Understanding the intricacies of rainfall dynamics using entropy measures

Climate change has multifaceted impacts, influencing not only the quantitative aspects but also the frequency and spatial distribution of rainfall occurrences (López et al. 2023; Tamm et al. 2023). In addition, it substantially impacts hydrological cycles, precipitation patterns, evaporation rates, and overall water circulation (Han et al. 2023). The convoluted interplay involving phases like evaporation, condensation, and precipitation is integral to maintaining water balance and irrigation system (Canet-Martí et al. 2023; Singh, S., Kumar, D. et al. 2023, Singh, S., Kumar, N. et al. 2023; Zekrifa et al. 2023). These variations significantly impact the ecosystem, society, weather patterns, global water availability, and distribution (Konapala et al. 2020; Ansley et al. 2023). Understanding this complex relationship of hydrological cycles is crucial for the sustainable management of water resources, ensuring adaptability to changing weather patterns (Ficklin et al. 2022; Dingle et al. 2023; Sukanya & Joseph 2023). It is recognized that analyzing hydro-meteorological variables is fundamental to understanding and effectively managing water resources amidst evolving climate dynamics (Kumar 2012; Biao 2017). Rainfall, functioning as the linchpin of the hydrological cycle, coordinates the exchange of mass and energy between the terrestrial environment and the atmosphere (Mishra & Tiwari 2023). However, these processes also affect the timing, magnitude, and duration of water-related disasters, such as floods and droughts, concurrently influencing water quality. Understanding the precipitation distribution is essential to interpreting the elements of the hydrological cycle, climate interaction, allocating water resources, seasonal rainfall modeling, and lessening the effects of floods and droughts (Mukherjee & Mishra 2022; Rautela et al. 2023; Singh et al. 2024). The variability in rainfall, particularly during the monsoon season, significantly impacts India's agricultural productivity, contributing around 22% to the country's gross domestic product (GDP) (Krishnakumar et al. 2004). This reliance highlights how fluctuations in rainfall patterns, timing, intensity, and distribution across regions directly affect crop growth and yield. Numerous research studies have demonstrated that the intensity of precipitation rises as global warming increases (Lal 2003; Alexander 2016). In most scenarios, the latest report of the Intergovernmental Panel on Climate Change (IPCC) suggests that ENSO (El Niño–Southern Oscillation) rainfall changes are expected to increase by the second half of the 21st century (IPCC 2021). Chauhan et al. (2022b) assessed the influence of ENSO on vegetation and monsoon rainfall in Haryana State; they found high variability in the western agro-climatic zone of Haryana. However, examining regional precipitation variation poses challenges due to its potential for heightened unpredictability and significant spatiotemporal fluctuations within the same area (Ahmad et al. 2018). This complexity necessitates a deeper exploration and analysis to navigate the intricacies of regional precipitation patterns amid changing climates.

In recent times, Shannon's entropy theory, introduced in 1948, has gained considerable attention in the field of hydrology. In addition, the entropy approach has been applied to examine spatial and temporal precipitation variability at global (Sreeparvathy & Srinivas 2022) and regional scales (Guhathakurta & Rajeevan 2008; Krishnakumar et al. 2009; Kumar et al. 2010; Huang et al. 2014; Hong et al. 2015; Chandniha et al. 2017; Singh & Kumar 2020; Guntu et al. 2020b; Ghorbani et al. 2021; Patel et al. 2021; Singh, S., Kumar, D. et al. 2023, Singh, S., Kumar, N. et al. 2023). Kawachi et al. (2001) suggested Shannon entropy for determining the temporal variability of precipitation and categorized various zones of water resources in Japan. As metrics of information regarding rainfall variability, Maruyama et al. (2005) defined apportionment entropy (AE) and intensity entropy (IE). Ascertaining probability is the most critical element in the entropy calculation process. Mishra et al. (2009) used an index-based entropy approach to find the variability in rainfall patterns at decadal, annual, seasonal, and monthly time-periods. The disorder index was used by various studies to find the variability (Zhao et al. 2011; Zhang et al. 2016; Cheng et al. 2017; Roushangar et al. 2018; Singh & Kumar 2021). However, the majority of the aforementioned studies estimated rainfall variability using non-normalized indices. The shortcoming is that the results cannot be compared across timescales and data lengths. Guntu et al. (2020b) proposed a standardized variability index (SVI) taking the rainfall data from 1901 to 2013 to overcome this limitation. The range of SVI values varied from 0 (no variation) to 1 (high variation). Bharti et al. (2023) employed entropy to investigate the complex network of groundwater. Prajapati et al. (2024) used entropy to monitor the precipitation network in Bihar, India. The significant advantage of this theory is that it operates without relying on prior assumptions regarding the probability distribution or statistical properties of the data (Koutsoyiannis 2005; Agarwal et al. 2016). Haryana State is known for its substantial contributions of paddy during Kharif and wheat during Rabi. The rainfall patterns fluctuate substantially over relative ranges due to local topography, land use, and other geographical aspects (Buytaert et al. 2006; Pielke et al. 2007) and their detrimental effects from various perspectives, including soil degradation, farming, and water supply system (Mutekwa 2009; Nhemachena et al. 2020). Further, the changes in rainfall seasonality, extreme rainfall in short duration followed by a prolonged dry spell, disrupt crop production and infrastructure, and cause biodiversity loss (Chandol et al. 2021; Buheji & Muhorakeye 2023; Niyonsenga et al. 2024). In a study made on the Ghaghara River basin situated in Haryana it was found that heavy rain in certain parts of the basin, along with water runoff from the middle, causes floods, specific areas such as Patiala and Ambala are at risk, and annual rainfall is substantially impacted by southwest monsoon (Gorai et al. 2021). Therefore, it is essential to conduct seasonal studies, which need attention due to frequent extreme events, to identify areas more susceptible to precipitation variations as they influence soil moisture (Ganeshi et al. 2020) and water resources in that region. This proactive approach allows for timely resource allocation, readiness for extreme weather events like droughts and floods, and the sustainable management of water resources year-round (Dey & Mujumdar 2019). Keeping all these studies and gaps in view, the author found that seasonal study needs more attention in this area. Therefore, this study aims to address the gap in seasonal rainfall analysis, identifying specific months potentially responsible for inconsistencies in rainfall amounts and rainy days. Moreover, it seeks to comprehensively measure rainfall variability and its implications for agriculture in Haryana.

The MK test detects a monotonic trend in a time series. Nevertheless, when analyzing hydro-climatic time-series data, it is also crucial to identify the initial time-period of significant trends. Furthermore, determining changes in the trend over time is essential in trend detection studies. For this purpose, the sequential Mann–Kendall (SMK) test is very valuable as it facilitates change detection analysis. It also estimates the approximate year when a significant trend begins. Additionally, in any trend detection study, it is crucial to determine changes in trends over time, and this test proves to be particularly valuable for conducting change detection analysis.

The SMK generates two series: a progressive (forward) series R(t) and a retrograde (backward) series R’(t). The intersection between the forward and backward series marks a potential trend-turning point in the time series. Suppose the progressive or retrograde row exceeds predetermined limits before and after the crossing point. In that case, the null hypothesis of no change points is rejected, indicating a significant trend at a specified significance level. In such cases, the trend-turning point may hold significance at a specific significance level, such as a 5% significance level.

The SMK test involves the following steps:

• (1) The yearly mean time-series values of X_j (j = 1, …, n) are compared with X_i (i = 1, …, j − 1), and the number of times X_j > X_i is recorded as n_j.

• (2) The test statistic t_k is determined by the following equation:

  

  (5)
• (3) The mean of the test statistic is calculated by the following equation:

  

  (6)
• (4) The variance of is determined by the following equation:

  

  (7)
• (5) Furthermore, the sequential values of the statistic are computed by the following equation:

  

  (8)

In the same way, the values of R′(t) are calculated in reverse order, starting from the end of the series.

In addition, the total rainfall amount can describe the typical pattern of rainy days or vice versa, as both are interrelated to each other. For instance, if total rainfall amounts are high, that can be achieved through frequent light rain over many days or intense heavy rainfall over fewer days. However, it is different for different climates where both rainfall amounts and rainy days may be higher. This means that it describes the distribution of rainfall intensity. The present study observed that variation in rainy days was greater compared with rainfall amount, indicating that the distribution of rainfall intensity was highly fluctuating within the year.

Trend analysis methods, such as MMK and SMK tests, were used for SVI_AE and SVI_IE at various time-scales to determine the trend, change point, and significant year. In addition, Theil–Sen's slope (TSS) estimator was also employed to determine the magnitude of the trend of the SVI_AE and SVI_IE series. The SMK test was not conducted for those stations that did not have a significant trend.

Based on the results of the SMK test, as shown in Table 1, the change over the years and the significantly increasing trend in daily SVI_IE were more pronounced than in daily SVI_AE. The change year of SVI_IE was found in the years 1970 and 1986 for the Jind district, and the change year of SVI_AE for the Panchkula district was found in 1999, and for SVI_IE, it was found in 1998. Only these stations exhibited a significantly increasing trend on a daily scale. A significantly increasing trend for SVI_IE was seen during 2006–2007 and 2012–2019 for the Jind district. For Panchkula district, a significantly increasing trend in SVI_AE was seen from 2008 to 2020, and in SVI_IE, it was during 2006–2020.

The inverse distance weighing (IDW) method was employed to quantify spatial variability at different monthly, seasonal, and annual time-scales. The spatial variability was determined based on an SVI of marginal entropy (SVI_ME), which describes the inter-variability across these time scales. In addition, colors (light green, blue, pink, and red) were used to display the inter-variability of rainfall based on SVI_ME. The light green color indicates low variability, whereas the red color indicates high variability of rainfall across the districts of Haryana, and their respective ranges were displayed on the map itself.

The range of SVI_ME for February varied from 0.099 to 0.208. Panchkula, Ambala, and Yamunanagar showed low variability in rainfall, ranging between 0.099 and 0.126. Kurukshetra, the eastern part of Kaithal and Karnal showed medium variation in rainfall, ranging between 0.126 and 0.153. Bhiwani, Fatehabad, Gurugram, Hisar, Jhajjar, Jind, Kaithal, Panipat, Rohtak, Sirsa, and Sonipat showed a moderate variation of rainfall ranging between 0.153 and 0.180. Charkhi Dadri, Faridabad, Mahendragarh, Nuh, Palwal, and Rewari showed high variability of rainfall within a range from 0.180 to 0.208.

The range of SVI_ME for March varied from 0.116 to 0.243. Panchkula, Kurukshetra, Kaithal, Karnal, Ambala, and Yamunanagar showed low variability in rainfall ranging between 0.116 and 0.146. Fatehabad, Hisar, Jind, Panipat, Sonipat, Rohtak, and Bhiwani showed medium variation in rainfall ranging between 0.146 and 0.180. Charkhi Dadri, Faridabad, Jhajjar, Mahendragarh, Rewari, and Sirsa showed moderate variation of rainfall ranging between 0.180 and 0.211. Gurugram, Nuh, and Palwal showed high variability of rainfall within a range of 0.211–0.243.

The range of SVI_ME for April varied from 0.124 to 0.286. Panchkula, Ambala, and Yamunanagar showed low variability in rainfall, ranging between 0.124 and 0.165. Kurukshetra, Bhiwani, Charkhi Dadri, Rohtak, and Mahendragarh showed medium variations in rainfall, ranging between 0.165 and 0.205. Faridabad, Fatehabad, Hisar, Jhajjar, Jind, Kaithal, Karnal, Rewari, Sirsa, and Sonipat showed moderate rainfall variations ranging between 0.205 and 0.245. Nuh, Palwal, the central region of Panipat, Fatehbad, and Gurugram showed high variability of rainfall within a range of 0.245–0.286.

Figure 13(b) illustrates the spatial distribution of SVI_ME at a monthly time-scale for May, June, July, and August. The range of SVI_ME for May varied from 0.088 to 0.177. Panchkula, Kurukshetra, Ambala, and Yamunanagar showed low variability in rainfall, ranging between 0.088 and 0.111. Bhiwani, Fatehabad, Hisar, Jind, Kaithal, Karnal, Mahendragarh, Panipat, Rohtak, and Sonipat showed medium variation in rainfall ranging between 0.111 and 0.133. Sirsa, Charkhi Dadri, Jhajjar, Gurugram, Rewari, and Faridabad showed moderate variation of rainfall ranging between 0.133 and 0.155. Nuh and Palwal showed high rainfall variability from 0.155 to 0.177.

The range of SVI_ME for June varied from 0.041 to 0.094. Panchkula, Ambala, Kurukshetra, Karnal, and the central part of Sonipat showed low variability in rainfall, ranging between 0.041 and 0.054. Bhiwani, Fatehabad, Hisar, Jind, Kaithal, Panipat, Rewari, Rohtak, Sonipat, and Yamunanagar showed medium variation in rainfall ranging between 0.054 and 0.067. Sirsa, Charkhi Dadri, Jhajjar, and Gurugram showed moderate rainfall variation between 0.067 and 0.081. Mahendragarh, Faridabad, Nuh, and Palwal showed high rainfall variability within a range of 0.081–0.094.

The range of SVI_ME for July varied from 0.017 to 0.044. Panchkula, Ambala, Kurukshetra, and Yamunanagar showed low variability in rainfall, ranging between 0.017 and 0.024. Some regions of Kurukshetra and Karnal showed medium variations in rainfall, ranging between 0.024 and 0.030. Kaithal, Mahendragarh, Nuh, and Palwal showed moderate variation in rainfall, ranging between 0.030 and 0.037. Bhiwani, Charkhi Dadri, Faridabad, Fatehabad, Gurugram, Hisar, Jhajjar, Jind, Kaithal, Panipat, Rewari, Rohtak, Sirsa, and Sonipat showed high variability of rainfall within a range of 0.037–0.044.

The range of SVI_ME for August varied from 0.018 to 0.061. Panchkula, Ambala, and Yamunanagar showed low variability in rainfall, ranging between 0.018 and 0.028. Kurukshetra, Kaithal, Karnal, and Nuh, and the western part of Gurugram showed medium variations in rainfall, ranging between 0.028 and 0.039. Bhiwani, Charkhi Dadri, Faridabad, Fatehabad, the eastern part of Gurugram, Hisar, Jhajjar, Jind, Mahendragarh, Palwal, Panipat, Rewari, Rohtak and Sonipat showed moderate variation of rainfall ranging between 0.039 and 0.050. Sirsa and some Fatehabad regions showed high rainfall variability within a range of 0.050–0.061.

Figure 13(c) illustrates the spatial distribution of SVI_ME at the monthly time-scale for the months of September, October, November, and December. The range of SVI_ME for the month of September varied from 0.051 to 0.147. Panchkula, Ambala, Gurugram, Nuh, Kurukshetra, and Yamunanagar showed low variability in rainfall ranging between 0.051 and 0.075. Bhiwani, Charkhi Dadri, Faridabad, Jhajjar, Kaithal, Karnal, Mahendragarh, Nuh, Palwal, Panipat, Rewari, Rohtak and Sonipat showed medium variation in rainfall ranging between 0.075 and 0.098. Fatehabad, Hisar, and Jind showed moderate rainfall variations between 0.098 and 0.123. Sirsa showed high variability of rainfall within a range of 0.123–0.147.

The range of SVI_ME for October varied from 0.251 to 0.404. Ambala, Panchkula, Yamunanagar, Kurukshetra, Karnal, Sonipat, Rohtak, Charkhi Dadri, Gurugram and Mahendragarh showed low variability in rainfall ranging between 0.251 and 0.289. Bhiwani, Faridabad, Hisar, Jhajjar, Jind, Kaithal, Nuh, Panipat, and Rewari showed medium variation in rainfall ranging between 0.289 and 0.328. Fatehabad and the central region of Palwal showed a moderate rainfall variation ranging between 0.328 and 0.366. Sirsa showed high variability of rainfall within a range of 0.366–0.404.

The range of SVI_ME for the month of November varied from 0.274 to 0.386. Panchkula, Ambala, Yamunanagar, and the central part of Bhiwani showed low variability in rainfall, ranging between 0.274 and 0.302. Charkhi Dadri, Faridabad, Hisar, Jhajjar, Jind, Kaithal, Karnal, Kurukshetra, Mahendragarh, Nuh, Rewari, Rohtak, and Sonipat showed medium variation in rainfall ranging between 0.302 and 0.330. Panipat, Sirsa, Gurugram, and Palwal showed moderate rainfall variation between 0.330 and 0.358. Fatehabad, the central part of Panipat, Sirsa, Gurugram, and Palwal showed high rainfall variability within a range of 0.358–0.386.

The range of SVI_ME for December varied from 0.168 to 0.228. Panchkula, Ambala, Yamunanagar, Kaithal, Karnal, and Kurukshetra showed low variability in rainfall ranging between 0.168 and 0.228. Sirsa, Fatehabad, Hisar, Jind, Panipat, Rohtak, and Sonipat showed medium variations in rainfall, ranging between 0.228 and 0.287. Bhiwani, Charkhi Dadri, Faridabad, Jhajjar, Mahendragarh, and Rewari showed moderate rainfall variations, ranging between 0.287 and 0.345. Nuh, Palwal, and the eastern part of Gurugram showed high rainfall variability within a range of 0.345–0.404.

The range of SVI_ME for the monsoon season varied from 0.008 to 0.022. Ambala, Panchkula, Yamunanagar, Kurukshetra, and Karnal showed low variability in rainfall, ranging between 0.008 and 0.012. Kaithal, the southern portion of Karnal, Faridabad, Gurugram, Nuh, Panipat, and Sonipat showed medium variation in rainfall ranging between 0.012 and 0.015. Rohtak, Palwal, Charkhi Dadri, Mahendragarh, Bhiwani, Hisar, Jhajjar, Jind, and Rewari showed a moderate rainfall variation ranging between 0.015 and 0.019. Fatehabad, the central part of Mahendragarh and Sirsa showed high rainfall variability from 0.019 to 0.022.

The range of SVI_ME for the post-monsoon season varied from 0.115 to 0.223. Panchkula, Ambala, Yamunanagar, and Kurukshetra showed low variability in rainfall, ranging between 0.115 and 0.143. Karnal, Sonipat, Mahendragarh, Bhiwani, Faridabad, Jhajjar, Jind, Kaithal, Rohtak, Charkhi Dadri, and Rewari showed medium variation in rainfall ranging between 0.143 and 0.171. Fatehabad, Faridabad, Hisar, Panipat, Gurugram, and Nuh showed moderate rainfall variation between 0.171 and 0.198. Sirsaand Palwal showed high rainfall variability within a range of 0.198–0.223. These variations or inconsistencies in rainfall patterns at monthly, seasonal, and annual scales may be due to the topography of the regions (Reda et al. 2015), changes in the atmospheric circulation patterns, and the effect of ENSO (Donat et al. 2014; McGree et al. 2014; Chauhan et al. 2022b) and may be due to anthropogenic activity in the local region.

A few districts in Haryana, such as Fatehabad, Faridabad, and Sirsa, experienced rainfall variations. Another study also observed variations in the district of Fatehabad, which verified the reliability of the study's outcomes (Chauhan et al. 2022b). Additionally, the post-monsoon season exhibited high rainfall inconsistency, followed by the winter season, primarily due to variations in October, followed by December (Choobeh et al. 2024). Similarly, the post-monsoon and winter seasons were responsible for the annual variation in rainfall patterns (Guntu et al. 2020b).

The present study primarily focuses on quantifying the variability of rainfall dynamics across multiple time-periods and various entropy measures. Daily gridded rainfall data spanning 70 years, from 1951 to 2020, were utilized to study Haryana. Additionally, the MMK, Sen's slope, and SMK tests were employed to identify trends in the entropy indices. The IDW method was applied to ascertain the spatial variability of rainfall based on SVI_ME.

The comprehensive analysis of rainfall variability in Haryana provides a valuable understanding of this hydro-climatological parameter that governs precipitation in this state. One of the key observations from this study is the consistent trend of increasing intra-variability in rainfall amounts (SVI_AE) and rainy days (SVI_IE) with a prolonged time-scale. This trend implies that the precipitation patterns in Haryana are becoming more unpredictable and variable over time; for the agriculture sector, this poses a challenge as farmers need to adapt to this heightened variability, possibly through adopting resilient crop varieties, efficient water-management practices, and exploring alternative livelihood options. Moreover, this study observed the distinct patterns in rainfall patterns across different regions and time periods. The interrelation between total rainfall amounts and rainy days demonstrates that total rainfall amounts can result from either frequent light rains over many days or intense heavy rainfall over fewer days. However, this relationship varies across different climates. The study found that the variation in rainy days was greater than the variation in total rainfall amounts, indicating a highly fluctuating distribution of rainfall intensity throughout the year.

The study also highlights the spatial variability during different seasons, offering valuable insights into the distinct patterns observed across districts. For instance, during the post-monsoon season, Sirsa and Palwal districts exhibit high spatial variability, driven by variations occurring in October and December. This spatial heterogeneity imposes a district-specific approach to planning and resource allocation. Urban planning and disaster vigilance attempts should consider these variations to enhance resilience and adaptive capacity, especially in vulnerable districts. On the annual scale, the study recognizes some districts (Fatehabad and Mahendragarh) with high variability due to variations in winter followed by monsoon season.

In contrast, Panchkula, Ambala, and Yamunanagar show low variability, reflecting consistent variations across all seasons. This annual-scale variability has implications for water resource management, indicating the need for sustainable strategies that account for the dynamic nature of precipitation throughout the year. The repercussions of these outcomes are helpful across various sectors. The need for adaptive measures to cope with increasing intra-variability is evident in agriculture. Crop diversification, precision agriculture, and improved water-use efficiency are strategies that farmers may consider to mitigate the impacts of unpredictable precipitation patterns. Considering the intensified spatial variability during specific seasons, society should focus on urban planning and disaster preparedness, particularly in vulnerable districts. Considering the annual-scale variability observed in different districts, water resource management strategies must be dynamic.

This study sets the stage for future research in capturing autocorrelation in data through machine-learning methods like recurrent neural networks (RNNs) or long short-term memory (LSTM), as these models can capture intricate patterns and dependencies in the data, making them more powerful in understanding the autocorrelation structure of time-series data and potentially providing better predictive performance in specific scenarios. In addition, integrating climatic models and forecasting techniques into rainfall variability studies can provide a forward-looking perspective, enabling stakeholders to anticipate and prepare for future challenges.

We are deeply grateful to the India Meteorological Department (IMD) for providing gridded rainfall data. Also, we are thankful to anonymous reviewers and the editor for their insightful comments/suggestions on the earlier draft of this work.

All authors agree to publish the manuscript.

The authors express their consent to publish the research work.

Each author made a substantial contribution to the current research. Particularly, S. S. prepared the methodology, data collection, and analysis part; A. K. rendered support in core findings, conclusions, and supervision.

All relevant data are available from an online repository or repositories: https://www.imdpune.gov.in/cmpg/Griddata/Rainfall_25_NetCDF.html.

The authors declare there is no conflict.