Climate simulations for a given emissions scenario s, initial condition e, simulated by model m can be formally represented as tensors V1,…,Vnlat,lon,t,s,m,e, where each variable Vi (e.g., temperature, precipitation) is defined at spatial coordinates (lat, lon) and time t. To investigate and quantify the development of the regional climate, we look at time series IM_t,s,m of indicators IM, defined as functions IM=FV1,…,Vn, and aggregated over a specific region r. An indicator IM may represent a direct climate variable, such as monthly temperature; an impact-relevant compound indicator, such as the wet bulb globe temperature; or an impact indicator derived from an external impact model (e.g., a hydrological model), which is forced with the climate variables simulated by the ESM and which outputs quantities such as projected flood depths. The aggregation typically consists of a weighted average or sum over the relevant grid points within a region over a time period. A region may represent a country, an administrative unit, or a single grid cell. The resulting time series describes how a model simulates climate conditions or their impact in a region under the specified scenario.

RIME-X is designed to probabilistically emulate the response of IM to arbitrary emission scenarios s as a distribution quantifying the associated uncertainties. Formally, RIME-X aims to approximate all quantiles Vp of the distribution of the regional indicator IM, defined by Ps,yIM≤Vp=p, where Ps,y denotes the quantile function of IM under emissions scenario s, in year y and p∈[0,1] is the corresponding quantile level. The resulting distribution incorporates (i) global climate response uncertainty represented by an ensemble of SCM simulations and (ii) model and natural variability in the mapping from GMT to IM derived from ESM simulations in a MIP e.g. CMIP or ISIMIP;, referred to as the training dataset. The estimation of the quantile Vp in RIME-X is based on all values of IM simulated in the training dataset and all plausible GMT trajectories generated by the SCM ensemble for the target scenario s. RIME-X proceeds through the following three steps, utilizing a discretization of the GMT variable as described in Sect. :

Sampling GMT: draw GMT samples from a discretized prior GMT distribution for the scenario s and year y with probabilities Ps,y(GMT), derived from the SCM ensemble as detailed in Sect. .

The RIME-X emulator is implemented as an extension of the existing rime Python package . It includes a preprocessing pipeline designed to facilitate the data handling for emulation tasks, with built-in support for downloading simulation output from the ISIMIP repository. The pipeline can also be applied to other MIP datasets, provided they are manually preprocessed into a similar structure as the downloaded ISIMIP dataset. In the first step, the required data is downloaded and converted into NetCDF files containing the target indicator IM in a gridded format, at the minimum necessary temporal resolution. This gridded data is then aggregated regionally using predefined spatial masks. By default, the package includes masks for continents, countries, provinces, and scientifically relevant regions like the IPCC regions , with aggregation weighted by area, GDP, or population. Users may also supply custom regional masks. The regionally aggregated data is transformed into NetCDF files, we call quantile maps with dimensions for GMT level, quantile, and region, encoding the conditional distributions described in Sect. . Alternatively, the package supports transformation into grid-point-level quantile maps, with dimensions GMT level, quantile, and spatial coordinates. In this work, we use 101 equally spaced quantile levels (corresponding to the integer percentiles 0–100) and 0.1° temperature increments. For quantile emulation, the package provides a dedicated function. It takes as input an ensemble simulation of GMT for a given scenario, a list of years and quantiles of interest, and the regional conditional distribution data preprocessed in the described format. The function returns the requested quantiles of the indicator IM for the specified scenario, years, and regions. In practice, this function first samples a GMT value from the SCM ensemble and a quantile from the quantile map. It then obtains the regional indicator sample by linearly interpolating between the quantile values of the indicator distributions conditional on the two nearest GMT values to the GMT sample in the quantile map, so no temperature binning is needed during prediction.

Here we present selected examples of emulations produced by RIME-X to illustrate its capabilities in capturing the shift in distributions of regional climate indicators. We focus on the Climate Action Tracker (CAT) Current Policies scenario, which exhibits a strong climate signal across various indicators due to its high degree of projected warming. The GMT distribution used as input for the emulations is derived from an ensemble simulation using the simple climate model MAGICC. To highlight the capabilities of RIME-X, we present the emulated outputs using a variety of visualization formats. Figure  provides a grid-point-level representation of selected percentiles of the emulated distributions of the annual maximum daily maximum temperature indicator (TXx), shown relative to the 2020 median. The upper panel presents the 5th, 50th, and 95th percentiles for the year 2050. The lower panel focuses on the 95th percentile in 2050, 2075, and 2100, highlighting temporal shifts in extreme heat conditions. In Fig. , we focus on the temporal evolution of regional indicator distributions. On the left-hand side, we display how the density functions of distributions of selected indicators shift from 2020 to 2100 under the CAT Current Policies scenario. These distributions are approximated by applying a Gaussian kernel density estimate (KDE) to each quantile emulated by RIME-X. On the right-hand side, we show the evolution of the emulated quantiles over time. We highlight the median and the 90 % confidence interval. To emphasize the flexibility of RIME-X, we selected a diverse set of emulation cases. Results are shown for a variety of spatial region types and locations (e.g., countries such as Germany and Mexico, provinces such as Guangdong, and IPCC regions such as Southeastern Africa); various indicator types, including indicators taken directly from the ISIMIP data (mean annual temperature), indicators derived from ISIMIP data (cooling degree days and extreme precipitation), and indicators generated using impact models (maize yield change). The underlying indicator simulations were obtained from the ISIMIP3 repository, using all available simulations (including secondary input) and preprocessed with a 21-year running mean to reduce the effect of natural variability, leading to smoother distributions representing mainly the climatic trend in the indicator. A more detailed description of the specific indicators used can be found in Appendix , and an evaluation of the RIME-X emulator for the indicators is presented in Appendix .

To develop a validation strategy for RIME-X, we first identify potential conceptual limitations in its methodology. The first step of RIME-X – translating an emissions scenario into a GMT distribution using SCMs – follows established climate science as applied in recent IPCC assessments e.g.. SCMs are highly simplified representations of the climate system that are calibrated against historical observations and have been shown to reproduce the historical period well. Comparisons of SCM projections with ESM projections – our current best tools to project the evolution of the future climate – find no major structural limitations for the GMT variable relevant to the RIME-X framework. However, such evaluations primarily assess consistency with ESM behavior rather than the absolute accuracy of SCM projections and underlying ESM behavior, which means that such validation still cannot guarantee that their application to future climate states delivers adequate projections . SCMs guard against overconfidence in projections by simulating large ensembles based on systematically generated, historically constrained probabilistic parameter sets. The parameterization process samples a wide range of plausible responses in key physical processes – such as land–atmosphere interactions and ocean carbon uptake – typically designed to encompass the IPCC-assessed uncertainty ranges . These uncertainties are explicitly propagated within RIME-X to inform regional impact projections across multiple emissions scenarios. While we acknowledge that SCMs have inherent limitations, such as unresolved or potentially misrepresented processes following from the strong simplifications in the model structure, they currently provide the best available probabilistic estimates of GMT responses to emissions. The use of probabilistic ensembles that cover IPCC-assessed ranges also guards against introducing bias in any single direction. Given their extensive validation and assessment elsewhere, we consider validation of SCM-based projections to be beyond the scope of this study and treat SCM-based projections as established climate science.

The second step – deriving GMT-conditioned regional indicator distributions from MIP datasets – presents the main methodological challenge. Here, regional indicator values are binned by GMT to extract conditional distributions, assuming path independence, meaning that changes of regional indicator distributions from climate change depend entirely on the level of GMT, while the trajectory taken to reach that level of GMT and other regional influences like aerosol levels or deforestation are not taken into account. Though supported and applied in the literature for many indicators , this assumption can introduce biases, particularly for overshoot or stabilization scenarios, and in regions or indicators affected by pattern effects, high regional aerosol concentrations, changing circulation patterns, land–atmosphere feedbacks, SST-driven pattern effects, monsoon dynamics, or changes in regional land use . These limitations also restrict the applicability of our approach to idealized model experiments deploying solar radiation modification. However, when all scenarios and years of a training dataset are included in the calibration, any loss of predictability due to time lag or related effects will manifest as increased uncertainty (i.e., wider error bars) rather than as an overconfident estimate, as long as the emulated scenario is not a strongly out-of-sample case with respect to the training data, for example, when emulating scenarios involving high-overshoot pathways using a training dataset that does not include such processes. Additional limitations stem from the data: MIP ensembles offer a limited number of simulations, often developed using models that share components and are best viewed as “ensembles of opportunity” . While RIME-X can mitigate some of these limitations through model weighting, the underlying epistemic uncertainties persist . Moreover, many models in the MIP dataset do not reach higher levels of warming, meaning that the regional indicator distributions conditioned on these higher GMT levels are based on an increasingly small and selective subset of models that can provide data at these temperatures. This effect is partly mitigated by the fact that global climate response uncertainty increases with higher emission levels, leading to a wider range of simulated GMT outcomes. As a result, higher-emission scenarios generally sample a broader span of GMT levels, which counteracts – at least to some extent – the reduction in available samples for any specific high-GMT bin. The inclusion of high-warming scenarios such as SSP5-8.5 in the training data also provides additional coverage at high-GMT levels. Despite these limitations, MIPs remain the most holistic and flexible datasets for linking global and regional climate responses and are thus the best available foundation for RIME-X’s emulation approach, which is designed to be flexible across regions, scenarios, and indicators. The third and final step of RIME-X is a direct application of the law of total probability.

In summary, the first and third steps of RIME-X either rely on well-established methodologies or are mathematically valid. Accordingly, our empirical validation in Sect.  focuses on the second step: evaluating whether GMT-conditioned regional distributions extracted from MIP datasets can reliably reproduce simulated regional outcomes. This analysis does not, and cannot, address the epistemic limitations of MIP data. Instead, we treat MIP-derived distributions as ground truth and assess whether they can be reconstructed based solely on their dependence on GMT.

To investigate how reliably RIME-X emulates simulated distributions using GMT-conditioned regional indicator distributions, we adopt a holdout validation framework. The emulator is calibrated on a subset of simulations from CMIP6 , withholding all simulations of the test scenario. As test scenarios, we use SSP1-2.6, SSP2-4.5, and SSP3-7.0, which we aim to emulate. For both calibration and ground truth, we use data from the CMIP6 next-generation archive , which has undergone centralized preprocessing, including interpolation to a common 2.5°×2.5° latitude-longitude grid. This archive provides over 1500 simulations from 48 models (see Appendix ) covering the historical period and the emissions scenarios SSP1-2.6, SSP2-4.5, SSP4-6.0, SSP3-7.0, and SSP5-8.5, enabling robust calibration of the RIME-X emulator, including in our holdout validation framework. The archive additionally offers more than 150 simulations of each of the test scenarios from the same set of models, which allow us to extract statistically meaningful ground truth quantiles from actual simulations for comparison with our emulated quantiles. For emulator calibration and ground truth preprocessing, the CMIP6 simulations are spatially aggregated to the country level using masks from the ISIMIP archive and area-based grid cell weights. Temporal aggregation is then performed by applying a 21-year running mean to the regional averages in order to reduce sample uncertainty in the ground-truth distribution and make it comparable to the emulated distribution. This aggregation reduces the impact of outliers arising from natural variability, as the ground-truth distribution is estimated from significantly fewer samples per year than the emulated distribution. Countries are chosen as regions for the evaluation due to their heterogeneous sizes, diverse climate zones, and global landmass coverage, thus testing the method’s applicability across varied regional characteristics. Only simulations from the scenarios SSP1-2.6, SSP2-4.5, SSP4-6.0, SSP3-7.0, and SSP5-8.5 that are not simulations of the test scenario and share a model and initial condition ensemble with a simulation of the test scenario (including their respective historical simulations) are used to calibrate the RIME-X emulator. We give equal weight to every simulation (see Sect. ). All available CMIP6 simulations for the test scenarios are used to derive the ground truth distributions. To ensure comparability between emulated and simulated distributions, we use the actual global warming levels from the test scenario simulations to extract the scenario-dependent GMT distribution instead of using an SCM. This departs from the standard RIME-X methodology, which would otherwise rely on warming levels from SCM ensemble simulations (see Sect. ). However, SCM ensembles are designed to simulate realistic warming trajectories by assuming observation-based values for key metrics like the transient climate response , whereas many CMIP6 models tend to overestimate transient warming rates . Using the same warming levels as the ground truth simulations ensures that deviations between emulated and simulated distributions arise solely from the second step of the emulation methodology, rather than differences in warming trajectories. Consequently, if the emulated distributions closely match the simulated ground truth, we can conclude that RIME-X can use the relationship between GMT levels and regional indicators extracted from the CMIP6 simulations to produce meaningful regional indicator distributions for the unseen scenario.