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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
4.6 Statistical Testing of Estimated Parameters
Statistical testing was performed on each solution. The smallest
critical value computed for the global test was F(0.01:8,876) —
2.60 (occurring for load 2 for the LMS-Z210) at a level of
significance of 1%. The smallest computed test statistic (Eq. 3)
was 185.9 for the LMS-Z210 during load case 6. All models
satisfied their respective critical values. Therefore, no parameter
removal was necessary given these statistical test results.
The test was extended to individual parameter testing. The
smallest of all critical values used for comparison was 6.80. The
smallest computed test statistic (Eq. 4) was 20.2 (parameter by,
from the dead load epoch for the LMS-Z210). Analysis of the
results revealed that each parameter statistic grew larger as
testing went on. As the beam experiences increasing amounts of
deflection, the individual parameters are required to model the
additional curvature that, in turn, increases their significance in
the model. In all cases, the intercept terms (ag, and by)
exhibited much greater influence in the models than any other
parameter. None of the estimated parameters were deemed
insignificant at the 1% significance level for any of the models
and no parameter elimination is required using this test. The
developed models are therefore considered appropriate. The
statistical testing supplies confidence towards the derivation of
the models since they were developed from first principles of
beam deflection by integration and that each parameter
theoretically has a sound physical basis for inclusion in the
model.
4.7 Vertical Deflections
Vertical deflections were derived using the estimated models for
each of the eight load epochs. The x and y coordinates of each
of the 13 photogrammetric targets (constituting the top row of
targets on the beam) were passed into the estimated models to
compute a z-coordinate. Only the top row of targets was used
because they were the closest to the beam top. The z-
coordinates were then used to determine vertical deflections
between epochs. Each TLS set of vertical deflections (i.e. for
the Cyrax 2500 and the LMS-Z210) was compared to the
vertical deflections produced by the photogrammetry and the
differences are shown in Table 1.
RMS of Differences (mm)
Cyrax 2500 LMS-Z210
Nominal Vertical
Deflection (mm)
5 +0.12 +3.6
10 +0.14 +41
15 +0.47 +32
20 +0.26 312.3
25 0.24 +5.0
30 +0.27 +5.0
35 30.30 +17
40 +0.34 +1.4
Total RMS +0,29 +3.6
Table |. RMS of differences between TLS-derived and
photogrammetry-derived vertical deflections using
13 targets per deflection case.
Table | indicates that the estimated models using Cyrax 2500
data, compared to the benchmark photogrammetry, give an
overall RMS of differences of +0.29mm. The largest RMS of
differences is +0.47mm for the 15mm deflection case. The
957
overall RMS of differences for the LMS-Z210 is +3.6mm. The
maximum RMS is +5.0mm for the 25mm deflection case. The
overall RMS values represent a factor of improvement (in
precision) of 21 times for the Cyrax 2500 and 7 times for the
LMS-Z210 over the coordinate precision of each TLS.
The linear term, ay, was used to model rotation about the x-
axis. In adjustments undertaken without the y-term, plots of
residuals versus y-axis coordinates for the Cyrax 2500 shows a
distinct tilt of approximately 1.7? and a tilt of 2.7? for the LMS-
Z210 indicating that the beam top was not horizontal in the
reference coordinate system for all cases. Analysis of the aj,
parameter shows that it was consistently the same size for the
Cyrax 2500 dataset (-0.030 30.001) but fluctuated in the LMS-
Z210 results (-0.047 x0.017). This was primarily due to the
sparsity of data in the y-direction of the LMS-Z210 compared to
the Cyrax 2500. The uncertainty in the determination of the aj,
parameter caused vertical deflection measurements to be worse
for the LMS-Z210 when compared to results where y-term was
omitted (overall RMS of differences £2. Imm for all cases where
the y-term was omitted). Cyrax 2500 results were better with the
y-term included and were worse without it (overall RMS of
differences +0.46mm without the y-term).
5. EXPERIMENT II: CONCRETE BEAM
The second major experiment conducted to test the analytical
modelling strategy involved an ‘L-shaped’ (in cross section)
7.2m x 0.5m x 0.5m reinforced concrete beam that was loaded
until failure. The beam was formerly part of an old bridge.
which had been dismantled for the purpose of controlled
laboratory testing. The beam was placed in a heavy-duty
outdoor testing frame and supported at each end. The two load
points were near the centre of the beam.
The beam was loaded in increments up to 240kN
(approximately 13mm of vertical deflection), at which point the
load was relaxed (epoch seven). This permitted the zero-datum
of the contact sensors to be redefined. Loading resumed and
continued in increments until the beam failed (490kN). The
purpose of this loading schedule was to ascertain the elastic
properties of the concrete beam.
5.1 Set Up and Targeting
An LMS-Z210 was situated 7m from the beam and directly in
front of the test frame (see Figure 2). It was set up as high as
possible on the tripod enabling acquisition of points from the
top surface of the beam. Similar to the first experiment, the
position and orientation of the TLS was determined by
resection. Photogrammetry was used to benchmark the
experiment and the photogrammetric coordinate system
provided the reference frame for the experiment.
5.2 Data Collection
A total of 13 measurement epochs were acquired during the
period of testing. À dead load epoch was acquired at epoch zero
and also at epoch seven (where the load on the beam was
relaxed). The final measurement epoch where the beam was
intact was epoch 12 but contact sensors were removed prior to
this (after recording epoch 10) because failure was imminent.
The impending specimen failure did not affect the remote
measurement techniques (i.e. photogrammetry and TLS)
highlighting, through practice, the advantage of a remote
