The DSQM method, which morphs both the intensity and spatial structure of observed rainfall fields driven by temperature changes, is described in detail by ; only a summary of the method is provided here. At its core, DSQM modifies four key properties of rainfall fields at each time step: the mean rainfall over the entire domain, the rainfall-affected area (i.e., the rainfall spatial extent), the magnitude of the spatial variation of rainfall within the domain (i.e., the rainfall spatial coefficient of variation), and the most intense portion of rainfall (e.g., the 99th percentile) in the domain. The following steps are taken:

The scaling relationship between rainfall and temperature is estimated using the quantile regression method . The scaling is first derived for the domain-averaged rainfall intensity and then extended to other rainfall properties, conditioned on rainfall magnitude.

An example of the method is presented in Fig. , where an hourly rainfall field recorded over Beijing is morphed to represent a 2 °C regional warming level, assuming a hypothetical 7 % °C⁻¹ increase in mean areal rainfall intensity and a 8 % °C⁻¹ decrease in the rainfall-affected area, demonstrating that the regional storm becomes more intense and localized with temperature warming.

SST is a rainfall frequency analysis method that estimates spatially distributed rainfall extremes by resampling and transposing observed rainfall fields within a defined domain . It enables computation of IDF curves for a given location or region while explicitly considering the rainfall spatial structure, without manipulating or interpolating observed data obtained from point sources e.g., from multiple rain gauges;. Furthermore, SST can produce robust rainfall frequency analysis even with relatively limited rainfall data i.e., less than the recommended 30 years,, making it suitable for applications where only a few years of high-resolution gridded rainfall data from weather radar are available. The resampling and transposition of storms within the domain effectively mimic the natural spatial variability of heavy rainfall, ensuring its adequate representation in the resulting IDF curves . An overview of the development and applications of the SST method is provided by .

In our framework, we have implemented the SST method through the RainyDay model , which follows a four-step procedure:

First, define the area of interest (e.g., city) where rainfall IDFs are to be estimated. In addition, define a transposition domain, which is a larger geographic region surrounding the area of interest and is characterized by climatology and topography similar to the area of interest, such that extreme rainfall properties are approximately homogeneous.

The observed and DSQM-morphed storms are both used as input to the SST model. Figure illustrates how the SST generates stochastic storm trajectories by assigning random starting position while keeping velocity unchanged. The figure shows an observed storm from the archive corresponding to a specific rainfall return level, and for the same storm after morphing it with the DSQM (now heavier, with enhanced spatial heterogeneity).

Our framework employs the coupled hydrological-hydrodynamic AUTOSHED model for simulating pluvial floods . AUTOSHED has been applied to various real flood events and thus is suitable for transforming warming-conditioned rainfall forcings to surface inundation fields. Regarding to its physical mechanism, AUTOSHED implements the rain-on-grid approach where surface runoff generation and routing processes are sequentially simulated on triangular-shaped meshes under spatially distributed rainfall forcings. To achieve a detailed representation of surface micro-topography, road and river boundaries are incorporated as break-lines during mesh generation , enabling mesh edges to align adaptively with these features and accurately capture their elevation profiles. At each time step, AUTOSHED first performs distributed rainfall-runoff simulations on each triangular unit. Each unit is divided into impervious and pervious parts according to the empirical impervious ratio, and AUTOSHED calculates the total surface runoff as the area-weighted sum of each component. The rainfall-runoff simulation includes typical urban hydrological processes such as building interception, vegetation interception, and soil infiltration, where related parameterization details can be found in and and in Sect. S1 in the Supplement.

After obtaining surface runoff, AUTOSHED solves the full two-dimensional shallow water equations to simulate the flood routing process, where all boundaries are specified as open boundaries . The equations are solved by the Godunov-type finite volume method further details are discussed by. The time integration is solved by the explicit first-order scheme and an appropriate time step is dynamically adjusted by a Courant-Friedrichs-Lewy (CFL) value of 0.5 to maintain numerical stability. For computational efficiency, AUTOSHED uses a multi-core parallel architecture of GPUs .

Beijing Municipal Administrative Center (BMC, Fig. b) is located within the large metropolitan area of Beijing (Fig. a). It serves as the eastern gateway to China's national capital, where numerous local government departments have been relocated since 2015 to reduce excessive congestion in the central city. BMC's area is approximately 300 km² and is relatively flat with an average slope of 1.5 %. It exhibits a humid continental climate, with an average annual rainfall of 536 mm and an average annual temperature of 13 °C. As one of the world's largest megacities, Beijing is particularly susceptible to pluvial flooding from extreme storms; this vulnerability is underscored by observed increases in sub-daily precipitation extremes across North China, where reported positive and accelerated trends in six extreme sub-daily precipitation indexes during 1971–2022, especially after 2001. Studies have shown that thermodynamical factors, such as high temperatures and the urban heat island effect, have played a dominant role in the intensification of local extreme rainfall events . At the same time, dynamic factors, such as monsoon circulation and topography influences, may also affect summer extreme rainfall in the Beijing region . However, the dynamic contributions are not explicitly accounted for in the present study. Furthermore, storms are projected to become smaller and more concentrated with higher intensity in the future , increasing the probability of the occurrence of severe flood disasters. Two notable unprecedented storms were recorded in the last decade: the first one on 21 July 2012 (areal rainfall of 215 mm in 20 h), and the second one hitting Beijing on 31 July 2023 (331 mm in 83 h), resulted in heavy floods and 79 and 33 deaths, with economic losses of 1.4 billion and EUR 60 million, respectively . These events underscore the urgent need to evaluate the future evolution of rainfall fields with the corresponding response of flood hazards to better inform disaster risk management.

Considering data, we used gridded hourly rainfall fields at 1 × 1 km resolution, downscaled from the 8 × 8 km CMORPH product for the period from 1998 to 2019 . Hourly and 25 × 25 km near-surface air temperature data for the same period were obtained from the ERA5 climate reanalysis . These datasets were used to derive the rainfall-temperature scalings and prepare the present and future rainfall archives for the SST. In addition, we collected multi-source geographical data to set up the urban pluvial flood model, including: a 0.5 m digital elevation model (Fig. c) that was generated from an airborne Light Detection and Ranging (LiDAR) with a sampling density of 100 points per m² and a mean absolute height error of 0.05 m (Fig. d); a 10 m land use map from the WorldCover 2021 product (Fig. b), a 250 m soil hydraulic properties from the HiHydroSoil v2.0 database , a 250 m leaf area index data from the MODIS MOD15A2H product , and building footprint and road network layers from the Baidu map service.

Furthermore, we collected observational data related to a recent torrential rainfall event that occurred in August 2024 to evaluate the model's performance. Radar-based quantitative precipitation estimations (QPE), covering the period between 21:00, 8 August, and 09:00, 10 August (local time), were acquired from a local agency with a spatiotemporal resolution of 500 m and 5 min (see Fig. S1). In addition, the maximum inundation depth records at six underpasses as a result of the storm were obtained from the local agency as well.

In the following, we analyzed changes in rainfall extremes as a result of regional 1, 3, and 5 °C warming in comparison with the 1998–2019 period. Projected changes in return levels of averaged rainfall over the BMC, based on 24 h storm events, are shown in Fig. a. The average increase in extreme rainfall intensity for different return levels is 3 %, 11 %, and 18 % under the 1, 3, and 5 °C regional warming levels, respectively. To highlight the mean shifts in average extreme rainfall over BMC, we further computed the natural climate variability in the present climate obtained from 100 storm realizations and represent it with the green shaded area in Fig. a. While the mean return levels for the 1 and 3 °C warming still fall within the natural variability of the present return levels, 5 °C warming is falling beyond the variability. To exemplify the high variability of the extreme rainfall both in space and time, we further illustrated three 24 h storm events randomly selected from the simulated pool (Fig. b).

The projected changes in urban pluvial floods can be estimated from the ensemble of AUTOSHED simulations driven by multiple storm realizations associated with specified return levels and regional warming levels. Using storm events at the 100-year return level as an example, results of the current-climate simulations demonstrate that the median inundation depth, calculated as the pixel-wise medians from the maximum inundation depth maps under 50 storms, is reaching up to 2 m depths within the BMC (Fig. a), with pronounced flooding beneath underpasses and within riparian parks in the city center (Fig. b). Under warming levels of 1, 3, and 5 °C, median maximum flood depth increases by 4 %, 7 %, and 8 %, respectively, while the maximum increase reaches 28 % under +5 °C (Fig. c). Figure d–f illustrate the spatial distribution of median differences in inundation depths, revealing that inundation depths increase by 5 %–25 % in both major roads and residential areas, with the most extensive impacts under +5 °C extending across almost the entire urban area.

The flood inundation duration of the current climate, defined as the time during which water depth exceeds 0.4 m for the 100-year rainfall return period, is shown in Fig. a. The close-up to the center area (Fig. b) highlights the persistent water accumulation up to 12 h along major roads and in low-lying residential areas. In contrast to the distribution of the inundation depth, the distribution of the current inundation duration exhibits a more pronounced long-tail behavior, with the increase rate in maximum duration reaching 125 % (Fig. c) for 1 °C warming. This indicates that some regions currently flooded for a short duration will face prolonged inundation under increasing temperatures. As presented in the maps in Fig. d–f, substantial relative changes in inundation duration can be observed at the fringes of the inundated regions. When evaluating the overall tendency, the median inundation duration increases by approximately 9 %, 11 %, and 8 % under warming levels of 1, 3, and 5 °C, respectively, indicating that further warming beyond 3 °C will not necessarily result in further increases of overall inundation duration. This can be attributed to the spatiotemporal changes in extreme rainfall structures: while higher temperature raises the mean areal rainfall, it also leads to a reduction in storm area. Such localized rainfall intensification with fast movement can contribute to higher instantaneous inundation depth; however, its influence on inundation duration may be minimal or even negative due to counteracting effects between accelerated surface flow accumulation (as presented next).

We further investigated how surface flow velocities change with increasing warming. In the current climate, peak flow velocity for a 100-year return period storm generally remains below 0.5 and increases up to 1 m s⁻¹ along major roads and underpasses (Fig. a, b). Under 1, 3, and 5 °C warming, the distributions of the peak flow velocities are positively skewed with medians increasing by 3 % (the upper tail reaches 17 %), 6 % (22 %), and 8 % (28 %), respectively (Fig. c). Detailed views of the center area (Fig. d–f) reveal considerable velocity increases (> 10 %) that are initially located along major roads and gradually expand into low-lying residential areas through narrow streets. Such increased velocity indicates strengthened momentum transfer and further leads to possibly faster recession and shorter inundation duration.

In summary, for major areas in the BMC, climate warming tends to increase both inundation depth and surface flow velocity, whereas its influence on inundation duration remains unclear due to counterbalancing hydrodynamic mechanisms between higher runoff volumes that are accumulating faster and subsequently leading to faster recession time.

We last evaluated the impact of the changes in 100-year storms on pluvial flood hazards. Flood hazards are categorized by maximum inundation depths following local government standards,: low hazard (0.15–0.27 m), moderate hazard (0.27–0.4 m), moderate-high hazard (0.4–0.6 m), and high-hazard (> 0.6 m) (Fig. a, b). Under current climatic conditions, moderate-high and high hazard areas are predominantly confined to riparian parks and major roads, especially at low-lying underpasses with insufficient drainage capacities (Fig. b).

Analysis of the associated inundation duration (Fig. S2) reveals that these areas can also experience long-lasting inundation for more than 12 h, severely disrupting local transportation networks and leading to substantial economic losses and potential fatalities due to the prolonged blocking of evacuation routes . Furthermore, it should be highlighted that moderate-high and high hazard zones increase by 15 %–20 % compared to current conditions, while low and moderate hazard areas expand by up to 5 %–10 % (Fig. c). These results demonstrate the higher sensitivity of high hazard zones to climate change, with already severely inundated areas, projected to experience more significant increases, emphasizing the urgent need for targeted adaptation measures in flood-prone locations.

One of the key strengths of the proposed framework over most existing approaches lies in its capacity to assess changes in both spatial variability and areal intensity of rainfall fields with increasing temperatures, and to evaluate their implications on urban pluvial flooding. In the BMC, for example, we examine the spatio-temporal characteristics of 50 simulated 24 h storms representing the 100-year return level under three regional warming levels. We find that while the mean rainfall over the domain is projected to increase by 0.73 mm h⁻¹ °C⁻¹ (black lines in Fig. ), the maximum rainfall over the domain from each 24 h storm (blue line) and the heaviest part of each storm events (i.e., the 99th percentile when analyzing the grid scale, red line) increase sharply with rates of 1.89 and 2.48 mm h⁻¹ °C⁻¹, respectively. In contrast, the rainfall area exhibits a slight decline of -1 % °C⁻¹ under regional warming levels (green dashed line in Fig. ).

These results indicate that future storms in the BMC are likely to become more intense yet more spatially concentrated under warming. For urban environments, this can still trigger more severe surface inundation when intense rainfall is concentrated over impervious and hydraulically connected parts of the city, such as major roads, underpasses, and low-lying residential areas. Under these conditions, runoff is generated and routed more rapidly, leading to higher inundation depths (Fig. ) and stronger flow velocities (Fig. ). By contrast, the reduced storm footprint may partly limit the duration or spatial spread of flooding, which is consistent with the weaker and less monotonic changes found here for inundation duration (Fig. ).

Overall, the combination of higher intensities and reduced spatial coverage is likely to exacerbate urban pluvial flood risks . While in the BMC, the change in storm size under climate warming is relatively minor and does not offset the stronger increase in rainfall intensity. However, larger negative trends have been documented elsewhere, such as the tropics and parts of the United States , which could, in some cases, reduce flood impacts as smaller storms may have a lower probability of city-wide inundation due to constrained trajectories . To conclude, the proposed framework provides an ideal tool to investigate flood responses to such changes in rainfall characteristics.

While rainfall was simulated at a 1 km spatial resolution, a substantially finer resolution of 5 m is adopted for the pluvial flood simulations. This raises the question of whether such refinement is warranted, given the additional computational time and resources required, as well as the effort involved in obtaining a finer DEM. Hence, we further explored how sensitive the surface inundation response is to the spatial resolution of the model. We did so by selecting 22 locations that are frequently affected by flooding (see Fig. S3) and re-running the model using the fine resolution of 5 m, and a coarser resolution of 30 and 90 m that can be obtained from multiple freely available sources e.g., the SRTM product. Across those locations, warming-driven increases in flood depth, inundation time, and velocity are markedly larger at finer grid resolutions: at 5 m resolution, increase of 1 °C yields roughly 1.4 cm higher depth, 0.22 h longer inundation, and a 0.017 m s⁻¹ rise in peak velocity, whereas at 30 m (90 m) these sensitivities fall to 0.4 cm (0.3 cm), 0.12 h (0.17 h), and 0.005 m s⁻¹ (0.003 m s⁻¹). The reduced sensitivity at coarser spatial resolutions underscores the importance of high-resolution hydrodynamic simulations for robust assessments of urban pluvial flood risks under climate change .

Although rainfall is applied at 1 km and hourly resolutions in our case study (reflecting data availability constraints), using finer resolutions, on the order of 100 m and 5 min, is recommended for pluvial flood applications . This recommendation is due to the high spatial variability of extreme rainfall in urban environments and the rapid hydrological response characteristic of urban catchments . Such high-resolution datasets, unfortunately, remain scarce. However, the proposed framework is flexible with respect to the spatial and temporal resolution of the rainfall fields. Therefore, if rainfall inputs with finer spatial and temporal resolution are available, they can be used directly within the framework, without requiring further methodological modifications.

Additional potential parameterizations of the framework, beyond those presented in the case study, warrant discussion. First, there is an ongoing debate regarding the appropriate approach to estimate rainfall-temperature scaling, specifically whether to use near-surface air temperature or dewpoint temperature, as the latter explicitly accounts for humidity . Additionally, there is discussion on whether hourly or daily temperatures better represent the atmospheric processes leading to rainfall events . While hourly near-surface air temperature is used in our case study, the framework is flexible and the scalings can be implemented with either temperature variable and at different temporal resolutions.

Defining an appropriate transposition domain for the SST is critical as it influences the composition of the storm catalog and sampling of rainfall extremes. Here, we define the homogeneous transposition domain based on the similarity of the topography and rainfall extremes with the BMC area, yet different criteria, such as using the mean rainfall depth, cloud-to-ground lightning, and storm counts can be applied. Further work on systematic delineation of transposition domains is underway . More importantly, instead of using a “uniform” transposition process , we have used a “non-uniform” one in our case study, which is based on historical frequency and location of regional storms. This can be particularly relevant in urban and mountainous areas, where storm trajectories may have preferred routes .

There are several advantages associated with the proposed framework, beyond the clear benefit of enabling application in cities lacking high-resolution rainfall data from climate models: (i) changes in rainfall properties are derived directly from the well-established thermodynamic relationship between rainfall and temperature , particularly relevant for short-duration extreme (convective) events ; (ii) the framework modifies not only rainfall intensity but also the spatial coverage of storms, which can play a more significant role in urban flooding dynamics ; (iii) future changes are projected according to prescribed levels of regional warming e.g.,, rather than through specific emission scenarios. This approach reduces some of the uncertainties associated with IDF projections stemming from both climate model and emission scenario discrepancies ; (iv) the space-time stochasticity of extreme rainfall (also referred to as natural variability or chaotic storm evolution), which constitutes the dominant source of uncertainty in rainfall intensity-frequency relationships , is explicitly represented within the framework; and (v) the AUTOSHED rain-on-grid model uses GPU-accelerated ensemble hydrodynamic simulations to efficiently convert high-resolution rainfall fields into detailed flood hazard maps, including inundation and flow velocities needed for targeted disaster management .

Notably, the framework primarily considers thermodynamic effects on extreme rainfall , but dynamic factors such as atmospheric circulation and urban areas that can affect storm frequency and intensity are not explicitly taken into account. Conducting the analysis for specific rainfall types and explicitly considering their potential future frequency changes, or applying the framework to different large-scale atmospheric modes (e.g., El Niño, La Niña) and embedding their projected shifts based on information from climate models, can partially account for changes in the dynamic component. However, fully implementing the dynamic factor affecting rainfall into the framework remains a subject for future development.

Although the proposed framework is developed for short-duration convective rainfall extremes and urban pluvial flooding, its rainfall-generation component could be used more broadly. The rainfall–temperature scaling is most appropriate at sub-daily durations , whereas the SST module can also generate rainfall extremes at longer durations . This opens the possibility of using the generated rainfall fields as input to catchment-scale flood models. For fluvial applications, however, the rainfall module would need to be coupled with a hydrological–hydrodynamic model that properly represents soil water storage, groundwater dynamics, and their interactions with channel routing. Such processes are particularly important in large semi-humid catchments like Beijing, where complex runoff generation mechanisms strongly condition flood response . In addition, the present framework modifies rainfall spatial structure but does not explicitly account for warming-induced changes in storm temporal structure, which may also affect flood regimes .

Furthermore, a substantially larger number of hydroclimatic scenarios would be needed for a more comprehensive stochastic flood-frequency analysis, and GPU parallelization of numerical simulations may not be sufficient to support such computational demands. A promising avenue for future work is therefore to integrate the present workflow with emerging physics-informed surrogate models, such as super-resolution models that use deep neural networks to transform computationally efficient coarser-resolution inundation fields into practically useful fine-scale flood maps . Such hybrid approaches may offer a practical way to expand the ensemble size while preserving the spatial detail needed for urban flood-hazard mapping.

In summary, our proposed framework, based on open-source code and leveraging GPU-based high-performance computation techniques, offers an efficient and accessible approach to investigate how warming-conditioned changes in rainfall intensity, spatial extent, and spatial variability can alter future urban pluvial flood hazards. It introduces an efficient method for projecting short-duration rainfall under climate change, which is simple to implement and relies exclusively on observed rainfall and reanalysis temperature data, thus removing the dependency on high-resolution climate model simulations. Applied here to the BMC region, the framework demonstrates its potential to address flood risks associated with summer convective rainfall extremes. It can be readily implemented in other urban areas where similar rainfall types contribute to pluvial flooding and should be further evaluated for its applicability to other types of extreme rainfall, although no fundamental limitations are expected.