Comprehensive analysis of trends in mean temperature over the Cauvery River Basin, India using various statistical methods and discrete wavelet transform

Climate change (CC) and global warming are widely acknowledged as the most crucial environmental problems faced by the contemporary world. Climate scientists are showing keen interest in CC and global warming due to the concern expressed by governments, non-governmental organizations, and the worldwide community. This has led to the conduction of several studies on the detection of climate trends at the global, hemispherical, and regional scales (Safari 2012).

Rahmstorf & Coumou (2011) observed an increase in extreme heat events, especially in recent decades, as a result of the ongoing warming trend. Extreme temperatures primarily lead to heatwaves, cold spells, and droughts – meteorological events that pose significant threats to human life, ecosystems, and agriculture (Aksu 2021). Possible changes in the frequency of extreme events are gaining significant attention alongside global warming as these extremes have a direct effect on human society and the economy. For most socially relevant extremes and variations, analyzing daily data is essential (Yan et al. 2002).

Surface air temperature variations brought by variations in climate have a notable impact on the surface energy budget and the hydrological cycle, which, in turn, affects water resources (Nalley et al. 2013). Temperature is a crucial atmospheric factor that directly and significantly affects almost all hydrological variates. One of the key variables influencing biological systems is temperature, which plays a crucial role in crop growth. Temperature has been taken into consideration more and more than in the past under the conditions of CC and global warming. Temperature plays a key role in detecting CC brought on by industry and urbanization (Kousari et al. 2013).

According to the sixth IPCC Assessment Report, global mean surface temperatures have risen since 1970. Information about CC at the basin scale is essential for planning, development, and utilization of water (Verma & Kale 2018). The rise in global and local T_mean along with increased human interventions has significantly impacted the hydrology of CRB, leading to floods and droughts and disrupting the natural functioning of the ecosystem (Gowri et al. 2021). Increasing temperature has many other impacts like season shifting, hot spells, and increasing drought severity frequencies. Global warming is expected to have an impact on species ranges, population levels, and community composition, according to Sparks & Menzel (2002). Temperature changes impact various climate system parameters, including the onset and length of seasons, seasonal extension, frost duration, the number of frost days, tropical days, and the duration of heat and cold waves, among other factors (Aksu 2022). Kuglitsch et al. (2010) observed a significant increase in the temperature of hot summer days and nights, along with the number, length, and intensity of heat waves since the 1960s. Hence, it is very important to analyze the trends in T_mean at the basin scale for the CRB.

Assessing the magnitude of the trend in T_mean is necessary for quantifying the extent of CC, which aids in developing mitigation and adaptation strategies. An essential part of every statistical test is the assessment of its assumptions. If the data do not support the assumptions made in the statistical test, the test results may be meaningless. This is because there would be serious errors in the significance level calculations. Whether a test statistic differs significantly from the range of values that would typically occur under the null hypothesis is indicated by the level of significance (Kundzewicz & Robson 2004). Historical data that exhibit monotonic (MT) and steadily increasing patterns cause planning, management, and operating procedures used by atmospheric researchers, climatologists, meteorologists, hydrologists, and economists to be modified. Thus, it is important to attempt to detect potential monotonous trend components in any given time series before making any future forecasts (Şen 2017). When adequate spatially distributed data are available, it is recommended for regional research to use a field significance rather than significance for the individual locations (Kundzewicz & Radziejewski 2006).

Therefore, trend analysis (TA) of T_mean data over the CRB by assessing assumptions regarding data before the application of statistical testing and assessment of four aspects, viz. magnitude, significance, nature, start, and end of trend along with the evaluation of regional significance of trends are necessary. Alongside examining fundamental trend characteristics, it is essential to recognize sequential changes in trends and evaluate the periodicity of hydroclimatic variables (Adarsh & Janga Reddy 2014). For this purpose, both continuous wavelet transform (CWT) and discrete wavelet transform (DWT) can be used.

It is preferable to use DWT over CWT, as CWT produces information in a two-dimensional format rather than a time series. If the DWT is selected for analysis, the transformation process is made simpler, fewer calculations are needed and still an extremely accurate and efficient analysis is executed. This is because the DWT often relies on a ‘dyadic’ computation of a signal's position and scale (level) (Adarsh & Janga Reddy 2014).

Various studies have been carried out on the topic of TA corresponding to non-basin scale at various locations for T_mean data. Some studies have analyzed the magnitude and significance of trends in T_mean data (Ray et al. 2019; Meshram et al. 2020; Sharafi & Mir Karim 2020; Monforte & Ragusa 2022). Some studies have assessed the nature of trends along with the magnitude and significance of trends in T_mean (Alemu & Dioha 2020; Yenice & Yaqub 2022). Start and endpoints of trends in T_mean are also determined along with magnitude and significance by some of the reviewed studies (Feidas et al. 2004; Mohsin & Gough 2010; Hadi & Tombul 2018). Mondal et al. (2015) and Chakraborty et al. (2017) have determined break points in trends in T_mean along with the magnitude and significance of trends. Javanshiri et al. (2021) have assessed the magnitude, significance, and regional significance of trends in T_mean in Iran.

Some of the studies have analyzed the trends in T_mean on the basin scale. Long-term TA for major climate variables including T_mean of the Yellow River Basin, China, was carried out (Xu et al. 2007). Wang & Zhang (2012) have carried out long-term TA for annual and seasonal T_mean of the Jinsha River Basin, China. Addisu et al. (2015) have conducted TA in rainfall and T_mean of the Tana Sub-basin of Lake Tana, Ethiopia. For the Sutlej River Basin in India, spatial and temporal changes in surface temperature parameters including T_mean were assessed (Singh et al. 2015). Cui et al. (2017) have evaluated the magnitude, significance, and nature of trends over the Yangtze River Basin in China. The magnitude and significance of trends in rainfall and T_mean data over the Woleka sub-basin, Ethiopia, were assessed (Asfaw et al. 2018). Ceribasi et al. (2021) have assessed the magnitude and nature of trends in T_mean data over the Susurluk Basin, Turkey.

Based on the literature review, the study's novelties can be summarized as follows: (1) Analysis of trends in T_mean data over the entire CRB: the study analyzed trends in T_mean data specifically over the entire CRB. This fills a gap identified in the literature, where no reviewed study has undertaken such an analysis for this particular area. (2) Comprehensive assessment of trends in T_mean data over the entire CRB: the study evaluated four key aspects of trends in T_mean, namely magnitude, significance, nature (direction), and start and endpoints. Additionally, it examined the regional significance of these trends using the FDR test. This comprehensive approach sets it apart from the reviewed studies that typically assessed only one or two aspects of the trends. (3) Evaluation of statistical test presumptions: unlike most reviewed studies, which often overlook the assessment of assumptions required for the selection of suitable statistical tests, this study addresses this limitation by evaluating the assumption of independence of data needed for the selection of appropriate non-parametric statistical tests. This is performed through the utilization of a correlogram for the selection of suitable statistical tests. (4) Utilization of DWT for time-series decomposition: in contrast to limited existing reviewed studies that have not explored the decomposition of T_mean time-series data to identify the time scales influencing trends, this study employed DWT for the analysis of time scales and identification of time scales that have the most significant influence on the observed trends in T_mean data. These novelty statements highlight the unique contributions and methodological advancements of the study compared to the reviewed research in the field.

According to the data of the Central Water Commission (2014), the climate in the CRB is predominantly tropical and sub-tropical. In the upper reaches of Kerala and Karnataka, temperature variation is minimal. The mean monthly temperature across the basin fluctuates between 22.98 and 28.43 °C, with lower temperatures typically observed in the northern regions compared to the southern parts. During the monsoon and post-monsoon seasons, temperatures remain moderate, ranging from 25 to 27.5 °C. On average, the maximum temperature recorded in the CRB is 30.56 °C, while the mean minimum temperature stands at 20.21 °C. The Cauvery basin is predominantly affected by the South-West monsoon in Karnataka and Kerala, and the North-East monsoon in Tamil Nadu. The heaviest rainfall typically occurs in July or early August, with the average annual rainfall being approximately 1,075.23 mm.

In this study, 1̊×1̊ gridded daily maximum and minimum temperature data for the period 1970 to 2022 are obtained from the website of the India Meteorological Department (IMD) (Srivastava et al. 2009; https://www.imdpune.gov.in/lrfindex.php). The daily T_mean data at each grid point are calculated by taking the mean of the daily maximum and minimum temperature data at the respective grid. A total of 16 grids are selected for the analysis, as depicted in Figure 1. The annual T_mean time series at every grid is prepared by averaging daily T_mean data of every year of the analysis period. Annual T_mean time-series data at each grid consist of 53 data points. The monthly T_mean time series is prepared using monthly data of January to December of every year. Monthly T_mean data are prepared by averaging the daily T_mean data of the given month. In the present study, the monthly time series is not prepared for the individual months of the year. Therefore, the monthly time series consists of 636 (12 × 53) data points.

The gridded datasets developed by the IMD were validated against the observed station data before being released to the users. These datasets have undergone a comprehensive set of quality assurance procedures which were found to have a high correlation with other global gridded datasets (Sharma et al. 2016). Therefore, gridded data are used in the present study as these data are quality-checked. Additionally, there are a substantial number of missing values in the station data but in the case of gridded data, there is no gap in the data series. Therefore, gridded data are utilized in the current study.

In the present study, four aspects of the trends in T_mean data over the CRB are evaluated, viz. magnitude, significance, nature, start, and endpoint along with regional significance evaluation of trends in T_mean data over the CRB for the period 1970–2022. Additionally, the time scales that are influencing trends in T_mean data the most are determined using the decomposition technique. The presumption of independence of data is evaluated with the help of an autocorrelation plot (Kundzewicz & Robson 2000). Sen's slope (SS) test (Sharma et al. 2016) is used to determine the magnitude of the trends, while the significance was evaluated with the MK test (Khaliq et al. 2009) for independent data, and the MKBBS test (Khaliq et al. 2009) for dependent data. The nature of trends, whether MT or non-monotonic (NMT), was assessed using the ITA plot (Kale 2016). The SQMK test (Sonali & Kumar 2013) was employed to identify the start and endpoints of trends within the analysis period and the regional significance of the trends was determined using the FDR test (Verma & Kale 2018).

As the above-mentioned statistical tests are described in most of the research articles and also due to page constraints, these are not described in detail here. However, a description of the method used for the decomposition of the time series to determine the most influential time scales affecting trends is given by very few reviewed studies. In this study, DWT is used to decompose time series and it is explained in the following section.

Fourier analysis laid the foundation for development of the wavelet analysis hypothesis. While Fourier analysis decomposes a signal into smooth, continuous sinusoids of infinite duration, wavelet is the mathematical function which is capable of localizing a function in both time and space scales. Unlike Fourier transform, which only provides frequency information, wavelet transform enables the simultaneous acquisition of time, location, and frequency information of a signal (Pandey et al. 2017).

In Equation (2), wavelet coefficients are segregated into detailed (or high-frequency) coefficients (⁠⁠) at levels L = 1, 2, …, n using a high-pass filter and an approximation (or low-frequency) coefficient (⁠⁠) at level n with the help of a low-pass filter. The approximation coefficient reflects background information of the original signal, while D₁, D₂, D₃, …, and D_n comprise detailed information about the original signal, including periodicity, breaks, and jumps (Adarsh & Janga Reddy 2014).

DWT coefficients at various decomposition levels could be utilized for non-parametric trend tests. Since the time-series decomposition occurs on dyadic scales (e.g., 2, 4, 8, etc.), the time series of DWT coefficients exhibit variations on seasonal, annual, and interannual scales over various periods. This aids in identifying the predominant periodicities responsible for the trend observed in the series (Adarsh & Janga Reddy 2014).

The step-by-step procedure followed for the analysis in this study is as follows:

• (1) Preparation of annual, monthly, and seasonal (winter, pre-monsoon, monsoon, and post-monsoon) T_mean time series from the daily T_mean data for all the considered grids over the CRB corresponding to the period 1970–2022.

• (2) Application of SS test to estimate the magnitude of the trends in T_mean data over the CRB.

• (3) Checking the presumption of data independence using an autocorrelation plot, which is required to select a suitable statistical test.

• (4) Application of the MK test, if the data are serially uncorrelated and the MKBBS test, if the data are serially correlated, to assess the statistical significance of the trend.

• (5) Assessment of the nature of the trend, i.e., whether the trend is MT or NMT by employing an ITA plot.

• (6) Detecting start and endpoints of trend within the considered analysis period, i.e., 1970–2022 by employing the SQMK test. The start of the trend is indicated by the first intersection point of the progressive and retrograde series, whereas the last intersection point indicates the end of the trend. Intermediate intersection points show the progress of the trend (Kale 2016).

• (7) Evaluation of the regional significance of trends using the FDR test.

• (8) Selecting an appropriate wavelet function and determining the number of decomposition levels.

• (9) Decomposition of all the time series, viz. annual, monthly, and seasonal into a set of approximation and detailed coefficients using a suitable algorithm. The original signal is then divided into several low-resolution coefficients by repeating the decomposition procedure with successive approximations.

• (10) Reconstruction of the signal utilizing appropriate combinations of detailed and approximation series.

• (11) Determination of z-values from the MK/MKBBS test for various time series of wavelet combinations to identify the combination that yields a z-value closest to that of the original time series.

• (12) Compare the progressive sequential values from the SQMK test for the original time series with those from each wavelet combination time series. Calculate the root mean square error (RMSE) and coefficient of determination (⁠⁠) for each wavelet combination, then identify the combination with the lowest RMSE and highest values.

• (13) Plot the progressive sequential values obtained from the SQMK test for various time series of wavelet combinations against the original time series. Determine which wavelet combination time series matches best with the original series.

• (14) Determine which periodicity is the most prominent using the criteria outlined in steps (11)–(13).

• (15) The final step is the interpretation and presentation of the results.

The gridded daily T_mean data over the CRB for 1970–2022 is used to prepare annual, monthly, and seasonal (winter, pre-monsoon, monsoon, and post-monsoon) time series. The methodology shown in Figure 2 is applied for each time series separately.

Figure 6 illustrates the spatial variation in the magnitude of trends across the entire CRB. Specifically, in the gridded annual T_mean data, a decrement in the magnitude of trends is observed from the upper to lower portion of the basin. Gridded monthly T_mean data indicated a lower magnitude of trends compared to gridded T_mean data of other temporal scales with relatively consistent SS values across all grids in the CRB. Notably, during the winter season, the magnitude of trends is maximal throughout the basin except for some southern portions. Conversely, the gridded pre-monsoon T_mean data displayed a lower magnitude of trends. During the monsoon season, the magnitude of trends varies across the basin, with higher values in most of the upper portion and lower values in the lower portion. Gridded post-monsoon T_mean data exhibited the highest trend magnitudes over the entire basin. From Figure 7, it is evident that winter and post-monsoon seasons exhibited the highest magnitude of trends, while gridded monthly T_mean data displayed the least magnitude of trends compared to other gridded data.

The trend's nature, whether MT or NMT, is determined through ITA plots prepared for each grid corresponding to all timescales. The results indicated that the trends are predominantly MT in nature across all grids and temporal scales with the exception of grid 3 for the pre-monsoon season.

The start and endpoints of the trends in T_mean data are identified through the utilization of SQMK plots. Progressive (u(t)) and retrograde (u'(t)) values of the data are plotted to locate the intersection points. The initial intersection point signifies the commencement of the trend, while the final intersection point indicates the end of the trend within the specified period (Kale 2016). It is observed from the results that trends in gridded T_mean data started in the 1970s at most of the grids for each temporal scale. In the majority of cases, endpoints are not found in the considered analysis period and these might have occurred after the analysis period.

Regional significance is assessed using the FDR test, which evaluated the significance of the trends in T_mean data for the basin as a whole corresponding to each temporal scale. The analysis revealed that trends are regionally significant only for winter and monsoon seasons. Even though gridded annual T_mean data had shown the highest number of grids having significant trends, regional significance is not observed for the given data. Interestingly, during the winter season, where only six grids exhibited significant trends had shown regionally significant trends.

After the decomposition of the original time series, z-values for the original and each wavelet combination time series are determined using the MK test (if data are independent) or the MKBBS test (if data are dependent). The z-values obtained for annual and monthly T_mean time series are given in Tables 4 and 5, respectively. The progressive sequential values from the SQMK test applied to the original series and each wavelet combination time series are determined. The RMSE and values for all wavelet combination time series relative to the original time series are also computed and these values are presented in Tables 4 and 5. The results like z-values (Tables 4 and 5) are then compared with the corresponding original time series.

Table 4

The RMSE, _, and z-values for the original series and each wavelet combination time series corresponding to gridded annual T_mean data

View Large

Table 5

The RMSE, _, and z-values for the original series and each wavelet combination time series corresponding to gridded monthly T_mean data

View Large

The wavelet combination time series which has the nearest z-value, least RMSE, and highest value corresponding to the original series should be considered as the most influencing timescale. In ambiguous circumstances, the values of RMSE, are prioritized over z-values. In some cases, the highest and lowest RMSE values are found in different timescales. Then, progressive sequential values obtained from the SQMK test are referred to decide the most influential timescale. The wavelet combinations found to be the most influential on the corresponding trend in T_mean are identified based on aforesaid criteria for each grid and these wavelet combinations are shown with blue colour fill in Tables 4 and 5.

The original time series is decomposed into detailed and approximation coefficients with the help of DWT to identify the timescales that influenced the trends most. The z-value, RMSE and values are determined for each wavelet combination time series corresponding to the original series at all grids. Also, plots of progressive sequential values obtained by the SQMK test for various wavelet combination time series are prepared along with that for the corresponding original time series. It can be observed from the results that closeness with the original time series is not shown by individual detailed series. However, in all instances, the combination of detailed and approximation series is closest to the original time series. Hence, it is found that trend components are mainly carried by the approximation component. In the case of the annual series, the d1 + a3 component (2-year periodicity) is identified as the most influential component on the corresponding trends (except at grids 4 and 7). It indicated that d2 (4-year) and d3 (8-year) components have not contributed much to the trends. For the monthly series, for 8 out of 16 grids, the d2 + a7 (4-month) component is observed to be the most contributing and for the remaining eight grids it was d3 + a7 (8-month) component. Similarly, for seasonal series, the d1 + a3 (2-year) component is the most contributing. During all the seasons, at each grid, the 2-year component is the most contributing except at grids 2, 4, 7, and 11 during pre-monsoon season. The most influencing timescales for trends in gridded T_mean data corresponding to each temporal scale are given in Table 7. In Table 7, ‘Y’ and ‘M’ indicate year and month, respectively.

The warming trends in T_mean observed over the CRB between 1970 and 2022 are in line with results from other regions. For instance, Mondal et al. (2015), who have also analyzed T_mean in addition to temperature and precipitation extremes across India, reported significant increases in mean and extreme temperatures, similar to the trends identified in this study. Furthermore, Ray et al. (2019) analyzed trends in mean and extreme temperatures over parts of India, emphasizing the rising temperature trends, which are consistent with the significant warming seen over the CRB. Kale (2016) also analyzed the trends in regional T_mean data over the Tapi Basin, Gujarat, and India, and obtained the increasing trends for winter and annual temporal scales.

On a global scale, Javanshiri et al. (2021) documented rising temperature extremes and T_mean in Iran, which aligns with the increasing temperature patterns and periodic fluctuations found in this analysis. Additionally, Sanderson et al. (2017) linked the recent increase in global temperatures to a rise in the intensity and frequency of heatwaves, which corroborates the warming trends observed in the Cauvery River Basin (CRB) as part of a broader global phenomenon. These trends also align with Aksu (2021), who noted similar temperature increases and seasonal shifts across Turkey. Therefore, the results of this study are in agreement with both regional and global findings,

In this study, four aspects of trend, viz. magnitude, significance, nature, start, and end of the trend are evaluated along with regional significance evaluation of the trends for T_mean data corresponding to the period of 53 years for six temporal scales, viz. annual, monthly, and seasonal (winter, pre-monsoon, monsoon, and post-monsoon) over the CRB using various statistical tests. Furthermore, in conjunction with the MK/MKBBS test and SQMK test, DWT is applied to decompose the time series to determine the dominant periodicities affecting trends in T_mean data over the CRB corresponding to each temporal scale.

The following conclusions are derived from the study:

• Consistent warming trends are observed in gridded T_mean data over the basin for all the temporal scales with positive SS values.

• On average, half of the considered grids over the basin corresponding to various temporal scales have shown significant trends among which annual and monsoon gridded T_mean data exhibited a maximum number of grids (10) having significant increasing trends.

• All gridded datasets corresponding to each temporal scale have shown MT trends throughout the basin with the exception of the pre-monsoon T_mean time series at grid 3.

• The start of trends was observed in the 1970s for the majority of trends corresponding to all temporal scales and similar results were also shown by Safari (2012).

• Only winter and monsoon seasons have shown regionally significant trends over the CRB for the considered analysis period. Hence, it is necessary to attribute the CC detected in gridded T_mean data over the CRB corresponding to winter and monsoon temporal scales.

• The trends in T_mean data are primarily influenced by periodic patterns lasting under ten years, or, generally, 2 and 4 years for annual and seasonal scales and 4 and 8 months for monthly temporal scales, according to the results obtained from the decomposition of time series. Short-term periodicities can be linked to El Niño-Southern Oscillation (ENSO) and North Atlantic Oscillation (NAO) teleconnections (Burić et al. 2024).

The present study is helpful for hydrologists and environmental engineers to understand temperature trends and dominant periodicities dominating the trends in order to efficiently manage the region's water resources. This methodology can be used anywhere in the world to detect the CC in terms of significant trends and identify dominant periodicities influencing the significant trends for any meteorological data. This study can be extended to include attribution of the regionally significant trends in future.

The authors are thankful to the India Meteorological Department, Pune for providing the data required in the present study. We express our sincere thanks to the editor-in-chief and anonymous reviewers for their precious time and for providing us with useful suggestions and constructive comments.

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

M.C.H. contributed to data acquisition, trend analysis of T_mean data, manuscript writing, and submission. G.D.K. contributed to conceptualization, supervision, and editing.

All authors have read, understood, and have complied as applicable with the statement on ‘Ethical responsibilities of Authors’ as found in the Instructions for Authors.

All relevant data are available from an online repository or repositories.

The authors declare there is no conflict.