Arithmetical Operation 
17 
c) Processing of 19 years of gauge observations 
The astronomical rudiments of tidal theory have been sufficiently described 
(Horn, 1948). From these derives, amongst other things, the demand for the 
simultaneous analysis of respective time periods of at least 19 years. Table 6 in 
the appendix shows the beginning (31 Dec 1990, 23:45 UTC) and the end 
(31 Dec 2009, 18:15 UTC) of the time period of vertex observations at the Cux 
haven gauging station, transit times of the Moon and syntheses. 
The observed vertex times and levels (Table 6, Columns 5 and 7) must first be 
paired with the Moon transits (Columns 1 and 2). For this, definite mapping can 
be achieved for calm weather conditions (Müller-Navarra, 2009). In the event of 
strong storms, when onset times deviate from HWT and LWT by several hours, 
amends are required. Yet this, too, may be largely automated. The data set from 
the Cuxhaven gauging station - the basis for these analyses - ranges from 
high water after lower culmination (1=14467, k=3) to low water after upper cul 
mination (1=21172, k=2). 
In the case of the hydrograph Illustration, for the same period of time, each 
transit at upper culmination Is assigned 96 subsequent sampling points (see 
above). 
d) Filtering of data 
Since the water level measured at coastal gauging stations proves to be very 
susceptible to wind, cumulative stormy years with many storm surges or long 
periods of lowered water levels can “spoil” the forecast, especially as regards 
the long-periodic elements. Filtering of the 19-year time series has proven itself 
in practice, excluding all individual cases deviating by more than 3 standard 
deviations from the mean value. Table 3 shows mean values and standard devi 
ations for the observation, the filtered data set and the vertex value forecast. Fil 
tering cancels out only 0.1 % of data as regards the onset times and approxi 
mately 1.5% of observations as regards the water levels. 
In the case of hydrograph processing, the mean values and the standard devi 
ation (SD) are represented in a graph (Figure 1). Here, the numbers of cases m 
average at 6623 and, after filtering, at 6563 of 6689 possible numbers. It is con 
spicuous that the greatest standard deviations occur approximately 2 h after 
LWT. This Is precisely where the main problem of hydrograph forecasting for 
tidal rivers lies. The further upstream the tide gauge, the shorter the flood dura 
tion and the faster the water level rises. In extreme cases, sometimes also only 
at spring tide, this can result in phenomena akin to tidal bores with flood dura 
tions being virtually zero.