To diagnose aerosol forcing and associated parametric uncertainty, we conducted two PPEs for PD and PI aerosol conditions. Each PPE consists of 383 simulations with different parameter combinations (Sect. ). We selected the earliest 12-month period (1 August 2024 to 31 July 2025) for which new observations from both the PACE and EarthCARE satellites are available. The PD and PI ensembles differ only in prescribed anthropogenic and biomass burning aerosol emissions. Hence, the PI and PD simulations are driven by the same boundary conditions over the simulating period, including the greenhouse gases and atmospheric oxidants concentration from the Shared Socioeconomic Pathways SSP245 scenario as well as monthly sea surface temperature and sea ice cover from the ERA5 reanalysis. Both PI and PD horizontal winds throughout the atmospheric column are nudged towards ERA5 winds in the PD period, with a relaxation timescale of 6 h. To our knowledge, this is the first study applying nudging in ICON-A-(HAM). Evaluation of the nudging setup indicates that it achieves a nudging strength similar to that used in ECHAM–HAM. We also compared the aerosol forcing in twin simulations with full-column and above-2 km nudging and found that the global-mean ERFaer is insensitive to nudging extent (Fig. S1 in the Supplement), consistent with previous studies , while over regions with strong biomass-burning emissions, allowing the lower atmosphere to respond to aerosol absorption leads to a slightly more positive ERFari.

The PD ensemble uses anthropogenic emissions from the Community Emissions Data System (CEDS v2025_04_18) and biomass burning emissions from the Global Fire Assimilation System (GFAS) for the simulation period, whereas the PI ensemble uses CMIP6 emissions for the year 1850. Since CEDS data are only available through 2023, emissions from 2022–2023 are assumed to represent also the 2024–2025 period in the PD simulations. This approximation is not expected to substantially affect the results, because emissions are unlikely to change significantly over two years based on recent rates of change in anthropogenic emissions , and moreover our PPE framework is able to scale emissions accordingly.

Prior to the main PPE simulations, a 1-year run with the default model setup (CTRL) was performed to spin up the model. The PPE runs were then initialized from the end of the 1-year CTRL run. Each PPE member underwent an additional 3-month spin-up followed by a 12-month run used for the final analysis.

ERFaer is calculated as the difference in net radiative flux at the top-of-atmosphere (TOA) between the PD and PI simulations. Following , ERFaer is further decomposed into ERFaci and ERFari, through double calls to the radiation calculation with and without aerosols.

Perturbed parameters, their ranges, and probability distribution functions (PDFs) were initially determined based on the elicitation of model domain experts and prior PPE studies , and subsequently refined through preliminary screening tests. First, we conducted 3-month one-at-a-time perturbation experiments to screen out parameters with little impact on key aerosol and cloud properties and hence ERFaer (not shown). Then, we perform a 3 d PPE to make sure that the parameter combinations from the initial ranges would not lead to a large fraction of failed simulations. The screening revealed that failures occur when conv_cmfctop and conv_entrscv are simultaneously high; their upper bounds were therefore reduced. Following this screening procedure, a total of 42 parameters and their ranges were determined for perturbation in the PPEs, including 26 aerosol-related parameters (Table ) and 16 cloud-related parameters (Table ).

The aerosol-related parameters represent processes/properties such as aerosol emissions, nucleation, aging, deposition, chemistry, and optical and hygroscopic properties (Table ). These parameters are largely consistent with those used in the ECHAM–HAM aerosol PPE by , but we extend the perturbations to better represent size- and species-dependent effects. Specifically, aerosol emissions are perturbed by sector (anthropogenic and biomass burning) and by species (SO2, BC, and OC), and sea-salt emissions are perturbed by mode (accumulation and coarse). In addition to scaling overall emission fluxes, we perturb key parameters in the sea-salt and dust emission parameterisations (emi_ss_expo and emi_du_fricv), which control not only the emission flux but also the size of emitted particles. The perturbed aerosol parameters affect aerosol sources and sinks, as well as their ability to absorb and scatter radiation and to serve as CCN. Cloud-related parameters describe processes/properties such as droplet activation, cloud microphysics, convection, cloud cover, cloud optical properties, and turbulence (Table ). These parameters influence cloud properties and the boundary-layer humidity distribution, and are key controls on the model's radiation balance. Together, the aerosol and cloud parameters are expected to modulate ERFaer through ARI and ACI pathways, or both.

Maximin Latin Hypercube Sampling (LHS) was employed to create parameter combinations spanning the 42-dimensional parameter space . In this study, two parameter distributions are input to LHS for different purposes: one to generate the original PPEs to train emulators (Fig. S2) and another to construct the emulated PPEs (Fig. S3). For the original PPEs, most parameters are sampled from uniform distributions, while a subset follows trapezoidal distributions with a uniform central region and sloped edges representing relatively unlikely parameter ranges (Fig. S2). This design preserves dense, uniform-like sampling in likely parameter space while efficiently extending the parameter ranges with minimal additional computational cost. For the emulated PPEs that are used for quantifying the parametric uncertainty, the trapezoidal distributions are adopted (Tables  and , Fig. S3). Compared to Gaussian distributions, this approach applies less tight expert-elicited constraints, avoids central oversampling, thereby preserving a broader uncertainty range. Note that the ranges displayed in Figs. S2 and S3 are the physical values, whereas the LHS operates on normalized ranges to determine the parameter combinations.

In total 400 parameter combinations were designed using LHS of the 42 perturbed parameters. Of these, 383 full-year ensemble members were successfully produced for both PI and PD periods: one corresponds to the default ICON–HAM configuration (CTRL), while the remaining 382 reflect the sampled parameter combinations. This yields a simulations-to-parameters ratio (rsp) of approximately 9, which is relatively high compared to previous PPE studies. For instance, , , and used rsp values of around 6, while and adopted a higher value of 10. The choice of rsp is crucial because it directly affects both computational cost and emulator accuracy, yet its optimal values, balancing accuracy, computational cost, storage requirements, remain unclear. This question will be addressed in Sect. .

Emulator performance is assessed using a split-sample validation, with 80 % of simulations (307) used for training and 20 % (76) for testing. Figure  shows globally averaged comparisons of GP predictions against truth, for key aerosol (AOD and AE) and cloud (CDNC, LWP, and cloud fraction (CF)) properties governing ARI and ACI, as well as ERFaer. Overall, the GP predictions closely follow the 1:1 reference line. The best performance is found for aerosol properties, with slopes of 0.89 for AOD and 0.85 for AE. Cloud properties show slightly lower slopes (< 0.85), while ERFaer deviates most strongly from the 1:1 line, with a slope of 0.67. The coefficients of determination (R2) follow a similar behaviour. Root mean square error (RMSE) is small relative to the magnitude of each variable, demonstrating good absolute accuracy. The progressive degradation in emulator performance from aerosol properties to cloud properties and ultimately to ERFaer likely reflects the increasing influence of nonlinear processes. For ERFaer in particular, spatial differences between PI and PD emissions further complicate the response because global parameter perturbations cannot capture this structure. Nevertheless, even for ERFaer, an R2 of 0.76 still indicates robust predictive skill.

Figure  shows 2-D maps of comparison between predicted and reference AOD as an example; similar behaviour is found for other variables (not shown). Overall, the emulator faithfully reproduces the spatial patterns of all quantities. Despite visible regional variations of absolute and relative differences, the overall biases remain very small, with global-mean relative biases below 1.5 % and local maxima under 4 %, which are substantially smaller than typical model and observational uncertainties . It is interesting to note that regions with lower R2 align closely with areas of strong convective precipitation, particularly over the tropical oceans near the Intertropical Convergence Zone, the Amazon, southern Africa, and South Asia over land. The association between reduced emulator performance and strong convective precipitation implies that the emulator may struggle to capture the strong nonlinear interactions among processes, such as emissions, convection, and wet scavenging. Nevertheless, this has only a limited impact on the overall emulator performance, with the minimum R2 remaining high (> 0.7).

The excellent emulator performance of our PPEs is not surprising given the relatively high simulations-to-parameters ratio in our PPE (rsp ∼ 9). However, such a large rsp also implies substantial computational cost and data-storage demands, posing challenges for large PPEs or multi-model PPE (MMPPE) intercomparison project. Identifying the optimal rsp, defined as the minimum value that yields acceptable performance (within the uncertainty range of metrics at rsp=8), is therefore essential for guiding future PPE design.

Following the procedure illustrated in Fig. , we assess emulator performance as a function of rsp for key aerosol and cloud properties, as well as aerosol forcing (Fig. ). Consistent with Fig. , the emulator performs best for aerosol quantities (AOD and AE), followed by cloud quantities (CF, LWP, and CDNC), with lower skill for aerosol forcing. This ranking is robust across different rsp choices and performance metrics. Note that the relatively limited performance of ERFaer mainly stems from ACI. Emulating ERFaer is challenging because it highly depends on spatial differences between PI and PD emissions, as well as on the minimum CDNC (CDNCmin) imposed in the model . This CDNCmin sets a hard bound on several processes (activation, autoconversion, entrainment mixing, etc.), thereby introducing pronounced nonlinear interactions between model parameters and outputs that are difficult to capture in a PPE .

Overall, the performance metrics (particularly the slope and R2) first increase with rsp and then begin to level off beyond a certain threshold. This threshold is around 5 for aerosol and cloud properties and ERFari, but higher (∼ 8) for ERFaer and ERFaci. Nevertheless, even for ERFaer and ERFaci, the metric values at rsp=5 (yellow star) remain within the uncertainty range of those at rsp=8, and the emulator bias and RMSE decrease by only 0.01 and 0.02 Wm-2 when increasing rsp from 5 to 8 (Fig. S5). This indicates that rsp=5 already achieves near-optimal emulator performance while substantially reducing computational cost by 40 %. We therefore consider rsp=5 as a compromise between accuracy and efficiency, even for ACI-related analyses. Further improvements in emulator performance should therefore focus on refining the machine-learning model (e.g., kernel tuning) rather than simply increasing the ensemble size.

Figure  shows probability density functions (PDFs) of global mean ERFaer, ERFaci and ERFari over 1850–2025, derived from 200 000 emulated variants generated based on the expert-elicited parameter distributions shown in Fig. S3. The mean ERFaer is -1.04 Wm-2, with a 5–95th percentile spanning -1.42 to -0.65 Wm-2 (Fig. a). The ACI component, ERFaci, shows a very similar estimate of -1.03 [-1.37 to -0.67] Wm-2 with a slightly narrower range (Fig. b), indicating that most of the uncertainty in ERFaer stems from ACI. In contrast, the ARI component ERFari is small with a mean of 0.04 Wm-2 but a somewhat wide range of -0.13 to 0.23 Wm-2 spanning both negative and positive values (Fig. c). Unlike ERFaci, which is uncertain only in magnitude, ERFari is uncertain in both magnitude and sign.

To isolate the contributions from the aerosol and cloud parameter groups, we use the emulator to vary the parameters in one group over their uncertainty ranges while fixing all parameters in the other group at their prior medians (Fig. ). Perturbing aerosol parameters only (AER; pink) results in a 5–95th percentile range of [-1.43 to -0.71] Wm-2, close to the total uncertainty range, whilst perturbing cloud parameters only (CLD; blue) yields a narrower range of [-1.18 to -0.9] Wm-2, with the width reduced by 64 % relative to total range, suggesting that aerosol parameters dominate the overall uncertainty. Comparing ACI and ARI (Fig. b,c) shows that cloud parameters influence ACI more strongly, although their effect remains weaker than that of aerosol parameters. The specific process-level sources of uncertainty will be identified in Sect. .

Figure  also compares the ICON–HAM PPE results with other PPEs using different climate models, multi-model ensembles (MMEs), and IPCC assessments. Comparison with different PPE studies shows that forcing estimates may vary across models depending on model structure, parameter perturbations considered, experimental setup (e.g., nudged versus free runs and the simulation period). For ECHAM–HAM, which perturbs a similar set of parameters (but only aerosol parameters) within a comparable parameterisation design (both using the HAM aerosol module) and utilises same nudging strategy (the whole column wind nudging), the resulting forcing range is very similar to that of our PPE but slightly narrower . This is partly because cloud-related uncertainty is not sampled, and also because the ECHAM–HAM PPE adopts Gaussian parameter distributions, which sample the central region more densely than the trapezoidal distributions used in our study, leading to a narrower range. The somewhat weaker ERFari in our PPE may result from the inclusion of above-cloud aerosols in the forcing calculation, which influences the balance between ACI and ARI .

Models with greater structural differences from ICON–HAM show larger discrepancies in forcing estimates. Four PPE experiments with HadGEM–UKCA show similar magnitudes and uncertainty ranges but yield more negative ERFaer than the ICON–HAM PPE. In addition to differences in model structure and parameter perturbations, the stronger forcing may also arise from the different PD reference period used, as a more polluted year (2008) was adopted in those experiments. According to the IPCC AR6 , ERFari was 50 % stronger in 2008 than in 2019, and ERFaci was 21 % stronger. Although all uncertainties shown here are from PPEs that are not yet constrained using observations, different parameter sampling strategies applied to the emulator implicitly lead to different levels of expert elicitation. ICON–HAM, HadGEM–UKCA, and ECHAM–HAM use expert-elicited parameter distributions (trapezoidal for ICON–HAM and HadGEM–UKCA, Gaussian for ECHAM–HAM), which represent prior expert beliefs about parameter likelihoods. In contrast, CAM6 and UKESM1–A use uniform distributions based only on expert-specified parameter ranges, which sample the full parameter space uniformly and therefore produce much wider uncertainty ranges. Note that the winds in ICON–HAM and ECHAM–HAM PPEs are more tightly nudged compared with the free-running (CAM6) and above-boundary-layer nudging (UKESM1-A and HadGEM–UKCA) PPEs, thereby limiting atmospheric adjustments and resulting in weaker forcing (particularly for ACI). Comparisons among PPEs suggest that different PPEs are not readily comparable. This highlights the need for a coordinated multi-model PPE (MMPPE) with a standardized protocol across models, including common perturbed parameters, parameter distributions, and experimental setups, to better disentangle structural and parametric uncertainties .

The spread in MMEs (CMIP5, CMIP6, ACCMIP and AeroCom) mainly reflects structural diversity and internal variability, because each model uses its own tuned parameter set without sampling parameter uncertainty within the model. Interestingly, the ERFaer uncertainty ranges from MMEs and PPEs are of similar magnitude, possibly implying a comparable contribution from structural uncertainty and internal variability versus from parametric uncertainty. A more robust attribution, however, would require further coordinated MMPPE efforts. IPCC assessments consider a broader range of evidence than MMEs alone, including individual model studies, uncertainties from missing or poorly represented processes, independent observational, and energy-balance constraints , thus resulting in broader ERFaer uncertainty ranges.

Figure  shows the spatial distributions of ERFaer, ERFaci, and ERFari and their corresponding uncertainties, along with the relative contributions of aerosol parameters to these uncertainties. Similar to the global-mean results (Fig. ), ERFaer and ERFaci exhibit very similar spatial patterns, indicating ERFaer is primarily driven by ACI. The strongest forcings (< -5 Wm-2) are located in industrialized continental regions, particularly Asia, Europe, and North America, followed by marine stratocumulus regions over the Atlantic, North Pacific, and the southeastern Pacific off the west coast of South America. The spatial patterns of ERFaer and ERFaci broadly agree with those from the previous ECHAM–HAM PPE , which uses similar aerosol and cloud parameterisations and comparable aerosol parameter perturbations. Our results, however, show weaker forcing over China and stronger forcing over India, likely due to anthropogenic emission reductions in China and increases in India between the respective simulation years (2010 for ECHAM–HAM and 2024–2025 for ICON–HAM). Notably, ERFaer and ERFaci turn positive over the UK because its anthropogenic aerosol emissions in 2024–2025 have dropped below 1850 levels (Fig. S6). Overall, regions exhibiting strong ERFaer and ERFaci also tend to show larger associated uncertainties (Fig. d and e).

The near-zero global mean ERFari is a result of the compensation of negative and positive regional contributions (Fig. c). Negative ERFari is prominent over the northern Indian Ocean, influenced by the transport of light-scattering sulfate aerosols from India. Positive ERFari occurs primarily where absorbing carbonaceous aerosols are densely emitted, or where they are transported over high-albedo cloud decks, including the northern India, South America, and North Atlantic. Similar effects of the above-cloud aerosols on ERFari were also identified in the HadGEM PPE , albeit with different spatial patterns. Additionally, a positive ERFari is also seen over the Saharan region but through a different mechanism, namely a reduction in dust aerosols resulting from their interactions with sulfate . Sulfate can coat dust particles, transferring them from insoluble to soluble modes and thereby enhancing their wet deposition, which more strongly reduces PD dust. At the same time, dust provides surfaces for sulfate condensation, suppressing new particle formation. Together, these processes contribute to a positive ERFari over dust-dominated regions. Similar positive ERFari induced by sulfate–dust interactions has been reported in other models, including GFDL , GISS , and ECHAM–HAM . The weak ERFari over China likely results from the partial cancellation between warming from absorbing BC aerosols and cooling from scattering sulfate aerosols, both of which have substantial anthropogenic emissions in this region . This is further supported by the sensitivity analysis presented later (Fig. S19), where ERFari shows comparable but opposite sensitivities to their emission parameters over land (-2.6 to emi_antns_so2 and +2.4 to emi_ant_bc). Substantial uncertainty is found in regions strongly influenced by industrial and biomass-burning emissions, including Asia, the boreal regions of North America and their downwind oceans, as well as parts of South America and Africa (Fig. f).

The lower panels of Fig. g–i illustrates the relative contribution of aerosol parameters to total uncertainty, with cloud parameters accounting for the remainder. Brownish colors denote aerosol control of uncertainty, and greenish colors cloud control. For ERFaer and ERFaci, aerosol parameters dominate over land and across the Northern Hemisphere, where aerosol loadings are higher, whereas the contribution of cloud parameters is limited to tropical regions characterized by intense convection. In contrast, ERFari is largely aerosol-dominated across the globe, with only a modest contribution from cloud parameters in convective regions, likely linked to wet scavenging. A detailed process-level attribution is provided in the next section.

Figure  presents the global and regional percentage contributions of individual parameters to uncertainties in ERFaer, ERFaci, and ERFari, with region abbreviations defined in the caption and Fig. . The corresponding contributions to uncertainties in global-mean forcing are provided in Table .

Globally, the ERFaer uncertainty is dominated by emission parameters. Anthropogenic SO2 (emi_antns_so2), natural DMS (emi_dms), and emitted particle size from biomass burning and fossil fuel sources (emi_cmr_bb and emi_cmr_ff) contribute similarly (25 %, 27 %, and 28 %), together accounting for nearly 80 % of the total uncertainty. This agrees with results from HadGEM and ECHAM–HAM PPEs, but shows a larger influence of emitted-size parameters, particularly for biomass burning, likely due to stronger boreal fires in our simulation year . It should also be noted that a key difference in the representation of ACI between this study and previous studies lies in the droplet activation. The HadGEM and ECHAM–HAM PPEs employed the default ARG scheme , whereas we uses a modified version by . The higher sensitivity of the revised ARG to Aitken mode could also help explain the increased sensitivity to particle size found here. The parameters emi_antns_so2 and emi_dms cause the uncertainty by controlling the magnitude of the PI–PD aerosol change. In particular, the role of DMS highlights the importance of the PI aerosol background. Similarly, uncertainties in biogenic volatile organic compound (VOC) emissions and processing may affect the PI aerosol background, too, but are not explored here because full chemistry is not activated in the current model configuration. For a given emitted mass, emi_cmr_bb and emi_cmr_ff determine the emitted particle number, thereby influencing CCN concentrations and cloud properties, and hence ACI. In comparison, the convection parameter conv_cprcon, controlling the conversion of convective cloud water to rain, contributes a smaller fraction of 6 % (Table ). It does not directly influence the anthropogenic aerosol perturbation but modifies the cloud radiative state by altering cloud amount, which sets the background cloud state for aerosol–cloud interactions.

It is unsurprising that the emission parameters dominate the ERFaer uncertainty, given their first-order control on PD–PI aerosol differences. Once anthropogenic, biomass burning, and DMS emissions are fixed at their prior medians, cloud parameters dominate the remaining uncertainty (76 %; Fig. S10), primarily through convection (conv_cprcon, conv_entrpen, and conv_cmfctop), cloud cover (cld_cov_crs), and auto-conversion (micro_ccsaut). The remaining uncertainty mainly from aerosol properties (refractive indices) and processes (SO2 chemistry and wet deposition).

Because of the short lifetime of tropospheric aerosols, the contributions of these parameters to ERFaer uncertainty vary substantially across regions. In the main industrial regions (NAM, EUR, and ASI) and their outflow regions (ARC, NPO, and NAO), where the influence of biomass burning aerosols is also present, emi_cmr_bb appears to play a dominant role (Figs. a and S7), with contributions ranging from about 25 % in NAM to as high as 55 % in EUR. In contrast, in biomass-burning affected regions (AFR, SAM, TAO and SPO) in the Southern Hemisphere, where anthropogenic sulfate is less abundant, emi_cmr_bb's contribution is much smaller (< 10 %). This contrast reflects the role of aerosol aging: abundant soluble materials (e.g., sulfate) can coat insoluble biomass-burning particles, transforming them into more hygroscopic particles with higher CCN efficiency and thereby modulating ACI forcing. Anthropogenic SO2 emissions (emi_antns_so2) affect ERFaer uncertainty in most regions, except in the very pristine Antarctic Ocean. Its contribution is generally larger over land (10 %–40 %). Over the oceans, however, other processes dominate: natural DMS emissions (emi_dms) exert a stronger influence over the pristine Southern Ocean (SPO, SAO, and SIO), contributing about 30 %–55 %; while convection and entrainment parameters (conv_entrpen, conv_cprcon, and conv_cmfctop) contribute more substantially over tropical oceans (TPO, TAO, and TIO) where deep convection is active, accounting for 30 %–47 %.

ERFaci exhibits a similar uncertainty structure to ERFaer (Fig. b) in most regions, although the relative contributions differ slightly. Compared with ERFaer, uncertainty in ERFaci is more strongly influenced by natural emissions (emi_dms; 32 %) and emitted particle size (emi_cmr_bb and emi_cmr_ff; 34 %), while the contribution from anthropogenic SO2 emissions (emi_antns_so2) is smaller (8 %). Convection parameters (conv_cprcon and conv_entrpen) also contribute more strongly to ERFaci uncertainty than to ERFari. Sea-salt emissions were found to be the dominant natural aerosol source of ERFaci uncertainty in the HadGEM and UKESM1–A PPEs, whereas they show almost no influence in our PPE and only a similarly weak effect in the ECHAM–HAM PPE . This difference may be partially explained by an underestimation of marine stratocumulus clouds in ICON–HAM and ECHAM–HAM , where sea-salt is expected to exert an influence. These discrepancies highlight the need for MMPPE efforts to better separate parametric and structural uncertainties across models with substantially different structures.

In contrast, ERFari shows a distinctly different uncertainty pattern (Fig. c). Besides anthropogenic SO2 emissions (emi_antns_so2; 18 %), most of the uncertainty stems from parameters controlling aerosol absorption, including anthropogenic BC emissions (emi_ant_bb; 12 %) and refractive indices of BC (rad_bc_ni; 23 %), OC (rad_oc_ni; 26 %), and dust (rad_du_ni; 5 %). The stronger role of absorption compared with other PPE studies likely results from enhanced biomass-burning emissions and lower SO2 levels in our simulation year 2024–2025 , as well as the inclusion of above-cloud aerosol effects , which are not included in all PPEs, together increasing the relative importance of aerosol absorption.

ERFari shows substantial regional variations in parameter contributions (Fig. c). The parameter emi_antns_so2 contributes strongly to ERFari uncertainty in regions with strongly negative ERFari (Figs. c and S9), including the industrial regions EUR, ASI, and AFR and their outflow oceans (NPO, TPO, and TIO). Strong sulfate transport from India leads to the largest emi_antns_so2 contribution in TIO, reaching about 60 %. In biomass-burning source regions (NAM, ARC, and SAM) and their downwind regions (EUR, ARCO, and NAO), uncertainty is primarily driven by the refractive index of OC (rad_oc_ni), contributing 23 %–70 %. The refractive index of BC (rad_bc_ni) contributes to ERFari uncertainty in nearly all regions, reflecting the efficient long-range transport of small BC to remote areas. Associated with this transport, the coating (so4_coating) and in-cloud wet deposition (wetdep_ic) parameters also contribute to the uncertainty over the Southern Ocean (SAO, SPO, and SIO), by changing the absorption efficiency and by controlling the removal of BC from the atmosphere, accounting for about 10 % and 20 %, respectively. Other parameters related to hygroscopicity (kappa_so4) and chemistry (chem_so4), although identified as major contributors in the ECHAM–HAM PPE, do not appear to be important here.

In addition to the parametric uncertainty, another source of the ERFaer uncertainty arises from the prescribed PI emission baseline. In existing PPEs, the spatial pattern and reference period of PI emissions are fixed, while the emission scaling parameters are perturbed in the same way as in the PD runs. Different assumptions about PI biomass-burning emissions and the choice of PI period could substantially alter the PI aerosol background and consequently the magnitude and regional pattern of PI-to-PD aerosol forcing . The PI baseline may further affect the relative contributions of individual parameters presented here. For example, an emission perturbation may exert a stronger effect under a cleaner PI background than under a more polluted background because of the nonlinear response of clouds and radiation to aerosol perturbations . Although scaling natural aerosol emissions provides some means of constraining the PI aerosol background, uncertainties associated with the prescribed PI emissions are not captured by the current PPE design and cannot be constrained by PD observations alone.

Because the PI atmosphere is not directly observable, constraints on PI–PD aerosol forcing must rely on PD observations . Identifying the processes controlling key PD aerosol and cloud properties helps elucidate whether and how these quantities can constrain the forcing uncertainty. Here, we focus on three aerosol properties observable from space (Fig. ), including AOD (total aerosol amount), AE (aerosol size) and AAOD (aerosol absorption), which are closely linked to aerosol extinction and CCN activity. The role of cloud properties (CDNC, LWP, and CFliq; Fig. ) is discussed separately in the following Sect. .

Figure a indicates that natural emission parameters dominate global AOD uncertainty, accounting for nearly 50 % of the variance. The main contributors include DMS emissions (emi_dms; 17 %), sea-salt emissions (emi_ss_expo and emi_ss_acc; 21 %), and dust emissions (emi_du_fricv; 8 %). Their influence, however, varies strongly by region (Figs. a and S11). DMS and sea-salt parameters primarily affect oceanic regions but also influence nearby continents such as OCE and ANTA, with contributions increasing toward the Southern Ocean. Although dust emissions contribute less than 10 % to global AOD uncertainty, they exert a substantial influence in the regions where its has a significant source (77 % in AFR and 35 % in ASI) and in downwind regions (20 % in TAO and 30 % in EUR). Similarly, despite their small global contributions, biomass burning OC emissions (emi_bb_oc) can explain around 45 % of the uncertainty in NAM and ARC due to boreal fires, and up to 20 % in SAM due to tropical fires, while anthropogenic SO2 emissions (emi_ant_so2) and sulfate hygroscopicity (kappa_so4) collectively contribute around 30 % of the uncertainties in EUR and ASI due to intense human activities. These strong regional variations suggest that observational constraints should be applied on a regional basis, as different regions are characterised by distinct processes driving the uncertainty . In addition, parameters affecting aerosol removal, e.g., conversion of convective cloud water to rain (conv_cprcon) and in-cloud wet deposition (wetdep_ic), contribute relatively uniformly to global AOD uncertainty (40 %), with slightly weaker influence over the Southern Ocean (Figs. a and S11).

As a component of total AOD, absorbing aerosol optical depth (AAOD) shares three common sources of uncertainty with AOD (emi_du_fricv, wedep_ic, conv_cprcon; Fig. a and b). Overall, AAOD uncertainty is dominated by absorption-related parameters, which together explain about 80 % of the variance, in which emission parameters for absorbing aerosols (BC, OC, and dust) account for about 33 %, while their optical property parameters (rad_bc_ni, rad_oc_ni, and rad_du_ni) play an even larger role, contributing ∼ 45 %. These optical property parameters control the imaginary refractive index of BC, OC, and dust, directly governing the absorption efficiency and influencing AAOD, and were also identified as key contributors to single-scattering albedo in the ECHAM–HAM PPE . Although so4_coating has little impact on global AAOD uncertainty, it can affect regions where both coating material (sulfate) and coating cores (biomass-burning or dust particles) are abundant (Fig. S12), by enhancing aerosol absorption through internal mixing of strongly and negligibly absorbing compositions .

Similar to AOD, the Ångström exponent (AE) uncertainty is also dominated by natural emission parameters, explaining an even larger fraction of the variance (about 74 %) than for AOD (Fig. c). Parameters associated with coarse-mode emissions (emi_ss_coarse and emi_du_fricv) play a particularly important role. Over land, dust emissions dominate the uncertainty, accounting for about 60 %–70 % in dust-influenced regions (EUR, ASI, and AFR) and ∼ 20 % elsewhere. In oceanic regions, coarse-mode sea-salt emissions dominate the uncertainty (∼ 35 %), with dust additionally contributing in the Northern Hemisphere through long-range transport (Figs. b and S13) . Although the wind-exponent parameter in the sea-salt emission scheme (emi_ss_expo) is not directly linked to coarse-mode production, stronger winds preferentially generate coarse sea-salt particles, shifting the emitted size distribution toward larger particles and thus influencing AE.

In comparison, fine-mode emission parameters have only a limited influence on AE uncertainty. DMS emissions cause the AE uncertainty through sulfate formation in most regions (∼ 16 % globally), while anthropogenic SO2 mainly affect their source regions contributing < 5 % globally and < 10 % regionally. Of course, other parameters controlling aerosol sink contribute to AE uncertainty, too, but through different mechanisms. For example, in dust-rich regions, wetdep_bc and so4_coating each contribute about 10 % of the AE variance by modifying precipitation scavenging and dust hygroscopicity through aerosol aging, respectively. Moreover, convection-related parameters (conv_entrpen and conv_cprcon) together cause ∼ 9 % of global uncertainty, by affecting wet removal and aerosol growth through cloud processing .

Overall, our uncertainty attribution reveals that present-day aerosol properties share several driving parameters with aerosol forcing, indicating their potential to constrain forcing. For example, emi_dms and conv_cprcon influence AOD, AE, and ERFaci, while rad_bc_ni, rad_oc_ni, rad_du_ni, and emi_ant_bc affect both AAOD and ERFari. The utility of different observations to constrain aerosol forcing is discussed further in Sect. .

In comparison with the ECHAM–HAM PPE , we identify several common parameters related to aerosol emissions, deposition, and optical properties, but their contributions differ substantially. These differences partly reflect the different expert-elicited parameter distributions used in the two studies, and likely also the inclusion of cloud-related parameters in our PPE, on top of structural differences. By perturbing both aerosol and cloud processes simultaneously, our PPE allows stronger interactions among parameters, which alter the relative contributions of individual parameters. For instance, the convection parameter conv_cprcon alone explains about 20 % of the global AOD uncertainty and causes regional uncertainty of up to ± 0.03 in subtropical regions (Fig. S11), by affecting aerosol removal and regeneration from cloud processing. This highlights the importance of including cloud perturbations even in an aerosol-focused PPE study.

Cloud droplet number concentration (CDNC) is central to aerosol–cloud interactions, linking aerosol perturbations to responses of cloud microphysical (LWP) and macrophysical (CFliq) properties. In climate models, CDNC is primarily determined by CCN concentrations, updraft velocity, and activation treatment. The activation treatment governs how efficiently aerosol particles activate into cloud droplets for a given updraft. Its uncertainty in our PPE is represented by the parameter activ_aero, whose range reflects the diversity among commonly used activation schemes .

Figure a shows that global CDNC uncertainty arises almost equally from parameters controlling CCN concentrations (35 %; emi_dms, emi_ss_expo, drydep_acc, and chem_so2) and activation scheme (33 %; activ_aero), followed by updraft velocity (14 %; activ_w), with activ_aero being the largest single contributor. For CCN-related parameters, DMS emissions dominate globally; however, the relative importance of individual parameters varies strongly by region (Figs. a and S15). For example, anthropogenic (emi_antns_so2, emi_ant_oc, and emi_cmr_ff) and biomass-burning (emi_bb_oc and emi_cmr_bb) emission parameters dominate over land and also contribute substantially to ERFaer uncertainty (Fig. ), implying strong potential for CDNC observations to constrain ERFaer, while over oceans the uncertainty is mainly driven by natural emissions (emi_dms, emi_ss_acc, and emi_ss_expo). The parameters drydep_acc and chem_so2 each cause about ∼ 5 % uncertainty in most regions (except for the Southern Ocean) by altering aerosol lifetime through dry deposition and sulfate production through SO2 oxidation, respectively.

Comparing the three parameter categories (CCN, updraft, and activation), it is interesting to see that CCN-related parameters play a larger role in the cleaner, aerosol-limited Southern Ocean than in the more polluted, updraft-limited Northern Hemisphere. This suggests that ICON–HAM reasonably captures the regime-dependent activation behaviour reported in previous studies using cloud parcel models and satellite observations . Beyond these three categories, the convection parameter conv_cmfctop contributes about 17 % uncertainty, but only in tropical regions (TPO, TNO, and TIO), by influencing moisture transport and thus cloud formation .

The parameters dominating CDNC uncertainty also influence LWP uncertainty, but with much smaller contributions, decreasing from 82 % for CDNC to only 18 % for LWP (Fig. b). Instead, LWP uncertainty is dominated by parameters related to cloud microphysics and convection. Globally, auto-conversion rates from cloud water to rain in stratocumulus (micro_ccraut) and convective (conv_cprcon) clouds explain 31 % and 24 % of the LWP uncertainty, respectively. Although convective mass-flux transport (conv_cmfctop) and entrainment mixing (conv_entrpen) account for only 10 % and 6 % of the global uncertainty, they can have a strong impact in tropical regions (AFR, SAM, OCE, TPO, TAO, and TIO), where deep convection is more active. The parameter conv_cmfctop regulates the strength of convective mass-flux transport that redistributes moisture vertically, whereas conv_entrpen controls entrainment mixing between convective updrafts and surrounding air, both affecting the amount of condensed cloud water and thus LWP (also CFliq; Fig. b and c).

The model diagnoses CFliq solely from relative humidity ; therefore, the parameters dominating CDNC have little influence on CFliq (Fig. c). Uncertainty attribution shows that convection-related parameters (conv_cmfctop, conv_cprcon, and conv_entrscv) account for most of the uncertainty (about 90 %), with the remainder associated with the cloud-cover scheme (cld_cov_crs). Among these, conv_cmfctop is the largest contributor, explaining around 66 % of the CFliq uncertainty by regulating convective mass-flux upward transport and thereby modifying the moisture available in the boundary layer where liquid clouds form . conv_entrscv contributes about 8 % by influencing the mass-flux transport more indirectly through entrainment in shallow convection, which modifies the buoyancy of convective updrafts. In addition, conv_cprcon explains about 12 % of the variance by altering convective precipitation efficiency via auto-conversion and thus the persistence of liquid clouds. Only 5 % of the uncertainty arises from the non-convection parameter cld_cov_crs, which determines the critical RH threshold for cloud formation. These parameters produce similar levels of uncertainty across oceanic regions, where liquid clouds are most prevalent (Figs. c and S17).

Beyond these parameters, stratiform cloud microphysics parameters controlling phase transitions between cloud water components and precipitation cause the CFliq uncertainty over land, accounting for about ∼ 10 % from micro_ccraut and about ∼ 30 % from micro_ccsaut. The parameter micro_ccraut affects liquid cloud formation directly by altering rain formation via auto-conversion. In contrast, micro_ccsaut (snow formation via aggregation) affects the CFliq mainly by changing the partitioning of liquid and ice water in mixed phase clouds through the Wegener-Bergeron-Findeisen process, in which cloud ice grows through the depletion of liquid droplets . This parameter has an even stronger influence on ice and total cloud cover (figure not shown).

To conclude, cloud properties are predominantly controlled by cloud-related parameters (80 % for LWP and 90 % for CFliq), except for CDNC, for which aerosol and cloud parameters contribute roughly equally. This uncertainty attribution also provides insight into the processes governing aerosol–cloud interactions. The much smaller aerosol contribution to macrophysical properties implies lower LWP and CF susceptibilities to aerosol perturbation compared to CDNC, consistent with the relatively weak adjustments in LWP and CFliq relative to the Twomey effect reported in climate models . This, however, contrasts with observational evidence that suggests that cloud adjustments (particularly CF adjustment) can be comparable to or even stronger than the Twomey effect . Such discrepancies may point to structural deficiencies in the model, i.e., missing or poorly represented processes, such as aerosol effects in cloud fraction parameterisation and turbulence–microphysics interactions in cloud-top entrainment .

Figure  shows annual mean AOD, AAOD, and AE from the ICON–HAM default run against SPEXone retrievals. Overall, the model appears to underestimate AOD compared to observations (Fig. a–c), with a global mean bias of -0.036. The underestimation of AOD is also common in other AeroCom models . Spatially, the largest differences are primarily found over land, particularly in biomass burning regions over Southern America and Africa tropical rain forests, where the model exhibits pronounced negative biases (Fig. c). A similar, though weaker, negative bias occurs in northern boreal regions. Similar bias patterns have also been reported in ECHAM-HAM and an earlier version of ICON–HAM, and have been attributed to too low biomass burning emissions and too strong wet deposition by convective clouds . With the PPE, we suggest the dominant role of biomass-burning emissions, as the spatial pattern of the negative bias closely aligns with the parametric uncertainty caused by emi_bb_oc (Fig. S11). In-cloud wet deposition (wetdep_ic) and convective cloud-to-rain conversion rate (conv_cprcon) also contribute to the bias in these regions, albeit to a lesser extent (Fig. S11). In addition, negative biases of AOD are evident in major industrial regions (e.g., China and India) as well. These biases are likely linked to the high uncertainty associated with emissions of anthropogenic aerosols (emi_antns_so2) and their hygroscopicity (kappa_so4) in these regions (Fig. S11). The positive biases, in contrast, occur primarily in the Sahara and its outflow regions, whose spatial pattern is fairly compatible with that of the uncertainty related to dust emissions (emi_du_fricv; Fig. S11). Consistent with its sister model ECHAM–HAM , ICON–HAM overestimates AOD over subtropical oceans, but less strongly . This bias may be explained by relatively large parametric uncertainties caused by both sources (emi_dms, emi_ss_acc, emi_ss_coarse, emi_ss_expo) and sinks (wetdep_ic and conv_cprcon) of natural aerosols (Fig. S11). Overall, regions with larger biases often coincide with high parametric uncertainty (green isolines; Fig. c), implying that part of these biases may be reduced through parameter optimisation.

AAOD largely mirrors the AOD bias map, with a global mean bias of -0.003 (Fig. d–f). There are, however, two notable exceptions. First, the positive bias over subtropical oceans seen in AOD disappears in AAOD, as these regions are dominated by scattering aerosols (DMS-derived sulfate and sea-salt). Second, the negative AOD bias over China and India shifts to a positive bias in AAOD, likely reflecting large uncertainties in atmospheric absorption, e.g., anthropogenic emissions of BC (emi_ant_bc) and its refractive index (rad_bc_ni) (Fig. S12). Figure f illustrates that virtually all regions with large biases closely correspond to areas of high parametric uncertainty, suggesting that parameter optimisation offers also an effective pathway to reduce the AAOD bias.

AE exhibits an overall positive bias relative to SPEXone observations (Fig. g–i), with a mean bias of 0.19, pointing to too small particle sizes in ICON–HAM. Over land, the largest positive biases occur in dust source regions in the western US, where the model fails to reproduce the lower AE compared to the eastern US observed in the data. A similarly strong positive bias is found over the Tibetan Plateau, likely attributed to limitations in simulating aerosol transport over complex terrain at high altitudes, as well as reduced retrieval accuracy under complex terrain and low aerosol. Such issues are also seen in AeroCom models . Another continental region showing positive bias is South America. As shown in Fig. S13, the origin of the AE bias over land is complex due to the interplay between fine- and coarse-mode aerosols across multiple processes, including emissions (emi_dms and emi_du_fricv), particle aging (so4_coating), and wet deposition (wetdep_ic and wetdep_bc). Although rad_oc_ni also contributes to AE uncertainty over land, this does not reflect changes in particle size. Instead, changes in the OC absorption can alter aerosol extinction differently across wavelengths and thereby affect AE .

Strong positive AE biases are broadly found over oceans, particularly over large-scale convergence zones, including the Intertropical Convergence Zone (ITCZ), the South Pacific Convergence Zone (SPCZ), the Indian Ocean Convergence Zone (IOCZ), and the South Atlantic Convergence Zone (SACZ) (Fig. i). These regions are characterised by intense moisture transport, active convective cloud formation, and strong precipitation, where precipitation is often biased in models . Notably, the spatial patten of the positive bias agrees well with that of the parametric uncertainties associated with emissions of DMS (emi_dms) and sea-salt (emi_ss_coarse and emi_ss_expo), as well as convective processes such as cloud-to-rain conversion (conv_cprcon) and entrainment in deep convection (conv_entrpen) (Fig. S13). This coincidence, together with the AOD underestimation over oceans (Fig. c), suggests that insufficient coarse-mode sea-salt emissions and overly strong wet scavenging in convective clouds contribute to the AE overestimation. Comparing model bias with parametric uncertainty (Fig. i), the green isolines largely enclose regions of high parametric uncertainty, highlighting the potential for reducing AE bias through parameter optimisation in most areas. Nonetheless, part of the convergence zones has a positive bias that cannot be explained by the parametric uncertainty alone, pointing to possible structural deficiencies in the convection parameterisation.

To provide a more direct overview of the parameters driving uncertainty in regions with both large biases and high parametric uncertainty, we repeat the analyses of Fig.  considering only locations within the green isolines (Fig. S14). It is shown that the dominant parameters are overall consistent with those inferred from the spatial patterns discussed above, although regional averaging can obscure subregional signals, such as the role of emi_bb_oc over central Africa. These results can inform which parameters may be most useful to constrain observationally for reducing model biases in each region.

Before comparing the model with MODIS, it is worth noting that MODIS retrievals of cloud microphysics (CDNC and LWP) are less reliable over polar regions and land than over mid-to-low-latitude oceanic regions .

Figure  compares model diagnostics and MODIS observations processed consistently using the COSP simulator. ICON–HAM exhibits positive CDNC biases in most regions, except in polar and some land areas where MODIS cloud retrievals are subject to larger errors . Over oceans, the largest positive biases occur in high-latitude regions, followed by subtropical eastern ocean stratocumulus decks (Fig. a–c). This bias pattern is very similar to the spatial distribution of parametric uncertainties associated with natural emissions of DMS (emi_dms) and sea-salt (emi_ss_acc and emi_ss_expo), as well as activation (activ_aero) and vertical velocity (activ_cwtrub), as shown in Fig. S15. Over land, a pronounced CDNC overestimation is found in China and India, despite an underestimation of AOD (Fig. c). This positive bias may be thus attributed to too many fine particles from anthropogenic emissions (emi_cmr_ff) and/or too efficient activation (activ_aero) in the model (Fig. S15). Moreover, the good spatial consistency between mode bias and parametric uncertainty (Fig. c), suggests that most of the CDNC bias would be reduced by constraining these uncertain parameters.

For LWP, ICON–HAM shows a large negative bias over high-latitude continents (Fig. d–f), yet no parameters can explain this bias patten (Fig. S16). This is most likely attributable to large retrieval errors caused by high solar zenith angles and the presence of mixed-phase clouds . In contrast, in the remaining regions where LWP retrievals are considered more reliable, the model generally overestimates LWP, particularly in the ITCZ. As shown in Fig. S16, processes of auto-conversion in both stratocumulus and convective clouds and entrainment in deep convection, might be responsible for this bias. Again, most regions with large biases coincide with areas of high parametric uncertainty, whereas the remaining regions are likely influenced by retrieval biases and/or model structural uncertainties. For the locations within the green isolines, the parameter contributions to LWP and CDNC to uncertainties in each region (Fig. S18) are broadly consistent with the spatial correspondence discussed above.

CFliq is underestimated globally by around 25 % (Fig. g–i), which is particularly pronounced in regions with persistent low-level cloud decks over subtropical and polar oceans. This may partly explain the weaker ERFaci in offshore stratocumulus regions in ICON–HAM than in other AeroCom and CMIP6 models . Unlike other properties, the CFliq bias cannot be associated to parametric uncertainty, as evidenced by the little spatial overlap between regions of large bias and those of high uncertainty (Fig. i). Instead, structural improvements in cloud parameterisation are required, for example, moving from a RH-based scheme to a prognostic cloud cover scheme . The underestimation of cloud cover in stratocumulus region has been identified as an inherent issue in ICON 2.6.4 , and is expected to be addressed in upcoming versions.