Introduction

Climate change has increased the occurrence frequency of weather and climate extremes, including heatwaves, heavy precipitation and droughts1,2,3, which are increasingly likely to converge and result in compound extreme events4,5,6,7. For a given location, compound extreme events can be classified into two categories8: (1) sequential compound extreme events, which occur in rapid sequence, such as sequential heatwave and precipitation (SHP) events, and (2) concurrent compound extreme events, which occur simultaneously, such as concurrent drought and heatwave (CDH) events. Compound extreme events can amplify the impacts of individual hazards, even if a single event is not extreme9. The prior or concurrent events could increase a system’s vulnerability to the sequential or co-occurring hazards of another extreme10, possibly exceeding human coping capacities11. China is one of the countries that has been most severely affected by both SHP and CDH events12,13,14,15. SHP events, driven by increased atmospheric instability from extreme heat, can trigger flash floods and other secondary disasters16,17,18, causing widespread damage to water quality, crop yields and human livelihoods19. As for CDH events, the summer of 2022 witnessed simultaneous droughts and heatwaves that struck the Sichuan-Chongqing region, leading to a cascade of socio-economic consequences. The severe drought greatly reduced hydropower generation, causing electricity shortage that restricted the ability to cope with the high temperature during heatwaves20.

Despite the severe consequences, the future spatiotemporal variations of the two types of compound extreme events under climate change remain poorly understood. Specifically, most previous studies only focused on one type of compound extreme events with more attention on CDH events. While it is generally agreed that CDH events will dramatically increase globally21,22,23,24, the future changes in China are not consistent among different studies. Some project significant increases in CDH events across China9,14,25, while others suggest that the increases in CDH events in China are quite limited due to projected rising precipitation26. Recently, SHP events have also received increasing attention. The fractions of heavy precipitation events preceded by heatwaves are found to have increased in the past few decades across the globe27,28,29 and in China19,30,31,32,33, while the future changes of SHP events in China under different emission scenarios remain underexplored. Furthermore, research on future changes in compound events has generally relied on univariate bias correction methods to adjust the raw projections from the general circulation models (GCMs)29,34,35. This widely adopted approach, however, neglects the dependence between variables36, which could limit the prediction accuracy of compound extreme events and associated socio-economic exposure, thus hindering the formulation of corresponding adaptive management strategies.

Here, we provided a comprehensive analysis of the spatiotemporal variations of two typical compound extreme events (i.e., SHP and CDH events) across China during the historical (1961–2014) and future period (2015–2100). For the future projections, we adopted a bivariate bias correction method that accounted for the precipitation and temperature dependence to adjust the raw projections from five GCMs under three different scenarios (SSP1-2.6, SSP3-7.0, and SSP5-8.5). Subsequently, we examined the future changes of the return period for the historical 50-year compound extreme events and investigated the underlying drivers using a copula-based bivariate framework. Finally, we assessed the future socio-economic exposure related to SHP and CDH events for China’s ten major urban clusters in terms of gross domestic product (GDP) and population (Supplementary Table 1), which could help providing guidance for adaptive measures in response to compound extreme events in a changing climate.

Results

Historical patterns of SHP and CDH characteristics

We first explore the spatial distribution of SHP and CDH characteristics based on frequency, duration and magnitude index during 1961–2014 (“Methods”). SHP events mainly occur in eastern China (Fig. 1a–c), closely resembling the distribution of heavy precipitation (Supplementary Fig. 1a–c), with the southeastern coastal regions and the middle and lower Yangtze River basin being most severely affected. Furthermore, we examine the characteristics of SHP events in China’s ten major urban clusters (Fig. 1g). Guangdong-Hong Kong-Macao Greater Bay Area (GHM), Shandong Peninsula (SDP), Western Coast of the Strait (WCS), Chengdu-Chongqing (CY), and Central and Southern Liaoning (CSL) are found to be the five most severely affected urban clusters (Fig. 1h and Supplementary Table 2). Averaged across China, SHP events mainly start and end in July. While in the aforementioned most severely affected urban clusters, the occurrence time of SHP events tends to be longer, with the end date extending into August (Supplementary Fig. 2a, b).

Fig. 1: Spatial patterns of compound events during the historical period.
Fig. 1: Spatial patterns of compound events during the historical period.
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a–c The spatial patterns of frequency, duration and magnitude index of SHP events during 1961–2014, with white areas indicate no events occurred. d–f The same as a–c, but for CDH events. g distribution of the ten major urban clusters in China, including Central and Southern Liaoning (CSL), Beijing–Tianjin–Hebei (JJJ), Shandong Peninsula (SDP), Central Plains (CP), Guanzhong Plain (GZP), Yangtze River Delta (YRD), Middle Yangtze River (MYR), Chengdu-Chongqing (CY), Western Coast of the Strait (WCS) and Guangdong-Hong Kong-Macao Guangdong-Hong Kong-Macao Greater Bay Area (GHM). The listed hotspots represent the five urban clusters that were most severely affected by SHP and CDH events. h Multi-year average characteristics of SHP events in five most severely affected urban clusters, ranked in increasing order based on the three characteristics. i The same as h but for CDH events.

In contrast, CDH events occur in most regions except the Tibetan Plateau and the Tianshan Mountains (Fig. 1d–f), which are similar to the spatial patterns of heatwaves (Supplementary Fig. 3a–c). The top five hotspots most severely affected by CDH events are GHM, WCS, Central Plains (CP), Yangtze River Delta (YRD) and SDP, respectively. Each of these urban clusters experiences an average occurrence frequency of more than twice a year (Fig. 1i and Supplementary Table 2). CDH events primarily start in July across China, except for CP and SDP, whose onsets tend to be earlier, usually occurring in June (Supplementary Fig. 2c). The significant differences in the end date indicate larger threats in the southern regions of China, where the latest CDH events occur in August in the historical period (Supplementary Fig. 2d).

Future changes in SHP and CDH characteristics

The spatiotemporal changes of SHP and CDH characteristics in the future period (2015–2100) relative to the historical period (1961–2014) are also evaluated in terms of frequency, duration, and magnitude. SHP indicators are projected to increase at all grids, with higher emission scenarios showing greater increases (Fig. 2a–c and Supplementary Fig. 4). Notably, the frequency and magnitude of SHP events for both Middle Yangtze River (MYR) and YRD are projected to increase by over 400% and 200% in the future under SSP5-8.5 scenario (Fig. 2a–c and Supplementary Table 3), much greater than the increasing rates of heatwaves and heavy precipitation events alone (Supplementary Figs. 1 and 3). For the remaining urban clusters, the main concern is that the duration is expected to increase by more than three times (Fig. 2b and Supplementary Table 3), with the start date in GHM and WCS extending earlier into June (Supplementary Fig. 2m). For CDH events, despite alleviated droughts in most parts of China (Supplementary Fig. 5), CDH indicators are expected to increase at most grids (Fig. 2d–f and Supplementary Fig. 6), with durations doubling for China’s most urban clusters (Supplementary Table 3). When considering changes in the spatial extent experiencing compound events, ~50% and 75% of the area in China are expected to experience SHP and CDH events, respectively by the 2050 s, with these proportions projected to further increase by ~5% for both events by the 2090 s under SSP5-8.5 scenario (Supplementary Fig. 7).

Fig. 2: Future spatiotemporal variations in compound event characteristics.
Fig. 2: Future spatiotemporal variations in compound event characteristics.
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a–c Spatial distribution of SHP change in the future period (2015–2100) under SSP5-8.5 scenario relative to the historical period (1961–2014) in terms of frequency, duration and magnitude, respectively. d–f The same as a–c, but for CDH. g–i Annual time series of frequency, duration and magnitude of SHP events averaged across China during the historical (1961–2014) and future period (2015–2100) under three scenarios, with the shades indicating ±1 SD among the five GCMs. j–l The same as g–i, but for CDH events.

From the temporal variations of compound event characteristics averaged across China, the future changes in SHP events show temporally increasing divergence among scenarios. The rising trend persists into the end of the century under SSP3-7.0 and SSP5-8.5 scenarios. In contrast, under SSP1-2.6 scenario, the increasing trend is slower, with characteristics showing slight declines after 2060 (Fig. 2g–i and Supplementary Fig. 8a, b). For CDH events, the differences among scenarios are similar to those for SHP events. However, the frequency of CDH events stabilizes in the latter half of the century under all three scenarios, while the duration and magnitude continue to increase under higher emission scenarios, indicating that a single CDH event in China will be longer and more intense if effective measures to control carbon emissions are not taken (Fig. 2j–l and Supplementary Fig. 8c, d).

Projected changes in return periods and the driving factors

To quantify the future changes of compound extreme events, we examined the future changes of the return periods (RPs) for the historical 50-year compound events and investigated the driving factors within a copula-based bivariate framework. The changes can be attributed to three parts: the marginal distribution of individual events and their copula dependence structure. In the future climate, we expect shortened RP for SHP events across China. In the historical five hotspots (GHM, SDP, WCS, CY, CSL), the RP will be shortened to ~10 years (Fig. 3a and Supplementary Table 4). The factors driving these changes show consistent results across different scenarios. Heatwaves are the primary factors that shorten the RP (Fig. 3c and Supplementary Fig. 9c, d). The contribution of the copula dependency structure illustrates that elevated emissions will lead to stronger interactions between heavy precipitation and heatwaves, resulting in SHP events with the same magnitudes as the historical ones occurring more frequently in the future (Fig. 3g and Supplementary Fig. 9g, h).

Fig. 3: Projected future RPs for the historical 50-year compound events under SSP5-8.5 scenario and the driving factors of the future changes.
Fig. 3: Projected future RPs for the historical 50-year compound events under SSP5-8.5 scenario and the driving factors of the future changes.
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a, b Spatial patterns of future RP for SHP and CDH events, with probability density for different RPs shown in the bottom left corner. c, d Relative contribution of marginal distribution of heatwave for SHP and CDH events. e The same as c but showing relative contribution of heavy precipitation’s marginal distribution. f The same as d but showing that of drought’s marginal distribution. g, h The same as c, d but showing that of copula dependence between variables.

Regarding CDH events, the RPs for historical 50-year events are also expected to be shortened in most areas in the future, with the RP shorter than 10 years accounting for the largest proportion. Nevertheless, under SSP1-2.6 scenario, the RPs at some grids will increase and the average RPs in YRD and WCS are ~60 years owing to mitigated droughts (Supplementary Fig. 10a and Supplementary Table 4). The area proportions with prolonged RP are smaller under higher emission scenarios, and the average RPs of all urban clusters are expected to be shortened (Fig. 3b, Supplementary Fig. 10b and Supplementary Table 4). As for the attribution results, heatwaves are regarded as the dominant factor for RP change, which is similar to SHP events, although the spatial differences in contributions are smaller than those for SHP events (Fig. 3d and Supplementary Fig. 10c, d). The contributions of droughts are smaller compared to those of heatwaves, but for areas with prolonged RPs, the positive contributions of droughts are expected to outweigh the negative contributions of heatwaves (Fig. 3f and Supplementary Fig. 10e, f). The dependency structure shows a negative contribution at almost all grids, revealing a mutual reinforcement effect between droughts and heatwaves in the future (Fig. 3h and Supplementary Fig. 10g, h).

Projected socio-economic exposure to SHP and CDH hazards

The rising SHP and CDH hazards across China put human livelihoods and property safety at higher risks. Here we assessed the potential socio-economic impacts of SHP and CDH events by measuring exposure in terms of GDP and population. In China, as GDP shows a strong upward trend (Supplementary Fig. 11a), GDP exposure to both events is expected to increase in the future, especially under higher emission scenarios (Fig. 4a, b). For population exposure, the most significant growth is expected under SSP3-7.0 scenario, while under the low emission scenario (SSP1-2.6), limited exposure increases are expected. Under SSP5-8.5 scenario, unlike the rapid increase in GDP exposure, dramatic decreases in population (Supplementary Fig. 11b) are projected to result in lower population exposure to compound extreme events by the end of the century compared to SSP3-7.0 scenario (Fig. 4c, d).

Fig. 4: Temporal changes of GDP and population exposure to SHP and CDH hazards and attribution results in China’s major urban clusters.
Fig. 4: Temporal changes of GDP and population exposure to SHP and CDH hazards and attribution results in China’s major urban clusters.
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a–d annual time series of GDP and population exposure to SHP and CDH events averaged across China during the historical (2000–2014) and future period (2015–2100) under three scenarios, with the shades indicating ±1 SD among the five GCMs. The unit for GDP exposure is a hundred million (RMB/km2) × days/y, and the unit for population exposure is (persons/km2) × days/y. e–h Changes in GDP and population exposure to SHP and CDH events in the future period (2015–2100) compared with the baseline period (2000–2014) in the urban clusters that have been identified as the top five hotspots in the historical period. The total exposure change (TC) can be further divided into GDP or population change effect (GE or PE), climate hazard change effect (CE), and interaction effect (IE).

We further examined the future exposure changes in China’s major urban clusters using 2000–2014 as the baseline period, given significant GDP and population growth during the historical period. In the urban clusters identified as historical hotspots, the GDP and population exposure to SHP and CDH hazards are all expected to increase in the future (Fig. 4e–h and Supplementary Figs. 12–15). This is most evident in GHM under SSP5-8.5 scenario, where GDP exposure to SHP and CDH will surge 62 and 43 times, and population exposure will increase by 1.50 and 0.77 times, respectively (Supplementary Figs. 12–15). The total exposure change (TC) can be further divided into GDP or population change effect (GE or PE), climate hazard change effect (CE), and interaction effect (IE) (Methods). For exposure to both events in GHM, CE was minimal for GDP, resulting in only about a 2-fold increase compared with an 18-fold increase contributed by GE. The large increase in GDP exposure caused by IE can be attributed to the concurrent rapid GDP growth and increasing occurrence of compound events under selected future scenarios. While for population exposure, CE plays a dominant role in the increase, in contrast to the negative contributions from the other two effects (Fig. 4e–h and Supplementary Figs. 12–15). Similar results are also observed in most other urban clusters, where PE leads to negative population exposure changes, resulting in negative IE, while the positive CE tend to outweigh the negative PE and IE and lead to an overall increase in population exposure to compound hazards in the future.

Discussion

Most previous studies on compound extreme events conducted bias correction for each variable separately29,34,35, which fails to account for the variable dependence. To investigate the differences in projections caused by considering or ignoring precipitation and temperature dependence, we compared the characteristics and exposure of SHP and CDH events estimated from the bivariate bias correction method adopted by this study (i.e., multivariate bias correction using N-dimensional probability density function transform, MBCn) with the results from the univariate bias correction method adopted by most previous studies (i.e., Quantile Delta Mapping, QDM). In the historical period, the spatiotemporal variations of both compound events identified by bias-corrected GCM data using both MBCn and QDM method closely resemble those of the observation-based results (Fig. 1a–f and Supplementary Figs. 16–18). For future changes in SHP events, high emission scenario amplifies the differences between the two methods across all characteristics, with the MBCn method leading to higher projection results (Supplementary Fig. 19). For future changes in CDH events, the underestimation caused by univariate bias correction is more pronounced (Supplementary Fig. 20), with the frequency underestimated by ~20% across all three scenarios in the 2090 s. The socio-economic exposures to both events would also be underestimated if ignoring the precipitation and temperature dependence (Supplementary Figs. 21 and 22). The univariate methods, like QDM, fail to maintain the inter-variable relationships when correcting biases in marginal distributions37. Nevertheless, these relationships are crucial in compound events38. For SHP events, atmospheric instability from heatwaves can trigger or intensify localized precipitation19,39,40, which could terminate the preceding heatwave through strong evaporative cooling16,40,41, thus forming sequential events. For CDH events, there are several mechanisms that contribute to the coupling between heatwaves and droughts, e.g., high temperatures can increase the sensible heat and atmospheric aridity through land-atmosphere interactions, creating self-reinforcing CDH conditions21,42,43.These mechanisms highlight the importance of maintaining the variable dependence when projecting the future changes of compound events.

Our results show that, regardless of bias correction methods, climate change has intensified compound events. However, most previous studies have focused on a single type of event, primarily CDH events, with less attention on SHP events. Here we investigated the future changes of these two types of events in China within a unified analytical framework. Regarding their distribution, regions in the east of Hu’s Line face severe threats from both events, while the northwest regions are only affected by CDH events. More specifically, both events are expected to significantly jeopardize the middle and lower Yangtze River basin, and the onset of SHP events is expected as early as June in the future. But interestingly, CDH events generally start in July in this region, later than the start date of CDH events in other regions of China (Supplementary Fig. 2). This anomaly could be attributed to the Mei-Yu season, which predominantly occurs in June in the Yangtze River basin, and leads to heavy precipitation that mitigates the drought occurrences to some extent44. The start of the Mei-Yu season is expected to advance in the future, while the end date shows no significant changes45. Consequently, SHP events in the Yangtze River basin are expected to begin earlier, but the onset of CDH events will remain largely unchanged.

When considering the socio-economic impacts of compound hazards, it is crucial to focus on historical hotspots of both SHP and CDH events, i.e., GHM, SDP, and WCS (Fig. 1g). While the hotspots for SHP and CDH events show some differences, these three regions consistently rank among the top five most affected urban clusters in China. Future climate change and socio-economic development are expected to significantly increase exposure in these hotspot regions, especially in GHM, which is China’s fastest-growing economic and high-tech center with the highest GDP (Supplementary Figs. 12–15). Besides, the regions along the Yangtze River (i.e., CY, MYR, and YRD), which heavily rely on hydropower generation and play a seminal role in China’s power supply network, are particularly vulnerable to CDH events. Such events can cause hydropower deficits as in the summer of 2022, increase reliance on thermal power, and consequently, lead to higher carbon emissions. Under SSP3-7.0 scenario that represents the shared socio-economic pathway characterized by regional rivalry, the most critical challenge is the substantial increase in population exposure (Fig. 4c, d). Despite the projected population decline34,46, the rapid aging of the population47 along with longer-duration CDH events and floods triggered by SHP could lead to higher mortality. Fortunately, under the low-emission and sustainable development SSP1-2.6 scenario, GDP and population exposure are expected to increase slowly in the coming decades and projected to decline in the latter half of the 21st century (Fig. 4). Our results underscore the necessity of strengthening adaptive capacities and resilience to mitigate the impacts of compound hazards under climate change, and highlight the importance of controlling greenhouse gas emissions to reduce compound extreme events and associated socio-economic exposure for the sake of sustainable development.

Methods

Datasets

This study adopts the CN05.1 daily gridded dataset as the observational data, which provide precipitation and maximum temperature data at a 0.25° resolution for the historical period from 1961 to 2014. The CN05.1 dataset is constructed using the “anomaly approach” during the interpolation but with more station observations (~2400) in China48. In the “anomaly approach”, a gridded climatology is first calculated, and then a gridded daily anomaly is added to the climatology to obtain the final dataset. We resample the resolution to 0.5° using arithmetic averaging.

For the future climate scenarios, precipitation and maximum temperature data from the Intersectoral Impact Model Intercomparison Project 3b (ISIMIP3b) are collected49, including five GCMs (GFDL-ESM4, IPSL-CM6A-LR, MPI-ESM1-2-HR, MRI-ESM2-0, UKESM1-0-LL) that participate in the sixth Coupled Model Intercomparison Project (CMIP6). The models cover a historical period (1961–2014) and a future period (2015–2100) under three shared socio-economic pathway (SSP) scenarios (SSP1-2.6, SSP3-7.0, SSP5-8.5) at a spatial resolution of 0.5°. SSP1-2.6 represents a combination of the sustainability socio-economic pathway and a low representative concentration pathway 2.6 (RCP2.6), SSP3-7.0 combines regional rivalry with medium-high emissions (RCP7.0), and SSP5-8.5 pairs high fossil-fuel development with high emissions (RCP8.5)50.The three scenarios adopted by ISMIP3b can reflect the effects of different greenhouse gas emission levels, and have been widely used for assessing the future changes in hydroclimatic extremes24,51.

Gridded GDP and population data are used to calculate socio-economic exposure, with the baseline period for historical data from 2000–2014 and the future projection period from 2015 to 2100. Historical GDP52 and population53 data at a 1 km resolution are available for the years 2000, 2005, 2010, and 2015, and annual data for the intervening years are obtained through linear interpolation. As for future period, GDP data are available at a 0.5° resolution on an annual basis46. Population data are derived from the Global 1-km Downscaled Population Base Year and Projection Grids Based on the SSPs, version 1.01 (2000–2100)54, with data available at 10-year intervals and annual results obtained through linear interpolation. The GDP and population for each grid are recalculated to match a 0.5° resolution. SSP1, SSP3, and SSP5 scenarios are selected to match the SSP1-2.6, SSP3-7.0, and SSP5-8.5 scenarios, respectively.

Bias correction of general circulation models (GCMs)

The Multivariate Bias Correction using N-dimensional probability density function transform (MBCn) method is adopted for bias correction of GCMs, accounting for the dependence between precipitation and temperature. It’s a flexible multivariate extension of Quantile Delta Mapping (QDM), integrating QDM and the N-dimensional probability density function transform (N-pdft) as outlined by Cannon36. This method is implemented in the MBCn algorithm, available as the MBC package in R.

To investigate the differences in projections caused by variable dependence in bias correction, the univariate bias correction method, QDM, is also applied for comparison with MBCn, preserving absolute changes in quantiles55. For variables where relative changes are more meaningful, such as precipitation (a ratio variable with an absolute zero), the transfer function can be adjusted by using multiplication and division operators instead of addition and subtraction. The formulas for QDM are:

$$\varDelta (i)={x}_{p}(i)-{F}_{S}^{-1}({F}_{p}({x}_{p}(i))$$
(1)
$${\hat{x}}_{p}\left(i\right)={F}_{T}^{-1}\left({F}_{p}\left({x}_{p}\left(i\right)\right)\right)+\varDelta \left(i\right)$$
(2)

where \(\varDelta (i)\) represents the bias correction term, \({x}_{p}(i)\) is the original data, \({F}_{S}^{-1}\) and \({F}_{T}^{-1}\) are the inverse cumulative distribution functions for the source and target distributions, respectively, and \({\hat{x}}_{p}(i)\) is the corrected data.

Identification of SHP and CDH events

The connected components in 3D (CC3D) algorithm56 is used to identify events for each grid, ensuring their spatial and temporal continuity. The CC3D algorithm employs 26-connectivity, meaning that for any given grid, an event is considered continuous if it is connected to its 8 neighboring cells on the same day and its 9 neighboring cells on the preceding and following days57. Unlike simple grid-scale recognition, CC3D considers both spatial and temporal dimensions. This means that CC3D can identify a long-term event as continuous even if there is a brief interruption (e.g., one day) in one grid while the surrounding cells remain part of the same event, thus avoiding the overestimation of event frequency. The CC3D algorithm is implemented in Python and is available through the connectedcomponents-3d package.

Two types of compound events are identified, i.e., SHP (sequential heatwave and heavy precipitation) events and CDH (concurrent drought and heatwave) events. SHP events are defined as occurrences of heavy precipitation starting within 7 days following the end of a heatwave, and CDH events are defined as the co-occurrence of drought and heatwave. In this study, a heatwave is characterized as a period of at least three consecutive days where the daily maximum temperature exceeds a threshold. This threshold is determined for each grid cell based on the higher value between the 90th percentile of the daily maximum temperature during the warm season (May to October) from 1961 to 2014 and 30 °C44. Following the classification criteria of China Meteorological Administration, heavy precipitation events are defined as days with daily precipitation exceeding 50 mm33. In this study, we only consider meteorological droughts identified by standardized precipitation index (SPI), calculated based on monthly precipitation data and fitted to a gamma distribution. A month is considered to be experiencing a drought if the SPI is less than −158,59,60, with the assumption that the SPI remains constant for each day of the month9.

The characteristics of SHP and CDH events are analyzed based on their frequency (number of events per year per grid), duration (for SHP, the combined duration of heatwave and precipitation days; for CDH, the number of days experiencing concurrent drought and heatwave9,61,62), and the earliest occurrence and end dates within a year. We also use magnitude index (MI), typically calculated by normalizing the individual event variables (such as temperature and precipitation) using specific percentiles (e.g., the 25th or 75th percentile) or distributions to get the MI of heatwave (HWMI) or precipitation (PRMI), and then multiplying two MI21,24,29,63,64,65. We get HWMI and PRMI using the empirical Gringorten plotting position formula66, which was widely used in previous research25. The MI are calculated as HWMI multiplied by PRMI for SHP events and HWMI multiplied by DMI (negative SPI) for CDH events, as follows:

$${{SHPMI}}=\left(\mathop{\sum}\limits_{1}^{{{DU}}_{{hw}}}{T}^{* }\right)\,* \left(\mathop{\sum}\limits_{1}^{{{DU}}_{p}}{P}^{* }\right)$$
(3)
$${{CDHMI}}=\left(\mathop{\sum}\limits_{1}^{{{DU}}_{{hw}}}{T}^{* }\right)\,* \left(\mathop{\sum}\limits_{1}^{{{DU}}_{d}}\left(-{{SPI}}\right)\right)$$
(4)

where SHPMI and CDHMI represent the magnitude indices for SHP and CDH events. \({T}^{* }\) and \({P}^{* }\) are the normalized values of daily maximum temperature and precipitation, respectively. \({{DU}}_{{hw}}\), \({{DU}}_{p}\) and \({{DU}}_{d}\) represent the number of days of heatwave, heavy precipitation, and drought, respectively. For CDH events, the number of heatwave and drought days are the same. The abovementioned event identification and analysis framework can successfully identify typical compound events in the historical period (Supplementary Fig. 23), including the 1998 SHP in the Yangtze River Basin67 and the 2013 CDH in Southern China68,69. It should be noted that the selected time interval between heatwave and following heavy precipitation (7 days) for SHP events are also widely adopted by previous studies8,19,28,63,70. We also repeated our analysis by using 3-day and 5-day intervals, and found that the choice of the time interval of SHP events does not affect the main conclusions of the study (Supplementary Fig. 24).

Return period calculation of compound events and change attribution

To calculate the return period (RP) of compound events (SHP and CDH) based on a copula bivariate framework, we perform the following steps. First, we estimate the marginal distributions of the variables involved (HWMI and PRMI for SHP events, HWMI and DMI for CDH events). We consider various parametric and non-parametric distributions, including Normal, Lognormal, Weibull, Exponential, Gamma, Generalized Extreme Value, tLocationScale, Extreme Value, as well as Gringorten plotting position formula and kernel density estimation. The best-fit marginal distributions are selected based on the Kolmogorov–Smirnov (KS) criterion71 and the significance of the P-value (when P > 0.05, the marginal distribution is considered suitable). Subsequently, we construct the joint distribution of these variables, with five commonly used copula families as candidates: Gaussian, t, Gumbel, Clayton, and Frank. The best-fitting copula is selected based on the Akaike Information Criterion (AIC)72.In this study, the bivariate RP is based on the “AND” criterion73, meaning the probability of an event that both variables exceed given thresholds. The formula is74:

$${RP}=\frac{E}{1-{F}_{x}-{F}_{y}-C({F}_{x},{F}_{y})}$$
(5)

where \({F}_{x}\) and \({F}_{y}\) are the marginal distributions of the variables, \(C({F}_{x},{F}_{y})\) is the copula function representing their joint distribution and E is the average inter-arrival time between compound events.

We quantify the contributions to changes of the return period (RP) for historically 50-year compound events in future period (2015–2100). According to previous studies24,29,75,76, changes in return periods can be divided into three parts: (1) changes due to the marginal distribution of heatwaves (HWMI), (2) changes due to the marginal distribution of heavy precipitation (PRMI for SHP events) or drought (DMI for CDH events), and (3) changes due to the copula dependence structure between the variables. The contribution of each driver can be calculated by:

$$\Delta {{CF}}_{i}\left( \% \right)=100\times \frac{\Delta {{RP}}_{i}}{\left|\Delta {{RP}}_{1}\right|+\left|\Delta {{RP}}_{2}\right|+\left|\Delta {{RP}}_{3}\right|}$$
(6)

where \(\Delta {{CF}}\) (%) represents the contribution fraction of change in the joint return period due to driver i, ranging from −100% to 100%. \(\Delta {{RP}}_{1}\) is obtained by changing the marginal distribution of heatwaves and other distributions are the same as historical periods, \(\Delta {{RP}}_{2}\) by only changing the marginal distribution of heavy precipitation or drought, and \(\Delta {{RP}}_{3}\) by only changing the copula function between the variables.

Assessment of socio-economic exposure

The socio-economic costs caused by extreme events are influenced by both the event hazards and the socio-economic context77. We assess the exposure of GDP and population to compound events (SHP and CDH) using the same method as previous studies47,78. This assessment is conducted for each grid individually, with exposure calculated by multiplying the number of event days by the GDP density or population density in that grid29. The unit for GDP exposure is a hundred million (RMB/km2) × days/yr, and the unit for population exposure is (persons/km2) × days/y.

The changes in exposure can be decomposed into three components: GDP or population change effect (GE or PE), climate hazard change effect (CE), and interaction effect (IE)34,63,79. The equations for these changes are given as:

$$\Delta {E}_{{GDP}}=\Delta {{GDP}}\times {{CE}}_{{his}}+{{GDP}}_{{his}}\times \Delta {{CE}}+\Delta {{GDP}}\times \Delta {{CE}}$$
(7)
$$\Delta {E}_{{POP}}=\Delta {{POP}}\times {{CE}}_{{his}}+{{POP}}_{{his}}\times \Delta {{CE}}+\Delta {{POP}}\times \Delta {{CE}}$$
(8)

where \(\Delta {E}_{{GDP}}\) and \(\Delta {E}_{{POP}}\) represent changes of GDP and population exposure in the future period (2015–2100) compared with the historical period (2000–2014), respectively. \(\Delta {{GDP}}\) and \(\Delta {{POP}}\) denote changes in GDP and population, and \({{GDP}}_{{his}}\) and \({{POP}}_{{his}}\) is the GDP and population during the historical period. \({{CE}}_{{his}}\) is the historical number of event days for compound events (SHP and CDH), and \(\Delta {{CE}}\) is the change in the days of events.