Figure 8. East residual for tracks 1-5
Figure 9. North residual for tracks 1-5
Figure 10. Horizontal residual for tracks 1-5
From the graphs presented, it is visually apparent that the train
starts on track 2 and after 55 s changes to track 3. The large
change in the east, north and horizontal residuals of
approximately 2 m indicates the change of track. Therefore, the
data was broken into two sections and each section was analysed
individually. The change of track is evident in the LOM test,
whereby the null hypothesis for track 2 is rejected after 55 s and
the null hypothesis for track 3 is accepted. This is also confirmed
in the GOM test, where within 4 s, it is confirmed at 90%
probability that the train has changed track (Table 3).
Table 2 contains the number of times each predetermined
probability was met, divided by the number of points within that
section of track. When Table 2 and Table 3 are analysed together,
it can be seen that once a certain probability is met, this
probability is retained for the remainder of the section.
Probability %
90
95
99
99.9
99.99
99.999
Section 1
51/55
50/55
48/55
45/55
43/55
41/55
Section 2
43/46
42/46
41/46
40/46
38/46
37/46
Table 2. The number of times each predetermined probability
was met, divided by the number of points in the section
Probability %
90
95
99
99.9
99.99
99.999
Section 1
5
6
9
11
13
15
Section 2
4
5
6
7
9
10
Table 3. Time (s) taken to achieve predetermined probabilities
The residuals in Figure 8 and Figure 9 (55 - 101 s) that appear
parabolic in shape, are due to inadequate modelling of the track.
It is assumed that track between two nodes is straight, however,
in practice this is seldom the case. The parabolic errors associated
with generalising the track to a series of nodes are a maximum
half way between two nodes and a minimum near the map nodes.
To reduce the magnitude of these errors, the track needs to be
mapped at more regular intervals (Figure 11). However, one must
consider the increase in file size associated with a decrease in the
distance between two map nodes.
Figure 11. The effect of reducing the distance between two
nodes. Note the reduction in parabolic errors using option b).
In Figure 8, the east residual for all five tracks (88 - 101 s)
appears to converge. To determine the residual between the
filtered position and the track, a perpendicular offset from each
track is calculated using simple trigonometry. From this offset, a
map matched position on the track is determined. This assumes
that there is no long-track error associated within the filtered
position. Therefore, when the train is travelling in an easterly
(88 - 101 s) or westerly direction, there will be minimal, if any,
east or west residual. Likewise, when the train is travelling in a
northerly or southerly direction, the respective residuals will be
minimal. However, when the train is travelling between two
cardinals, i.e. in a north-easterly direction, a residual in both the
east and north component would be expected (0 - 55 s). As the
train travels closer to a cardinal, that cardinals residual will
diminish and the other cardinals residual will increase (55 - 88 s).
