1374 
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/ 
Cuxhaven, Steubenhöft Hamburg, St. Pauli 
Figure 12. Observations and two predictions for the tide gauge Cux- Figure 13. Observations and two predictions for the tide gauge 
haven, Steubenhoft. The first 10 d of June 2016 are shown. Hamburg, St. Pauli. The first 10 d of June 2016 are shown. 
squares fit with the following model function: 
L 
Tharm = ¿0 + z [Hi • cos (V,(fo)+ &>,?-£,)]. (7) 
/=i 
We use the 68 partial tides (with angular velocities &>/) 
from Foreman (1977). These are also the default constituents 
in the MATLAB packages t_tide and UTide (Pawlowicz 
et al., 2002; Codiga, 2011), which have become widely ac 
cepted standard implementations of the harmonic method. 
Since the data records exceed 18.6 years, we add the par 
tial 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 to of the time series 
and the astronomical argument V/(to) is calculated for each 
partial tide using the expressions for the fundamental astro 
nomical arguments as published by the International Earth 
Rotation and Reference Systems Service (2010, Sect. 5.7). 
The analysis is applied in two iterations with a 3er clipping 
in-between to remove outliers in the observations. 
We show in Figs. 12 and 13 the predictions and obser 
vations 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). 
6.2 Evaluation of residuals 
As in Sect. 5, 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 
Table 7. Residuals of predicted and observed times and heights of 
high and low water: mean and standard deviation a. 
Times (min) 
Gauge name 
HRoI (39 p.t.) 
Harmonic pred. 
ß 
a 
ß 
a 
Cuxhaven 
1.0 
9.0 
12.0 
12.9 
Hamburg 
-7.7 
10.3 
11.8 
15.1 
Heights (m) 
Gauge name 
HRoI (39 p.t.) 
Harmonic pred. 
ß 
a 
ß 
a 
Cuxhaven 
0.03 
0.27 
0.05 
0.29 
Hamburg 
-0.11 
0.31 
-0.10 
0.37 
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 7. The differences for the heights 
are within a few centimetres. For the times, the standard de 
viations are approximately 4-5 min larger in the case of the 
harmonic method. The residuals for the times are also shown 
in Fig. 14, 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 pre 
diction 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 applica 
tion of the HRoI is especially useful for tide gauge locations 
in rivers.