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A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions 
Ocean Sei., 15,1363-1379,2019 
www.ocean-sci.net/15/1363/2019/ 
to have one comprehensive set of constituents that can be 
used for all tide gauges under investigation. Horn (1960) pre 
sented 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 (cf. 
Table 2 in Sect. 2). To our knowledge, no documentation of 
the methods and specific water level records that were used 
to prepare these lists of angular velocities exists. 
The objective of this work is to review the set of partial 
tides used with the HRoI by determining the most impor 
tant 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. 3. The analysis of high- 
and low-water time series is described in Sect. 4. In Sect. 5, 
tidal predictions based on an existing list of partial tides and 
predictions based on the new set are compared with observed 
water levels. The article closes with a comparison of predic 
tions made with the HRoI and the harmonic method for two 
sites (Sect. 6) and the conclusions (Sect. 7). 
2 Harmonic representation of inequalities 
The harmonic representation of inequalities (HRoI) is a 
derivative of the non-harmonic method by essentially trans 
lating 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 waters are cal 
culated by adding mean lunitidal intervals and correspond 
ing inequalities to the times of lunar transits. Likewise, the 
heights of high and low waters are determined by adding cor 
responding 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). 
The original implementation of the HRoI, as introduced by 
Horn (1948, 1960), can be used to calculate vertices of tide 
curves, i.e. high-water time, high-water height, low-water 
time and low-water height. In this form the method is tailored 
to semi-diurnal tides. Miiller-Navarra (2013) shows how the 
HRoI may be generalized to predict tidal heights at equidis 
tant fractions of the mean lunar day. This generalization al 
lows for the determination of the full tidal curve at a chosen 
sampling interval. Here, we focus only on the application of 
calculating the times and heights of high and low waters. 
According to Horn (1960), the HRoI combines the best 
from the harmonic and the non-harmonic methods: the ana 
lytical procedure of the first method and the principle of cal 
culating isolated values directly, which a is characteristic of 
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 shallower waters, such as the German Bight, and in 
rivers. As the HRoI uses only observed times and/or heights 
Table 1. The high and low waters are classified into four types 
(event index k). 
k Description 
1 high water assigned to upper transit 
2 low water assigned to upper transit 
3 high water assigned to lower transit 
4 low water assigned to lower transit 
of high and/or low waters, the method can also be applied 
when a record of the full tidal curve is not available (e.g. his 
toric data) or when a tide gauge runs dry around low water 
(e.g. analysis of only high waters). 
Let (tj,hj), j = 1,..., /, be a time series of length J of 
high- and low-water heights hj recorded at times tj. All 
times need to be given in UTC. The HRoI method is based 
on the assumption that the variations in the individual heights 
and lunitidal intervals around their respective mean values 
can be described by sums of harmonic functions. The luni 
tidal interval is the time difference between the time tj 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 Fig. 1 in Sect. 3 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 tran 
sit on 31 December 1949, which has been arbitrarily set to 
iii — 0 to- 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 
Table 1. The differentiation between upper and lower tran 
sits allows for changes in the moon’s declination which al 
ternately advance and retard times, and increase and decrease 
the heights of successive tides (diurnal inequality). 
A full tidal analysis with the HRoI comprises the inves 
tigation of eight time series (heights and lunitidal intervals 
of the four event types listed in Table 1). Each time series is 
described by a model function, y, of the following form: 
L 
y(n t ) — a 0 + ^ [a t cos(&>;n t ) + a i+L sin(&>;n t )] . (1) 
l=\