Hydrologic regionalization of non-stationary intensity–duration–frequency relationships for Indian mainland

India is a country which is highly vulnerable to climate change as its average climate is structured by the Indian monsoon and its diverse geographical features vary from highly elevated mountains to adjoining oceans on its three sides. Furthermore, India's mean temperature increased by nearly 0.7 °C between 1901 and 2018 and the impact of climate change has already begun to alter its seasons experiencing both more persistent dry spells and more acute wet spells (Krishnan et al. 2020). In addition, he noted that throughout Central India, the frequency of daily rainfall (RF) extremes with intensities above 150 mm/day increased by around 75% between 1950 and 2015. According to research, short-duration RF events that last less than a day are more likely to occur again, which could enhance the intensity and frequency of RF-derived flash floods. In order to support rural and urban planning policies and the building of flood protection infrastructure, it is crucial to comprehend the changing behaviour and implications of extreme RF. The identification of the intensity–duration–frequency (IDF) relationship of extreme precipitation events aids in the development of more effective design standards for the construction of infrastructure, thereby lowering the hazards involved. More than 80% of India's population, according to Mohanty & Wadhawan (2021), resides in regions that are extremely vulnerable to extreme weather occurrences.

The development of RF IDF curves in the past frequently took into account the stationarity assumption, which may not be realistic in the present or the future given the existing and changing environmental conditions. Since IDF curves are utilized in the design, planning, and administration of water infrastructure systems, they must be developed with the utmost care. The IDF curves are developed based on historical RF time series data by fitting a theoretical probability distribution of annual maximum series (AMS) or partial duration series (PDS). In AMS, only the annual maximum value per year is considered ignoring the fact that secondary events in a year might be exceeding the annual maxima of other years. This limitation is overcome by PDS where all events above a certain threshold are considered. In general, the AMS method is most commonly used due to its easiness. In the case of PDS, there are mainly two difficulties while extracting the peaks: the first one is maintaining the independence of consecutive peak events and the second one is the appropriate selection of threshold value to extract maximum information. Infrastructures like reservoirs and stormwater channels are designed based on the extreme rainfall or flood estimates and also under a non-stationary (NS) environment, hence, the magnitude of design rainfall will be more than that computed by stationary (S) assumption. This underestimation may result in the failure of hydraulic infrastructure (Sarhadi & Soulis 2017). In a scenario where the environment is changing, standard design philosophies must be modified, and the concepts of the return period (RP) and risk must be reconsidered (Salas & Obeysekera 2014; Mondal & Daniel 2018). In the past decade, numerous scholars conducted investigations to create the NS IDF curves in various parts of the world (Simonovic & Peck 2009; Yilmaz & Perera 2013; Cheng & AghaKouchak 2014; Yilmaz et al. 2014; Eckerston 2016; Wi et al. 2016; Agilan & Umamahesh 2017b; Ganguli & Coulibali 2017, 2019; Sarhadi & Soulis 2017; Yilmaz et al. 2017; Nwaogazie & Sam 2020; Silva et al. 2021).

In the Indian context, the majority of studies pertaining to the development of NS IDFs focused on urban regions. Mondal & Mujumdar (2015) performed the NS modelling of extreme rainfall over India by varying the parameters of extreme value distributions with physical covariates like the El-Nino Southern Oscillation (ENSO) index, global average temperature, and local mean temperature. They used daily gridded data from the 1901–2004 period at 1° × 1° resolution for the study. The dataset demonstrates non-stationarity as a result of several factors, and there is no spatially consistent pattern in the changes in them across the nation. The best-fit models were also used to examine the 100-year extreme rainfall event. Agilan & Umamahesh (2016a) developed the covariate-based NS rainfall IDF curves of Hyderabad city and compared the same with two future time periods (2015–2056 and 2057–2098) IDF curves. The future time period IDF curves were developed using 24 global climate models' (GCMs') simulations and ‘K’-nearest neighbour (KNN) weather generator-based downscaling method. The results indicated that the return period of extreme RF of Hyderabad city follows a decreasing trend. Additionally, it was determined that IDF curves created with the trend in observed rainfall are a suitable option for designing Hyderabad city infrastructures having a design life of almost 50 years. Agilan & Umamahesh (2016b) developed NS IDF curves using Multi-Objective Genetic Algorithm (MOGA) and compared with S IDF curves for Wilmington city, USA and Hyderabad city, India. By modelling the series' non-linear trend, MOGA was used to create NS GEV models with low bias and high quality. The S GEV model and the linear trend NS GEV model were also compared with the suggested GEV model, and the best model for each duration RF series was selected based on the corrected Akaike Information Criterion (AIC) value. The rainfall intensity–duration connections for various return times were created based on the best NS models. When S and NS IDF curves were compared, it became clear that the S models' IDF curves significantly underestimated the extreme events of the Wilmington and Hyderabad annual maximum rainfall series for all chosen durations. Additionally, it was observed that the duration of a heavy downpour in Hyderabad and Wilmington was shortening. Furthermore, Hyderabad's NS and S IDF curves differed less from Wilmington's than from Hyderabad.

Most of the studies in India were focused on developing the IDF curves for selected locations where significant non-stationarity is expected. It's crucial to pinpoint the areas or regions of India that are vulnerable to NS precipitation events. Even if the effects of those changes are currently mild in certain regions, they could exacerbate in the near future because of changing climate. It is, therefore, highly important to identify such places a priori to alleviate the adverse impact of hydrologic disasters. By regionalizing India in accordance with how vulnerable various areas are to NS RF intensities, it may be possible to reduce the danger of precipitation extremes. In order to gain additional knowledge about potential sites that experience substantial NS climatic extremes, this study intends to generate a geographical pattern using NS intensity values of rainfall. The specific goals of this study are (i) to prepare the IDF curves for six selected locations in India using annual maximum hourly data for S and NS cases and (ii) to develop IDF curves for every grid point in India using annual maximum daily RF data for S and NS cases in a regionalization perspective. The next section elaborates the study area and data collection. In Section Three, the methodology of the study is briefly outlined. The results of the developed IDF curves are presented in Section Four along with important comments. Finally, Section Five presents the study's main conclusions.

Annual maximum rainfall data are corresponding to sub-daily durations with a data length higher than 30 years being the desired type of data for developing IDF curves. Even though the collection of hourly data is sparse in the Indian subcontinent, the station data at an hourly time scale is highly desirable for developing IDF curves, as the extreme rainfalls are showing more non-stationarity on a sub-daily basis. Hence, in this study, NS RF IDF relationships and associated curves were developed for two types of datasets, the hourly RF dataset (Deshpande et al. 2012) and the daily gridded RF data (1° × 1°) collected from the Indian Meteorological Department (IMD), Pune.

As the hourly data collection is very sparse in India, the use of daily gridded RF data is used as a proxy for much of the hydrological analysis but appropriate rainfall disaggregation methods are to be followed for developing IDF curves (Adarsh & Janga Reddy 2018). The daily gridded rainfall data over a period of 65 years (1951–2015) with a 1° × 1° spatial resolution was collected from IMD. It includes daily rainfall data for 357 points which covers the geographical extent of the entire India. The dataset was originally prepared by Rajeevan et al. (2006) for the 1951–2003 period considering 1,803 stations in India, where a minimum of 90% of the data are available for the analysis period. They have followed Shepard's interpolation method (Shepard 1968) based on the weights calculated from the distance between the station and the grids and also the directional effects. They performed statistical quality control measures before interpolation analysis, compared the data with many global gridded data products and reported the superiority in the accurate representation of spatial rainfall variation (Rajeevan et al. 2006). An updated version of this data product is used in the present study.

S and NS rainfall intensities were evaluated for the 357 grid points corresponding to all durations for all five NS model variants outlined in the Methodology using the data obtained after RF disaggregation of yearly maximum hourly RF data. The properties of the RF data during acquisition and after processing (extraction and disaggregation) are given in Table 2.

For the stationary model, t= 0. Hence, μ(t) = μ₀ and σ(t) =σ₀.

The trend in the location parameter due to the effect of the covariate is represented by the slope parameters μ₁. The slope parameter σ₁ represents the trend in the scale parameter due to changes in time. The exponential function ensures the positive value of the scale parameter.

This section summarizes the findings from IDF curves created for six stations using hourly data and those created for various grids around India using daily gridded data, along with pertinent remarks.

From the IDF curves developed, it was found that differences in intensity values are more for shorter durations. Different stations were showing different trends in NS rainfall estimates according to the variation of RPs. In some stations like Kozhikode, Cuddalore, and Tiruchirappalli, differences in intensity values were more for higher RPs, which indicate a higher risk for structures with longer design life at these locations. AIC values for the stationary models are −12.1, −11.5, −38.71, 13.1, 9.8, and −9.4, respectively, for Thiruvananthapuram, Kozhikode, Coimbatore, Cuddalore, Hisar, and Tiruchirappalli, respectively. By comparing the AIC values, it was evident that the best-fitted NS model was way better than the S model for the Kozhikode station. It was also found that corresponding to some of the durations the underestimation of S rainfall intensities was as high as 50%. It indicates the necessity of consideration of NS IDF curves in the Kozhikode station. The Tiruchirappalli station was also found to have significant NS (about 35%) according to the best-fitted model. The Hisar station had also shown significant NS (>15%) for smaller RPs. Even though the stations like Thiruvananthapuram, Coimbatore, and Cuddalore do not show much non-stationarity according to their best-fitted NS model, Thiruvananthapuram is showing considerable NS (>30%) behaviour according to other models fitted to the data, which have only slight difference in their AIC values with the best model. After comparing S and NS intensity values, the highest percentage deviation of each station was identified from the best-fitted model. Those critical durations and RPs corresponding to the highest deviations are given in Table 5.

From an overall analysis of the results obtained, the number of grid points, which were showing the different extents of non-stationarity is given in Table 7. For calculating the percentage change for each grid point, the mean of percentage variation of different combinations of durations and RPs were considered together. Thereby, the grid points which were consistently showing significant NS behaviour were identified.

From Table 7, it is evident that about 23% of the grid points were showing a non-stationarity of 15% or more. Among those about 7% of grid points were showing a non-stationarity of 25% or more. Very high levels of non-stationarity (>35%) were shown by 3% of grid points, which comprises about 1 × 10⁵ km² of area across the whole of India. For shorter return periods, the percentage deviation of S and NS intensity values was found to be comparatively smaller. In other words, percentage deviations in intensity values are higher for longer RPs. So, for structures having large design life such as dams and reservoirs, the non-stationary effects should be seriously considered. In different combinations of durations and RPs, the south-east coastal region of Tamil Nadu shows significant NS behaviour (>25%), so the design of infrastructures in this region should definitely follow the NS models. In general, coastal regions are showing higher non-stationarity compared to central India. Some parts of Gujarat are showing significant non-stationarity (sometimes even >35%), and also consistent levels of non-stationarity were identified near the borders of Andhra Pradesh and Odisha. Eastern regions of India (including Meghalaya, Arunachal Pradesh, and Assam) are experiencing higher levels of non-stationarity in rainfall extremes in almost all combinations. Meghalaya was showing higher variations in intensity values (>30%) in all combinations.

For examining the non-stationary rainfall, the trend of the annual maxima series of each grid and stations were determined using the MK test. On considering the punctual (station) data, the MK values were found to be 2.12, 1.97, and 2.32 for Kozhikode, Hisar, and Tiruchirappalli, which displays significant trends in the datasets of these three stations. The NS was also found to be significant in these stations (see the previous section). Table 8 shows the locations having a significantly increasing (Z > 1.96 at a 5% significance level) trend according to the MK test and also represents more than 15% underestimation of rainfall extremes while using S analysis. It was found that most of the stations for which significant NS behaviour was detected also showed an increasing trend in rainfall extremes according to the MK test.

This study developed IDF for six stations in India using the variation of GEV distribution parameters considering the best model among the five combination models. Furthermore, the study investigated the spatial pattern of intensities for the S and NS scenarios and evaluated the percentage differences in order to identify the regions of significant non-stationarity in climate extremes. Such regionalization may help to identify the places which demand prior attention for the design of critical hydraulic infrastructure in such regions. The findings of the study are actually pointing to the necessity for adopting immediate actions to mitigate the potentially disastrous effects of non-stationary rainfall extremes in the future. Even though the study provides a broad overview of critical locations, the detailed study is to be conducted using station-wise data within and in the vicinity of critical regions/grids which were found to experience significant NS behaviour in precipitation extremes. Also, studies considering future precipitation by different candidate general circulation models, considering appropriate covariates, uncertainties in parameter estimation, the influence of data length on the distribution parameters, and types of non-stationary models, need to be solicited in order to draw more comprehensive inferences.

In this work, the yearly maximum rainfall data generated from the hourly data of six sites are initially used to build the NS IDF curves. The five models used to develop the NS IDFs were compared, and the best model was chosen. The best NS model is then used to determine the RF intensity values for 357 grid locations over India using the daily gridded data and rainfall segregation. It was determined from the results and their interpretations that there are considerable NS behaviours at various sites around the nation. The following findings are reached after carrying out the analysis:

• For several combinations of durations and RPs, the Kozhikode station displayed non-stationarity with at least 50% fluctuation.

• Stations like Kozhikode, Cuddalore, and Tiruchirappalli and also most of the grid points revealed higher discrepancies in NSRF intensity estimates for larger RPs. So, at these locations, proper modelling of non-stationarity must be carried out before designing structures having longer life spans.

• Approximately 1 × 10⁵ km² of the entire country of India is experiencing extremely high levels of NS behaviour during intense rainfall (>35% underestimate on average for various combinations of durations and RPs with regard to S rainfall intensities).

• For various pairings of durations and RPs, about 23% of grid points showed a deviation of over 15%.

• Higher levels of non-stationarity were observed in Coastal regions and Eastern parts of India when compared to the central parts.

• A significantly increasing trend was spotted for 32 grid points where an increase of more than 15% in NS rainfall intensity estimates was also noticed.

The authors would like to extend our gratitude to Nayana Deshpande, Scientist, Indian Institute of Tropical Meteorology (IITM), Pune, India for providing hourly rainfall data for selected locations for promoting non-commercial scientific research. Authors also thank IMD, Pune for supplying the daily gridded rainfall dataset needed to carry out this research. The Matlab^®2020b, R software and its packages ‘ismev’ are used in this study. M.G.M. acknowledges DST-SERB for the fellowship support.

The work is supported by the project entitled ‘Developing Non-Stationary Frequency Relationships of Hydro-climatic Extremes of Kerala Meteorological Subdivision under Changing Climate Scenario’ (File Number: CRG/2021/003688) funded by the Science and Engineering Research Board, Department of Science and Technology (DST-SERB) under the Government of India.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.