The fractional energy balance equation  for climate projections through 2100

Procyk, Roman; Lovejoy, Shaun; Hébert, Raphael

doi:https://doi.org/10.5194/esd-13-81-2022

Articles | Volume 13, issue 1

https://doi.org/10.5194/esd-13-81-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/esd-13-81-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 13, issue 1

Research article

|

19 Jan 2022

Research article |

| 19 Jan 2022

The fractional energy balance equation for climate projections through 2100

Roman Procyk, Shaun Lovejoy, and Raphael Hébert

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Final revised paper (published on 19 Jan 2022)
Preprint (discussion started on 04 Aug 2020)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Review of "The fractional energy balance equation for climate projections" by Roman Procyk et al', Anonymous Referee #1, 11 Sep 2020
- AC1: 'Reply on RC1', Roman Procyk, 18 Mar 2021
RC2: 'Review for esd-2020-48', Anonymous Referee #2, 19 Jan 2021
- AC2: 'Reply on RC2', Roman Procyk, 18 Mar 2021
RC3: 'Interactive comment on “The Fractional Energy Balance Equation for Climate projections through 2100” by Roman Procyk et al.', Anonymous Referee #3, 20 Jan 2021
- AC3: 'Reply on RC3', Roman Procyk, 18 Mar 2021

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

ED: Reconsider after major revisions (07 Apr 2021) by Valerio Lucarini

AR by Roman Procyk on behalf of the Authors (21 May 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (24 Sep 2021) by Valerio Lucarini

RR by Anonymous Referee #3 (08 Oct 2021)

Suggestions for revision or reasons for rejection

Summary of the paper:

The authors apply a fractional energy balance equation to model temperature variations. They formulate the model in a Bayesian framework. This approach is used to perform temperature projections, and estimate the equilibrium climate sensitivity and the transient climate response. These are compared with the CMIP5 and CMIP6 projections.

The paper has seen substantial improvement over the previous iteration, the authors have better contextualized their work and provided a more detailed description of their statistical methodology. Nevertheless, I would still recommend major revisions be made before the manuscript could be acceptable for publication.

I have the following major comments:

- This paper addresses the same problems as Hebert et al. (2021), but uses the FEBE model instead of a truncated power law. I would like to see some more comments on the strengths and weaknesses between these two models, and some discussion on what this paper adds to Hébert et al. (2021) to justify this work as a standalone paper.

- In line 298 the paper argues that the fractional Gaussian noise approximation of the residuals allows the model to take into account the strong power law correlations, but it’s accuracy is weaker on low frequencies which only weakly influence the likelihood function. I would like some clarification on what motivates the FEBE (on the applications considered in this study) instead of simply using a power law.

- There is currently very little description on the limitations of the FEBE. The manuscript would benefit from further discussion on where the model is suitable and where it is not.

- In the estimation of the model parameters the forced response is subtracted from the data before the residuals are fitted using Mathematica. In my understanding this is done by first sampling parameters from the prior distribution, then by removing the corresponding forced response (based on the simulated parameters) before fitting a fractional Gaussian noise process to the data. However, since the removed forced response also depends on the same parameters which are to be estimated, this component is not fitted to the data before it is removed and hence its shape is determined entirely by the priors. This could make the model more sensitive to the choice of priors.

In my opinion, it would be better if both the forced response and the residuals were to be fitted simultaneously. This can be achieved by e.g. a hierarchical Bayesian modeling approach. This framework is also able to incorporate non-Gaussian priors, which could possibly remove the need for approximating the joint posterior.

- I would like to see a comment added to the text which ensures that the Gaussian approximation of the joint posterior (Eq. (21)) is indeed accurate.

- Gaussian priors imply a non-zero probability of negative values, which could cause e.g. scaling parameters to be negative. I would like a comment that addresses if/how the authors have constrained the model parameters.

- In line 467 the authors state that they have performed 500 Monte Carlo simulations of the projections. Has it been verified that the accuracy is sufficient? A comment clarifying this would be welcome. Furthermore, would it be computationally feasible to increase this number, if needed?

Other than this I have some minor/technical comments and suggestions:

- In the Bayesian framework one should use “credible intervals” instead of “confidence intervals”.

- Appropriate punctuation after equations:
(3), (4), (11), (20), (22)

- Figure 13: Caption states that CMIP5/6 MME is represented by black, but in the figure the color is gray.

- Line 693: “Latter” is used when the preceding sentence only has one object

- Line 712: “Projections through to 2100”

- Line 726: “The FEBE could be also” to “The FEBE could also be”

- Line 731: citation should be parenthetical

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RR by Anonymous Referee #4 (14 Oct 2021)

Suggestions for revision or reasons for rejection

The authors described a recently proposed fractional energy balance model to investigate temperature variations and to perform projections through 2100. These are then discussed in comparison with CMIP simulations for two different scenarios.
I think the paper is well written and it matches ESD topics. I find the manuscript worthy of publication after some minor refinements.

1. I would suggest to take care of some typos, to close some parentheses, correct figure captions when describing lines or symbols, and to fix some issue as capital letters without any punctuation before. This especially occurs after equations.

2. Line 238: “We consider the standard assumption about internal variability that it is forced by a Gaussian “delta correlated” white noise”. Could the authors add a reference to this? Why not to use a red noise spectrum that is also generally related to the internal noise?

3. Figure 10: I would like to suggest the authors to comment more on this interesting result. In particular:
what is the range of scales over which the scaling exponent is evaluated?
if, as expected, in the macroweather regime temperature fluctuations decrease, I was wondering why the authors do not use the full range of scales belonging to this regime?
If using the (full) accessible macroweather regime to compute h, how to explain the difference with observations? I would expect h~0 for observations.

4. Figure 12: it seems to me that FEBE lowered projection uncertainties but also directly projections. How the authors explain this? Could be due to the parametric uncertainty that differs from structural one of MME?

5. Figure 13: how the authors explain the oscillations observed for the RCP 2.6/SSP 1-26 scenario for FEBE? It seems to me that they are 10-yr period, is there any relation with the transition for scaling laws?

6. Figure 13: it seems to me that there is a good agreement with RCP 8.5/SSP 5-85 scenario for FEBE and MME (although shifted, why?), while a different slope of the probability is found (steeper for FEBE than MME). Could the authors argument on this? Is it related to the intrinsic parametric uncertainty?

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ED: Publish subject to minor revisions (review by editor) (24 Oct 2021) by Valerio Lucarini

AR by Roman Procyk on behalf of the Authors (10 Nov 2021) Author's response Author's tracked changes Manuscript

ED: Publish as is (11 Nov 2021) by Valerio Lucarini

AR by Roman Procyk on behalf of the Authors (18 Nov 2021) Manuscript

Short summary

This paper presents a new class of energy balance model that accounts for the long memory within the Earth's energy storage. The model is calibrated on instrumental temperature records and the historical energy budget of the Earth using an error model predicted by the model itself. Our equilibrium climate sensitivity and future temperature projection estimates are consistent with those estimated by complex climate models.