|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