A Framework to Develop High-Resolution Intensity–Duration–Frequency Curves: Historical Analysis and Scaling Relationships Integrating Gauge and Gridded Rainfall Products

A Framework to Develop High-Resolution Intensity–Duration–Frequency Curves: Historical Analysis and Scaling Relationships Integrating Gauge and Gridded Rainfall Products | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Framework to Develop High-Resolution Intensity–Duration–Frequency Curves: Historical Analysis and Scaling Relationships Integrating Gauge and Gridded Rainfall Products Yaswanth Pulipati, Balaji Narasimhan, C. Balaji This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9699065/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Advances in satellite rainfall retrieval have made remote sensing estimates increasingly reliable for infrastructure design and planning. Intensity-Duration-Frequency (IDF) curves remain one of the most widely used statistical methods for estimating design rainfall in water resources engineering. In developing countries, meteorological observatories typically have access to daily data and sometimes lack long archives of hourly rainfall data. The current study provides a framework for developing high-resolution IDF curves with durations ranging from 1 to 72 hours at various return periods, integrating gauge and gridded rainfall products (GRPs). The Duration-dependent Generalized Extreme Value (d-GEV) and Gumbel distributions were evaluated for modeling annual maximum rainfall series (AMS) at stations with hourly observations, with the Gumbel distribution emerging as the better choice. Four gridded rainfall products (GPM, GSMaP, PERSIANN, and MSWEP) are evaluated at daily and sub-daily time scales by employing the Gumbel distribution on annual maximum rainfall series to derive the IDF curves. In the first stage, return levels are estimated for 24-, 48-, and 72-hour durations by merging a dense network of daily rainfall data from non-recording rain gauge stations with gridded products using machine learning regression techniques. The scale invariance theory of rainfall is investigated using hourly rainfall data from self-recording rain gauge stations (SRRG) to derive rainfall intensities for sub-daily durations. Scaling behavior was assessed at each SRRG station using non-central moments (NCMs) and was found to perform well for modeling rainfall extremes. The derived scaling exponents are spatially interpolated using potential covariates to develop IDF curves at 10-km resolution. The framework is implemented for the Indian subcontinent, which shows significant variability in rainfall spatial and temporal patterns. The research outputs are available through a user-friendly open web platform, where users can select a location on an interactive map to view IDF curves and confidence intervals via interactive graphs and tables. Hydrology Hydrology Design rainfall Intensity-duration-frequency curves Scale invariance Remote sensing precipitation products Random Forest Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 1. Introduction Hydrological extremes are rare yet cause severe social and economic losses, making it vital to understand them for effective water resource management. Designing critical infrastructure such as stormwater drains, reservoir spillways, and flood protection measures requires quantitative estimates of extreme rainfall. Intensity-Duration-Frequency (IDF) curves (Rodríguez et al., 2014 ) remain the most widely used statistical tool in hydrology and engineering applications. IDF curves depict the relationship between rainfall intensity (i), duration (d), and return period (T), providing valuable inputs for hydrologic design across various timescales and exceedance probabilities. Traditionally, IDF curve construction has relied on dense rain gauge networks. A notable example is NOAA Atlas 14 (Perica et al., 2013 ), which utilized data from 2,846 daily stations (with an average of 63 years) and 994 hourly stations (with an average of 40 years) to provide spatially distributed IDF curves. In regions with sparse gauge networks, constructing IDF curves is more challenging. For instance, IMD produced Isopluvial maps using Annual Maxima Series from 120 self-recording rain gauge (SRRG) stations (Sahoo and Kumar Yadav, 2022 ). Yet, due to heterogeneity in rainfall caused by topography, orography, and distance from the coast, these maps fail to adequately capture spatial variability. Point measurements cannot adequately capture the spatial structure of precipitation, and therefore the quality and accuracy of gauge-based IDF rainfall representations progressively deteriorate as the distance from rain gauge stations increases (Ochoa-Rodriguez et al., 2019 ). Sub-daily IDF derivation is crucial for applications such as urban drainage. Approaches include rainfall disaggregation (Rodríguez-Solà et al., 2017 ), stochastic simulations (Kossieris et al., 2016 ; Li et al., 2017 ), areal reduction factors (ARFs) (Ombadi et al., 2018 ), and regionalization/interpolation (Noor et al., 2021 ). These methods introduce uncertainties. For example, ARFs vary with geography, storm type, and spatial correlation (Ombadi et al., 2018 ). Regionalization, although widely adopted, assumes rainfall homogeneity within defined regions, an assumption that may not hold for short-duration extremes or in complex terrain. Capturing local rainfall variability is often better achieved by interpolating distribution parameters rather than regionalization (Deidda et al., 2021 ). An alternative is the scale-invariance theory, in which the statistical properties of annual maximum rainfall remain consistent across durations. Scaling-based approaches, including mono- and multifractal frameworks, exploit the scale invariance of rainfall extremes to derive sub-daily Intensity–Duration–Frequency (IDF) relationships from coarser temporal data, offering an alternative in regions with limited high-resolution observations (Chandra et al., 2015 ; Rodríguez-Solà et al., 2017 ; Lanciotti et al., 2022 ; Casas-Castillo et al., 2025 ). These methods effectively capture the spatial and climatic variability of rainfall extremes and provide physically consistent IDF estimates across durations and return periods, thereby improving hydrological design and risk assessment in data-scarce and climatically heterogeneous regions. Gridded rainfall products (GRPs) now provide high-resolution data and potential covariates for spatial interpolation. Recent studies (Zhang et al., 2021a ; Lei et al., 2022 ) have employed a Machine Learning (ML) approach that combines satellite rainfall estimates and topographic variables with gauge data to produce merged rainfall estimates. While geostatistical methods like Kriging assume stationarity and isotropy (Chen et al., 2021 ; Zou et al., 2021 ), ML approaches are data-driven, avoid subjective assumptions, and enhance reproducibility (Zhang et al. , 2021b). Several recent studies (Lei et al., 2022 ; Nwaila et al., 2024 ) have further demonstrated that machine learning frameworks can outperform traditional geostatistical techniques in capturing nonlinear precipitation dynamics and reducing interpolation error, particularly in regions with complex terrain or heterogeneous climatic conditions. Despite this promise, only a limited number of studies have explored the derivation of IDF from GRPs. This study develops high-resolution (10 km) IDF curves across daily to sub-daily scales using rainfall frequency analysis, integrating satellite-derived estimates and gauge data via ML regression to improve precision. Section 2 outlines the study framework, area, and extreme value distributions. Section 3 presents parameter estimation using gauges and GRPs, along with scaling-based derivations. Section 4 discusses the results. Section 5 summarizes the findings. 2. Data and Methods 2.1 Study area The study area selected is the region of the Indian sub-continent extending from 66°E to 100°E and from 6°N to 39°N. The climate in India is divided broadly into four seasons, namely (1) Winter (January and February), (2) Pre-monsoon (March to May), (3) Southwest monsoon (June to September), also known as summer monsoon, (4) Post-monsoon (October to December), also known as Northeast monsoon. The vast area of the country receives more than 80% of the annual rainfall during the southwest monsoon season (Sooraj et al., 2020 ). However, the Southeast Peninsula receives substantial rainfall during the retreating (Northeast monsoon) season. The study region comprises diverse climatic regions, with annual average rainfall ranging from 300 mm to 6,000 mm. Due to diverse climatic regions and topography, rainfall distribution in India exhibits notable spatial heterogeneity (Deshpande et al., 2012 ; Jha et al., 2021 ). Deriving high-resolution IDF curves across India is essential in hydro-meteorological and hydraulic design studies. In most regions of India, accessing rainfall IDF curves poses a challenge due to the scarcity of hourly rainfall records and the limited availability of existing curves. The following framework is implemented for the Indian sub-continent to construct IDF curves for 2-, 5-, 10-, 25-, 50-, and 100-year return periods for rainfall intensities obtained over 1-, 3-, 6-, 9-, 12-, 15-, 24-, 48-, and 72-hour durations. 2.2 Framework The framework proposed in the present study for deriving IDF curves includes several stages. The first stage involves deriving parameters of the extreme value distribution with durations of 24, 48, and 72 hours at a 10-km resolution by integrating the properties of multiple gridded rainfall products and a dense network of non-recording rain gauge stations. The availability of daily rainfall data from a dense network of non-recording gauges is a key advantage for frequency analysis. Since rainfall at non-recording stations is recorded at fixed daily intervals, the fixed-window maxima may underestimate the true moving-window design rainfall. To correct this discretization error, the clock hour correction factor (CHCF), defined as the ratio of the sliding window to the fixed window (IS 5542:2003), is applied. CHCF adjusts fixed-interval daily, 2-day, and 3-day maxima to equivalent moving-window maxima (24-, 48-, 72-hour), ensuring corrected values represent true maxima before probability distribution fitting (Ghate and Timbadiya, 2021 ). Return levels are estimated for the 24-, 48-, and 72-hour Durations at 10-km resolution using the derived parameters. The second stage includes deriving the scaling exponents at SRRG station locations using the scale-invariance property of AMS rainfall across the durations. The third stage involves interpolating derived scaling exponents to 10-km resolution across the region using machine learning regression techniques. The final stage entails applying the scaling exponents to derive sub-daily return levels from the 24-hour duration return levels. The framework developed to derive IDF curves is shown in Fig. 1 . 2.3 Gauge observations Historical time series of rainfall observations are obtained from the India Meteorological Department (IMD) for the years 1971 to 2021 (51 years), based on data from 1900 non-recording rain gauges (Fig. 2 a) at the daily temporal scale. The available rain gauge network is denser in the southern peninsular region than in the rest of the country. The IMD daily rainfall records are daily values recorded at 8:30 AM Indian Standard Time, indicating the total rainfall accumulated over 24 hours from the previous day at 8:30 AM. Although daily station data is available from many gauges, the sub-daily data is recorded by a sparse network of automatic rain gauge stations. Hourly rainfall data from 188 self-recording rain gauge stations (SRRG) were obtained from the India Meteorological Department. In the first stage, the station records were screened based on record length and data quality. Quality assessment was performed through internal consistency checks and spatial and temporal consistency analyses. The data is screened for quality based on i) internal consistency check and ii) quality checking in the spatial and temporal domain. A significant amount of rainfall in India occurs from June to December, while summer convective events typically occur in May. We noticed a considerable amount of missing rainfall data during the dry season, particularly from February to April. Since these months generally record negligible precipitation, the missing values are excluded from the analysis. Focusing on the remaining months, if more than 100 hours of rainfall data are missing, the year is considered incomplete and excluded from further analysis. Furthermore, we excluded stations with more than three consecutive missing years. Following this screening, valid data from each SRRG station spanned 10 to 40 years. Of the available 188 stations with hourly rainfall data, 91 stations have satisfied all the above conditions with a minimum of 20 years of data availability, which are shown in Fig. 2 b. The station number, duration of the data available, information about missing data, and the geographic locations of the stations are given in Supplementary Information (Table S1). 2.4 Past methods for IDF curve derivation in India The regional IDF equations developed for hydrological design practice in the Indian subcontinent are described here. Babu et al. ( 1979 ) have provided the IDF curve equation for several Indian cities as follows: $$\:{I}_{t}^{T}=\:\frac{K{T}^{x}}{{(t+b)}^{n}}$$ 1 where, \(\:I\) is the intensity in mm/hr, \(\:T\) denotes the return period, \(\:t\) indicates the storm duration in hours (assumed as equal to time of concentration), \(\:K,\:x,\:b,\:n\) are constants derived for various cities/stations of India. Kothyari and Garde (1992) suggested the following IDF equation: $$\:{I}_{t}^{T}=\:\frac{C{T}^{0.20}\:{\left({R}_{24}^{2}\right)}^{0.33}}{{\left(t\right)}^{0.71}}$$ 2 where \(\:{I}_{t}^{T}\) denotes the intensity in mm/hr, \(\:T\) is the return period in years, \(\:t\) indicates the design storm duration in hours (assumed as equal to time of concentration); C is a constant provided for different geographical regions of India, \(\:{R}_{24}^{2}\) is the magnitude of rainfall in mm for a 2-year, 24-hour duration. The IDF curves developed in this study were validated against estimates derived from Equations 1 and 2 at various geographical locations across India. 2.5 Gridded Rainfall Products from Satellite Retrievals The gridded satellite rainfall products used in this study are Integrated Multi-Satellite Retrievals (IMERG), Global Satellite Mapping of Precipitation (GSMaP), PERSIANN-CCS-CDR (0.04° × 0.04°, 3-hourly), and Multi-Source Weighted-Ensemble Precipitation (MSWEP). The Gridded Rainfall Products (GRPs) from different sources used in the present study are shown in Table 1 , and particulars of these products are provided in the supplementary information (S1): Table 1 Gridded rainfall products selected in the current study Dataset Representation Spatial and temporal resolution Source IMERG Final precipitation GPM 0.1 ◦ ×0.1 ◦ /half-hourly GES DISC https://disc.gsfc.nasa.gov/ GSMaP gauge adjusted GSMaP 0.1 ◦ ×0.1 ◦ /hourly JAXA https://hokusai.eorc.jaxa.jp PERSIANN PCCSCDR 0.04 ◦ ×0.04 ◦ /3 hourly https://doi.org/10.11572/P24W2F MSWEP MSWEP 0.1 ◦ ×0.1 ◦ /3 hourly http://www.gloh2o.org/ 2.6 Return Level Estimation Two common approaches for modeling extremes in hydrology are the block maxima and peaks-over-threshold (PoT) methods. While PoT requires defining thresholds and de-clustering, which is problematic in regions with diverse climatological conditions, the block maxima approach produces an annual maximum series (AMS) that ensures independence and facilitates comparison of rainfall datasets. The Generalized Extreme Value (GEV) distribution (Papalexiou and Koutsoyiannis, 2013 ), which unifies the Gumbel, Fréchet, and Weibull families, is traditionally used in AMS to construct IDF curves. The distribution function of GEV is given as $$\:{F}_{GEV}\left(z;\:\mu\:,\sigma\:,\:\xi\:\right)=\text{exp}\left[-{\left\{1+\xi\:\left(\frac{z-\mu\:}{\sigma\:}\right)\right\}}_{+}^{-\frac{1}{\xi\:}}\right]\:\:\:\:\:\:\:\:\:\left(3\right)$$ for \(\:z\:ϵ\:R\) and \(\:{y}_{+}=\text{m}\text{a}\text{x}(y,0)\) for \(\:y=1+\:\xi\:\:.\:\frac{z-\mu\:}{\sigma\:}\:ϵ\:R\) and µ, σ, and ξ are the location, scale, and shape parameters, respectively. For ξ > 0 and ξ < 0, the GEV distribution reduces to the Fréchet and Max-Weibull distributions, respectively (Ragulina and Reitan, 2017 ). The GEV distribution reduces to the Gumbel distribution when the shape parameter, ξ, equals 0. The AMS \(\:({z}_{1},{z}_{2},{z}_{3}\dots\:{z}_{T})\) for a specific \(\:d\) duration is obtained by summing hourly rainfall over a sliding window of width \(\:d\) and selecting the maximum value for each year. The shape parameter (ξ) of the GEV distribution plays a critical role in determining the tail behavior of the distribution, and this becomes more uncertain with shorter data records, leading to less reliable return level estimates in regions with limited data (Papalexiou and Koutsoyiannis 2013 ). Therefore, the current study employs the Gumbel distribution to model the annual maximum rainfall. The return levels for a \(\:d\) duration maxima are obtained from the following equation. $$\:{z}_{T}=\mu\:-\sigma\:\text{log}\left\{-\text{log}\left(1-\frac{1}{T}\right)\right\}\:$$ 4 where \(\:{z}_{T}\) is the return level for a return period \(\:T\) . This study compared return level estimates at SRRG stations using the Gumbel (GEV1) and Fréchet (GEV2) distributions. Due to limited record length, the Gumbel distribution is used provisionally. However, this may underestimate return levels for very long return periods (≥ 100 years). Studies (Papalexiou and Koutsoyiannis, 2013 ) indicate that estimation of positive shape parameters generally requires much longer records, making the Gumbel assumption a stable alternative. This effect is illustrated through return-level comparisons at the SRRG stations. 2.6.1 Duration-dependent GEV The present study also considered the Duration-dependent GEV (d-GEV) model formulated by Koutsoyiannis et al. ( 1998 ), which describes the complete IDF relation for various return periods over different durations. The underlying concept of d-GEV is the efficient parameterization of the GEV distribution using three to five parameters. The five-parameter d-GEV shown in Eq. 5 is applied to derive duration-dependent IDF curves. $$\:G\left(z,d;\stackrel{\sim}{\mu\:},{\sigma\:}_{o},\:\xi\:,\:\theta\:,\:{n}_{d}\right)=exp\left\{-{\left[\xi\:\:\left(\frac{z}{{\sigma\:}_{o}\left({\left(d+\:\theta\:\right)}^{-{n}_{d}}\right)}-\:\stackrel{\sim}{\mu\:}\right)\right]}^{\frac{-1}{\xi\:}}\right\}\:\:\:\:\:\left(5\right)$$ Here \(\:\stackrel{\sim}{\mu\:}\) represents the re-parameterized location parameter, \(\:\theta\:,\:\) and \(\:{n}_{d}\) represent duration offset and duration exponent, respectively. The scale parameter \(\:\sigma\:\) is assumed to follow a power law relation \(\:\:\sigma\:={\sigma\:}_{o}\:{\left(d+\:\theta\:\right)}^{-{n}_{d}}\) with duration \(\:d\) and scale offset \(\:{\sigma\:}_{o}\) remain constant across all the durations. 2.7 Parameter estimation Several established methods exist for estimating parameters of the generalized extreme value (GEV) distribution fitted to annual maximum rainfall series, including Bayesian approaches, maximum likelihood estimation (MLE), generalized maximum likelihood estimation (GMLE), and L-moments. Among these, the L-moments method (Hosking, 1990 ; Papalexiou and Koutsoyiannis, 2013 ) is particularly robust and often preferred for small to moderate sample sizes (n < 100), as it provides more stable and less biased parameter estimates compared to likelihood-based methods, especially for the shape parameter (Hosking, 1990 ; Papalexiou and Koutsoyiannis, 2013 ). Given the typical record lengths in this study (30–50 years), which fall well within this range where L-moments outperform MLE in terms of bias and variance, the L-moments approach was selected for parameter estimation of the GEV distribution applied to the annual maximum series (AMS) data. 2.8 Uncertainty considering the record length In this study, the bootstrap resampling procedure proposed by Marra et al. ( 2017 ) was adopted to assess the uncertainty associated with the record length of AMS. In this bootstrap method, each of the original annual maximum rainfall data series was resampled without replacement to generate multiple bootstrap samples. These bootstrap samples are then used to estimate return levels for various return periods. By repeating this sampling process 1000 times, a distribution of return levels was obtained, allowing uncertainty estimation. The 5th -95th quantile interval of the bootstrap sampling is used to quantify the uncertainty associated with the undersampling of rainfall climatology caused by short data sets. Both Gumbel and d-GEV models were applied across 91 SRRG stations. The selection of the most suitable model was determined by evaluating performance indicators and accounting for uncertainties arising from record length. 2.9 Evaluation of GRP's distribution parameters against gauge data Numerous studies (Marra et al., 2019; Noor et al., 2021 ) have recently examined satellite-based rainfall products, and these studies consistently report random and systematic errors in satellite estimates that are influenced by resolution, retrieval algorithms, topography, and climatology. It is therefore essential to evaluate each product's performance across the study region. In this study, the location and scale parameters of the AMS distribution were derived at gauge stations and collocated GRP grids to assess the ability of GRPs to reproduce observed parameters. As the gauge data period does not always coincide with the gridded data, the comparison primarily aims to evaluate the degree of agreement between distribution parameters and quantify the quality of spatial matching. The accuracy of the fitted extreme value distributions and the consistency between gauge-based and satellite-derived Annual Maximum Series (AMS) rainfall were evaluated using a suite of standard performance metrics, including ME, RMSE, PBias, MAE, CC, IA, and KGE. Additionally, the statistical validity of the models was confirmed through the Anderson–Darling and Kolmogorov–Smirnov goodness-of-fit tests(De Michele and Avanzi, 2018 ). These metrics evaluate bias, accuracy, and distributional conformity (see Supplementary Information S2 for equations and procedures). 2.10 Developing the location and scale parameters for a 24-hour duration at 10 km resolution across India Probability distributions were fitted to rainfall maxima after applying the clock-hour correction factor per IS 5442:2003 to convert fixed-interval daily, 2-day, and 3-day rainfall to equivalent 24-, 48-, and 72-hour moving-window maxima. Gumbel distributions were fitted to data from 1,900 IMD gauge locations to estimate location and scale parameters. These were spatially interpolated to 10-km resolution using Random Forest (RF) regression incorporating geographically relevant covariates. Similarly, distribution parameters for GPM, GSMaP, and MSWEP were derived at 10-km resolution, with PCCSCDR values mapped to the same grid for consistency. The 24-hour rainfall intensity parameters at each gauge station serve as predictands, while the predictors include gridded parameters and auxiliary variables. The Random Forest (RF) algorithm constructs an ensemble of decision trees trained on bootstrap samples, with model performance internally assessed using out-of-bag (OOB) data (Zhang et al. , 2021b). The RF model requires specification of several hyperparameters, including the number of trees ( \(\:ntree\) ), the maximum tree depth, and the number of features considered at each split ( \(\:mtry\) ). In this study, the number of features at each split was fixed to one-third of the total predictors (Zhang et al. , 2021b), and trees were grown to full depth without explicit pruning. Among the hyperparameters, only the number of trees was tuned, which was optimized using k-fold cross-validation to ensure stable and generalized performance. After identifying the optimal ntree, the final RF model was retrained using the complete dataset. Predictors include latitude, longitude, elevation, and distance to the coast. Additional covariates are derived from ETCCDI indices: the median Simple Daily Intensity Index (M_SDII) and annual maximum 1-day rainfall (M_AMS) from each gridded product. Including an excessive number of predictors in ML can increase model complexity, slow down the training process, and introduce noise. To mitigate this, the RF model identifies the most relevant features using Permutation Feature Importance (Zhang et al., 2021). To account for high spatial variability, parameters were interpolated within homogeneous areas delineated through K-means clustering approach based on extreme rainfall characteristics. RF regression was applied within each cluster to ensure variables are tailored to regional variability. The resulting 24-hour distribution parameters are termed merged parameters as they integrate properties from several datasets. 2.11 Developing IDF curves at 10 km resolution 2.11.1 Applying the property of scale-invariance to derive d-duration return levels Scaling implies that the same underlying relationship governs the statistical characteristics of a process across different temporal scales (Pöschmann et al., 2021 ). In the context of extreme rainfall, scale invariance indicates that the statistical properties of annual maximum rainfall intensities at shorter durations can be reliably derived from those at longer durations through a power-law scaling relationship. This concept has been successfully applied in several studies (Chandra et al., 2015 ; Bairwa et al., 2016 ; Ghanmi et al., 2016 ) to develop IDF relationships from daily rainfall data. In the present study, scale invariance for 1–24 hour durations was confirmed through the linear behavior of non-central moments in log–log space and the linear dependence of scaling exponents on moment order. Assessing whether rainfall exhibits simple or multi-scaling behavior is necessary before constructing IDF curves using the invariance approach. While multi-scaling behavior has been reported in earlier studies (Fauer et al., 2021 ) at sub-hourly scales (finer than 1-hour), a single power-law relationship often sufficiently characterizes the statistical moments of extreme rainfall in the hourly-to-daily regime (Chandra et al., 2015 ; Maity and Maity, 2022 ). The theoretical details and formulation of the scale-invariance method employed here are provided in the supplementary document (Section S4). In the present analysis, scaling diagnostics were evaluated over the considered time range (1–24 hours), and the simple scaling assumption was found adequate. When departures from simple scaling are identified, a piecewise or multi-scaling framework is typically adopted, in which distinct scaling exponents are estimated for short-duration events (e.g., < 1 hour), thereby preserving regime-dependent scaling behavior when deriving sub-daily intensities from daily extremes (Chandra et al., 2015 ). The return levels estimated using the scaling method are compared with those from the Gumbel distribution at SRRG stations with historical hourly rainfall records. This helps in evaluating the performance of the invariance method in reproducing the observed IDF curves at SRRG stations. 2.12 Estimation of scaling exponent and return level magnitudes at 10 km resolution for IDF generation Estimating return levels at ungauged locations requires transferring scaling exponents using interpolation techniques. The scaling exponents from the hourly gauge stations are interpolated across the study region using appropriate explanatory/auxiliary variables. The analysis revealed that the scaling exponents exhibit strong correlations with the location and scale parameters of the 24-hour Gumbel distribution parameters. The integration of gridded rainfall products, as detailed in the preceding section, enables the derivation of location and scale parameters at a consistent 10-km resolution across the Indian subcontinent. During the study, it is also noted that the duration exponent \(\:{n}_{d}\) obtained by fitting the D-GEV distribution (Eq. 5) has characteristics similar to the scaling exponents \(\:k\left(q\right)\) acquired by the scale invariance theory. Hence, \(\:{n}_{d}\) obtained from all four gridded rainfall products are also used as predictors, assuming this variable can act as a potential covariate. The scaling exponents available at SRRG stations are thus interpolated to the 10-km resolution using appropriate covariates. Then, the return level magnitudes at 10 km resolution are obtained by applying the scaling exponents to the 24-hour-duration estimated rainfall intensities. 2.13 Estimation of confidence intervals for return level estimates The study uses parametric bootstrapping to derive confidence intervals (CIs) for IDF curves at each grid point. Bootstrap methods are widely used when analytical CIs are challenging to obtain (Carpenter and Bithell, 2000 ; Huang et al., 2019 ). As a resampling approach, the bootstrap uses computer simulations to estimate uncertainty in distribution parameters and quantiles. Prior studies have shown that the parametric bootstrap often outperforms the non-parametric approach (Kyselý, 2008 ). While the non-parametric version relies on resampling with replacement, the parametric version generates random samples from a fitted distribution. When an appropriate model is known, data are limited, and tail behavior is critical (e.g., IDF estimation), the parametric bootstrap is preferred (Kyselý 2008 ). In this study, pdf parameters are first estimated using scaling exponents at 10-km resolution, and parametric bootstrapping is then applied to compute CIs for return levels. 3. Results and Discussion 3.1 D-GEV vs GEV1 (Gumbel) IDF curves were developed at 91 locations using hourly rainfall data with both the Gumbel and D-GEV distributions. Model performance was evaluated using Q-Q plots generated with the Weibull plotting position formula (Smitha et al., 2018), alongside Percent Bias and Mean Absolute Error (MAE) to quantify deviations between simulated quantiles and observed Annual Maximum Series (AMS) values. Analysis of Mean Error (ME) and Root Mean Square Error (RMSE) across all stations (Fig. 3 ) initially indicated that the Gumbel (GEV1) model yielded lower ME values, suggesting better fit to the observed data. Given the limited record lengths available from both gauge stations and GRPs, assessing the uncertainty associated with return level estimates is critical, as shorter time series can significantly influence the reliability of extreme value analysis. To address the impact of limited record lengths on extreme value reliability, a bootstrap resampling analysis (999 repetitions) was conducted. Using the 5th–95th quantile interval to quantify sampling uncertainty, results from six representative stations (Fig. S1) showed that for records of 20 years, the Gumbel distribution had narrower uncertainty for return periods between 2 and 75 years. Consequently, based on the return-level estimates and favorable Q-Q plot results, the study adopted the Gumbel distribution. Its statistical adequacy was further confirmed across all stations using Anderson–Darling and Kolmogorov–Smirnov tests. 3.2 Deriving Homogeneous Regions Using Extreme Rainfall Characteristics The median annual maximum 1-day rainfall (M_AMS) and Simple Daily Intensity Index (M_SDII) for 24-hour duration were computed from four gridded rainfall products: IMERG Final, GSMaP Gauge Adjusted, PERSIANN, and MSWEP (Fig. 4 ). Both M_AMS and M_SDII show prominently high values over the Western Ghats and northeastern India, with markedly lower values in the rain-shadow regions of the southern peninsula. These indices, along with mean annual rainfall, served as input variables in the clustering algorithm to delineate homogeneous rainfall regions across the country. The Silhouette Score (Fig. S2) indicates that the optimal k-value is nine, resulting in the delineation of nine homogeneous zones, as depicted in Fig. 5 . The outcomes of the cluster analysis align reasonably well with homogeneous rainfall zones identified in previous studies (Falga and Wang, 2022 ; Sahoo and Kumar Yadav, 2022 ), supporting the physical consistency of the regionalization. The delineation of homogeneous rainfall regions is not only used for characterization but also provides the basis for region-specific modeling. Given the strong spatial heterogeneity in rainfall-generating mechanisms across India ranging from orographic enhancement in the Western Ghats to convective and monsoon-core processes in central and northeastern India and rain-shadow effects in the peninsular interiors the modelling framework was implemented independently for each identified cluster. This cluster-wise approach enabled the capture of localized statistical relationships between the predictors and the characteristics of extreme rainfall. To convert the 1-day observed maximum rainfall to the 24-hour true cumulative rainfall, IS 5542:2003 advises using a CHCF value of 1.15 across the entire Indian region. However, it is recommended to use site-specific CHCF values for reliable estimates of extreme rainfall. Hence, in the present study, CHCF values are derived at each SRRG station for 24-, 48-, and 72-hour durations, and the mean CHCF values are subsequently determined for each cluster. Table S2 summarizes the number of available daily rain gauge stations per cluster, the mean values of the Gumbel location and scale parameters (for 24-hour duration), and the CHCF values for 24-, 48-, and 72-hour durations. 3.3 Evaluation of GRPs with rain-gauge data Recent studies (Lin and Huybers, 2019 ; Pradhan and Indu, 2021 ) have demonstrated the capability of satellite rainfall estimates to effectively capture the spatial distribution of rainfall in the Indian domain. Figure 6 and Fig. S3 present the 2-year return level maps of GRPs for 3-hour and 24-hour durations, respectively. The coastal and northeast regions exhibit higher return-level magnitudes across all GRPs. The inland areas of the southern peninsular regions exhibit similar patterns across all GRPs. Overall, the spatial distribution of return-level magnitude matches the pattern of extreme rainfall characteristics (Fig. 4 ) derived from gridded rainfall data. 3.3.1 Evaluation of location and scale parameters of GRPs The location and scale parameters of the Gumbel distribution estimated from the GRPs are compared with those derived from SRRG station data using standard performance metrics. Table 2 and Table S3 present the performance evaluation of location and scale parameters, respectively. Higher IA and CC values between GRP and SRRG stations in Table 2 indicate that GRPs reasonably capture the spatial gradient of extreme rainfall. GRP performance is notably better in humid regions than in arid regions, consistent with findings of Pradhan and Indu ( 2021 ). MSWEP consistently outperforms other products (GPM, PCCSCDR, GSMaP) in estimating Gumbel location and scale parameters across clusters, showing the lowest average bias (ME), highest IA and CC, and lowest RMSE. Collectively, these results show that GRP accuracy improves markedly with increasing duration. Yet, MSWEP consistently yields the highest agreement and lowest error in representing the spatial variability of Gumbel parameters across diverse climatic regimes. GRP performance was further assessed at SRRG locations by comparing distribution parameters and return levels from collocated datasets. Detailed evaluation results and statistical summaries are provided in Supplementary Section S5 (Supplementary Figures. S4–S7). Table 2 Performance assessment of the location parameter obtained from GRPs against the gauge station for 24-hour duration in different homogenous regions. Cluster region ME. IA CC RMSE Cluster region ME. IA CC RMSE C1_GPM -0.82 0.81 0.75 1.52 C5_GPM -0.63 0.65 0.62 0.56 C1_PCCSCDR 0.57 0.50 0.48 1.80 C5_PCCSCDR -1.28 0.53 0.49 0.56 C1_MSWEP -0.34 0.88 0.80 1.39 C5_MSWEP 0.02 0.71 0.65 0.64 C1_GSMaP 0.55 0.71 0.57 1.71 C5_GSMaP -1.91 0.74 0.68 0.60 C2_GPM -0.06 0.73 0.55 0.45 C6_GPM -0.61 0.36 0.24 0.84 C2_PCCSCDR -0.71 0.54 0.41 0.89 C6_PCCSCDR -1.05 0.40 0.22 1.24 C2_MSWEP 0.42 0.65 0.50 0.64 C6_MSWEP 0.38 0.52 0.45 0.70 C2_GSMaP -0.36 0.60 0.42 0.63 C6_GSMaP -0.12 0.65 0.48 0.61 C3_GPM -0.60 0.66 0.57 0.88 C7_GPM -0.76 0.52 0.41 1.17 C3_PCCSCDR -1.22 0.54 0.57 1.42 C7_PCCSCDR -1.15 0.64 0.67 1.39 C3_MSWEP 0.02 0.78 0.65 0.56 C7_MSWEP 0.32 0.67 0.47 0.92 C3_GSMaP -1.82 0.51 0.61 2.01 C7_GSMaP -0.02 0.76 0.58 0.79 C4_GPM -0.76 0.64 0.44 1.06 C8_GPM -1.37 0.48 0.35 1.52 C4_PCCSCDR -0.61 0.49 0.25 1.09 C8_PCCSCDR -0.87 0.47 0.32 1.16 C4_MSWEP 0.52 0.65 0.48 0.90 C8_MSWEP 0.20 0.60 0.39 0.60 C4_GSMaP -0.32 0.51 0.38 1.04 C8_GSMaP -0.57 0.39 0.01 0.98 C9_GPM -1.23 0.67 0.66 1.82 C9_PCCSCDR -0.78 0.60 0.44 1.81 C9_MSWEP 0.25 0.82 0.71 1.27 C9_GSMaP 0.78 0.66 0.54 1.69 3.4 Developing location and scale parameters for 24-, 48-, and 72-Hour Rainfall duration at 10 km resolution using ML regression technique The location \(\:{(\mu\:}_{24-merged},{\mu\:}_{48-merged},{\mu\:}_{72-merged})\) and scale parameters \(\:({\sigma\:}_{24-merged},{\sigma\:}_{48-merged}\) , \(\:{\sigma\:}_{72-merged}\) ) are derived at 10 km resolution for each cluster using appropriate covariates as input to the RF regression model, as explained in section 2.10. Table S6 presents the Permutation Feature Importance (PFI) metrics for predictors in the Random Forest (RF) framework used to estimate 24-hour Gumbel distribution parameters. High PFI values indicate strong influence, with elevation and distance to the coast as key contributors in specific clusters. Predictors with PFI values close to zero are excluded. Consistency across 48-hour and 72-hour durations suggests identical governing factors. Given the high performance of MSWEP, the M_SD11 and M_AMS indices are provided as model inputs. The final predictor combinations are in Tables S7 and S8. Cluster-specific RF configurations were implemented with optimized ntree values reported in Table S9. The predictive performance of the RF model was evaluated using the statistical metrics summarized in Tables S10 through S12. Higher CC and IA values across these tables indicate strong agreement between RF-derived parameters and station data for all durations. Location and scale parameter scatter plots (Fig. S8) show IA values from 0.83 to 0.97 for training and 0.82 to 0.95 for testing, indicating good agreement. Mean Error values (Table S10) show a notable reduction compared to gridded products (Table 2 ), highlighting improved reliability. For brevity, 24-hour location and scale parameters at 10-km resolution are shown in Fig. S9. These parameters exhibit a spatial pattern consistent with the observed extreme rainfall climatology over India. 3.5 Verification of the scale-invariance property The scale invariance of rainfall was assessed over durations ranging from 1 to 24 hours. As illustrated in Fig. 7 a, the log-transformed first three non-central moments (NCM1, NCM2, and NCM3) exhibit a distinct linear relationship with the log of rainfall duration, confirming that the statistical moments scale consistently across the temporal domain. Furthermore, we observed that the regression relationship between log-transformed values has a coefficient of determination (R²) ≥ 0.96 across all SRRG stations, indicating the strength of the linear relationship. Furthermore, the scaling exponents, derived from the slopes of the linear fits in Fig. 7 a, exhibit a linear dependence on the moment order (q), as shown in Fig. 7 b. The findings confirm a simple scaling (monofractal) regime for the 1–24 hour interval. These findings are consistent with earlier studies (Maity and Maity, 2022 ), who also reported a strong scaling relationship between hourly and daily precipitation extremes across the Indian region, confirming the presence of simple scaling behavior in rainfall intensity–duration relationships. The spatial distribution of the scaling exponents for the 1–24 hour duration is presented in Fig. 8 . The exponents exhibit a regional gradient across India. Lower values predominate along the Western Ghats, eastern coastal regions, and Northeast, reflecting rainfall characteristics associated with orographic enhancement and convective processes during the monsoon. In contrast, higher exponents characterize the central and northwestern arid-to-semiarid zones, with rainfall regimes in rain-shadow and continental interior areas. Figure S10 provides a visual comparison between 100-year return period rainfall intensities estimated via scale-invariance theory and those derived from conventional Gumbel analysis at two selected rain gauge stations. The generated IDF curves closely match the observed ones. Figure S11 presents a comparative performance assessment of rainfall intensity estimates derived from the scale-invariance theory and those obtained from the Gumbel distribution across 91 SRRG stations. Panels (a) and (b) depict the Kling-Gupta Efficiency (KGE) for 1-hour and 3-hour durations, respectively, while panels (c) and (d) illustrate the Mean Absolute Error (MAE) for the same durations. Higher KGE values, together with lower MAE, indicate that the scale-invariance model effectively reproduces observed rainfall intensities, supporting its reliability for deriving IDF curves. 3.6 Interpolation of scaling exponents to the 10 km resolution The methodology for interpolating station-based scaling exponents onto a 10-km grid and its application for estimating short-duration return levels are outlined here. Correlation analysis (Table 3 ) shows strong relationships between scaling exponent, 24-hour parameters, and the duration exponent ( \(\:{n}_{d}\) ) from the D-GEV distribution, aligning with the findings of Menabde et al. ( 1999 ). Consequently, these variables were selected as potential predictors for the Random Forest model, given their influence on the distribution parameters. The RF regression is used to interpolate the scaling exponent at 10-km resolution, using covariates selected from collocated GRP pixels corresponding to the 91 station locations. The regression equation using RF can be expressed in functional form as: $$\:k\left(q\right)=f\left({\mu\:}_{24-merged},{n}_{d}\left(GPM\right),{n}_{d}\left(MSWEP\right),{n}_{d}\left(PCCSCDR\right),{n}_{d}\left(GSMap\right)\right)$$ The data is split into 70% for RF model training and 30% for model testing. We ensured that training and test data contained stations from different clusters at this study point. Table 3 Correlation values between scaling exponents \(\:k\left(q\right)\) obtained from scale invariance theory approach with selected covariates Parameter \(\:\varvec{k}\left(\varvec{q}\right)\) (SRRG) µ 24-hour merged η d GPM η d MSWEP η d PCCSCDR η d GSMaP \(\:\varvec{k}\left(\varvec{q}\right)\) (SRRG) 1.00 µ 24hour merged -0.76 1.00 η d GPM 0.86 -0.75 1.00 η d MSWEP 0.87 -0.76 0.85 1.00 η d PCCSCDR 0.70 -0.61 0.53 0.54 1.00 η d GSMaP 0.73 -0.64 0.58 0.57 0.51 1.00 The performance of the Random Forest regression model in capturing observed scaling exponents is illustrated in Figure S12. The interpolated values demonstrate a high degree of agreement with station-based observations, with most data points clustering closely along the 1:1 reference line, indicating minimal bias in the spatial scaling framework. To mitigate the risk of overfitting, a k-fold cross-validation framework was utilized, providing a more reliable estimate of model performance than a single train-test split. The data from 91 stations are partitioned into 4 folds (22, 22, 22, and 25 samples, respectively). The RF model is trained on three folds and validated on the remaining fold using a four-fold cross-validation approach. The results of k-fold cross-validation (Fig. S13) show that the RF regression model can reasonably predict the data without overfitting. The predicted scaling exponents across the study region at 10-km resolution are shown in Fig. S14. Building on the 24-hour return levels derived in the previous section, the interpolated scaling exponents based on scale invariance theory were applied to estimate short-duration (1-, 3-, 6-, 9-, 12-, and 18-hour) return levels. The rainfall intensities for d-duration, estimated using the scale-invariance approach, were subsequently compared with those derived from SRRG station data based on the Gumbel distribution. Furthermore, the rainfall intensities from parametric relationships suggested by Ram Babu et al. ( 1979 ) and Kothyari and Garde (1992) for the Indian region are also evaluated against those obtained from the Gumbel-based analysis. The IDF curve derived from the proposed methodology at the grid point closest to the corresponding SRRG station is selected for comparative evaluation. Figure 9 illustrates the 100-year return period IDF curves derived using the present approach, along with at-site estimates and traditional empirical formulations by Ram Babu et al. ( 1979 ) and Kothyari and Garde (1992). The comparison indicates that the proposed method closely follows the at-site IDF curves across durations, demonstrating its capability to reliably reproduce observed rainfall intensities. Additionally, the current estimates are slightly higher than the at-site curves, particularly at shorter durations, suggesting a conservative bias. From a design perspective, this is advantageous as it ensures a safer margin against the empirical formulations, which tend to underestimate intensities. In contrast, traditional techniques significantly underestimate rainfall intensity. The performance assessment is conducted using RMSE, and the results are shown for 10-year and 50-year return period curves at SRRG stations. Figure 10 presents the spatial distribution of Mean Error (ME) in estimated rainfall intensity (mm/h) across India for the proposed current procedure and two widely used traditional empirical methods. As shown in Fig. 10 (a), the current procedure demonstrates superior performance, with mean error (ME) values exhibiting only a slight positive bias ranging from 0 to + 4 mm/h. From a structural design perspective, this modest positive bias is highly desirable, as it naturally incorporates a safety margin against underestimation of peak rainfall intensities. In contrast, the traditional methods proposed by Ram Babu et al. (Fig. 10 (b)) and Kothyari and Garde (Fig. 10 (c)) exhibit consistently large negative ME values, reaching as low as − 14 mm/h in several regions. This systematic underestimation indicates that these empirical formulae tend to under-predict short-duration, high-intensity rainfall events over much of the Indian subcontinent. Consequently, infrastructure designed using these traditional approaches may be inadequately sized, potentially compromising the safety and resilience of drainage systems. Figure 11 presents the 10 km spatial distribution of 1-hour rainfall intensities over India, showing high intensities in the Western Ghats and Northeast states and lower intensities in inland regions. Figures S15 and S16 show that the 95% confidence interval (CI) width is greater for shorter durations and higher return periods. These CIs reveal spatial and duration-dependent patterns where the widest 1-hour CIs (typically > 15 mm) are concentrated along the Western Ghats, Northeast India, and the eastern coastal belt, reflecting high estimation uncertainty in orographic and intense convective rainfall regimes. In contrast, narrower CIs dominate the central plains, northwestern arid zones, and the southern interior. For practical applications, engineers should adopt conservative estimates, such as the upper 95% CI bound or a safety factor, particularly in high-uncertainty orographic zones or for long-return-period designs. While current estimates may be sufficient for preliminary assessments, critical infrastructure should incorporate upper CI or safety-adjusted values to ensure resilience against extreme rainfall events. To demonstrate the practical implications of the proposed framework, Figure S17 presents the derived IDF curves for two cities located in contrasting climatic regimes of India. The coastal setting of Chennai (Fig. S17a) exhibits higher rainfall intensities, while comparatively lower intensities are evident over the inland city of Bangalore (Fig. S17b). The proposed framework effectively captures the spatial gradients in rainfall intensities across the study region. It can be extended to other geographical regions and is particularly suitable for areas where dense sub-hourly or hourly rainfall observations are unavailable. 4. Discussion and conclusion This research develops a methodological framework to generate high-resolution IDF curves by integrating gauge and satellite-retrieved rainfall, addressing the lack of dense, long-term in situ records and the limitations of sparse, unevenly distributed rain gauge networks. By capturing spatial rainfall variability through the combination of gridded rainfall products and gauges, the approach provides reliable estimates of extreme rainfall and a feasible solution for deriving sub-daily to daily IDF curves for hydrological design and climate risk assessment. The Gumbel (GEV1) and D-GEV distributions were tested on AMS data from SRRG stations, with the Gumbel distribution found to be most suitable. Initially, the study estimated the 24-hour distribution parameters at a 10 km spatial resolution using a Random Forest regression model, with predictors derived from gridded rainfall datasets and topographic variables including latitude, longitude, and elevation. Then, scale-invariance theory (Rodríguez-Solà et al. 2017 ) was applied to obtain d-hour return levels. Interpolating 24-hour parameters using homogeneous regions proved effective, since the influence of covariates varies with factors such as elevation and coastal proximity. Overall, the results confirm that scale-invariance theory, combined with gauge–satellite integration, enables the reliable generation of high-resolution IDF curves, providing a practical tool for hydrological applications in data-sparse regions. The current framework (with spatial interpolation and disaggregation) already provides a slight conservative overestimation of design rainfall, with the mean error found to be positive at most locations, in contrast to the significant negative bias exhibited by the traditional methods; this positive bias is preferable from an engineering safety perspective as it naturally incorporates an additional safety margin. Evidence from long record analyses suggests that extreme rainfall often exhibits a weakly positive shape parameter, implying a Fréchet distribution rather than a bounded upper tail. Consistent with Papalexiou and Koutsoyiannis ( 2013 ), the Gumbel assumption remains a stable alternative when reliable estimation of ξ is constrained by record length. They argue that a bounded upper tail is physically inconsistent for rainfall extremes and recommend using Gumbel or a fixed positive shape parameter of 0.114 for safety when negative estimates arise, noting that the slow convergence to asymptotic behavior explains the historical preference for Gumbel despite its limitations. In the present study, Gumbel was adopted for return level estimation to provide conservative results given the high uncertainty in shape parameter estimation from limited records (30–48 years), while bootstrap estimates and their confidence intervals were analyzed to assess the plausibility of ξ ≈ 0 (Figures S18a, S18b; Table S13). The spatial distribution of the parameters reveals values clustered near zero or slightly positive across most regions. However, given the limited record lengths available at individual SRRG stations, the associated confidence intervals frequently encompass zero. The spatial distribution of differences in return level estimates (GEV2 − current approach) at SRRG station locations is shown in Fig. S19. Panels (a) and (b) present results for the 1-hour duration at 10-year and 100-year return periods, respectively, while panels (c) and (d) show the corresponding differences for the 24-hour duration. Differences between the two approaches remain modest across most stations and return periods. The current framework (with spatial interpolation and disaggregation) already provides a slight conservative overestimation of design rainfall, which is preferable from an engineering safety perspective. Even when a positive shape parameter is used in GEV2, the return levels show limited deviation from the current estimates. In contrast, a negative shape parameter in GEV2 leads to underestimation of extremes, an outcome that is neither physically realistic nor recommended in the literature. These findings indicate that adopting a more complex GEV2 formulation offers negligible improvement for practical design purposes under the present framework. While the current study focuses on the application of Random Forest for spatial mapping, future research will incorporate a comparative assessment with alternative spatial estimation techniques, including geostatistical approaches such as Kriging, to further evaluate their relative performance across India's diverse climatic zones. Although the current framework applies cluster-specific modelling to represent regional hydroclimatic heterogeneity, additional assessment through spatial cross-validation strategies, such as leave-region-out experiments, is recommended to quantify model transferability under ungauged conditions. Building on this framework, further work will also evaluate Bayesian estimation with weakly informative priors, regional frequency analysis, and the use of extended homogenized records to constrain positive shape contributions better and address non-stationarity. Recent studies (Dargham and Andraos, 2026 ) have also highlighted the potential of machine learning and deep learning approaches to improve IDF curve estimation, demonstrating reliability under nonstationary rainfall conditions and offering promising directions for future research. The results of the present study are accessible through a publicly available web-based tool ( https://www.idf-iitmadras.in/ ). Users can select their desired location on the interactive map, and the tool generates IDF curves along with their corresponding confidence intervals. The results in this web-based tool are displayed in both tabular and graphical formats. Declarations This manuscript is a preprint currently under peer review at the International Journal of Climatology. Supporting Information Additional figures, tables, and methodological details supporting this study are provided in the supplementary document. Acknowledgements The authors would like to acknowledge and thank the scholarship support provided by the Ministry of Human Resource Development (MHRD), Government of India. The authors would also like to thank Vikash Raaj for his support in developing the web-based visualization platform used for disseminating the study results. Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. Competing interests The author declares no known competing financial interests or personal relationships that could have influenced the work reported in this paper. Author contributions Balaji Narasimhan, Yaswanth and C Balaji , contributed to the study conception and design. Yaswanth P conducted material preparation, analysis, performance evaluation and derivation. The first draft of the manuscript was prepared by Yaswanth P and all authors commented on previous versions of the manuscript. Balaji Narasimhan, Yaswanth P and C Balaji approved the final manuscript. Funding This research received funding from the Department of Science and Technology, Ministry of Science and Technology, Government of India, under Grant/Award Number: DST/CCP/CoE/141/2018(G). Availability of data and materials The results of the present study are accessible through a publicly available web-based tool (https://www.idf-iitmadras.in/). References Babu R, Tejwani KG, Agarwal MC, Bhushan LS (1979) Rainfall intensity-duration-return period equations and nomographs of India. 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Int J Climatol 40(12):5223–5245. https://doi.org/10.1002/joc.6516 Zhang L, Li X, Zheng D, Zhang K, Ma Q, Zhao Y, Ge Y (2021a) Merging multiple satellite-based precipitation products and gauge observations using a novel double machine learning approach. Journal of Hydrology . Elsevier B.V., 594(November 2020): 125969. https://doi.org/10.1016/j.jhydrol.2021.125969 Zou Wyue, Yin S, qing, Wang W (2021) ting. Spatial interpolation of the extreme hourly precipitation at different return levels in the Haihe River basin. Journal of Hydrology . Elsevier B.V., 598: 126273. https://doi.org/10.1016/j.jhydrol.2021.126273 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9699065","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":639443149,"identity":"cea95f9b-0c1e-41cb-9fa2-1f04886f4fa1","order_by":0,"name":"Yaswanth Pulipati","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVRIiWNgGAWjYBCDBAYG5oYDQAYPP5hbQFCHAVALI0iLAY9kA0iLAZFaQAwGgwNgATxqjx9/JvFzx588c4nExsMFNX9kjM+vTvzwwIBBnl/sAHYtZ3LMJHvPGBRbzkhsODzjmAGP2Y23myWADjOcOTsBu5YDOWw3eNsMEjfcAGrhYQNpObsBpCXB4DYOLeefP7v5F67lnwGP8Yyzm3/g1XIjwew23BYgg8eAv3cbXlskb7wx/y3bZpy4s+dhw+GZfcY8Ejd4t1kkGEjg9Avf+fTHhm/b5BK3sycf/lzwTc6ev//s5ps/Kmzk+aWxa1E4AHMhEDODWRJglRJYlYOAfAOGFv4DONSOglEwCkbBSAUAem9p8TpxN7QAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0001-6374-0266","institution":"Indian Institute of Technology Madras","correspondingAuthor":true,"prefix":"","firstName":"Yaswanth","middleName":"","lastName":"Pulipati","suffix":""},{"id":639496113,"identity":"d1c9c112-856f-45f9-ae02-43671df6c9ce","order_by":1,"name":"Balaji Narasimhan","email":"","orcid":"","institution":"Indian Institute of Technology Madras","correspondingAuthor":false,"prefix":"","firstName":"Balaji","middleName":"","lastName":"Narasimhan","suffix":""},{"id":639498147,"identity":"62b107e9-d33f-4137-9fa8-3338edf631fd","order_by":2,"name":"C. Balaji","email":"","orcid":"","institution":"Indian Institute of Technology Madras","correspondingAuthor":false,"prefix":"","firstName":"C.","middleName":"","lastName":"Balaji","suffix":""}],"badges":[],"createdAt":"2026-05-13 05:57:53","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9699065/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9699065/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109247101,"identity":"006852ac-9f61-4f0c-bc6b-29a8adfba62b","added_by":"auto","created_at":"2026-05-14 08:18:40","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":261021,"visible":true,"origin":"","legend":"\u003cp\u003eMethodological framework of the study (CHCF represents clock-hour correction factor)\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/47d88c6c1c9b391dfa49fbde.png"},{"id":109405194,"identity":"43baffee-2a72-45e6-ba67-0131510b038e","added_by":"auto","created_at":"2026-05-17 13:00:40","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":121621,"visible":true,"origin":"","legend":"\u003cp\u003ea) Locations showing IMD non-recording rain gauge stations providing daily rainfall data. b) Locations showing IMD self-recording rain gauge (SRRG) station providing hourly rainfall data.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/69bcf20aae651aa9348a2359.png"},{"id":109249509,"identity":"6c30b7ce-0e35-4c18-a77a-6dfb60c7e245","added_by":"auto","created_at":"2026-05-14 08:54:50","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":57987,"visible":true,"origin":"","legend":"\u003cp\u003eBoxplots showing the performance of Gumbel and duration-dependent GEV (D-GEV)\u003c/p\u003e\n\u003cp\u003eDistributions with mean error (ME) across different durations (1-, 3-, 12-, and 24-hour)\u003c/p\u003e\n\u003cp\u003eat 91 stations.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/28dd756f4bb21ca14de825de.png"},{"id":109249593,"identity":"84e544d2-e995-43d0-8601-65217cfc4604","added_by":"auto","created_at":"2026-05-14 08:57:14","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":478342,"visible":true,"origin":"","legend":"\u003cp\u003eSpatial distribution of (a–d) median annual maximum 1-day rainfall (M_AMS) and\u003c/p\u003e\n\u003cp\u003e(e–h) Simple Daily Intensity Index (M_SDII) for a 24-hour duration across India. The\u003c/p\u003e\n\u003cp\u003emaps are derived from four gridded rainfall products: (a, e) IMERG Final, (b, f) GSMaP\u003c/p\u003e\n\u003cp\u003eGauge Adjusted, (c, g) PERSIANN, and (d, h) MSWEP.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/bc46f0c558ef4ab0d8af2c7d.png"},{"id":109249682,"identity":"82d821a0-b3c1-49c9-b5aa-a6a80c4e3505","added_by":"auto","created_at":"2026-05-14 08:58:55","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":58794,"visible":true,"origin":"","legend":"\u003cp\u003eHomogeneous regions identified using the K-means clustering approach\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/fde066385d2915fae81f31a0.png"},{"id":109249579,"identity":"75ac5d4e-89a3-4908-9c16-ef6c564f68f2","added_by":"auto","created_at":"2026-05-14 08:56:49","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":783011,"visible":true,"origin":"","legend":"\u003cp\u003eSpatial distribution of rainfall intensity (mm h⁻¹) corresponding to the 2-year return period for 3-hour duration over India, derived from (a) GPM, (b) PCCSCDR, (c) MSWEP, and (d) GSMaP datasets.\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/f84694a72271f829927b6e40.png"},{"id":109249520,"identity":"1a02636d-92d6-4c4e-8709-430cc4c292ed","added_by":"auto","created_at":"2026-05-14 08:55:07","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":95873,"visible":true,"origin":"","legend":"\u003cp\u003ea) Log-Log Plot of Non-Central Moments vs Rainfall Duration at Delhi. b) Scaling exponent of moments as a function of moment order.\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/41078905090f75d268ddea0b.png"},{"id":109247106,"identity":"a9594b8f-b2bc-48e7-989c-54b3dd16adaa","added_by":"auto","created_at":"2026-05-14 08:18:40","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":297281,"visible":true,"origin":"","legend":"\u003cp\u003eSpatial distribution of the scaling exponent (\u003cem\u003eK(q)\u003c/em\u003e) derived from the 1–24 hour duration range using the first three non-central moments (\u003cem\u003eq\u003c/em\u003e = 1,2,3).\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/52a5ed0d502819550adbb50a.png"},{"id":109252465,"identity":"8fc0699b-41e1-4bec-9d1e-75e3233dad95","added_by":"auto","created_at":"2026-05-14 09:26:51","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":185720,"visible":true,"origin":"","legend":"\u003cp\u003eIDF curve for 100-year return period obtained in the current method, derived using Rambabu et.al (1979) and Kothyari and Garde (1992) at two cities in the study region.\u003c/p\u003e","description":"","filename":"image9.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/4f495efa1376dad4ee165208.png"},{"id":109405259,"identity":"595dabf9-202d-4e72-a4b1-18a80fc45968","added_by":"auto","created_at":"2026-05-17 13:14:09","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":254921,"visible":true,"origin":"","legend":"\u003cp\u003eMean Error values in mm/hour for 10-year return period rainfall intensities considering all the durations: a) Current study, b) Ram Babu, c) Kothyari and Garde.\u003c/p\u003e","description":"","filename":"image10.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/ba645664d7bc5aaee442bd54.png"},{"id":109249591,"identity":"fe76b288-e170-4875-88b0-8182c4f03ec2","added_by":"auto","created_at":"2026-05-14 08:57:11","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":632478,"visible":true,"origin":"","legend":"\u003cp\u003eRainfall intensities for 1-hour duration at return periods: a) 2-year, b) 10-year, c) 25-year, d)50-year.\u003c/p\u003e","description":"","filename":"image11.png","url":"https://assets-eu.researchsquare.com/files/rs-9699065/v1/59394a5dd0a3679d5c05c57a.png"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Framework to Develop High-Resolution Intensity–Duration–Frequency Curves: Historical Analysis and Scaling Relationships Integrating Gauge and Gridded Rainfall Products\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eHydrological extremes are rare yet cause severe social and economic losses, making it vital to understand them for effective water resource management. Designing critical infrastructure such as stormwater drains, reservoir spillways, and flood protection measures requires quantitative estimates of extreme rainfall. Intensity-Duration-Frequency (IDF) curves (Rodr\u0026iacute;guez et al., \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) remain the most widely used statistical tool in hydrology and engineering applications. IDF curves depict the relationship between rainfall intensity (i), duration (d), and return period (T), providing valuable inputs for hydrologic design across various timescales and exceedance probabilities.\u003c/p\u003e \u003cp\u003eTraditionally, IDF curve construction has relied on dense rain gauge networks. A notable example is NOAA Atlas 14 (Perica et al., \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), which utilized data from 2,846 daily stations (with an average of 63 years) and 994 hourly stations (with an average of 40 years) to provide spatially distributed IDF curves. In regions with sparse gauge networks, constructing IDF curves is more challenging. For instance, IMD produced Isopluvial maps using Annual Maxima Series from 120 self-recording rain gauge (SRRG) stations (Sahoo and Kumar Yadav, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Yet, due to heterogeneity in rainfall caused by topography, orography, and distance from the coast, these maps fail to adequately capture spatial variability. Point measurements cannot adequately capture the spatial structure of precipitation, and therefore the quality and accuracy of gauge-based IDF rainfall representations progressively deteriorate as the distance from rain gauge stations increases (Ochoa-Rodriguez et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Sub-daily IDF derivation is crucial for applications such as urban drainage. Approaches include rainfall disaggregation (Rodr\u0026iacute;guez-Sol\u0026agrave; et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), stochastic simulations (Kossieris et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Li et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), areal reduction factors (ARFs) (Ombadi et al., \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), and regionalization/interpolation (Noor et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). These methods introduce uncertainties. For example, ARFs vary with geography, storm type, and spatial correlation (Ombadi et al., \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Regionalization, although widely adopted, assumes rainfall homogeneity within defined regions, an assumption that may not hold for short-duration extremes or in complex terrain. Capturing local rainfall variability is often better achieved by interpolating distribution parameters rather than regionalization (Deidda et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). An alternative is the scale-invariance theory, in which the statistical properties of annual maximum rainfall remain consistent across durations. Scaling-based approaches, including mono- and multifractal frameworks, exploit the scale invariance of rainfall extremes to derive sub-daily Intensity\u0026ndash;Duration\u0026ndash;Frequency (IDF) relationships from coarser temporal data, offering an alternative in regions with limited high-resolution observations (Chandra et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Rodr\u0026iacute;guez-Sol\u0026agrave; et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Lanciotti et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Casas-Castillo et al., \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). These methods effectively capture the spatial and climatic variability of rainfall extremes and provide physically consistent IDF estimates across durations and return periods, thereby improving hydrological design and risk assessment in data-scarce and climatically heterogeneous regions. Gridded rainfall products (GRPs) now provide high-resolution data and potential covariates for spatial interpolation. Recent studies (Zhang et al., \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2021a\u003c/span\u003e; Lei et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) have employed a Machine Learning (ML) approach that combines satellite rainfall estimates and topographic variables with gauge data to produce merged rainfall estimates. While geostatistical methods like Kriging assume stationarity and isotropy (Chen et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Zou et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), ML approaches are data-driven, avoid subjective assumptions, and enhance reproducibility (Zhang \u003cem\u003eet al.\u003c/em\u003e, 2021b). Several recent studies (Lei et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Nwaila et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) have further demonstrated that machine learning frameworks can outperform traditional geostatistical techniques in capturing nonlinear precipitation dynamics and reducing interpolation error, particularly in regions with complex terrain or heterogeneous climatic conditions. Despite this promise, only a limited number of studies have explored the derivation of IDF from GRPs.\u003c/p\u003e \u003cp\u003eThis study develops high-resolution (10 km) IDF curves across daily to sub-daily scales using rainfall frequency analysis, integrating satellite-derived estimates and gauge data via ML regression to improve precision. Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e outlines the study framework, area, and extreme value distributions. Section 3 presents parameter estimation using gauges and GRPs, along with scaling-based derivations. Section \u003cspan refid=\"Sec24\" class=\"InternalRef\"\u003e4\u003c/span\u003e discusses the results. Section 5 summarizes the findings.\u003c/p\u003e"},{"header":"2. Data and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Study area\u003c/h2\u003e \u003cp\u003eThe study area selected is the region of the Indian sub-continent extending from 66\u0026deg;E to 100\u0026deg;E and from 6\u0026deg;N to 39\u0026deg;N. The climate in India is divided broadly into four seasons, namely (1) Winter (January and February), (2) Pre-monsoon (March to May), (3) Southwest monsoon (June to September), also known as summer monsoon, (4) Post-monsoon (October to December), also known as Northeast monsoon. The vast area of the country receives more than 80% of the annual rainfall during the southwest monsoon season (Sooraj et al., \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). However, the Southeast Peninsula receives substantial rainfall during the retreating (Northeast monsoon) season.\u003c/p\u003e \u003cp\u003eThe study region comprises diverse climatic regions, with annual average rainfall ranging from 300 mm to 6,000 mm. Due to diverse climatic regions and topography, rainfall distribution in India exhibits notable spatial heterogeneity (Deshpande et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Jha et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Deriving high-resolution IDF curves across India is essential in hydro-meteorological and hydraulic design studies. In most regions of India, accessing rainfall IDF curves poses a challenge due to the scarcity of hourly rainfall records and the limited availability of existing curves. The following framework is implemented for the Indian sub-continent to construct IDF curves for 2-, 5-, 10-, 25-, 50-, and 100-year return periods for rainfall intensities obtained over 1-, 3-, 6-, 9-, 12-, 15-, 24-, 48-, and 72-hour durations.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Framework\u003c/h2\u003e \u003cp\u003eThe framework proposed in the present study for deriving IDF curves includes several stages. The first stage involves deriving parameters of the extreme value distribution with durations of 24, 48, and 72 hours at a 10-km resolution by integrating the properties of multiple gridded rainfall products and a dense network of non-recording rain gauge stations. The availability of daily rainfall data from a dense network of non-recording gauges is a key advantage for frequency analysis. Since rainfall at non-recording stations is recorded at fixed daily intervals, the fixed-window maxima may underestimate the true moving-window design rainfall. To correct this discretization error, the clock hour correction factor (CHCF), defined as the ratio of the sliding window to the fixed window (IS 5542:2003), is applied. CHCF adjusts fixed-interval daily, 2-day, and 3-day maxima to equivalent moving-window maxima (24-, 48-, 72-hour), ensuring corrected values represent true maxima before probability distribution fitting (Ghate and Timbadiya, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Return levels are estimated for the 24-, 48-, and 72-hour Durations at 10-km resolution using the derived parameters. The second stage includes deriving the scaling exponents at SRRG station locations using the scale-invariance property of AMS rainfall across the durations. The third stage involves interpolating derived scaling exponents to 10-km resolution across the region using machine learning regression techniques. The final stage entails applying the scaling exponents to derive sub-daily return levels from the 24-hour duration return levels. The framework developed to derive IDF curves is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Gauge observations\u003c/h2\u003e \u003cp\u003eHistorical time series of rainfall observations are obtained from the India Meteorological Department (IMD) for the years 1971 to 2021 (51 years), based on data from 1900 non-recording rain gauges (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea) at the daily temporal scale. The available rain gauge network is denser in the southern peninsular region than in the rest of the country. The IMD daily rainfall records are daily values recorded at 8:30 AM Indian Standard Time, indicating the total rainfall accumulated over 24 hours from the previous day at 8:30 AM. Although daily station data is available from many gauges, the sub-daily data is recorded by a sparse network of automatic rain gauge stations.\u003c/p\u003e \u003cp\u003eHourly rainfall data from 188 self-recording rain gauge stations (SRRG) were obtained from the India Meteorological Department. In the first stage, the station records were screened based on record length and data quality. Quality assessment was performed through internal consistency checks and spatial and temporal consistency analyses. The data is screened for quality based on i) internal consistency check and ii) quality checking in the spatial and temporal domain. A significant amount of rainfall in India occurs from June to December, while summer convective events typically occur in May. We noticed a considerable amount of missing rainfall data during the dry season, particularly from February to April. Since these months generally record negligible precipitation, the missing values are excluded from the analysis. Focusing on the remaining months, if more than 100 hours of rainfall data are missing, the year is considered incomplete and excluded from further analysis.\u003c/p\u003e \u003cp\u003eFurthermore, we excluded stations with more than three consecutive missing years. Following this screening, valid data from each SRRG station spanned 10 to 40 years. Of the available 188 stations with hourly rainfall data, 91 stations have satisfied all the above conditions with a minimum of 20 years of data availability, which are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb. The station number, duration of the data available, information about missing data, and the geographic locations of the stations are given in Supplementary Information (Table S1).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Past methods for IDF curve derivation in India\u003c/h2\u003e \u003cp\u003eThe regional IDF equations developed for hydrological design practice in the Indian subcontinent are described here. Babu et al. (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1979\u003c/span\u003e) have provided the IDF curve equation for several Indian cities as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{I}_{t}^{T}=\\:\\frac{K{T}^{x}}{{(t+b)}^{n}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:I\\)\u003c/span\u003e\u003c/span\u003e is the intensity in mm/hr, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:T\\)\u003c/span\u003e\u003c/span\u003e denotes the return period, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:t\\)\u003c/span\u003e\u003c/span\u003e indicates the storm duration in hours (assumed as equal to time of concentration), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:K,\\:x,\\:b,\\:n\\)\u003c/span\u003e\u003c/span\u003e are constants derived for various cities/stations of India.\u003c/p\u003e \u003cp\u003eKothyari and Garde (1992) suggested the following IDF equation:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{I}_{t}^{T}=\\:\\frac{C{T}^{0.20}\\:{\\left({R}_{24}^{2}\\right)}^{0.33}}{{\\left(t\\right)}^{0.71}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{I}_{t}^{T}\\)\u003c/span\u003e\u003c/span\u003e denotes the intensity in mm/hr, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:T\\)\u003c/span\u003e\u003c/span\u003e is the return period in years, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:t\\)\u003c/span\u003e\u003c/span\u003e indicates the design storm duration in hours (assumed as equal to time of concentration); C is a constant provided for different geographical regions of India, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{24}^{2}\\)\u003c/span\u003e\u003c/span\u003e is the magnitude of rainfall in mm for a 2-year, 24-hour duration. The IDF curves developed in this study were validated against estimates derived from Equations \u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e at various geographical locations across India.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Gridded Rainfall Products from Satellite Retrievals\u003c/h2\u003e \u003cp\u003eThe gridded satellite rainfall products used in this study are Integrated Multi-Satellite Retrievals (IMERG), Global Satellite Mapping of Precipitation (GSMaP), PERSIANN-CCS-CDR (0.04\u0026deg; \u0026times; 0.04\u0026deg;, 3-hourly), and Multi-Source Weighted-Ensemble Precipitation (MSWEP). The Gridded Rainfall Products (GRPs) from different sources used in the present study are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, and particulars of these products are provided in the supplementary information (S1):\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGridded rainfall products selected in the current study\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDataset\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRepresentation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSpatial and temporal resolution\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSource\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIMERG Final precipitation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGPM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003csup\u003e◦\u003c/sup\u003e\u0026times;0.1\u003csup\u003e◦\u003c/sup\u003e/half-hourly\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGES DISC\u003c/p\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://disc.gsfc.nasa.gov/\u003c/span\u003e\u003cspan address=\"https://disc.gsfc.nasa.gov/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGSMaP gauge adjusted\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGSMaP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003csup\u003e◦\u003c/sup\u003e\u0026times;0.1\u003csup\u003e◦\u003c/sup\u003e/hourly\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eJAXA\u003c/p\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://hokusai.eorc.jaxa.jp\u003c/span\u003e\u003cspan address=\"https://hokusai.eorc.jaxa.jp\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePERSIANN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePCCSCDR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.04\u003csup\u003e◦\u003c/sup\u003e\u0026times;0.04\u003csup\u003e◦\u003c/sup\u003e/3 hourly\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.11572/P24W2F\u003c/span\u003e\u003cspan address=\"10.11572/P24W2F\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMSWEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMSWEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003csup\u003e◦\u003c/sup\u003e\u0026times;0.1\u003csup\u003e◦\u003c/sup\u003e/3 hourly\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://www.gloh2o.org/\u003c/span\u003e\u003cspan address=\"http://www.gloh2o.org/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.6 Return Level Estimation\u003c/h2\u003e \u003cp\u003eTwo common approaches for modeling extremes in hydrology are the block maxima and peaks-over-threshold (PoT) methods. While PoT requires defining thresholds and de-clustering, which is problematic in regions with diverse climatological conditions, the block maxima approach produces an annual maximum series (AMS) that ensures independence and facilitates comparison of rainfall datasets. The Generalized Extreme Value (GEV) distribution (Papalexiou and Koutsoyiannis, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), which unifies the Gumbel, Fr\u0026eacute;chet, and Weibull families, is traditionally used in AMS to construct IDF curves. The distribution function of GEV is given as\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{F}_{GEV}\\left(z;\\:\\mu\\:,\\sigma\\:,\\:\\xi\\:\\right)=\\text{exp}\\left[-{\\left\\{1+\\xi\\:\\left(\\frac{z-\\mu\\:}{\\sigma\\:}\\right)\\right\\}}_{+}^{-\\frac{1}{\\xi\\:}}\\right]\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(3\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003efor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:z\\:ϵ\\:R\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{+}=\\text{m}\\text{a}\\text{x}(y,0)\\)\u003c/span\u003e\u003c/span\u003e for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:y=1+\\:\\xi\\:\\:.\\:\\frac{z-\\mu\\:}{\\sigma\\:}\\:ϵ\\:R\\)\u003c/span\u003e\u003c/span\u003e and \u0026micro;, σ, and ξ are the location, scale, and shape parameters, respectively. For ξ\u0026thinsp;\u0026gt;\u0026thinsp;0 and ξ\u0026thinsp;\u0026lt;\u0026thinsp;0, the GEV distribution reduces to the Fr\u0026eacute;chet and Max-Weibull distributions, respectively (Ragulina and Reitan, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). The GEV distribution reduces to the Gumbel distribution when the shape parameter, ξ, equals 0. The AMS \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({z}_{1},{z}_{2},{z}_{3}\\dots\\:{z}_{T})\\)\u003c/span\u003e\u003c/span\u003e for a specific \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:d\\)\u003c/span\u003e\u003c/span\u003e duration is obtained by summing hourly rainfall over a sliding window of width \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:d\\)\u003c/span\u003e\u003c/span\u003e and selecting the maximum value for each year. The shape parameter (ξ) of the GEV distribution plays a critical role in determining the tail behavior of the distribution, and this becomes more uncertain with shorter data records, leading to less reliable return level estimates in regions with limited data (Papalexiou and Koutsoyiannis \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Therefore, the current study employs the Gumbel distribution to model the annual maximum rainfall. The return levels for a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:d\\)\u003c/span\u003e\u003c/span\u003e duration maxima are obtained from the following equation.\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{z}_{T}=\\mu\\:-\\sigma\\:\\text{log}\\left\\{-\\text{log}\\left(1-\\frac{1}{T}\\right)\\right\\}\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{z}_{T}\\)\u003c/span\u003e\u003c/span\u003e is the return level for a return period \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:T\\)\u003c/span\u003e\u003c/span\u003e. This study compared return level estimates at SRRG stations using the Gumbel (GEV1) and Fr\u0026eacute;chet (GEV2) distributions. Due to limited record length, the Gumbel distribution is used provisionally. However, this may underestimate return levels for very long return periods (\u0026ge;\u0026thinsp;100 years). Studies (Papalexiou and Koutsoyiannis, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) indicate that estimation of positive shape parameters generally requires much longer records, making the Gumbel assumption a stable alternative. This effect is illustrated through return-level comparisons at the SRRG stations.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.6.1 Duration-dependent GEV\u003c/h2\u003e \u003cp\u003eThe present study also considered the Duration-dependent GEV (d-GEV) model formulated by Koutsoyiannis et al. (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e1998\u003c/span\u003e), which describes the complete IDF relation for various return periods over different durations. The underlying concept of d-GEV is the efficient parameterization of the GEV distribution using three to five parameters. The five-parameter d-GEV shown in Eq.\u0026nbsp;5 is applied to derive duration-dependent IDF curves.\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:G\\left(z,d;\\stackrel{\\sim}{\\mu\\:},{\\sigma\\:}_{o},\\:\\xi\\:,\\:\\theta\\:,\\:{n}_{d}\\right)=exp\\left\\{-{\\left[\\xi\\:\\:\\left(\\frac{z}{{\\sigma\\:}_{o}\\left({\\left(d+\\:\\theta\\:\\right)}^{-{n}_{d}}\\right)}-\\:\\stackrel{\\sim}{\\mu\\:}\\right)\\right]}^{\\frac{-1}{\\xi\\:}}\\right\\}\\:\\:\\:\\:\\:\\left(5\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{\\sim}{\\mu\\:}\\)\u003c/span\u003e\u003c/span\u003e represents the re-parameterized location parameter, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\theta\\:,\\:\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{d}\\)\u003c/span\u003e\u003c/span\u003e represent duration offset and duration exponent, respectively. The scale parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sigma\\:\\)\u003c/span\u003e\u003c/span\u003e is assumed to follow a power law relation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\sigma\\:={\\sigma\\:}_{o}\\:{\\left(d+\\:\\theta\\:\\right)}^{-{n}_{d}}\\)\u003c/span\u003e\u003c/span\u003e with duration \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:d\\)\u003c/span\u003e\u003c/span\u003e and scale offset \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{o}\\)\u003c/span\u003e\u003c/span\u003e remain constant across all the durations.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e2.7 Parameter estimation\u003c/h2\u003e \u003cp\u003eSeveral established methods exist for estimating parameters of the generalized extreme value (GEV) distribution fitted to annual maximum rainfall series, including Bayesian approaches, maximum likelihood estimation (MLE), generalized maximum likelihood estimation (GMLE), and L-moments. Among these, the L-moments method (Hosking, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1990\u003c/span\u003e; Papalexiou and Koutsoyiannis, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) is particularly robust and often preferred for small to moderate sample sizes (n\u0026thinsp;\u0026lt;\u0026thinsp;100), as it provides more stable and less biased parameter estimates compared to likelihood-based methods, especially for the shape parameter (Hosking, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1990\u003c/span\u003e; Papalexiou and Koutsoyiannis, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Given the typical record lengths in this study (30\u0026ndash;50 years), which fall well within this range where L-moments outperform MLE in terms of bias and variance, the L-moments approach was selected for parameter estimation of the GEV distribution applied to the annual maximum series (AMS) data.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e2.8 Uncertainty considering the record length\u003c/h2\u003e \u003cp\u003eIn this study, the bootstrap resampling procedure proposed by Marra et al. (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) was adopted to assess the uncertainty associated with the record length of AMS. In this bootstrap method, each of the original annual maximum rainfall data series was resampled without replacement to generate multiple bootstrap samples. These bootstrap samples are then used to estimate return levels for various return periods. By repeating this sampling process 1000 times, a distribution of return levels was obtained, allowing uncertainty estimation. The 5th -95th quantile interval of the bootstrap sampling is used to quantify the uncertainty associated with the undersampling of rainfall climatology caused by short data sets. Both Gumbel and d-GEV models were applied across 91 SRRG stations. The selection of the most suitable model was determined by evaluating performance indicators and accounting for uncertainties arising from record length.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e2.9 Evaluation of GRP's distribution parameters against gauge data\u003c/h2\u003e \u003cp\u003eNumerous studies (Marra et al., 2019; Noor et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) have recently examined satellite-based rainfall products, and these studies consistently report random and systematic errors in satellite estimates that are influenced by resolution, retrieval algorithms, topography, and climatology. It is therefore essential to evaluate each product's performance across the study region. In this study, the location and scale parameters of the AMS distribution were derived at gauge stations and collocated GRP grids to assess the ability of GRPs to reproduce observed parameters. As the gauge data period does not always coincide with the gridded data, the comparison primarily aims to evaluate the degree of agreement between distribution parameters and quantify the quality of spatial matching. The accuracy of the fitted extreme value distributions and the consistency between gauge-based and satellite-derived Annual Maximum Series (AMS) rainfall were evaluated using a suite of standard performance metrics, including ME, RMSE, PBias, MAE, CC, IA, and KGE. Additionally, the statistical validity of the models was confirmed through the Anderson\u0026ndash;Darling and Kolmogorov\u0026ndash;Smirnov goodness-of-fit tests(De Michele and Avanzi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). These metrics evaluate bias, accuracy, and distributional conformity (see Supplementary Information S2 for equations and procedures).\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.10 Developing the location and scale parameters for a 24-hour duration at 10 km resolution across India\u003c/b\u003e \u003c/p\u003e \u003cp\u003eProbability distributions were fitted to rainfall maxima after applying the clock-hour correction factor per IS 5442:2003 to convert fixed-interval daily, 2-day, and 3-day rainfall to equivalent 24-, 48-, and 72-hour moving-window maxima. Gumbel distributions were fitted to data from 1,900 IMD gauge locations to estimate location and scale parameters. These were spatially interpolated to 10-km resolution using Random Forest (RF) regression incorporating geographically relevant covariates. Similarly, distribution parameters for GPM, GSMaP, and MSWEP were derived at 10-km resolution, with PCCSCDR values mapped to the same grid for consistency.\u003c/p\u003e \u003cp\u003eThe 24-hour rainfall intensity parameters at each gauge station serve as predictands, while the predictors include gridded parameters and auxiliary variables. The Random Forest (RF) algorithm constructs an ensemble of decision trees trained on bootstrap samples, with model performance internally assessed using out-of-bag (OOB) data (Zhang \u003cem\u003eet al.\u003c/em\u003e, 2021b). The RF model requires specification of several hyperparameters, including the number of trees (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:ntree\\)\u003c/span\u003e\u003c/span\u003e), the maximum tree depth, and the number of features considered at each split (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:mtry\\)\u003c/span\u003e\u003c/span\u003e). In this study, the number of features at each split was fixed to one-third of the total predictors (Zhang \u003cem\u003eet al.\u003c/em\u003e, 2021b), and trees were grown to full depth without explicit pruning. Among the hyperparameters, only the number of trees was tuned, which was optimized using k-fold cross-validation to ensure stable and generalized performance. After identifying the optimal ntree, the final RF model was retrained using the complete dataset. Predictors include latitude, longitude, elevation, and distance to the coast. Additional covariates are derived from ETCCDI indices: the median Simple Daily Intensity Index (M_SDII) and annual maximum 1-day rainfall (M_AMS) from each gridded product. Including an excessive number of predictors in ML can increase model complexity, slow down the training process, and introduce noise. To mitigate this, the RF model identifies the most relevant features using Permutation Feature Importance (Zhang et al., 2021).\u003c/p\u003e \u003cp\u003eTo account for high spatial variability, parameters were interpolated within homogeneous areas delineated through K-means clustering approach based on extreme rainfall characteristics. RF regression was applied within each cluster to ensure variables are tailored to regional variability. The resulting 24-hour distribution parameters are termed merged parameters as they integrate properties from several datasets.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e2.11 Developing IDF curves at 10 km resolution\u003c/h2\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e2.11.1 Applying the property of scale-invariance to derive d-duration return levels\u003c/h2\u003e \u003cp\u003eScaling implies that the same underlying relationship governs the statistical characteristics of a process across different temporal scales (P\u0026ouml;schmann et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In the context of extreme rainfall, scale invariance indicates that the statistical properties of annual maximum rainfall intensities at shorter durations can be reliably derived from those at longer durations through a power-law scaling relationship. This concept has been successfully applied in several studies (Chandra et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Bairwa et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Ghanmi et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) to develop IDF relationships from daily rainfall data. In the present study, scale invariance for 1\u0026ndash;24 hour durations was confirmed through the linear behavior of non-central moments in log\u0026ndash;log space and the linear dependence of scaling exponents on moment order. Assessing whether rainfall exhibits simple or multi-scaling behavior is necessary before constructing IDF curves using the invariance approach. While multi-scaling behavior has been reported in earlier studies (Fauer et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) at sub-hourly scales (finer than 1-hour), a single power-law relationship often sufficiently characterizes the statistical moments of extreme rainfall in the hourly-to-daily regime (Chandra et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Maity and Maity, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). The theoretical details and formulation of the scale-invariance method employed here are provided in the supplementary document (Section S4). In the present analysis, scaling diagnostics were evaluated over the considered time range (1\u0026ndash;24 hours), and the simple scaling assumption was found adequate. When departures from simple scaling are identified, a piecewise or multi-scaling framework is typically adopted, in which distinct scaling exponents are estimated for short-duration events (e.g., \u0026lt;\u0026thinsp;1 hour), thereby preserving regime-dependent scaling behavior when deriving sub-daily intensities from daily extremes (Chandra et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). The return levels estimated using the scaling method are compared with those from the Gumbel distribution at SRRG stations with historical hourly rainfall records. This helps in evaluating the performance of the invariance method in reproducing the observed IDF curves at SRRG stations.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e2.12 Estimation of scaling exponent and return level magnitudes at 10 km resolution for IDF generation\u003c/h2\u003e \u003cp\u003eEstimating return levels at ungauged locations requires transferring scaling exponents using interpolation techniques. The scaling exponents from the hourly gauge stations are interpolated across the study region using appropriate explanatory/auxiliary variables. The analysis revealed that the scaling exponents exhibit strong correlations with the location and scale parameters of the 24-hour Gumbel distribution parameters. The integration of gridded rainfall products, as detailed in the preceding section, enables the derivation of location and scale parameters at a consistent 10-km resolution across the Indian subcontinent. During the study, it is also noted that the duration exponent \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{d}\\)\u003c/span\u003e\u003c/span\u003e obtained by fitting the D-GEV distribution (Eq.\u0026nbsp;5) has characteristics similar to the scaling exponents \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:k\\left(q\\right)\\)\u003c/span\u003e\u003c/span\u003e acquired by the scale invariance theory. Hence, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{d}\\)\u003c/span\u003e\u003c/span\u003e obtained from all four gridded rainfall products are also used as predictors, assuming this variable can act as a potential covariate. The scaling exponents available at SRRG stations are thus interpolated to the 10-km resolution using appropriate covariates. Then, the return level magnitudes at 10 km resolution are obtained by applying the scaling exponents to the 24-hour-duration estimated rainfall intensities.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e2.13 Estimation of confidence intervals for return level estimates\u003c/h2\u003e \u003cp\u003eThe study uses parametric bootstrapping to derive confidence intervals (CIs) for IDF curves at each grid point. Bootstrap methods are widely used when analytical CIs are challenging to obtain (Carpenter and Bithell, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Huang et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). As a resampling approach, the bootstrap uses computer simulations to estimate uncertainty in distribution parameters and quantiles. Prior studies have shown that the parametric bootstrap often outperforms the non-parametric approach (Kysel\u0026yacute;, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). While the non-parametric version relies on resampling with replacement, the parametric version generates random samples from a fitted distribution. When an appropriate model is known, data are limited, and tail behavior is critical (e.g., IDF estimation), the parametric bootstrap is preferred (Kysel\u0026yacute; \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). In this study, pdf parameters are first estimated using scaling exponents at 10-km resolution, and parametric bootstrapping is then applied to compute CIs for return levels.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1 D-GEV vs GEV1 (Gumbel)\u003c/h2\u003e\n\u003cp\u003eIDF curves were developed at 91 locations using hourly rainfall data with both the Gumbel and D-GEV distributions. Model performance was evaluated using Q-Q plots generated with the Weibull plotting position formula (Smitha et al., 2018), alongside Percent Bias and Mean Absolute Error (MAE) to quantify deviations between simulated quantiles and observed Annual Maximum Series (AMS) values. Analysis of Mean Error (ME) and Root Mean Square Error (RMSE) across all stations (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) initially indicated that the Gumbel (GEV1) model yielded lower ME values, suggesting better fit to the observed data. Given the limited record lengths available from both gauge stations and GRPs, assessing the uncertainty associated with return level estimates is critical, as shorter time series can significantly influence the reliability of extreme value analysis. To address the impact of limited record lengths on extreme value reliability, a bootstrap resampling analysis (999 repetitions) was conducted. Using the 5th\u0026ndash;95th quantile interval to quantify sampling uncertainty, results from six representative stations (Fig. S1) showed that for records of 20 years, the Gumbel distribution had narrower uncertainty for return periods between 2 and 75 years. Consequently, based on the return-level estimates and favorable Q-Q plot results, the study adopted the Gumbel distribution. Its statistical adequacy was further confirmed across all stations using Anderson\u0026ndash;Darling and Kolmogorov\u0026ndash;Smirnov tests.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec19\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2 Deriving Homogeneous Regions Using Extreme Rainfall Characteristics\u003c/h2\u003e\n\u003cp\u003eThe median annual maximum 1-day rainfall (M_AMS) and Simple Daily Intensity Index (M_SDII) for 24-hour duration were computed from four gridded rainfall products: IMERG Final, GSMaP Gauge Adjusted, PERSIANN, and MSWEP (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e). Both M_AMS and M_SDII show prominently high values over the Western Ghats and northeastern India, with markedly lower values in the rain-shadow regions of the southern peninsula. These indices, along with mean annual rainfall, served as input variables in the clustering algorithm to delineate homogeneous rainfall regions across the country. The Silhouette Score (Fig. S2) indicates that the optimal k-value is nine, resulting in the delineation of nine homogeneous zones, as depicted in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. The outcomes of the cluster analysis align reasonably well with homogeneous rainfall zones identified in previous studies (Falga and Wang, \u003cspan class=\"CitationRef\"\u003e2022\u003c/span\u003e; Sahoo and Kumar Yadav, \u003cspan class=\"CitationRef\"\u003e2022\u003c/span\u003e), supporting the physical consistency of the regionalization. The delineation of homogeneous rainfall regions is not only used for characterization but also provides the basis for region-specific modeling. Given the strong spatial heterogeneity in rainfall-generating mechanisms across India ranging from orographic enhancement in the Western Ghats to convective and monsoon-core processes in central and northeastern India and rain-shadow effects in the peninsular interiors the modelling framework was implemented independently for each identified cluster. This cluster-wise approach enabled the capture of localized statistical relationships between the predictors and the characteristics of extreme rainfall.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTo convert the 1-day observed maximum rainfall to the 24-hour true cumulative rainfall, IS 5542:2003 advises using a CHCF value of 1.15 across the entire Indian region. However, it is recommended to use site-specific CHCF values for reliable estimates of extreme rainfall. Hence, in the present study, CHCF values are derived at each SRRG station for 24-, 48-, and 72-hour durations, and the mean CHCF values are subsequently determined for each cluster. Table S2 summarizes the number of available daily rain gauge stations per cluster, the mean values of the Gumbel location and scale parameters (for 24-hour duration), and the CHCF values for 24-, 48-, and 72-hour durations.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3 Evaluation of GRPs with rain-gauge data\u003c/h2\u003e\n\u003cp\u003eRecent studies (Lin and Huybers, \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e; Pradhan and Indu, \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e) have demonstrated the capability of satellite rainfall estimates to effectively capture the spatial distribution of rainfall in the Indian domain. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e and Fig. S3 present the 2-year return level maps of GRPs for 3-hour and 24-hour durations, respectively. The coastal and northeast regions exhibit higher return-level magnitudes across all GRPs. The inland areas of the southern peninsular regions exhibit similar patterns across all GRPs. Overall, the spatial distribution of return-level magnitude matches the pattern of extreme rainfall characteristics (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e) derived from gridded rainfall data.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cdiv id=\"Sec21\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.1 Evaluation of location and scale parameters of GRPs\u003c/h2\u003e\n\u003cp\u003eThe location and scale parameters of the Gumbel distribution estimated from the GRPs are compared with those derived from SRRG station data using standard performance metrics. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e and Table S3 present the performance evaluation of location and scale parameters, respectively. Higher IA and CC values between GRP and SRRG stations in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e indicate that GRPs reasonably capture the spatial gradient of extreme rainfall. GRP performance is notably better in humid regions than in arid regions, consistent with findings of Pradhan and Indu (\u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e). MSWEP consistently outperforms other products (GPM, PCCSCDR, GSMaP) in estimating Gumbel location and scale parameters across clusters, showing the lowest average bias (ME), highest IA and CC, and lowest RMSE. Collectively, these results show that GRP accuracy improves markedly with increasing duration. Yet, MSWEP consistently yields the highest agreement and lowest error in representing the spatial variability of Gumbel parameters across diverse climatic regimes. GRP performance was further assessed at SRRG locations by comparing distribution parameters and return levels from collocated datasets. Detailed evaluation results and statistical summaries are provided in Supplementary Section S5 (Supplementary Figures. S4\u0026ndash;S7).\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003ePerformance assessment of the location parameter obtained from GRPs against the gauge station for 24-hour duration in different homogenous regions.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCluster region\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eME.\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eIA\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCC\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eRMSE\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCluster region\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eME.\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eIA\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCC\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eRMSE\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC1_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.82\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.81\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC5_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.63\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.62\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.56\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC1_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.50\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC5_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.28\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.53\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.49\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.56\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC1_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.34\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.88\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC5_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.64\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC1_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.55\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC5_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.91\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.74\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC2_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.06\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.55\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.45\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC6_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.24\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.84\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC2_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.41\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.89\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC6_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.05\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.40\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.22\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.24\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC2_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.42\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.50\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC6_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.38\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.45\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.70\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC2_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.42\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.63\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC6_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC3_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.88\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC7_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.41\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.17\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC3_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.22\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.42\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC7_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.67\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.39\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC3_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.78\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.56\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC7_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.67\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.92\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC3_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.82\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.51\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC7_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.58\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.79\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC4_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.06\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC8_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.37\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.35\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.52\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC4_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.49\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.09\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC8_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.87\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.16\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC4_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.90\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC8_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.20\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC4_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.51\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.38\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.04\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC8_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.98\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC9_GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.23\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.67\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.82\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC9_PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.78\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.81\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC9_MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.82\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.27\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eC9_GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.78\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.69\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003e3.4 Developing location and scale parameters for 24-, 48-, and 72-Hour Rainfall duration at 10 km resolution using ML regression technique\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe location \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{(\\mu\\:}_{24-merged},{\\mu\\:}_{48-merged},{\\mu\\:}_{72-merged})\\)\u003c/span\u003e\u003c/span\u003e and scale parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({\\sigma\\:}_{24-merged},{\\sigma\\:}_{48-merged}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{72-merged}\\)\u003c/span\u003e\u003c/span\u003e) are derived at 10 km resolution for each cluster using appropriate covariates as input to the RF regression model, as explained in section 2.10. Table S6 presents the Permutation Feature Importance (PFI) metrics for predictors in the Random Forest (RF) framework used to estimate 24-hour Gumbel distribution parameters. High PFI values indicate strong influence, with elevation and distance to the coast as key contributors in specific clusters. Predictors with PFI values close to zero are excluded. Consistency across 48-hour and 72-hour durations suggests identical governing factors. Given the high performance of MSWEP, the M_SD11 and M_AMS indices are provided as model inputs. The final predictor combinations are in Tables S7 and S8. Cluster-specific RF configurations were implemented with optimized ntree values reported in Table S9. The predictive performance of the RF model was evaluated using the statistical metrics summarized in Tables S10 through S12. Higher CC and IA values across these tables indicate strong agreement between RF-derived parameters and station data for all durations. Location and scale parameter scatter plots (Fig. S8) show IA values from 0.83 to 0.97 for training and 0.82 to 0.95 for testing, indicating good agreement. Mean Error values (Table S10) show a notable reduction compared to gridded products (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e), highlighting improved reliability. For brevity, 24-hour location and scale parameters at 10-km resolution are shown in Fig. S9. These parameters exhibit a spatial pattern consistent with the observed extreme rainfall climatology over India.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e\n\u003ch2\u003e3.5 Verification of the scale-invariance property\u003c/h2\u003e\n\u003cp\u003eThe scale invariance of rainfall was assessed over durations ranging from 1 to 24 hours. As illustrated in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea, the log-transformed first three non-central moments (NCM1, NCM2, and NCM3) exhibit a distinct linear relationship with the log of rainfall duration, confirming that the statistical moments scale consistently across the temporal domain. Furthermore, we observed that the regression relationship between log-transformed values has a coefficient of determination (R\u0026sup2;)\u0026thinsp;\u0026ge;\u0026thinsp;0.96 across all SRRG stations, indicating the strength of the linear relationship. Furthermore, the scaling exponents, derived from the slopes of the linear fits in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea, exhibit a linear dependence on the moment order (q), as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003eb. The findings confirm a simple scaling (monofractal) regime for the 1\u0026ndash;24 hour interval. These findings are consistent with earlier studies (Maity and Maity, \u003cspan class=\"CitationRef\"\u003e2022\u003c/span\u003e), who also reported a strong scaling relationship between hourly and daily precipitation extremes across the Indian region, confirming the presence of simple scaling behavior in rainfall intensity\u0026ndash;duration relationships. The spatial distribution of the scaling exponents for the 1\u0026ndash;24 hour duration is presented in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e. The exponents exhibit a regional gradient across India. Lower values predominate along the Western Ghats, eastern coastal regions, and Northeast, reflecting rainfall characteristics associated with orographic enhancement and convective processes during the monsoon. In contrast, higher exponents characterize the central and northwestern arid-to-semiarid zones, with rainfall regimes in rain-shadow and continental interior areas.\u003c/p\u003e\n\u003cp\u003eFigure S10 provides a visual comparison between 100-year return period rainfall intensities estimated via scale-invariance theory and those derived from conventional Gumbel analysis at two selected rain gauge stations. The generated IDF curves closely match the observed ones. Figure S11 presents a comparative performance assessment of rainfall intensity estimates derived from the scale-invariance theory and those obtained from the Gumbel distribution across 91 SRRG stations. Panels (a) and (b) depict the Kling-Gupta Efficiency (KGE) for 1-hour and 3-hour durations, respectively, while panels (c) and (d) illustrate the Mean Absolute Error (MAE) for the same durations. Higher KGE values, together with lower MAE, indicate that the scale-invariance model effectively reproduces observed rainfall intensities, supporting its reliability for deriving IDF curves.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec23\" class=\"Section2\"\u003e\n\u003ch2\u003e3.6 Interpolation of scaling exponents to the 10 km resolution\u003c/h2\u003e\n\u003cp\u003eThe methodology for interpolating station-based scaling exponents onto a 10-km grid and its application for estimating short-duration return levels are outlined here. Correlation analysis (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) shows strong relationships between scaling exponent, 24-hour parameters, and the duration exponent (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{d}\\)\u003c/span\u003e\u003c/span\u003e) from the D-GEV distribution, aligning with the findings of Menabde et al. (\u003cspan class=\"CitationRef\"\u003e1999\u003c/span\u003e). Consequently, these variables were selected as potential predictors for the Random Forest model, given their influence on the distribution parameters. The RF regression is used to interpolate the scaling exponent at 10-km resolution, using covariates selected from collocated GRP pixels corresponding to the 91 station locations. The regression equation using RF can be expressed in functional form as:\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$\\:k\\left(q\\right)=f\\left({\\mu\\:}_{24-merged},{n}_{d}\\left(GPM\\right),{n}_{d}\\left(MSWEP\\right),{n}_{d}\\left(PCCSCDR\\right),{n}_{d}\\left(GSMap\\right)\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe data is split into 70% for RF model training and 30% for model testing. We ensured that training and test data contained stations from different clusters at this study point.\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eCorrelation values between scaling exponents \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:k\\left(q\\right)\\)\u003c/span\u003e\u003c/span\u003e obtained from scale invariance theory approach with selected covariates\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameter\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{k}\\left(\\varvec{q}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e(SRRG)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003c/p\u003e\n\u003cp\u003e24-hour merged\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e GPM\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e MSWEP\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e\u003c/p\u003e\n\u003cp\u003ePCCSCDR\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e GSMaP\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{k}\\left(\\varvec{q}\\right)\\)\u003c/span\u003e\u003c/span\u003e (SRRG)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026micro; 24hour merged\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e GPM\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.86\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e MSWEP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.87\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.85\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e PCCSCDR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.70\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.53\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026eta;\u003csub\u003ed\u003c/sub\u003e GSMaP\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-0.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.58\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.51\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.00\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe performance of the Random Forest regression model in capturing observed scaling exponents is illustrated in Figure S12. The interpolated values demonstrate a high degree of agreement with station-based observations, with most data points clustering closely along the 1:1 reference line, indicating minimal bias in the spatial scaling framework. To mitigate the risk of overfitting, a k-fold cross-validation framework was utilized, providing a more reliable estimate of model performance than a single train-test split. The data from 91 stations are partitioned into 4 folds (22, 22, 22, and 25 samples, respectively). The RF model is trained on three folds and validated on the remaining fold using a four-fold cross-validation approach. The results of k-fold cross-validation (Fig. S13) show that the RF regression model can reasonably predict the data without overfitting. The predicted scaling exponents across the study region at 10-km resolution are shown in Fig. S14. Building on the 24-hour return levels derived in the previous section, the interpolated scaling exponents based on scale invariance theory were applied to estimate short-duration (1-, 3-, 6-, 9-, 12-, and 18-hour) return levels. The rainfall intensities for d-duration, estimated using the scale-invariance approach, were subsequently compared with those derived from SRRG station data based on the Gumbel distribution. Furthermore, the rainfall intensities from parametric relationships suggested by Ram Babu et al. (\u003cspan class=\"CitationRef\"\u003e1979\u003c/span\u003e) and Kothyari and Garde (1992) for the Indian region are also evaluated against those obtained from the Gumbel-based analysis. The IDF curve derived from the proposed methodology at the grid point closest to the corresponding SRRG station is selected for comparative evaluation. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e illustrates the 100-year return period IDF curves derived using the present approach, along with at-site estimates and traditional empirical formulations by Ram Babu et al. (\u003cspan class=\"CitationRef\"\u003e1979\u003c/span\u003e) and Kothyari and Garde (1992). The comparison indicates that the proposed method closely follows the at-site IDF curves across durations, demonstrating its capability to reliably reproduce observed rainfall intensities. Additionally, the current estimates are slightly higher than the at-site curves, particularly at shorter durations, suggesting a conservative bias. From a design perspective, this is advantageous as it ensures a safer margin against the empirical formulations, which tend to underestimate intensities. In contrast, traditional techniques significantly underestimate rainfall intensity. The performance assessment is conducted using RMSE, and the results are shown for 10-year and 50-year return period curves at SRRG stations. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e presents the spatial distribution of Mean Error (ME) in estimated rainfall intensity (mm/h) across India for the proposed current procedure and two widely used traditional empirical methods. As shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e(a), the current procedure demonstrates superior performance, with mean error (ME) values exhibiting only a slight positive bias ranging from 0 to +\u0026thinsp;4 mm/h. From a structural design perspective, this modest positive bias is highly desirable, as it naturally incorporates a safety margin against underestimation of peak rainfall intensities. In contrast, the traditional methods proposed by Ram Babu et al. (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e(b)) and Kothyari and Garde (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e(c)) exhibit consistently large negative ME values, reaching as low as \u0026minus;\u0026thinsp;14 mm/h in several regions. This systematic underestimation indicates that these empirical formulae tend to under-predict short-duration, high-intensity rainfall events over much of the Indian subcontinent. Consequently, infrastructure designed using these traditional approaches may be inadequately sized, potentially compromising the safety and resilience of drainage systems.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e presents the 10 km spatial distribution of 1-hour rainfall intensities over India, showing high intensities in the Western Ghats and Northeast states and lower intensities in inland regions. Figures S15 and S16 show that the 95% confidence interval (CI) width is greater for shorter durations and higher return periods. These CIs reveal spatial and duration-dependent patterns where the widest 1-hour CIs (typically\u0026thinsp;\u0026gt;\u0026thinsp;15 mm) are concentrated along the Western Ghats, Northeast India, and the eastern coastal belt, reflecting high estimation uncertainty in orographic and intense convective rainfall regimes. In contrast, narrower CIs dominate the central plains, northwestern arid zones, and the southern interior. For practical applications, engineers should adopt conservative estimates, such as the upper 95% CI bound or a safety factor, particularly in high-uncertainty orographic zones or for long-return-period designs. While current estimates may be sufficient for preliminary assessments, critical infrastructure should incorporate upper CI or safety-adjusted values to ensure resilience against extreme rainfall events.\u003c/p\u003e\n\u003cp\u003eTo demonstrate the practical implications of the proposed framework, Figure S17 presents the derived IDF curves for two cities located in contrasting climatic regimes of India. The coastal setting of Chennai (Fig. S17a) exhibits higher rainfall intensities, while comparatively lower intensities are evident over the inland city of Bangalore (Fig. S17b). The proposed framework effectively captures the spatial gradients in rainfall intensities across the study region. It can be extended to other geographical regions and is particularly suitable for areas where dense sub-hourly or hourly rainfall observations are unavailable.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Discussion and conclusion","content":"\u003cp\u003eThis research develops a methodological framework to generate high-resolution IDF curves by integrating gauge and satellite-retrieved rainfall, addressing the lack of dense, long-term in situ records and the limitations of sparse, unevenly distributed rain gauge networks. By capturing spatial rainfall variability through the combination of gridded rainfall products and gauges, the approach provides reliable estimates of extreme rainfall and a feasible solution for deriving sub-daily to daily IDF curves for hydrological design and climate risk assessment. The Gumbel (GEV1) and D-GEV distributions were tested on AMS data from SRRG stations, with the Gumbel distribution found to be most suitable. Initially, the study estimated the 24-hour distribution parameters at a 10 km spatial resolution using a Random Forest regression model, with predictors derived from gridded rainfall datasets and topographic variables including latitude, longitude, and elevation. Then, scale-invariance theory (Rodr\u0026iacute;guez-Sol\u0026agrave; et al. \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) was applied to obtain d-hour return levels. Interpolating 24-hour parameters using homogeneous regions proved effective, since the influence of covariates varies with factors such as elevation and coastal proximity. Overall, the results confirm that scale-invariance theory, combined with gauge\u0026ndash;satellite integration, enables the reliable generation of high-resolution IDF curves, providing a practical tool for hydrological applications in data-sparse regions. The current framework (with spatial interpolation and disaggregation) already provides a slight conservative overestimation of design rainfall, with the mean error found to be positive at most locations, in contrast to the significant negative bias exhibited by the traditional methods; this positive bias is preferable from an engineering safety perspective as it naturally incorporates an additional safety margin.\u003c/p\u003e \u003cp\u003eEvidence from long record analyses suggests that extreme rainfall often exhibits a weakly positive shape parameter, implying a Fr\u0026eacute;chet distribution rather than a bounded upper tail. Consistent with Papalexiou and Koutsoyiannis (\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), the Gumbel assumption remains a stable alternative when reliable estimation of ξ is constrained by record length. They argue that a bounded upper tail is physically inconsistent for rainfall extremes and recommend using Gumbel or a fixed positive shape parameter of 0.114 for safety when negative estimates arise, noting that the slow convergence to asymptotic behavior explains the historical preference for Gumbel despite its limitations. In the present study, Gumbel was adopted for return level estimation to provide conservative results given the high uncertainty in shape parameter estimation from limited records (30\u0026ndash;48 years), while bootstrap estimates and their confidence intervals were analyzed to assess the plausibility of ξ\u0026thinsp;\u0026asymp;\u0026thinsp;0 (Figures S18a, S18b; Table S13). The spatial distribution of the parameters reveals values clustered near zero or slightly positive across most regions. However, given the limited record lengths available at individual SRRG stations, the associated confidence intervals frequently encompass zero. The spatial distribution of differences in return level estimates (GEV2\u0026thinsp;\u0026minus;\u0026thinsp;current approach) at SRRG station locations is shown in Fig. S19. Panels (a) and (b) present results for the 1-hour duration at 10-year and 100-year return periods, respectively, while panels (c) and (d) show the corresponding differences for the 24-hour duration.\u003c/p\u003e \u003cp\u003eDifferences between the two approaches remain modest across most stations and return periods. The current framework (with spatial interpolation and disaggregation) already provides a slight conservative overestimation of design rainfall, which is preferable from an engineering safety perspective. Even when a positive shape parameter is used in GEV2, the return levels show limited deviation from the current estimates. In contrast, a negative shape parameter in GEV2 leads to underestimation of extremes, an outcome that is neither physically realistic nor recommended in the literature. These findings indicate that adopting a more complex GEV2 formulation offers negligible improvement for practical design purposes under the present framework.\u003c/p\u003e \u003cp\u003eWhile the current study focuses on the application of Random Forest for spatial mapping, future research will incorporate a comparative assessment with alternative spatial estimation techniques, including geostatistical approaches such as Kriging, to further evaluate their relative performance across India's diverse climatic zones. Although the current framework applies cluster-specific modelling to represent regional hydroclimatic heterogeneity, additional assessment through spatial cross-validation strategies, such as leave-region-out experiments, is recommended to quantify model transferability under ungauged conditions. Building on this framework, further work will also evaluate Bayesian estimation with weakly informative priors, regional frequency analysis, and the use of extended homogenized records to constrain positive shape contributions better and address non-stationarity. Recent studies (Dargham and Andraos, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2026\u003c/span\u003e) have also highlighted the potential of machine learning and deep learning approaches to improve IDF curve estimation, demonstrating reliability under nonstationary rainfall conditions and offering promising directions for future research.\u003c/p\u003e \u003cp\u003eThe results of the present study are accessible through a publicly available web-based tool (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.idf-iitmadras.in/\u003c/span\u003e\u003cspan address=\"https://www.idf-iitmadras.in/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e). Users can select their desired location on the interactive map, and the tool generates IDF curves along with their corresponding confidence intervals. The results in this web-based tool are displayed in both tabular and graphical formats.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003e\u003cem\u003eThis manuscript is a preprint currently under peer review at the International Journal of Climatology.\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSupporting Information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAdditional figures, tables, and methodological details supporting this study are provided in the supplementary document.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors would like to acknowledge and thank the scholarship support provided by the Ministry of Human Resource Development (MHRD), Government of India. The authors would also like to thank Vikash Raaj for his support in developing the web-based visualization platform used for disseminating the study results.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares no known competing financial interests or personal relationships that could have influenced the work reported in this paper.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eBalaji Narasimhan, Yaswanth and C Balaji , contributed to the study conception and design. Yaswanth P conducted material preparation, analysis, performance evaluation and derivation. The first draft of the manuscript was prepared by Yaswanth P and all authors commented on previous versions of the manuscript. Balaji Narasimhan, \u0026nbsp;Yaswanth P and C Balaji \u0026nbsp;approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research received funding from the Department of Science and Technology, Ministry of Science and Technology, Government of India, under Grant/Award Number: DST/CCP/CoE/141/2018(G).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe results of the present study are accessible through a publicly available web-based tool (https://www.idf-iitmadras.in/).\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBabu R, Tejwani KG, Agarwal MC, Bhushan LS (1979) Rainfall intensity-duration-return period equations and nomographs of India. 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Elsevier B.V., 598: 126273. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.jhydrol.2021.126273\u003c/span\u003e\u003cspan address=\"10.1016/j.jhydrol.2021.126273\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Indian Institute of Technology Madras","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":false,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Design rainfall, Intensity-duration-frequency curves, Scale invariance, Remote sensing precipitation products, Random Forest","lastPublishedDoi":"10.21203/rs.3.rs-9699065/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9699065/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAdvances in satellite rainfall retrieval have made remote sensing estimates increasingly reliable for infrastructure design and planning. Intensity-Duration-Frequency (IDF) curves remain one of the most widely used statistical methods for estimating design rainfall in water resources engineering. In developing countries, meteorological observatories typically have access to daily data and sometimes lack long archives of hourly rainfall data. The current study provides a framework for developing high-resolution IDF curves with durations ranging from 1 to 72 hours at various return periods, integrating gauge and gridded rainfall products (GRPs). The Duration-dependent Generalized Extreme Value (d-GEV) and Gumbel distributions were evaluated for modeling annual maximum rainfall series (AMS) at stations with hourly observations, with the Gumbel distribution emerging as the better choice. Four gridded rainfall products (GPM, GSMaP, PERSIANN, and MSWEP) are evaluated at daily and sub-daily time scales by employing the Gumbel distribution on annual maximum rainfall series to derive the IDF curves. In the first stage, return levels are estimated for 24-, 48-, and 72-hour durations by merging a dense network of daily rainfall data from non-recording rain gauge stations with gridded products using machine learning regression techniques. The scale invariance theory of rainfall is investigated using hourly rainfall data from self-recording rain gauge stations (SRRG) to derive rainfall intensities for sub-daily durations. Scaling behavior was assessed at each SRRG station using non-central moments (NCMs) and was found to perform well for modeling rainfall extremes. The derived scaling exponents are spatially interpolated using potential covariates to develop IDF curves at 10-km resolution. The framework is implemented for the Indian subcontinent, which shows significant variability in rainfall spatial and temporal patterns. The research outputs are available through a user-friendly open web platform, where users can select a location on an interactive map to view IDF curves and confidence intervals via interactive graphs and tables.\u003c/p\u003e","manuscriptTitle":"A Framework to Develop High-Resolution Intensity–Duration–Frequency Curves: Historical Analysis and Scaling Relationships Integrating Gauge and Gridded Rainfall Products","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-14 08:18:31","doi":"10.21203/rs.3.rs-9699065/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"068e6ed1-e0ea-41ee-bb52-fa600bf3a1dd","owner":[],"postedDate":"May 14th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":68130937,"name":"Hydrology"},{"id":68130938,"name":"Hydrology"}],"tags":[],"updatedAt":"2026-05-14T08:18:31+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-14 08:18:31","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9699065","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9699065","identity":"rs-9699065","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}