Figure 1: Trilateration Network Diagram
Fixed
(y only)
Legend
À Fixed Station
O Unknown Station
2 Dimensional Horizontal Network
Figure 2: Empirical Residual Sampling Distribution for
Figure 1
0.8 0.8 0.8
0.6 0.6 0.6
1 2 3
0.4 0.4 0.4
o Lx ica o EX Ade o I SS | mm
-1 o 1 -1 o 1 -1 o 1
0.8 0.8 0.8
0.6 0.6 0.6
4 5 6
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0.2 Il 0.2 0.2 | |
o DL o Dic o xcd cl
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$
E 0.8 0.8 0.8
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S 7 8 9
= 0.4 m 0.4 0.4
g
0.2 0.2 0.2
S o L3 dc lr o Cn Io L Ten
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a
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10 11 12
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o: 5 x o wem JN wem Ó Td Lac ok
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13 14 15
0.4 ii 0.4 0.4
0.2 | | 0.2 [T] 0.2
gb mem qe cies 9 CE
ed o 1 = o 1 1 o 1
Residual Magnitude (meters)
Figure 3: Nonnormality of Li Residuals
T T T T T
0.03} |
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3
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Residual Magnitude (meters)
40
Normality was tested by fitting a Gaussian function to
the Monte Carlo based sampling distribution and examin-
ing the fit. In this case, the empirical sampling distribution
has fatter tails and a higher peak than the Gaussian dis-
tribution. The Gaussian curve is denoted by solid line in
Figure 3. In addition to the high peak and fatter tails, the
spike (which has been removed from Figure 3) also deviates
sharply from the normality assumption. These departures
from normality prevent the engineer from making statis-
tical inference based on the common distributions men-
Instead, accurate statistical inference
can be achieved using the Monte Carlo based sampling dis-
tributions described earlier in this section. This statistical
tioned previously.
inference may include critical value computation which is
discussed in section 6. It should be emphasized that such
an approach yields critical values which are problem de-
pendent and even data dependent.
5 Preanalysis of Networks
5.1 Reliability
Preanalysis of networks is an important design tool used to
evaluate network precision and sensitivity prior to execut-
ing fieldwork. Baarda has derived several reliability mea-
sures based on normally distributed random errors for the
L5 norm mathematical model (Baarda, 1968). Although
counterparts to many of these L2 reliability measures may
exist under Li, these derivations have not been completed
at the time of this writing. In lieu of such derivations,
the Monte Carlo method is used to evaluate network re-
liability. First, network internal reliability is examined
to determine if a blunder can be detected under any cir-
cumstances, and secondly, network external reliability is
examined to visualize the effects of a nonunique estimator
on parameter estimates.
5.2 Lj Internal Reliability
The underlying concept behind internal reliability is to de-
termine how large an erroneous observation must be before
it will be statistically detected through examination of the
In this section, this concept is modified such
that interest lies in detecting the erroneous observation
under any conditions at all, statistically or otherwise. To
residuals.
examine the behavior of the L; norm under this scenario,
the Monte Carlo method is used in conjunction with a
geodetic trilateration network. A single observation is re-
moved from the network to study its impact on blunder
detection.
The initial trilateration network is presented in Fig-
ure 1 and has a total of 15 observations. After 100,000
Monte Carlo simulations, the sampling distribution of each
residual is described by the histograms in Figure 2. No-
tice that the sampling distributions fluctuate in height and
shape. The primary cause of these fluctuations is network
geometry since the a priori à = 0.220 m is identical for all
observations as mentioned previously. The L2 norm resid-
ual sampling distribution also experiences these fluctua-
tions in height as a result of network geometry. However,
when observation 15 is removed from the trilateration net-
work and a new set of Monte Carlo simulations are com-
puted, the residual sampling distribution changes substan-
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
Figure 4: R
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norm holds m
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5.3 Exter
External relia
vation has on
ied by many
Experimentat
indicates that
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work in Figu
where the L4
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Figure 6. Wh
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