Estimating extreme sea levels using a copula joint probability method

Koh, Zhi Yang; Grandey, Benjamin S.; Dauwels, Justin; Horton, Benjamin P.; Chew, Lock Yue

doi:10.5194/egusphere-2026-2118

Preprints

https://doi.org/10.5194/egusphere-2026-2118

Preprints

22 Apr 2026

| 22 Apr 2026

Estimating extreme sea levels using a copula joint probability method

Zhi Yang Koh, Benjamin S. Grandey, Justin Dauwels, Benjamin P. Horton, and Lock Yue Chew

Abstract. Extreme sea levels pose significant risks to coastal communities and infrastructure. Joint probability methods are widely used to estimate return levels of extreme sea levels by combining tidal and non-tidal components, but most implementations assume independence between tide and surge. This assumption is not always valid, and neglecting correlation can affect the estimation of return levels and their associated uncertainties. Here, we introduce a Copula Joint Probability Method (CJPM) that explicitly accounts for non-linear correlation between peak tides and skew surges, generalising the Skew Surge Joint Probability Method (SSJPM). Using long tide gauge records (≥100 years) from 23 locations, we assess how incorporating correlation affects both central estimates and confidence intervals of estimated return levels. We find that accounting for this correlation can shift estimated return levels by up to approximately 10 cm at some locations. Importantly, uncertainty in the peak tide–skew surge correlation can be a statistically significant contributor to the width of confidence intervals, in some cases exceeding the contribution from uncertainty in the extreme skew surge distribution. At other locations, correlation has a negligible effect, and CJPM and SSJPM estimates are indistinguishable. These results demonstrate that explicitly representing correlation and its uncertainty provides a more complete quantification of return levels and their associated confidence intervals, and helps determine whether correlation materially affects return level estimation. The CJPM provides a flexible framework that can be applied across a wide range of settings without requiring assumptions about the strength or cause of any correlation.

Received: 14 Apr 2026 – Discussion started: 22 Apr 2026

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Zhi Yang Koh, Benjamin S. Grandey, Justin Dauwels, Benjamin P. Horton, and Lock Yue Chew

Status: final response (author comments only)

RC1: 'Comment on egusphere-2026-2118', Joanne Williams, 08 May 2026

Thank-you to the authors for this contribution to developing joint-probability methods. A clearly set out and well written paper, and I particularly appreciate the use of the schematic to explain the method. The background material was presented appropriately and without excessive detail. Throughout the paper almost every point I questioned or was uncertain on whilst reading was addressed within the next page, so I have few comments left and am happy to recommend the paper for publication.

There are a few minor points to address:

On the whole you have been very careful to distinguish between skew surge and surge residual but there are a few places where there is ambiguity. Please check every use of "surge" and ensure that "skew surge" or "surge residual" is specified. Eg line 10, line 83, line 470, line 425, there may be others.

Line 39 - mean sea-level is a bit ambiguous as there are other sea-level components such as seasonal changes and sea-level rise that don't fall under surge residuals either, I think given your later work this is "annual mean sea-level" and anything else is disregarded or bundled into tidal-residuals. Please clarify.

Line 128 - hopefully your code also includes a minimum interval (eg 8 hours) between consecutive low waters or else it will break in places with double Low Water or where shallow water causes a delay around low water.

Line 178 - it is unfortunate that the criteria of being long enough for 4-way self comparison means there's only one location (Key West) in common with W2016 or SA&V2018. It would be good to test your method against the findings in those papers, which would mean using shorter series, have you tried this yet?

Fig 2 - Is the copula diagram on the right a real one or just constructed for the benefit of this schematic? Are the copula diagrams useful diagnostically? Can they be used to identify the source of non-linearity in the TLL. (eg if it occurs at LW it may not be a concern.) I'd like to see some examples (in 2d rather than 3d, though the 3d is useful for the schematic)

Sec 2.4 What is the sensitivity of the method to gaps or errors in the data? What happens if there are missing values at peaks?

Sec 3 Results - I suggest putting in some subheadings

Sec 3 Results, Fig 3. At Brest, you find that the SSJPM is estimating high, and CJPM covers the data, so conclude that tide and surge are not independent (I agree) and that this is not seasonality. I'm less convinced of the second point. Extreme skew surges at Brest are very seasonal, 20-40 cm higher in December, Jan, Feb than September, and much smaller in summer. (This is in model hindcast, I don't have the data handy. But it agrees with Newlyn data too). There is also weak seasonality in tide, with the biggest tides by ~20cm being in September and February. Can you separate Fig 4b by season to see if this is also true in the data?

Line 323 and 364 - If the CJPM does improve performance over the SSJPM but it is not due to seasonality (I'm still not sure of this) then what is the mechanism for the improvement? What dependence is being modelled by the copula?
It would be valuable (I don't insist upon it for this paper, possibly another) to test this on an artificially constructed data set with a known dependence and return period). This could be much longer than 100 yr.

Line 347 "We expect" is this tested?

Fig 4 - another useful diagram thank-you.

Line 454 - Can you quantify this? How much data does the TLL need? Does it set a limit on the minimum useful record length?

I have not validated the code used, but the zenodo link works.

Citation: https://doi.org/10.5194/egusphere-2026-2118-RC1
RC2:
'Comment on egusphere-2026-2118', Karen Palmer, 17 Jun 2026
General comments:
This work presents a copula design to account for potential interactions between tide and surge variables in joint probability estimation of extreme sea levels from long-term tide gauge records. The article is generally well-written and draws upon relevant literature and impressively long tide gauge records to present the key arguments. Overall, I believe the study presents a useful technique and analysis of the potential benefits of the copula approach that complements existing alternative approaches.
Specific comments:
The justification for the approach is to address the potentially false assumption of independence between tide and surge. In lines 60-76, three weaknesses of the early JPM (Pugh & Vassie) are stated. The first and third weaknesses could be considered two aspects of one weakness; possible dependence between tides and surges (autocorrelation and tide-surge interaction). A review on the same topic by Haigh et al. (2010; doi:10.1016/j.coastaleng.2010.04.002) also identified three inadequacies: assuming independence between hourly tide and surge levels, assuming adequate representation of extremes in the tail of the surge distribution, and the inability to compute confidence intervals. This review is not mentioned in the paper but perhaps should be. It seems that the central justification of the CJPM is not to address limitations of the JPM but those of the SSJPM, i.e., specifically tide-surge interactions. To this end, I think the introduction could be re-worded to better explain the importance of this interaction (i.e., expanding on the cited work of Williams et al. 2016) and directly compare the CJPM with alternative approaches which account for seasonality such as the JPMM (Palmer et al., 2024) or D’Arcy et al., (2023).

As the authors clearly acknowledge in the discussion (lines 367-378) there are several drawbacks for applying the copula approach. The first is the need for very long records, which have several disadvantages, including those mentioned (spatial coverage, number in existence, etc). Another disadvantage is the tendency for older records to be less reliable (e.g., Brest datum discontinuity as per Woppelman et al., 2008) which is highlighted in this study by the comparison between results derived from different records from the same location. From this point of view, 23 unique locations is a limited sample, and this number could be mentioned earlier (e.g., line 95). Secondly, I appreciate the authors acknowledging the possible rationale for choosing a more complex model and the potential issues related to non-stationarity in tides and surges. Another limitation that I think should be addressed is related to the increased complexity and agnostic characteristics of the copula approach and the presented application to only two variables. Variability in MSL, including relative MSL associated with vertical land motion, and other factors such as coastal trapped waves and estuary interactions, are significant contributors to extreme sea levels. These would seem to particularly relevant to Brest and Thunder Bay where the differences between CJPM and SSJPM estimates were greatest (lines 337-339).

In line 379 the authors state that “quantifying the differences between the results of the SSJPM and the CJPM may be of scientific interest”. In presenting the CJPM as an alternative approach, it would be useful the authors could provide not only the quantified differences but also a comparison with other techniques which address the potential dependence between tide and skew surge. The estimates of return levels being “in closer visual and numerical agreement” (line 328) and the training errors and justification for using the TLL copula (line 428) are somewhat subjective. Can this evidence be strengthened?

Technical corrections:
The discussion is quite lengthy and contains some very long paragraphs. It would benefit from structuring with sub-headings and editing.

In some places, the statements of performance are broad and require quantification.

Later in the paper, the focus appears to shift to an analysis of potential collider bias. I suggest reframing the abstract and introduction if this is a key objective of the paper.
Citation: https://doi.org/10.5194/egusphere-2026-2118-RC2

Zhi Yang Koh, Benjamin S. Grandey, Justin Dauwels, Benjamin P. Horton, and Lock Yue Chew

Model code and software

Analysis code for "Estimating extreme sea levels using a copula joint probability method" Zhi Yang Koh https://doi.org/10.5281/zenodo.19312731

Zhi Yang Koh, Benjamin S. Grandey, Justin Dauwels, Benjamin P. Horton, and Lock Yue Chew

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Short summary

Extreme sea levels pose a risk to coastal communities. We present a joint probability method that uses a copula to account for non-linear correlation between peak tides and skew surges. Accounting for this correlation can shift central estimates of return levels by up to about 10 cm. We further quantify how uncertainty in this correlation affects confidence intervals, allowing assessment of whether correlation plays a statistically significant role in return level estimation.


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