Trend and homogeneity test analysis for rainfall over a 121-year time period in the desert district of Western Rajasthan, India

Socio-economic systems and our way of life face many difficulties due to climatic changes and the earth's warming. The process of climatic fluctuation has a relatively long time horizon. Long-term changes in temperature, precipitation, wind speed, evaporation, humidity, and other parameters reflect the substantial climatic variability observed in a region (Mehta & Yadav 2021a). Temperature is rising worldwide, but the weather condition is still not predictable. Precipitation shows various patterns like stable, upward, or downward trends. For example, rainfall shows no trend in some places, but in others, it shows upward or downward trends. A significant variation in worldwide precipitation is observed in all scenarios (Koutsoyiannis 2020). One of the primary climatic variables affecting water availability patterns is rainfall. Rainfall is an essential variable of the hydrological system, and its trend significantly and directly impacts the water supply system. Hydrologists and researchers are now concerned about the changing rainfall pattern due to climate change (Krishn et al. 2022; Kumar 2023). It is hard to identify, assess, and predict rainfall trends and their impacts on the river flow system due to climate change (Taxak et al. 2014; Krishn et al. 2022). Rainfall trend analysis is helpful for rainfall forecasting, water resources planning, irrigation practices, and construction of water storage structures. The rainfall data can also help industrial development, water supply systems, and climatic disaster management in the present and future (Patakamuri et al. 2020). Gridded data are preferred for model validation because the model outputs are generated at fixed grid points. However, depending on the application, the resolution of the gridded data can be different (Pai et al. 2014).

Low rainfall is a critical problem for India because its economy is based on agriculture and is heavily dependent on monsoon rainfall. India's climate change is too significant compared to the overall climatic variability. The monsoon of India is a crucial part of the global climatic system. Daily rainfall data over a more extended period is essential to understand the elements and functions of the Indian monsoon (Mitra et al. 2013).

The selection process of crops and ecological changes in an area are significantly influenced by rainfall and other precipitation levels. Precipitation trends can have a significant impact on a nation's economic growth in the future. The Indian economy is significantly influenced by rainfall variability and extreme rainfall events that cause drought and floods, affecting India's GDP and food security (Ahmad et al. 2015).

During years of drought and unusual weather, food production, especially in Rajasthan, plummets. Compared to the previous good monsoon year, severe drought decreased crop grain production in western Rajasthan by 57% in 2004–05, 50% in 2002–03, and 70% in 1987–88 (Narain et al. 2006).

The 32 million ha of hot, arid land in Rajasthan and Gujarat experience exceptionally high temperatures and infrequent rainfall (Machiwal et al. 2021). The main arid zone of India is located in western Rajasthan, which contains about 62% of India's arid zone (covering about 20 million ha of area). Disastrous droughts frequently strike Rajasthan's arid region, causing enormous economic losses and harming the area's natural resources (Narain et al. 2006). Droughts will likely occur in Jaisalmer and Barmer districts approximately every 2–3 years. The district of Jaisalmer is particularly vulnerable to drought. Between 1901 and 2005, there were 70 years of agricultural drought in the area, of which 45 years saw severe drought and 25 years saw moderate drought, which significantly impacted crop and fodder production. The Barmer district adjacent to Jaisalmer had a severe drought for 32 years and a mild drought for 18 years (Narain et al. 2006).

Similar studies have already been conducted on climatic variables. Phulpagar and Kale have conducted a study on Ajmer division, revealing an increasing trend in pre-monsoon and monsoon seasons from 1994 to 2018 (Phulpagar & Kale 2023). Long-term rainfall trend analysis study has been conducted for a period of 102 (1901–2002) years by Mehta and Yadav on Barmer which revealed a rising trend for pre-monsoon, post-monsoon, south-west monsoon, and annual rainfall (Mehta & Yadav 2021a). Two other rainfall studies revealed an increasing annual rainfall trend in western Rajasthan (Mehta & Yadav 2020, 2021b). A trend analysis study conducted for Jodhpur has observed a significantly increasing trend at eight stations adjacent to the Jaisalmer district (Kumar & Kumar 2020). Meena et al. studied the trend for 55 years (1957–2011) time series and found a rising trend for Jaisalmer annual rainfall (Meena et al. 2019). Saini et al. conducted a study on Rajasthan state for a time series from 1961 to 2017 and found a significant rise in the rainfall events and annual rainfall of the Jaisalmer district. Annual rainfall has risen by 2.17 mm/year for the Jaisalmer district (Saini et al. 2022).

In the present study, rainfall trend analysis has been conducted using the Modified Mann–Kendall (MMK) test and Sen's Slope (SS) estimator. The trend-changing year has also been detected in this study. Similar studies on climatic variables are conducted in the state of Rajasthan. However, this type of study has not been conducted in the district to identify the rainfall trend and change of year of the trend for such a long time series.

Jaisalmer is the old commerce hub located on the western side of the Indian state of Rajasthan, in the middle of the Thar Desert. Jaisalmer serves as a border guard for both India and western Rajasthan. Jaisalmer has geological importance. Aakal's Wood Fossil Park is around 15 km from the city. Here, one can learn about and follow the geologic disasters in the Thar Desert that took place 180 million years ago. In the vicinity of Jaisalmer, the Indian government began conducting departmental oil prospecting in 1955–1956. Oil India Limited found natural gas in 1988 in the Jaisalmer basin.

The region experiences a desert environment characterized by dry air, wide temperature variations, and irregular rainfall. The temperature swings significantly from day to night in both summer and winter. During summer, the minimum temperature is 25 °C, and the highest is approximately 49 °C. Typically, the coldest temperature during winter is −5 °C, and the maximum temperature is around 23.6 °C. The highest ever recorded temperature is 50.0 °C, and the lowest is −5.9 °C.

• To study the trend and variability of rainfall for the Jaisalmer district by the MMK test.

• To determine the magnitude of the trend using SS estimator.

• To determine the change of point for rainfall trend based on rainfall data.

In the current article, rainfall trend analysis and trend change point identification are conducted. According to the Indian Meteorological Department (IMD), there are four distinct meteorological seasons: March to May is pre-monsoon, June to October is south-west monsoon, November to December is post-monsoon, and January to February is winter. Annual mean rainfall of 16 grids, district seasonal rainfall, and annual rainfall data with a high spatial resolution (0.25° × 0.25°) from 1901 to 2021 have been analyzed (Pai et al. 2014). Although there are many research articles on climatic variables (rainfall) at different spatial and temporal scales, none aim to comprehend the trends and change of points for the Jaisalmer district. The temporal variation of the rainfall pattern has been examined using a variety of methodologies and techniques for rainfall data analysis (Thomas et al. 2016) (Figure 2).

Daily rainfall data have been used in the current analysis. Statistical parameters, including coefficient of variation (CV), mean, standard deviation, minimum, and maximum have been calculated for annual rainfall. The monsoon is the only factor that influences the seasonality of the climate in Rajasthan's south-west region's arid and semi-arid environment.

The rank-based non-parametric MMK test and slope-based SS estimator have been used to evaluate the rainfall trend for the 121-year time series. Both of these techniques presuppose that the time series has a linear trend. Rainfall is the dependent variable, and time is the independent variable in a regression analysis.

The MK tests adopt the alternative hypothesis (H_a) and the null hypothesis (H₀). The MK statistic is used to test the null hypothesis that there is no trend in time series data values. The null hypothesis cannot be ruled out if the number exceeds the significance level (α). A different hypothesis is adopted, and the null hypothesis is rejected if the statistic value is less than the significance level (α). The significance level (α) determines how certain a result must be before it can be considered as proof of a trend. For time series of rainfall, a two-tailed test was conducted with a 95% degree of confidence.

Kendall's tau is a statistic that can be derived after performing the MK test. This correlation metric evaluates how closely two factors are related. Tau value depends on the ranks of data, so it can have values between −1 and +1, with a positive association signifying that the ranks of both factors rise concurrently. In contrast, a negative value signifies that if one variable's rank increases, the other variable's rank decreases. Kendall's tau is an important statistic to remember when determining if two variables are related.

The original MK test is based on the assumption of independent and identically distributed rainfall values of time series. Dimitriadis et al. 2021 reported long-term autocorrelation functions for rainfall (Dimitriadis et al. 2021). Autocorrelation is vital to identify trends in a time series analysis. Even if there are no significant trends, autocorrelation makes it more likely to find them, and vice versa.

When autocorrelation exists in the data, the empirical significance levels of the original MK trend test are significantly off from nominal significance levels. The proposed modified MK test's empirical significance values are significantly closer to the correct nominal significance levels (Hamed & Ramachandra Rao 1998).

The non-parametric (Pettitt 1979) test makes no assumptions regarding the distribution of the data. Pettitt's test modifies the Mann–Whitney test based on ranks to determine the shift's time of occurrence.

The contrasting and null hypotheses are revised as follows:

H₀: One or more distributions with the same location value are followed by the T variables.

Two-tailed test: H_a: There is a time t from which the location parameter's variables change.

Left-tailed test: H_a: There is a time t from which the variable's location is diminished by D.

Left-tailed test: H_a: There is a time t from which the variable's location is increased by D.

The Petitt's statistic for the various alternative hypotheses (H_a) is stated as follows: ⁠; ⁠; for the right - tailed case.

To identify a change in a set of rainfall data, Alexandersson (1986) and Alexandersson & Moberg (1997) created the SNHT (Standard Normal Homogeneity Test). The SNHT is applied to a number of ratios that contrast a measuring station's data with the mean of multiple stations. After that, the percentages are standardized. Here, the standardized ratios are represented by the sequence of x_i. The following factors decide the null and alternate hypotheses:

H₀: The T variables x_i follow an N(0,1) distribution.

H_a: Between times 1 and n, the variables follow an N(μ1, 1) distribution, and between n + 1 and T they follow an N(μ₂, 1) distribution.

The likelihood of the two different models is compared in the computation that yields the T₀ statistic. In order to determine the n parameter with the highest likelihood, the model corresponding to H_a means that ì₁ and ì₂ are estimated.

The Buishand's test (Buishand 1982) can be applied on variables with any distribution. But the normal case has received special attention in studies of its properties. Although Buishand concentrates on the two-tailed test case in his article, one-sided cases are also feasible for the Q statistic that is shown below. There is only room for a bilateral theory for the second statistic R that Buishand created. The null and alternative theories for the Q statistic are provided by:

H₀: The T variables adhere to one or more distributions with a common mean.

Two-tailed test: H_a: There is a time t from which the mean of variable change.

Left-tailed test: H_a: There is a time t from which the mean of variable is diminished by Δ.

Left-tailed test: H_a: There is a time t from which the mean of variable is increased by Δ.

The alternative and null hypotheses are assumed as follows:

H₀: The T variables adhere to one or more distributions with a common mean.

Two-sided test: H_a: The T variables are not uniform in terms of their mean.

Statistics of rainfall for the time series period of 121 years from 1901 to 2021 have been prepared based on the annual mean rainfall for 16 grids, as shown in Table 2. The mean of Jaisalmer's annual rainfall ranges from 178.216 to 205.849 mm, whereas the standard deviation ranges from 106.503 to 116.875 mm. The maximum mean annual rainfall in any grid of Jaisalmer is 612.634 mm in a 121-year period.

Table 3 shows the type of rainfall event according to the value of the percent coefficient of variance.

As shown In Table 2, CV ranges from 53.48 to 60.96%. Hence, rainfall event is very high.

The location-wise trend analysis is presented in Table 4. The values of Kendall's tau represent trend interpretation, whereas SS represents test interpretation. The analysis gives positive values of Kendall's tau for mean annual rainfall, indicating that the trend is rising for all grids of Jaisalmer. The negative tau value represents a falling trend that has not been found at any of the 16 grids. Sen's positive slope values imply an upward trend in all the grids' mean yearly precipitation data, as shown in Table 4. p-value is not more than 0.05 in the annual mean rainfall data of all 11 grids except grids 1, 2, 3, 4, and 13. The minimum % of change found at grids 2 and 13 are 32.47 and 31.97%, respectively. The highest percentage of change was found at grid 7, which is 54.03%.

Table 5 represents MMK test results for the whole Jaisalmer district. A significant increasing trend has been recorded for the annual mean rainfall of the Jaisalmer district. 34.24% change has been recorded for the annual mean rainfall of the Jaisalmer district.

According to this study, there are two common grids and eight other grids that identify 1992 as the change year. However, Pettitt's test reveals that the change year is 1991. Because it shows four times 1992 as the change year, and six times show the change year as 1991 out of 16 grids. Ten grids show the change year as 1992 after applying any of the three tests, hence it is concluded that the change year is 1992. μ2 is greater than μ1, meaning a rise in mean annual rainfall has been noticed after the change of point/year. The homogeneity test results are shown in Table 6.

The value of percentage coefficient of variance ranges from 53.48 to 60.96%, which falls in the range of 40–70. Hence, it can be concluded that rainfall events are very high in the time series of 121 years. The mean annual rainfall of the Jaisalmer district for the study period is 190.8 mm. It has been found that all the values of Kendall's tau and SS are positive, which indicate that trend and test interpretation are increasing. A total of 11 grids have a p-value less than 0.05, and 5 grids have more than 0.05, which represent a significant increasing trend at all 11 grids except grids 1, 2, 3, 4, and 13. A significant increasing trend has been recorded for the annual mean rainfall of the Jaisalmer district. A change of 34.24% has been recorded for the annual mean rainfall for the study area.

Based on the study, it is concluded that rainfall has increased significantly all over Jaisalmer district during the past three decades. The study area is already in an arid zone with low annual rainfall, so it is a good indication for the stakeholders and policymakers to plan their strategies accordingly. A further study considering other climatic variables using advanced technology of assessment, measurement, and analysis (e.g., innovative trend analysis) may reveal more specific outcomes, which will help water resource management in this desert area.

Project is not funded by any external agency.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.