Appl. Sci. 2023, 13, 1872 
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Table 3. Night-time signals used for the position and time estimation in green with the ”h” flag. 
Signals in red with the “uh” flag were not used. 
Hoburg Rozewie Holmsjö Helgoland Hammerodde Groß Mohrdorf 
CWI1 CW2 CW1 CW2 CW1 CW2 CWI CW2 CW1 CW2 CW1 CW2 
The R-Mode position solution was compared with the GNSS RITK reference track, as 
shown in Figure 8a in a local east-north coordinate system. It can be noted that the ship 
kept its position almost stable overnight. Therefore, we can think of this test as a semi-static 
case with a very limited position variation of the ship, in the order of 20m, and induced by 
the wind and waves. 
dm 
£ 
© 
> 
150 
00h 
50 
. 
—50 
-—-- R-Mode 
—— RTK 
LO0- 
E 
50! 
o 
£ 
t 
£ 
-100: 
- In I 
5050-100 -50 0 50 100 150 
East [m] 
EA EN 
A 
—100 -50 0 50 100 
Fast [m] 
= 
250: 
S 
€ = 
n 
3 
2150 
bo 
5 100. 
5 100 
59' 
050 160 ; aka | 
—150 -100 -50 0 50 100 150 
Clock bias [ns] 
‘CC! 
Figure 8. Night-time results. (a) Reference RTK (red circles) and R-Mode (blue crosses) tracks, 
(b) Scatterplot of the R-Mode positioning error in the east and north. (c) Histogram of the estimated 
teceiver clock bias. 
The scatter of the R-Mode solution is presented in Figure 8b, where it is clear that the 
DOP plays an important role in this case. In fact, as mentioned before, only three stations 
were used to solve the navigation equation, and this increased the overall DOP. The east 
DOP and north DOP increased to 1.3 and 4.9, respectively, whereas the HDOP reached a 
value of 5.0. Thus, a loss in the performance level is expected. The 95th quantile accuracy 
for this data set is 55.3 m, which again satisfies the requirements for coastal navigation, as 
stated in the previous section. Last but not least, the accuracy of the time estimation is 
given in the histogram represented in Figure 8c. By comparing this histogram with the one 
in Figure 7c, we can clearly observe the decrease in the timing accuracy. In particular, the 
overall bias of 2.3 ns improves, but the standard deviation of 32.3 ns increases by a factor of 
3, approximately. 
In order to see the sky-wave effect on the ranging performance, the unbiased nor- 
malized distribution of the range error for the CW1 of all the transmitters is shown in 
Figure 9, for daytime and night-time. Looking at the Groß Mohrdorf plot (Figure 9a), a 
small difference between the daytime and night-time distributions is visible. Surprisingly, 
the standard deviation for the daytime range error is slightly higher (3.0 m) than the night- 
time one (1.7m). Our hypothesis to explain this unexpected result is that the daytime 
accuracy is somewhat degraded by the AGDF impact. Additionally, there might be a dif- 
ference between daytime and night-time atmospheric noise, which can change the carrier 
to noise-density ratio and, therefore, the accuracy of the phase estimation. Unfortunately, 
che carrier to noise-density ratio estimation was not available at the time of the campaign. 
Nevertheless, no sky-wave was expected to be visible for that radio beacon due to the 
distance of only 62 km. 
For Hammerodde (Figure 9b), there is a difference between the daytime standard 
deviation, 5.9 m, and the night-time one of 8.3 m. In this case, a decrease in the accuracy was 
expected due to the greater distance of about 212 km between the receiver and transmitter.