Coincidence flood risk due to the simultaneous flood occurrences on both mainstream and its tributary results in downstream inundation of a confluence. Therefore, this study was taken up for coincidence flood risk analysis of Mahanadi River basin considering the annual maximum (AM) and the peak over threshold (POT) series. In this study, the Mann–Kendall trend test was performed to analyze the trend in flood magnitudes, while circular statistics was used to analyze the persistence in flood timing. The joint distributions between the streams were established using bivariate copula functions considering flood magnitudes and occurrence dates as variables. The results of MK test revealed a mixture of significant and insignificant trends for AM series for the selected stations, while the trends were insignificant for POT series. Additionally, the flood occurrence dates showed a high level of persistence. It is evident from the results that the coincidence risk is high for Seorinarayan–Bamnidhi confluence point with a risk value of 7.63 × 10−3. The coincidence risk increases mostly from late July to mid-September, and the coincidence risk is high for more frequently occurring flood events. The obtained results will help in prioritizing flood hazard zones for effective flood mitigation strategies in Mahanadi River basin.

  • The mean flood dates for both the annual maximum (AM) and peak over the threshold (POT) series tended to concentrate around mid-August.

  • Majority of the stations showed a delay in the occurrence timing for both the AM and POT series, possibly influenced by changing rainfall patterns and water control structures.

  • Coincidence risk was high in the monsoon season, highlighting vulnerability during monsoon months.

Floods result from a complex interaction among hydrological processes, climate dynamics, and human interventions, with the significance of each factor varying across different types of floods and geographical regions. Climate change causes alterations in the frequency and intensity of precipitation extremes (Reddy & Ray 2024a), and river flow patterns (Zaghloul et al. 2022) and causes subsequent floods. This, in turn, amplifies the vulnerability of areas prone to flooding (Dankers et al. 2014; Arnell & Gosling 2016). As a result, flash floods and large river floods are the prevalent extreme events (Ashley & Ashley 2008; Di Baldassarre et al. 2010), that present the highest risk of flooding to the general population. The floods can vary temporally as well as spatially, which in turn makes it riskier for the people. The analysis of temporal variability in floods provides valuable insights into long-term trends, and flood occurrence patterns at a local scale. The analysis of any trends existing in flood magnitudes helps to identify whether there is any long-term increase or decrease in the intensity of flooding over time and is essential for assessing the severity of future flood events. In addition to this, analyzing the persistence of flood occurrence can help in identifying the seasons when the floods are more likely to happen thereby allowing for early preparedness of the population. Various works of literature are available on temporal variations of flood intensity and consistency in its timing at watershed scales (Panda et al. 2013; Bawden et al. 2014; Jena et al. 2014), assessment of trends, (Hall et al. 2014; Matti et al. 2017; Mangini et al. 2018), unexpected shifts in floods (Villarini et al. 2009; Nka et al. 2015) and seasonal variations in flood regimes (Cunderlik & Ouarda 2009; Burn et al. 2010, 2016; Burn & Whitfield 2018). However, the coincidence analysis of trends in floods and persistence in their occurrence will help in identifying potential hotspots, thereby providing information for planning any risk management measures, cascade reservoir operations (Reddy & Nagesh Kumar 2007) and also aids in flood risk analysis.

River floods occur when the water level in a river or a stream exceeds its normal capacity and overflows to the surrounding area. One unique type of river flooding is confluence point flooding where flooding occurs at the meeting point of two or more rivers. In such case, evaluating a flood event at a specific location alone may not be sufficient (Tosunoglu & Singh 2018) because the peak discharge can be the result of a combination of discharge from the mainstream as well as tributaries located upstream (Favre et al. 2004). This simultaneous occurrence of floods leads to large floods as the flood peak and volume get superimposed, causing damage to the downstream area (Chen et al. 2012; Wang 2016). Hence, it is important to study the coincidence of floods in the mainstream and its tributaries to propose any flood mitigation strategies in downstream. Recently, copula models have been incorporated in the flood coincidence analysis (Jianping et al. 2018; Muthuvel & Mahesha 2021). For instance, Favre et al. (2004) analyzed the coincidence risk of flooding of a mainstream and its tributary using copula function and estimated the conditional probability of the volume for a specific flow value. Bender et al. (2016) and Schulte & Schumann (2016) evaluated the flood occurrence at the confluence by introducing a multivariate copula function. Peng et al. (2017) used the Monte Carlo method of Copula to estimate the flood risk at the confluence and to control the flood downstream of a reservoir. Gilja et al. (2018) used bivariate copula to analyze flood hazards at river confluences. However, all of this research considered only flood peak magnitudes and ignored the occurrence time of the floods. The flood coincidence in different rivers is significant only if the occurrence time of the flood in both the rivers is in close proximity and the flood magnitudes exceed the design flood value. Hence, the inclusion of occurrence time in addition to flood magnitude appears to be crucial when assessing flood coincidence. This provides valuable insights into the temporal synchronicity of flood events across different rivers. Therefore, coincidence flood risk analysis is performed in this analysis for the Mahanadi river basin (MRB) of India.

MRB is acknowledged as one of the climatologically vulnerable regions of India that experiences frequent floods and cyclones due to its proximity to the Bay of Bengal. As a hotspot for tropical cyclone genesis, the Bay of Bengal frequently produces cyclones that make landfall along India's eastern coast, including the MRB (Rajeev & Mishra 2022). Over the past century, the sea level in the Bay of Bengal has been steadily rising (Ghosh et al. 2017). Concurrently, there has been an increase in the Bay of Bengal sea surface temperatures (Sheehan et al. 2023). This dual phenomenon contributes to the intensification of extreme weather events such as heavy precipitation and cyclones, leading to heightened flooding and erosion along coastal areas. This increases the basin's exposure to cyclonic storms, resulting in heavy rainfall and strong winds with subsequent intense flooding (Rajeev & Mishra 2023; Reddy & Ray 2024b). Additionally, the Bay's shallow bathymetry and coastal terrain amplify cyclone impacts. The flat topography and close proximity to the coast make the MRB more susceptible to inundation and flooding from storm surges and heavy rainfall. The basin has experienced numerous catastrophic floods in its history, including significant inundation events in 1855, 1933, 1937, 1955, 1980, 1982, 2003, 2008, and 2011 (Alam & Muzzammil 2012). Recent floods in the MRB show a significant increasing trend and were caused due to a surge in extreme precipitation (Jena et al. 2014). Climate models project a rise in the likelihood of moderate to heavy precipitation events, accompanied by a decrease in lower precipitation occurrences (Pattanayak et al. 2017; Wasson 2018). Pandey et al. (2022) predict that a substantial rise in streamflow is expected to occur which may lead to floods in the long future in MRB. However, to the best of the author's knowledge, the combined analysis of the trend in flood magnitude and persistence in its timing has not been done so far in the MRB. In addition to that, most of the works have used annual maximum (AM) flood events for such analysis. The use of the Peak Over Threshold (POT) method can give better coincidence risk values as it gives more events even in a single year and even low-range floods can be taken into analysis. In this sense, the present study tries to answer the following objectives: (i) To examine the trends in peak flood values and the persistence in the timing of flood occurrences. (ii) To analyze the coincidence flood risk and quantify the joint probabilities of flooding at various confluence points of the mainstream and its tributaries in the MRB using copula functions. Such information can be essential for effective flood risk assessment, floodplain management, and the design of mitigation measures to reduce the impacts of coincident flood events on communities and infrastructure.

Study area

The present study is carried out in the MRB (Figure 1). It extends between the latitudes of 19°8′ & 23°32′ N and the longitudes of 80°23′ & 86°43′ E. The origin of the Mahanadi River is in the Dhamtari district of Chhattisgarh and drains into the Bay of Bengal. The drainage area of this river basin is 141,589 km2 and covers Chhattisgarh, Odisha and smaller parts of Jharkhand, Maharashtra and Madhya Pradesh. The river stretches approximately 851 km from its origin to its outfall into the Bay of Bengal, with 357 km located in Chhattisgarh and the remaining 494 km in Odisha. The annual average precipitation of this basin varies between 1,200 and 1,600 mm and receives most of the precipitation in the monsoon season. During winter, the mean daily temperature ranges from 13 to 20 °C and in summer, the mean daily temperature ranges from 30 to 37 °C.
Figure 1

Location map of the MRB.

Figure 1

Location map of the MRB.

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Data used

This study is conducted using four Gauge and Discharge (G&D) sites namely, Simga, Jondhra, Seorinarayan, and Basantpur from mainstream and four other G&D sites namely, Andhiyarkhore, Rajim, Bamnidhi, and Kurubhata from different tributaries of Mahanadi River. The daily stream flow data for the selected G&D sites are collected from India-WRIS official website (https://indiawris.gov.in/wris/). The study period varies for different confluence points depending on the availability of observed data (Supplementary material, Table S1). The study period spans from 1977 to 2018 for Andhiyar Khore and Simga, from 1978 to 2018 for Basantpur and Kurubhata, from 1979 to 2018 for Jondhra and Rajim, and from 1985 to 2018 for Bamnidhi and Seorinarayan.

The methodology outlined for this study is shown in Figure 2. The present study is conducted in two approaches, i.e., (1) AM approach and (2) POT approach. The trends are detected using Mann–Kendall (M–K) test. The persistence analysis in flood occurrence dates is carried out using circular statistics. The joint probability of flood magnitude and its occurrence dates for mainstream and tributary is performed using Copula. The coincidence risks are then calculated for flood magnitudes and occurrence dates. This risk estimation provides insights into the joint probability of floods of specific magnitudes occurring during particular time periods at a specific confluence point.
Figure 2

Detailed methodology of the proposed study.

Figure 2

Detailed methodology of the proposed study.

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Extraction of AM series and POT series

The AM and POT time series are extracted from the daily stream flow series at each G&D station. The threshold for POT series is chosen as 98.5th percentile (on an average of three events per year) arbitrarily (Svensson et al. 2005; Ganguli et al. 2023). The smaller events in the successive events which occur within a fixed time span are neglected and the higher ones are selected to ensure the events are independent and identically distributed (iid) (Ganguli et al. 2020). The fixed time span is 5 days for catchment areas less than 45,000 km2 and 10 days for the catchment areas between 45,000 and 100,000 km2, respectively.

Trend and persistence analysis

The trends existing in flood magnitudes are examined using MK trend test at a 5% significance level. The persistence in flood occurrence was also analyzed using circular statistics, where the data will be represented over a circle of unit radius.

MK test

Analyzing the patterns of flood events over a time period can help to assess the flood risk locally and facilitate the development of mitigation strategies. Here, in this study, the existence of a trend in peak values is detected using non-parametric MK test (Mann 1945; Helsel et al. 2020). The null hypothesis assumes no existence of a trend in the series at a 5% significance level while the existence of a trend is the alternate hypothesis. The quantitative change in trend is measured using Theil–Sen slope estimator (Sen 1968; Theil 1992). More details of the MK and Theil–Sen slope estimator tests can be found in various literature (Yadav et al. 2014; Ray et al. 2019; Ray & Goel 2021; Zaghloul et al. 2022; Shawky et al. 2023).

Persistence in flood occurrence timing

The temporal variation in the occurrence of flood events can be assessed by the persistence in flood dates. This helps in knowing how well the events are clustered over a particular time period and can prioritize the peak time of flood risk. For this study, the continuity in flood occurrence data is assessed using circular statistics, which is the most used method to describe periodic directional data (Burn & Whitfield 2018). Here, the date (in the form of Julian date) is taken as a directional variable and converted to angular value (λi) using Equation (1).
(1)
JDi indicates Julian date in the form of 1 for Jan 1 to 365 or 366 for Dec 31 for a non-leap year or a leap year and l(yr)– length of the year. Then the x-coordinate () and y-coordinate () of the mean event date are calculated using Equation (2), and the angular mean date () of the flood in a year is found using Equation (3).
(2)
where qi is the flow value of each year.
(3)
The mean flood date (δ) is calculated using the Equation (4).
(4)
The persistence in flood occurrence is determined using the mean resultant length () using Equation (5).
(5)

The value of as 0 indicates no persistence in flood timing while the value of 1 indicates high persistence i.e., all floods occur on the same day of the year.

The temporal variability in peak discharge events can be assessed by calculating circular variance (σ2) given by Equation (6).
(6)
The adjusted Theil–Sen slope estimator is used to estimate the trend in flood timing (β) by finding the median of the difference in flood dates for all combinations of years (i and j) using Equations (7)–(9).
(7)
(8)
(9)
where is the average number of days per year.

Joint distribution using copula

The copula function plays a significant role in establishing joint distribution between two correlated random variables as it eliminates the assumption of two random variables following the same marginal distribution. The copula function helps in formulating multivariate distributions by incorporating a correlation between the random variables with the given marginal distribution (Nelson 1999). If the marginal distributions of n random variables X1, X2, … , Xn can be represented as , then the joint multivariate distribution F(x1, x2, … , xn)can be given as
(10)
where Cθ() is a copula function and θ is the parameter of copula function.

In the present study, copulas from Archimedean copula (Gumbel, Frank and Clayton copulas) and Elliptical copula (Gaussian and t copulas) families are used which are the most commonly used copula families in hydrology (Nelson 1999). Supplementary material, Figure S1 shows the pictorial representation of copula. Supplementary material, Table S2 shows the copula functions used in this study and their parameter ranges.

The best-fitted copula is selected based on Akaike Information Criteria (AIC) values (Akaike 1974). The AIC of each copula is calculated from Equation (11). AIC is a statistical tool to compare different models and find the best-fitting one based on the maximum likelihood method. It tries to strike a balance between the goodness of fit (Log likelihood of copula function (Cθ)) and the complexity of the model (K refers to the Number of parameters) and the best-fitting model is the one with the lowest AIC value.
(11)

Marginal distribution of flood peak and its occurrence dates

In order to get joint distribution using the copula function, a suitable marginal distribution has to be selected for the flood peak magnitudes and dates. Various parametric distributions namely Gamma, Gumbel, Lognormal, Pearson type –III, and General extreme value (GEV) distributions are used in the study for flood peak values of AM series. Generalized Pareto distribution is used for the POT series. As the flood occurrence dates are periodic in nature, they can be represented as directional variables using circular statistics in which Von Mises is the most fitted distribution for angles or directions, time, etc. A mixture of Von Mises distribution is used for the flood occurrence dates in the AM and POT series. In general, mixed von Mises distribution combines multiple von Mises distributions having its own mean direction and concentration parameter based on the proportion of each component to be in the overall distribution. Supplementary material, Table S3 represents the probability density functions and parameters of the selected marginal distributions used in the study.

In the present study, three goodness of fit tests namely Kolmogorov–Smirnov test (KS), Chi-squared (χ2) test and L-Moment plots are used to assess the performance of fitted marginal distributions. The detailed information on these tests can be seen in Peel et al. (2001) and Ramachandran & Tsokos (2015). The KS and χ2 tests are analyzed at the 5% significance level. If the obtained p-value is lesser than the significance level, then the null hypothesis is rejected. If the p-value is greater than the significance level, the null hypothesis is not rejected. The selection of the best distribution is based on the consensus of majority of the best-fitted distributions of the three tests.

Estimation of coincidence risk of floods

For analysis of coincidence events, establishing joint distribution is essential for assessing the probability of flood events occurring together. Here, the joint distributions of flood peak magnitudes for each confluence point are computed using copula functions. The exceedance coincidence risk of flood magnitudes () in mainstream and its tributary surpassing the design flood value for different return periods is estimated using Equation (12)
(12)
where and refer to the peak discharges of design floods at different return periods (Po). QM and QT are flood peaks of mainstream and tributary, respectively.
Similarly, the coincidence risk of flood date occurring on the same day on any ith day is given by Equation (13)
(13)
where TM and TT represent the occurrence dates of floods in mainstream and tributary, respectively.
The coincidence risk of flood occurrence dates (PT) is given by the Equation (14):
(14)
The coincidence of flood magnitude exceeding the design flood value in mainstream and tributary becomes insignificant if it occurs with a large occurrence interval. Thus, the coincidence risk of dates must be considered for the analysis of the coincidence of flood magnitude. The coincidence risk of floods (Prisk) is calculated using Equation (15), assuming that flood occurrence dates and flood magnitudes are independent.
(15)

Trend in peak flow and persistence in timing

The existence of trends in the peak flow series for all eight stations was obtained by performing non-parametric MK test at a 5% significance level. Figure 3(a) and 3(b) indicated the trends in the AM and POT series for all eight stations, respectively. For AM series, Kurubhata (0.09 cumecs/year) and Seorinarayan (0.812 cumecs/year) showed a significant increasing trend and Bamnidhi (−0.67 cumecs/year) showed a significant decreasing trend. From Figure 3(b), it is evident that none of the stations showed a statistically significant trend for the POT series. The observed variation in trend over the basin can potentially be due to changing rainfall patterns, as streamflow and rainfall are positively correlated (Panda et al. 2013). Furthermore, the decreasing trend may be influenced by the construction of water control structures like weirs and barrages.
Figure 3

Trend in flood magnitude exhibited by the stations for (a) AM and (b) POT series (at 5% significance level).

Figure 3

Trend in flood magnitude exhibited by the stations for (a) AM and (b) POT series (at 5% significance level).

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Circular statistics were employed to examine the persistence of the flood occurrence dates. The analysis involved evaluating the mean flood date, persistence, and temporal variability of each station in both the AM and POT series. Figure 4 indicates the persistence in flood timing of mean flood date in AM and POT series. It is evident from Figure 4 that the mean flood dates of all the stations tend to occur in the middle of August for both the AM and POT series. Figure 5(a) shows the mean flood dates of each station and Figure 5(b) shows the temporal variability of each station. The tendency of concentration of mean flood dates occurring around the middle of August was high in both the AM and POT series. Hence, it can be inferred that the flood occurrence is highly persistent at the selected stations, which means every year the flood occurs consistently in the same period. Additionally, Andhiyarkhore, Bamnidhi and Kurubhata experience high temporal variability in the case of the POT series while the remaining stations show high temporal variability for the AM series. The flow in the streams of MRB is mostly occurring in the monsoon period as the basin experiences monsoonal rainfall. This result agrees with the work done by (Burn & Whitfield 2018) across Canada and the northern US and concludes that the flow regime due to precipitation shows high persistence in peak discharge timing.
Figure 4

Persistence in flood timing of AM and POT series.

Figure 4

Persistence in flood timing of AM and POT series.

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Figure 5

Bar plots showing (a) mean flood dates and (b) temporal variability in flood dates.

Figure 5

Bar plots showing (a) mean flood dates and (b) temporal variability in flood dates.

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To assess any shifts in the timing of the flood events at each station, the adjusted Theil–Sen estimator was used. The estimated shifts in mean dates of all eight stations are shown in Figure 6. In Figure 6, the pink-colored square box indicates an early occurrence of flood events (negative values) while the green square box indicates a delayed occurrence of flood events. The grey color box represents a station that did not have any shift in the occurrence. Out of eight stations, Andhiyarkhore, Rajim, Seorinarayan, Bamnidhi, Simga and Kurubhata showed a delayed flood occurrence and Jondhra showed an earlier occurrence for both AM and POT series. In contrast, Basantpur did not exhibit any shift in flood occurrence for both the AM and POT series. Simga showed an early occurrence in the POT series, in contrast to a delayed occurrence in the AM series. Overall, the majority of the stations showed a delay in the occurrence timing for both the sampling methods. This can likely be attributed to the change in rainfall patterns and the construction of numerous water structures in recent decades.
Figure 6

Shifting in flood timing expressed in days per decade for AM and POT series.

Figure 6

Shifting in flood timing expressed in days per decade for AM and POT series.

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Table 1 shows the combined result of either increasing or decreasing trends in peak flows with delayed or early occurrence of floods for each station. On analyzing the coincidence of the trend in flood peak magnitude and shift in its timing, for AM series, Andhiyarkhore, Kurubhata, and Seorinarayan stations showed an increasing trend with a delay in occurrence. On the other hand, Bamnidhi, Rajim, and Simga showed a decreasing trend with delayed occurrence. From the position of these stations, many weirs and barrages are constructed upstream of these stations and this can be the reason for the decreasing trend. Jondhra showed a decreasing trend with early occurrence. For the POT series, Simga transitioned from a decreasing trend with delayed occurrence to early occurrence. Andhiyarkhore showed a decreasing trend with delayed occurrence Bamnidhi showed an increasing trend with delayed occurrence and the remaining stations exhibited the same results as in the AM series. The more vulnerable sites are those with an early occurrence of flood as the population may not be adequately prepared for such events, leading to more severe damages.

Table 1

Combined result of trend in flood peak and shift in flood occurrence

StationPOT
POT
trendShift (days per decade)trendShift (days per decade)
Andhiyar khore Increase (2.38 cumecs/year) Delay (1) Decrease (−0.18 cumecs/year) Delay (2) 
Bamnidhi Decreasea(−0.67 cumecs /yearDelay (7) Increase (0.62 cumecs/year) Delay (7) 
Basantpur Decrease (−0.01 cumecs/year) No shift (0) Increase (0.55 cumecs/year) No shift (0) 
Jondhra Decrease (−0.32 cumecs/year) Early (−3) Decrease (−0.06 cumecs/year) Early (−1) 
Kurubhata Increasea (0.09 cumecs/yearDelay (1) Increase (0.29 cumecs/year) Delay (3) 
Rajim Decrease (−0.54 cumecs/year) Delay (6) Decrease (−0.06 cumecs/year) Delay (1) 
Seorinarayan Increasea (0.812 cumecs/yearDelay (8) Increase (0.19 cumecs/year) Delay (2) 
Simga Decrease (−0.482 cumecs/year) Delay (5) Decrease (−0.07 cumecs/year) Early (−1) 
StationPOT
POT
trendShift (days per decade)trendShift (days per decade)
Andhiyar khore Increase (2.38 cumecs/year) Delay (1) Decrease (−0.18 cumecs/year) Delay (2) 
Bamnidhi Decreasea(−0.67 cumecs /yearDelay (7) Increase (0.62 cumecs/year) Delay (7) 
Basantpur Decrease (−0.01 cumecs/year) No shift (0) Increase (0.55 cumecs/year) No shift (0) 
Jondhra Decrease (−0.32 cumecs/year) Early (−3) Decrease (−0.06 cumecs/year) Early (−1) 
Kurubhata Increasea (0.09 cumecs/yearDelay (1) Increase (0.29 cumecs/year) Delay (3) 
Rajim Decrease (−0.54 cumecs/year) Delay (6) Decrease (−0.06 cumecs/year) Delay (1) 
Seorinarayan Increasea (0.812 cumecs/yearDelay (8) Increase (0.19 cumecs/year) Delay (2) 
Simga Decrease (−0.482 cumecs/year) Delay (5) Decrease (−0.07 cumecs/year) Early (−1) 

aSignificant increase or decrease.

Coincidence risk analysis

Marginal distribution

In the present study, Pearson type-III, GEV, Lognormal, Gumbel and Gamma distributions were used to fit the flood peak values of the AM series, and General Pareto distribution was used for the POT series. On the other hand, the Mixed Von Mises distribution was used to fit the flood occurrence dates from AM and POT series. The parameters of the Von Mises distribution were estimated using the Maximum Likelihood method while for other distributions, the parameters were estimated using L-Moment method. Goodness of Fit tests namely KS test and Chi-square test and L-moment plots were used to identify the best-fitting distribution of flood peak values. The statistical results of KS test and Chi-square test are shown in Supplementary material, Table S4. Figure 7, indicates the L-moment ratio plots at each station. The selection of the best-fitted distribution was based on the consensus of the majority of these three tests. Supplementary material, Table S5 shows the selected best-fitting distribution at each station. Both Pearson type-III and GEV distributions were identified as best-fitting distributions for the flow values of the AM series.
Figure 7

L-moment ratio plots of each station.

Figure 7

L-moment ratio plots of each station.

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The CDF curves of empirical distribution and best-fitted distribution for flood peaks and its occurrence dates of AM series for confluence point 1 are shown in Figures 8 and 9, respectively. The results show that the selected distributions fit the AM flood peak values reasonably well. Similarly, the CDF curves between empirical distribution and best-fitted distribution for flood peak values and occurrence dates of POT series for confluence point 1 are shown in Figures 10 and 11, respectively.
Figure 8

CDF plots between fitted distribution and empirical CDF of AM flood peaks for confluence point 1 (Andhiyarkhore and Simga).

Figure 8

CDF plots between fitted distribution and empirical CDF of AM flood peaks for confluence point 1 (Andhiyarkhore and Simga).

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Figure 9

CDF plot of Mixed Von Mises distribution fitted to occurrence dates for confluence point 1 (Andhiyarkhore and Simga).

Figure 9

CDF plot of Mixed Von Mises distribution fitted to occurrence dates for confluence point 1 (Andhiyarkhore and Simga).

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Figure 10

CDF plots between fitted distribution and empirical CDF of POT flood peaks for confluence point 1 (Andhiyarkhore and Simga).

Figure 10

CDF plots between fitted distribution and empirical CDF of POT flood peaks for confluence point 1 (Andhiyarkhore and Simga).

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Figure 11

CDF plot of Mixed Von Mises distribution fitted to occurrence dates of POT series for confluence point 1 (Andhiyarkhore and Simga).

Figure 11

CDF plot of Mixed Von Mises distribution fitted to occurrence dates of POT series for confluence point 1 (Andhiyarkhore and Simga).

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Joint distribution

The joint distribution helps in understanding the joint behavior of multiple random variables. Thus, copula functions can be used to establish joint distribution as it describes the dependence structure between the variables with the help of Kendall's Tau. Here, Kendall's Tau values were estimated between the mainstream and the tributary for flood magnitudes and occurrence dates. Table 2 shows the estimated Kendall Tau values from both the Am and POT series for both flood peak and occurrence dates. For AM series, except for the flood magnitudes of the Basantpur–Kurubhata point, the remaining confluence points showed a positive correlation. In the case of the POT series, Baasantpur–Kurubhata point showed a negative correlation, Andhiyarkhore–Simga and Bamnidhi–Seorinarayan points showed a positive correlation for flood peak while the correlation was negative for occurrence dates and Jondhra–Rajim point, the flood peak showed negative correlation and occurrence dates showed positive correlation.

Table 2

Kendall τ values for the flood peak value and its occurrence dates of AM and POT series

Andhiyar khore & Simga
Bamnidhi & Seorinarayan
Basantpur & Kurubhata
Jondhra & Rajim
AMPOTAMPOTAMPOTAMPOT
Flood peak 0.117 0.102 0.134 0.209 −0.007 −0.078 0.136 −0.226 
Occurrence date 0.343 −0.025 0.273 −0.051 0.269 −0.024 0.173 0.124 
Andhiyar khore & Simga
Bamnidhi & Seorinarayan
Basantpur & Kurubhata
Jondhra & Rajim
AMPOTAMPOTAMPOTAMPOT
Flood peak 0.117 0.102 0.134 0.209 −0.007 −0.078 0.136 −0.226 
Occurrence date 0.343 −0.025 0.273 −0.051 0.269 −0.024 0.173 0.124 

Three copulas from Archimedean family (Clayton, Gumbel and Frank) and two copulas from elliptical family (Gaussian and Student's t) were used to formulate bivariate joint probability distributions. The parameters of these copulas were calculated using maximum likelihood estimation method. The most appropriate copula for each confluence point was selected based on AIC (Akaike Information Criterion) values. Supplementary material (Tables S6 and S7) shows the AIC values calculated for each copula at each confluence point. It can be seen that the best-fitting copula varies for both flood magnitude and occurrence dates at each confluence. Figures 12 and 13 depict the plots between the joint CDF of the best-fitting copulas and the empirical joint CDF for AM series and POT series, respectively. These plots provide a visual comparison between the fitted copulas and the observed data. The surface plots of joint distribution obtained through copula for AM and POT series of Andhiyarkhore–Simga confluence point (confluence point 1) are represented in Figure 14, which gives a graphical representation of the joint behavior of the variables.
Figure 12

Joint CDF of flood peak for AM series.

Figure 12

Joint CDF of flood peak for AM series.

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Figure 13

Joint CDF of flood peak for POT series.

Figure 13

Joint CDF of flood peak for POT series.

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Figure 14

Joint distribution of confluence point 1 for flood magnitude of both AM and POT series.

Figure 14

Joint distribution of confluence point 1 for flood magnitude of both AM and POT series.

Close modal

Coincidence risk analysis

The coincidence risk of flood occurrence dates was calculated for the period from May 15 to November 31 using Equation (12). Figures 15 and 16 represent the resulting coincidence plots of flood occurrence dates for AM and POT series, respectively. The coincidence risk of flood occurring dates is high around the mid of September for Andhiyarkhore–Simga, Bamnidhi–Seorinarayan and Basantpur–Kurubhata confluence points while it is high around the start of the August month for Jondhra–Rajim confluence point for AM series. In the case of POT series, the coincidence risk reaches a high level at the start of August month for Andhiyarkhore–Simga and Bamnidhi–Seorinarayan confluence points. The risk is high at the mid of August for Jondhra–Rajim confluence point. The risk peaks at the august start and again at September start for Basantpr –Kurubhata confluence point. The coincidence risk values are low in the case of POT series compared to that of AM series. From Figure 15 it is evident that, for AM series, Andhiyarkhore–Simga, Bamnidhi–Seorinarayan and Basantpur–Kurubhata confluence points are bimodal (i.e., have two peaks). Similarly, for POT series (Figure 16), Andhiyarkhore–Simga and Bamnidhi–Seorinarayan confluence points are unimodal in nature for POT series by having a single peak and Basantpur–Kurubhata confluence point is bimodal. Jondhra–Rajim confluence point is unimodal for both AM and POT series. At the Basantpur-Kurubhata confluence point, two coincidence peaks occur on July 31 and September 14, with risk values of 3.19 × 10−4 and 4.01 × 10−4, respectively. Similarly, the Andhiyarkhore–Simga confluence point encounters coincidence peaks on August 11 and September 14, with risk values of 2.29 × 10−4 and 5.18 × 10−4, respectively. The Bamnidhi–Seorinarayan confluence point exhibits coincidence peaks on August 2 and September 16, with associated risk values of 3.59 × 10−4 and 4.39 × 10−4, respectively. Lastly, a single coincidence peak is observed for the Jondhra–Rajim confluence point on August 7, carrying a risk value of 3.34 × 10−4 for AM events. For POT series, Basantpur–Kurubhata confluence point has two coincidence peaks occurring on August 16 and September 11 with risk values of 1.97 × 10−4 and 1.87 × 10−4. Andhiyarkhore–Simga has a single coincidence peak occurring on August 12 with a risk value of 2.43 × 10−4. Bamnidhi–Seorinarayan with a single peak on August 8 with a risk of 2.06 × 10−4. Jondhra–Rajim with a single peak on August 18 with a risk of 2.95 × 10−4. The chance of coincidence of floods in both the mainstream and the tributary at each confluence point starts increasing from July, peaks at mid of August and September and starts decreasing after September mid while it remains almost zero before June and after October. This result of coincidence floods occurring during August and September is in accordance with the actual flooding situation in the basin which agrees with the results of the work done by Sahu et al. (2020) where they have concluded that the MRB is subjected to increased floods in the monsoon period.
Figure 15

Coincidence plots of occurrence dates of AM series.

Figure 15

Coincidence plots of occurrence dates of AM series.

Close modal
Figure 16

Coincidence plots of occurrence dates of POT series.

Figure 16

Coincidence plots of occurrence dates of POT series.

Close modal

The coincidence risk of flood magnitudes was calculated using Equation (11) for different design floods of the return period of 2, 5, 10, 25, 50, and 100 years. Table 3 shows the estimated coincidence risk values for flood magnitudes. The analysis indicates that the coincidence risk of flood magnitudes decreases as the return period increases. For 2-year return period, the coincidence risk is high for the Seorinarayan–Bamnidhi confluence point with a value of 0.336 and the lowest is for Basantpur–Kurubhata confluence point with a risk value of 0.267 for AM series. For POT series, the risk is high for the same point as in AM series but the lowest is for Rajim–Jondhra confluence point with a risk value of 0.199.

Table 3

Coincidence risk values of flood peaks for AM and POT series

StationCoincidence risk of flood magnitudes
2-year
5-year
10-year
25-year
50-year
100-year
AMPOTAMPOTAMPOTAMPOTAMPOTAMPOT
Simga 0.321 0.276 0.079 0.057 0.033 0.013 0.020 4.65 × 10−3 6.92 × 10−3 2.28 × 10−3 5.03 × 10−3 3.64 × 10−4 
Andhiyar khore 
Basantpur 0.267 0.215 0.047 0.049 7.75 × 10−3 3.28 × 10−3 4.45 × 10−3 1.18 × 10−3 2.03 × 10−3 8.19 × 10−4 2.03 × 10−3 5.46 × 10−4 
Kurubhata 
Seorinarayan 0.336 0.314 0.083 0.080 0.034 0.042 0.016 0.017 0.011 7.04 × 10−3 0.01 7.04 × 10−3 
Bamnidhi 
Rajim 0.288 0.199 0.102 0.031 0.054 4.69 × 10−3 9.48 × 10−3 1.35 × 10−3 5.68 × 10−3 4.26 × 10−3 6.0 × 10−3 2.84 × 10−4 
Jondhra 
StationCoincidence risk of flood magnitudes
2-year
5-year
10-year
25-year
50-year
100-year
AMPOTAMPOTAMPOTAMPOTAMPOTAMPOT
Simga 0.321 0.276 0.079 0.057 0.033 0.013 0.020 4.65 × 10−3 6.92 × 10−3 2.28 × 10−3 5.03 × 10−3 3.64 × 10−4 
Andhiyar khore 
Basantpur 0.267 0.215 0.047 0.049 7.75 × 10−3 3.28 × 10−3 4.45 × 10−3 1.18 × 10−3 2.03 × 10−3 8.19 × 10−4 2.03 × 10−3 5.46 × 10−4 
Kurubhata 
Seorinarayan 0.336 0.314 0.083 0.080 0.034 0.042 0.016 0.017 0.011 7.04 × 10−3 0.01 7.04 × 10−3 
Bamnidhi 
Rajim 0.288 0.199 0.102 0.031 0.054 4.69 × 10−3 9.48 × 10−3 1.35 × 10−3 5.68 × 10−3 4.26 × 10−3 6.0 × 10−3 2.84 × 10−4 
Jondhra 

The coincidence risk of floods was calculated using Equation (15) for different design floods of the return period of 2, 5, 10, 25, 50, and 100 years. Table 4 shows the estimated coincidence risk values for flood dates. The analysis indicates that the coincidence risk values decrease as the return period increases (i.e., Low frequency) indicating lower coincidence risk for rare extreme events. For 2-year return period, the coincidence risk is high for the point Seorinarayan–Bamnidhi confluence point with value of 7.63 × 10−3 and the lowest is for Rajim–Jondhra confluence point with a risk value of 4.47 × 10−3 for AM series. For the POT series, the risk is high for the confluence point as in AM series but the lowest is for Basantpur–Kurubhata confluence point with a risk value of 2.36 × 10−3. Except for the 10-year return period in AM results, Seorinarayan–Bamnidhi confluence point experiences the highest coincidence risk for both AM and POT series. This may be due to the close proximity of these two stations, leading to a higher likelihood of experiencing coincident floods.

Table 4

Coincidence risk values of flood peakdates for AM and POT series

StationFlood coincidence risk (×10−3)
2-year
5-year
10-year
25-year
50-year
100-year
AMPOTAMPOTAMPOTAMPOTAMPOTAMPOT
Simga 4.59 3.02 1.13 0.628 0.47 0.139 0.29 0.051 0.099 0.025 0.072 0.004 
Andhiyar khore             
Basantpur 4.49 2.36 0.79 0.538 0.13 0.036 0.048 0.013 0.034 0.009 0.034 0.006 
Kurubhata             
Seorinarayan 7.63 3.21 1.88 0.821 0.77 0.434 0.36 0.171 0.238 0.072 0.227 0.072 
Bamnidhi             
Rajim 4.47 2.81 1.58 0.438 0.84 0.066 0.147 0.019 0.088 0.006 0.093 0.004 
Jondhra             
StationFlood coincidence risk (×10−3)
2-year
5-year
10-year
25-year
50-year
100-year
AMPOTAMPOTAMPOTAMPOTAMPOTAMPOT
Simga 4.59 3.02 1.13 0.628 0.47 0.139 0.29 0.051 0.099 0.025 0.072 0.004 
Andhiyar khore             
Basantpur 4.49 2.36 0.79 0.538 0.13 0.036 0.048 0.013 0.034 0.009 0.034 0.006 
Kurubhata             
Seorinarayan 7.63 3.21 1.88 0.821 0.77 0.434 0.36 0.171 0.238 0.072 0.227 0.072 
Bamnidhi             
Rajim 4.47 2.81 1.58 0.438 0.84 0.066 0.147 0.019 0.088 0.006 0.093 0.004 
Jondhra             

Climate change causes shifts in the patterns of precipitation and extreme weather events, thereby affecting flood characteristics (Swain 2014). It significantly affects the hydrological conditions of the river basins and water resources. The increasing concerns about the impact of climate change on floods necessitate the understanding of variations in flood behavior and estimating flood risk. There are various methods to assess flood risk namely hydrological and hydraulic modeling, GIS analysis, floodplain mapping and probabilistic analysis. By examining the recent observations of flood data, trends and changes over time can be identified. Here, trend analysis and persistence analysis of flood events were carried out to give a prior understanding of flood behavior in recent decades. The results revealed significant (increase and decrease) trends in certain stations and other stations showed statistically insignificant trends. This variation in trends can be due to the construction of a few major dams that regulate the flow of water, thereby affecting the flood trends over time (Ganguli et al. 2023). Additionally, persistence analysis was carried out using circular statistics to analyze periodic data (Cunderlik et al. 2004; Villarini 2016). This revealed that the persistence of flood timing was high in the basin around the middle of August, which means that the flood mostly occurs in the monsoon season i.e., June, July and August (Sahu et al. 2020), and most of the stations showed a delayed flood occurrence, which can be attributed to the recent construction of weirs and dams. The results of the persistence analysis obtained are in line with the findings of Ganguli et al. (2020) where the mean flood dates were highly concentrated in August in MRB. These findings provide insights into the impact of changing rainfall patterns and the influence of water structures on flood management in the study area as the surrounding areas of the stations considered are croplands.

Various studies (Swain 2014; Jain et al. 2017) show that the future prediction of flood events reveals that there may be an increase in flood occurrence in MRB. Thus, there is a need to assess flood risk to tackle flood events by planning effective mitigation measures. Hence, the coincidence risk analysis was performed for different design floods with various return periods using copula functions. Since flood behavior is complex to understand due to the effect of various factors, a multidimensional approach considering the effect of various influencing factors together has to be used. Therefore, in this study, copula functions have been used to estimate coincidence flood risk analysis in MRB. As mentioned earlier, though POT series can give more flood events than AM series (Bezak et al. 2014), however, the risk values were lower in the case of the POT series due to the absence of coincident events in the two rivers of each confluence points. This point is supported by the negative correlation between the dates of these events which has been incorporated in estimating the coincidence risk values. This analysis of coincidence risk indicates that as return periods increase, the risk values approach zero and the chance of coincidence flooding is less likely to occur for large flood events (Li et al. 2022). Seorinarayan–Bamnidhi confluence point consistently showed the highest coincidence risk for both AM and POT series, possibly due to their close proximity. These coincidence risks were mostly high in the monsoon season of the basin, thereby highlighting the vulnerability of the basin to floods during this season. On analyzing the results of coincidence risk, all the confluence points are positioned successively in the basin and experience coincidence risks almost at the same time around mid of august. Thus, implementing effective flood control measures at upstream confluence points can eventually reduce the flood risk at downstream points. Further, on analyzing the stream flow values of mainstream and tributaries, we found that the main stream has a considerably higher stream flow than tributaries for confluence points 1, 2, and 3. Hence, implementing appropriate flood control measures at Simga, Seorinarayan and Basantpur (which are on the main stream), can reduce the coincidence risk at the respective confluence points (Feng et al. 2020). For Confluence Point 4, since both the stations experience a similar range of flows, control measures have to be implemented in both mainstream and tributary to alleviate flood risk at that point.

Understanding the coincidence risk can aid in water resource planning, reservoir operations (Peng et al. 2017), flood control measures, and the design and location of infrastructure projects in flood-prone areas. Further, past studies (Shawky et al. 2023; Reddy & Ray 2024b) indicate that a comprehensive analysis, encompassing factors such as wind speed, land use and land cover characteristics, and land surface temperatures, is essential for gaining a deeper understanding of flood risk at the basin level. In their study, Shawky et al. (2023) discovered a significant alteration in the land surface temperature (LST) and a decrease in the cropland area across the MRB regions. Changes in land use, such as reduced cropland area and deforestation, can produce variations in the LST of the MRB regions. These variations can either cool or raise the average temperature and have the potential to alter the flow patterns. Due to the reduction in agricultural land, the elevated water levels will rapidly approach the confluence locations, resulting in a heightened probability of floods.

The future scope of the present study can be extended further by mapping the coincidence risks and zoning the high-risk areas using software's like ArcGIS, HEC-RAS, etc. Also, the concept of non-stationarity can be incorporated in predicting future coincidence risks. The findings of this study can be incorporated into flood forecasting systems, early warning systems and decision support tools to develop targeted mitigation measures and improve emergency response planning in the MRB.

Limitations of the study

Temporal changes in land use and land cover patterns, which significantly influence flood risk, are not fully integrated into the analysis (Shrestha 2019; Emmanuel et al. 2020). Further, assumptions of independence among flood events may not hold, potentially skewing the understanding of flood risk and its underlying drivers (Mzava et al. 2021). Additionally, although the Mann–Kendall trend test is a non-parametric method, it has limitations concerning serial correlation in the data (Zaghloul et al. 2022). Thus, the MK trend test should be applied cautiously to datasets characterized by seasonal regimes.

Examining the variations in flood peak magnitude and shifts in occurrence can provide valuable insights into water availability, facilitating storage planning and effective water management. Additionally, this analysis aids in assessing the risk of coinciding flood events. In this paper, the trend in flood magnitude and shift in its timing have been assessed and its coincidence has been analyzed for around eight gauge stations using two different flood event sampling methods (AM and POT). After assessing such trends in the flood variables (peak magnitude and its occurrence dates), coincidence risk analysis has been performed at the confluence points using the copula function. The key findings from this study include the following:

  • (1) The middle reach of MRB shows a mixture of significant decreasing and increasing trends with downward being predominant for AM series. For POT series, both the upper and the middle regions do not show any statistically significant trend.

  • (2) From the persistence analysis of the flood occurrence dates using both AM and POT series, the mean date of floods is highly concentrated in the mid of the August. More than half of the considered stations show a delay in flood occurrence dates except for Basantpur where it concurs with the mean occurrence date.

  • (3) From the trend in flood peak magnitude and shift in its timing, for AM series, Andhiyarkhore, Kurubhata and Seorinarayan show an increasing trend with delayed occurrence, Bamnidhi, Rajim and Simga show a decreasing trend with delayed occurrence and Jondhra station shows a decreasing trend with early occurrence.

  • (4) Jondhra shows a decreasing trend with early occurrence for POT series while it shows a delayed occurrence for AM series. Basantpur, Kurubhata, Rajim and Simga show the same result as like AM series except for Andhiyarkhore (decreasing with delayed occurrence) and Bamnidhi (increasing with delayed occurrence).

  • (5) Compared to Lognormal, Gamma, Gumbel, P-III, and GEV distributions, P-III and GEV distributions seem to be best fitting the observed flood magnitude of AM series while it is GP distribution for POT series. For occurrence dates of both the sampling methods, the Mixed Von Mises distribution fits the observed dates reasonably well.

  • (6) The coincidence risk values of flood magnitudes of the four confluence points 1, 2, 3, and 4 for 2-year return period are 4.59 × 10−3, 7.63 × 10−3, 4.49 × 10−3, and 4.47 × 10−3, respectively for AM series. For the POT series, the coincidence risks of flood magnitudes of four confluence points for 2-year return period are 3.02 × 10−3, 3.21 × 10−3, 2.36 × 10−3, and 2.81 × 10−3, respectively. The coincidence risk values decrease as the return period increases indicating that the coincidence risk is high for more frequent events.

Of the two sampling methods, the coincidence risk value is less for POT series and this can be due to the negative correlation of the dates of POT series as they are considered for coincidence risk of flood magnitude. Almost for all the confluence points, the coincidence risk peaks between July end and August mid while for a few points that are bimodal, the second coincidence peak occurs around September mid. Thus, the river banks and the areas around these four confluence points can be highly vulnerable to flooding during the monsoon period in MRB. Therefore, by planning effective flood mitigation measures at upstream points, the flood risk at downstream points can be reduced. High coincidence flood risk may even expose areas that are typically less prone to flooding as the population may not be adequately prepared to deal with such superimposed floods. Knowledge of this coincidence risk assists in the planning and management of water resources and allows for the development of strategies to balance water allocation, reservoir operations, and flood control measures during peak coincidence flood time. It also helps in designing and locating infrastructure and urban development projects in flood-prone areas. This knowledge can guide the implementation of appropriate flood protection measures, such as levees, floodwalls, or establishing floodplain zoning and also can contribute to accurate flood risk assessments.

The authors wish to acknowledge Dineshkumar Muthuvel for his professional assistance during the work. We are thankful to the editor and all the anonymous reviewers, who helped immensely by improving the quality of the paper by providing their valuable suggestions.

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

The authors have no competing interests to declare that are relevant to the content of this article.

All authors contributed to the conceptualization and design of the study and formal analysis. Data extraction and software analysis were performed by S.R.S. and V.M.R. The original draft of the manuscript was written by S.R.S. L.K.R. supervised the work, reviewed and edited the writing. All authors read and approved the final manuscript.

The daily streamflow data used in this study is freely available from the India-WRIS official website: https://indiawris.gov.in/wris/

The authors declare there is no conflict.

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