Reassessment of long-period constituents for tidal predictions along the German North Sea coast and its tidally inﬂuenced rivers

. The Harmonic Representation of Inequalities is a method (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) procedure (cid:58) for tidal analysis and prediction (cid:58)(cid:58)(cid:58) that (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) combines (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) aspects (cid:58)(cid:58) of (cid:58)(cid:58)(cid:58) the (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) non-harmonic (cid:58)(cid:58)(cid:58) and (cid:58)(cid:58)(cid:58) the (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) harmonic (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) method. With this technique, the deviations of heights and lunitidal intervals, especially of high and low waters, from their respective mean values are represented by superpositions of long-period tidal constituents. This study (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) article (cid:58) documents the preparation of a constituents list for the operational application of the Harmonic Representation of Inequalities. Frequency analyses of observed heights and lunitidal intervals of high and low water from 111 5 tide gauges along the German North Sea coast and its tidally inﬂuenced rivers have been carried out using the generalized Lomb-Scargle periodogram. One comprehensive list of partial tides is realized by combining the separate frequency analyses and by applying subsequent improvements, e.g. through manual inspections of long-time (cid:58)(cid:58)(cid:58) long (cid:58)(cid:58)(cid:58)(cid:58) time (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) series (cid:58) data. The new set of 39 partial tides largely conﬁrms the previously used set with 43 partial tides. Nine constituents are added and 13 partial tides, mostly in close neighbourhood of strong spectral components, are removed. The effect of these changes has been studied by 10 comparing predictions with observations from 98 tide gauges. Using the new set of constituents, the standard deviations of the residuals are reduced (cid:58)(cid:58) on (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) average (cid:58) by 2.41% (times) and 2.30% (heights) for the year 2016. The new set of constituents is (cid:58)(cid:58)(cid:58) will (cid:58)(cid:58) be used for tidal analyses and predictions starting with the German tide tables for the year 2020.

>> The recorded sampling rate of the tide gauges is 1 minute (for about the last 20 years, depending on the individual tide gauge; previously only high and low water data were available).The presented prediction method uses time series of times and heights of high and low water (and predicts only times and heights of high and low water).
3) "What is the maximum gap you observed in the tide gauges time series?And the longest continuous time series?">> The longest continuous time series are from Cuxhaven and Hamburg, each with 115 years and a maximum data gap of half a day, which means that no more than one high or low water is missing at a time.The maximum gaps can be longer than 10 years but the 60% criterion ensures that a sufficient amount of data is available from each tide gauge.4) "The last sentence, page 10, is important.The fact that this relates to parameters introduced in table 2, the fundamental variables, could be added in a note or in bracket.">> Thank you for this remark.We will add a note that the parameters, for which we define the ranges on page 11, are related to the parameters in Table 2. Furthermore, we will rework this part of Sect.4.3 to improve the readability (see also item 5, 6, 16).5) "Fig.4 displays the periodogram of lunitidal intervals (L) after normalization.I understand that the normal variable (value) is the maximum value of lunitidal intervals.Fig. 4, I think it is useful to add in legend, the normalization variable in order to define what is the reference variable used for normalization.It's simple note but it drives the results and plots; Same suggestion for height variable.">> see answer to comment 16. 6) "I understand from ms, mh, mp, mN' fundamental parameters limits selection specific for this study (p 11) that these expressions are introduced in the text because they are useful for functions arguments development.But, this development is not included in the paper, nor cited/referenced.I'd say that if this part can't be used in the paper to help understanding the discussion, results or development, it could be removed from the text.But, if I'm wrong and if these expressions shouldn't be removed from text, then (1) addition of equations where these parameters are used would be useful for understanding or (2) one sentence could be added to say how it's useful to know the what type of selection have been done on ms, mh, mp, mN'." >> We will rework this part of Sect.4.3 (see also item 4, 5, 16) and include information on the calculation of the angular velocities (and add references to Section 2 and Table 2).The definition of the ranges of the linear coefficients m should stay in the manuscript as it sets the limits for the assignment of possible partial tides.4: To get a short analysis of the percentage presented in this table, what is the importance of the main partial tide?Could you precise in table 4 legend, that the results of the most influencing tidal wave is a synthesis from all the selected tide gauges?>> The importance of the main partial tide (half synodic month) can be derived from table 3, where the average line intensities from the periodograms are listed.A quantitative statement (in physical units) is difficult, because of the averaging of data from different tide gauges and the normalization of the generalized Lomb Scargle periodogram.We will make it clear in the legend of table 4 (and table 3) that the results are a synthesis from all the selected tide gauges 8) "P 17: Could you confirm that the results (fig.7, p17) are residuals representative of both HW, LW?I think yes if I'm referring to fig 8. and 9, later in the paper.Fig. 7 validates the method in the frame of HW, LW prediction.Writing the residual mathematical formula is needed, I think, to sustain the text above fig.7." >> Yes, Fig. 7 shows the results for both high water and low water.Residuals are the difference between observed and predicted vertices (high or low water) with the same transit number and event index k.We will add the information on how the residuals are calculated in the revised manuscript.9) "HW/LW prediction improvement percentage presented in tables could be completed by few words to provide some elements of analysis and understanding, to follow the reassessment.">> We will add the formula for calculation of the changes and a few sentences about the contents (especially the extreme bins) of Fig. 8 and 9. 10) "For my interest, I'd like to see a result based on harmonic analysis and least squares minimization for the region of interest, in order to be able to compare its capacity to solve tidal dynamics to the HRoi method presented in the paper (for example in section 5).But therefore, I understand that the authors would have to make some other computations using tools and different methods from those which are presented and used here.So it's more a point for future discussion.Is the paper the first publishing for HRoi of investigations on long period constituents, as it is written in the paper?I'm not aware about the previous HRoi investigations publications for long time period constituents.">> This is the first publication in which the investigation of long term constituents for the HRoI is described.Older publications on the HRoI, as cited in the manuscript, only present the list of constituents without details about its preparation.We will add a sentence in the conclusions about the need to conduct a comparison study with the harmonic analysis to gain more insight on the relative performances.11) "P 2: angular velocities: After the word in text "omega", the mathematics notation !i could be introduced, because it's used later, as the first reference in the text.">> We will add the mathematics notation "omega" after "angular velocities" on page 2. 12) "P 3: -y ˆ to be define in legend (equation 1) (predicted value I think, with y for height or lunitidal interval ).I think adding units in equation 1 is needed.Eq.2: symbol L for partial tide: Doodson reference for Eq.1 and Eq. 2 should be cited.They are derived from Laplace and Doodson theory and from harmonic analysis technics.Particularly, Doodson number is in the first column in tables 2 and 3 and is more generally a number currently used in tidal studies."

7) P 12: Table
>> We add some information on variable names and units to better guide the reader and include the references to the alphanumeric Doodson number.The symbol J (not L) is correct in Eq. 2 as the sum runs over the J data points.13) "P 4: In table 2, I think there is a need to write the thinking who makes you remove the fundamental parameter tau ( ), respective to first letter in Doodson notation?, s, h, p, N' for fundamental parameters to describe tides.to refer to hour angle of "mean" Moon.Just a suggestion: If you think it's relevant, I'd move table 2 in annex for it to play its role of quantitative reference (Tab 2 section 2)." >> By construction of the method (HRoI) only long-period constituents need to be used, i.e. the parameter belonging to tau is always equal to zero.We will revise the corresponding sentence.We think that Table 2 is more than a quantitative reference, but a central element of the paper that connects our work with previous studies and presents fundamental information for all constituents.Therefore, we would like to keep the table in the main part of the paper instead of moving it to the appendix.14) "P 5: Could you give a clear distinction between tn symbol used (p 4, p 5) and nt =lunar transit number (p3)?Reading page 5 and section 4.2 (page 7), could you write tn versus nt? It'd ease the reading and ease the comparison withp3, when transit number is introduced.Its formal symbol should be written (first sentence below table 1)." >> The symbol "tn" is the unit symbol for transit number (such as "h" for hour).The symbol "n_t" is the variable that stands for a value of the transit number (such as "t" for time).We move the introduction of the unit symbol to the sentence with the definition of the transit number.This gives a better distinction between the two notations.
15) "P 7: section 4.1 Data preparation: For both, Â´n lunar transit number", Â´nthe calculation of lunitidal intervals" my opinion is that adding symbols would be benefit for reading.nt and y ˆ (I suggest).">> The symbol for the transit number is added in the corresponding sentence.The lunitidal intervals do not have a unique symbol (the symbol "y" can stand for lunitidal intervals or heights) and is therefore not included in the sentence.Instead, we included the symbol "y" in the first sentence in Sect.4.2.16) "P 10: May I ask you to add slight modification to Li, Lh expression adding legend and adding units in these 2 expressions.I think it could be good to read the units ex: of angular velocity degree per h (cf table 2)?Degree per tn and L units.">> The expressions Li and Lh are unitless as the generalized Lomb-Scargle periodogram is normalized to unity.A value of 0 indicates no improvement of the fit and a value of 1 a "perfect" fit (see Zechmeister and Kürster 2009, full reference in the manuscript).We will add this information on page 10. 17) "P 12:  3 shows the results from the initial analysis based on the defined rules.Table 4 shows the final results, after manual adjustments have been made to the selection of partial tides.Keeping these two tables (which also belong to different sections) separate makes the procedure more transparent.18) "Fig.6 (p 16): I appreciate the synthetic view of figure 6. Suggestion: could you add if possible, the explanation of number above the figure (relating the upper points of partial tides [rank])." >> The numbers at the top of the figure are just the counts of partial tides (number of circles) in each "column".This is mentioned in the caption.
19) "If possible, it could be interesting to see on map fig. 1, the location of Borkum tide gauge, Cuxhaven, Steubenhöft and Emden, Große Seeschleuse tide gauges used in the paper to highlight results" >> Thanks for this great suggestions.These three stations (and Hamburg) will be highlighted in Fig. 1 in the revised manuscript.
Thank you very much for your time and effort to review our manuscript.Please find below our replies to your comments.The different items from the review report are first cited, followed by our responses.1) "They asserted the slight improvement using the new set of constituents through just one year (2016) verification.I recommend that the authors should conduct additional two year (2017)(2018) comparison between prediction and observation to clearly show the improvement.">> The main focus of the manuscript is the preparation of the list of tidal constituents as used by the operational tidal forecasting service.We think that in this context the verification over one year is sufficient.The old sets of constituents have proven over several decades to deliver good results and no major differences were expected.The presented comparison shows that the new set of constituents can be expected to work equally well or even better because several frequencies in the residuals are now removed (see Fig. 10 and 11).We argue that an additional comparison over two years is not likely to show any significantly different results, but rather inflate the manuscript unnecessarily.Furthermore, quality-checked times and heights of high and low waters are not yet available from the respective authorities for several tide gauges for the year 2018.This would limit the comparability of the different verifications.We hope that the referee understands our arguments and does not insist on further verification studies.
2) "As the authors mentioned, the HRoI is not widely used in comparison with a 'standard harmonic analysis and prediction (HAP)' method even if it has the better computational efficiency.Is it because that the HRoI is not open to the public or inconvenient to use?Additionally, I wonder if tidal prediction accuracy for the HRoI is better than that of the HAP.Can it predict tides at any time interval like the HAP?I think that the authors need to explain the additional reason why the HRoI is still used at BSH but most of countries have not used it.What are the advantages of using this method?">> We are not in a position to judge why the HRoI is not used more widely.The method has been published (as referenced in the manuscript) and it is fairly easy to use.The original implementation of the HRoI, as described in Sect.2, uses recorded time series of times and heights of high and low waters in order to predict times and heights of high and low waters.The concept of the HRoI can be generalized to determine the full tidal curve based on equally spaced water level records (e.g. 10 minute intervals).This generalized concept is explained in Müller-Navarra (2013; full reference in the manuscript) and is not subject of the analysis presented in the manuscript.The characteristics (including advantages) of the HRoI are mentioned in Sect. 1 and 2, but we agree that this aspect is scattered throughout the two sections and should be cleaned up and expanded.In the revised manuscript, we will remove the last paragraph of Sect. 2 and insert it after the second paragraph of Sect. 2. The paragraph will also be expanded to address the advantages in more detail.The comparison of the HRoI with other methods (e.g. the harmonic method) is not the subject of this paper and would be beyond its scope.Reliable harmonic constants exist only for a few German tide gauges and a comparison study of this kind would need a lot of resources that are not available at present.We agree that this testing is interesting and invite others to use the HRoI for their applications and comparisons.
3) "On p. 2 line 4: 44 angular velocities -> 45 angular velocities (Need to check it)" >> The list of partial tides published by Horn (1960) consists of 44 angular velocities.These 44 angular velocities are also marked in our Table 2.The sentence in the manuscript is correct.4) "In Table 3 and Table 4, angular velocity (!) should be expressed more than seven decimal places.">> We will add one decimal place to the angular velocities in Tables 3 and 4 in the revised manuscript; also to make it consistent with the angular velocities in Table 2 and the operational usage.More decimal places would be beyond the uncertainty estimate which is of the order of 1e-7 degrees/transit number.5) "On p. 6 line 21: The authors need to explain how to determine the criteria of 60% of high and low waters in more details.It seems to me that the value is low.As shown in Table A1, there are a lot of data sets with more than 90% completeness.">> This selection criterion ensures that only data from tide gauges that record both high and low water are included in the analysis.Some tide gauges fall dry at low water and do not record meaningful tidal data during this time (and no low water height and time is included in the quality-checked time series).These tide gauges have a data completeness of 50% at most.The threshold at 60% is rather conservative in this regard.6) "On p. 7 line 12: What is 'tidal events'?" >> High water and low water are referred to as "tidal events".We will clarify the language in the revised manuscript.7) "On p. 19 lines 1 and 2: in the residua -> in the residual (?); the two residua -> the two residual (?)" >> Yes, this is a typo.The sentence should read "The change of constituents has an influence on the remaining periodicities in the residuals."This will be corrected in the revised manuscript.8) "In Figure 7: The authors need to explain how to determine a bin width for time and height differences.">> The number and the width of the bins are chosen in such a way that the central bin is centred at the origin.9) "The authors need to use the subscript in expressing name of tidal constituents throughout the manuscript.That is, Sa -> S_a (subscript a)" >> We followed the naming scheme of the "Standard list of Tidal Constituents" by the IHO which does not use subscripts.We will add this information on page 6, line 2 in the revised manuscript, but prefer to keep the naming as it is if the reviewer does not have any objections.

Authors' response to Editor Comment #1
Dear Phil, Thank you very much for your additional comments to our manuscript.Please find below our replies to your comments.We will upload the revised manuscript as soon as possible.
1) "One is that paper does have the feel of a highly-technical internal report and it might help to have an introductory paragraph in Section 6 (perhaps) to show that you know that there have been other methods for analysing HL waters in the past.">> We will add some more references to other methods of tidal predictions in the revised manuscript.Thank you for your literature suggestions.
2) "Another is the comment by R1 about comparing the method used here to more standard harmonic methods, which you replied to in your paragraph (3) saying this was work in progress.But surely a tidal agency like the BSH is called on to produce hourly (or similar) tidal values for use in science or practical applications and you must have those data sets to hand.As regards the present paper it would not take much work to make a comparison for one or two places (say Cuxhaven).Last year I picked up a leaflet at the BSH which says'complete predictions of water level curves at Cuxhaven have been available on the internet since May 2010'.">> The tidal information service from BSH does not provide hourly predictions on an operational basis (yet).As we start to have 19 years of 1-minute data from more and more tide gauges, we are currently setting up the programs to use this high resolution data in our routine tidal predictions.For previous years, only the HL water recordings were saved for most tide gauges.As the request for a comparison with the more widely used harmonic methods has been expressed in all review reports, we will include a short comparison for two stations (probably Cuxhaven and Hamburg) in the revised manuscript.The leaflet that you are referring to probably covers the water level and storm surge forecasting service (not the tidal information service).These water level curves (with and without surge) are produced by different methods, in which the tidal data calculated with the HRoI (times and heights of high and low waters) is used as an input.
3) "I understand the method for a particular station of course, but the rankings must be different for different stations so I was unclear how you arrived at the final choice.Could you make that clearer?" >> The rankings displayed in Table 4 are a synthesis from the data of all analysed tide gauges.This will be mentioned in the caption of the table and will be made clearer in the corresponding paragraph.The goal of Table 4 is to produce one comprehensive list that reflects the information from all tide gauges in the area under investigation.A tailored analysis for an individual tide gauge is of course possible (and needed), if the general list does not lead to good results.4) Most of the other remarks have been directly incorporated into the revised manuscript (thanks for all the details).Here are the answers to your questions: 4a) "I suspect that when most agencies produce tidal constants for a particular year they do not remove big storms; they are part of the sea level climatology, leading inevitably to ambiguity as to what defines the tide.So, in your case does this storm surge removal make any difference to the results?" >> We try to predict water levels considering past long-term meteorological conditions (as good as this is possible).A single extreme event, like a severe storm surge, is not representative of the tidal behaviour at a site (and cannot be forecasted in the framework of tidal predictions).The model function (sum of harmonics) is also not made to properly account for these extreme events and the least squares method is likely (depends on the number of storm surges) to give results that lead to slightly higher heights at all times, if the storms are nor removed.We do not have numbers at hand on how much this storm surge removal influences the results (this will also depend on the number of extreme events in an individual time series).Part of this topic is the fundamental question on how to define the (astronomical) tide.4b) "Tables 5 and 6 -is it necessary to have gauge number in these tables" >> The numbers are necessary, because the short names for the tide gauges are not always unique, e.g.Borkum (Fischerbalje) vs. Borkum (Südstrand).Since 1954, a method named Harmonic Representation of Inequalities (HRoI) is ::: has ::::: been used at BSH to calculate tidal predictions for tide gauge locations along the German North Sea coast and its tidally influenced rivers (Horn, 1948(Horn, , 1960;;Müller-Navarra, 2013).This technique allows analysing the deviations of times and heights, especially at high and low water, from their respective mean values.In contrast to the widely used harmonic method :::::::::::::::::::::::::::::::::: (e.g.Parker, 2007, and references therein) , the HRoI utilizes only long-period partial tides.This reduction in frequency space allows for a computationally efficient way to calculate times and heights of high and low water.:::: Other ::::::::: techniques ::: for :::: tidal ::::::: analysis :: of :::: high :::: and ::: low :::::: waters ::: are ::::::::: described, constituents.: The HRoI has proven to be especially useful for predicting semi-diurnal tides in shallow waters where the harmonic method would need a large number ( 60) of constituents or could even fail to produce adequate results.The fundamentals of the HRoI are summarized in Sect. 2 for completeness.
An important aspect of tidal prediction is the selection of relevant partial tides (angular velocities, :: ω) to be included in the underlying analysis of water level records.While it is possible to determine these partial tides individually for each single tidal analysis, it is desirable in an operational service to have one comprehensive set of constituents that can be used for all tide gauges under investigation.Horn (1960) presented a list of 44 angular velocities that were used with the HRoI.This selection of partial tides was probably utilized until the year 1969 when the set was slightly modified (compare Tab. 2 in Sect.2).To our knowledge, no documentation exists of the methods and specific water level records that were used to prepare these lists of angular velocities.

Harmonic Representation of Inequalities
The Harmonic Representation of Inequalities (HRoI) is a derivative of the non-harmonic method by essentially translating it into an analytical form.The non-harmonic method has been used for a long time, e.g. by Lubbock (1831) for the analysis of tides in the port of London.With the non-harmonic method, the times of high and low water ::::: waters : are calculated by adding mean lunitidal intervals and corresponding inequalities to the times of lunar transits.Likewise, the heights of high and low water ::::: waters : are determined by adding corresponding inequalities to the respective mean heights.The inequalities are corrections for the relative positions of earth, moon and sun (e.g.semi-monthly, parallactic, declination).
Let (t j , h j ), j = 1, ..., J, be a time series of length J of high and low water heights h j recorded at times t j .All times need to be given in UTC.The HRoI method is based on the assumption that the variations of the individual heights and lunitidal intervals around their respective mean values can be described by sums of harmonic functions.The lunitidal interval is the time difference between the time t j and the corresponding lunar transit at Greenwich.As a general rule, the daily higher high water and the following low water are assigned to the previous upper :::: lunar : transit, and the daily lower high water and the following low water are assigned to the previous lower transit.For example, in the year 2018, the mean lunitidal interval for high (low) water was determined to be 9 h 4 min (16 h 5 min) for Borkum and 15 h 22 min (22 h 32 min) for Hamburg.::: See ::: for ::: the :::::::: locations :: of ::::: these ::: two ::::: sites.
A convenient method to organize high and low waters of semi-diurnal tides is the lunar transit number n t (Müller-Navarra, 2009).It counts the :::::: number :: of : upper lunar transits :::: (unit ::::::: symbol: :: tn) : at the Greenwich meridian since the transit on December 31, 1949, which has been arbitrarily set to n t = 0 :: tn.A lower transit always has the same transit number as the preceding upper transit.Each high and low water is uniquely identified by using the number n t of the assigned lunar transit and an additional event index k as defined in Tab. 1.The differentiation between upper and lower transit allows for changes in the Moon's declination which alternately advance and retard times, and increase and decrease the heights of successive tides (diurnal inequality).
A ::: full tidal analysis with the HRoI comprises the investigation of eight times ::: time : series (heights and lunitidal intervals of the four event types listed in Tab. 1).Each time series is described by a model function : ŷ of the following form: The parameters a l , l = 0, ..., 2L are determined from a least-squares fit, i.e.
where y j are the observed heights or lunitidal intervals.The angular velocities ω l [ ::: • /tn] are taken from a previously defined set of L partial tides.In Tab. 2, we list two sets of partial tides that have been used in the past at BSH and the new set that is the result of this work.All tidal constituents considered here have angular velocities that are linear combinations of the rate of change of four fundamental astronomical variables :::::::: arguments: the mean longitude of the moon (s), the mean longitude of the sun (h), the mean longitude of the lunar perigee (p) and the negative of the longitude of the moon's ascending node (N ).The second to fifth column in Tab. 2 give the respective linear coefficients m.The ::: two ::::: other :::::::: arguments :::: that ::: one ::::::::: encounters ::::: using ::: the :::::::: harmonic is ::::: given :: in ::: the :::: first ::::::: column :::::::::::::::::::::::::: (Doodson, 1921;Simon, 2013) .:::: The eighth column states the commonly used names 1 .A mark in one of the last three columns indicates whether the angular velocity is included in the respective constituents list for usage with the HRoI.
According to Horn (1960) , the HRoI combines the best from the harmonic and the non-harmonic method: the analytical procedure of the first method, and the principle of calculating isolated values directly which is characteristic for the second.The strength of the HRoI lies in the prediction of times and heights of high and low water when the full tidal curve is considerably non-sinusoidal.This is frequently the case in shallow waters and rivers.
3 Tide gauge data The tide gauges at the German coast and :: in rivers are operated by different federal and state authorities.These agencies provide BSH with quality checked ::::::::::::: quality-checked : water level records of high and low waters (times and heights).readily available for tide gauges :: at Cuxhaven (BSH gauge number DE :::: 506P) and Hamburg (DE :::: 508P) for which data since the year 1901 is used.We are aware that the tidal regime can change over such a long time, but include all available data in the analysis to maximize the achievable spectral resolution.
Only tide gauges with more than 19 years of data are included in order to cover the period of rotation of the lunar node (18.6 years) in the frequency analysis.In addition, we use only tide gauges where more than 60% of high and low waters are recorded during the gauge's data period.This criterion excludes gauges for which no low water observations are available.The 111 gauges that fulfill :::: fulfil : these two criteria are marked in the column labelled "used for analysis" in Tab.A1.The locations of the :: all : tide gauges are shown on the map in Fig. 1.

Analysis of high water and low water time series
The following analysis is applied to the water level records of all 111 tide gauges that are marked in the seventh column of Tab.A1 in the appendix.

Data preparation
Data preparation includes the assignment of lunar transit numbers :: n t and the calculation of lunitidal intervals as described in Sect. 2 for each record of high or low water.The lunar transit times are calculated using :::::::: following : the algorithm by Meeus (1998)  The observed water levels include extreme events, such as storm surges.These events are not representative for the tidal behaviour at the site of a tide gauge and need to be ::: are removed from the data set.We apply a 3-sigma-clipping separately for the eight times series analysed with the HRoI (see Sect. 2).Only those data points are used in the analysis, for which the height and the lunitidal interval are within the range of three times the respective standard deviation.
The tidal events ::::::::: Artefacts from spectral leakage pose a major problem when identifying peaks in a periodogram.They arise from the finite length of the time series(e.g.Press et al., 1992) .This effect can be reduced by applying an appodization function :::::::::: apodization ::::::: function, ::: i.e. :::::::::: multiplying ::: the :::: data ::::: with : a ::::::: suitable ::::::: window :::::::: function, : that smoothly brings the recorded values to zero at the 5 beginning and the end of the sampled time series ::::::::::::::::::::::::::::::: (e.g.Press et al., 1992;Prabhu, 2014) .We apply a Hanning window to the data which gives a good compromise between reducing side lobes and preserving the spectral resolution.For each tide gauge, periodograms are calculated for the eight time series that are analysed with the HRoI.In the upper panels of Fig. 2 and 3, we show periodograms of the lunitidal intervals and heights (of high waters assigned to an upper transit, ::::: event :::: index : k = 1) for the tide gauge Cuxhaven.Cuxhaven (together with Hamburg) provides by far the longest times :::: time series that is used in the analysis (compare Tab. A1).In these figures, the vertical axis is normalized to the strongest peak and the horizontal axis is converted to degrees per transit number for better comparison with Tab. 2. The periodogram for the lunitidal intervals reveals many more strong spectral lines above the noise floor as compared to the periodogram for the heights.A frequency depended :::::::::::::::: frequency-dependent : noise level can clearly be seen in Fig. 3 (noise level increases towards lower angular velocities).The lower panels of Fig. 2 and 3 show a small extract of the respective upper periodograms.Additionally, data for tide gauge Emden is included for illustration of the differences in spectral line width.The time series from Emden is about four times shorter than the one from Cuxhaven.This leads to broader spectral lines in the periodogram and it can be expected that some weaker lines are unresolvable.

Identifying relevant partial tides
We aim to find all local maxima in a periodogram that are above a noise threshold.This threshold is calculated in a two-step process that is described in the following.
In The strongest lines are removed from the periodogram (grey vs. green lines : ; ::: first :::: step :: as ::::::: described :: in :::: Sect.::: 4.3) and an exponential function (dashed red curve) is fitted to selected points (blue; :::::: second ::: step :: as ::::::: described :: in :::: Sect.A partial tide from the precalculated list is assigned uniquely to the closest peak in the periodogram if the difference in angular velocity is less than half the spectral resolution.The spectral resolution r is defined as with T being the length of the time series in transit numbers.For example, the spectral resolution of a time series of 19 years is where τ = 1.03505013 d/tn is the length of the mean lunar day. For each identified partial tide, we calculate (i) the percentage of periodograms in which the partial tide has been detected, 10 separately for lunitidal interval (N i ) and height (N h ); and (ii) the average intensity in the periodograms, separately for lunitidal interval (I i ) and height (I h ).In order to be considered relevant, a partial tide with angular velocity ω must meet the following criteria: N i ≥ 33% and I i (ω)>L i (ω), or N h ≥ 33% and I h (ω)>L h (ω).All partial tides that meet these selection criteria are listed Tab. 3.

Adjustment of constituent list and ranking
In this section, we describe adjustments made to the list of partial tides based on manual inspections of certain periodograms and other considerations for an operational application.These adjustments lead to the set of partial tides in Tab. 4.
The periodograms calculated from longer time series offer a higher spectral resolution and contain more spectral information compared to the periodograms of shorter time series.For example, this can be seen :::: This : is :::::::::::: demonstrated in the lower panels of Fig. 2 and 3 with periodograms based on time series from tide gauge Cuxhaven (115 years) and Emden (27 years).The higher information content from longer water level records needs to be appreciated and incorporated adequately.Therefore, the periodograms of Cuxhaven and Hamburg have been inspected manually to find partial tides that appear in the data of these two tide gauges and might not be detectable in other periodograms.Six partial tides with the following Doodson numbers were identified and added to the list: ZAZZAZ (ZAZZZZ), ZBXZYZ (ZBXZZZ), ZBZXZZ, ZBZZAZ (ZBZZZZ), ZCXZZZ and ZDXZAZ (ZDXZZZ).The Doodson numbers in parenthesis are partial tides from Tab. 3 that differ only by ∆m N = ±1.For these pairs, long time series are needed to clearly see two separate spectral lines in the periodograms.
The noise in the periodograms increases towards lower angular velocities and the identification of partial tides below 1 • /tn becomes less clear.For this reason, and after inspecting several periodograms manually, the partial tide ZZAXZZ is considered to be a misidentification and has been removed from the list.The other way round ::::::::: Conversely, the partial tide ZZBXZZ has been added to the list, because of its importance for tide gauges located upstream in the Elbe river.Finally, we decided to cut the list after the eighth synodic month to keep the range of angular velocities consistent with previously used lists of partial tides (compare Tab. 2).
The final set of long-term ::::::::: long-period : partial tides from our analysis is listed in Tab. 4. In the last column, each partial tide is assigned a number R indicating its overall importance (in decreasing order).The rank R is ::::: based ::: on ::: the :::::::: combined ::::::::: evaluation :: of ::: data ::::: from :: all :::: tide :::::: gauges ::: and :: is : calculated by the following procedure: where the function norm() returns normalized values in the range [0,1] and the function rank() returns the position of a list element, if the list were sorted in increasing order.In Eq. 5, the results from lunitidal intervals are weighted higher ::::::: stronger because the noise level is lower in the respective periodograms.The rank R can be used to select a sublist of partial tides when performing a tidal analysis of water levels with less than 18.6 years of data.This is important, because not all partial tides can be resolved against each other for shorter time series.The minimum difference in angular velocity is given by the resolution criterion (Eq.3). in ::: the ::: list : cannot be included for time series shorter than about nine years.For : a tidal analysis of time series shorter than nine years, it is therefore often better to perform a reference analysis: 19 years of data are used from a different tide gauge with a similar tidal behaviour (e.g.similar course of the semi-monthly inequality) and the results are translated to the original gauge by applying the respective differences of the mean lunitidal intervals and mean heights.This way, nodal corrections can be avoided which come with their own assumptions and approximations (e.g.Godin, 1986) :::::::::::::::::::::::::: (e.g.Godin, 1986;Amin, 1987) .
5 Comparison of predictions with observations : : :::: two ::::::: different :::: lists :: of ::::::::::: constituents ::: for ::: the :::::: HRoI For verification of the new constituent list, tidal predictions (i) based on an existing list of partial tides and (ii) based on the new set are compared with observations.The predictions are made for the year 2016 and are compared with tide gauge observations from the same year.
Appendix A: 4c) "Figure 8-11.It might be good to make 8 and 10 into 8(a,b) and 9 and 11 into a new 9(a,b).">>We would like to keep these figures separated, as they cover slightly different aspects of the residual analysis.
the German Bight are calculated at the Federal Maritime and Hydrographic Agency (Bundesamt für Seeschifffahrt und Hydrographie, BSH) and are published in official tide tables each year.The preparation of tidal predictions has a long tradition at BSH and its predecessor institutions: the first tide tables by the German Imperial Admiralty were issued for the year 1879.

Figure 2 .
Figure 2. Upper panel: Normalized periodogram of the lunitidal intervals of high waters (assigned to upper lunar transits) for the tide gauge Cuxhaven.Notice the :::: upper ::: part :: of ::: the logarithmic scale which is truncated at 0.1 for better visibility of weak lines.Lower panel: Zoom into the region with the spectral line corresponding to half a tropical month (Mf) at 27.2764618 • /tn.The longer time series for Cuxhaven leads to narrower spectral lines (solid blue curve) compared to Emden (dashed green line).

Figure 3 .
Figure 3. Upper panel: Normalized periodogram of the heights of high waters (assigned to upper lunar transits) for the tide gauge Cuxhaven.Notice the logarithmic scale.Lower panel: Zoom into the region with the spectral line corresponding to half a tropical month (Mf) at 27.2764618 • /tn.The longer time series for Cuxhaven leads to narrower spectral lines (solid blue curve) compared to Emden (dashed green line).

Figure 4 .
Figure 4. Determination of the noise threshold for the tidal interval (high water, upper transit) at tide gauge Borkum, Fischerbalje: the : .::: ::: 4.3).The noise threshold (thick red line) is shifted up by one standard deviation.See text for more details.

Figure 5 .
Figure 5. Same as Fig. 4 but for the heights at tide gauge Borkum, Fischerbalje.

Figure 6 .
Figure 6.The partial tides (identified by their rank : R : from Tab. 4) that can be resolved as a function of the (minimum) length of the time series.If two partial tides cannot be resolved against each other, the one with the lower rank is dropped.Note the logarithmic time axis from 0.2 to 20 years.The numbers at the top are the counts of partial tides.

Figure 8 .
Figure 8. Histogram of the change in the standard deviation of the residuals of high and low water times for all 98 tide gauges.

Figure 9 .
Figure 9. Histogram of the change in the standard deviation of the residuals of high and low water heights for all 98 tide gauges.
Table 4 could be inserted in table 3, by adding: column R (table 4) after column Nh [%] in table 3, column description/name (table 4) in table 3" >> We think that table 3 and table 4 should be kept separate.Table

Table 1 .
The high and low waters are classified into four types (event index k). :::::::

Table 2 .
Sets of angular velocities that have been used with the HRoI.See Sect. 2 for a description of the columns.

Table 3 .
The most important partial tides that were identified in the periodograms, based on the combined evaluation of data from all tide gauges.See Sect.4.3 for information on selection criteria and Ii, Ih, Ni and Nh.

Table 4 .
The modified and adopted new list of long-period partial tides.The rank R indicates the importance of a partial tide for tidal analysis, based on the combined evaluation of data from all tide gauges.

Table 5 .
Residuals of predicted and observed times of high and low water: mean µ and standard deviation σ in minutes.

Table 6 .
Residuals of predicted and observed heights of high and low water: mean µ and standard deviation σ in meters.

Table A1 .
137 German tide gauges which deliver water level observations on a regular basis and for which tidal predictions were published in BSH tide tables (Gezeitentafeln) or tide calendar (Gezeitenkalender) for the year 2018.The data from the tide gauges are observed times and heights of high and low water.The tide gauges are operated by different federal and state agencies which provide tide gauge records to BSH.Abbreviations in the third column correspond to the following agencies: E: Emden Waterways and Shipping Authority (Wasserstraßen-und Schifffahrtsamt Emden, WSA Emden), BH: WSA Bremerhaven, B: WSA Bremen, C: WSA Cuxhaven, T: WSA Tönning, HPA: Hamburg Competing interests.The authors declare that they have no conflict of interest.