**1. Introduction**

With the increasing human consumption rate of water around the world, and especially in highly arid regions that have experienced water depletion, the demand for water resources has increased dramatically [1]. Furthermore, changing patterns of global climate have further resulted as an impediment to human survival [2]. Thus, the human population and other species on Earth are subject to more and more droughts. In arid as well as semi-arid regions, droughts are common and repeating. Drastic change in the

**Citation:** Kumar, P.; Shah, S.F.; Uqaili, M.A.; Kumar, L.; Zafar, R.F. Forecasting of Drought: A Case Study of Water-Stressed Region of Pakistan. *Atmosphere* **2021**, *12*, 1248. https://doi.org/10.3390/ atmos12101248

Academic Editors: Andrzej Walega and Agnieszka Ziernicka-Wojtaszek

Received: 11 August 2021 Accepted: 22 September 2021 Published: 26 September 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

projection of floods and droughts has been seen in the 21st century compared to the 20th century. Mainly, the effect of prolonged droughts on natural ecosystems has highly deteriorated regional agriculture, water resources and the environment [3–5]. In such complex situations, the unavailability of proper evaluation of drought may result in wrong decisions and actions by policy makers and monitors [6,7]. For this reason, scientists divide drought conditions into four main categories, namely meteorology [8], agriculture [9], hydrology and socioeconomics [10]. In order to describe these drought categories, drought detection and monitoring indicators have different natural variables over the required time period, such as Precipitation (PCPN), soil moisture, Potential Evapotranspiration (PET), vegetation condition, and ground water along with surface water. In simple words, a drought index measures the actual features and their correlated effects [11]. The prominent attributes or qualities that must be focused on in the index are the time span, intensity, magnitude and spatial extent of the drought; nevertheless, incorporating all these features into one index is highly an arduous task. For this, numerous indices have been put forward by researchers. Some of them are, the Palme Drought Severity Index (PDSI) [12], Surface Water Supply Index (SWSI) [13], Palfai Aridity Index (PAI) [14], Standardized Precipitation Index (SPI) [15,16], and Standardized Precipitation Evapotranspiration Index (SPEI). The examination through any of above indices is subject to the essence of the index, local surroundings, accessibility of facts and rationality [17]. The advantages and disadvantages of drought indices are based upon the clarity and provisional flexibilities of their administration Standardized Precipitation Index (SPI) has been endorsed as a standard drought index by the world meteorological organization that is likely due to its simple calculations, as precipitation data is itself enough to direct and test without the demand of statistical barriers. Furthermore, the SPI is also capable of showing high performance in finding and computing drought potency [18,19]. Despite of all this ease, the SPI still has some issues regarding water balance. This led to the development of SPEI and eliminated these problems in the SPI release. SPEI focuses on PDSI sensitivity to PET mutations and the multifaceted compound of SPI [20]. Both SPEI and SPI have almost the same evaluation procedure. In this way, the use of climate water balance by SPEI is the differentiation between the two indices [21,22]. For this purpose, now days, SPEI has been widely used as a relevant index to observe the drought in various regions worldwide [23].

At present, the crucial challenge in the research of droughts is in the development of reasonable techniques and methods to forecast the start and end points of droughts. Thus, the significance of drought prediction advances from the reduction of drought effects [24]. It has been attempted multiple times to utilize statistical models in meteorological drought forecasting, depending upon time series methods such as ARIMA models, exponential smoothing and neural networks [25]. ARIMA is a classic model for statistical evaluation that makes use of time series data to foresee subsequent tendencies in meteorological variables such as annual and monthly temperature and precipitation can also be estimated through ARIMA models [26]. In a meteorological time series, the ARIMA model approach can exceed multiple new models such as exponential smoothing and neural network. Thus, due to its statistical properties together with effective methodology in establishing the model, ARIMA has relative advantage to the other models [27].

A lot of research has focused on contemporary drought predictions. It uses a new imitation hybrid wavelet-Bayesian regression model to develop a meteorological time series for extended lead-time drought prediction [28], which is a combined model using wavelet and fuzzy logic [29]. A group of authors also discussed the application of a wavelike fuzzy logic model based on meteorological variables to PDSI measurement [30]. Mishra also provides a drought prediction model that uses hybrid, single-neural and synthetic randomized Artificial Neural Network (ANN) models to predict droughts based on SPI. All these efforts are fruitful and sufficient in predicting time-series droughts [31]. The Standardized Precipitation Index (SPI) is currently widely utilized in both scholarly and operational applications around the world. The SPI at short time scales is lower limited, referring to non-normally distributed for arid climates or those with a distinct dry season

when zero values are typical. The SPI is always greater than a particular number in certain instances, and thus fails to signal the onset of a drought. The non-normality rates appear to be closely associated to local precipitation climates. Herein, the authors present the Standardized Precipitation Evapotranspiration Index (SPEI) as an enhanced drought indicator that is particularly well suited for research of the effect of global warming on drought severity. The SPEI evaluates the effect of reference evapotranspiration on drought severity, similar to the Palmer Drought Severity Index (PDSI), but its multi-scalar nature allows identification of distinct drought types and drought consequences on various systems [19].

Thus, the SPEI, like the standardized precipitation index, has the sensitivity of the PDSI in measuring evapotranspiration demand (induced by variations and trends in climatic variables other than precipitation), is easier to compute, and is multi-scalar (SPI). Detailed discussions of the SPEI's theory, computations, and comparisons to other widely used drought indicators including the PDSI and SPI are given by [32]. The SPEI is calculated in the same way as the SPI is calculated. However, rather than using precipitation as an input, the SPEI employs "climatic water balance," which is the difference between precipitation and reference Potential evapotranspiration. At various time frames, the climatic water balance is estimated (i.e., over three month, six months, nine months, twelve months, and 24 months). Despite the fact that the SPEI was just recently developed, it has already been employed in a number of research looking at drought variability.

The first step in the SPI calculation is to determine the probability density function (PDF), which describes the long-term observed precipitation. It also allows the SPI to be computed at any location and at any number of time scales, depending upon the impacts of interest to the user. Ratios of drought on the basis of analysis of stations across Colorado are given in Table 1 [33].


**Table 1.** Ratio of drought in Colorado [33].

With the help of normal distribution of SPI above percentage are expected. The cumulative probability is then applied to the inverse normal (Gaussian) function, yielding the SPI. This approach is a transformation of equivalence. The equiprobability transformation's key feature is that the probability of being less than a particular value of the produced cumulative probability should be the same as the probability of being less than the normal distribution's equivalent value [33]. Based on the time series of drought monitoring findings of the Vegetation Temperature Condition Index (VTCI), Autoregressive Integrated Moving Average (ARIMA) models were developed. From the erecting stage to the maturity stage of winter wheat (early March to late May in each year at a ten-day interval) of the years 2000 to 2009, about 90 VTCI pictures produced from Advanced Very High-Resolution Radiometer (AVHRR) data were selected to create the ARIMA models. The ARIMA models' category drought predictions findings in April 2009 are more severe in the northeast of the Plain, which accord well with the monitoring results. The AR(1) models have smaller absolute errors than the SARIMA models, both in terms of frequency distributions and statistic findings. SARIMA models, on the other hand, are better at detecting changes in the drought situation than AR(1) models. These findings suggest that ARIMA models can better predict the type and extent of droughts, and that they can be used to predict droughts in the plain [34]. Time series forecasting has been extensively used and has emerged as a key method for drought forecasting. The Autoregressive Integrated Moving Average (ARIMA) model is one of the most extensively used time series

models. The ARIMA model's wide application owes to its flexibility and methodical search (identification, estimation, and diagnostic check) for an acceptable model at each stage. The ARIMA model offers various advantages over other approaches, such as moving average, exponential smoothing, and neural networks, including predicting and more information on time-related changes. Hydrologic time series have also been analyzed and modelled using ARIMA models [35]. From the literature review and retrospective analysis, it has been found that the SPEI was not utilized as a drought index previously. Therefore, the main focus of this research investigation is to establish the stochastic model ARIMA to lay and predict the SPEI series on discrete time series. In addition, it also provides a subjective method to deal with climate-related parameters for 14 years of drought, i.e., (2005–2019).
