1. Summary

In this paper, Lutringer et al., 2022 suggested a new method to estimate erosion on the landward slope of an earthen dike submitted to wave overtopping in considering uncertainties. This approach considers the potential compound effects from waves and sea-level height in using the copula method and links it to the wave overtopping and erosion. To calculate the related uncertainties, a sensitivity analysis based on the Sobol index is done which also permits to assess the importance of the drivers carrying uncertainties. This is a case study located on the site: Salin-de-Giraud, Camargue (France).

2. General comment

I find this paper very interesting. The proposed method to assess such processes in considering uncertainties and highlighting the importance of each physical driver is innovative and relevant. However, there are some concerning aspects that the authors should address before this paper can be considered for publication in NHESS. I will list them here, together with some minor/technical corrections. 

3. Specific comment

• Article structure

I find it confusing to follow the general structure of the paper. Both the data, methods and results are overlapping and should be clearly separated. To me, a discussion section is missing where limitations within this study could be presented.

• Introduction

When expressing some aspects of Climate Change and looking into future scenarios, it is important to express the related uncertainty the scientific community addresses as expressed in the IPCC (lines 16-18 for example, how likely is this “increase in storm intensity” in this region?). Some text clarification is needed (line 45 for example: what is an “ “and” return period ”).

• Data

It is not clearly expressed which type of data we are referring to, is-it reanalysis data? Modelled data? In-situ observations data? Also, some references about it would be appreciated if possible. Also, the water level data is said to be taken “quite far off the actual place”, it is hard both to understand how far it is (maybe adding it on the map of Figure 2 can help) but also and mainly, a justification on the possibility to use such data far from the site location is needed (literature, sea-level analysis, …).

• Bivariate Joint Distribution

It is not clearly written how the “extremes” are identified, it looks like some Peaks over threshold method has been used but it is hard to know, and, if so, we do not know how such threshold value has been chosen. Also, 2 sets of coupled data (Vi, Wi) should be identified if either the peak values Vi is chosen on the sea-level dataset OR the wave dataset, clarification is needed. Concerning the GEV fit, it is hard to know which block length is chosen to apply such distribution and literature would be appreciated. It seems that no correlation-test is made on the 2 sets before applying the copula method as commonly proposed in the literature. Also, recent literature highlighted the need of testing different copula as it is often case dependent. Here, the choice of the copula approach (type of scenario) is not explained and the choice of the Gumbel copula is motivated by “Brummer et al. (2016)” (line 114) which studies a copula approach on flood peaks and river discharge in Switzerland, far from being a coastal region where dynamics can be quite different. Moreover, this paper does not conclude on which copula is best to use in general but stipulates that it is case specific and investigates the choice of the approach which “has a significant influence on the outcome of the design variable quantiles”.  

4. Technical correction

I would specify that this analysis is based on a specific site in the title.

Figure 1 should be part of the section 2, not of the introduction. Also, the figures are not always well presented. For example, the colorbar in Fig. 1 does not present any units. It would be interesting and relevant, I think, to highlight the location of the site and the sea-level tide gauge on figure 2. Using “a” and “b” instead of “bottom” or “top” would, I think, help the reader.

The manuscript needs in-depth proofreading, some sentence constructs are hard to follow, and there are few typos throughout the paper (I provide some examples below, but the list is not exhaustive).

Title: “wave overtoppings” should be “wave overtopping”

Line 20: it should be “quite” not “quiet”

Line 77: To clarify the text, I would replace “(see Figure 2)” by “(Figure 2)” or even “(fig. 2)”

Line 81: I think the sentence could be simpler: “The water level, noted N, is then extracted…” could be “The water level (N) is extracted…”

Line 94: “part 3.1” should be “part2.1”

Line 352: I think there is a “0” missing in the number of points proposed and those sentences are hard to understand as it is not clearly written that we are talking about computational time (if I understood correctly the goal of this sentence).

Legend Fig. 10: it should be “at”, not “a” and there is a “the” missing before “landward slope”

Summary and general comments:

This paper suggests a new method that combines different equations and techniques to provide the return period of erosional damage on a set dyke. The method takes into account the uncertainty of the input parameters, while the global sensitivity identifies those parameters that by varying the most, modify the modelling outputs.  

The paper is interesting and fairly easy to follow. However, it is not very clear the research gap that the authors aim to fill.  The literature review does not clearly reveal why this new method is necessary now and why it has not been attempted before. The uncertainty analysis requires some computational power and the advantages to use it compared to the field reports should be clearly stated, given that both methods provide similar results. Furthermore, the comparison between the field reports and the modelled results is not clear, mostly because the authors do not describe how the return period from the field reports (around 2 years) is calculated. I would recommend to emphasise the relevance of the method suggested, discuss more in detail the results and provide some examples of how the method and/or results could be useful for practitioners. I would thus invite the authors to address the following suggestions.  

Specific comments:

Research gap not clear: the introduction presents a literature review that is mostly a list of authors that used the various techniques introduced in the paper. I would recommend to clearly state why combining the different techniques has not been attempted before and what advantage it could bring (e.g. prediction of future conditions given climatic changes; having estimates of different return periods avoiding expensive field work etc.).

The paper is not well structured. The manuscript starts with the data description but it is fractioned into various sub-section (e.g. from sections 2.2 onwards) which also include methodological steps. The entire section becomes unclear. I recommend to delete all the sub-indices and describe the data first and then the method in another section (e.g. Peak selection). This will also prevent mentioning data that are described in the later sections (see line 94). I would also suggest to add a figure that represents the methodological framework in all its stages, given that it is the core of this paper. In the same figure, the innovative steps and potential outputs should be highlighted. Once the method has been described, the results should be reported in a separate section (now all incorporated in the section 5: ‘sensitivity analysis’) together with their critical discussion (which are now mostly in the conclusion section). Also, what are the limitation of this study?

Data sources: After explaining the combined copula + terminal velocity method, the authors mention that 2m/s is a critical velocity for a dyke like the one observed. This value seems quite crucial, but it is not clear how it was obtained, together with its return period (from the formulas just described?), weakening the observations made after. The calculation becomes then clearer in the conclusions, where the authors mention that the return period is calculated using some average values – does it mean average values of the parameters reported in Table 2? The authors here need to be more specific, also highlighting if this averaging method is the one usually employed in practice.  

Data uncertainty: the authors report in Table 2 the parameters that will be subjected to the uncertainty analysis. I guess these parameters are empirically chosen. Given that no other global sensitivity analysis (GSA) has been reported in the literature review, I would expect that this is the first GSA applied in this context. The choice of the parameters under analysis needs therefore to be justified, if not by previous studies, by an expert opinion.  I also assume that the probability distributions used for the Monte Carlo sampling are all uniform – this aspect should be specified, given that different distributions can lead to very different modelling output (and thus GSA results).

The discussion should be improved: for example, inserting the limitations of this work and the comparison of the results with previous studies (if present). The difference between the results associated with the 1^st Sobol and the total Sobol indices should be better discussed, potentially with some physical interpretation. In the section where the return period distribution is presented, more should be said on its potential use. The sentence: “it appears that bad design of a dyke, resulting in a small return period, can be more easily reached than a good one” is overly simplistic given that an infrastructure is designed for a given return period, generally including a cost-benefit analysis. I would suggest to better contextualise the discussion, also including potential recommendation for the design practice that considers the GSA performed. Also the authors mention in the abstract that the method suggested could improve over time but they do not discuss this point and how it could get achieved (e.g. by varying only the important parameters found in the GSA in the next application)

The grammar and readability need to be improved throughout the text.

Technical corrections

The abstract should be clear before reading the rest of the text. These are the questions I noted when reading it for the first time: “The mean of the return periods is slightly higher with an average return period of six years but the peak of the distribution is located around the two years mark” – this sentence at this point of the text is a bit unclear also because up to this point you have not mentioned that you are going to provide a distribution of return periods. The phrase “suggest that using more robust methods at their corresponding steps could improve the reliability of the framework” is also a bit unclear. “Their corresponding steps” means where the empirical parameters appear in the methods used in this paper? To re-phrase it

Figure 1 does not have unit of measure in the side bar. It would be good to associate the figure of the bathymetry with a picture of the dyke (currently reported in the supplementary material)

L 89: is dataset A, a general dataset? Or the dataset of H or N? I would be more specific when it is possible in order to understand what V and W represent and make the methodological steps easier to read. Same holds for L 116 (u and v, e.g. θ representing the relation between wave height and water level)

Figure 3: in the description the top and bottom should be exchanged (assuming that H is the wave height as previously mentioned)

Figure 5: the label of the x axis needs to be adjusted

L 134, L 138, L 164, L 229, L 297 (and potentially others): delete parenthesis

L 190 and Figure 8: the unit of measurement of the velocity needs to be specified

L 327: re-phare the sentence because by its own, it does not have a clear meaning

Figure 10: the meaning of shape, scale and loc reported in the legend should be briefly specified in the caption for the readers not familiar with the terminology.

Generally, the authors should check for typos and missing capital letters throughout the text.