General comments

This study presents a new technique for extreme value analysis of coastal tide gauge records called the TMAX method. The TMAX method primarily improves upon the AMAX method of Gumbel & Lieblein (1954) by using a subset of the highest total water level values of each tidal day, rather than annual maxima. The author applies the TMAX method to the same 35 UK tide gauge records used by Batstone et al. (2013) for development of the skew surge joint probability method, and results are compared to extreme water level estimates derived from the SSJPM and the AMAX method of Gumbel & Lieblein (1954).

The author describes several valid advantages of this method, including: 1) the method is simpler than the SSJPM because it does not require harmonic analysis or fitting a probability distribution; and 2) using the highest water level of each tidal data, rather than annual maxima, enables more elegant treatment of incomplete time series.

However, I disagree with one of the author’s motivations for developing the TMAX method. They write: “SSJPM is complex, often requiring manual intervention.” It is true that the SSJPM is complex, but papers such as Batstone et al. (2013) and Baranes et al. (2020) clearly demonstrate that application of joint probability methods in regions where tides are large relative to surge (and thus large surge events may not be included in the highest recorded total water levels) provides more precise and stable return level estimates with a narrowed uncertainty range. It would be more appropriate to apply the TMAX method in an area with smaller tides – or, perhaps, to apply the TMAX method to calculating skew surge statistics, and then convolving the TMAX probabilities with the tide probability distribution. In terms of the manual intervention, I interpreted that part of Batstone et al. (2013) as finding that manual intervention with the SSJPM was necessary in two distinct geographic regions (the Severn and James estuaries). To me, this points to a geographically linked phenomenon as a potential challenge (such as nonlinear tide-surge interaction or river influence), rather than the statistical method itself.

I interpret the primary conclusions of this paper as the TMAX method 1) giving “a significantly better internal fit and reduced variance” compared to the AMAX method, and 2) being “at least as accurate as the AMAX and SSJPM methods.” In this paper’s current form, I do not think that these conclusions are sufficiently supported.

• In general, there is little analysis provided of the results. In particular, it is important to discuss how geography, record length, and tidal range affect the results.
• I think “accuracy” is determined by consistency with the SSJPM, yet it seems that for 9 of the 35 locations, TMAX and SSJPM results are significantly different, even at the 20-year return period level. It is difficult to interpret this result without further discussion of factors such as record length and geography.
• It’s a bit unclear how “internal fit” is compared, is it a comparison of the analyses that extend through 2009 to the ones that extend through 2018? If so…
  • On average, the TMAX-derived return water levels increase when the analysis is extended from 2009 to 2018 (Table 4). This is likely due to detrending the entire dataset using a constant assumed rate of RSLR that likely increased between 2009 and 2018. To actually assess something like stability (what I am interpreting “internal fit” to mean), it would make more sense to use something like a moving 365-day window to remove sea level rise and variability, then compare the results over the two time periods. Baranes et al. (2020) showed that application of a joint probability method significantly improved stability compared to fitting a probability distribution to total water level using a Monte Carlo validation method and by comparing statistical fits over two time periods. A similar sort of analysis could be used here to support the author’s conclusion.
  • If I am interpreting “internal fit” correctly, the only comparison of the AMAX and TMAX internal fits are in Table 3, which only show mean values.

Additional comments:

1. Fitting a constant sea level rise rate for the “AMAX to 2009” and “TMAX to 2009” analysis may facilitate comparison with the results in Batstone et al. (2013), but it will yield less robust statistics compared to using something like a moving 365-day mean, as rates of RSLR are variable, and interannual sea level variability may significantly affect the results. This should be discussed in the manuscript.
2. The use of a tidal day, rather than 1 year, is described as a novel approach. However, once the tidal days are reduced to a subset of the largest tidal days, I interpret this method as becoming equivalent to what Batstone et al. (2013) does in applying a peaks-over-threshold approach to skew surge (although this paper uses total water level). In other words, selecting a number of peak tidal day water levels that is equivalent to 5 times the record length is roughly equivalent to using the top 0.7%. This is thus equivalent to the innovation of the peaks-over-threshold approach, and the novelty here is overstated.
3. In the specific comments below, I highlight several parts of the introduction with questionable descriptions of published studies, the methods description is confusing, or the results are inconsistent.

Specific comments

Lines 24-26: While tide gauge-based extreme value analysis is valuable for many reasons, the spatially varying nature flood return levels that you mention (lines 24-25), along with the fact that tide gauges are generally purposefully installed in wave-sheltered locations, make it somewhat rare that they are the only tool used to determine design elevations for coastal defense structures. Perhaps you could modify the text to describe alternative applications, such as for determining boundary conditions and/or validating numerical models used for coastal planning.

Lines 29-31: This definition of HAT could use some clarifying. Do you mean that HAT assumes average conditions for the meteorological component of tidal height (or perhaps of total water level)? This might be clearer than calling the meteorological component of water level “noise.”

Lines 42-43: I’m curious why this is being highlighted as a particular weakness of the JPM when none of the extreme value analysis methods you discuss in the introduction provide flood duration information.

Lines 45-46: This is not quite right. The timing of the actual astronomical tide is shifted compared to the predicted astronomical tide (and the predicted tide is what’s used to calculate the non-tidal residual).

Lines 49-51: It’s not that the “difference” is uncorrelated; it’s that skew surge is uncorrelated with measured high water (see Williams, 2016).

Lines 51-53: The SSJPM fitting a GPD to skew surges is not a reflection of there being fewer skew surge values compared to non-tidal residual values for a time series of the same length. Fitting the GPD (or any extreme value distribution), as opposed to an empirical distribution, has the advantage of providing probabilities for values that exceed the maximum observed value. In fact, the Revised Joint Probability Method (Tawn & Vassie 1989; Tawn, 1992) made this improvement by fitting a GEV to the non-tidal residuals (rather than an empirical distribution).

Lines 69-71: I would recommend defining these terms earlier in the introduction and using one consistent term for measured minus predicted water level. I recommend “residual,” rather than storm surge (because the residual is often not storm surge) or random noise (because there are deterministic components of the residual).

Lines 54-57: See general comments above about the Batstone et al. (2013) manual intervention

Lines 128-134: This paragraph is confusing in a couple of places:

• Lines 129-130: Do you mean the reverse? i.e. that the extreme values are more difficult to determine?
• Lines 81-83 (and Equation 1) show Gumbel ranking in ascending order, but you say “substituting for Gumbel’s descending rank with an ascending rank”

Lines 209-214: Why not fit a spline to the hourly data? Or you could show that it’s not important to do this by comparing high waters over time periods with 15-minute data to time periods with hourly data.

Table 3:

• Is the “mean difference” the mean of the difference between AMAX or TMAX and Batstone et al. (2013) across all 35 stations? Which is subtracted from which? And is the standard deviation the standard deviation of the difference?
• I would suggest showing these results on a map for each individual gauge – especially because Batstone et al. (2013) discusses individual sites where the GPD fit to the skew surge distribution was not physically plausible. Essentially, show the information in Table 5 on a map.
• Why don’t the means and standard deviations in Table 5 match the mean differences in the “TMAX to 2009” row of Table 3?

Figure 4: This is difficult to interpret the way the bins are labeled and without geographic information. It should be shown on a map with exact values reported. The same should be done for AMAX (and AMAX should somehow be compared to TMAX) to support the conclusion that one method provides more stable estimates than the other (see also general comments above).

Results presented in Table 5: There are 9 sites that have differences greater than or equal to 10 cm at the 20-year return period level, compared to the SSJPM (Avonmouth, Dover, Hinkley Point, Immingham, Newlyn, North Shields, Port Ellen, Tobermory, Workington). This seems like a relatively large difference, but it is difficult to interpret without information on geography, record length, or tidal range

Conclusions: I interpret the primary conclusions as 1) the TMAX method giving “a significantly better internal fit and reduced variance” compared to the AMAX method, and 2) the TMAX method being “at least as accurate as the MAX and SSJPM methods.”  These conclusions should be described in an expanded and quantitative discussion section that points to the specific results and/or analyses that support the conclusions. Topics such as how geography, tidal range, and record length impact the results should be discussed.

Technical corrections

Lines 48-49: Perhaps revise to “… difference between the maxima of measured and predicted water level for each tidal cycle…”

Batstone et al. (2013) is sometimes referred to as “Batstone” and sometimes referred to as “Batstone 2013.”

The equations used in the TMAX method should be more clearly and concisely stated.

Figure 5: I recommend not using red and green for colorblindness

This paper is another contribution based on the application of extreme value theory (EVT) to se level extremes. EVT is a wide field and in recent times since the early 21st century has been widely used in environmental studies , especially for extreme events and any modifications have been suggested thereof. Similarly this paper touches on a specific aspect of the EVT and suggests some new developments to the Gumbel type I AMAX approach and looks at the highest tide each tidal day and verifies the approach with comparison against  SSJPM.

The methodology and the results are sound. However a consideration of the following points may benefit the manuscript:

• The Most important point would be regarding a more detailed treatment of the comparison of the TMAX method with the r-largest method , which arises as a limiting distribution to the GEV distribution , see Coles (2021). The r-largest method is well established, theoretically sound, and implemented in R packages with maximum likelihood estimation. Many studies implementing the r -largest method usually stick to top 3/top 5 annual maxima sea level.  Within the TMAX method as well the author infers that n=5 leads to minimum variance. So More detail is required on how the TMAX is an added value over the r-largest method. Following this the results derived from comparison with the SSJPM is definitely meaningful.
• Following the above point , the introduction could be improved to include more about the r-largest method and how the proposed TMAX method is an added value.