the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Aug 2026
Towards Resolving Equifinality in Aerosol-Cloud Radiative Forcing Through Process-Level Constraints
Abstract. Equifinality—where multiple parameter combinations produce indistinguishable climate states—is a fundamental obstacle to calibrating Earth system models (ESMs). Parameter sets that agree well with the observed mean state can produce dramatically different responses to external perturbations, reflecting a mathematical non-uniqueness that standard calibration approaches cannot resolve. This poses a critical challenge for constraining the effective radiative forcing due to aerosol-cloud interactions (ERFACI), which depends sensitively on cloud responses to aerosol perturbations. We use a perturbed parameter ensemble that, as a minimal working example, varies only two parameters related to warm-rain formation (autoconversion and accretion) and constrain on synthetic observations in a perfect-model setup. Even under ideal conditions, radiation-only constraints produce a fundamentally degenerate, bimodal ERFACI posterior. Incorporating the observable cloud state (liquid water path, 𝔏) for warm marine stratocumulus clouds can resolve this bimodality, but its utility diminishes with large observational uncertainty. A far more robust constraint is achieved by directly targeting the underlying physical process rates that modulate 𝔏. Process-rate constraints effectively eliminate bimodality and dramatically reduce posterior uncertainty even under large assumed observational uncertainties. As process rates are not directly observable, we test their observable proxies as a tractable alternative, finding they offer only limited additional value over cloud state due to their emergent behavior. These results establish a critical principle for ESM calibration: to achieve robust constraints on ERFACI and beyond, it is necessary to constrain both the emergent climate state and the physical pathways that govern its formation.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-4488', Jamie Knight, 06 Oct 2026
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RC2: 'Comment on egusphere-2026-4488', Anonymous Referee #2, 08 Oct 2026
General comments
This manuscript addresses the issue of parameter equifinality in Earth system model (ESM) parameter calibration using a simplified version of an emerging, but typically more complex, workflow. The work begins with a perturbed parameter ensemble (PPE) with two microphysical parameters perturbed in the atmosphere component of the E3SM ESM. This PPE is used to train Gaussian processes (GPs) to cheaply emulate the expensive ESM. Bayesian parameter inference is then performed with the GPs using the default model settings to generate target “observations.” Inference is repeated with different combinations of constraining metrics, including radiation and cloud mean state properties as well as process rates and process rate proxies. Beyond different metric combinations, the experiments also include different uncertainty assumptions. With this simplified but non-trivial setup, the manuscript explores how parameter equifinality after constraint impacts uncertainty in aerosol-cloud interaction quantities of interest like ERF_ACI.
This work represents a substantial, novel contribution to scientific progress within the scope of ACP because it combines a process-based approach and a simplified known truth setting to present a clear story that will help future work include observational constraints with greater capacity to reduce parameter equifinality. Before this manuscript can be accepted for publication, the manuscript must address some shortcomings of the 5% error experiments (described in detail in major issue 1 below). Beyond this issue, the manuscript has excellent presentation quality, its methodology is well supported by robust appendices, and its scientific approach is consistent with the literature and referenced appropriately.
Major issue 1: Unaddressed shortcomings in 5% error experiments
One major benefit of the known truth framework is knowing what answer we should recover. Figure 2a and Figure 3a reveal the posterior has very little probability near the “truth” for parameter values (3a) or ERF_ACI (2a) in the 5% error experiments even with the process rate observations. Right now, this disconnect is not addressed at all in the text and is somewhat obscured by Figure 2 and Figure 6 (which extends Figure 2) not including the known truth’s ERF_ACI value. The manuscript needs to address this inconsistency and ideally resolve it. My naïve first hypothesis on why the truth is more difficult to recover at 5% vs 20% observational error would be emulator uncertainty estimation.
Specific comments
Ln 45: Would suggest clarifying here that this is a PPE of the atmosphere component of E3SM (rather than waiting until ln 96)
Ln 101-103: what year(s) of MERRA2 fields used?
Section 2.2: It might be worth mentioning somewhere (Section 4?) that autoconversion and accretion do not only impact warm Sc. I can guess why this focus was chosen for ERF estimation goals, but it might be worth including a sentence in the discussion section acknowledging what potential issues might emerge by focusing parameter inference on only specific elements of a parameterized process rather than the process comprehensively.
Section 2.4: State explicitly whether constraints are applied only to PD? Or PI too? (I assume not, but the manuscript is not clear.)
Ln 143: For the L_adj pathology, I wonder if this could be reduced by emulating L_PD and L_PI separately then calculating L_adj?
Ln 195: Is Figure A4 the basis for R in the covariance matrix (equation 6)? This might be stated explicitly if so, around the introduction of R.
Figure 2 (and 6): Would be helpful to include markers for the known truth ERF_ACI.
Figure 4: Would be helpful to include a marker for the true parameters across panels.
Ln 257: “The intersection of these two broad surfaces [of radiation only and L only in Figure 4] explains the shape of the radiation + L posterior in Fig. 3b.” Figure 4 panels appear to show that the radiation-only surfaces are entirely contained in LWP-only surfaces, so I struggle to follow this sentence.
Figure D2: It would be helpful to mark known truth values on these panels.
Citation: https://doi.org/10.5194/egusphere-2026-4488-RC2
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Overall, I thought this was a very good paper and really enjoyed reading it. The study provides a clear and interesting exploration of equifinality in climate model calibration, using aerosol-cloud radiative forcing as a case study. The authors demonstrate convincingly that constraining the model using cloud-top radiative properties alone is not sufficient, because different parameter combinations can produce similar radiative states through different underlying cloud and precipitation pathways. The analysis then shows that constraints targeting the cloud state and, more directly, the physical processes controlling that state can substantially reduce this degeneracy. I found this a valuable and well-presented contribution, and my comments below are mainly intended to clarify and strengthen the methodology and interpretation.
Comments:
(1) The discussion of aerosol-cloud interactions (ACIs) in the second paragraph of the introduction seems somewhat muddled. In particular, the text appears to conflate ACIs themselves with the aerosol-cloud interaction component of effective radiative forcing (ERF_ACI). ACIs describe the physical interactions between aerosol and clouds more generally, whereas ERF_ACI specifically quantifies the radiative effect associated with the change in anthropogenic aerosol between the pre-industrial and present-day states. I suggest clarifying this distinction. I am also not sure that the sign of ERF_ACI is generally considered uncertain; for example, see the range of estimates reported in the latest IPCC assessment.
(2) The penultimate paragraph of the introduction reads somewhat like an abstract or conclusions paragraph, as it summarises the main results before they are presented.
(3) Section 2.1 may benefit from some brief discussion of the implications of the nudging approach for the ERF_ACI calculations. Nudging horizontal winds may suppress or alter some rapid adjustments, and therefore I am not sure that these should be described as strict ERF calculation. Nevertheless, wind-only nudging has been shown to preserve aerosol indirect effects reasonably well (e.g., Zhang et al., 2014), so I think the calculations proved a useful estimate of the radiative forcing associate with the aerosol perturbation.
(4) Am I correct in understanding the precipitation amplification frequency (PAF) as:
PAF = P(P > 0.1 mm h-1 | P > 0.001 mm h-1)?
If so, I found this conditional-probability formulation considerable clearer than the current description and think it would be useful to state the definition explicitly.
(5) I found Section 2.3 difficult to follow. Am I correct in understanding the emulator procedure as follows: 40 of the 50 perturbed parameter ensemble (PPE) members are used to train the emulator, with the remaining 10 used for independent testing; following assessment of emulator performance, the final emulator is then trained using all 50 PPE members? If so, I suggest making this work flow more explicit.
(6) I had a question regarding the likelihood formulation in Eqs. (4)-(5). I am not an expert in Bayesian inference, so please disregard this comment if I have misunderstood the implementation. If the GP predictive variance varies with (\Phi), rather than being constant, would the Gaussian likelihood also require the corresponding normalisation term? For example, for Eq. (4), this would introduce a term of the form
−1/2ln[2π(σobs2 +σGP2 )]
If the GP uncertainty is small relative to the assumed observational uncertainty, this may have little practical effect.
(7) Throughout the manuscript, it could be made clearer that it is the combination of radiative, cloud-state, and process-level constraints that resolves the equifinality identified here. In places, the texts reads as though process-rate constraints are uniquely necessary to resolve the problem. However, as the authors note elsewhere, additional climate-state variables (e.g., clout-top CDNC) could provide further constraint, and this study represents a single case study focused on ACIs. I would suggest framing the process-rate constraints as particularly effective within the tested framework, rather than as uniquely required in general.
(8) The covariance structure used in the likelihood is based on Pearson correlations calculated from the PPE. Pearson correlation assumes linear relationships, whereas some of the relationships shown in the manuscript appear strongly nonlinear. Do the authors expect this assumption to have a significant impact on the likelihood and resulting posterior? Have alternative measures of dependence, such as Spearman rank correlation, been explored as a sensitivity test? I appreciate that Spearman correlation cannot simply be substituted into a Gaussian likelihood, such a comparison could nevertheless help assess the sensitivity of the results to the assumed correlation structure.
Technical corrections:
(1) “at 700 and 500 hPa” -> “at both 700 and 500 hPa”
(2) “the MCMC navigates” -> “the MCMC algorithm navigates”