Trivariate copula to design coastal structures

Orcel, Olivier; Sergent, Philippe; Ropert, François

doi:https://doi.org/10.5194/nhess-21-239-2021

Articles | Volume 21, issue 1

https://doi.org/10.5194/nhess-21-239-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Special issue:

Advances in extreme value analysis and application to natural...

https://doi.org/10.5194/nhess-21-239-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 21, issue 1

Research article

|

25 Jan 2021

Research article |

| 25 Jan 2021

Trivariate copula to design coastal structures

Olivier Orcel, Philippe Sergent, and François Ropert

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Final revised paper (published on 25 Jan 2021)
Preprint (discussion started on 19 Mar 2020)

Interactive discussion

Status: closed

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

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RC1: 'Review of Trivariate copula to design coastal structures', Anonymous Referee #1, 01 Apr 2020
- AC1: 'Revised manuscript', Philippe Sergent, 30 Apr 2020
RC2: 'Review of nhess-2020-80', Anonymous Referee #2, 05 May 2020
- AC2: 'Reply', Philippe Sergent, 16 Jun 2020
- AC3: 'Revised manuscript #2', Philippe Sergent, 16 Jun 2020

Peer-review completion

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

ED: Reconsider after major revisions (further review by editor and referees) (24 Jun 2020) by Thomas Wahl

AR by Philippe Sergent on behalf of the Authors (07 Jul 2020) Manuscript

ED: Referee Nomination & Report Request started (21 Jul 2020) by Thomas Wahl

RR by Anonymous Referee #1 (23 Jul 2020)

Suggestions for revision or reasons for rejection

The revised manuscript is an improvement on the original submission but in my opinion requires more major revisions before it warrants publication.
- Abstract is rather long. Consider removing “We select the copulas with the same tail dependence as data. In the event of an opposite tail dependence structure, we resort to the survival copula.” Also, with reference to L13-16 generally the choice of copula (family) is carried out and therefore described before estimation of the copula parameters.
- The Defra method is mentioned in the abstract but not in the introduction which seems strange.
- The justification for not using other methods such as those applied in Aas et al. (2009) and Gouldby et al. (2014) is poor and contains inaccuracies. For instance, it is stated that “Comparing different construction methods, he [Corbella (2013)] concludes that the Chakak and Koehler (1995) method that is based on bivariate conditional distribution is too complex and not robust enough” in fact Corbella (2013) states that “The Chakak and Koehler (1995) construction technique is the simplest and gave satisfactory model results”. Furthermore, Corbella (2013) did not test the methods in Aas et al. (2009) and Gouldby et al. (2014) and so his guidelines cannot be used to justify why they have not been implemented.
- This discussion does not currently provide a convincing evaluation of the strengths and weaknesses of the copula and conditional extreme models mentioned (see next point for examples of suitable references).
- Although many relevant references have been added since the original submission there remains a lack of referencing (see minor comments) and, in particular, contemporary references (only two of all the papers cited were published in the past 5 years). There are two papers in this special issue alone: Jane et al. (2020) and Tilloy et al (2020) which could be added to both remedy the lack of recently published papers and improve the discussion in the introduction. For instance, in Jane et al. (2020) PCC and the conditional extreme model used in Gouldby et al. (2014) are shown to outperform standard trivariate copulas. Tiloy et al. (2020) illustrated the importance of having a range of bivariate models on the table (including models already mentioned in the manuscript) when attempting to capture interrelations between pairs of hazards.
Minor comments
- L5: Consider replacing “global warming” with “climate change”.
- L17: ‘Clayton normal copula”, not sure this is a copula.
- L43: Definition of “all the pairs” not clear.
- L56: “using also conditional distributions” this model is referred to as a “conditional extreme model” in Tiloy et al. (2020). Consider changing to differentiate the method from the other conditional distribution method mentioned.
- L57: “They show that the fully nested method of creating hierarchical copulas provides the best results for their case study” compared to which other methods.
- L69: Define “isovalue lines”. Later isovalue lines are stated as “contours of equal joint exceedance probability” consider also using this phrase here.
- L93-94: “The independence assumption is not completely valid when two tuples per day are selected but that is an approximation commonly used”. Reference required.
- L111-112: “We then select the copula with the lowest minimum error.” I cannot see where a minimum error is calculated, perhaps it is the “minimum mean error”?
- L225 Typo. “tum”.
- Equation 35. Please define u1 and u2.
- Table 1. Please check Table 1 as it currently shows some discrepancies with Table 1 in Caillault and Guegan (2005).
- Poor grammar. Change “The simplified Defra method refers to univariate survival functions 𝐹̅𝐻 and 𝐹̅ 𝑆 of wave height and storm surge. The reason is that coastal engineers usually work with exceedance probability rather than with non-exceedance probability” to “The simplified Defra method refers to univariate survival functions 𝐹̅𝐻 and 𝐹̅ 𝑆 rather than cumulative distribution functions of wave height and storm surge as coastal engineers usually work with exceedance probability rather than with non-exceedance probability”.
- L276: “The equation (41) is used to determine the table 4.15 of Rock Manual (Ciria et al., 2007).” Not clear, needs a more detailed explanation.
- Table 2. The text states that the “[tail dependence] result will condition the choice of the copula” but then all the copulas are fitted to the data regardless of whether they possess the same type of tail dependence as the sample. Perhaps in Table 2 the copulas with the same types of tail dependence as the sample and therefore candidates to model the dataset could be highlighted.
- Table 4. Results in the table should be discussed e.g. how parameter changes along the coast or moved to appendix.
- L411-420 belong in the introduction.
- L485: Grammar. Change “limit” to “limitations”
References (not included in the manuscript):
Caillault, C., and Guegan, D.: Empirical Estimation of Tail Dependence Using Copulas. Application to Asian Markets. Quantitative Finance, Taylor & Francis (Routledge), 5, 489 – 501, 2005.
Jane, R., Cadavid, L., Obeysekera, J., and Wahl, T.: Multivariate statistical modelling of the drivers of compound flood events in South Florida, Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2020-82, in review, 2020.
Tilloy, A., Malamud, B. D., Winter, H., and Joly-Laugel, A.: Evaluating the efficacy of bivariate extreme modelling approaches for multi-hazard scenarios, Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2020-28, in review, 2020.

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RR by Anonymous Referee #2 (24 Aug 2020)

ED: Reconsider after major revisions (further review by editor and referees) (26 Aug 2020) by Thomas Wahl

AR by Philippe Sergent on behalf of the Authors (29 Oct 2020) Author's response Manuscript

ED: Referee Nomination & Report Request started (02 Nov 2020) by Thomas Wahl

RR by Anonymous Referee #1 (03 Nov 2020)

Suggestions for revision or reasons for rejection

I would like to thank the authors for their responses to my previous comments. Compared with the last version of the manuscript the introduction provides far more context for the work and the description of the methodology is much more detailed. I believe that once all of the minor comments below are addressed the manuscript will be ready for acceptance in NHESS.

General comments

Are the sea level time series detrended to ensure the independent and identically distributed (i.i.d.) assumption is met?

More explanation of the complete Defra method would be desirable on line 302 since it is referred to later on in the text i.e. is the Gauss copula used to estimate the dependence between wave height and storm surge?

The explanation on lines 321-325 is not at all clear. I would also be tempted to restate “In the following section, we use survival copula 𝐶̅ and survival function 𝑢̅. Upper tail dependence and lower tail dependence will be inverted.” (lines 260-261) here too, to remind the reader that since survival probabilities are being assessed lower tail dependence corresponds to high wave heights and water levels.

Be careful to differentiate between the class of tail dependence and the ‘actual’ tail dependence. For instance, the Gumbel survival copula has the same class of tail dependence as the Clayton copula (i.e. only lower tail dependence) but the tail dependence is not necessarily the same since it depends on the value of the copula parameters.

The survival Gumbel copula is also referred to as the Gumbel survival copula. Be consistent!

Equations need to be form part of a sentence. Often this correction can be achieved by moving the full stops to after the equation.

Specific comments

Line 12: Grammar. Perhaps change to “joint distribution functions to their”.
Line 32: I think they are “reliable return period estimates” rather than “reliable return periods”.
Line 34: Grammar. Add “the” before “simplified”.
Lines 46-57: The text on all these lines can form a single paragraph.
Line 50: “Copulas generally aggregate only two random variables.” I suggest stating more explicitly that this is only the case because most of the studies carried out ‘to date’ have considered two variables.
Line 51: Grammar. Remove “a” before “specific”.
Line 57: As stated by the other reviewer the “conditional mixture” model needs additional explanation.
Line 61: “The pair copula construction (PCC)” change to “PCC” as the acronym has already been defined.
Lines 58-70: The text on all these lines can form a single paragraph.
Line 71: Grammar. Change “to a” to “for a”. Also, a bad result is still a result; consider rephrasing the next part of the sentence too.
Line 73: “best results” as compared to which other approaches/models?
Line 74-82: The text on all these lines can form a single paragraph. Also, add an opening sentence summarizing the paragraph e.g. “The paper is divided into three parts.”.
Line 81: Consider changing “our practical applications of coastal engineering” to “our coastal engineering based applications” or similar.
Line 86: Replace “he” with “is denoted by” or similar.
Line 163 &166: Consider changing “two-to-two” to “pairwise”.
Line 245: Replace “it” with “tail dependence”. The next sentence does not make sense; consider removing.
Lines 256 & 260: Replace “it” with “the copuIa”.
Table 1 and Table A1: Typo. Change “Franck” to “Frank”.
Line 265: “Gauss” is also an abbreviation requiring a definition.
Line 266: “deal with” is too colloquial.
Line 306: Add “dependence factor “ before “FD”. As stated by the other reviewer what do the values of 25 and 20 correspond to e.g. Weak dependence? Which corresponds to stronger dependence? Why does Kergadallan (2013) recommend a minimum value of 25?
Line 268-280: The text on all these lines can form a single paragraph.
Line 334: Clarification required here. “as the Gumbel copula has an upper tail dependence, the use of its survival copula is recommended” By recommended do you mean recommended over the Clayton copula? Or recommended as another copula which maybe appropriate given the observed tail dependence?
Line 319: “Since the sample has a tail dependence, it should be known whether it has a lower tail dependence or an upper tail dependence.” Not clear!
Line 341: Clarification required here. “We will select the survival copula with the largest likelihood among those which possess the same class of tail dependence as the sample.” Is this more accurate?
Line 344: Consider ending the sentence after Student and discussing the lower tail dependence of the AMH in the next sentence. The text currently implies that all of the listed copulas only have lower tail dependence when their parameter(s) are close to 1.
Line 346: Remove “correct” before “tail”.
Line 361: “Gumbel survival copula is as large as the log-likelihood of the Clayton copula.” Not true, it is smaller! Consider adding “approximately” or similar before large.
Line 364: “is therefore as suitable as the Clayton copula.” Again, not sure this is accurate as in the next section it is shown that the Clayton is the most suitable. Perhaps say “potentially as suitable”.
Line 386-399: Condense the text on these lines currently there is too much repetition of the line “The points obtained by the XX copula come close to the bisector.”.
Line 399: Grammar. “We therefore reestablish a right tail dependence which gives correct results.” Rephrase and as stated in the general comments it is the appropriate class of tail dependence.
Figure 8: Shorten caption e.g. Joint exceedance probability obtained with a) Clayton copula (0.38) for Saint Malo b) with Clayton copula (0.74) for Le Havre with tide for return periods of 10, 100 and 1000 years.
Line 438: Remove “a” before “lower”.
Line 438: Reorder. “We can therefore conclude that the Clayton copula is the most appropriate copula for our application. For this purpose, the Table 4 gives the parameters of the different sites.” Move these two sentences to the end of the paragraph.
Line 446: Grammar. “There are areas” not clear.
Line 453: The use of the term “parameters” is a little confusing as parameter refers to the parameter of the copula elsewhere in the manuscript. Could it be “bivariate copula with a single”? A bivariate copula by definition models the dependence between two variables.
Line 454: “Unlike Corbella (2013) we introduce two parameters”. More explanation required.
Line 462: “For all three combinations, the Clayton copula still has the largest maximum likelihood value. In addition, we find that for the combination (H, T) the log-likelihood is significantly higher. The parameters (H, T) are therefore the most correlated parameters.” The survival Gumbel copula has a highest maximum likelihood for the (H,T) case. Also, the loglikelihood does not most correlated parameters.
Line 488-500: Shorten and arrange into a single paragraph.
Line 505: “generally good”. Too vague. Do you mean the best of the four fitted trivariate copulas?
Line 508: Rephrase e.g. “As expected, the fit of the single parameter Archimedean copula is …”.
Table 8: Define “KHI-2” and “KS”.
Line 525: “In present practice, “. Please make it clear that joint probabilities are not calculated like this in all cases.
Line 659: “site specific”. Given the next sentence it seems they are more “region specific”?
Line 482: “There exist four families”. There are more than four families of copulas. Perhaps “Four of commonly applied copulas families are the …” Also, the “Archimedeans, Elliptics” do not need to be plural.
Table A1: Remove an erroneous bracket in the first column of the Joe copula row. Also, please double check the formulae in the Frank copula row.

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ED: Publish subject to minor revisions (review by editor) (03 Nov 2020) by Thomas Wahl

AR by Philippe Sergent on behalf of the Authors (13 Nov 2020) Author's response Manuscript

ED: Publish as is (18 Nov 2020) by Thomas Wahl

AR by Philippe Sergent on behalf of the Authors (27 Nov 2020)

Short summary

Coastal structures subjected to the actions of waves must be redesigned due to rising sea levels. Their design requires an estimate of the long return period of wave height, wave period, storm surge and more specifically their joint exceedance probabilities. We confirm that the best results are obtained by first aggregating the most correlated variables: wave height and wave period. Nevertheless, the choice of method of aggregation is much less important than the choice of the copula.