A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions
1372
Ocean Sei., 15,1363-1379,2019
www.ocean-sci.net/15/1363/2019/
-60 -40 -20 0 20 40 60 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Observation - prediction [min] Observation - prediction [m]
Figure 7. Histograms of the residuals for tide gauge Cuxhaven and
year 2016. 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), (a) Time differences with
a bin width of 4 min. (b) Height differences with a bin width of
0.04 m.
Table 5. Residuals of predicted and observed times of high and low
water: mean and standard deviation a in minutes.
Gauge
number
Gauge
name
43 partial tides
39 partial tides
ß
O
ß
O
101P
Borkum
-2.7
11.2
-2.1
11.0
103P
Bremerhaven
-6.4
10.4
-4.6
10.1
111P
Norderney
1.5
10.9
0.7
10.6
502P
Bremen
-7.3
10.8
-5.5
10.5
505P
Büsum
3.4
17.5
4.6
17.4
506P
Cuxhaven
-0.1
9.6
1.0
9.0
507P
Emden
-9.2
13.8
-8.2
13.3
508P
Hamburg
-10.1
10.4
-7.7
10.3
509A
Helgoland
-2.4
7.8
-2.6
7.7
51 OP
Husum
-5.1
12.0
-4.3
11.7
512P
Wilhelmshaven
-3.2
10.0
-2.7
9.6
Table 6. Residuals of predicted and observed heights of high and
low water: mean and standard deviation a in metres.
ing frequencies. 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
and event index k.
Figure 7 shows histograms of the residuals for the tide
gauge Cuxhaven. The panels on the left and on the right dis
play histograms for the times and heights, respectively. Each
panel contains one histogram for the tidal prediction with 43
partial tides (red) and one histogram for the tidal prediction
with 39 partial tides (yellow). Using the new set of partial
tides, the standard deviation of the residuals decreases from
9.6 to 9.0 min for the times and from 0.28 to 0.27 m for the
heights.
In the same way as for Cuxhaven, residuals are calculated
for the data of all 98 tide gauges included in the verifica
tion. The mean values, /i, and standard deviations, er, of the
residuals for 11 major ports are summarized in Table 5 for
the times and in Table 6 for the heights. Based on the re
sults from all tide gauges, the average standard deviation of
the residuals is 13.2 min or 0.28 m, respectively, using the set
of 39 tidal constituents. In most cases, the new set of partial
tides gives small improvements in /i and a.
The percentage changes. Act, in the standard deviations
are presented in the histograms in Fig. 8 (times) and Fig. 9
(heights) for all 98 tide gauges. The percentage change is
calculated in the following way:
Act [%] = 100 % • g39 pi ~ g43 pi , (6)
043 p.t.
where <739 pt . and <743 pt . are the standard deviations of the
residuals using the predictions with 39 partial tides (p.t.) and
43 partial tides, respectively. The average reductions of the
standard deviations are 2.41 % (times) and 2.30% (heights).
The two tide gauges with the largest improvements in this
Gauge
number
Gauge
name
43 partial tides
39 partial tides
ß
O
ß
O
101P
Borkum
0.05
0.24
0.03
0.24
103P
Bremerhaven
0.06
0.28
0.04
0.28
111P
Norderney
0.03
0.25
-0.01
0.24
502P
Bremen
0.01
0.27
-0.01
0.26
505P
Büsum
0.04
0.29
0.01
0.28
506P
Cuxhaven
0.05
0.28
0.04
0.27
507P
Emden
-0.01
0.28
-0.02
0.27
508P
Hamburg
-0.08
0.33
-0.12
0.31
509A
Helgoland
0.03
0.25
0.02
0.24
510P
Husum
0.02
0.29
0.00
0.28
512P
Wilhelmshaven
0.05
0.27
0.03
0.27
study, both with regard to times and heights, are Holmer Siel
(BSH gauge number 649B) at the North Frisian coast and
Bremervörde (687P) in the river Oste. Further tide gauges
with improved standard deviations are located all around the
area of investigation. Regarding the times (Fig. 8), the four
tide gauges with increased standard deviations are Wester
land (620P) and Hörnum (624P), located on the North Frisian
island of Sylt, and Dove-Elbe (727P) and Bunthaus (729P),
located upstream of the river Elbe. Regarding the heights
(Fig. 9), the five tide gauges with the largest positive changes
are also located upstream in the Elbe, namely Geesthacht
(732D), Altengamme (732A), Zollenspieker (73IP), Ilmenau
(730A) and Fahrenholz (730C). The water levels in this part
of the Elbe are partly influenced by river discharge, which
can lead to deviations from the tidal predictions.
The change in constituents has an influence on the remain
ing periodicities in the residuals. Periodograms are calcu
lated for the two sets of residuals (times and heights) for
each tide gauge. The 98 periodograms of each type are aver-