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A. Boesch and S. Müller-Navarra: Reassessment of long-period constituents for tidal predictions 
Ocean Sci., 15,1363-1379,2019 
www.ocean-sci.net/15/1363/2019/ 
Height (high water, upper transit) 
40 60 80 
Angular velocity [7tn] 
120 
Figure 3. (a) Normalized periodogram of the heights of high wa 
ters (assigned to upper lunar transits) for the tide gauge Cuxhaven. 
Notice the logarithmic scale, (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 Cuxhaven leads 
to narrower spectral lines (solid blue curve) compared to Emden 
(dashed green line). 
4.3 Identifying relevant partial tides 
We aim to find all local maxima in a periodogram that are 
above a noise threshold. This threshold is calculated in a two- 
step process that is described in the following. 
In the first step, the strongest spectral lines are removed 
from the periodogram. The values above the 99.5th percentile 
are removed from the data set and a histogram is calculated 
from the remaining values p (100 bins with central values 
Xbin)- The histogram shows an exponential trend from a large 
number of data points with low periodogram values to a few 
points that fall into the bins at the upper end. An exponential 
curve, ybin = fl • exp(—.tbin/i’). is fitted to the histogram with 
fit parameters a and b. The process of removing data points 
above the 99.5th percentile from the periodogram is repeated 
0.0010 
0.0008 
£ 
ÍÜ 
cn 
■O 0.0006 
o 
o¡ 
Q. 
'S 0.0004 
(T3 
£ 
ê 0.0002 
0.0000 
Angular velocity [°/tn] 
Figure 4. Determination of the noise threshold for the tidal interval 
(high water, upper transit) at tide gauge Borkum, Fischerbalje. The 
strongest lines are removed from the periodogram (grey vs. green 
lines; first step as described in Sect. 4.3) and an exponential function 
(dashed red curve) is fitted to selected points (blue; second step as 
described in Sect. 4.3 ). The noise threshold (thick red line) is shifted 
up by 1 standard deviation. 
Borkum - tidal interval (high water, upper transit) 
— Periodogram (complete) 
— Periodogram (strong lines removed) 
— Noise threshold 
— - Fit to selected points 
20 40 60 80 100 120 
until the ratio max(p)/b falls below the value of 30. This 
value is based on experience. 
In the second step, the noise threshold is determined using 
a set of remaining points in the periodogram that represent 
a continuum above the noise level. The result is illustrated 
in Figs. 4 and 5 for lunitidal intervals and heights at the tide 
gauge Borkum. For this procedure, the periodogram is split 
into 25 sections with the same number of data points. The 
data point at the 99.5th percentile is selected in each section 
and an exponential function is fitted to these 25 points. The 
fit is repeated after a ler clipping. The noise threshold corre 
sponds to the resulting exponential function plus 1 standard 
deviation (solid red line in Figs. 4 and 5). 
In preparation for the following combined evaluation of 
the results from all tide gauges, the noise threshold functions 
from the different periodograms are averaged; this is sepa 
rately for lunitidal intervals (¿¡) and heights (L|,): 
Li(o)) = 0.0004816 • exp(—0.0101045 tn/° •«), 
L h (m) = 0.0024472 • exp(-0.0149899tn/° • «). 
These two functions represent mean lower-intensity bound 
aries for the selection of significant peaks. The expressions 
L{ and ¿h are unitless, due to the normalization of the Lomb- 
Scargle periodogram (Zechmeister and Ktirster, 2009). 
In addition to the intensity of the local maxima, the num 
ber of their occurrences in the different periodograms and 
their assignment to the partial tides determine their inclusion 
into the list of constituents for the HRoI. The local maxima 
must match the theoretically expected partial tides that have 
well-known angular velocities computable from the linear 
combinations of the rate of change of the four fundamen-