Arithmetical Operation 
13 
4 Arithmetical Operation 
HRol can be used for the calculation both of vertices (high water time HWT, 
high water height HWH, low water time LWT, low water height LWH) and of tide 
hydrographs. The vertices require 8 least square fits, since one has to differen 
tiate between transits over the prime meridian In the upper culmination and 
over 180° west longitude In the lower culmination. In the case of the 
hydrographs, the number of sampling points Is determined on the basis of the 
time step. For Instance, the choice of a time step of 15 Moon minutes 
(=15.5257515 min) necessitates 96 least square fits. A larger time step does 
not seem useful given the rather fast rise speeds at some tide gauges at the 
onset of the flood current, whereas a smaller one may well be useful, provided 
the availability of sufficient computing capacities. 
Syntheses of vertex times and analogously of vertex levels (Table 1) are 
effected by equation (1). The synthesis regarding onset of Individual HW and 
LW produces the time lag after the respective preceding mean transits t. (see 
below). 
ZK, - i cosr:; •' • c ', . - sin " -0; ^ = 1 4;/ = 1,2 (1) 
j=i 
m^i = N start ,...,N end . 
Table 1: Inequalities In time (1=1) and height (l=2). (k= 1,2: upper culmination, 
k=3,4: lower culmination) 
Inequalities 
1=1 
1=2 
k= 1 
w u.THWT-( 
w 12 .=HWH. 
k=2 
n-,,=LWH. 
k=3 
"'urHWT-i, 
tv u =HWH. 
k=4 
w 4U =LWT\-/. 
w 42 ,=LWK 
Table 2: Significance of constant parts from (1). (k= 
k=3,4: lower culmination) 
1,2: upper culmination, 
Significance 
1=1 
1 = 2 
k= 1 
c II0 =MHWI 
c^= MHW 
k=2 
c, ;o =MLWI 
c„ 0 =MLW 
k=3 
c, 10 =MHWI 
c,, 9 =MHW 
k=4 
c 4 0 =MLWI 
c 4 , 0 =MLW