A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions 
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Ocean Sei., 15,1363-1379, 2019 
Figure 5. Same as Fig. 4 but for the heights at tide gauge Borkum, 
Fischerbalje. 
tal astronomical arguments s, h, p and N' (see Sect. 2). The 
angular velocities of 1268 partial tides have been precalcu 
lated using, again, the expressions for the fundamental astro 
nomical arguments as published by the International Earth 
Rotation and Reference Systems Service (2010, Sect. 5.7). 
The ranges of the linear coefficients m are chosen based on 
experience (cf. Table 2): 
m s —0 8 
m h — -8 3 
nip — —2 3 if ((m s — 0 and mi, > 0) 
or (m s > 0 and w/, > —m s — 1)), 
_ 0, 1 if m s — 0 and mi, — 0 and m p — 0 
mN —1,0,1 if m s ^ 0 or mi, ^ 0 or m p ^ 0 
A partial tide from the precalculated list is assigned 
uniquely to the closest peak in the periodogram if the dif 
ference in angular velocity is less than half the spectral reso 
lution. The spectral resolution r is defined as 
r — 360° /T, (3) 
with T being the length of the time series in transit num 
bers. For example, the spectral resolution of a time series of 
19 years is 
360° 
19yr-365.25dyr 1 ■ r 
0.05° tn“ 1 . 
(4) 
where r = 1.03505013 dtn -1 is the length of the mean lunar 
day. 
For each identified partial tide, we calculate (i) the percent 
age of periodograms in which the partial tide has been de 
tected, separately for lunitidal interval (A;) and height (Ah), 
and (ii) the average intensity in the periodograms, separately 
for lunitidal interval (/¡) and height (/h). In order to be con 
sidered relevant, a partial tide with angular velocity o> must 
meet the following criteria: A; > 33 % and /;(&>) > ¿¡(tu) or 
Ah > 33 % and 4(m) > L^w). All partial tides that meet 
these selection criteria are listed Table 3. 
4.4 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 peri 
odograms and other considerations for an operational appli 
cation. These adjustments lead to the set of partial tides in 
Table 4. 
The periodograms calculated from longer time series offer 
a higher spectral resolution and contain more spectral infor 
mation compared to the periodograms of shorter time series. 
This is demonstrated in Figs. 2b and 3b 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 in 
corporated adequately. Therefore, the periodograms of Cux 
haven 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 par 
tial tides with the following Doodson numbers were identi 
fied 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 3 that differ only by Am,/y- = ± 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 an 
gular velocities and the identification of partial tides below 
1° tn -1 becomes less clear. For this reason, and after inspect 
ing 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 2). 
The final set of long-period partial tides from our analy 
sis is listed in Table 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 eval 
uation of data from all tide gauges and is calculated by the 
following procedure: 
Ri — rank(norm(/;(&>) — ¿¡(tu)) • A;), 
Rh = rank(norm(/h(<w) — ¿h( ft) )) • Ah), 
R = rank((3Ri + R h )/4), (5) 
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