A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions 
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Ocean Sci., 15,1363-1379, 2019 
Change in standard deviation [%] 
Figure 8. Histogram of the change in the standard deviation of the 
residuals of high- and low-water times for all 98 tide gauges. 
Change in standard deviation [%] 
Figure 9. Histogram of the change in the standard deviation of the 
residuals of high- and low-water heights for all 98 tide gauges. 
aged. The resulting mean periodograms are shown in Figs. 10 
and 11. In both figures, the strongest peaks are located at 
very low angular velocities (<1° tn -1 ). As mentioned before, 
the unambiguous identification of partial tides at these peri 
ods is difficult and consequently no major improvements are 
achieved in reducing the (average) residual periodicities in 
this range. Four further strong peaks are visible in both fig 
ures at about 15, 25, 52 and 64° tn -1 for the prediction with 
43 partial tides. These peaks are clearly reduced with the new 
predictions (39 partial tides). 
6 Comparison of predictions with observations: the 
HRoI and the harmonic method 
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 
Figure 10. 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 ) 
Angular velocity [°/tn] 
Figure 11. Same as Fig. 10 but for the high- and low-water heights. 
demonstrate the respective capabilities. The comparison is 
done for the two tide gauges at Cuxhaven, Steubenhoft, 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. 
6.1 Tidal analysis and prediction 
The predictions with the HRoI (39 partial tides) are the same 
as in Sect. 5. The harmonic analysis is based on continuous 
observations from the years 1996 to 2014 at 10 min intervals. 
The harmonic constituents (amplitudes H and phases g) and 
the constant vertical offset, Zq, are determined from a least-