Data preparation includes the assignment of lunar transit numbers, nt, and the calculation of lunitidal intervals as described in Sect.  for each record of high or low water. The lunar transit times are calculated following the algorithm by Chap. 15 with the modification of direct calculation of lunar coordinates using the periodic terms given in the work by .

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 are removed from the data set. We apply a 3σ clipping separately for the eight time series analysed with the HRoI (see Sect. ). Only those data points for which the height and the lunitidal interval are within the range of 3 times the respective standard deviation are used in the analysis.

The observed heights and lunitidal intervals (y) can be understood as being functions of the assigned transit number (nt). We calculate periodograms for the heights and tidal intervals using the corresponding frequency scale per transit number.

The occurrences of high and low waters are irregularly spaced in time. Additionally, there are many longer data gaps which cannot be interpolated. This excludes the fast Fourier transform (FFT) as a spectral analysis technique. Instead, we use the generalized Lomb–Scargle periodogram as defined by , including their normalization if not mentioned otherwise. The frequency scale covers the range from 0.0001 to 2 tn-1 with an interval of 0.01999 tn-1 (100 000 points in the periodogram). This corresponds to approximately 0.0057–114.5916∘ tn-1 or 0.0002–4.6130∘ h-1. The upper limit corresponds to twice the mean sampling interval (Nyquist criterion).

Artefacts from spectral leakage pose a major problem when identifying peaks in a periodogram. They arise from the finite length of the time series. This effect can be reduced by applying an apodization function, i.e. multiplying the data with a suitable window function, that smoothly brings the recorded values to zero at the beginning and the end of the sampled time series e.g.. 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 Figs. a and a, 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 time series that is used in the analysis (cf. Table ). 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 Table . The periodogram for the lunitidal intervals reveals many more strong spectral lines above the noise level as compared to the periodogram for the heights. A frequency-dependent noise level can clearly be seen in Fig. a (noise level increases towards lower angular velocities). Figs. b and b show a small extract of the respective upper periodograms. Additionally, data for tide gauge Emden are included for illustration of the differences in spectral line width. The time series from Emden is about 4 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.

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 first step, the strongest spectral lines are removed from the periodogram. The values above the 99.5th percentile are removed from the data set and a histogram is calculated from the remaining values p (100 bins with central values xbin). The histogram shows an exponential trend from a large number of data points with low periodogram values to a few points that fall into the bins at the upper end. An exponential curve, ybin=a⋅exp⁡(-xbin/b), is fitted to the histogram with fit parameters a and b. The process of removing data points above the 99.5th percentile from the periodogram is repeated until the ratio max⁡(p)/b falls below the value of 30. This value is based on experience.

In the second step, the noise threshold is determined using a set of remaining points in the periodogram that represent a continuum above the noise level. The result is illustrated in Figs.  and for lunitidal intervals and heights at the tide gauge Borkum. For this procedure, the periodogram is split into 25 sections with the same number of data points. The data point at the 99.5th percentile is selected in each section and an exponential function is fitted to these 25 points. The fit is repeated after a 1σ clipping. The noise threshold corresponds to the resulting exponential function plus 1 standard deviation (solid red line in Figs.  and ).

In preparation for the following combined evaluation of the results from all tide gauges, the noise threshold functions from the different periodograms are averaged; this is separately for lunitidal intervals (Li) and heights (Lh): Li(ω)=0.0004816⋅exp⁡(-0.0101045tn/∘⋅ω),Lh(ω)=0.0024472⋅exp⁡(-0.0149899tn/∘⋅ω). These two functions represent mean lower-intensity boundaries for the selection of significant peaks. The expressions Li and Lh are unitless, due to the normalization of the Lomb–Scargle periodogram .

In addition to the intensity of the local maxima, the number of their occurrences in the different periodograms and their assignment to the partial tides determine their inclusion into the list of constituents for the HRoI. The local maxima must match the theoretically expected partial tides that have well-known angular velocities computable from the linear combinations of the rate of change of the four fundamental astronomical arguments s, h, p and N′ (see Sect. ). The angular velocities of 1268 partial tides have been precalculated using, again, the expressions for the fundamental astronomical arguments as published by the Sect. 5.7. The ranges of the linear coefficients m are chosen based on experience (cf. Table ): ms=0,…,8mh=-8,…,3mp=-2,…,3if((ms=0andmh≥0)or(ms>0andmh≥-ms-1)),mN′=0,1ifms=0andmh=0andmp=0-1,0,1ifms≠0ormh≠0ormp≠0

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 3r=360∘/T, 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 4r=360∘19yr⋅365.25dyr-1⋅τ=0.05∘tn-1, where τ=1.03505013 d tn-1 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, separately for lunitidal interval (Ni) and height (Nh), and (ii) the average intensity in the periodograms, separately for lunitidal interval (Ii) and height (Ih). In order to be considered relevant, a partial tide with angular velocity ω must meet the following criteria: Ni≥33 % and Ii(ω)>Li(ω) or Nh≥33 % and Ih(ω)>Lh(ω). All partial tides that meet these selection criteria are listed Table .

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 Table .

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. This is demonstrated in Figs. b and b with periodograms based on time series from tide gauges 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 Table  that differ only by ΔmN′=±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-1 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. 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 (cf. Table ).

The final set of long-period partial tides from our analysis is listed in Table . 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: Ri=rank(norm(Ii(ω)-Li(ω))⋅Ni),Rh=rank(norm(Ih(ω)-Lh(ω))⋅Nh),5R=rank((3Ri+Rh)/4), 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. (), the results from lunitidal intervals are weighted stronger because the noise level is lower in the respective periodograms.

The rank R can be used to select a sub-list 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. ). Figure  illustrates the resolvable partial tides as a function of the length of the data record. Note that the high-ranked partial tide representing the tropical month which occurs at R=4 in the list cannot be included for time series shorter than about 9 years. For a tidal analysis of time series shorter than 9 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 in the mean lunitidal intervals and mean heights. This way, nodal corrections can be avoided which come with their own assumptions and approximations e.g..

We calculate tidal predictions (times and heights of high and low waters) with the HRoI using (i) the 43 partial tides from Set 2 in Table  and (ii) the 39 partial tides derived from our analysis. The data and software are otherwise identical. The predictions are based on amplitudes am (see Eq. ) that are determined from a tidal analysis of water level records from 1995 to 2013 (19 years). Data are only used from tide gauges that have delivered enough observations in this time period to include all partial tides in the analysis. Additionally, the tide gauges must have delivered observations for the year 2016. The 98 tide gauges that fulfil these criteria are marked in the column “Used for verif.” in Table . All tide gauge data are prepared as described in Sect. , including the removal of extreme events from the observations that are used for comparison. The analysis is applied in two iterations with a 3σ clipping in-between.

In this section, we present results from the analysis of the residuals in the following order: the distributions of residuals for the tide gauge Cuxhaven, the means and standard deviations for some major ports, the changes in the standard deviations for all tide gauges, and the changes in the remaining frequencies. The residuals are the differences between the observed and the predicted vertices (times and heights of high and low waters) with the same assigned transit number and event index k.

Figure  shows histograms of the residuals for the tide gauge Cuxhaven. The panels on the left and on the right display histograms for the times and heights, respectively. Each panel contains one histogram for the tidal prediction with 43 partial tides (red) and one histogram for the tidal prediction with 39 partial tides (yellow). Using the new set of partial tides, the standard deviation of the residuals decreases from 9.6 to 9.0 min for the times and from 0.28 to 0.27 m for the heights.

In the same way as for Cuxhaven, residuals are calculated for the data of all 98 tide gauges included in the verification. The mean values, μ, and standard deviations, σ, of the residuals for 11 major ports are summarized in Table  for the times and in Table  for the heights. Based on the results from all tide gauges, the average standard deviation of the residuals is 13.2 min or 0.28 m, respectively, using the set of 39 tidal constituents. In most cases, the new set of partial tides gives small improvements in μ and σ.

The percentage changes, Δσ, in the standard deviations are presented in the histograms in Fig.  (times) and Fig.  (heights) for all 98 tide gauges. The percentage change is calculated in the following way: 6Δσ[%]=100%⋅σ39 p.t.-σ43 p.t.σ43 p.t., where σ39 p.t. and σ43 p.t. are the standard deviations of the residuals using the predictions with 39 partial tides (p.t.) and 43 partial tides, respectively. The average reductions of the standard deviations are 2.41 % (times) and 2.30 % (heights). The two tide gauges with the largest improvements in this study, both with regard to times and heights, are Holmer Siel (BSH gauge number 649B) at the North Frisian coast and Bremervörde (687P) in the river Oste. Further tide gauges with improved standard deviations are located all around the area of investigation. Regarding the times (Fig. ), the four tide gauges with increased standard deviations are Westerland (620P) and Hörnum (624P), located on the North Frisian island of Sylt, and Dove-Elbe (727P) and Bunthaus (729P), located upstream of the river Elbe. Regarding the heights (Fig. ), the five tide gauges with the largest positive changes are also located upstream in the Elbe, namely Geesthacht (732D), Altengamme (732A), Zollenspieker (731P), Ilmenau (730A) and Fahrenholz (730C). The water levels in this part of the Elbe are partly influenced by river discharge, which can lead to deviations from the tidal predictions.

The change in constituents has an influence on the remaining periodicities in the residuals. Periodograms are calculated for the two sets of residuals (times and heights) for each tide gauge. The 98 periodograms of each type are averaged. The resulting mean periodograms are shown in Figs.  and . In both figures, the strongest peaks are located at very low angular velocities (≲1∘ tn-1). As mentioned before, the unambiguous identification of partial tides at these periods is difficult and consequently no major improvements are achieved in reducing the (average) residual periodicities in this range. Four further strong peaks are visible in both figures at about 15, 25, 52 and 64∘ tn-1 for the prediction with 43 partial tides. These peaks are clearly reduced with the new predictions (39 partial tides).

The predictions with the HRoI (39 partial tides) are the same as in Sect. . The harmonic analysis is based on continuous observations from the years 1996 to 2014 at 10 min intervals. The harmonic constituents (amplitudes H and phases g) and the constant vertical offset, Z0, are determined from a least-squares fit with the following model function: 7y^harm=Z0+∑l=1LHl⋅cos⁡(Vl(t0)+ωlt-gl).

We use the 68 partial tides (with angular velocities ωl) from . These are also the default constituents in the MATLAB packages t_tide and UTide , which have become widely accepted standard implementations of the harmonic method. Since the data records exceed 18.6 years, we add the partial tide with the angular velocity of the lunar node and omit nodal corrections. This gives a total of L=69 constituents. The time t is referenced to the midpoint t0 of the time series and the astronomical argument Vl(t0) is calculated for each partial tide using the expressions for the fundamental astronomical arguments as published by the Sect. 5.7. The analysis is applied in two iterations with a 3σ clipping in-between to remove outliers in the observations.

We show in Figs.  and the predictions and observations for Cuxhaven and Hamburg, respectively. Only 10 d in June 2016 are shown from the complete time series for better visibility of the individual high and low waters. The two curves in each figure are the observed water levels (dark blue) and the harmonic prediction (light green). The high and low waters are marked separately for observations (red circles), vertices determined from the harmonic prediction (green squares) and predictions made with the HRoI (yellow triangles).

As in Sect. , the residuals are the differences between the observed and the predicted vertices (times and heights of high and low waters) with the same assigned transit number and event index k. We calculate the means and the standard deviations of the residuals regarding times and heights. The results are shown in Table . The differences for the heights are within a few centimetres. For the times, the standard deviations are approximately 4–5 min larger in the case of the harmonic method. The residuals for the times are also shown in Fig. , where the curves for the harmonic method (blue) suggest that long-period periodicities could be present in the residuals which are not covered by the predictions. Based on the calculated parameters, the deviations of the harmonic prediction from the observations (and also from the prediction made with the HRoI) are larger for Hamburg as compared to Cuxhaven. This supports the assumptions that the application of the HRoI is especially useful for tide gauge locations in rivers.