A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions 
1367 
www.ocean-sci.net/15/1363/2019/ 
Ocean Sei., 15,1363-1379, 2019 
4.1 Data preparation 
Data preparation includes the assignment of lunar transit 
numbers, n t , and the calculation of lunitidal intervals as de 
scribed in Sect. 2 for each record of high or low water. The 
lunar transit times are calculated following the algorithm by 
Meeus (1998, Chap. 15) with the modification of direct cal 
culation of lunar coordinates using the periodic terms given 
in the work by Chapront-Touze and Chapront (1991). 
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 3er clipping separately for the eight 
time series analysed with the HRoI (see Sect. 2). 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. 
4.2 Frequency analysis 
The observed heights and lunitidal intervals (y) can be under 
stood as being functions of the assigned transit number (n,). 
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. In 
stead, we use the generalized Lomb-Scargle periodogram as 
defined by Zechmeister and Klirster (2009), including their 
normalization if not mentioned otherwise. The frequency 
scale covers the range from 0.0001 to 2tn _1 with an in 
terval of 0.01999 tn -1 (100000 points in the periodogram). 
This corresponds to approximately 0.0057-114.5916° tn -1 
or 0.0002^1.6130° 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. Press et al., 1992; Prabhu, 2014). 
We apply a Planning 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 PlRoI. In Figs. 2a 
and 3 a, we show periodograms of the lunitidal intervals and 
heights (of high waters assigned to an upper transit, event in 
dex k — 1) for the tide gauge Cuxhaven. Cuxhaven (together 
with Plamburg) provides by far the longest time series that is 
used in the analysis (cf. Table Al). In these figures, the verti 
cal axis is normalized to the strongest peak and the horizon 
tal axis is converted to degrees per transit number for better 
(b) 
Angular velocity [°/tn] 
0.08 
0.07 
E 
(o 0.06 
CD 
O 
o 0.05 
a> 
■o 0.04 
<v 
N 
1 0.03 
o 
2 0.02 
0.01 
0. 
— Cuxhaven, Steubenhöft 
— Emden, Große Seeschleuse 
it 
05 27.10 27.15 27.20 27.25 27.30 27.35 27.40 27.45 
Angular velocity [°/tn] 
Figure 2. (a) 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 is trun 
cated at 0.1 for better visibility of weak lines, (b) Zoomed-in view 
of the region with the spectral line corresponding to half a tropical 
month (Mf) at 27.2764618° tn -1 . The longer time series for Cux 
haven leads to narrower spectral lines (solid blue curve) compared 
to Emden (dashed green line). 
comparison with Table 2. 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. 3a (noise level increases towards lower angular veloc 
ities). Figs. 2b and 3b 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.