17 Mar 2021
17 Mar 2021
Correlation of wind waves and sea level variations on the coast of the seasonally icecovered Gulf of Finland
 Finnish Meteorological Institute, P.O. Box 503, FI00101, Helsinki, Finland
 Finnish Meteorological Institute, P.O. Box 503, FI00101, Helsinki, Finland
Abstract. Both sea level variations and windgenerated waves affect coastal flooding risks. The correlation of these two phenomena complicates the estimates of their joint effect on the exceedance levels for the continuous water mass. In the northern Baltic Sea the seasonal occurrence of sea ice further influences the situation. We analysed this correlation with 28 years (1992–2019) of sea level data, and four years (2016–2019) of wave buoy measurements from a coastal location outside the City of Helsinki, Gulf of Finland. The wave observations were complemented by 28 years of simulations with a parametric wave model. The sea levels and waves at this location show strongest positive correlation (τ = 0.5) for southwesterly winds, while for northeasterly winds the correlation is negative (−0.3). The results were qualitatively similar when only the open water period was considered, or when the ice season was included either with zero wave heights or hypothetical noice wave heights. We calculated the observed probability distribution of the sum of the sea level and the highest individual wave crest from the simultaneous time series. Compared to this, a probability distribution of the sum calculated by assuming that the two variables are independent underestimates the total water levels corresponding to one hour per 100 years by 0.1–1.2 m. We tested three Archimedean copulas, of which the Gumbel copula best accounted for the mutual dependence between the two variables.
Milla M. Johansson et al.
Status: open (until 28 Apr 2021)

RC1: 'Comment on nhess202155', Anonymous Referee #1, 01 Apr 2021
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This manuscript uses observations of sea level, wind and waves to infer their relationships in the Gulf of Finland. The authors compute rank correlations and bivariate distributions (with and without dependence) between sea level and significant wave heights.
I have some concerns regarding the methods applied and the interpretation of the results, that I outline below, with some suggestions for improvement. Some of these suggestions may require big changes in the manuscript, though. Some moderate to minor comments follow.
First, when fitting the marginal distributions, in section 2.8, exponential functions are used for sea level and wave height. The exponential function has one parameter to fit, so I assume that the 2 parameters here refer to the two marginal CCDFs (one parameter each). I am not convinced that exponential functions are the most appropriate for extreme value analyses, which are the values the authors intend to extrapolate. Statistically, they may indicate better performance when using tests as AIC or BIC simply because the number of parameters to be fitted is smaller (one vs 2 or 3 in GEV, etc). But this does not mean that the fitting is better. In fact, extreme distributions do converge to these families of distributions (provided they fulfil the hypotheses). This choice is based on the results by Leijala (2018) which are in turn based on Särkkä (2017). The only reason provided there to use the exponential function was the number of parameters, but I do not think this is justified enough. Also, to me, one major issue here is why are the authors fitting the entire dataset to a distribution if the focus is on extremes. I think it would be better to select only those values considered as extremes in either one or another variable. That would make the rest of the computations easier too.
In the results on the correlations (section 3.1) I disagree with the statement that Hs and sea level are correlated. The correlation is not apparent from figure 3, since values of Hs exceeding 1 m may occur with nearly any value of sea level. The analysis provided later looking at correlations as a function of wind direction is more meaningful.
Actually, Figure 5 is very illustrative. However, I do not understand the results. When wind blows from the SW then there exists correlation between Hs and sea level at the buoy in Suomenlinna; but this buoy has some islets in its SW so it should be on their shadow. Am I missing something? or perhaps I am misinterpreting the angles. In any case it would be useful to specify the convention used for the directions in the text.
Still in section 3.1, I do not think that computing the correlations between spectra is the way to calculate how the correlation changes for different timescales in the time series. Crossspectra provides that information partly. The alternative is to filter the original records using bandpass filters of the desired frequencies and then correlate each of them. Also, when correlations are stated, it is necessary to provide confidence intervals. I very much doubt that 0.2 is a significant correlation, for example. I am not sure that this part of the section is meaningful, at least not in its present form.
In section 3.2, when using all hourly values within the time series, any of the copulas seem to fit the observations satisfactorily, as shown in figure 7. Given the variability of the correlations with wind (and thus wave) direction, it would make sense to restrict the analyses to those periods of time when the two variables do show a coherent behaviour. This would surely improve the estimates.
In summary, I think the authors are working with ta data set for which it is worth exploring the dependences. In my opinion this should be done differently though. First, the authors should consider using only extreme for both variables. This reduces the number of data but the relationship is clearer. Second, extreme should be fitted with a suitable distribution for extremes. Third, events of either sea level or waves caused by different wind directions are very likely to belong to different families of distribution, since they probably arise from different atmospheric perturbations (e.g. they travel in distinct directions). This implies that the data should be treated and analysed separately. Copula functions should be also fitted for every subset in terms of direction and using the corresponding rank correlation. There are statistical tests to select the best copula fit, in case they show similar performance when compared to the observations. The comparison to the independent case is useful but cannot be taken as realistic if the Kendall correlation is high. Finally, I would suggest to remove the correlations of the spectra.
Other comments:
Page 2, lines 1719: there are many others: Wahl et al (2015) (https://doi.org/10.1038/NCLIMATE2736) for rain and storm surges; Arns et al (2017) (https://doi.org/10.1038/srep40171) for surges and waves; Marcos et al (2019) (10.1029/2019GL082599) for surges and waves too but globally. And references therein...
 equation 1 in page 7: this is instantaneous water level at a water depth of around 20 m, where waves are measured/modelled. Sea level maxima are generally defined over periods longer than just a few seconds, so the exact meaning of z must be clearly specified to avoid misinterpretations.
p. 8, l. 20: please, provide the full name of the library used
p.8, l. 22: units of the frequency are missing (I guess h^1).
p.10, l.34: normalization will not impact the Pearson correlations
p.10, l.5: weaker>weaken?
p. 15, l.89: the bias of the model in extreme waves is not discussed enough. This is an important shortcoming of this work
p. 17,l.33: the observed sea levels correlate> actually, this is true only for particular prevailing wind conditions.
p.18, l. 23: “including hypothetical noice wave heights during the ice season did not markedly alter the correlation,”. This is the expectation, right? why would this change if the relationship wind and waves remain the same?

RC2: 'Comment on nhess202155', Anonymous Referee #2, 12 Apr 2021
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This paper assesses the role of sea level and wind waves in generating total water level events in the Gulf of Finland, which is covered by ice during certain times of the year, making it a more challenging but also very interesting analysis. I think the content of the manuscript is novel and deserves publication with NHESS. While most of the analysis is technically sound and wellpresented there are some aspects which I think require a bit more work before the paper can be recommended for publication. I summarize these below split into one major general comment and several (mostly minor) specific comments.
General comment:
I find the copula analysis part of the paper pretty weak. The authors decide to only use three copulas, without providing convincing arguments for that selection other than pointing to previous literature. In the past especially people using Matlab ended up focusing on the Archimedean copulas as those were implemented and easy to use. However, the development of the MvCAT copula toolbox by Sadegh et al. (2017) has made it much easier to draw from a larger set of copulas. In R there are also many more copulas readily available to use. More important than using a larger set of copulas would be to show whether or not the copulas that are used at the moment are actually capable of capturing the dependence structure of the observations. There are many different goodnessoffit tests available. Without any such tests blindly applying a random set of copulas does not provide relevant insight. I strongly encourage the authors to invest a little bit more time in strengthening this part of the analysis, as I find it to be an important component (otherwise it could just be left out but that puts a hole into the analysis).
Specific comments
P1, l6 make clear in the abstract that Hs is used to represent waves
P1, l11 the one hour per 100 years sounds a bit strange; I understand the reasoning but isn’t it just a 100year event in the end?
P1, l18 It would be good to already mention here when defining total water levels that tides are negligible, not everyone will know that and wonder about the definition
P1, l21 “in many locations”
P2, l25 why not say “decimeters”?
P3, l29 consider replacing “pathed” by “inferred”
Fig. 1 mention in caption what the contours are
P4, l5 is the trend also statistically insignificant?
Sect. 2.2 mention somewhere early on why temperature is needed (it’s not a typical variable one would use in most areas; recalling that it is used to determine icefree periods would be helpful)
P6, l2 not sure if “long waves” is a good term here as it is basically reserved for long period waves (not travel distance)
P7, l16 I had a hard time following this definition, if Hs is the average of the 33% highest waves how is the highest single wave lower than that?
Eq. 3: using theta which is special to the Archimedean (and one parameter) copulas makes that an alteration of Sklar’s theorem
P8, l19 first spell out and then introduce the abbreviation CCDFs
Fig. 3 when describing the results it would be helpful to recall that sim I has wave heights set to zero
P10, l5 “sea level variations are weaker”
Fig. 5 caption: the values for tau from the entire data are not in table 3 (the one for observations of 0.2 is included but just not rounded, but the 0.16 is not)
Fig. 7 & 8: make clear that it yaxis shows exceedance probability per hour (or translate values to exceedance probability per year)
Fig. 8 It would be good to show the convolution results from figure 7c as well for direct comparison (only the one for 1992 to 2019); also see general comment above about testing which (or any) of the copulas are actually a good choice
P15, l8 “is the highest”
P16, l6 “quite” is a better term then “peaceful”
P16, l68 that sentence wasn’t clear to me please explain better what it’s supposed to tell the reader and why it’s relevant
P17, l2225 In the discussion further up the authors correctly point to the fact that the observed data is way too short to infer information for longer return periods (I would have commented on that if I hadn’t seen that remark); why not focus (or at least add) results for a more reasonable return period, such as 10 years or so? At least we know the results would be more robust.
Milla M. Johansson et al.
Milla M. Johansson et al.
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