Spatiotemporal Rainfall Variability and Trend Analysis of Shimsha River Basin, India

Karnataka state has the second highest rainfed agricultural land in India, where agricultural output relies heavily on rainfall. The Shimsha basin, a sub-basin of Cauvery in the state, comes under a semi-arid region and predominantly consists of rainfed agricultural land. Rainfall patterns have changed dramatically with time resulting in frequent floods and droughts. Understanding the spatiotemporal distribution of rainfall and its change patterns in the area would benefit sustainable agriculture planning and water resources management practices. The current study aims to determine the variability and trend in rainfall. The daily rainfall data of the Shimsha basin from 1989 to 2018 is collected, and the annual, seasonal, and monthly rainfall totals and the number of rainy days are derived. All the time series are subjected to statistical methods to examine rainfall variability and trend. The mean, standard deviation, coefficient of variation (CV), and Standardized Anomaly Index are used for the preliminary and variability analysis, while the coefficient of skewness and kurtosis are used to understand the rainfall distribution characteristics. The homogenous and serially independent series are identified by homogeneity and serial correlation tests. The trend in the homogenous and serially independent series is identified by Mann–Kendall and Spearman’s rank correlation tests, while the magnitude of the trend is quantified using the Sen’s slope technique, and the trend change point is evaluated using the sequential Mann–Kendall test. Based on the study, the average rainfall in the study area is 801.86 mm, with CV ranging from 43.3 to 22.27%. The southwest monsoon (SWM) season brings the greatest rain to the basin, followed by the post-monsoon (PM), summer, and winter seasons. In the annual time frame, except one station, all other stations have shown significant or insignificant increasing trends. The seasonal rainfall has shown insignificant rising trends during the summer and winter seasons while insignificant increasing and decreasing trends during the PM season. The SWM season has indicated significant increasing trends, insignificant increasing and decreasing trends. Overall, the study area has noticed an increased annual and seasonal rainfall except for the post-monsoon season, during which the rainfall showed a considerable decline. The findings of the study are helpful in water resource management, agricultural planning, and socioeconomic development in the study area.


Introduction
Rainfall is the most vital component of hydrological processes, and fluctuations in its spatiotemporal distribution may significantly influence our ecosystems (Karpouzos et al. 2010;Machiwal and Jha 2012).Rainfall patterns have transformed dramatically during the twentieth century, resulting in frequent floods and droughts, a loss of biodiversity, and disruption of the river environment (Ahmad et al. 2015;Harka et al. 2021).Due to human influences, rainfall-induced hazards have become stronger and more frequent in recent decades.Changes in global precipitation patterns concern agriculture and water management (Animashaun et al. 2020;Mathew et al. 2021;Singh et al. 2021), and water resource planners face several challenges, including the effects of changing rainfall patterns on water resources.Rainfall is the primary water source for agriculture and governs our ecosystems.The proper allocation of water across various water-use sectors has become difficult for decision-makers due to climate change, economic growth, and population growth (Ahmad et al. 2015;Khalil 2020).The effects of climate change on smaller river basins are expected to be more severe over a shorter timescale (Mathew et al. 2021).It has been acknowledged worldwide that analyzing historical climate factors at a continental or regional scale is less than adequate for planning at the local or river basin scale (Guhathakurta and Saji 2013;Elsanabary and Gan 2015).As a result, a better knowledge of rainfall trends, distribution, and variations at regional or basin scales is critical in order to manage water resources and practice sustainable agriculture (Guhathakurta and Saji 2013;Mathew et al. 2021).
The magnitude and distribution of rainfall greatly influence water availability and ecological changes in a region.Understanding changes in the occurrence and distribution of rainfall becomes crucial in a country like India, where agriculture is predominantly rainfed and the rainfall variation is enormous (Patakamuri et al. 2020;Singh et al. 2021).Accurate knowledge of the rainfall regime is necessary for successful water resource planning and management.Computing precipitation trends accurately may help with rainfall forecasting, water resource management, water storage structure design, drinking water supply, irrigation practices, crop choices, disaster management, and sustainable watershed management (Ahmad et al. 2015;Khalil 2020).For example, developing sustainable agrotechnological solutions for the nation's food security, as well as other types of impact on social and economic development activities, precise trend identification, and future forecast, are critical in protecting lives and property (Rahman et al. 2017;Asfaw et al. 2018a;Getahun et al. 2021).
These findings inspired research on climate variable patterns in the country, and several researchers have attempted to examine the rainfall trend.Rajeevan et al. (2006) used 1803 stations to develop high-resolution gridded rainfall data for the country from 1951 to 2003 to identify the break and active periods during the monsoon season.The authors further analyzed the inconsistency in the trends of severe rainfall events over 104 years .They found the hypothesis supporting the rising trend of severe rainfall events and concluded that changes are associated with rising surface latent heat flux and surface temperature of the ocean.Goswami et al. (2006) found significant increasing trends in the frequency and intensity of extreme rainfall events over central India throughout the monsoon season.Kumar et al. (2010a) determined changes in monthly seasonal and annual rainfall in thirty sub-divisions in the country over 135 years  and observed a rising annual rainfall trend in half of the sub-divisions and a reverse trend in the remaining.The authors have indicated an insignificant decreasing trend in annual and monsoon season but an increased rainfall trend during other seasons over the country.Adarsh and Janga Reddy (2015) analyzed the long-term rainfall trends in Kerala, Telangana, Tamil Nadu, and Karnataka.The authors have found a rising trend in all three regions except in Kerala, where annual rainfall showed a declining trend.Nengzouzam et al. (2020) used a high-resolution data of 0.25° × 0.25° data to examine the spatial and temporal variations in rainfall trends from 1901 to 2013 and observed a significant rising trend in the SWM season.Sridhara et al. (2020) have attempted to understand the rainfall trend for 41 years  in every district of Karnataka state.The authors concluded that although the monsoon, winter, and annual rainfall increased, the premonsoon and post-monsoon rainfall decreased.Samanth et al. (2022) investigated spatiotemporal rainfall variability over Karnataka state for 56 years  and found a decreasing trend in annual and southwest monsoon (SWM) rainfall in the taluks of the northern and western parts and an increasing trend in the southern interior parts.Several researchers have also tried to analyze the rainfall trend over specific parts of Karnataka.Kumar et al. (2017) attempted to analyze the rainfall trend in Uttara Kannada, Dakshina Kannada, and Udupi, the three coastal districts of Karnataka, for 34 years (1980Karnataka, for 34 years ( -2013)).In Dakshina Kannada district, the annual rainfall has significantly decreased, while the other two districts have seen an increase.Research by Chandrashekar and Shetty (2018) used gridded data (0.25° × 0.25°) for 113 years  to analyze the trend of severe rainfall occurrences in coastal Karnataka and could observe a considerable declining trend in heavy rainfall events as well as a statistically significant increase in the frequency of severe rainfall events.Using IMD gridded data (0.25° × 0.25°) for a period of 69 years , Harishnaika et al. (2022a, b) examined the rainfall trend in the dry districts of Karnataka, like Kolar and Chikkaballapura, and found an increasing annual trend.Similar investigations are also carried out by many other researchers (Bharathkumar and Mohammed-Aslam 2018;Pathak and Dodamani 2020) in various regions of Karnataka and observed a rising trend in annual rainfall over the past several years.The findings of these studies show substantial spatiotemporal variability in the annual and seasonal rainfall across India and Karnataka (Kumar et al. 2010b;Adarsh and Janga Reddy 2015;Sridhara et al. 2020;Samanth et al. 2022).
Since regional trends are often averaged in large-scale rainfall studies, understanding regional rainfall patterns is essential to address the concerns of changing rainfall extremes at local levels (Pathak and Dodamani 2020;Mehta and Yadav 2021;Samanth et al. 2022).Furthermore, the applicability of information about rainfall variability depends on the circumstances; for instance, developing a unified rainfall zone for a particular application might not be suitable for another (Samanth et al. 2022).Karnataka state has the second highest rainfed agricultural land in India, where agricultural productivity is highly dependent on rainfall.Furthermore, the second largest proportion of geographical areas experiencing drought in India is in Karnataka, next to Rajasthan (Samanth et al. 2022;N Harishnaika et al. 2022a, b).Shimsha basin is one of the semi-arid regions, dominated by rainfed agricultural land.As studied related to rainfall trend analysis has not been reported on the Shimsha basin, understanding the spatiotemporal distribution of rainfall and its changing patterns is critical.
Most studies on trend analysis use nonparametric techniques to identify trends and measure their magnitude.However, these studies lack the proper understanding of nonparametric methods and fail to follow a systematic approach to trend analysis.To perform trend analysis using the absolute approach, the time series must be homogenous free from serial correlation.Absolute approaches analyze the trend in each station separately, while relative approaches utilize neighboring stations (Patakamuri et al. 2020;N Harishnaika et al. 2022a, b).
The present study attempted to identify (i) rainfall variability and (ii) rainfall trend as well as (iii) its features (magnitude and change point) in the Shimsha basin.This information would benefit crop adaptation and water resources management planning (Mathew et al. 2021).

Study area and data used
The Shimsha basin, a sub-basin of the Cauvery River basin, in Karnataka state, India, lies between 76°14′43″ E and 77°20′53″ E longitude and 12°17′18″ N and 13°33′51″ N latitude.It originates on the southern part of Devarayanadurga hill in the Tumkur district and traverses 221 km to join the Cauvery River.The basin has a catchment area of 8694.61 km 2 and an elevation ranging from 314 to 1194 m MSL.The major rainfall seasons in the area are South-west monsoon from June to September, post-monsoon from October to November, winter from December to February, and summer from March to May.The daily rainfall data (gridded: 0.25° × 0.25°) procured from the Indian Meteorological Department (IMD) website (https:// imdpu ne.gov.in/) for 30 years  is used for this study.The study area includes nine grid stations, as seen in Fig. 1, and daily rainfall data of these stations are analyzed in various timescales.

Methodology
This work uses several statistical approaches to analyze rainfall variability and trend.The preliminary and variability analysis utilizes the mean, standard deviation, skewness, kurtosis, coefficient of variation (CV), and Standardized Anomaly Index (SAI).Kurtosis and skewness are used to characterize the characteristics of rainfall distribution (Ahmad et al. 2015;Animashaun et al. 2020;Singh et al. 2021).Standard normal homogeneity (SNH) test and Buishand's range (BR) tests are applied at 5% significant level to examine the homogeneity of the time series (Wijngaard et al. 2003;Patakamuri et al. 2020).Then the lag-1 serial correlation method is applied to check for autocorrelation in the series (Onyutha et al. 2016;Ali and Abubaker 2019;Mehta and Yadav 2021).The series without any autocorrelation are subjected to Mann-Kendall (MK) and Spearman's rank correlation (SRC) tests to determine the existence of the trend and its direction (Mallick et al. 2022;Patakamuri et al. 2020;Mohamed et al. 2021;Mehta et al. 2022).The magnitude and change point of the trend is determined by Sen's slope estimator (SSE) and sequential Mann-Kendall (SQMK) tests, respectively (Partal and Kahya 2006;Karpouzos et al. 2010;Bari et al. 2016;Rahman et al. 2017) (Fig. 2).
The homogeneity tests are performed with the R-package "trend," while the nonparametric MK and SRC tests are performed with the R-package "modifiedmk."For change-point analysis, another R-package called "trendchange" is utilized (Khalil 2020;Patakamuri et al. 2020).The following parts go through the statistical methodologies in depth.

Coefficient of variation (CV)
The CV is employed to determine the degree of variability in the rainfall series.Greater CV values suggest more significant variability in the rainfall time series data (Machiwal and Jha 2012;Ahmad et al. 2015), and it is calculated using the formula where σ and μ represent the standard deviation and average of the time series, respectively.The values of CV are classified as very high (CV > 40%), high (CV > 30%), moderate (20% < CV < 30%), and less (CV < 20%) (Asfaw et al. 2018a;Getahun et al. 2021).

Standardized Anomaly Index (SAI)
The SAI used to investigate the nature of trends and identify dry and wet years is another technique for measuring rainfall variability (Harka et al. 2021;Mohamed et al. 2021).It is calculated as where x is the annual rainfall, σ and μ represent the standard deviation and average of time series, respectively.Compared to the specified reference period, a negative value of SAI shows a drought, while a positive number suggests a wet environment.Extremely dry (Z < − 2), severely dry (− 1.5 > Z > − 1.99), moderately dry (− 1.0 > Z > − 1.49), near normal (0.99 > Z > − 0.99), moderately wet (1.49 > Z > 1.0), very wet (1.9 > Z > 1.5), and extremely wet (Z > 2) are the different classifications for Z values (Asfaw et al. 2018a;Harka et al. 2021).

Coefficient of skewness and coefficient of kurtosis
Skewness is a dimensionless value determining a distribution's asymmetry (Animashaun et al. 2020).It essentially calculates the size difference between the two tails.The coefficient of skewness (CS) is a popular skewness measurement; positive and negative CS indicate skewed distribution to the right and left, respectively (Good 1994;Animashaun et al. 2020;Maskooni et al. 2020).Generally, hydrometeorological data are skewed, and strongly skewed data make hypothesis testing that presumes a normal distribution impossible.As a result, it is critical to assess the data's skewness.
Kurtosis is another dimensionless measure representing the total weight of the tails compared to the rest of the distribution.It is used to measure the peakedness or flatness of a curve.Positive kurtosis indicates more weight, while negative kurtosis indicates less weight in the tails than a dataset with a normal distribution (Good 1994;Animashaun et al. 2020;Maskooni et al. 2020).If the values of CS and CK are 0 and 3, respectively, then a time series is said to have a normal distribution (Ahmad et al. 2015;Animashaun et al. 2020).

Homogeneity test
Non-climatic influences on climate time series data, like the change of instruments, observation procedures, station locations, and station surroundings, are assessed using homogeneity tests (Khalil 2020;Patakamuri et al. 2020).The SNH (Alexandersson 1986) and the BR tests (Buishand 1982) are employed to analyze the homogeneity of the all-time series of nine stations.The R-package "trend" is used to perform these tests (Khalil 2020;Pohlert 2020).For a data sample of size 30 and 5% significance level, the rainfall data are judged homogenous if the SNH test statistic (T o ) is less than 7.747 (Alexandersson 1986) and the BR test statistic (R∕ √ n) is less than 1.50 (Buishand 1982).

Serial correlation effect
The MK test requires the rainfall time series (x 1 , x 2 , x 3 , …, x n ) to be serially independent.If the data has a serial correlation (also known as autocorrelation), the MK test's significance level will be underestimated or overestimated, depending on whether the serial correlation is negative or positive.The MK test is performed on the dataset only if the lag-1 correlation coefficient (r) is not substantial at the 95% confidence level.Modified variants of the MK-test are used if r is substantial (Dahmen and Hall 1990a, b;Khaliq and Ouarda 2007;Luo et al. 2008;Sa'adi et al. 2019).

Mann-Kendall (MK) trend test
The statistic S of the MK test is determined as ( Kendall 1975;Mann 1945) where x i and x j are the ith and jth (j > i) observations in the time series, respectively, n is the number of observations, and sgn(x j − x i ) is the sign function computed as For series with sample size n > 10, the test statistic S is asymptotically normal, with mean E(S) and variance Var(S) as follows: When n is bigger than 10, and there is a chance of a tie in the value of x, the distribution of statistics S approaches to normality (Kendall 1975); therefore, the variance is estimated as where t k is the number of extent k and m is the number of tied groups.The standard normal test statistic Z, which is used to detect a significant trend, is The positive and negative values of Z MK display a rising and declining trend.If |Z|> Z 1−α/2 , (H o ) is rejected, and the hydrologic time series shows a significant trend.The critical value of Z 1−α/2 is ± 1.96 from the standard normal table for a sample size of 30 observations and a p-value of 0.05.
The magnitude of the monotonic connection between x and y is measured by tau.Kendall's tau correlation coefficient is calculated as follows:

Spearman's rank correlation test
The SRC test is a rank-based nonparametric approach implemented for trend identification and is employed in contrast with the MK test.In this test, the null hypothesis (H o ) implies the absence of a trend, assuming that the time series is independent and uniformly distributed, whereas the alternative hypothesis (H 1 ) implies the existence of a trend (Ahmad et al. 2015;Rahman et al. 2017).The standardized statistics Z sp and the test statistics R sp are defined as E(S) = 0, and  (Dahmen and Hall 1990a, b).

Sen's slope estimator (SSE)
SSE is a nonparametric technique for estimating the slope of a trend in hydro-climatic time series (Sen 1968).The slope where Qt is the slope, while B is a constant.
Initially, the slopes (value) of the full time series data are computed to get the slope estimation (Q i ) as where x j and x k are the data values at times j and k (j > k), respectively.
We receive as many N = n(n-1)/2 slope estimates Q i if there are n values xj in the time series.The median of these N values of Q i is SSE.The N values of Q i are ranked from the smallest to the largest, and the SSE is Finally, using a nonparametric model, Q med is computed to determine the trend and slope magnitude.Q i 's positive and negative values imply a rising and declining trend, whereas the value "zero" implies no trend.The amount of slope in original units per year would be the unit of resulting Q i .

Sequential Mann-Kendall test
To assess the approximate year of the commencement of a significant trend, the SQMK test is utilized.This test also demonstrates how the trend changes over time.This test creates two series: progressive u(t) and retrograde u′(t).
There is a statistically significant change point if they cross each other and diverge above a specified threshold value, i.e., 95% confidence level.The year the trend starts is approximated by the moment they cross each other.To determine u(t) and u′(t), the following procedures are used:

Summary of descriptive statistics and variability analysis
The annual and seasonal rainfall data from the nine IMD stations in the Shimsha basin from 1989 to 2018 and the corresponding descriptive statistical metrics such as mean, SD, CS, and CK and are shown in Table 1.The maximum and minimum mean annual rainfall in the Shimsha basin is 895.16 mm and 582.61 mm at station 2 and station 9, respectively.The study area's average annual rainfall, calculated by the Thiessen polygon method, is 801.86 mm.The maximum rainfall in the basin is observed during the SWM season and the minimum during the winter season.The SWM season is the predominant rainy season in the study region, contributing about 52.8% of total annual rainfall, followed by the PM season (25.5%), summer season (19.7%), and winter season (2%).The October months receives the highest rainfall (19.3%), followed by September (17%), August (14.3%),July (12%), May (12%), and June (10%), and nearly 84.6% of total rainfall is received during these 6 months.The average rainfall of all the nine stations for the three decades 1989-1998, 1999-2008, and 2009-2018 are 721.9mm, 774.2 mm, and 885.5 mm, respectively.Similar observations are reported by several researchers (Kumar . 2017;Sridhara 2020;Samanth et al. 2022;Harishnaika et al. 2022a, b).Though the average annual rainfall has been increasing over the past few decades, which is a positive sign for the study area, understanding its temporal distribution is crucial.The coefficient of variation ranges from 43.3 (S-6) to 22.27% (S-5) for annual rainfall and from 69.59 (S-2) to 28.13% (S-7), 67.45 (S-9) to 42.48% (S-4), 205.52 (S-8) to 94.8% (S-5), and 67.13 (S-9) to 39.27% (S-3) for seasonal rainfall in SWM, PM, winter, and summer season, respectively.The variability range is maximum during winter (205.52%),followed by SWM (69.59%),PM (67.45%), and summer (67.13%).The annual rainfall at stations S-1, S-2, S-6, S-8, and S-9 showed a higher CV of above 30%.The average CV value for annual series across the study area is 32.7% which represents moderate to slightly high rainfall variation.The average CV values in SWM, PM, winter, and summer seasons are 44.5%,51.2%, 135.7%, and 51%, respectively, indicating significant variations in seasonal rainfall, with CV above 30% in all the stations.The variation in the study area's seasonal rainfall is very high (> 40%), which may be attributed to climate change (Konapala et al. 2020).The variation in seasonal rainfall adversely affects the cropping practices, patterns, and productivity in the study area due to the dominance of the rainfed agricultural land.Failure of SWM and PM season rainfall in the study area in any year may severely affect the livelihood of farmers and water availability (Guhathakurta et al. 2017;Sah et al. 2021).
The spatial distribution of rainfall and CV for all the time series from 1989 to 2018 is depicted in Fig. 3.The upstream region experiences the highest annual rainfall, whereas the downstream region experiences the lowest due to the orographic effect of the Western Ghats, as explained in other studies (Francis and Gadgil 2006;Tawde and Singh 2015;Chandrashekar and Shetty 2018).The study area experiences more rainfall in its western part than eastern part due to its proximity to the Western Ghats and the significant contribution of rainfall during the SWM season (Chandrashekar and Shetty 2018;Singh et al. 2021).Furthermore, the variation in CV is more during the winter season, followed by SWM, PM, and summer seasons.Similar findings are reported in research carried out in the southern part of India and Karnataka (Chandrashekar and Shetty 2018;Pathak and Dodamani 2020;Samanth et al. 2022).
The coefficient of skewness for annual rainfall ranges from 0.83 (S-9) to − 0.5 (S-7).The time series may be classified as moderate to highly skewed based on the coefficient of skewness values.The S-1 and S-7 time series are the left-hand skewed (CS < 0), and the remaining are the righthand skewed (CS > 0).The coefficient of kurtosis for annual rainfall ranges from 13.95 (S-6) to − 0.73 (S-5).Based on the kurtosis coefficient, the time series could have low and high peaks.Only the S-6 time series show a high peak (CK > 3), The SAI may predict floods and droughts (Gocic and Trajkovic 2013;Koudahe et al. 2017;Harka et al. 2021).The SAI values of all stations and their relevance to water availability are shown in Table 2. SAI values indicate that the study area had received good rainfall except for a few years when some stations experienced moderate, severe, and arid conditions; similar observations are made in earlier studies (Guhathakurta et al. 2017;Siddharam et al. 2020;Harishnaika et al. 2022a, b).

Summary of trend analysis
There are 18 series at each station (one annual, four seasonal, twelve-monthly, and one rainy-day series), and various statistical tests are performed on 162 time series produced from 9 stations (9 × 18), the results of which are presented in this section.The results of the homogeneity analysis and trend test are reported at 95% confidence intervals.Trend change points are provided whenever a significant positive or negative trend is noticed in the time series.

Homogeneity of time series
The homogeneity test results are classified into three groups: • "Useful" if the null hypothesis is not rejected by any of the homogeneity tests • "Doubtful" if any test rejects the null hypothesis • "Suspect" if the null hypothesis is rejected by all tests A similar categorization may be found in studies carried out by Wijngaard et al. (2003) and Patakamuri et al. (2020).The details of homogeneity test results are highlighted in Table 3; out of 162 series, 128 time series are found "useful," 23 time series are found "doubtful," and 11 time series are found "suspect."Suspect series are regarded as inhomogeneous and are not subjected to trend analysis.

Serial correlation analysis
The confidence interval employed in statistical tests significantly impacts the serial correlation test.The results of this investigation show that none of the 162 series are serially associated at a 95% confidence interval.For a series with 30 observations, the critical value ranges from + 0.323 to − 0.392.As serial correlation does not affect time series, all the time series except inhomogeneous series are subjected to trend identification.

Mann-Kendall test and Spearman's rank correlation test
The findings of the trend analysis on annual, seasonal, monthly, and rainy-day series, performed using MK and SRC tests, are identical.Table 4 shows the findings of serially independent annual and seasonal rainfall alone.The rainfall trends at various stations in a seasonal and annual time frame are shown in Fig. 4.
In SWM season, stations 1 and 2 are non-homogeneous; station 9 has shown a significant increasing trend; stations 4, 6, and 7 have shown an insignificant increasing trend, and stations 3, 5, and 8 have shown an insignificant decreasing trend.In the PM season, stations 6, 7, and 9 have shown an insignificant increasing trend, while stations 1, 2, 3, 4, 5, and 8 have shown an insignificant decreasing trend.In the winter season, station 4 has shown a significant increasing trend, while the remaining stations have shown an insignificant increasing trend.In the summer season, stations 2, 7, and 9 are non-homogeneous; station 3 has shown a significant increasing trend; and stations 1, 4, 5, 6, and 8 have shown an insignificant increasing trend.The annual time series demonstrates non-homogeneous rainfall in station 2; a significant increasing trend in stations 1 and 9; an insignificant increasing trend in stations 3, 4, 5, 6, and 7; and an insignificant decreasing trend in station 8.The rainy-day series also indicates non-homogeneous rainfall in station 2; a significant increasing trend in stations 1, 7, and 9; an insignificant increasing trend in stations 3, 4, 6, and 8; and an insignificant decreasing trend in station 5.The trend in the monthly timescale of each station, as shown in Fig. 5, indicates significant increasing trends in March (stations 1, 2, 3, 4, 6, and 7), May (stations 7 and 9), June (station 4), August (station 1), September (station 1), and December (stations 2 and 4) months.Among nine stations, stations 1 and 4 demonstrated a significant increasing trend in 3 months, stations 2 and 7 over 2 months, and stations 3, 6, and 9 in just a month, whereas stations 5 and 8 have not shown any significant trend of increase or decrease in any of the months.rising trend with a magnitude of 0.33 mm/year, while all other stations have an insignificant trend ranging from 0.1 to 0.35 mm/year.Similarly, in the summer season, station 3 has shown a significant rising trend with a magnitude of 3.84 mm/year, while all other stations have an insignificant trend ranging from 0.84 to 3.55 mm/year.In the annual time series, stations 1 and 9 have shown a significant rising trend with a magnitude of 20.41 mm/ year and 10.71 mm/year, respectively, while all other stations have an insignificant trend ranging from − 5.08 to 8.03 mm/year.In the case of the rainy-day series, stations 1, 7, and 9 have shown a significant rising trend with a magnitude of 1.5, 0.77, and 0.53 days/year, respectively, while all other stations have an insignificant trend  extent of rainfall among all the seasons, whereas the PM season saw the most prominent negative trend; similar findings are reported in other studies conducted in the Karnataka region (Sridhara et al. 2020;Samanth et al. 2022).The increase in summer and SWM rainfall is a positive sign, while the decreasing trend in the PM rainfall could threaten agriculture productivity if the magnitude increases in the future.On a monthly timeline, station 1 displays a significant increasing trend, with magnitudes of 0.721 (March), 4.217 (August), and 3.255 (September) mm/year, and station 2 shows a significant rising trend of 0.5 (March) and 0.1 (December) mm/year.Stations 3 and 4 also exhibit substantial rising trends, with a rate of 0.58 mm/ year in March in the former station and rates of 0.65 (March), 2.72 (June), and 0.15 (December) mm/year in the latter station.Similarly, increasing trend with a rate of 0.32 mm/year in March in station 6, rates of 0.4 (March) and 3.744 (May) mm/year in station 7, and a rate of 2.47 mm/year in May in station 9 are observed.All the stations have shown an increasing rainfall trend in March, among which most of the stations have shown a significant trend.The increasing rainfall trend in March and May justifies the increasing trend of summer rainfall.In the SWM season, all stations have shown an increasing trend in August, while mixed trends in other months.In October, except for one station, which has shown a negligible increasing trend, all other stations have shown a decreasing trend.The declining rainfall trend in October contributes to reduced PM rainfall in the study area.The increased winter rainfall is contributed by the increasing trend observed in December.Overall, the study area is noticing a temporal shift in the distribution of rainfall, which may have an impact on the timing of sowing and crop yield.

Trend change-point analysis
The change-point analysis is carried out to identify the most probable time of occurrence of a significant change in the series by applying the SQMK test in all the series.Furthermore, the time series plots with a significant trend are highlighted and discussed in this section.Figures 7 and  8 illustrate SQMK plots for major trend shift points based on prograde and retrograde series interaction.The large intersection in the graph denotes the likely year of the trend shift, and the intersection of the prograde series with the threshold lines denotes the year of each significant trend.In the absence of a significant trend, series overlaps at multiple points.The annual SQMK plot of station 1 (Fig. 7A) shows a rising trend that started in 2013 and became significant in 2017.In Fig. 7B, station 3 summer series showed a clear rising trend from 2003 and became significant in 2008.Similarly, Fig. 7C demonstrates the station 4 winter series, with a rising trend from 2012 and becoming significant in 2018.In Fig. 7D, the station 9 annual series showed an increasing trend from 2003, which became significant in 2008.Figure 7E represents station 9 SWM series wherein an increasing trend is observed from 1995, which became significant in 2004.Figure 7F represents station 1 rainyday series with an increasing trend from 2013 and became significant in 2015.Figure 7G demonstrates the station 7 rainy-day series, which showed a rising trend from 2003 and Fig. 7 SQMK test statistics for seasonal, annual, and rainy-day series became significant in 2010.In Fig. 7H, the station 9 rainyday series shows a rising trend that became significant in 2009.Figure 8 depicts the trend change point of the monthly time series that exhibited a significant trend.Table 6 lists the change point and significant change point years for all the series which exhibited significant trends.
SQMK results on the annual scale demonstrate an increasing trend in all stations except station 8, whereas stations 1 and 9 exhibited a significant trend.In the SWM season, station 9 has shown a significant increasing trend, while all other stations have cyclically shown insignificant trends.Furthermore, none of the stations showed a significant trend but insignificant trend in a cyclic pattern throughout the PM station.In the winter, all the stations showed an insignificant increasing trend, whereas station 4 showed a significant increasing trend, while in the summer season, a similar case is noticed; all the stations showed an insignificant increasing trend, whereas station 3 showed a significant increasing trend.The SQMK plots of the rainyday series indicate a significant increasing trend in stations 1, 7, and 9. From the SQMK analysis, it is observed that the rainfall patterns vary from insignificant to significant in all the time series.Similar observations are made in the monthly series SQMK plots, which correlate with the seasonal rainfall trends.Overall, SQMK analysis reveals that most of the time series shows significant trends in recent years, denoting an increase in rainfall in the study area in recent times.

Conclusions
Shimsha basin, a sub-basin of Cauvery, plays a vital role in the Tumkur and Mandya districts and is dominated by rainfed agriculture.Hence, understanding rainfall variability, distribution, and trend are essential for developing future agricultural plans, water resources management plans, and policies.This study investigated the rainfall variability and trends on annual, seasonal, and monthly timeframe, as well as the number of rainy-day series for the Shimsha basin by MK, SRC, and SSE tests for 30 years .The average annual rainfall of the study area is 801.86 mm, with the highest rainfall recorded during the SWM season, followed by PM, summer, and winter seasons, respectively.The SWM and PM rainfall contributes 78% of the total annual rainfall, which significantly affects water availability and agriculture productivity.
Moreover, the highest rainfall is recorded in October, which is 19.3% of total annual rainfall, followed by August and September.It is also noticed that the study area receives an increased amount of annual rainfall on a decadal basis.Trend analysis is carried out to understand its temporal variation pattern on an annual, seasonal, and monthly basis.A consistent trend results are observed in both MK and SRC tests.The study revealed that 11 time series out of 162 series are non-homogeneous.Out of 151 homogeneous series, 21 have shown a significant increasing trend, 99 have shown an insignificant increasing trend, and 31 showed a decreasing trend.However, no significant downward trend is observed in any of the time series.All the stations have shown an insignificant to significant trend, except for station 8, which has shown only insignificant decreasing trend.The spatial distribution and rainfall trend during the SWM season is consistent with the annual rainfall, indicating the considerable contribution of SWM rainfall to the annual rainfall.The study area has noticed an increasing trend in annual, SWM, and summer timeframes.The most significant positive trend is observed in the summer season due to increased contributions from March to May, whereas the PM season noticed the most prominent negative trend due to the declining rainfall trend in October.
The findings of the study quantify the extent of seasonal and annual rainfall trends and variability in the Shimsha basin.The agricultural productivity of the study area will be impacted by the change in rainfall pattern, particularly during the PM season.
The information on the rise in summer and SWM rainfall and its periodicity may be helpful for irrigation planners in making decisions on effective cropping patterns as well as the efficient use of water resources.The methodology adopted in this study to analyze rainfall trend in time series is applicable to any other region in the world.The obtained results about rainfall and rainy-day trends can be analyzed to comprehend the fluctuation of surface and ground water in the study area.This study may be further extended to correlate with other hydrometeorological data.Future research might analyze the problems with trend attribution and the causes of the periodic components influencing the trend.The study's findings, such as trend and variability analyses and spatial variation maps, would be valuable for sub-basin-level water management and planning effective use of water resources.

Fig. 1
Fig. 1 Location of Shimsha basin and IMD stations

Fig. 4
Fig. 4 Rainfall trend for seasonal, annual, and rainy-day series

Fig. 8
Fig. 8 SQMK test statistics for monthly series the rank of the ith observation, I is the chronological order number, n is the entire length of the time series data, and Z SP is the Student's t-distribution with (n − 2) degrees of freedom.Positive Z SP values indicate a rising trend in the hydrologic time series, whereas negative values indicate a declining trend.(n−2,1−α/2) is the crucial value of t at a 0.05 significance level in the Student's t-distribution table.If |Z SP |> (n−2,1−α/2) , (H o ) is rejected, and the hydrologic time series shows a significant trend.The critical value of Z SP is ± 2.046 from the standard normal table for a sample size of 30 observations and a p-value of 0.05 is even 1.The yearly mean time series values of x j (j = 1, …, n) are related to x i (i = 1, …, j − 1).The number of times x j > x i is considered for each comparison and is marked by n j .2. The test statistics t j is then determined by the equation 3. The mean and variance of the test statistics are 4.The statistics u(t) sequential values are then calculated as Similarly, u′(t) values are computed backward from the end of the series.

Table 1
Summary of statistical analysis of seasonal and annual rainfall SD standard deviation, CV coefficient of variation, CS coefficient skewness, CK coefficient kurtosis, RS right skewed, LS left skewed, PK platykurtic, LK leptokurtic

Table 2
Summary of SAI values and drought condition

Table 4
Summary of MK, SRC, and SS tests

Table 5
Sen's slope values on a monthly timescale