Introduction

Droughts and heatwaves often occur together due to the influence of both local land-atmosphere feedbacks1 and large-scale ocean-atmosphere circulations2,3. These feedbacks lead to an increase in surface sensible heat from droughts and a subsequent rise in temperature, which can be propagated downwind through heat advection4,5. With an increase in the frequency of both heatwaves and droughts6,7, and stronger land-atmosphere feedback triggered by atmospheric warming8, compound hot-dry events are becoming more common and severe across different regions of the world9,10. These events have substantial adverse impacts on ecosystems, agriculture, water resources, and human health11,12,13.

To develop effective strategies for adapting to and mitigating the impacts of these events, it is critical to understand and project their future characteristics. However, defining compound hot-dry events as a new topic in climate change research is a complex task. The choice of dry indicators can drastically impact the projected changes in dry events14,15,16, potentially altering the magnitude and even the sign of the projected changes in compound hot-dry events. Such discrepancies can have substantial implications for adaptation and mitigation strategies, as different definitions may require different types of infrastructure investments and spatial planning to prepare for and mitigate the impacts of these events17. Therefore, evaluating the impact of different dry indicators on projections of compound events is crucial to ensure that decision-makers have access to accurate and reliable information for effective planning and decision-making.

Dry events in compound hot-dry events in climate change studies have typically been defined based solely on precipitation18,19,20,21,22, but this may oversimplify the underlying complexities of such events. To better capture these complexities, some studies have incorporated additional variables, such as runoff and soil moisture8,23. Recent studies comparing different definitions based on precipitation, runoff, and soil moisture have highlighted large differences in the projected changes in compound hot-dry events and associated uncertainties24,25. However, since these variables are interrelated components of the same process25, it is essential to consider their interactions for accurate projections of compound hot-dry events26. Unfortunately, global-scale compound hot-dry event projections that incorporate the interconnections between low precipitation, soil moisture, and runoff are currently lacking.

Properly understanding the uncertainty in future projections of compound hot-dry events is crucial for interpreting the impacts of climate change and making informed policy decisions to mitigate the associated risks27. However, while traditional sources of uncertainty in climate change projections such as climate models and scenarios have been well studied28,29,30,31, less attention has been given to uncertainties associated with other factors, including hazard definition. Although recent studies have highlighted the importance of including dry event definitions in uncertainty analyses32,33, its relative significance for compound hot-dry event projections compared with other sources of uncertainty remains unclear. Specifically, there is a lack of understanding regarding how much projected changes in compound hot-dry events can vary depending on dry indicators. Addressing this issue can help quantify the amount of uncertainty that can be reduced by refining the event definition.

To bridge these knowledge gaps, this study aims to project global compound hot-dry events using 22 global climate models (GCMs) from the Coupled Model Intercomparison Project Phase 6 (CMIP6) under four Shared Socioeconomic Pathways (SSPs). The study characterizes compound hot and dry precipitation, soil moisture, and runoff extremes, as well as compound hot and multivariate dry extremes. The study then compares the uncertainties arising from different definitions of dry conditions for compound hot-dry events with those arising from GCMs and SSPs.

Results

Having established that the CMIP6 GCMs have undergone comprehensive evaluations for simulating compound hot-dry events in prior studies21,24,34,35, our focus remains on investigating how different dry indicators impact projections of compound hot-dry events (Fig. 1). A comparison of changes in the probability of compound hot-dry events, with dry events characterized based on precipitation [HDI-P], runoff [HDI-R], and soil moisture [HDI-S], reveals a similar spatial distribution (Fig. 1). Across all indices, the probability of compound events is projected to increase worldwide by the end of this century under different scenarios. The most substantial increase is anticipated in most of South America, Central America, southern North America, southern and northern Africa, southern and central Europe, western Asia, and Australia. However, the magnitude of the increase in compound event probability noticeably varies among the different indices. Generally, the increase in probability for HDI-P is lower than that for HDI-R and HDI-S, and this disparity is more pronounced in regions expected to experience the most severe impacts from compound events. Based on the CMIP6 ensemble median, the global maximum increase in compound event probability differs noticeably across the various indices, with values ranging from 0.12 to 0.26 for HDI-P, 0.26 to 0.60 for HDI-R, and 0.31 to 0.59 for HDI-S, depending on the scenarios. Although the difference between the increase in probability for HDI-R and HDI-S is small, the increase in probability for HDI-R and HDI-S is consistently 2-3 times larger compared with HDI-P.

Fig. 1: CMIP6 ensemble median changes in the probability of compound hot-dry events over the long term (2061–2100) relative to the baseline period (1971–2010) under different scenarios.
figure 1

The figure depicts changes in the probability of compound hot-dry events defined using HDI-P, HDI-R, HDI-S, and HDI-MSDI under the scenarios a–d SSP1-2.6, e–h SSP2-4.5, i–l SSP3-7.0, and m–p SSP5-8.5. HDI-P, HDI-R, and HDI-S represent the standardized compound hot-dry event index that utilizes respectively precipitation (P), runoff (R), and soil moisture (S) to define dry events. HDI-MSDI represents the standardized compound hot-dry event index that uses all precipitation, runoff, and soil moisture based on a multivariate standardized drought index (MSDI) to define dry events. A compound hot-dry event was defined when HDI < −0.8.

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In addition to defining dry conditions for compound hot-dry events by linking temperature with various hydrological variables, an alternative approach involves defining compound events based on the combination of these variables, referred to as the HDI-MSDI (Fig. 1). When compound hot-dry events are defined using HDI-MSDI, the projected changes exhibit collective variations in the different hydrological variables, evident in both spatial distribution and magnitude. The spatial pattern of projected changes in the probability of HDI-MSDI closely resembles that of HDI-P, HDI-R, and HDI-S, with similar regions expected to be most impacted. In terms of magnitude, the expected increase in the probability of HDI-MSDI is greater than that of HDI-P, but smaller than that of HDI-R and HDI-S.

The projected changes based on HDI-P, HDI-R, HDI-S, and HDI-MSDI are globally significant at the 5% level (Supplementary Fig. 1). Even when reducing the significance level to 1% and 0.1%, the substantial changes obtained by HDI-P continue to encompass 100% of the global land areas. However, the coverage marginally decreases for the other indices, spanning 95-100% of the land area. Notably, HDI-P yields the highest signal-to-noise ratio (S/N), followed by HDI-MSDI, while the lowest ratios are observed for HDI-R and HDI-S. This S/N pattern is opposite to the pattern found for the changes.

A regional analysis of the changes reveals that HDI-R exhibits the largest increase in the probability of compound events in 12, 9, 8, and 9 regions out of the total 20 regions under SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5, respectively (Fig. 2). This is followed by HDI-S, which leads to the greatest increase in the probability of compound events in 8, 10, 11, and 10 regions under the respective scenarios. Conversely, compound events defined based on HDI-P result in the smallest increase in the probability of events in 11, 11, 10, and 11 regions under SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5, respectively.

Fig. 2: Regional comparisons of changes in the probability of compound hot-dry events defined based on HDI-P, HDI-R, HDI-S, and HDI-MSDI under different scenarios.
figure 2

The changes are for the long term (2061–2100) relative to the baseline period (1971–2010) based on the CMIP6 ensemble median for the scenarios a SSP1-2.6, b SSP2-4.5, c SSP3-7.0, and d SSP5-8.5. A compound hot-dry event was defined when HDI < –0.8. Figure 2e displays the continental and subcontinental land regions of the globe. ALA: Alaska; AMZ: Amazon basin; AUS: Australia; CAM: Central America; CAS: central Asia; CNA: central North America; EAF: eastern Africa; EAS: east Asia; ENA: eastern North America; MED: Mediterranean basin; NAS: north Asia; NEU: northern Europe; SAF: southern Africa; SAH: Sahara; SAS: south Asia; SEA: southeast Asia; SSA: southern South America; TIB: Tibet; WAF: western Africa; WNA: western North America.

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Overall, the Amazon, Australia, Central America, Mediterranean, southern Africa, South Asia, and Southeast Asia regions are anticipated to experience the most substantial increases in the likelihood of compound hot-dry events (Fig. 2). In the worst-case scenario (SSP5-8.5), the Mediterranean and Amazon regions may experience a potential increase of up to 0.19 and 0.18, respectively, in the probability of compound events. While the magnitude of the increase varies with different scenarios, this growth is however spatially heterogeneous. For example, the Mediterranean region is initially ranked as the fifth most important hotspot region for future compound hot-dry events under SSP1-2.6, but climbs to become the fourth, first, and first most important hotspot under SSP2-4.5, SSP3-7.0, and SSP5-8.5, respectively. Conversely, southern Africa drops from being the second most important hotspot region under SSP1-2.6 to the fourth most important hotspot under SSP5-8.5.

Following the global pattern, significant changes are observed in all regions using all indices; however, HDI-P may not necessarily yield the largest S/N in every region (Fig. 3). When comparing regional S/N across different indices, HDI-P exhibits the highest S/N in 17, 18, 19, and 19 regions out of the total 20 regions under SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5, respectively. In contrast, either HDI-R or HDI-S exhibits the lowest S/N in 75-85% of the regions. For all scenarios, southern Africa shows the largest S/N, even though it is not among the top three hotspot regions in terms of the magnitude of projected changes in compound hot-dry events, highlighting the importance of projection uncertainty. Another example is the Amazon region, which ranks among the top two hotspot regions based on the magnitude of changes but is not even among the top five hotspot regions in terms of S/N.

Fig. 3: Regional comparisons of signal-to-noise ratio (S/N) for the probability of compound hot-dry events defined based on HDI-P, HDI-R, HDI-S, and HDI-MSDI under different scenarios.
figure 3

The changes are for the long term (2061–2100) relative to the baseline period (1971–2010) based on the CMIP6 ensemble median for the scenarios a SSP1-2.6, b SSP2-4.5, c SSP3-7.0, and d SSP5-8.5. A compound hot-dry event was defined when HDI < −0.8. See Fig. 2 for the definition of the regions.

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To account for the sensitivity of the results to the chosen threshold defining compound hot-dry events, the analysis is replicated using two alternative thresholds of –0.9 and –1 within the same threshold range associated with a moderate compound hot-dry condition, in addition to the original −0.8. A comparison of the results reveals similar spatial patterns of changes across different thresholds (Fig. 1; Supplementary Figs. 2 and 3). However, in terms of magnitude, changes tend to decrease when the threshold is increased from –0.8 to –1. Regionally, an increase in the magnitude of the change is observed in all regions for all indices (Fig. 4). Specifically, for SSP5-8.5, the change magnitude for the –0.8 threshold is 46–101% larger than that for the –1 threshold. This breaks down to 51–76% for HDI-P, 46–67% for HDI-R, 51–101% for HDI-P, and 50–86% for HDI-MSDI. This pattern remains consistent across other scenarios (Supplementary Figs. 4–6), where the change magnitude for the –0.8 threshold is 45–97% larger than that for the –1 threshold. It is worth noting that the regions experiencing the most substantial changes (e.g., the Mediterranean and Amazon) exhibit the lowest sensitivity to the threshold.

Fig. 4: Sensitivity of the results to the chosen threshold for defining a compound hot-dry event under the SSP5-8.5 scenario.
figure 4

The figure compares projected changes in the probability of compound events defined using a HDI-P, b HDI-R, c HDI-S, and d HDI-MSDI based on three thresholds: −0.8, –0.9, and –1. These changes are evaluated for the long term (2061–2100) relative to the baseline period (1971–2010) using the CMIP6 ensemble median. Refer to Fig. 2 for the definition of the regions.

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To assess the relative importance of dry indicators for future compound hot-dry event projections, the uncertainty arising from the choice of dry indicators is compared with that stemming from GCMs and scenarios (Fig. 5). GCM uncertainty emerges as the dominant source of uncertainty in 17 out of 20 regions, accounting for an average of 48% across the regions. Scenario uncertainty takes precedence in Central America and the Mediterranean while holding a slight margin in eastern North America: 40.3% for scenario uncertainty compared with 39.8% for GCM uncertainty. In these three regions, the magnitude of the increase in compound event probability escalates rapidly across scenarios, with the increase under SSP5-8.5 being 2-3 times larger than that under SSP1-2.6. On average across the regions, the uncertainty associated with scenarios is 33%, while the uncertainty related to dry indicators for compound events averages at 19%. Notably, the dry indicator uncertainty surpasses the scenario uncertainty in four regions: east and west Africa, South Asia, and Tibet. Additionally, a noticeable uncertainty related to dry indicators (i.e., 27%) is seen in central and north Asia regions.

Fig. 5: Fractional contribution of individual sources to total uncertainty in projected changes in the probability of compound hot-dry events (HDI < −0.8) in the continental and subcontinental regions by 2061–2100 relative to 1971–2010.
figure 5

Dry indicator uncertainty represents the uncertainties associated with the choice of a dry indicator for defining compound hot-dry events. The continental and subcontinental land regions of the globe are shown in Fig. 2e.

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GCM uncertainty dominates the total uncertainty of compound hot-dry event projections, being the dominant source in 66% of the global land area (Fig. 6). GCM uncertainty of > 40% in compound hot-dry event projections are observed in 69% of the global land area, with 30% of the area showing GCM uncertainty of > 50% (Fig. 7). However, in regions where the expected increases in the likelihood of compound hot-dry events are large, scenario uncertainty is dominant, accounting for 22% of the global land area (Fig. 6). The uncertainty associated with the choice of dry indicators for compound hot-dry events is largest in the Middle East and parts of central Asia and central Africa, accounting for 12% of the global land area (Fig. 6). Scenario uncertainty of > 40% in compound hot-dry event projections is observed in 27% of the global land area, while dry indicator uncertainty of > 40% is seen in 11% of the global land area (Fig. 7).

Fig. 6: Dry indicator uncertainty for compound hot-dry event projections and its importance in relation to other sources of uncertainty.
figure 6

The figure depicts a the spatial distribution of absolute uncertainty associated with the choice of dry indicators for compound hot-dry events (HDI < −0.8) and b its relative importance compared to uncertainties in GCMs and scenarios. Uncertainties were calculated using the VD-SSS method.

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Fig. 7: Spatial distribution of fractional uncertainties for different sources.
figure 7

The uncertainties associated with a GCMs, b dry indicators for defining compound hot-dry events (HDI < −0.8), and c scenarios were calculated using the VD-SSS method.

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Discussion

The discernible intensification of compound hot-dry events, as evidenced through various station, reanalysis, and satellite-based observational datasets, is well-documented in recent studies3,10,34,36,37,38,39,40,41. This growing trend underscores the imperative of scrutinizing these events for their future implications. In this study, we contribute to this line of research by conducting an analysis to examine the evolving likelihood of compound hot-dry events under various future scenarios. Our findings are consistent with prior research, supporting the notion that compound hot-dry events are expected to intensify due to climate change42,43,44. This intensification is particularly evident in regions projected to face severe droughts15,43,45,46. When the impacts of dry events combine with the negative effects of hot events, resulting in compound hot-dry events, the overall consequences become even more severe. This means that the negative impacts of hot and dry events reinforce each other, leading to increased risks and potentially more pronounced environmental, ecological, and socioeconomic consequences. The interrelated nature of the physical processes responsible for dry and hot extreme events leads to cascading effects47, with distinct timescales of interaction, and sensitivity to changes in the soil-plant-atmosphere continuum hydroclimatic drivers, and background aridity48.

A key focus of this study was the issue of defining dry conditions for compound hot-dry events, which has received limited attention due to being a relatively new frontier in research. In contrast to recent studies that rely on individual hydrological variables to characterize dry events24,26, we explored the interactions between different hydrological variables and their influence on projections of compound hot-dry events. Our findings demonstrate that using a multivariate index that incorporates precipitation, runoff, and soil moisture to link with air temperature yields different results compared with cases where individual hydrological variables are used. In fact, compound events defined using the multivariate index demonstrate larger changes compared with those defined solely based on precipitation, yet they exhibit smaller changes than those relying on runoff and soil moisture. It is crucial to include these interactions in future projections of compound hot-dry events, given that land-atmosphere interactions are expected to strengthen under warming conditions49.

Furthermore, our study identified limitations in drought characterization when relying solely on precipitation, such as the Standardized Precipitation Index (SPI). Such an approach overlooks critical land surface and soil conditions that play a substantial role in drought development. Additionally, certain plant processes can influence how precipitation droughts manifest in the soil, such as the reduction in plant stomatal conductance during drought stress or the increased vegetation total water use efficiency under elevated atmospheric CO250,51. These processes create feedback loops in the near-surface climate that may vary geographically due to variations in vegetation (e.g., phenology, land cover) and land surface properties (e.g., soil moisture content, soil type, topography)52. The SPI’s inability to incorporate the influencing factors restricts its ability to offer a comprehensive understanding of drought dynamics, especially in regions where these additional variables play a crucial role33,53. This becomes particularly relevant in cases where precipitation changes in the opposite direction compared with soil moisture15,54.

In regions with modest or low climatological precipitation, precipitation alone may not suffice as a measure of drought55. Moreover, the precipitation deficit is not a reliable indicator of extreme drought56. Extreme drought is often determined by drought intensity, primarily driven by temperature57. Higher air temperatures can intensify evaporative demand, leading to increased evaporative losses from the surface and higher soil moisture deficits58. A study using GCMs demonstrated that up to 80% of surface rainfall can be lost through evaporation and transpiration, underscoring the substantial impact of temperature on drought severity59. Even for characterizing atmospheric water deficit, precipitation may not be the most suitable variable. Due to the limited availability of water for evaporation over land compared with the ocean, land surfaces warm approximately 50% more than ocean surfaces60, and water vapor content cannot increase rapidly enough to overcome this deficit. Consequently, further drying over land is anticipated as the climate continues to warm61, leading to an increasing disparity between actual and saturation water vapor concentrations. Therefore, the vapor pressure deficit (VPD), which also governs evapotranspiration demand, is expected to increase much faster by percentage than changes in other hydrological variables, such as precipitation61,62. Indices like the Standardized VPD Drought Index (SVDI)63 may be more appropriate in these situations.

Hence, we caution against relying solely on precipitation-based dry indicators, as it may lead to biased climate change impacts on compound hot-dry events. Instead, we recommend the inclusion of all pertinent variables in characterizing dry events and considering the interactions between them. Using a multivariate index to characterize dry conditions in compound hot-dry events is advisable, by directly analyzing climate model outputs rather than using a separate offline impact model which is potentially subject to large biases. Compared to precipitation, runoff, and soil moisture can more explicitly represent the core physical processes for drought assessments50,64. Our findings indicate that quantifying dry events based on soil moisture yields results closest to those derived from multivariate indices that comprehensively account for interactions between precipitation, soil moisture, and runoff.

Using precipitation to define dry conditions for compound hot-dry events, however, results in lower projection uncertainty and thus a larger S/N compared with the dry event quantification based on runoff, soil moisture, and a multivariate index. This is attributed to the lower uncertainty in precipitation projections compared with runoff and soil moisture projections15. In simulating soil and runoff, climate models are required to incorporate soil, landscape, and vegetation attributes alongside climate processes, potentially introducing an additional layer of uncertainty to their projections65. Our results show that the compound hot-dry events defined based on HDI-R, HDI-S, and HDI-MSDI inherit this uncertainty, leading to a smaller signal-to-noise ratio compared with the compound hot-dry events defined based on HDI-P.

To assess the relative importance of the characterization of dry conditions for compound hot-dry event projections, we compared the uncertainty arising from the choice of dry indicators with that originating from GCMs and scenarios. Consistent with previous research on extreme events27,66,67, we found that GCMs remained the primary source of uncertainty in our study. However, we also observed that the uncertainty associated with the event definition was substantial, reaching up to 30% in certain regions and even surpassing the uncertainty attributed to scenarios. Our study highlights the importance of giving careful attention to the choice of dry indicators when projecting compound hot-dry events, particularly in Africa (east and west Africa) and Asia (north, central, and south Asia, and Tibet). It emphasizes that this aspect should not be overlooked but rather acknowledged as a substantial source of uncertainty that can impact the findings and implications of studies concerning compound hot-dry events.

In addition to the primary focus of this paper, which explores the sensitivity of future compound hot-dry event projections to dry indicators, we also investigated their sensitivity to the chosen threshold for defining such events. Our findings reveal that future compound hot-dry event projections exhibit sensitivity to the chosen threshold. Determining the appropriate threshold entails striking a balance between statistical inference and event extremity68. A higher threshold leads to the selection of more extreme events but results in a smaller sample size, introducing bias into statistical analyses. Conversely, a lower threshold detects a greater number of events but leads to less extreme events being included. Hence, selecting the appropriate threshold for defining compound events should be guided by the objectives of impact modeling (e.g., crop modeling, hydrological modeling), while also considering potential statistical biases in the analyses.

While our study offers a quantitative and spatial assessment of the relative impact of the characterization of dry events for compound hot-dry event analyses compared with other sources of uncertainty, there are opportunities for future studies to expand upon this work. Although climate models generally depict historical trends in compound hot-dry events accurately35,69, their future projections remain subject to uncertainties. Important modeling uncertainties exist for several underlying processes that drive compound extremes, including changes in aridity70, land-atmosphere interactions71, dynamics of snow-atmospheric coupling72, natural vegetation51, and precipitation73. In our research, we primarily focused on investigating the sensitivity of compound hot-dry event projections and uncertainty to dry indicators. We employed climate model outputs, which directly capture core physical processes like soil moisture and runoff, to characterize dry conditions, enabling a more accurate assessment of the impacts of climate change on dry events50,64. However, the World Meteorological Organization (WMO) identified over 50 drought indices based on varying drought indicators (e.g., precipitation, temperature, evapotranspiration)74. It would be intriguing to explore how employing commonly used offline impact models, which may be prone to large biases64, such as the Standardized Precipitation Evapotranspiration Index (SPEI75) or the Palmer drought severity index76, to define dry events would influence the results of compound hot-dry events. Comparing outcomes based on climate model outputs, which directly capture core physical processes, with those derived from offline impact models could offer a more comprehensive understanding of the implications of different dry indicators and the potential biases introduced by using separate offline impact models.

Additionally, future research should explore the effects of defining hot events based on different variables, such as air temperature, dew temperature, and humidity. Different indices that capture these variables could be utilized to define hot events and investigate their influence on the outcomes of compound hot-dry event projections. This would contribute to a more comprehensive understanding of the interplay between different variables and indices in defining compound events. Nonetheless, our study has provided valuable insights into the significance of the dry condition definition for compound hot-dry events and the consideration of interactions between different hydrological variables to capture underlying physical processes and the complexities of events. We have demonstrated that neglecting or underestimating the significance of dry indicators can result in misleading assessments of the potential impacts of climate change on compound hot-dry events.

Materials and methods

Data

Our analysis incorporated precipitation flux (including both liquid and solid phases), total runoff (which includes drainage through the base of the soil model), total soil moisture content (summed water in all phases across all soil layers), and air temperature. To conduct our analysis, we utilized 22 CMIP6 GCMs, including ACCESS-CM2, ACCESS-ESM1-5, BCC-CSM2-MR, CAMS-CSM1-0, CanESM5, CESM2-WACCM, CNRM-CM6-1, CNRM-ESM2-1, FGOALS-f3-L, FGOALS-g3, GFDL-ESM4, INM-CM4-8, INM-CM5-0, IPSL-CM6A-LR, MCM-UA-1-0, MIROC6, MPI-ESM1-2-HR, MPI-ESM1-2-LR, MRI-ESM2-0, NorESM2-LM, NorESM2-MM, and TaiESM1. We analyzed monthly data for precipitation, runoff, soil moisture, and air temperature from these models for both historical simulations and four future tier 1 scenarios (SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5) covering the period 1971-2100 (historical+SSP). These scenarios represent plausible pathways towards different levels of radiative forcing and alternative socioeconomic developments, ranging from sustainable to fossil-fueled development77. Our study encompassed the entire global land area grid cells except Greenland due to its perennial snow and ice cover. While compound hot-dry events may not have immediate practical relevance for deserts and hyper-arid regions, they are still useful in providing insights into the potential impacts of climate change.

Quantification of dry events

We characterized dry events using precipitation, runoff, and soil moisture, as well as a multivariate index that combines these variables. For this purpose, we utilized the standardized precipitation index (SPI78), the standardized runoff index (SRI79), and the standardized soil moisture index (SSI80) as dry indicators. The standardized indices were computed for the 3-month time scale, representing seasonal droughts. We employed the Gringorten plotting position to determine the empirical probability of each variable for every month. The resulting probabilities were standardized using the inverse normal transformation to obtain SPI, SRI, and SSI values. Negative SPI, SRI, and SSI values indicate a dry climate condition (drought), whereas positive values signify a wet climate condition. A value close to zero indicates normal climate conditions.

To develop a multivariate standardized drought index (MSDI80), for each grid cell of the CMIP6 GCMs, we used copula functions to estimate the joint distribution of precipitation, runoff, and soil moisture. The MSDI is a trivariate index for the 3-month scale of the variables and is expressed as:

$$H\left({x}_{1},{x}_{2},{x}_{3}\right)=C\left({F}_{1}\left({x}_{1}\right),{F}_{2}\left({x}_{2}\right),{F}_{3}\left({x}_{3}\right)\right)=C\left({u}_{1},{u}_{2},{u}_{3}\right)$$
(1)

Here, H is a three-dimensional distribution function of random variables x1 (e.g., precipitation), x2 (e.g., runoff), and x3 (e.g., soil moisture), C is a copula function, and u1, u2, and u3 are variables generated by marginal distribution functions F1, F2, and F3, respectively. Since constructing three-dimensional functions using copulas can be challenging, we used two bivariate copulas to build a three-dimensional function as:

$$C\left({u}_{1},{u}_{2},{u}_{3}\right)={C}_{2}\left({{C}_{1}}(u_{1},{u}_{2}),{u}_{3}\right)=p$$
(2)

Here, \({C}_{1}\) is the first bivariate copula function that corresponds to variables \({u}_{1}\) and \({u}_{2}\), \({C}_{2}\) is the second bivariate copula function that corresponds to variables \({{C}_{1}}(u_{1},{u}_{2})\) and \({u}_{3}\), and p is the joint probability. \({C}_{1}\) and \({C}_{2}\) are of the same type of copula function. We used the Gaussian copula in this study with its parameters estimated by the maximum likelihood method. Previous studies also reported that the differences resulting from the use of different copula functions were negligible when considering changes in drought characteristics and compound hot-dry events at large scales25,81.

Finally, we computed the MSDI for each grid cell based on the joint probability p (Eq. 2) as:

$${MSDI}={\varphi }^{-1}\left(p\right)$$
(3)

where φ is the standard normal distribution function. Similar to SPI, SRI, and SSI, a negative MSDI points to a dry climate condition (drought), while a positive value indicates a wet climate, with an MSDI close to zero denoting normal climate conditions.

Quantification of compound hot-dry events

After characterizing dry events using various methods, we proceeded to quantify compound hot-dry events. To define hot events, we utilized the standardized temperature index (STI82). For the detection of compound hot-dry events, we employed bivariate copula functions. We derived the joint distribution probability between the 3-month dry event index, separately for each of the four indices (SPI, SRI, SSI, and MSDI), and monthly STI values83, using the Gaussian copula. By applying the standard normal distribution (Eq. 3) to transform the remapped joint probability obtained from copula functions, we obtained the standardized compound hot-dry event index (HDI), following the principles of drought indices. We refer to the HDI that uses precipitation (P), runoff (R), soil moisture (S), and MSDI to define dry events as HDI-P, HDI-R, HDI-S, and HDI-MSDI, respectively. To assess climate change impacts, we calculated the changes in the probability of occurrence during the historical (1971–2010) and future (2061–2100) periods. The probability of occurrence for each period was determined by the ratio of instances with a compound event (HDI < −0.8) to the total number of months in each period (40 years x 12 months = 480 months). The threshold of –0.8 corresponds to a moderate compound hot-dry condition, representing the 20th percentile84. In contrast to an approach where a compound event is defined only if both margins (dry index and hot index) exceed a specific threshold, our copula-based threshold procedure widens the event space. This procedure incorporates not only events that are jointly marginally extreme but also those that are extreme in the context of the bivariate distribution, even if not necessarily extreme in both margins85. Consequently, when both variables fall below an alarm threshold, HDI leads to a more severe extreme condition than either the dry or hot index alone80. To evaluate the impact of varying the threshold settings on the results, we also calculated the probability of compound events using two alternative thresholds of –0.9 and –1 within the same threshold range (–0.8 to –1.2) associated with a moderate compound hot-dry condition84.

To evaluate the robustness of projected changes in compound hot-dry events, we utilized the signal-to-noise ratio (\(S/N\)), as defined by Kendon et al.86:

$$S/N=\frac{\bar{y}-\bar{x}}{\sqrt{({\sigma }_{x}^{2}+{\sigma }_{y}^{2})/2}}$$
(4)

where an overbar denotes averaging over the ensemble members, and \({\sigma }_{x}^{2}\) and \({\sigma }_{y}^{2}\) are the variances across the ensemble in the historical (x) and future (y) periods, respectively.

To test the statistical significance of \(S/N\), we applied the t-test. In contrast to the approach of Aalbers et al.87, who employed a two-sided t-test for precipitation changes, accommodating both increases and decreases, we opted for a one-sided test for compound hot-dry events (HDI), projected to increase globally (see Fig. 1). The significance criterion is given by:

$$\frac{S}{N}\ge {t}_{n-1,0.05}\times \frac{1}{\sqrt{n-1}}$$
(5)

For our sample size of \(n=22\), the critical t-value \({t}_{n-{{{{\mathrm{1,0.05}}}}}}=1.72\) and a statistically significant change at the 5% level is indicated by \(\left|S/N\right|=0.38\). The critical values for the 1% and 0.1% significance levels are \(\left|S/N\right|=0.55\) and \(\left|S/N\right|=0.77\), respectively.

Uncertainty analysis

To assess the importance of dry indicators for compound hot-dry events relative to GCMs and SSPs, we used the variance decomposition-same sample size (VD-SSS) method27 to partition the cascade of uncertainties in compound event changes. This method overcomes the limitations of traditional VD methods, which artificially amplify uncertainty contributions from sources with larger sample sizes88. In the VD-SSS iterative sampling-theory based bootstrapping procedure, we used n to denote the smallest sample size among the uncertainty components, which included four sample sizes from dry indicators and SSPs. N represents the size of the largest uncertainty source, which was 22 for the GCM ensemble. After taking the median across the other uncertainty components, we randomly drew n from N and calculated the standard deviation across the bootstrap samples. We repeated this process multiple times (in our case, 1000 iterations) and calculated the median of the empirical bootstrap distribution of sample standard deviation to determine the uncertainty. For the uncertainty component of size n (dry indicator and SSP), we applied the traditional VD method. To determine the fractional uncertainties, we divided the uncertainty of each source by the total uncertainty, which is the sum of the uncertainty contributions (GCMs, SSPs, and dry indicators).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.