that distinguishes them from any other analysis. Understanding the previous trend of time series leads to mathematical modeling of the observed trend for monitoring the datasets and a precise prediction in the future as well (Shirmohammadi 2013). Nowadays, analysis of time series models is widely used. Time series analysis is employed for five main reasons: descriptive, explanation, forecasting, intervention analysis, and quality control. Time series have drawn researchers' attention throughout the world (Asakereh et al. 2013; Azad Talatapeh et al. 2013; Ghahraman 2013; Deng et al. 2015; Deng & Wang 2017).

The aims of this study are to focus on: (1) evaluation of occurrence of seasonal drought severity in Iran, using SPI index; (2) modeling and prediction of seasonal drought using time series models, from 2015 up to 2019 based on seasonal SPI index data series of 1967 to 2014; (3) considering the changes trend of drought (using nonparametric trend analysis techniques), termed as: (Spearman rho test) based on observed (1967–2014) and the combination of observed and predicted (1967–2019) seasonal SPI data series. Therefore, meteorological data of 38 synoptic stations in Iran with suitable spatial distribution and various climate conditions were collected.

MATERIALS

Study area and data collection

Iran is located geographically between 25° 03′ to 39° 47′ Northern latitude and 44° 05′ to 63° 18′ Eastern longitude, with an area of 1,648,198 square kilometers. The climate varies from sub-humid to hyper-arid in northern and southern parts of Iran, respectively (based on the United Nation Environmental Program aridity index). The annual precipitation of Iran varies from less than 100 to more than 2,000 millimeters but, in general, the average annual precipitation is about 250 millimeters. This implies there are various climates in Iran. The major amount of the annual precipitation in Iran falls in winter (Dinpashoh et al. 2004). In this study, precipitation data series required for calculating seasonal SPI index were collected from 38 synoptic stations (with different climate conditions) of the country (Figure 1), obtained from the meteorological organization service of Iran for the period 1967–2014 (http://www.irimo.ir/). Characteristics of synoptic stations are presented in Table 1.

Table 1

Geographical descriptions of the synoptic stations used in the study and monthly statistical parameters during the period 1967–2014

Station Latitude (N) Longitude (E) Elevation (meter from sea level) Average of precipitation (mm, year−1Average of ET0 (mm.day−1)a Aridity index Climate condition 
Abadan 30.367 48.250 6.25 158.48 6.82 0.065 Arid 
Ahvaz 31.333 48.667 22.34 234.76 6.79 0.096 Arid 
Arak 34.100 49.767 1,708.18 325.11 3.78 0.239 Semi-arid 
Babolsar 36.717 52.650 −21.31b 922.00 2.55 1.004 Humid 
Bandar Abbas 27.220 56.367 10.19 175.51 5.30 0.092 Arid 
Bandar Lenge 26.533 54.833 23.16 134.70 5.40 0.069 Arid 
Birjand 32.867 59.200 1491.45 163.76 5.23 0.087 Arid 
Bushehr 28.983 50.833 20.12 254.17 5.12 0.138 Arid 
Chabahar 25.283 60.617 8.24 113.66 4.21 0.075 Arid 
Dezful 32.400 48.383 143.25 392.62 4.77 0.229 Semi-arid 
Esfahan 32.617 51.667 1,550.39 130.98 4.36 0.083 Arid 
Fassa 28.967 53.683 1,288.48 147.20 4.69 0.087 Arid 
Ghazvin 36.250 50.050 1,279.13 321.27 3.79 0.235 Semi-arid 
Gorgan 36.850 54.267 13.37 573.11 2.81 0.567 Sub-humid 
Hamedan 35.200 48.717 1,697.22 321.88 4.09 0.219 Semi-arid 
Iran Shahr 27.200 60.700 591.24 111.53 6.28 0.049 Hyper-arid 
Kashan 33.983 51.450 982.31 136.41 3.40 0.111 Arid 
Kerman 30.250 56.967 1,753.47 139.34 5.41 0.072 Arid 
Kermanshah 34.350 47.150 1,318.47 447.09 4.35 0.285 Semi-arid 
Khoram Abad 33.433 48.283 1,147.32 499.73 4.09 0.339 Semi-arid 
Khoy 38.550 44.967 1,103.14 292.54 3.09 0.263 Semi-arid 
Mashhad 36.267 59.633 999.35 255.87 3.96 0.179 Arid 
Oroomieh 37.533 45.083 1,315.15 332.33 3.20 0.288 Semi-arid 
Ramsar 36.900 50.667 −20.25b 1208.89 2.24 1.499 Humid 
Rasht 37.250 49.600 −6.21b 1354.60 2.32 1.622 Humid 
Sabzevar 36.200 57.717 977.35 196.10 5.34 0.102 Arid 
Sanandaj 35.333 47.000 1,373.35 445.60 3.96 0.313 Semi-arid 
Semnan 35.583 53.550 1,130.42 142.10 4.10 0.096 Arid 
Shahre Kord 32.283 50.850 2,048.12 332.43 3.54 0.261 Semi-arid 
Shiraz 29.533 52.600 1,484.32 324.88 4.88 0.185 Arid 
Tabass 33.600 56.917 711.46 82.80 4.79 0.048 Hyper-arid 
Tabriz 38.083 46.283 1,361.29 272.80 4.18 0.181 Arid 
Tehran 35.683 51.317 1,190.36 239.76 4.85 0.137 Arid 
Torbat Hydarieh 35.267 59.217 1,450.19 270.39 3.89 0.193 Arid 
Yazd 31.900 54.283 1,237.28 58.89 5.50 0.030 Hyper-arid 
Zabol 31.033 61.483 489.31 56.21 8.62 0.018 Hyper-arid 
Zahedan 29.467 60.883 1,370.12 79.20 5.81 0.038 Hyper-arid 
Zanjan 36.683 48.483 1,663.18 306.68 3.47 0.246 Semi-arid 
Station Latitude (N) Longitude (E) Elevation (meter from sea level) Average of precipitation (mm, year−1Average of ET0 (mm.day−1)a Aridity index Climate condition 
Abadan 30.367 48.250 6.25 158.48 6.82 0.065 Arid 
Ahvaz 31.333 48.667 22.34 234.76 6.79 0.096 Arid 
Arak 34.100 49.767 1,708.18 325.11 3.78 0.239 Semi-arid 
Babolsar 36.717 52.650 −21.31b 922.00 2.55 1.004 Humid 
Bandar Abbas 27.220 56.367 10.19 175.51 5.30 0.092 Arid 
Bandar Lenge 26.533 54.833 23.16 134.70 5.40 0.069 Arid 
Birjand 32.867 59.200 1491.45 163.76 5.23 0.087 Arid 
Bushehr 28.983 50.833 20.12 254.17 5.12 0.138 Arid 
Chabahar 25.283 60.617 8.24 113.66 4.21 0.075 Arid 
Dezful 32.400 48.383 143.25 392.62 4.77 0.229 Semi-arid 
Esfahan 32.617 51.667 1,550.39 130.98 4.36 0.083 Arid 
Fassa 28.967 53.683 1,288.48 147.20 4.69 0.087 Arid 
Ghazvin 36.250 50.050 1,279.13 321.27 3.79 0.235 Semi-arid 
Gorgan 36.850 54.267 13.37 573.11 2.81 0.567 Sub-humid 
Hamedan 35.200 48.717 1,697.22 321.88 4.09 0.219 Semi-arid 
Iran Shahr 27.200 60.700 591.24 111.53 6.28 0.049 Hyper-arid 
Kashan 33.983 51.450 982.31 136.41 3.40 0.111 Arid 
Kerman 30.250 56.967 1,753.47 139.34 5.41 0.072 Arid 
Kermanshah 34.350 47.150 1,318.47 447.09 4.35 0.285 Semi-arid 
Khoram Abad 33.433 48.283 1,147.32 499.73 4.09 0.339 Semi-arid 
Khoy 38.550 44.967 1,103.14 292.54 3.09 0.263 Semi-arid 
Mashhad 36.267 59.633 999.35 255.87 3.96 0.179 Arid 
Oroomieh 37.533 45.083 1,315.15 332.33 3.20 0.288 Semi-arid 
Ramsar 36.900 50.667 −20.25b 1208.89 2.24 1.499 Humid 
Rasht 37.250 49.600 −6.21b 1354.60 2.32 1.622 Humid 
Sabzevar 36.200 57.717 977.35 196.10 5.34 0.102 Arid 
Sanandaj 35.333 47.000 1,373.35 445.60 3.96 0.313 Semi-arid 
Semnan 35.583 53.550 1,130.42 142.10 4.10 0.096 Arid 
Shahre Kord 32.283 50.850 2,048.12 332.43 3.54 0.261 Semi-arid 
Shiraz 29.533 52.600 1,484.32 324.88 4.88 0.185 Arid 
Tabass 33.600 56.917 711.46 82.80 4.79 0.048 Hyper-arid 
Tabriz 38.083 46.283 1,361.29 272.80 4.18 0.181 Arid 
Tehran 35.683 51.317 1,190.36 239.76 4.85 0.137 Arid 
Torbat Hydarieh 35.267 59.217 1,450.19 270.39 3.89 0.193 Arid 
Yazd 31.900 54.283 1,237.28 58.89 5.50 0.030 Hyper-arid 
Zabol 31.033 61.483 489.31 56.21 8.62 0.018 Hyper-arid 
Zahedan 29.467 60.883 1,370.12 79.20 5.81 0.038 Hyper-arid 
Zanjan 36.683 48.483 1,663.18 306.68 3.47 0.246 Semi-arid 

aET0 calculated based on FAO Penman–Monteith (FAO-56).

bMentioned synoptic stations located on the coasts of the Caspian Sea. The elevation of this area is lower than sea level.

Figure 1

Spatial distribution of the synoptic stations in the study area.

Figure 1

Spatial distribution of the synoptic stations in the study area.

METHODS

Standardized Precipitation Index

Mckee et al. (1993) reported the SPI index to monitor and evaluate drought. Fitting a gamma probability density to the total precipitation is essential, in order to assess SPI indices. The gamma distribution function (г(α)) is fitted to the dataset of precipitation involving a shape factor and a scale factor, termed as α and β, respectively, which require an estimation for each year and time scale, hence the maximum likelihood solutions are employed. If the amount of precipitation is represented by x, the cumulative probability (G(x)), can be assessed by: 
formula
(1)
If x is equal to zero, the gamma function is undefined and the precipitation distribution may contain zeros (Mckee et al. 1993), and the definition of cumulative probability (H(x)) changes to (Zarei & Mahmudi 2017): 
formula
(2)
where q is the probability of a zero. The cumulative probability is then transformed to the standard normal random variable Z with mean zero and variance of one, which is the value of SPI. SPI is categorized based on their range values as shown in Table 2.
Table 2

Drought classification of SPI (McKee et al. 1993; Bahrami et al. 2017)

SPI domain Drought class Class number 
+2 ≥ Extremely wet 
1.5 < SPI < 1.99 Very wet 
1.0 < SPI < 1.49 Moderately wet 
−0.99 < SPI < 0.99 Near normal 
−1.0 < SPI < −1.49 Moderately dry 
−1.5 < SPI < −1.99 Very dry 
−2 ≤ Extremely dry 
SPI domain Drought class Class number 
+2 ≥ Extremely wet 
1.5 < SPI < 1.99 Very wet 
1.0 < SPI < 1.49 Moderately wet 
−0.99 < SPI < 0.99 Near normal 
−1.0 < SPI < −1.49 Moderately dry 
−1.5 < SPI < −1.99 Very dry 
−2 ≤ Extremely dry 

Time series models

In general, time series models have three major models: Auto-Regressive (AR), Moving-Average (MA), and Auto-Regressive–Moving-Average (ARMA). AR model divides into Yule-Walker and Burg, and MA model divides into Hannan-Rissanen and innovations. Times series models, like any other model, have some variables (Mirzavand & Ghazavi 2015). The number of variables in AR and MA models are demonstrated by p and q indices, respectively. Various species of models for time series data exist that illustrate different non-deterministic processes. Modeling of time series typically occur based on a linear technique, for instance AR, MA, ARMA, and ARIMA (ARMA with more than one difference) models have a linear base (Klose et al. 2004). These models have been widely employed in various branches of science, such as meteorology sciences, water sciences, and others (Shabani et al. 2013). In this research, to assess time series model ITSM (Interactive Time Series Modeling) software was used. This software is an interactive Windows-based menu-driven software for time series modeling and forecasting (Brockwell & Davis 1991).

AR model

In series where persistency is present, that is the event outcome of (t+ 1)th period is dependent on the present tth period magnitude and those preceding values, then for such a series, the observed sequence X1, X2,, Xt is used to fit an AR model (Equation (3)): 
formula
(3)
where ,, … , are model parameters and coefficients, et is random error and Zt is the random component of the data that follows a normal distribution with mean 0 (Jahandideh & Shirvani 2011; Dastorani et al. 2016).

MA model

Moving average models are simple covariance stationary and ergodic models that can be used for a wide variety of autocorrelation patterns (Equation (4)): 
formula
(4)
where ,, … , are model parameters and coefficient, et is random error, and Zt is the random component of the data that follows a normal distribution with mean 0 (Jahandideh & Shirvani 2011; Dastorani et al. 2016).

ARMA model

The ARMA model is a synthesis of an AR and an MA model. ARMA models form a type of linear models which are widely applicable and parsimonious in parameterization (Equation (5)): 
formula
(5)
where is the stationary part of the ARMA model, indicates the th AR coefficient, is the th moving average coefficient, it shows the error part at time period , and refers to the value of drought observed or predicted at time period t (Jahandideh & Shirvani 2011; Dastorani et al. 2016; Zarei 2018).

ARIMA model

Autoregressive Integrated Moving Average (ARIMA) models are one of the well-known linear models for time series modeling and predicting. ARIMA models originated from the synthesis of AR and MA models. ARIMA is used to model time series data behavior and to make predictions. ARIMA modeling uses correlational methods and could be used to model arrays that may not be observable in plotted data. In ARIMA, the future amount of a parameter is assumed to be a linear function of past observations and random errors (Jahandideh & Shirvani 2011).

SARIMA model

A SARIMA model can be explained as ARIMA (,,) (,,), where (,,) is the non-seasonal component of the model and (,,) is the seasonal component of the model in which p is the order of non-seasonal auto-regression, d is the number of regular differencing, q is the order of nonseasonal moving average, P is the order of seasonal auto-regression, D is the number of seasonal differencing, Q is the order of seasonal moving average. The subscripted letter ‘s’ shows the length of the seasonal period. For example, in an hourly data time series s = 7, in a quarterly data s = 4, and in a monthly data s = 12 (Martinez et al. 2011).

Stationarity of SPI data series

For stationarity of calculated seasonal SPI indices, Box–Cox (Equation (6)) power (Shumway 1988) and difference conversion were used: 
formula
(6)
where zi is the initial value in time i, λ conversion parameter and ziλ is the converted value.

Time series model selection

The best time series models were selected based on autocorrelation functions (ACF), partial autocorrelation functions (PACF) and AICC indices. ACF with k lag as a function for expressing time depending on a time series structure is as follows: 
formula
(7)
where ρk is the value of time series autocorrelation function with k lags, zi and zi+k are the values of variables or time series data within phase time I, and phase with lag time k and is average value corresponding to variables.
Another approach for expressing time independently within a time series structure, is the definition of PACF (Equation (8)). If is the PACF of time series with lag time of k, the equation for PACF becomes: 
formula
(8)
where is the value of time series PACF with the lag of k.
The equation for computing AICC index depends upon the number of data used for calibration (n) and parameters used in modeling process (k) and the abbreviation form of mean square error as presented in Equation (9): 
formula
(9)
The less AICC index, the better the model fits the dataset.

Prediction and validation of SPI index

In this stage, the best time series models that fitted the seasonal SPI data series from 1967 to 2014 were used to predict seasonal SPI data series from 2015 to 2019 using ARAR method. (The ARAR algorithm is basically the process that applies memory shortening transformation and fitting the AR model to the transformed data. It is used to predict the future data from existing sequence data. The algorithm was introduced by Brockwell & Davis (1991) and it consists of three phases throughout the process.) To evaluate the accuracy of selected model prediction Ljung–Box p-value (if p-value of Ljung–Box index was more than 0.05 randomness of data series is significant at the 5% level), residual ACF, residual PACF (according to these tests, in both charts it should not be more than 5% of maximum number of lags (40) out of the zero band) and correlation coefficient between predicted and observed seasonal SPI data during 2010–2014 were used.

Trend analysis of seasonal SPI data series

In this section, after assessing the normality of seasonal SPI data series, the trends of these series were evaluated based on non-parametric statistical (Spearman's rho test) and parametric test (Pearson test). In this study, the changes trend of data series was evaluated in two stage: Stage 1, the changes trend of observed seasonal SPI data series (1967–2014) and Stage 2, changes trend of observed and predicted seasonal SPI data series (1967–2019).

RESULTS AND DISCUSSION

SPI index

In the present study, the SPI index in seasonal scale for 38 synoptic stations was calculated (Figure 2) based on the expressed approach by McKee et al. (1993). Results of SPI calculation showed that in all stations, drought severity with class 4 (normal class) and 3 (moderately wet) had the most occurrence frequency, and according to the results, extremely dry and very dry classes had the least occurrence frequency (Table 3).

Table 3

Frequency of each drought class for the calculated SPI (%)

Drought class 
Abadan 5.21 11.98 82.81 
Ahvaz 0.52 2.60 16.67 80.21 
Arak 0.52 1.04 12.50 64.06 21.88 
Babolsar 0.52 6.77 10.42 64.06 11.46 5.21 1.56 
Bandar Abbas 2.60 5.73 9.38 82.29 
Anzali 2.08 4.69 9.90 64.58 11.98 5.21 1.56 
Lenge 1.56 4.69 13.02 80.73 
Birjand 0.52 6.25 10.42 82.81 
Bushehr 1.04 3.13 14.58 81.25 
Chabahar 1.56 6.25 8.33 83.85 
… … … … … … … … 
Shiraz 0.52 5.21 12.50 62.50 19.27 
Tabas 1.04 5.21 8.33 85.42 
Tabriz 0.52 3.65 8.85 71.35 6.25 5.73 3.65 
Tehran 3.13 10.94 66.15 8.85 10.94 
Torbat 4.17 10.94 64.06 20.83 
Yazd 1.56 3.65 12.50 82.29 
Zabol 1.04 6.77 8.33 83.85 
Zahedan 1.56 4.17 11.98 82.29 
Zanjan 1.56 12.50 68.75 6.25 7.29 3.65 
Drought class 
Abadan 5.21 11.98 82.81 
Ahvaz 0.52 2.60 16.67 80.21 
Arak 0.52 1.04 12.50 64.06 21.88 
Babolsar 0.52 6.77 10.42 64.06 11.46 5.21 1.56 
Bandar Abbas 2.60 5.73 9.38 82.29 
Anzali 2.08 4.69 9.90 64.58 11.98 5.21 1.56 
Lenge 1.56 4.69 13.02 80.73 
Birjand 0.52 6.25 10.42 82.81 
Bushehr 1.04 3.13 14.58 81.25 
Chabahar 1.56 6.25 8.33 83.85 
… … … … … … … … 
Shiraz 0.52 5.21 12.50 62.50 19.27 
Tabas 1.04 5.21 8.33 85.42 
Tabriz 0.52 3.65 8.85 71.35 6.25 5.73 3.65 
Tehran 3.13 10.94 66.15 8.85 10.94 
Torbat 4.17 10.94 64.06 20.83 
Yazd 1.56 3.65 12.50 82.29 
Zabol 1.04 6.77 8.33 83.85 
Zahedan 1.56 4.17 11.98 82.29 
Zanjan 1.56 12.50 68.75 6.25 7.29 3.65 
Figure 2

Seasonal SPI data series in (a) Zanjan (with semi-arid climate condition) and (b) Ramsar (with humid climate condition).

Figure 2

Seasonal SPI data series in (a) Zanjan (with semi-arid climate condition) and (b) Ramsar (with humid climate condition).

Stationarity of SPI series

After calculation of SPI indices, for stationarity of seasonal data series, Box–Cox power conversion was used to immobilize the variances of datasets of seasonal SPI indices. Difference conversion was used for removal of trend in data series and to exert the frequency impact in the SPI indices dataset. Then the set of non-stationary data was converted to stationary data and was ready for modeling (Figure 3).

Figure 3

Seasonal SPI data series after making them stationary: (a) Zanjan (with semi-arid climate condition), (b) Yazd (with hyper-arid climate condition), and (c) Ramsar (with humid climate condition).

Figure 3

Seasonal SPI data series after making them stationary: (a) Zanjan (with semi-arid climate condition), (b) Yazd (with hyper-arid climate condition), and (c) Ramsar (with humid climate condition).

Selection of the best time series model

In this stage, various kinds of time series models were fitted to SPI datasets and the best fitted model was chosen based on having the least AICC index (Table 4). To determine order of time series models (p and q), ACF and PACF were used (Figure 4).

Table 4

The best time series fitted in seasonal SPI data series (for example)

Station AICC Index Method of time series model Non-significant θ or φ Fitted model 
Abadan 302.88 MA (5) Innovations Θ2 and θ3 X(t) = Z(t) −1.043 Z(t-1) −0.9407 Z(t-4) +1.002 Z(t-5) 
Bushehr 240.062 MA (5) Hannan-Rissanen Θ2 and θ3 X(t) = Z(t) −0.8941 Z(t-1) −0.9832 Z(t-4) +0.9123 Z(t-5) 
Bandar Abbas 385.08 MA (13) Innovations Θ2, Θ3, Θ6, Θ7, Θ8, Θ9, Θ10, Θ11 and Θ12 X(t) = Z(t) −1.364 Z(t-1) −1.038 Z(t-4) + 1.480 Z(t-5) −0.1564 Z(t-13) 
Hamedan 309.058 AR(16) Burg – X(t) = −0.9065 X(t-1) −0.6757 X(t-2) - 0.7055 X(t-3) −1.399 X(t-4) −1.236 X(t-5) - 0.9266 X(t-6) −1.031 X(t-7) −1.395 X(t-8) −1.011 X(t-9) −0.7304 X(t-10) −0.8228 X(t-11) −0.8379 X(t-12) − 0.4322 X(t-13) −0.2229 X(t-14) −0.2219 X(t-15) −0.2780 X(t-16) + Z(t) 
Ramsar 489.954 AR(12) Burg – X(t) = −0.8678 X(t-1) −0.6830 X(t-2) −0.4751 X(t-3) −1.081 X(t-4) −0.9604 X(t-5) −0.7525 X(t-6) −0.4821 X(t-7) −0.7516 X(t-8) −0.5709 X(t-9) −0.4051 X(t-10) −0.2051 X(t-11) −0.2698 X(t-12) +Z(t) 
Yazd 348.922 AR(14) Burg – X(t) = −0.8125 X(t-1) −0.5461 X(t-2) −0.4024 X(t-3) −1.230 X(t-4) −0.9642 X(t-5) −0.7978 X(t-6) −0.4816 X(t-7) −0.9558 X(t-8) −0.7731 X(t-9) −0.6326 X(t-10) −0.2937 X(t-11) −0.5050 X(t-12) −0.4143 X(t-13) −0.2123 X(t-4) +Z(t) 
Station AICC Index Method of time series model Non-significant θ or φ Fitted model 
Abadan 302.88 MA (5) Innovations Θ2 and θ3 X(t) = Z(t) −1.043 Z(t-1) −0.9407 Z(t-4) +1.002 Z(t-5) 
Bushehr 240.062 MA (5) Hannan-Rissanen Θ2 and θ3 X(t) = Z(t) −0.8941 Z(t-1) −0.9832 Z(t-4) +0.9123 Z(t-5) 
Bandar Abbas 385.08 MA (13) Innovations Θ2, Θ3, Θ6, Θ7, Θ8, Θ9, Θ10, Θ11 and Θ12 X(t) = Z(t) −1.364 Z(t-1) −1.038 Z(t-4) + 1.480 Z(t-5) −0.1564 Z(t-13) 
Hamedan 309.058 AR(16) Burg – X(t) = −0.9065 X(t-1) −0.6757 X(t-2) - 0.7055 X(t-3) −1.399 X(t-4) −1.236 X(t-5) - 0.9266 X(t-6) −1.031 X(t-7) −1.395 X(t-8) −1.011 X(t-9) −0.7304 X(t-10) −0.8228 X(t-11) −0.8379 X(t-12) − 0.4322 X(t-13) −0.2229 X(t-14) −0.2219 X(t-15) −0.2780 X(t-16) + Z(t) 
Ramsar 489.954 AR(12) Burg – X(t) = −0.8678 X(t-1) −0.6830 X(t-2) −0.4751 X(t-3) −1.081 X(t-4) −0.9604 X(t-5) −0.7525 X(t-6) −0.4821 X(t-7) −0.7516 X(t-8) −0.5709 X(t-9) −0.4051 X(t-10) −0.2051 X(t-11) −0.2698 X(t-12) +Z(t) 
Yazd 348.922 AR(14) Burg – X(t) = −0.8125 X(t-1) −0.5461 X(t-2) −0.4024 X(t-3) −1.230 X(t-4) −0.9642 X(t-5) −0.7978 X(t-6) −0.4816 X(t-7) −0.9558 X(t-8) −0.7731 X(t-9) −0.6326 X(t-10) −0.2937 X(t-11) −0.5050 X(t-12) −0.4143 X(t-13) −0.2123 X(t-4) +Z(t) 
Figure 4

Autocorrelation and partial ACF of the seasonal times series in (a) Zanjan and (b) Yazd stations.

Figure 4

Autocorrelation and partial ACF of the seasonal times series in (a) Zanjan and (b) Yazd stations.

According to the results of ACF and PACF, p and q orders equal to 5, 12, 13, 14, 16, 18, 22, and 25 were the best orders that were capable in the fitted time series models.

Results of time series model selection indicated that at Abadan, Arak, Bandar Lenge, Birjand, Dezful, Esfahan, Fasa, Qazvin, Gorgan, Iran Shahr, Kashan, Kermanshah, Khoramabad, Khoy, Oroomieh, Sabzevar, Sanandaj, Semnan, Tabriz, Tehran, Torbate Heydarieh, Zahedan, and Zanjan stations, MA (5) Innovations model was the best fitted time series model. At Ahvaz, Babolsar, Bushehr, Kerman, Sharekord, and Shiraz stations, MA (5) Hannan-Rissanen was the best fitted time series model. The only station fitted by MA (13) Innovations model is Bandar Abbas. Chabahar and Hamedam stations were fitted by AR (16) Burg model while at Rasht and Tabas stations, AR (18) Burg was the best fitted time series model. In Mashhad, Ramsar, Yazd and Zabol, the best fitted time series were AR (25) Burg, AR (12) Burg, AR (14) Burg, and AR (22) Burg, respectively.

Time series model validation

In order to validate the model dataset of seasonal SPI indices, five years (20 seasons) observed data were available, and were predicted using ARAR model. To compare predicted and observed seasonal SPI data, correlation coefficient was used (Table 5). According to the results, in all stations correlation coefficients between observed and predicted data series of seasonal SPI from 2010 to 2014 were significant at 5% level. In stations with humid, sub-humid, semi-humid, and semi-arid climate condition and stations with high relative humidity such as Chabahar, Gorgan, and Orumieh stations, R was less than the correlation coefficient in stations with arid and hyper-arid climate condition with low relative humidity such as Zabol, Yazd, and Tabas stations.

Table 5

Results of obtained correlation coefficient between observed and predicted data

Station Ra Station Ra Station Ra 
Abadan 0.713 Gorgan 0.623 Sanandaj 0.883 
Ahvaz 0.856 Hamedan 0.962 Semnan 0.790 
Arak 0.858 Iran Shahr 0.785 Sharekord 0.906 
Babolsar 0.777 Kashan 0.868 Shiraz 0.874 
Bandar Abbas 0.828 Kerman 0.845 Tabas 0.892 
Bandar Lenge 0.812 Kermanshah 0.824 Tabriz 0.774 
Birjand 0.913 Khoram Abad 0.812 Tehran 0.901 
Bushehr 0.868 Khoy 0.673 Torbate Heydarieh 0.920 
Chabahar 0.481 Mashhad 0.857 Yazd 0.817 
Dezful 0.756 Oroomieh 0.801 Zabol 0.824 
Esfahan 0.910 Ramsar 0.809 Zahedan 0.664 
Fasa 0.869 Rasht 0.857 Zanjan 0.860a 
Ghazvin 0.851 Sabzevar 0.885   
Station Ra Station Ra Station Ra 
Abadan 0.713 Gorgan 0.623 Sanandaj 0.883 
Ahvaz 0.856 Hamedan 0.962 Semnan 0.790 
Arak 0.858 Iran Shahr 0.785 Sharekord 0.906 
Babolsar 0.777 Kashan 0.868 Shiraz 0.874 
Bandar Abbas 0.828 Kerman 0.845 Tabas 0.892 
Bandar Lenge 0.812 Kermanshah 0.824 Tabriz 0.774 
Birjand 0.913 Khoram Abad 0.812 Tehran 0.901 
Bushehr 0.868 Khoy 0.673 Torbate Heydarieh 0.920 
Chabahar 0.481 Mashhad 0.857 Yazd 0.817 
Dezful 0.756 Oroomieh 0.801 Zabol 0.824 
Esfahan 0.910 Ramsar 0.809 Zahedan 0.664 
Fasa 0.869 Rasht 0.857 Zanjan 0.860a 
Ghazvin 0.851 Sabzevar 0.885   

aCorrelation coefficient is significant at 5% level.

Model prediction

In this stage, to predict seasonal SPI indices for five years after 2014 (from 2015 to 2019), the ARAR approach was employed. Predicted seasonal SPI indices and occurrence frequency are presented in Tables 6 and 7 and Figure 5.

Table 6

Predicted SPI index in different stations and different seasons

Season Station Abadan …. Tehran Yazd Zabol Zahedan Zanjan 
year Predicted seasonal SPI index 
Winter 2015 0.368 … 0.522 0.870 0.513 0.307 0.498 
2016 0.008 … 0.580 −0.118 −0.046 0.448 0.443 
2017 −0.422 … 0.463 −0.618 −0.502 −0.493 −0.897 
2018 0.714 … 0.525 0.171 0.373 0.102 0.661 
2019 0.348 … 0.534 0.958 0.906 0.785 0.407 
Spring 2015 −0.159 … −0.102 −0.046 0.084 0.427 0.257 
2016 −0.410 … −0.188 −0.600 −0.290 −0.257 −0.960 
2017 0.506 … −0.206 0.386 0.174 −0.299 0.698 
2018 0.518 … −0.152 0.836 0.646 0.631 0.400 
2019 −0.140 … −0.154 −0.013 0.029 0.224 0.282 
Summer 2015 −0.368 … −0.102 −0.537 −0.365 −0.435 −0.769 
2016 0.467 … −0.884 0.303 0.301 −0.184 0.569 
2017 0.422 … −0.775 0.814 0.704 0.450 0.350 
2018 −0.085 … −0.620 −0.016 0.125 0.420 0.266 
2019 −0.247 … −0.567 −0.505 −0.224 −0.247 −0.718 
Autumn 2015 0.506 … 0.386 0.318 0.256 −0.044 0.556 
2016 0.427 … 0.353 0.757 0.627 0.573 0.308 
2017 −0.139 … 0.309 −0.027 0.087 0.332 0.221 
2018 −0.231 … 0.315 −0.452 −0.244 −0.202 −0.670 
2019 0.452 … 0.307 0.292 0.275 −0.155 0.529 
Season Station Abadan …. Tehran Yazd Zabol Zahedan Zanjan 
year Predicted seasonal SPI index 
Winter 2015 0.368 … 0.522 0.870 0.513 0.307 0.498 
2016 0.008 … 0.580 −0.118 −0.046 0.448 0.443 
2017 −0.422 … 0.463 −0.618 −0.502 −0.493 −0.897 
2018 0.714 … 0.525 0.171 0.373 0.102 0.661 
2019 0.348 … 0.534 0.958 0.906 0.785 0.407 
Spring 2015 −0.159 … −0.102 −0.046 0.084 0.427 0.257 
2016 −0.410 … −0.188 −0.600 −0.290 −0.257 −0.960 
2017 0.506 … −0.206 0.386 0.174 −0.299 0.698 
2018 0.518 … −0.152 0.836 0.646 0.631 0.400 
2019 −0.140 … −0.154 −0.013 0.029 0.224 0.282 
Summer 2015 −0.368 … −0.102 −0.537 −0.365 −0.435 −0.769 
2016 0.467 … −0.884 0.303 0.301 −0.184 0.569 
2017 0.422 … −0.775 0.814 0.704 0.450 0.350 
2018 −0.085 … −0.620 −0.016 0.125 0.420 0.266 
2019 −0.247 … −0.567 −0.505 −0.224 −0.247 −0.718 
Autumn 2015 0.506 … 0.386 0.318 0.256 −0.044 0.556 
2016 0.427 … 0.353 0.757 0.627 0.573 0.308 
2017 −0.139 … 0.309 −0.027 0.087 0.332 0.221 
2018 −0.231 … 0.315 −0.452 −0.244 −0.202 −0.670 
2019 0.452 … 0.307 0.292 0.275 −0.155 0.529 
Table 7

Occurrence frequency of each predicted drought classes in percentage, from winter 2015 to autumn 2019

Drought class 
Abadan 100 
Arak 95 
Bandar abbas 95 
Anzali 20 60 20 
Esfahan 100 
Fasa 20 65 10 
Qazvin 85 15 
Hamedan 80 20 
Kerman 95 
Kermanshah 100 
Orumieh 80 20 
Ramsar 85 10 
Rasht 10 65 20 
Semnan 100 
Shahrekord 15 85 
Zanjan 100 
Drought class 
Abadan 100 
Arak 95 
Bandar abbas 95 
Anzali 20 60 20 
Esfahan 100 
Fasa 20 65 10 
Qazvin 85 15 
Hamedan 80 20 
Kerman 95 
Kermanshah 100 
Orumieh 80 20 
Ramsar 85 10 
Rasht 10 65 20 
Semnan 100 
Shahrekord 15 85 
Zanjan 100 
Figure 5

Observed, validated, and predicted seasonal SPI index in Zanjan (with semi-arid climate condition) and Yazd (with hyper-arid climate condition) stations.

Figure 5

Observed, validated, and predicted seasonal SPI index in Zanjan (with semi-arid climate condition) and Yazd (with hyper-arid climate condition) stations.

Results of observed, validated, and predicted seasonal SPI data series are indicated in Figure 5 for some stations.

According to the results, in all the synoptic stations, drought severity class 4 (normal) had the most frequency (in 71% of stations occurrence frequency of class 4 was 100%, in 21% of stations occurrence frequency of class 4 was more than 80%, occurrence frequency of class 4 in other stations was more than 60%). In addition, classes 1 and 7 (extremely wet and extremely dry) had the least occurrence frequency (Tables 6 and 7).

In the next stage, capability of time series models (based on ARAR method) for prediction of seasonal SPI indices were evaluated using Ljung–Box p-value and residual ACF and PACF methods (Figure 6 and Table 8).

Table 8

Ljung p-value of time series models to evaluate the capability of SPI predictions

Station Ljung p-value Station Ljung p-value Station Ljung p-value 
Abadan 0.702 Gorgan 0.133 Sanandaj 0.033 
Ahvaz 0.026 Hamedan 0.090 Semnan 0.743 
Arak 0.571 Iran Shahr 0.162 Sharekord 0.192 
Babolsar 0.065 Kashan 0.937 Shiraz 0.193 
Bandar Abbas 0.237 Kerman 0.672 Tabas 0.199 
Bandar Lenge 0.517 Kermanshah 0.326 Tabriz 0.155 
Birjand 0.055 Khoram Abad 0.874 Tehran 0.756 
Bushehr 0.041 Khoy 0.166 Torbate eydarieh 0.042 
Chabahar 0.096 Mashhad 0.020 Yazd 0.653 
Dezful 0.129 Oroomieh 0.533 Zabol 0.273 
Esfahan 0.012 Ramsar 0.201 Zahedan 0.011 
Fasa 0.019 Rasht 0.037 Zanjan 0.295 
Ghazvin 0.609 Sabzevar 0.147   
Station Ljung p-value Station Ljung p-value Station Ljung p-value 
Abadan 0.702 Gorgan 0.133 Sanandaj 0.033 
Ahvaz 0.026 Hamedan 0.090 Semnan 0.743 
Arak 0.571 Iran Shahr 0.162 Sharekord 0.192 
Babolsar 0.065 Kashan 0.937 Shiraz 0.193 
Bandar Abbas 0.237 Kerman 0.672 Tabas 0.199 
Bandar Lenge 0.517 Kermanshah 0.326 Tabriz 0.155 
Birjand 0.055 Khoram Abad 0.874 Tehran 0.756 
Bushehr 0.041 Khoy 0.166 Torbate eydarieh 0.042 
Chabahar 0.096 Mashhad 0.020 Yazd 0.653 
Dezful 0.129 Oroomieh 0.533 Zabol 0.273 
Esfahan 0.012 Ramsar 0.201 Zahedan 0.011 
Fasa 0.019 Rasht 0.037 Zanjan 0.295 
Ghazvin 0.609 Sabzevar 0.147   
Figure 6

Residual ACF and PACF for the seasonal time series in (a) Zanjan and (b) Yazd stations.

Figure 6

Residual ACF and PACF for the seasonal time series in (a) Zanjan and (b) Yazd stations.

Results of residual ACF and residual PACF demonstrated that in all the stations, capability of time series models was suitable at the 5% significance level because in both charts, residual ACF and PACF, no more than 5% of maximum number of lags (40) was out of the 95% confidence band (Figure 5). According to the Ljung–Box p-value, in 82% of stations randomness of data series was suitable at 5% significance level.

Trend analysis of seasonal SPI data series

Finally, trend of observed seasonal SPI data series from 1967 to 2014 and the whole observed and predicted seasonal SPI data series from 1967 to 2019 simultaneously was assessed statistically at 5% significance level, using Spearman's rho and Pearson, non-parametric and parametric tests, respectively (Table 9).

Table 9

Results of parametric and nonparametric correlation (changes of trend) based on observed (1967–2014) and observed plus predicted (1967–2019)

Station Observed Observed and predicted Station Observed Observed and predicted 
Abadan −0.061 −0.044 Khoramabad −0.111 −0.109 
Ahvaz −0.099 −0.068 Khoy −0.094 −0.081 
Arak −0.080 −0.087 Mashhad −0.054 −0.056 
Babolsar −0.002 −0.002 Orumieh −0.121 −0.119 
Bandar Abbas −0.076 −0.048 Ramsar 0.022 0.028 
Lenge −0.104 −0.051 Rasht −0.065 −0.050 
Birjand −0.083 −0.063 Sabzavar −0.046 −0.061 
Bushehr −0.014 −0.003 Sanandaj −0.169a −0.172a 
Chabahar 0.002 0.070 Semnan −0.004 −0.019 
Dezful −0.046 −0.030 Shahrekord −0.014 −0.010 
Esfahan 0.035 0.050 Shiraz −0.031 −0.023 
Fasa −0.065 −0.054 Tabas 0.100 0.120 
Qazvin 0.000 −0.049 Tabriz −0.155a −0.135a 
Gorgan −0.232a −0.231a Tehran −0.003 −0.036 
Hamedan −0.063 −0.113 Torbat −0.067 −0.073 
Iranshahr −0.082 −0.061 Yazd −0.069 −0.042 
Kashan 0.000 0.004 Zabol −0.076 −0.034 
Kerman −0.081 −0.063 Zahedan −0.066 −0.039 
Kermanshah −0.114 −0.104 Zanjan −0.063 −0.058 
Station Observed Observed and predicted Station Observed Observed and predicted 
Abadan −0.061 −0.044 Khoramabad −0.111 −0.109 
Ahvaz −0.099 −0.068 Khoy −0.094 −0.081 
Arak −0.080 −0.087 Mashhad −0.054 −0.056 
Babolsar −0.002 −0.002 Orumieh −0.121 −0.119 
Bandar Abbas −0.076 −0.048 Ramsar 0.022 0.028 
Lenge −0.104 −0.051 Rasht −0.065 −0.050 
Birjand −0.083 −0.063 Sabzavar −0.046 −0.061 
Bushehr −0.014 −0.003 Sanandaj −0.169a −0.172a 
Chabahar 0.002 0.070 Semnan −0.004 −0.019 
Dezful −0.046 −0.030 Shahrekord −0.014 −0.010 
Esfahan 0.035 0.050 Shiraz −0.031 −0.023 
Fasa −0.065 −0.054 Tabas 0.100 0.120 
Qazvin 0.000 −0.049 Tabriz −0.155a −0.135a 
Gorgan −0.232a −0.231a Tehran −0.003 −0.036 
Hamedan −0.063 −0.113 Torbat −0.067 −0.073 
Iranshahr −0.082 −0.061 Yazd −0.069 −0.042 
Kashan 0.000 0.004 Zabol −0.076 −0.034 
Kerman −0.081 −0.063 Zahedan −0.066 −0.039 
Kermanshah −0.114 −0.104 Zanjan −0.063 −0.058 

aCorrelation coefficient of trends is statistically significant at 5% level.

Based on the statistical test for observed seasonal SPI data series, in none of the stations except Gorgan, Sanandaj, and Tabriz was any significant trend observed. However, these three stations (Gorgan, Sanandaj, and Tabriz) had a significant decreasing trend in the values of their observed seasonal SPI data series. The same results were shown in the statistical test for the combination of observed and predicted seasonal SPI data series. Apart from having a significant trend, in all study stations except Chabahar, Esfahan, Qazvin, Kashan, Ramsar, and Tabas, a slight descending trend was observed in seasonal SPI data series. On the other hand, results of parametric and nonparametric test for the whole observed and predicted seasonal SPI data series from 1967 to 2014 indicated a gradual negative trend in the seasonal SPI data series in all stations, except Chabahar, Esfahan, Kashan, Ramsar, and Tabas. Decreasing trend in SPI index is due to the reduction of precipitation, which can be dangerous for different sections such as agriculture, ecosystems, water resources, and others.

CONCLUSIONS

Since drought affects human activities, agriculture, economy, environment, water resources management, evaluation and recognition of drought behavior will help society to overcome this disastrous phenomenon. In the present study, the seasonal SPI data series of 38 synoptic stations of Iran were calculated, modeled, and predicted. Finally, change trend of calculated seasonal SPI data series, based on non-parametric and parametric statistical test was evaluated. Results of this research demonstrated that in the majority of stations throughout the study, MA (5) Innovation and MA (5) Hannan-Rissenen were fitted better than other models with minimum AICC index. Results of predicted seasonal SPI indices indicated that drought severity class 4 (normal) had the most and classes 1 and 7 (extremely wet and extremely dry) had the least occurrence frequency. Results of validation indicated that the model prediction performed very cautiously and the lower band and upper band of the predicted SPI index and validated SPI index varies from the observed SPI index. Based on changes trend of data series test, in the vast majority of the synoptic stations in the study area, with various drought severity classes, a slight negative trend was seen in the seasonal SPI data series. In this study, changes trend of drought based on observed data (1967–2014) and changes trend of drought based on observed and predicted data (1967–2019) were evaluated, and the effect of SPI index from 2014 to 2019 in trend of changes in drought severity resulted (in other works mainly changes trend of observed data were evaluated). Zarei et al. (2016) employed the RDI index and time series model for modeling and prediction of the seasonal drought in Tehran. Their results demonstrated that MA (5) Hannan-Rissenen was the best fitted model, contrary to the obtained result of this study where MA (5) Innovation fitted the dataset as well as possible. The difference may be because of the various indices. Jahandideh & Shirvani (2011) used SPI and time series models to forecast drought in Fars province (Shiraz and Fasa synoptic stations) using precipitation data from 1967 to 2005. According to the results, the SARIMA model with the minimum AICC index was selected as the best model. However, the current research rejects their results in some cases. The difference may be because of the various time periods. According to the fact that the main occupation of Iranian people is agriculture and livestock farming, drought and its side effects have more detrimental effects on Iran. Therefore, drought assessment and drought prediction can be more helpful in reducing drought impacts. Finally, it is suggested that other research should be done using other models to predict drought index to confirm the results of this study. According to the results of validation, time series model prediction implemented very cautiously and the lower band and upper band of the predicted SPI index were close to the average of observed SPI data series.

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