Intensity–duration–frequency (IDF) curve is one of the important hydrologic tools used for the design of hydraulic infrastructure. The static return period assumption of precipitation extremes is invalid in a changing climate environment, and the underestimation of rainfall intensity may lead to the failure of infrastructure in extreme events. This study first developed the non-stationary (NS) IDF curves for six selected locations in India based on sub-daily station data based on time-dependent estimates of five combinations of Generalized Extreme Value (GEV) distribution parameters. Then, in order to identify the critical regions of rainfall non-stationarity, the IDF curves were developed for 357 grid points over India using the daily gridded data for the period 1951–2016 at 1° × 1° resolution. The comparison of spatial patterns of rainfall intensity estimates under stationary and non-stationary showed that about 23% of grids showed an overestimation of NS rainfall over their stationary counterparts by at least 15%. About 32 grid locations which showed at least 15% overestimation of rainfall under an NS case displayed a significantly increasing rainfall trend. The majority of the grids with larger deviation of non-stationary rainfall estimates over stationary values are located in India's eastern regions and coastal belts.

  • Developed non-stationary IDF curves for the whole of India.

  • Variation in location and scale parameters governs the fitting of the best model of over 85% of grids.

  • About 23% of regions showed high non-stationarity of rainfall.

  • Higher levels of non-stationarity were observed in Coastal regions and Eastern parts.

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.

Annual maximum sub-daily rainfall data for different stations

The National Data Centre (NDC) of IMD, Pune, India had prepared observed hourly rainfall data for 145 stations all over India (Deshpande et al. 2012). It was noted that stations near to coastal regions express higher non-stationary behaviour when compared to the inland regions in India (Deshpande et al. 2012). We have identified six stations in India, where non-stationarity in extreme rainfall events prevail. Thus, the hourly rainfall data for six stations, of which three (Thiruvananthapuram, Kozhikode, and Cuddalore) are located in southern coastal regions, two (Tiruchirappalli and Coimbatore) are located in southern India, and one (Hisar) is located in northern India were used for the present study. The time frames of data collected are displayed in Table 1. Also, the geographical locations of the stations selected for the analysis are shown in Figure 1.
Table 1

Details of the station-wise rainfall data

StationsData lengthType of data
Thiruvananthapuram 1969–2014 Annual maximum sub-daily data 
Kozhikode 1969–2014 Annual maximum sub-daily data 
Tiruchirappalli 1969–2014 Annual maximum sub-daily data 
Coimbatore 1969–2013 Annual maximum sub-daily data 
Cuddalore 1969–2013 Annual maximum sub-daily data 
Hisar 1970–2006 Annual maximum sub-daily data 
StationsData lengthType of data
Thiruvananthapuram 1969–2014 Annual maximum sub-daily data 
Kozhikode 1969–2014 Annual maximum sub-daily data 
Tiruchirappalli 1969–2014 Annual maximum sub-daily data 
Coimbatore 1969–2013 Annual maximum sub-daily data 
Cuddalore 1969–2013 Annual maximum sub-daily data 
Hisar 1970–2006 Annual maximum sub-daily data 
Figure 1

Stations selected for developing NS IDFs based on hourly data.

Figure 1

Stations selected for developing NS IDFs based on hourly data.

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Daily gridded rainfall data for the entire India

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.

The annual maximum RF data was extracted from daily data which in turn was de-aggregated to produce 1-, 3-, 6-, 12-, and 24 h for each grid point for hourly analysis. For the disaggregation, the popular and simple IMD reduction formulae (stated by Ramaseshan 1996; Chowdhury et al. 2007; Rashid et al. 2012) given in Equation (1) are followed.
(1)
where is the maximum precipitation for ‘t’ hour duration received in a year; is the maximum precipitation for 24-h duration received in a year.

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.

Table 2

Details of IMD gridded rainfall data

Properties of collected dataProperties of processed data
Data type Daily data Annual maximum sub-daily data 
No. of grid points 357 357 
Data length 1951–2015 1951–2015 
Duration 24 h 1, 3, 6, 12, and 24 h 
Spatial resolution 1° × 1° 1° × 1° 
Properties of collected dataProperties of processed data
Data type Daily data Annual maximum sub-daily data 
No. of grid points 357 357 
Data length 1951–2015 1951–2015 
Duration 24 h 1, 3, 6, 12, and 24 h 
Spatial resolution 1° × 1° 1° × 1° 

An IDF curve is a mathematical function that connects the intensity of rainfall with its duration and frequency of occurrence. IDF curves are frequently employed in both civil engineering and hydrology for the planning of urban drainage systems. The likelihood of the occurrence of specific rainfall intensity for a specific duration is graphically represented by the IDF curve. The IDF curves are represented by an equation whereby rainfall intensity, i(d,T), is directly proportional to RP, denoted as ‘T’ and indirectly proportional to the duration (d)
(2)
where a(T) depends on the RP, T and b(d) on the duration, d. The function b(d) is:
(3)
where θ and η are shape parameters subject to the inequality constraints that θ> 0 and 0<η< 1.
The function a(T) is completely determined from the distribution function of rainfall intensity, I(d). FI(d) (i;d) denotes the distribution function of I(d), then intensity rescaled by b(d), Y = I(d) b(d) will also be distributed as FI(d) (i;d) (i.e., . Since a(T) is the return level of I(d), the following expression is obtained:
(4)
The most common method for creating IDF curves is the annual maxima RF approach, and the annual maxima series for various durations are often created using hourly rainfall data. In earlier days, some researchers performed regional studies over India and presented empirical formulas, to get crude estimates of RF intensity values in the process of developing IDF curves. Although the extreme value I-based frequency factor is the simplest method for creating IDF curves, the use of three-parameter GEV distribution is broader and more realistic. Here, the parameters of the GEV distribution need to be estimated, for which the maximum likelihood (MLE) method or L-moment method are two possible alternatives. In this study, the MLE method is followed to compute the location (μ), scale (σ), and shape (γ) parameters, as it was reported that it can be easily extended to the non-stationary case (Coles 2001; Katz 2013; Wi et al. 2016). Moreover, some of the studies in the Indian context recommended the use of the MLE method even with lesser data length than used in the present study (Agilan & Umamahesh 2016a, 2017a). After getting the parameters, Equation (5) is utilized for calculating design rainfall intensities.
(5)
The aforementioned equation, in which XT stands for the intensity of the RF for a specific RP and duration, was derived from the cumulative distribution function (CDF) of the GEV distribution. These are predicated on the stationary assumption that the RP is constant and the parameters are constant (time-invariant). However, in a scenario with a changing environment, we must consider the distribution parameters are changing over time in order to create NS IDF curves. In this case, the shape parameter is held constant while the location and scale parameters are supposed to fluctuate over time (Wi et al. 2016). Therefore, the NS location and NS scale GEV parameters were introduced as time-dependent variables with unit step increments in values from t = 1 to N, where N is the number of years. By taking into account the following five models, NS RF intensities were calculated for all stations, corresponding to all durations:
  • (a)
    NS Model 1: Linear variation of location parameter
  • (b)
    NS Model 2: Linear variation of scale parameter
  • (c)
    NS Model 3: Exponential variation of scale parameter
  • (d)
    NS Model 4: Linear variation of both location and scale parameters
  • (e)
    NS Model 5: Linear variation of location parameter and exponential variation of scale parameter
where ‘t’ is the covariate representing time.

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.

The initial stage was to estimate the parameters using MLE, just like with stationary methods. Time-varying location and scale parameters are fixed as the 95th percentile of values obtained for (t) and σ(t), respectively (Cheng & Aghakouchak 2014). Design RF intensity values were estimated using these obtained values of the location, scale and shape parameters, identical to the stationary example. The best NS model can be identified after calculating AIC values (Akaike 1974). The best NS model corresponding to 24-h duration rainfall was considered the best model for the station. Using the measured design rainfall intensity values, IDF curves were generated. The overall framework used in this study is summarized in Figure 2. Furthermore, in order to investigate the role of trend on the NS intensity values, the trend is to be estimated, for which the most popular non-parametric Mann–Kendall (MK) trend test (Mann 1945; Kendall 1975) is used in this study.
Figure 2

Diagram illustrating the process used to create IDF curves.

Figure 2

Diagram illustrating the process used to create IDF curves.

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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.

Non-stationary IDF curves based on station-wise sub-daily rainfall data

S and NS IDF curves of six stations (Thiruvananthapuram, Kozhikode, Cuddalore, Tiruchirappalli, Hisar, and Coimbatore) are developed from annual maximum sub-daily RF data using the GEV distribution. The RPs of 2-, 5-, 10-, 25-, 50-, and 100 years for durations of 1-, 2-, 6-, 12-, 24-, 36-, 48-, 60-, and 72 h were considered in this study. Five different models are tried for developing the NS IDF curves. The most suitable model for each case is determined with the help of AIC. The percentage deviations of rainfall intensities between the S and NS estimates are calculated for all stations. S and NS RF intensities by five models for the Thiruvananthapuram station corresponding to different durations and RPs are given in Table 3.
Table 3

Stationary and non-stationary rainfall intensity estimates for the Thiruvananthapuram station (in cm/h)

T (Year)Duration (h)
1 h3 h6 h12 h18 h24 h36 h48 h60 h72 h
Stationary case 
 2 yr. 4.67 2.49 1.48 0.87 0.62 0.5 0.38 0.31 0.27 0.24 
 5 yr. 6.06 3.3 1.97 1.17 0.83 0.67 0.51 0.42 0.36 0.33 
 10 yr. 6.85 3.79 2.26 1.35 0.95 0.77 0.58 0.48 0.42 0.37 
 25 yr. 7.73 4.35 2.59 1.56 1.09 0.88 0.66 0.55 0.47 0.42 
 50 yr. 8.31 4.73 2.81 1.7 1.18 0.95 0.72 0.6 0.51 0.45 
 100 yr. 8.82 5.09 3.01 1.82 1.27 1.02 0.77 0.64 0.54 0.47 
NS Model 1 
 2 yr. 4.95 2.54 1.56 0.91 0.65 0.54 0.42 0.35 0.31 0.27 
 5 yr. 6.32 3.35 2.05 1.21 0.86 0.71 0.55 0.46 0.39 0.35 
 10 yr. 7.09 3.84 2.34 1.39 0.98 0.8 0.62 0.52 0.44 0.4 
 25 yr. 7.92 4.4 2.67 1.6 1.12 0.92 0.7 0.59 0.5 0.44 
 50 yr. 8.46 4.79 2.89 1.74 1.22 0.99 0.76 0.63 0.53 0.47 
 100 yr. 8.92 5.14 3.1 1.87 1.3 1.06 0.81 0.67 0.56 0.49 
NS Model 2 
 2 yr. 4.73 2.51 1.53 0.9 0.63 0.5 0.38 0.31 0.27 0.25 
 5 yr. 6.41 3.4 2.19 1.31 0.9 0.7 0.52 0.43 0.37 0.34 
 10 yr. 7.38 3.94 2.62 1.58 1.06 0.81 0.6 0.49 0.42 0.38 
 25 yr. 8.46 4.58 3.14 1.92 1.26 0.95 0.7 0.57 0.47 0.43 
 50 yr. 9.17 5.02 3.51 2.17 1.4 1.04 0.76 0.61 0.51 0.46 
 100 yr. 9.81 5.43 3.87 2.41 1.53 1.12 0.82 0.66 0.54 0.49 
NS Model 3 
 2 yr. 4.75 2.51 1.55 0.9 0.63 0.5 0.38 0.31 0.27 0.25 
 5 yr. 6.5 3.42 2.25 1.3 0.89 0.7 0.52 0.43 0.37 0.34 
 10 yr. 7.51 3.97 2.7 1.57 1.05 0.81 0.6 0.49 0.42 0.38 
 25 yr. 8.64 4.62 3.25 1.89 1.25 0.94 0.69 0.56 0.47 0.43 
 50 yr. 9.39 5.07 3.65 2.12 1.38 1.03 0.76 0.61 0.51 0.46 
 100 yr. 10.05 5.49 4.03 2.34 1.5 1.11 0.81 0.65 0.54 0.49 
NS Model 4 
 2 yr. 4.94 2.56 1.6 0.93 0.66 0.54 0.42 0.36 0.31 0.28 
 5 yr. 6.88 3.52 2.26 1.35 0.94 0.75 0.58 0.49 0.42 0.38 
 10 yr. 8.18 4.16 2.7 1.63 1.13 0.89 0.69 0.57 0.49 0.44 
 25 yr. 9.85 4.97 3.26 1.98 1.37 1.07 0.82 0.68 0.58 0.53 
 50 yr. 11.1 5.57 3.68 2.25 1.54 1.2 0.92 0.76 0.65 0.59 
 100 yr. 12.35 6.17 4.1 2.51 1.72 1.33 1.02 0.84 0.71 0.66 
NS Model 5 
 2 yr. 4.95 2.56 1.61 0.94 0.66 0.54 0.42 0.36 0.31 0.28 
 5 yr. 6.97 3.54 2.31 1.37 0.95 0.76 0.58 0.49 0.42 0.38 
 10 yr. 8.31 4.19 2.78 1.65 1.14 0.9 0.69 0.57 0.49 0.44 
 25 yr. 10.03 5.01 3.39 2.02 1.38 1.07 0.82 0.68 0.58 0.53 
 50 yr. 11.32 5.62 3.84 2.29 1.56 1.21 0.93 0.76 0.65 0.59 
 100 yr. 12.62 6.23 4.3 2.57 1.74 1.34 1.03 0.84 0.71 0.65 
T (Year)Duration (h)
1 h3 h6 h12 h18 h24 h36 h48 h60 h72 h
Stationary case 
 2 yr. 4.67 2.49 1.48 0.87 0.62 0.5 0.38 0.31 0.27 0.24 
 5 yr. 6.06 3.3 1.97 1.17 0.83 0.67 0.51 0.42 0.36 0.33 
 10 yr. 6.85 3.79 2.26 1.35 0.95 0.77 0.58 0.48 0.42 0.37 
 25 yr. 7.73 4.35 2.59 1.56 1.09 0.88 0.66 0.55 0.47 0.42 
 50 yr. 8.31 4.73 2.81 1.7 1.18 0.95 0.72 0.6 0.51 0.45 
 100 yr. 8.82 5.09 3.01 1.82 1.27 1.02 0.77 0.64 0.54 0.47 
NS Model 1 
 2 yr. 4.95 2.54 1.56 0.91 0.65 0.54 0.42 0.35 0.31 0.27 
 5 yr. 6.32 3.35 2.05 1.21 0.86 0.71 0.55 0.46 0.39 0.35 
 10 yr. 7.09 3.84 2.34 1.39 0.98 0.8 0.62 0.52 0.44 0.4 
 25 yr. 7.92 4.4 2.67 1.6 1.12 0.92 0.7 0.59 0.5 0.44 
 50 yr. 8.46 4.79 2.89 1.74 1.22 0.99 0.76 0.63 0.53 0.47 
 100 yr. 8.92 5.14 3.1 1.87 1.3 1.06 0.81 0.67 0.56 0.49 
NS Model 2 
 2 yr. 4.73 2.51 1.53 0.9 0.63 0.5 0.38 0.31 0.27 0.25 
 5 yr. 6.41 3.4 2.19 1.31 0.9 0.7 0.52 0.43 0.37 0.34 
 10 yr. 7.38 3.94 2.62 1.58 1.06 0.81 0.6 0.49 0.42 0.38 
 25 yr. 8.46 4.58 3.14 1.92 1.26 0.95 0.7 0.57 0.47 0.43 
 50 yr. 9.17 5.02 3.51 2.17 1.4 1.04 0.76 0.61 0.51 0.46 
 100 yr. 9.81 5.43 3.87 2.41 1.53 1.12 0.82 0.66 0.54 0.49 
NS Model 3 
 2 yr. 4.75 2.51 1.55 0.9 0.63 0.5 0.38 0.31 0.27 0.25 
 5 yr. 6.5 3.42 2.25 1.3 0.89 0.7 0.52 0.43 0.37 0.34 
 10 yr. 7.51 3.97 2.7 1.57 1.05 0.81 0.6 0.49 0.42 0.38 
 25 yr. 8.64 4.62 3.25 1.89 1.25 0.94 0.69 0.56 0.47 0.43 
 50 yr. 9.39 5.07 3.65 2.12 1.38 1.03 0.76 0.61 0.51 0.46 
 100 yr. 10.05 5.49 4.03 2.34 1.5 1.11 0.81 0.65 0.54 0.49 
NS Model 4 
 2 yr. 4.94 2.56 1.6 0.93 0.66 0.54 0.42 0.36 0.31 0.28 
 5 yr. 6.88 3.52 2.26 1.35 0.94 0.75 0.58 0.49 0.42 0.38 
 10 yr. 8.18 4.16 2.7 1.63 1.13 0.89 0.69 0.57 0.49 0.44 
 25 yr. 9.85 4.97 3.26 1.98 1.37 1.07 0.82 0.68 0.58 0.53 
 50 yr. 11.1 5.57 3.68 2.25 1.54 1.2 0.92 0.76 0.65 0.59 
 100 yr. 12.35 6.17 4.1 2.51 1.72 1.33 1.02 0.84 0.71 0.66 
NS Model 5 
 2 yr. 4.95 2.56 1.61 0.94 0.66 0.54 0.42 0.36 0.31 0.28 
 5 yr. 6.97 3.54 2.31 1.37 0.95 0.76 0.58 0.49 0.42 0.38 
 10 yr. 8.31 4.19 2.78 1.65 1.14 0.9 0.69 0.57 0.49 0.44 
 25 yr. 10.03 5.01 3.39 2.02 1.38 1.07 0.82 0.68 0.58 0.53 
 50 yr. 11.32 5.62 3.84 2.29 1.56 1.21 0.93 0.76 0.65 0.59 
 100 yr. 12.62 6.23 4.3 2.57 1.74 1.34 1.03 0.84 0.71 0.65 
Figure 3 shows the S (dashed lines) and NS (solid lines) IDF curves for the Thiruvananthapuram station corresponding to all NS models.
Figure 3

Comparison between stationary IDF and non-stationary IDF of Thiruvananthapuram city prepared by five different models for selected RPs (T = 5 years; T = 25 years; T = 50 years, and T = 100 years).

Figure 3

Comparison between stationary IDF and non-stationary IDF of Thiruvananthapuram city prepared by five different models for selected RPs (T = 5 years; T = 25 years; T = 50 years, and T = 100 years).

Close modal
It is evident from the IDF curves created for the Thiruvananthapuram station that for all models and duration–RP combinations, the NS RF intensities are higher than the S RF intensities. For NS Model 4 and NS Model 5, the rainfall intensities showcased more than a 40% increase from the S rainfall intensity estimates corresponding to some particular combinations of duration and RPs. It was discovered that estimates of NS intensity levels utilizing NS Model 4 and NS Model 5 are frequently found to be greater. Similar to the Thiruvananthapuram station, S and NS models were developed for all other five stations. The AIC values for each station were used to determine which NS model was the best. The best NS model corresponding to 24-h duration rainfall was considered as the best model for the station. AIC values for each model corresponding to 24-h duration rainfall data are given in Table 4. AIC is an estimator of predictor error thereby indicating the quality of the statistical model. As a result, the model with the lowest AIC value is the one that fits the data the best without overfitting. In this manner, the best model was identified from different models according to the AIC values. Figure 4 provides the IDF curves of five stations, developed for S and best-fitted NS cases.
Table 4

AIC values for different NS models for all stations for 24-h rainfall data

StationNS Model 1NS Model 2NS Model 3NS Model 4NS Model 5
Thiruvananthapuram − 17.8 −16.94 −16.92 −15.62 −15.6 
Kozhikode −15.11 −21.28 − 22.13 −18.87 −19.65 
Coimbatore − 48.15 −47.91 −47.91 −45.65 −45.65 
Cuddalore 8.80 8.65 8.64 11.24 11.22 
Hisar 5.23 6.33 6.39 7.88 7.94 
Tiruchirappalli −17.14 − 19.19 −19.00 −17.08 −16.85 
StationNS Model 1NS Model 2NS Model 3NS Model 4NS Model 5
Thiruvananthapuram − 17.8 −16.94 −16.92 −15.62 −15.6 
Kozhikode −15.11 −21.28 − 22.13 −18.87 −19.65 
Coimbatore − 48.15 −47.91 −47.91 −45.65 −45.65 
Cuddalore 8.80 8.65 8.64 11.24 11.22 
Hisar 5.23 6.33 6.39 7.88 7.94 
Tiruchirappalli −17.14 − 19.19 −19.00 −17.08 −16.85 

The bold numbers show a minimum AIC value.

Figure 4

Stationary IDF and best-fitted non-stationary IDFs of each station: (a) Kozhikode, (b) Coimbatore, (c) Cuddalore, (d) Hisar, and (e) Tiruchirappalli for selected RPs (T = 5 years; T = 25 years; T = 50 years, and T = 100 years).

Figure 4

Stationary IDF and best-fitted non-stationary IDFs of each station: (a) Kozhikode, (b) Coimbatore, (c) Cuddalore, (d) Hisar, and (e) Tiruchirappalli for selected RPs (T = 5 years; T = 25 years; T = 50 years, and T = 100 years).

Close modal

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.

Table 5

Highest percentage variation for each station according to the best-fitted NS model

StationHighest underestimation (cm/h)Largest variation (%)Duration (h)RP (years)
Thiruvananthapuram 2.9 13.1 48 
Kozhikode 13.4 50.3 12 100 
Coimbatore 1.1 5.6 60 
Cuddlier 7.7 11.63 72 100 
Hisar 7.6 18.9 
Tiruchirappalli 19.4 35.4 60 100 
StationHighest underestimation (cm/h)Largest variation (%)Duration (h)RP (years)
Thiruvananthapuram 2.9 13.1 48 
Kozhikode 13.4 50.3 12 100 
Coimbatore 1.1 5.6 60 
Cuddlier 7.7 11.63 72 100 
Hisar 7.6 18.9 
Tiruchirappalli 19.4 35.4 60 100 

Stationary and non-stationary rainfall estimates based on gridded rainfall data

S and NS rainfall intensities were estimated for all 357 grid points in regard to the best NS model among the five different NS models by comparing their AIC values (considering 24-h rainfall data as a representative). A consolidated picture of the percentage of grid points that best fit with each of the NS model is given in Table 6. From Table 6, it is evident that nearly half of the grid points were best fitted with NS Model 1. It implies that considering the linear variation of the location parameter, together with the assumption of time-invariant scale and shape parameter, is better for 46.5% of grid points. Similarly, NS Model 2 and NS Model 3 together (linear or exponential variation of scale parameter) were better for another 43.52%. NS models considering the variation of both location parameter and scale parameter together were best fitted only for 10.8% of grid points. From the table, it is also inferred that considering more and more parameters together is not always better for the modelling. Figures 58 display the plots of the S and NS intensities and the percentage variation of NS estimates for each grid point analogous to various combinations of durations and RPs. For the purpose of plotting, 1-, 6-, 12-, and 24-h durations and 2-, 5-, 10-, 25-, 50-, and 100 years RPs were considered.
Table 6

Percentages of grid points fitted to each NS model

Non-stationary modelPercentage of grid points
NS Model 1 46.5 
NS Model 2 21.29 
NS Model 3 22.13 
NS Model 4 5.04 
NS Model 5 5.04 
Non-stationary modelPercentage of grid points
NS Model 1 46.5 
NS Model 2 21.29 
NS Model 3 22.13 
NS Model 4 5.04 
NS Model 5 5.04 
Figure 5

Spatial distribution of RF intensity values at 1-h duration for S and NS cases along with the percentage change.

Figure 5

Spatial distribution of RF intensity values at 1-h duration for S and NS cases along with the percentage change.

Close modal
Figure 6

Spatial distribution of RF intensity values at 6-h duration for S and NS cases along with the percentage change.

Figure 6

Spatial distribution of RF intensity values at 6-h duration for S and NS cases along with the percentage change.

Close modal
Figure 7

Spatial distribution of RF intensity values at 12-h duration for S and NS cases along with the percentage change.

Figure 7

Spatial distribution of RF intensity values at 12-h duration for S and NS cases along with the percentage change.

Close modal
Figure 8

Spatial distribution of RF intensity values at 24-h duration for S and NS cases along with the percentage change.

Figure 8

Spatial distribution of RF intensity values at 24-h duration for S and NS cases along with the percentage change.

Close modal

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.

Table 7

Number of grid points showing different extents of non-stationarity

Percentage variation of non-stationarityPercentage of grid points
<15 77.03 
15–25 15.69 
25–35 4.20 
>35 3.08 
Percentage variation of non-stationarityPercentage of grid points
<15 77.03 
15–25 15.69 
25–35 4.20 
>35 3.08 

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.

Comparison of results using the Mann–Kendall trend test

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.

Table 8

Grid points with significantly increasing trend and more than 15% variation in rainfall intensity

Sl. No.LatitudeLongitudeZ-value% variation
33.5 76.5 2.13 25.47 
31.5 78.5 2.1 15.2 
28.5 72.5 2.67 23.88 
28.5 73.5 2.52 28.15 
27.5 93.5 2.03 17.23 
26.5 88.5 1.98 16.01 
26.5 90.5 2.58 17.22 
26.5 94.5 2.19 16.43 
25.5 70.5 3.77 35.18 
10 25.5 76.5 2.01 18.57 
11 25.5 92.5 3.54 43.83 
12 24.5 80.5 2.04 21.56 
13 24.5 84.5 2.5 18.03 
14 24.5 87.5 2.53 23.86 
15 24.5 88.5 2.61 19.42 
16 24.5 93.5 2.35 19.37 
17 23.5 69.5 2.32 29.87 
18 22.5 75.5 2.01 22.07 
19 22.5 79.5 2.6 21.73 
20 22.5 88.5 1.96 28.5 
21 21.5 77.5 2.64 24.38 
22 20.5 70.5 2.98 39.55 
23 19.5 81.5 2.12 17.07 
24 18.5 79.5 3.33 29.56 
25 17.5 80.5 2.46 19.4 
26 17.5 81.5 2.33 31.56 
27 17.5 82.5 2.6 30.96 
28 15.5 73.5 3.09 19.45 
29 15.5 78.5 2.15 25.29 
30 15.5 79.5 2.09 22.49 
31 10.5 77.5 2.07 21.21 
32 9.5 77.5 2.48 37.72 
Sl. No.LatitudeLongitudeZ-value% variation
33.5 76.5 2.13 25.47 
31.5 78.5 2.1 15.2 
28.5 72.5 2.67 23.88 
28.5 73.5 2.52 28.15 
27.5 93.5 2.03 17.23 
26.5 88.5 1.98 16.01 
26.5 90.5 2.58 17.22 
26.5 94.5 2.19 16.43 
25.5 70.5 3.77 35.18 
10 25.5 76.5 2.01 18.57 
11 25.5 92.5 3.54 43.83 
12 24.5 80.5 2.04 21.56 
13 24.5 84.5 2.5 18.03 
14 24.5 87.5 2.53 23.86 
15 24.5 88.5 2.61 19.42 
16 24.5 93.5 2.35 19.37 
17 23.5 69.5 2.32 29.87 
18 22.5 75.5 2.01 22.07 
19 22.5 79.5 2.6 21.73 
20 22.5 88.5 1.96 28.5 
21 21.5 77.5 2.64 24.38 
22 20.5 70.5 2.98 39.55 
23 19.5 81.5 2.12 17.07 
24 18.5 79.5 3.33 29.56 
25 17.5 80.5 2.46 19.4 
26 17.5 81.5 2.33 31.56 
27 17.5 82.5 2.6 30.96 
28 15.5 73.5 3.09 19.45 
29 15.5 78.5 2.15 25.29 
30 15.5 79.5 2.09 22.49 
31 10.5 77.5 2.07 21.21 
32 9.5 77.5 2.48 37.72 

From Table 8, it can be identified that 32 grid points are showing both significantly increasing trend according to the MK test and significantly higher NS values. It comprises 9% of the total grid points under consideration. The points which are showing an increasing trend in rainfall extremes according to the MK test and also show more than a 15% increase in NS rainfall estimates are shown in Figure 9. It has to be noted that the results from the MK test also support the findings of NS analysis. Among the grid points which were previously found to have more than 15% variation in NS rainfall intensity estimates, 75% of grid points are supported by the MK trend test also.
Figure 9

Grid points showing more than 15% non-stationarity and increasing trend according to the MK trend test.

Figure 9

Grid points showing more than 15% non-stationarity and increasing trend according to the MK trend test.

Close modal

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.

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