*3.2. Drought Frequency Variations*

The SPEI time series illustrates at variable time scales that covers the period from 2005 to 2019 for Tharparkar region shown in Figure 2. The outcome of this research manifests that the area of Tharparkar will face more and more drought in future. Figure 3. shows a clear idea that there is a continuous rise in the conditions of drought. In the starting years, the drought tendency proceeds towards the natural limits of close normal and reasonably. Wet scales with few years showing irregular non-typical values. These abnormal values are related to the shortage of rainfall, while on the contrary, in previous decade, the beginning of drought conditions with extreme dry patterns started to occur. All the localities of the region of Tharparkar depict time evolution, with slight deviations among others. This situation is the result of changing climatic patterns and their effect. The other parts of the world, e.g., Egypt [51], Turkey [52], Portugal [53], and China [54] also show such circumstances due to climatic change impact.

**Figure 3.** Time series plots of SPEI- 3, 6, 9, 12 and 24 are in right panel, and the left panel shows drought condition plots of SPEI- 3, 6, 9, 12 and 24.

The Figure 3 has two panels, in which the right side panel shows time series graphs of the given data, in which we can see the pattern of drought yearly; in the left side panel the color red shows the drought and color blue means no drought. The right panel shows time series graphs of SPEI-3, 6,9,12 and 24, showing the highest drought in the years on the x-axis, which is different in each scale of SPEI, but most of the SPEI shows in 2009 to 2014. Moreover, the time series and annual cycle of the precipitation and PET are shown in Figures 4–6, respectively. Our data source is Karachi Pakistan Meteorological Department (PMD) (https://www.pmd.gov.pk/en/, accessed on 21 September 2021) and the trend of drought is clearly shown, with the highest drought during the period of 2009 to 2014 and the lowest trend in the period of 2017 to 2019; the same period is also verified by the SPEI time series graph in Figure 3.

Additionally, as our selected region is desert area, there is no other source of water such as a canal or river and there is no source of water without rain for agriculture and other uses. Therefore, severe drought occurs, which can be seen from the annual precipitation graph (Figures 4–6) of the Mithi weather station.

## Unit Root Test

In the literature there are various tests but, in this research, we have selected the three most important and different tests on SPEI, given below as:


**Figure 4.** Precipitation and the highest rainfall during the period 2016 to 2019 and the lowest rain fall in the period of 2009 to 2014.

**Figure 5.** Annual cycle of the precipitation the input data starts from 2004 to 2019.

**Figure 6.** Annual cycle of the PET from 2004 to 2019 (Monthly variation).

Table 3 shows that the SPEI- 3, 6, 9, 12 and 24.have the 1st difference unit root test whereas the SPEI 24 has the 2nd difference unit root test.



Table 4 shows that the SPEI- 3, 6, 9, 12 and 24 have the 1st difference unit root test whereas the SPEI 24 has the 2nd difference unit root test.



Table 5 shows that the all SPEI- 3, 6, 9, 12 and 24 has the 1st difference unit root test. **Table 5.** KPSS test for level stationarity.


### *3.3. Estimation of Model Parameters*

In Tables 6–13, the model parameters are standard error, *p*-value and related significance value at a significance level less than 0.05 for Tharparkar region. In comparison with the parametric values it has been observed to be very small. The above proposition bears exclusion of the model parametric values of SPEI at the three-month time scale. Furthermore, at the significance level of less than 0.05, almost all ARIMA model parameters are significant. Therefore, these parameters ought to be incorporated in the model. Other models also showed identical results. Table 6 describes the generalized ARIMA seasonal and non-seasonal models for Tharparkar.

**Table 6.** Seasonal ARIMA and non-seasonal ARIMA models.


**Table 7.** SPEI-3 ARIMA (1,1,3)(0,0,0) for Seasonal.


**Table 8.** SPEI-3 ARIMA (1,1,3) for non-seasonal.


**Table 9.** SPEI-6 ARIMA (1,1,1)(0,0,2) for seasonal.


**Table 10.** SPEI-6 ARIMA (1,1,1) for non-seasonal.



**Table 11.** SPEI-9 ARIMA (1,1,1)(1,0,0) for seasonal.

**Table 12.** SPEI-12 ARIMA (0,1,0)(1,0,2) for Seasonal.


**Table 13.** SPEI-24 ARIMA (0,1,0)(1,0,2) for seasonal.


In the case of (1,1,3)(0,0,0) model estimation, the series is stationary and has no time dependence so the best prediction for this kind of series is the average of the series. In (1,1,3) model, A.R term is 1, difference/order of integration is 1 and moving average is 3, and in the model (0,0,0) whose A.R has 0 lag, difference/order of integration is 0 and moving average has also 0 lag.

In the case of (1,1,3) model estimation, the series is stationary, and prediction for this kind of series is the average of the series, whose A.R. term is 1 lag, difference/order of integration is 1 and moving average is 3 lags. This is the best ARIMA model at SPEI-3.

The (1,1,1)(0,0,2) model estimation, the series is stationary, and prediction for this kind of series is the average of the series, model (1,1,1) whose A.R. term is 1 lag, difference/order of integration is 1 and moving average is 1 lag, and the (0,0,2) model who's A.R has 0, difference/order of integration is 0 and moving average has 2 lags.

In the case of (1,1,1) model estimation, the series is stationary, and prediction for this kind of series is the average of the series, whose A.R. term is 1 lag, difference/order of integration is 1 and moving average is 1 lag. This is the best ARIMA model at SPEI-6.

For the (1,1,1)(1,0,0) model estimation, the series is stationary, and prediction for this kind of series is the average of the series, model (1,1,1) whose A.R. term is 1 lag, difference/order of integration is 1 and moving average is 1 lag, and the (1,0,0) model who's A.R has 1 lag, difference/order of integration is 0 and moving average has 0.

For the (0,1,0)(1,0,2) model estimation, the series is stationary and prediction for this kind of series is the average of the series, model (0,1,0) whose A.R. term is 0, difference/order of integration is 1 and moving average is 0, and the (1,0,2) model who's A.R has 1 lag, difference/order of integration is 0 and moving average has 2 lags.

#### *3.4. Diagnostic Checking of Residuals*

In order to test the authenticity of the model, diagnostic examination was carried out after the assessment of model parameters. Figure 7 depicts the ACF and PACF of the residuals at various time scales. All the values of the ACF and PACF are found within the limit of 0.01 range for all lags. Thus, no significant association is found between residuals in Figure 8 and the normally distributed histograms of residuals for the SPEI at varying time scales have been represented. This result for the shaped model is sufficient for the SPEI time series data and residuals to error terms. The greatest accuracy predicting models linked to every examined SPEI time scale with accuracy fit measures (ME, RMSE, MAE, MPE, MAPE, MASE and Theil's U) are shown in Table 14. In general, substantial results have been obtained for drought predicting with the help of ARIMA models in Tharparkar. In short explanation, the ARIMA models that have longer time scales shows profound ability of forecast and fit exactly with drought prediction in upcoming times.

**Figure 7.** ACF and PACF of SPEI- 3, 6, 9, 12 and 24 from this we can identified the data is stationary.

**Figure 8.** Residuals of SPEI- 3, 6, 9, 12 and 24 are normally distributed.


**Table 14.** Errors of SPEI-3, 6, 9, 12 and 24.

Almost familiar results have been shown across world and put into the SPI that forms the core of SPEI, i.e., China [54] and India [35]. These research studies have shown that the time series of SPEI for Mithi, Tharparkar has the same nature as of Figure 3. In addition to this, the time series of SPEI-12 and -24 has a similar trend, and likewise for the time series of SPEI-3 and -6, respectively. The identical order seems to be for the time series depending on 12 and 24 months however, the related 3- and 6-month scale do not show this result.

From the Figure 5 for PACF and ACF it is clearly shown that the selected data is stationary.

#### *3.5. Model Forecasting*

In fact, forecasting is one of the prime factors in decision making. It bears significant importance for the process about decision making and future planning. It assists in predicting the uncertain future by utilizing the behavior of past and ongoing experiments and observations. The forecasting that is performed using the ARIMA models lays out a sound basis for meteorological phenomena. The forecasting of drought is carried out by selecting the city of Tharparkar and then the data of that location has been utilized to foresee the data series of the SPEI at various time scales, from the 2005 to 2019 period to assess the agreement of data, where the examined and detected SPEI were plotted for its evaluation. Through the prism of comparison within predicted and observed data in Figures 9 and 10, high authenticity of forecasted data is observed. No doubt, with the increase in number of SPEI time series, the forecasting ability of model will be improved. This enhancement in the ability of model is due to rising number of SPEI time series that filters the final values, resulting in the decline of sudden shifts in the curve of SPEI.

The comparison of A.R and M.A coefficients suggests that the ARIMA models of 24-month time scale for Tharparkar is quite accurate. ARIMA models of 3-month time scale are similar in the surrounding regions of Tharparkar. The ARIMA models of 24-month time scale also showed the very accurate results. In Tables 7–13, the estimation of like parameters of developed ARIMA model has been shown. From the outcome in Table 14, the value of *p*, *d*, *q*, *P*, *D*, and *Q* received from the models shaped for Tharparkar are almost alike at the same time scales. Hence, the ARIMA model (0,1,0), (1,0,2) at 24-SPEI could be summarized for the whole region of Tharparkar. In addition to this, the ARIMA model (1,1,3)(0,0,0) at 3-SPEI is also applicable to the neighboring cities of Tharparkar as they are very close to it. In Tables 15–19, the point forecasted values of SPEI-3, 6, 9, 12 and 24 for five years model has been shown.

For the (0,1,0)(1,0,2) model estimation, the series is stationary and prediction for this kind of series is the average of the series, model (0,1,0) whose A.R. term is 0, difference/order of integration is 1 and moving average is 0, and the (1,0,2) model who's A.R has 1 lag, difference/order of integration is 0 and moving average has 2 lags.

#### *3.6. Comparison with Previous Study*

The comparative results of present and a previous study has been shown in Table 20. It is investigated that the present standard error and *t*-value of different SPEI (3, 6 and 24) have significant coherence with previous study [39].

**Figure 9.** The long-term projection of SPEI for 5 year.

**Figure 10.** The long-term projection of SPEI for 30 years.


**Table 15.** Point forecasted value of SPEI-3 for five years.

**Table 16.** Point forecasted value of SPEI-6 for five years.


**Table 17.** Point forecasted value of SPEI-9 for five years.


**Table 18.** Point forecasted value of SPEI-12 for five years.


#### **Table 19.** Point forecasted value of SPEI-24 for five years.



**Table 20.** A comparison of the present and previous study [39].

#### **4. Conclusions**

This research proves SPEI as a unique and powerful multi-scalar drought index for the examination of drought event variations in the region of Tharparkar, Sindh. The prime objective of this research work is categorized into two parts. The first part deals with the evaluation of climatic parameters and drought frequency on the basis of SPEI. This research concluded that the water crisis are a result of overlapping of two unfavorable factors—a decrease in precipitation amounts and an increase of temperature in the region of Tharparkar. Therefore, due to the deficiency of water, the likelihood of drought events has been greatly increased such situation is ultimately generating immense threat to available water resources. In addition to this, the variation in SPEI also shows unusual course of drought (extremely dry) since preceding decade. However, from these results, it is also evident that the situation of hyper-arid regions could be more alarming and eye-opening. It should be noted that the forecast of drought events is one of the most troublesome issues faced be meteorologists.

In this way, the second objective of this research was related to the development and test of ARIMA models for the forecast of drought by utilizing SPEI with 3, 6, 9, 12, and 24-month time scales. The identification of the ARIMA models was conducted on the basis of AIC and SBC values. The basic point for researchers is the credibility of forecasted values. Because the implementation of drought alleviation policies depends upon these forecasted values. In this way, a series of diagnostic checking tests were conducted after the inspection of the parameters of said models. The ARIMA model (0,1,0)(1,0,2) at 24-SPEI could be selected from other possible models for the region of Tharparkar. Additionally, the ARIMA model (1,1,3)(0,0,0) at 3-SPEI, the ARIMA model (1,1,1)(0,0,2) at 6-SPEI, the ARIMA model (1,1,1)(1,0,0) at 9-SPEI and the ARIMA model (0,1,0)(1,0,2) at 12-SPEI can be generalized for Tharparkar region. This is because other localities are very close to the Mithi region. It was also observed that the result obtained through the ARIMA model at the 24-SPEI time scale was the best forecasting model, that follows the lower values of ME, RMSE, MAE, MPE, MAPE and MASE. The ARIMA model at SPEI 3-time scale was found to be the worst model for the prediction of drought for the region of Tharparkar. The best ARIMA models represent profound accuracy in foretelling the droughts, as these can perform a very significant role for planners and water resources managers in measures for such regions as well as in view of drought.

In fact, the connectivity between climate change shown in droughts and the present water resources in Tharparkar is the need of the hour. Thus, in the Tharparkar region, it is very important to overcome the forecasted drought conditions and this should be considered as a significant future study. Additionally, unfortunately the Tharparkar region in Pakistan has only one meteorological station located at Mithi, which is the limitation of our study. Therefore, this study can be extended using different models and a larger set of data in future.

**Author Contributions:** Conceptualization, P.K. and S.F.S.; Data curation, P.K. and L.K.; Formal analysis, S.F.S., M.A.U., L.K. and R.F.Z.; Methodology, P.K. and R.F.Z.; Project administration, S.F.S. and M.A.U.; Software, P.K., L.K. and R.F.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported from the Higher Education Commission (HEC), Islamabad, Pakistan under the National Research Program for Universities (NRPU 6872).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

**Acknowledgments:** Authors acknowledge the support of the Higher Education Commission (HEC), Islamabad, Pakistan and the Mehran University of Engineering and Technology, Jamshoro, Pakistan. This research was supported from the Higher Education Commission (HEC), Islamabad, Pakistan under the National Research Program for Universities (NRPU-6872).

**Conflicts of Interest:** The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

#### **Nomenclature**



#### **References**

