The harmonic representation of inequalities (HRoI) is a procedure for tidal analysis and prediction that combines aspects of the non-harmonic and the harmonic 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 article 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 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 series 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 the close neighbourhood of strong spectral components, are removed. The effect of these changes has been studied by comparing predictions with observations from 98 tide gauges. Using the new set of constituents, the standard deviations of the residuals are reduced on average by 2.41 % (times) and 2.30 % (heights) for the year 2016. The new set of constituents will be used for tidal analyses and predictions starting with the German tide tables for the year 2020.

Tidal predictions for the German Bight are calculated at the Federal Maritime and Hydrographic Agency (

Since 1954, a method named harmonic representation of inequalities (HRoI) has been used at BSH to calculate tidal predictions for tide gauge locations along the German North Sea coast and its tidally influenced rivers

An important aspect of tidal prediction is the selection of relevant partial tides (angular velocities,

The objective of this work is to review the set of partial tides used with the HRoI by determining the most important long-period constituents for application in the German Bight. Therefore, we perform a spectral analysis of water level observations from 111 tide gauges. The available tide gauge data are presented in Sect.

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

The original implementation of the HRoI, as introduced by

According to

Let

A convenient method to organize high and low waters of semi-diurnal tides is the lunar transit number

The high and low waters are classified into four types (event index

A full tidal analysis with the HRoI comprises the investigation of eight time series (heights and lunitidal intervals of the four event types listed in Table

The parameters

Sets of angular velocities that have been used with the HRoI. See Sect.

All tidal constituents considered here have angular velocities that are linear combinations of the rate of change of four fundamental astronomical arguments: the mean longitude of the moon (

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 water level records of high and low waters (times and heights). Table

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 fulfil these two criteria are marked in the column labelled “Used for analysis” in Table

The locations of all tide gauges in the German Bight from Table

The following analysis is applied to the water level records of all 111 tide gauges that are marked in the seventh column of Table

Data preparation includes the assignment of lunar transit numbers,

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 3

The observed heights and lunitidal intervals (

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

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

For each tide gauge, periodograms are calculated for the eight time series that are analysed with the HRoI. In Figs.

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

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.

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 (

Determination of the noise threshold for the tidal interval (high water, upper transit) at tide gauge Borkum, Fischerbalje. The strongest lines are removed from the periodogram (grey vs. green lines; first step as described in Sect.

Same as Fig.

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

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

For each identified partial tide, we calculate (i) the percentage of periodograms in which the partial tide has been detected, separately for lunitidal interval (

The most important partial tides that were identified in the periodograms, based on the combined evaluation of data from all tide gauges. See Sect.

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.

The noise in the periodograms increases towards lower angular velocities and the identification of partial tides below 1

The final set of long-period partial tides from our analysis is listed in Table

The modified and adopted new list of long-period partial tides. The rank

The rank

The partial tides (identified by their rank

For verification of the new constituent list, tidal predictions based on an existing list of partial tides and 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.

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

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

Figure

Histograms of the residuals for tide gauge Cuxhaven and year 2016. The different colours indicate predictions based on the different sets of partial tides (red: predictions with 43 partial tides; yellow: predictions with 39 partial tides).

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,

Residuals of predicted and observed times of high and low water: mean

Residuals of predicted and observed heights of high and low water: mean

The percentage changes,

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

Histogram of the change in the standard deviation of the residuals of high- and low-water heights for all 98 tide gauges.

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.

Mean periodogram of residual high- and low-water times for all tide gauges used in the verification. The different colours indicate predictions based on the different sets of partial tides (red: predictions with 43 partial tides; yellow: predictions with 39 partial tides)

Same as Fig.

The harmonic method is the most widely used technique for tidal predictions. The following comparison of predictions calculated with the HRoI and with the harmonic methods will demonstrate the respective capabilities. The comparison is done for the two tide gauges at Cuxhaven, Steubenhöft, and Hamburg, St. Pauli. The first site is located at the mouth of the river Elbe, flowing into the North Sea, while the second is about 100 km upstream in the river Elbe. The predictions are compared with tide gauge observations from the year 2016.

The predictions with the HRoI (39 partial tides) are the same as in Sect.

We use the 68 partial tides (with angular velocities

We show in Figs.

Observations and two predictions for the tide gauge Cuxhaven, Steubenhöft. The first 10 d of June 2016 are shown.

Observations and two predictions for the tide gauge Hamburg, St. Pauli. The first 10 d of June 2016 are shown.

As in Sect.

Residuals of predicted and observed times and heights of high and low water: mean

The residuals of high- and low-water times for the tide gauges Cuxhaven

Time series of high- and low-water records from 111 German tide gauges were analysed to determine important long-period partial tides. Generalized Lomb–Scargle periodograms were calculated from lunitidal intervals and heights for all tide gauges, and spectral peaks were identified in these periodograms above noise thresholds. The separate analyses of lunitidal intervals and heights were combined to realize one comprehensive list of partial tides. An application is the usage of these constituents in tidal analyses and predictions with the HRoI.

The new set of 39 partial tides largely confirms the previously used set with 43 partial tides. It can be seen from Table

The verification based on observations from 98 tide gauges in the year 2016 suggests that the usage of the new constituents list can lead to slightly better predictions. In particular, the average standard deviations of the residuals are lower and four frequencies were reduced.

This study presents for the first time a thorough investigations of the long-period constituents used with the HRoI. The new list of constituents will be used in tidal analyses and predictions with the HRoI for German tide gauges starting with the BSH tide tables for the year 2020.

In future work, extensive comparison of the HRoI with the common harmonic method might provide more insights into the capabilities of both tidal analysis techniques. The German Bight would be an ideal area of investigation with its large network of tide gauges located both at the open North Sea and far within tidally influenced rivers.

The tide gauge observations used in this research are available from the respective authorities (cf. 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 (

Continued.

Continued.

Both authors designed the study and discussed the results. AB analysed the data and prepared the article with input from SMN.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE inter-journal SI)”. It is not associated with a conference.

This paper was edited by Philip Woodworth and reviewed by three anonymous referees.