This paper considers a class of runtime model corrections that seek to directly improve the climatological annual cycle of selected prognostic variables in an atmospheric climate model by applying cyclostationary corrections to their governing equations. Such state-independent Empirical Runtime Bias Corrections (ERBCs) are distinct from state-dependent runtime model corrections, which aim to improve aspects of the model formulation itself (e.g. Watt-Meyer et al., 2021) rather than the model’s basic state.

The paper focuses specifically on the derivation of the cyclostationary ERBC correction terms. At present, two approaches exist for deriving these bias corrections. The first is the “classical” nudging-based runtime bias correction method, initially derived by Guldberg et al. (2005). The second, referred to as Climatological Adaptive Bias Correction (CABCOR), was introduced more recently by Scinocca and Kharin (2024) and has been shown in direct comparisons to produce significantly larger bias reductions than the classical approach.

The present study investigates whether an iterative application of the classical approach “can lead to a more perfect bias reduction”. In addressing this question, the authors examine the relative roles of in-sample and out-of-sample validation in determining the number of iterations applied to the classical method. The study makes an interesting contribution to the ERBC literature. My recommendation is major revision primarily due to the need to more clearly relate the study’s main conclusions to the existing literature on both the classical and CABCOR approaches. My detailed comments follow.

John Scinocca

Major Points

1. Dependence of results on value of tau in the classical method

In Scinocca and Kharin (2024), hereafter referred to as SK24, a systematic analysis was presented in Section 3 of how the properties of the nudging applied in the N_0 calibration simulations, most notably the value of the nudging timescale tau (and, more generally, the spatial filtering of the nudging tendencies), affect the amount of bias reduction achieved in the corresponding ERBC model simulations (i.e. C_0). The results of that analysis were summarized in Fig. 1 of SK24.  For convenience, I reproduce that figure here. Panel a shows the global bias reduction of the prognostic fields corrected in the nudging runs (R-ADAPT in that study or N_0 here), while panel b shows the associated global bias reduction in the ERBC model simulations (R-ERBC in that study or C_0 here) as a function of tau and the spatial filtering applied to the nudging tendencies. As the present study does not employ spatial filtering, the relevant information corresponds to the information along a horizontal line at the top of this figure (i.e. grid-point nudging).

In the absence of spatial filtering, SK24 identified an optimal nudging timescale of approximately tau = 3d (for the N_0 runs) that yields the smallest biases in the corrected ERBC model simulations (C_0). As discussed in the final three paragraphs of Section 3 of SK24, this optimal value lies between weaker nudging, which limits the effectiveness of the bias correction, and stronger nudging, which introduces artifacts associated with the action of unbalanced motions. There, SK24 argued that, for the classical approach, a separation between balanced and unbalanced dynamics cannot be achieved in principle, and can only be approximated in practice through appropriate choices of the temporal (and spatial) properties of the relaxation applied in the nudging runs.

While the specific optimal value of tau will be model dependent, the analysis presented in SK24 suggests that there may be scope to further reduce biases in the C_0 correction runs of the present study through an adjustment of tau toward longer timescales. Given that the authors explicitly consider both the efficiency of the process (e.g. Section 4.5) and the pursuit of “more perfect” bias reductions, it would therefore seem important to identify the optimal value of tau for their configuration, denoted here as tau_opt. This could likely be accomplished with a relatively small number of additional experiments and would provide a useful baseline against which the performance of the iterative approach could be assessed.

Consideration of the role of tau_opt in this study raises several important questions:

a) Does the reduction of biases achieved using the iterative approach with tau = 1 d exceed the reduction obtained in a C_0 simulation using tau_opt?

b) If tau_opt were used instead of tau=1d within the iterative framework, would the approach still yield a meaningful additional improvement? The authors show in Fig. 1 that the effective strength of the nudging decreases with each successive calibration simulation N_i. Reducing tau from 1 d to the optimal value (approximately 3 d) similarly weakens the nudging. This raises the possibility that the iterative approach may, at least in part, represent a more computationally expensive means of approaching the same optimal bias reduction that could be achieved more directly through an appropriate choice of tau in the classical method.

Clarifying these issues would seem essential for assessing the practical utility of the iterative approach proposed in this study.

2. Out-of-sample validation.

An important aspect of the present study is the evaluation of the effectiveness of the ERBC model outside the period used to derive the bias correction. The authors have used this approach to identify the point at which bias reduction in the out-of-sample period ceases and iterations should be stopped – even if the correction continues to reduce biases when validated during the calibration period (e.g. ll 213-216).

If the whole system were stationary, then this out-of-sample validation approach would be appropriate.  However, the period 1981-2020 contains some of the strongest historical climate-change forcings.  So it is not strictly stationary.  For example, in Krinner et al. (2020), idealized experiments were performed to evaluate the efficacy of the ERBC approach under time-evolving climate change forcings.  There, it was found that the bias reduction achieved through ERBC evolved with time (e.g. Fig. 1 of that study).  In principle, the out-of-sample validation period (2021-2020) has more of a climate-change signal than the calibration period (1981-2000).  It is expected, therefore, that there will be some degradation of the impact of the ERBC during the validation period due to the change in external forcings.  While the climate-change signal might be small during this period, so too is the degradation of the bias reduction induced by the ERBC after 2-3 iterations.

This is a central aspect of the study’s method.  It is not clear how this can be evaluated within the study’s experimental setup.  At minimum, the authors should to include a discussion on this point and place caveats on their conclusions about terminating the number of iterations. 

3. The relationship of ERBC and Model tuning.

The final step in the development of all climate models is a tuning exercise in which all free model physics parameters are assigned values based on a process which generally minimizes model biases during the historical period.  ERBC is typically applied to a finalized model (what is called the free model in this study).  Different finalized models have different inherent annual-cycle climatological biases.  In this sense, the model physics was tuned to operate optimally in the presence of such biases.  When ERBC is applied to the model, say on u, v, and T, it reduces the biases in the finalized model by construction.  In principle, this will push the behaviour of physical parameterizations out of optimal performance.  In practice, the extent to which this is problematic depends on the magnitude of biases in the finalized model, their overlap with specific physical processes of interest, and the ability of the ERBC to significantly reduce those biases.

There is a large cold bias throughout the troposphere of free (finalized) LMDZ model (Fig. 5), which parameterizations such as convection have been tuned to compensate for.  Significantly reducing this bias by applying ERBC to T would throw such compensation out of balance.  It is not unexpected then, that the convection in runs of LMDZ with T correction would alter convective activity in a negative way as indicated by the authors (l 177).   Other models with less of a temperature bias in their finalized models might not suffer the same degradation in convective behaviour and so benefit from T runtime bias correction. 

This model tuning argument would seem to be the better explanation for why some models benefit from T runtime correction while others do not.  The authors, however, attempt to argue that there is a conceptual reason for limiting the runtime bias corrections to the dynamics (winds) to avoid interaction with the model physics.  Model physics formulations are often developed and validated offline against observed inputs of winds, temperature, and specific humidity.  The performance of the parameterizations should not inherently suffer when biases in such inputs are reduced.  They suffer because the values of their free parameters were set in the presence of model biases in the finalized model configuration.  The authors should offer up this explanation in Section 4.1 and where appropriate throughout the paper.

4. Relevance to SK24

ll 280–281. “While tests of the ‘direct compensation’ method recently proposed by Scinocca and Kharin (2024) yielded unsatisfying results with the LMDZ AGCM, …”. This statement is problematic.

The CABCOR method itself is not model dependent, in the same sense that the classical nudging-based method is not model dependent. Irrespective of the model, they both reduce annual-cycle climatological biases by construction.  In SK24, a systematic analysis demonstrated that, for a fixed climate model version, the CABCOR approach consistently produced significantly larger bias reductions in the ERBC model than the classical nudging approach, including in cases where the classical approach employed optimal relaxation parameters. This was shown to be a robust methodological result rather than a model-specific outcome.

Stating that the CABCOR approach yielded “unsatisfying results” relative to the classical approach for a single LMDZ model version, therefore, appears to be at odds with the primary conclusions of SK24. If the authors would like to refute the results of SK24 by asserting that CABCOR performs poorly relative to the classical method, this claim must be substantiated and explained in much greater detail. Alternatively, if the authors do not intend to refute the results of SK24, they should clearly clarify what is meant by “unsatisfying results,” how this assessment was made, and why it is not inconsistent with the findings reported in that study.

Minor Points

1. ll 52-53. For 20 year calibration runs, there will be significant sampling variability in G_0.  Some sort of smoothing of the averaged nudging tendencies is appropriate.  Was any smoothing used?  If so, please specify.  If not, this seems like an important source of sampling artifacts. 
2. l 70. How was tau=1d selected?  It differs from the optimal value identified in SK24 by a factor of 3.  (See Major point 1).
3. l 70. “The free-running corrected simulations”.  This seems to be mixing two model variants.  M is the free running model and C_i is the corrected model, and N_i is the nudged model.   Perhaps just define them as such and stick to those terms throughout.
4. ll 72-73. Why are there 2 out-of-sample corrected model runs but only 1 in-sample corrected model run?
5. l 84. “In particular, the temperature corrections induce a large-scale tropospheric warming”.  This would be better state as, “the temperature corrections significantly reduce a large-scale tropospheric cold bias in the freely running model” (i.e. see Major Point 3.).
6. ll 90-91.  The addition of global integrals of the absolute wind nudging tendencies would be helpful here in gauging the amplitude reduction with iteration number
7. l 94. It would be helpful to look at the contribution to the bias correction after each iteration - not just the total.  This would help identify when the iteration converges.
8. ll 113-120. Because the wind corrections act through the model dynamics, its influence on temperature is expected to arise primarily through dynamical balance (e.g. thermal wind, geostrophic and hydrostatic adjustment), which constrains horizontal temperature gradients rather than the global-mean temperature at a given pressure level. As a result, wind-only bias correction can reasonably be expected to improve regional temperature anomalies about the level-wise global mean while leaving the global-mean temperature largely unaffected, since the latter is controlled by the column-integrated energy balance and model physics. It may be helpful to clarify the physical interpretation of the temperature response in this way to motivate the decomposition of Temperature biases into the level mean and regional anomalies.
9. ll 140-141.  The caption for Table 2 says that the results are reported for 20 years (2001-2020) not 40.
10. The C_0 entry for 30-90°N, MAM in table 2 should also be bold.
11. ll 185-186.  There are two papers this year in Climate Dynamics on this topic:   Scinocca et al. (2025; https://doi.org/10.1007/s00382-025-07814-5) and Labonte et al. (2025; https://doi.org/10.1007/s00382-025-07911-5).  These provide a detailed analysis of the utility of driving regional climate models with the output of runtime bias corrected global model data.
12. ll 186-189 The discussion would benefit from explicitly distinguishing between the target application of the bias-corrected fields. If the corrected model is intended to be used as a standalone global LMDZ model, avoiding temperature bias correction is clearly justified given the strong impact on model physics documented (also see Major comment 3). However, if the purpose is to provide driving fields for regional downscaling, the relevant criterion is the quality of the lateral boundary conditions (u, v, T, q) rather than the internal physical behaviour of the global model. In that context, bias-correcting temperature could be appropriate even if it degrades the standalone AGCM, since an RCM uses different physics and is primarily sensitive to the climatological realism of the boundary fields. Explicitly articulating this distinction would help clarify the scope and implications of the conclusions.
13. ll 231-235. “However, an in-depth analysis of the reasons for regional bias persistence might be necessary in specific regional use cases.” The persistent regional biases in the global model are not really relevant to the dynamical downscaling problem.  Regional responses within RCM domains are not really sensitive to regional GCM behaviour in overlapping locations.  RCM regional responses are determined by their own physics packages and properties of the large-scale boundary driving, not regional GCM performance per se. The experimental design of RCM experiments includes boundary placement far from the region of interest, which deliberately filters out such interior GCM pathologies.
14. l 248.  “as a first simple approach one could try to simply use”  - redundant use of simple/simply.
15. Section 4.6.  Are these runs really that expensive that you could not have used the standard IPSL-CM6A-LR version of the model? The total number of AMIP simulated years is  360 (i.e. 18 20y simulations for this whole study, based on Table 1).  Had you used the standard version, would you have been more successful in the use of bias correction on T, as in Krinner et al. (2020)?
16. ll 276-278.  As discussed in the introduction portion of this review, the type of runtime bias correction investigated in this study is state-independent cyclostationary corrections to the model equations that directly target the correction of seasonal-cycle climatological biases.  The state-dependent machine-learning variants of runtime bias corrections (e.g. Watt-Meyer et al. 2021) directly target model formulation, which have indirect impact (both positive and negative) on climatological biases of the free model. This is an important distinction to make as discussed in the introduction to SK24. 
17. Also, The new approach to deriving the ERBC in SK24 has been described as the “direct compensation” method in this study.  It would be clearer if this study simply used the same terminology as SK24 – i.e. climatological adaptive bias correction method (or CABCOR).

See the PDF file in the Supplement. 

The authors present a study on an iterative bias reduction technique for GCMs based on traditional nudging tendencies. While determining and adding one pass of cyclostationary nudging tendencies can reduce biases ~30-50%, further iterations can further reduce biases, to a point. This new method, and its limits to some extent, are explored here. This study is very interesting and is worthy of publication. My main difficulty with the manuscript is that most of the metrics presented and parts of the methodologies are loosely defined textually, leaving significant room for uncertainty in interpretation. While I only have minor comments provided here, I recommend major revision to make it more clear and reproducible.

Overall comments:

Can you mention why you chose a nudging time scale of one day? This choice does seem reasonable. Still, one might expect qualitatively but not quantitatively similar results for a different choice of time scale, as dycores tend to fight nudging tendencies to different degrees depending on this time scale (e.g. Kruse et al. 2022). If you chose, say, a nudging time scale of half a day, do you think your results on variability (short-term in particular), would still hold?

Just a comment, not a request: It’d be very interesting to repeat this analysis using two different models, or versions of models, that have different mean states (i.e. are “biased” relative to each other). Much stronger statistics could be achieved. Conclusions on the influence of such a method on short-term, smaller-scale variability might be much stronger.

Minor Comments:

Line 52: It’d be very helpful to the readers if you define “cyclostationary time average”. Is G_0 a function of x,y,z,t then?

Line 92-94: There’s no cancellation between G_0, G_1, … ? Not too surprising, this but cannot be inferred from Fig. 1 since the absolute value is presented. If you were to further iterate, would you expect the sum of G_i to converge?

Fig. 2: are you plotting C_3/C_0 (i.e. total model tendencies?) or G_3/G_0? I assume the latter, but this is not completely clear from the text or figure caption. An inline equation would be very helpful.

Fig. 3: Are these biases in the corrected, but free-running runs (i.e. 2001-2020)?

Fig. 4: What is plotted is not stated in the figure caption nor the text! Is this RMSE 2001-2020 time averaged u and v between the bias-corrected, free-running runs  and ERA5? That is, the RMSE of a spatial bias?

Line 117: Do you mean middle and upper troposphere and not middle-atmosphere (i.e. stratosphere/mesosphere)?

Table 2: Again, some quantitative definition of these metrics would be helpful. I think I understand your metrics, but ideally there wouldn’t be uncertainty in interpretation.

Table 3: Again, it’s very difficult to be sure I understand all of the individual variability metrics. Could you please define these metrics (not ratios) quantitatively?

Reference:

Kruse, C. G., Bacmeister, J. T., Zarzycki, C. M., Larson, V. E., & Thayer-Calder, K. (2022). Do nudging tendencies depend on the nudging timescale chosen in atmospheric models? Journal of Advances in Modeling Earth Systems, 14, e2022MS003024. https://doi.org/10.1029/2022MS003024