**Study area.** The study region includes all of Bangladesh and spans latitudes of 20°34′ to 26°38′ N and longitudes of 88°01′ to 92°41′ E in South Asia. On the west, north, and northeast, Bangladesh has borders with India; on the south, it shares a border with Myanmar. The Bay of Bengal, with its extensive coastline, marks the southern border (Figure 1). The highest point in the country's northern region is 105 meters above sea level, however, the rest of Bangladesh is no higher than 10 meters.

Bangladesh is one of the countries that are most at risk from the escalating effects of climate change on a global scale. It is frequently struck by natural calamities, including flooding, drought, tornadoes, and tidal bores32. Bangladesh has recently faced drought on a frequent and regular basis; on average, at least once every 2.5 years33. Between 2011 and 2015, the frequency of droughts in all categories increased dramatically1. Despite the fact that drought is widespread in Bangladesh, the northwest region is particularly prone to it because of the region's high degree of rainfall variation11. Additionally, this region is relatively arid and characterized by sandy soils, receiving significantly less rainfall than the national average12. The moisture retention capacity of sandy soils is lower than that of other soil types, while infiltration is higher15. In addition, drought in the region has been exacerbated by the construction of the Farakka barrage in the Ganges River's upper reaches.

Stations from the Bangladesh Meteorological Department (BMD) are now grouped into seven hydrological areas29 to study drought situations throughout the nation, as illustrated in Figure 1, taking into account geography and land usage as well as anomalies in rainfall. The seven hydrological regions are referred to as: (i) Southwest (SW, 3 stations), (ii) Northeast (NE, 2 stations), (iii) North Central (NC, 2 stations), (iv) Northwest (NW, 5 stations) (v) South Central (SC, 4 stations), (vi) Southeast (SE, 7 stations), and (vii) Eastern Hills (EH, 6 stations)

**Observation Data.** Because of the country's subtropical monsoon climate, temperatures, rainfall, and humidity all vary greatly from season to season throughout the year in Bangladesh. From March to May, there is a hot, humid summer; from June to September, there is a wet, warm, and rainy monsoon season; from October to November, there is autumn; and from December to February, there is a dry winter. In Bangladesh, three separate crop seasons exist pre-Kharif (March–June), Kharif (July–October), and Rabi (November–February).

Bangladesh Meteorological Department (BMD) manages 35 meteorological stations throughout the country at the moment. Nonetheless, there are just 29 places with rainfall data going back more than 30 years. The 29 meteorological stations' monthly rainfall and temperature records from 1980 to 2018 were utilized to diagnose droughts in this study. Missing data is a key issue when using observational data. Averaged values from three surrounding sites were used to fill in the approximately 2% of data that was missing To obtain the spatial distribution of drought and pass the homogeneity test, we interpolated the data at a resolution of 1 km using inverse distance weighted (IDW) interpolation based on the placements of meteorological stations. Figure 1 depicts the research scope and the location of each site.

Between 1980 and 2018, there was a total of 2462.14 mm of rain recorded annually on the ground. Rainfall in Bangladesh peaks during the monsoon months (June to October) due to the weak humid depressions that are carried into Bangladesh by the moist monsoon winds9. One of the most notable aspects of Bangladesh's climate is the spatial and temporal unpredictability of the country's rainfall. Over the period 1980 to 2018, Bangladesh received rainfall ranging from approximately 1400 mm in the west to more than 4000 mm in the east. Meghalaya's elevation also contributes to the increased rainfall in the northeastern region.

**Calculation of Standardized Precipitation Index (SPI).** Drought conditions in the research area can be estimated using the Standardized Precipitation Index (SPI). SPI is an index that uses solely precipitation data. It is based on the chance of precipitation for a few consecutive months, and its major aim is to reflect the deficiency of precipitation across an area on several time scales relative to its climatology23. In spite of the fact that the SPI approach is not a drought forecasting tool, the SPI methodology has been used to identify dry or wet conditions and analyze their influence on water resources management. SPI can be calculated at several time intervals, including 1, 3, 6, 12, and 24 months34.

Unless researchers have a firm grasp on the required intervals, this strong feature of drought can generate an overwhelming amount of data35. The SPI is calculated mathematically using the cumulative likelihood of a certain rainfall event occurring at a given station. To begin, a two-parameter gamma density distribution function is used to fit the precipitation frequency of a meteorological station for each calendar month. The function of the gamma distribution is presented.

\(f\left(x\right)=\frac{1}{{\beta }^{\alpha }{\Gamma }\left(\alpha \right)}{x}^{\alpha -1}{e}^{-x/\beta }\) ………………………………………………………… (1)

Where, *𝛼* and *𝛽* are the factors defining the shape and scale, respectively. Monthly precipitation is denoted by X. The Thom method36 can be used to estimate the two parameters. Then, using the gamma cumulative distribution function, one may compute the cumulative probability G(x) at x. Finally, the inverse of the cumulative standard normal distribution function is used to turn G(x) into the SPI value. Lue et al.37 introduced a thorough computation of SPI and drought categorization in their study.

**Calculation of Standardized Precipitation Evapotranspiration Index (SPEI).** As a result of global warming, surface evaporation changes have become more sensitive to droughts. The original SPI calculation algorithm is used to calculate the SPEI. The SPI is calculated using monthly precipitation data as the input. Monthly differences in precipitation and PET are used to calculate the SPEI. Using multiple time scales, the SPEI can be determined using this simple climatic water balance.

To determine the value of SPEI, the difference in the water balance is normalized as a log-logistic probability distribution. The probability density function can be expressed using the following equation:

$$f\left(x\right)=\frac{\beta }{\alpha }\left(\frac{x-\gamma }{\alpha }\right){\left[1+\left(\frac{x-\gamma }{\alpha }\right)\right]}^{-2}$$

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Where the parameters scale, shape, and origin are denoted by *𝛼*, *𝛽*, and \(\gamma\), respectively. Thus, the probability distribution function can be described in terms of a probability density function.

$$F\left(x\right)={\left[1+{\left(\frac{\alpha }{x-\gamma }\right)}^{\beta }\right]}^{-1}$$

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Vicente-Serrano et al.34 defined the SPEI as follows:

$$SPEI=W-\frac{{C}_{0}+{C}_{1}W+{C}_{2}{W}^{2}}{1+{d}_{1}W+{d}_{2}{W}^{2}+{d}_{3}{W}^{3}}$$

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When *P≤0.5*, \(W=\sqrt{-2ln\left(P\right)}\) and when P>0.5, \(W=\sqrt{-2ln(1-P)}\), C0=2.5155, C1=0.8028, C2=0.0203, d1 = 1.4327, d2 = 0.1892, d3 = 0.0013.

Using the SPEI package for R38, the SPI and SPEI drought index were calculated for this article. It's a great research and practical tool for assessing drought conditions. SPI/SPEI values were used to classify the severity of the drought, as shown in Table 1. SPI/SPEI values with a reduction in rainfall are indicative of drought, while SPEI values with a rise in rainfall are indicative of wetter or more typical conditions.

Table 1

Classification of drought based on SPI/SPEI values

SPEI or SPI Values | Drought category |

0.99 to −0.99 | Normal |

−1.0 to −1.49 | Moderate drought |

−1.5 to -1.99 | Severe drought |

≤ −2 | Extreme drought |

**Runs theory**

Drought characteristics include the duration of the drought, its severity, and its frequency. SPI/SPEI under normal conditions (SPI/SPEI≥− 1) is also calculated from the general technique when the absolute value of SPI/SPEI is calculated, which has a substantial impact on drought assessment. Thus, we employed Yevjevich’s run theory39 to define the severity and frequency of droughts. The runs theory constructs a segment of the drought variable time series with all values less than or larger than the set threshold. This segment is referred to as a negative or positive run. The formula for calculating drought intensity is

$$S=\frac{\sum _{n=1}^{T}|{S}_{SPI/SPEI}-K|}{T}$$

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where drought intensity is denoted by S, \({S}_{SPI/SPEI}\) denotes an SPI or SPEI value less than or equal to the drought threshold, K denotes drought threshold, which in this study is set to be less than or equal to -1, indicating that the severity of the drought is greater than that of moderate drought, and T is the duration of the drought.

Drought frequency is a metric for determining how often a drought occurs in a certain area, and its formula is as follows:

$$DF=\frac{n}{N}\times 100$$

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where N is the time period during which the site was detected, and n denotes the number of droughts that occurred at the site during that time period.

**Conditional probability.** The term "conditional probability" refers to the likelihood of occurrence of a particular event A in the presence of another event B, i.e. Cp (A/B). However, Cp (SPI) in this study refers to the possibility of an of SPEIdrought occurring during an SPIdrought and vice versa for Cp(SPEI). The formula is as follows:

\(Cp \left(SPI\right)=\frac{{T}_{SPI/SPEI}}{{T}_{SPEI}}\) or \(Cp \left(SPEI\right)=\frac{{T}_{SPEI/SPI}}{{T}_{SPI}}\) (7)

whereas TSPI and TSPEI reflect drought periods in a certain location over time depending on the SPI/SPEI value, T SPI/SPEI and TSPEI/SPI indicate the times of droughts in an area based on the SPEI/SPI evaluation that drought has happened, while SPI/SPEI conducts a re-evaluation of the region's droughts.

**Trend Test.** The Modified Mann-Kendall test (MMK) was introduced by Hamed et al.40 to address the issue of serial correlation by the use of the variance correction approach. For computing, the MK test the modified variance (Var(S)) is applied41,42 and the subsequent Eqs. (8–11) are applied to compute the autocorrelation:

\(\text{V}\text{a}\text{r}\left(\text{S}\right)\text{⃰}=\text{V}\text{a}\text{r}\left(\text{S}\right)\times \left(\frac{\text{n}}{\text{n}\text{⃰}}\right)\) | (8) |

\(\left(\frac{\text{n}}{\text{n}\text{⃰}}\right)=1+\left(\frac{2}{\text{n}(\text{n}-1)(\text{n}-2)}\right)\times \sum _{\text{k}=1}^{\text{n}-1}{(\text{n}-\text{k})(\text{n}-\text{k}-1)(\text{n}-\text{k}-2)}^{\text{r}}.\text{k}\) | (10) |

\({\text{r}}_{\text{k} =}\frac{\left(\frac{1}{\text{n}-\text{k}}\right)\sum _{\text{i}=1}^{\text{n}-\text{k}}({\text{x}}_{\text{i}}-\text{x})({\text{x}}_{\text{i}+\text{k}}-\text{x})}{\left(\frac{1}{\text{n}}\right)\sum _{\text{i}=1}^{\text{n}} ({{\text{x}}_{\text{i}}-\text{x})}^{2} }\) | (11) |

where nn* and \({\text{r}}_{\text{k} }\) implies the modified coefficient of autocorrelated data and autocorrelation coefficient of k-th lag, respectively; x denotes the mean of the time series. The significance of the trend at a 95% confidence interval of the k-th lag can be estimated by Eq. (12):

\(\left(\frac{-1-1.96\sqrt{\text{n}=\text{k}-1}}{\text{n}-\text{k}}\right)\le {\text{r}}_{\text{k} }\left(95\text{\%}\right)\le \left(\frac{-1+1.96\sqrt{\text{n}=\text{k}-1}}{\text{n}-\text{k}}\right)\) | (12) |

The 95% confidence level is achieved if the \({\text{r}}_{\text{k} }\) satisfies the upper condition. Hence, the dependence of the data and influence of the autocorrelation between different time lags should be eliminated for estimating the trend.

In this study, we used the MMK test in place of the classical MK test because of its ability to eliminate the influence of autocorrelation on test significance. The traditional MK test does not consider autocorrelation, which is common in the hydro-climatological time series. The positive autocorrelation increases the likelihood of test significance and vice-versa for negative autocorrelation40.

**Sen’s slope estimator.** The nonparametric Sen’s slope (SS) technique43 was employed in this study to estimate the rate of trend magnitude In time-series datasets, this method was used to determine the magnitude of a trend. In comparison to other methods, the impact of an outlier on-trend outcomes is negligible using this strategy44. The Sen’s slope (SS) can be calculated by Eq. (13)

$$\beta =Median\left[\frac{{x}_{j}-{x}_{i}}{j-i}\right] all j>i$$

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Where xj denote the jth values and xi the ith values in observational data. A positive value of β denotes an increase whereas a negative value indicates a decreasing rate of change.

**Multivariable linear regression method.** The method of multivariable linear regression (MLR) is used to construct a multiple regression model in which meteorological variables affect their geographical interpolation components. We use precipitation (Y) as the dependent variable and altitude (H), maximum temperature (Tmax), minimum temperature (Tmim), longitude (Lo), and latitude (La) as the independent variables. We assume that each independent variable has a linear effect on the dependent variable and that the mean value of precipitation changes evenly when one independent variable changes while the others remain constant. The present study built a multiple regression model of precipitation (Y) for altitude (H), maximum temperature (Tmax), minimum temperature (Tmin), longitude (Lo), and latitude (La), and estimated the residuals using the following expression:

$$Y(H, Tmax,Tmin, Lo, La) =b0 + b1H + b2Tmax+ b3Tmin + b4Lo + b5La + \epsilon$$

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where b1, b2, b3, b4, b5 are the unknown coefficients; b0 is a constant and ɛ is the residual value 45