The paper presents a novel technique that aims to solve an intrinsic challenge existent when unravelling paleo-climate data from deep ice cores. The paper is well-written and structured, the research is original, and the conclusion provides a promising outlook in the journey towards restoring paleo-climate data from the oldest ice. I therefore recommend this paper for publication in The Cryosphere after some minor revisions have been addressed.

My primary concern is that it is unclear to me why the method described in Sec. 3.1 returns the P0(f) relation that can be used in deep ice diffusion estimates. As the authors also write on line 103, P0(f) is the PSD of the isotopic profile before diffusion. They then proceed to use ice core sections with ages more than 10 kyr which at a minimum have been subjected to firn diffusion which must have altered the initial signal. They argue that the time horizon that they assess is unaffected by firn diffusion, but firn diffusion has completed its alteration of a deposited snow layer within 70-200 years (Johnsen et al., 2000). So, I recommend the authors to extend their argumentation to include why firn diffusion is negligible/irrelevant to their P0(f) estimation methodology.

It is also unclear to me whether they calculate the diffusion lengths estimate on a time or depth basis. The text and figures indicate it is on a time domain but the tables and presented values are depth-domain estimates. Moreover, if the authors take a time-domain approach, then there are some complications that they have not covered. For instance, there is an inherent uncertainty associated with an ice core chronology that is not accounted for in time scale estimates. This makes time-domain diffusion length estimates more uncertain that depth-domain estimates. This can be accounted for in the PSD estimation, and I would expect that the Bayesian framework that the authors are adopting are suitable to handle such uncertainties. So, I’d encourage the authors to (1) clarify what types of diffusion length estimates that they are calculating, and (2) consider implementing chronology uncertainties in the methodology in case they estimate time-dependent P0(f) and diffusion length estimates, and (3) that they specify the conversion factors from time-dependent diffusion length estimates to the depth-dependent estimates that are presented in the table.

Finally, I suggest that the authors define what types of diffusion lengths that they refer to throughout the paper. Are they PSD-estimated, firn diffusion lengths, ice-equivalent diffusion lengths, etc.? There are occasionally references to modelling output estimates or other studies (e.g., lines 49-51), and it will help the reader to ensure that the same metric is being used when they compare magnitudes of diffusion.

Minor Comments

• Lines 178 – 189: I’d like an extended and visual assessment of how sensitive the diffusion length outputs are to changes in priors. Given your weakly defined priors, it seems to me that it isn’t that sensitive, but I think it would be valuable to emphasize this further given this is a new methodology. This could be in an appendix. Moreover, it would be valuable with some suggestions or guidelines for prior values to select for different climate regions like East Antarctica, West Antarctica and Greenland.
• Line 177 - please elaborate a bit on what N(0,02,0.07) means in terms of the gamma distribution. For instance, can you specify what N refers to in this case. Is it a normal, uniform or gamma distribution? I could assume the N(0.1, 1) and N(1.5, 1) refers to the shape and scale coefficients of the gamma distribution but please state it explicitly then.
• Line 177, you write that your fit undiffused climate spectra, but are these spectra really undiffused? I agree that they have been subjected to less diffusion that MIS 19 but I think this is something that should be addressed. See main comment.
• Line 190, as this is a seminal paper on a novel approach, then I would like to see the underlying figures in an appendix such that it is clear to the reader how the model converges from the a priori guess towards the underlying distributions. This will be helpful to future users when they are deploying your framework in practice.
• Figure 2. The power-law estimates don’t seem to fit the signal within the grey-shaded area that well (particularly fig. 2c for MIS 9). Why is that? The coefficients’ standard deviations seem small relative to the visual deviation, so perhaps you could update Figure 2 with the confidence intervals from the estimated parameters?
• Figure 5. Missing reference in Figure 5 caption

The authors use a novel approach to estimate the strength of the molecular diffusion in the ice dated back to MIS 19, the oldest interglacial recorded in an Antarctic ice core. The diffusion length is deduced from the analysis of the spectral properties of the isotopic time series with the use of a Bayesian approach. The only, but important, assumption is that the spectral properties of MIS 19 are similar as those of younger isotopic stages (MIS 1, MIS 5 and MIS 9).

The authors conclude that the diffusion length is 31 +/- 5 cm, which is much shorter than in the previous works (40-60 cm). It’s a very good news for the climatologists seeking for the Earth’s oldest ice, and this is one of the reasons why I like this manuscript.

I do not have major comments on the MS, only a couple of minor suggestions:

Line 61 – you have a citation of this dataset in the “Data availability” section, so it’s possible to remove it from here.

Table 1 – the depth range for MIS 1 starts from 7.755 m and the time range from 0 ka, but at the depth of 7.6 m the age a priori cannot be 0 years.

Line 87 – significantly affect?