Statistical Noise and Missing Forcing Limit Estimates of Earth's Feedback from Prescribed Sea-Surface Temperature Simulations

Gyuleva, Gergana; Knutti, Reto; Noyelle, Robin; Beyerle, Urs; Fischer, Erich; Sippel, Sebastian

doi:10.48550/arXiv.2608.13219

Preprints

https://doi.org/10.48550/arXiv.2608.13219

Preprints

21 Aug 2026

| 21 Aug 2026

Statistical Noise and Missing Forcing Limit Estimates of Earth's Feedback from Prescribed Sea-Surface Temperature Simulations

Gergana Gyuleva, Reto Knutti, Robin Noyelle, Urs Beyerle, Erich Fischer, and Sebastian Sippel

Abstract. Earth's feedback parameter measures how the Earth system responds to forcing and is inversely proportional to climate sensitivity. Sea-surface temperature (SST) patterns can modulate the value of the feedback parameter. Differences between observed and simulated SSTs have raised the question how the observed SSTs evolution impacts the global feedback. The standard method for estimating this effect uses observed SSTs prescribed to an atmospheric model with fixed pre-industrial atmospheric forcing. This method makes two assumptions: first, that the observed SSTs capture all relevant effects from the forcing, so that prescribing a time-varying forcing is unnecessary; second, that the temporal variations in the feedback parameter are driven by the evolving SST pattern and can be estimated via moving-window regressions. We test these assumptions by running controlled experiments in which SSTs from fully-coupled historical simulations are prescribed to an atmospheric model. We find that the prescribed-SST experiments fail to capture the coupled feedback evolution. This is explained by two effects: First, the absence of prescribed atmospheric forcing, and second, statistical noise arising from the computation of moving-window regressions. We find no evidence of any significant relationship between evolving SST patterns and changes in the feedback time series in a 4000-year pre-industrial control simulation. Any trends in the feedback parameter detected on timescales shorter than ~100 years are indistinguishable from statistical noise, making their attribution to the evolving SST pattern extremely difficult. Our results imply that prescribed SST simulations offer limited potential for inferring temporal changes in Earth's parameter over the observational period.

Received: 18 Aug 2026 – Discussion started: 21 Aug 2026

Gergana Gyuleva, Reto Knutti, Robin Noyelle, Urs Beyerle, Erich Fischer, and Sebastian Sippel

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-4925', Tim Andrews, 01 Oct 2026
Review of “Statistical noise and missing forcing limit estimates of Earth’s feedback from prescribed sea-surface temperature simulations” by authors Gyuleva et al.
This manuscript addresses an important question and contains some potentially valuable results. In particular, I found the analysis of omitted forcing and the resulting land-temperature differences between coupled and prescribed-SST simulations, with possible implications for radiative feedback, physically insightful. However, I found that the manuscript overreaches in its conclusions, and I do not believe that the analysis supports its central claim that prescribed-SST experiments have limited utility for inferring temporal changes in climate feedback over the historical period.
My recommendation is based on three major and fundamental concerns:
The CESM2 perfect-model framework does not test whether feedback variations associated with the observed SST evolution are detectable, which is the central scientific question motivating the amip-piForcing literature challenged by this study. Instead, it tests whether feedback variations associated with CESM2-simulated SST evolution are detectable. These are not equivalent questions, particularly given evidence that AOGCMs do not reproduce the magnitude and duration of relevant observed SST-pattern changes.

If the feedback evolution diagnosed in amip-piForcing is primarily statistical noise or a consequence of omitted forcing, as the manuscript argues, it is unclear why qualitatively similar feedback evolution emerges from several independent methods that do not rely on amip-piForcing moving-window regressions. The manuscript does not adequately reconcile its conclusions with this wider evidence base.

The manuscript does not sufficiently recognise that the amip-piForcing literature is already aware that feedback estimates calculated over short, approximately 30-year windows are noisy. The community has adopted methods to address this problem, including multi-model analysis and atmospheric initial-condition ensembles, neither of which is adequately tested here.

In my view addressing these fundamental issues would require a substantial reconceptualization of the manuscript rather than a traditional iterative revision process. I therefore recommend rejection, while encouraging the authors to take on board this review in the constructive manner it is intended, and submit a fundamentally reframed manuscript centred on the physically interesting results, such as the role of missing forcing, land warming and uncertainty estimates.
I elaborate on my major concerns below:

Major Comment 1: The perfect-model framework is not testing the question of interest
The manuscript demonstrates that prescribed-SST CAM6 simulations do not perfectly reproduce feedback evolution diagnosed in corresponding CESM2 historical simulations. However, the manuscript then extends this conclusion to argue against the broader utility of amip-piForcing experiments. I do not think this inference is justified.
A substantial literature has documented that coupled models generally fail to reproduce important aspects of the observed historical SST evolution, particularly in the tropical Pacific. Indeed, the manuscript itself discusses this issue in the introduction.
The critical question for much of the pattern-effect literature is whether the observed SST evolution generates a meaningful radiative response and evolution. Whereas the question addressed here is whether an atmosphere model can reproduce the feedback evolution of *individual* CESM2 historical members. These are not necessarily equivalent.
I am particularly surprised that the manuscript does not engage more directly with Figure 5 of Gregory et al. (2020; https://doi.org/10.1007/s00382-019-04991-y). Specifically, their grey shading and dotted black line (Gregory et al., 20202; Figure 5a) shows the 30-year rolling feedback evolution for individual CMIP5 coupled atmosphere-ocean historical simulations, and does not show any significant variation over time. This is consistent with the results presented in this manuscript. The solid black line however is the feedback variation of the ensemble mean (all models and simulations together). This does have a historical variation. Similar is seen with the orange lines (a large ensemble of MPI simulations). The dotted (individual members) shows very little variation, while solid (ensemble-mean) does exhibit variation.
I think this means we cannot expect to detect a forced variation in feedback from AOGCM historical simulations in any single integration, which is what this manuscript tests. These results suggest that a large ensemble may be required to discern the forced variation from internal variability. Hence the inability to recover feedback evolution in individual members is therefore not, by itself, evidence against feedback evolution. It may simply reflect the expected signal-to-noise problem, and therefore the CESM2 perfect-model test is simply reaffirming what we already suspect.
However, and more crucially, the single realised observed historical SST trajectory on which amip-piForcing is based produces a substantially larger and apparently coherent temporal variation in feedback across AGCM experiments. This variation is much *stronger* than that seen in individual AOGCM simulations. It also appears larger than internally generated atmosphere-only variability, consistent with the broadly similar feedback evolution across the amip-piForcing integrations (Fig 5c of Gregory et al., 2020; https://doi.org/10.1007/s00382-019-04991-y). The manuscript would need to engage much more directly with this existing result and acknowledge that CESM2 feedback variation is not the same as that seen in amip-piForcing with observed SST, before drawing broad conclusions about amip-piForcing experiments.
In summary, I believe the manuscript has demonstrated that feedback evolution is difficult to detect in individual CESM2 historical members, which is to be expected. However, it has not demonstrated that feedback evolution associated with the observed SST trajectory which is larger and more coherent than in AGOCMs is similarly undetectable. Since the latter is the central scientific question motivating the studies being criticised, I do not believe the perfect-model framework presented here is sufficient to support the manuscript's broader conclusions.

Major Comment 2: The manuscript does not reconcile its conclusions with the wider evidence and literature
A central claim of the manuscript is that much of the feedback variability diagnosed from amip-piForcing experiments can be explained by statistical artefacts associated with moving-window regressions and/or missing forcing agents. However, the manuscript does not adequately discuss how this interpretation can be reconciled with several independent lines of evidence supporting SST-pattern-dependent feedback evolution, particularly the result of strongly negative feedback over recent decades.
Examples include:
timeslice experiments in which SST patterns associated with strong- and weak-feedback periods in amip-piForcing reproduce the corresponding feedback states and are not subject to the 30 year rolling window regression criticisms. This is shown for example in Figure 2d of Gregory and Andrews (2016; https://doi.org/10.1002/2016GL068406). Here the horizontal lines are derived from timeslice experiments. In those experiments, an AGCM is forced separately with repeated cycles of SST and sea ice representative of two periods diagnosed as having lower and higher effective climate sensitivity, 1926–1955 and 1979–2008. The resulting feedbacks reproduce the difference inferred from the transient amip-piForcing moving-window analysis, despite using a methodology that is not subject to the same noise or rolling-regression criticism.

Historical feedback variations derived from machine learning approaches and observational records (e.g. Van Loon et al. 2026; https://doi.org/10.1029/2026GL122980), as well as reconstructions from Green’s function approaches (Dong et al. 2019; https://doi.org/10.1175/JCLI-D-18-0843.1)

Historical cloud feedback variations based on observational constraints of cloud radiative properties and cloud controlling factors (Myers et al. 2023; https://doi.org/10.1175/JCLI-D-22-0862.1)

Process understanding linking tropical Pacific SST patterns and deep convection to changes in global feedback variations and satellite observed fluxes (Fueglistaler and Silvers, 2021; https://doi.org/10.1029/2020JD033629);

Collectively, these studies provide independent evidence consistent with physically meaningful historical feedback evolution and SST-pattern sensitivity, using process understanding, observational constraints and methods that do not all share the statistical limitations examined here. Yet the manuscript appears to move repeatedly from identifying shortcomings in a particular approach to questioning a much broader body of evidence. I found this aspect of the manuscript particularly unsatisfactory. If the authors wish to argue that previously diagnosed feedback evolution is largely statistical artefact, they need to explain why these various independent lines of evidence converge on a similar interpretation.

Major Comment 3: The statistical-noise argument does not engage with how the community already addresses this problem
The manuscript's discussion of statistical noise repeatedly focuses on feedback estimates derived from individual simulations using 30-year moving regressions. The ensemble analysis includes an ensemble of different SST and sea-ice values. As far as I can tell, the manuscript does not perform an atmospheric initial-condition ensemble of amip-piForcing-type simulations, meaning simulations with identical SST and sea-ice boundary conditions but different atmospheric initial conditions.
The existing literature is already well aware that such estimates are noisy.
Indeed, previous work cited in this manuscript has explicitly adopted several approaches designed to reduce this problem, including:
multi-model analysis;

atmospheric initial-condition ensembles;

temporal averaging;

and timeslice experiments.

These approaches feature prominently in the literature being questioned by the present manuscript but are largely absent from the analysis presented here.
Moreover, the manuscript does not address how similar temporal evolution in feedback can be generated across multiple models with different atmospheric physics and different initial condition members. This raises an important question that the manuscript does not address: whether the coherence across models and initial-condition members indicates a boundary-condition-forced response that is substantially larger for the observed SST evolution than for the CESM2 trajectories considered here.
Related to this point – the authors might find the following pre-print useful (Cooper et al. in revision for JCLIM; https://eartharxiv.org/repository/view/14002/). I appreciate that this preprint may not have been available when the manuscript was prepared, but it provides relevant complementary analysis of uncertainty arising from atmospheric physics, internally generated variability, and the choice of SST and sea-ice dataset.

Final comment
I found the most interesting result of the paper to be different from its headline conclusion. The comparison between coupled and prescribed-SST simulations appears to highlight a potentially important role for forcing-mediated land responses that are absent from standard amip-piForcing experiments. I do not regard their absence as inherently a deficiency of the amip-piForcing design, which is intended to diagnose the radiative response to prescribed SST and sea-ice variations under fixed forcing. Nevertheless, identifying and quantifying those additional land-mediated responses is a potentially new and important result.
A reworked manuscript focused on missing forcing, land-temperature responses, and the precise quantities estimated by prescribed-SST and coupled experiments could make a valuable contribution. In contrast, I do not believe the analyses presented here convincingly demonstrate that amip-piForcing experiments are generally unable to provide useful information about evolving climate feedbacks.
For these reasons I recommend rejection, but encourage the authors to pursue a substantially reframed manuscript centred on the physically interesting results emerging from this study.

I have signed the review.
Tim Andrews.
Citation: https://doi.org/10.5194/egusphere-2026-4925-RC1
RC2: 'Comment on egusphere-2026-4925', Kyle Armour, 13 Oct 2026

About a decade ago, when I first started thinking about using prescribed SST/SIC simulations to study pattern effects, a student in my group did a similar test by prescribing SST/SIC anomalies taken from a coupled abrupt4xCO2 simulation in an atmosphere-only model and found that the feedbacks were similar to those diagnosed in the coupled simulation. This gave us some confidence that the prescribed SST/SIC approach of AMIP is reasonable to quantify feedbacks in coupled models. But we didn’t think to also evaluate this for the 30-year regression-based feedbacks which have become a common since then. I’m very glad the authors have done so.

This is a well written, interesting, and potentially useful paper. It could be publishable with some substantial revision including new analyses that, I suspect, could change the main conclusions. I have a number of concerns about methodology, comparison to previous literature, and interpretation of results that I’d like to see addressed. I outline them below.

Major:
A) The paper focuses primarily on biases in the use of 30-year regressions to diagnose historical radiative feedbacks. While 30-year regressions are indeed often used to illustrate historical feedback changes, it’s not the most relevant measure of historical pattern effects for the purposes of understanding how the historical effective climate sensitivity differs from ECS because of SST/SIC pattern differences. Indeed, Andrews et al. (2018), Sherwood et al. (2020), Forster et al. (2021), Andrews et al. (2022), Cooper et al. (2026) use either regression on a centennial timescale or anomalies from PI to estimate the historical effective feedback. I would like to see the analysis repeated for feedbacks defined from (i) regression over 1850-2014 and (ii) calculated as lambda = (N-F)/T with anomalies taken as mean over 2000-2014 relative mean over 1850-1900. Does any bias remain after doing so? My guess is that they would be substantially reduced, but that the exclusion of historical forcing could still be a source of bias worth worrying about.
Either way, clear statements should be made about the degree to which the amip-piForcing approach to calculating historical feedbacks and pattern effects over longer timescales is an adequate representation of the true feedbacks. As written, the paper gives the impression that the use of amip-piForcing for calculating feedbacks and linking to SST/SIC patterns fails in general, when in fact it’s probably just fine on longer timescales and with other feedback definitions.

B) Your interpretation is that feedbacks diagnosed from amip-piForcing and amip-histForcing-histSST are biased relative to the feedbacks in the corresponding coupled simulation primarily due to variability in N uncorrelated with the prescribed SST/SIC boundary conditions and thus have little to do with the SST/SIC boundary conditions applied. It seems like an easy way to check this interpretation would be to run multiple amip-piForcing and amip-histForcing-histSST ensemble members (with the same SST/SIC boundary conditions). Figure 3 of Cooper et al. (2026) shows exactly this for AMIP and ERSST datasets, and the results suggest that while there is some variability, the feedbacks in the different ensemble members tend to track each other fairly closely. This suggests that the feedbacks diagnosed from 30-year regression do indeed depend strongly on the SST/SIC boundary conditions, at odds with your interpretation.

C) I think another possible explanation for the results is that regression-based feedbacks will inherently be different between coupled and prescribed-SST/SIC simulations because they are measuring different things. An amip simulation provides a measure of how N responds to SST/SIC changes (and thus feedbacks) because it is uncoupled. However, N in a coupled simulation both responds to SST/SIC changes and drives SST/SIC changes. Thus, when you use regression to estimate feedbacks within a coupled model you need to worry about regression dilution. In Armour et al. (2024) we dealt with this by using Deming regression (a total least squares regression) to estimate 30-year regression feedbacks in coupled models in an attempt to isolate the N response to SST/SIC, which would then be comparable to amip feedbacks. Other approaches aiming to remove the influence of variability are (i) calculating feedbacks as the ratio of d(N-F)/dt and dT/dt (e.g., Tam et al. 2026) or (ii) using a 5-year smoothing of the N and T timeseries prior to calculating the feedback from regression (e.g., Cooper et al. 2026). Do any of these three approaches bring the coupled model feedbacks more in line with the amip feedbacks? You may also find Lin et al. (2025) useful for their analysis of how internal variability can bias feedback estimates.

D) I am having trouble understanding your findings that the 30-year regression feedback within the piControl simulation has little connection to SST/SIC trend patterns. In my own work (Armour et al. 2024) I’ve regressed the feedback over 1981-2014 diagnosed from all ensemble members of the coupled CESM LE onto SST patterns and found a strong correlation with patterns that make physical sense (higher effective climate sensitivity corresponding to larger rates of east Pacific and high latitude warming; see Fig. S2). Moreover, a Green’s Function applied to the SST trend patterns produce feedbacks that are highly correlated with the 30-year regression feedbacks (Deming method) diagnosed across ensemble members. This suggests to me that there is indeed a strong connection between feedbacks and SST/SIC patterns within coupled climate models (supported by other studies as well). I think there is an error in your piControl analysis that may explain the discrepancy. In particular, you plot SST trend maps for periods where the feedback (calculated from 30-year regression) is either increasing or decreasing. But the regression feedback depends on the SST trend within the 30-year period over which it is calculated, so the correct analysis would be to regress the feedback onto 30-year SST trends directly (as I did in Armour et al. 2024), or to composite SST trend patterns for periods of large and small feedback. I believe that either approach will show a strong and physically-meaningful connection between feedbacks and SST trend patterns.

E) I find the analysis and findings that AMIP-style simulations could produce biased estimates of feedbacks due to their lack of radiative forcing pretty compelling. I note that the finding that the feedbacks diagnosed with and without historical radiative forcing can be different from one another has also been found in other work – see Fig. 1 of Dong et al. 2021, comparing feedbacks diagnosed using amip (with forcing) and amip-piForcing (without forcing) simulations and regression over 1979-2014. There, amip produced a slightly less negative feedback than amip-piForcing, which seems consistent with your Fig. 5 over the same period. More importantly, I think more needs to said about the mechanism by which the exclusion of radiative forcing could bias feedback estimates, because it’s not obvious why subtracting an estimate of radiative forcing (from piClim-histall) from amip-histForcing-histSST wouldn’t result in the same radiative response as amip-piForcing-histSST. I suspect the reason has to do with nonlinearities in the land response, e.g., associated with vegetation and/or soil moisture responding nonlinearly to warming and/or CO2. Please clarify the mechanisms.

Minor:
- Page 4, paragraph 3: As noted above, you should be clear that use of 30-year regressions, while common, is not the main way we calculate historical pattern effects for purposes of comparing historical effective climate sensitivity to long-term ECS.
- Table 1: Is the amip-piForcing-piSST simulation 405 years or 410 years (as stated in the text)?
- Appendix A: What value of feedback (lambda) do you assume for the Hansen-style forcing correction?
- Figure 1 and Figure 4: If you plot the sum of anomalies in amip-histSST-piForcing and piClim-histall (for both surface temperature and TOA radiation fields), how closely do you reproduce the coupled model response? This might be worth illustrating as another way of making the comparison.
- Page 26, final paragraph: Of course, the SST trend pattern in observations really has been exceptional (Wills et al. 2022).
- Page 28, paragraph 2: You can of course check how well the Green’s Functions do in estimating the regression-based feedback compared to amip and coupled model simulations. It will have errors, of course, but I think it will capture the overall multidecadal feedback variability.
- Section 4.1 and Section 5: I think these conclusions are far too strong given the concerns I raise above.

References not already cited in the paper:
Cooper, Armour, Gregory, Andrews, Günther and Fan: Historical pattern effects and climate sensitivity revisited with novel constraints on past warming patterns. https://eartharxiv.org/repository/view/14002/
Lin, Cesana, Proistosescu, Zelinka, and Armour (2025) The relative importance of forced and unforced temperature patterns in driving the time variation of low-cloud feedback. Journal of Climate, 38, 513–529, doi: 10.1175/JCLI-D-24-0014.1
Tam, Myers, Zelinka, Proistosescu, Lin, and Marvel (2026) Meteorological drivers of the low-cloud radiative feedback pattern effect and its uncertainty. ACP, 26, 4289–4311, doi: 10.5194/acp-26-4289-2026

Sincerely,
Kyle Armour

Citation: https://doi.org/10.5194/egusphere-2026-4925-RC2

Gergana Gyuleva, Reto Knutti, Robin Noyelle, Urs Beyerle, Erich Fischer, and Sebastian Sippel

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Short summary

How much Earth warms depends on climate feedbacks, which change with sea surface temperatures. This effect is usually modelled by prescribing observed sea-surface temperatures to atmospheric models. We test this standard approach in idealised runs and find it fails to reproduce the true feedback. We also find that feedback variations can be fully explained via statistical noise. Our results suggest historical "pattern effect" estimates are not robust, increasing uncertainty in future warming.


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