De Gelder et al., present a novel inversion approach to estimate marine terrace formation parameters and sea level histories from marine terrace topographic data. They use a set of test scenarios to demonstrate the feasibility of the approach and apply it to well-studied marine terrace sequences in Santa Cruz and the Gulf of Corinth. The topic and scope of the study are very promising. Using inverse schemes to study marine terrace sequences is a logical next step for the community and will be of interest to many people. This study therefore holds a lot of promise and comes with nice figures, however, the authors omitted a lot of critical detail that is necessary when describing a new method.

Unfortunately, the authors uploaded a preprint without line numbers, therefore, I added some more descriptive text locations.

The authors measure the horizontal misfit with topography and therefore focus on the width of terraces. This is interesting because terrace width is probably an underused terrace parameter that holds information. At the same time, terrace width (and topography I general) is often highly variable along-strike. It would be interesting to include a real case, where uplift rate is roughly constant, and several topographic profiles of the same sequence are jointly inverted. This would show whether this approach is robust enough to cope with along-strike topographic variability.

The authors should include the equation of the forward model to make the methods easier to follow.

A supplementary table summarizing all the paleo-sea level ranges, including a justification, should be added to the supplement.

The methods require significantly more detail. Currently, the authors do not provide an adequate description of the algorithm. For instance, there is no mentioning of the type of sampler used in the MCMC. How many individual random walks are performed per scenario? Currently, only the number of forward simulations is mentioned. What acceptance ratio was aimed for? How was burn-in defined? What about autocorrelation? Effective sample size? ETC. This paper describes an exciting new inverse algorithm for marine terraces, and as such is lacking a lot of critical information. Citing an inverse problem review paper is not sufficient. This lack of detail also makes it harder to assess the presented study. None of the standard diagnostic plots of an MCMC inversion are presented, and it is therefore, hard to assess the performance of the algorithm.

Priors: The authors describe the model parameters, but do not specify how these are treated in the inversion. The authors refer to a prescribed range, and therefore, I assume, use flat priors. But this should be specified. Also, it would help to stick to common terms in Bayesian frameworks, such as priors.

The authors measure the difference between observed and modelled topography in the horizontal axis, putting much emphasis on the width of terraces. How do the presented results change for measuring the vertical misfit? Commonly, terrace studies focus on terrace elevations and not their width.

Why are so many model parameters fixed in the test scenarios? Would be interesting to explore additional scenarios, to see which information is required for a given terrace sequence and when the inversion is not able to recover the parameters. Another important question is, how the inversion performs when the problem gets under-constrained. Does the algorithm converge on a (fake-) solution in a local minimum or does the MCMC roam the parameter space as it should.

The authors do not define how they report inversion results. E.g., for the Santa Cruz inversion, they state the posterior range of parameters but do not explain, whether these are confidence intervals, a standard deviation around the mean, or similar. Also, the authors state that the model limits the uplift rate to 1.35-1.65 mm/yr, but this is the prior range and was therefore set beforehand. The word “limits” could also imply that the authors are in fact referring to the prior range. However, posterior range results are following, such as the range for initial slope. I know I am being nitpicky with this sentence, but here and elsewhere, imprecise language concerning aspects of the inversion creates confusion.

There seems to be a degree of circularity in the approach. Narrow uplift rate prior ranges are defined for the Corinth and Santa Cruz models, based on the elevation of dated marine terraces. These prior uplift rates ranges are then used to reproduce the stair-case morphology and invert for uplift rate, which was already an implicit input. Is this OK because the focus of the study is on the width of terraces and resolving parameters other than uplift? The authors should address this.

For the Corinth case, the posterior distributions of wave base depth in profile 2 & 3 have their maximum values at the boundary of the prior range, suggesting the algorithm would like to go to even deeper wave base depths. The authors, do not mention the results for the IS, ER, WB, UR parameters. These results should also be described and the implications of the posterior distribution ramping up against the prior boundary should be discussed, since this may be a problem.

Currently, the discussion ends with a lengthy paragraph of the Gulf of Corinth case. As a reader, this is a bit weird. Until here, the focus of this paper was the inversion method. However, the long Corinth section hangs at the end like an afternote. To improve flow and readability, I’d suggest to condense this section. The authors present an exciting new tool and there are many things that could be discussed, but currently are not. What about typical lateral variability of coastline morphology and its influence on inversion results? What prior knowledge is typically needed to recover reliable results? What parameter trade-offs typically exist? Etc.

Section 2, second paragraph: There seems to be a typo in the reference (REEF).

This manuscript is interesting, novel and well written, and I like it. It presents a new method that inverts the topography (mainly widths) of marine terrace staircases, simultaneously solving for sea-level history, uplift rate and some other parameters. The method seems to work pretty well on the three test datasets (one synthetic, two real).

I agree with the comments of Anonymous Referee #1 — notably that for a paper focused on methodology there are quite a few details lacking. It’s possible to find some information on inversion parameters in the code in the GitHub repo, but the repo is confusingly set up, with instructions focusing on how to run it on one HPC at one institution. I would recommend including discussion of the inversion parameters in the text and if possible, making the code more user friendly.

I also agree that it would be good to present posterior distributions for inverted parameters: perhaps a table showing prior ranges for uplift rate, wave base height etc., together with the 95.4%-confidence ranges from the posterior distributions. It would be also nice to put some 95% contours on Figure 3e-g.

A paragraph discussing the results of the Corinth inversion in terms of geological plausibility would strengthen the paper… Do the different posterior distributions on initial slope and erosion rate make sense given the local geomorphology of each profile swath? Relative differences in nearby sediment supply might influence erosion rates, for example.

I don’t have much to say about section 6.2, but I agree with Anonymous Referee #1 that it comes as a bit of a surprise (especially given the title) and could be shortened.

Finally, a couple of typos:

1. A colon is missing between “Andersen et al., 2010” and “De Gelder” in the caption of Figure 1.
2. There is a “d” missing in “and” in the 4^th line of page 6.

Anyway, nice paper!