Gaussian function to
ribution and examin-
sampling distribution
an the Gaussian dis-
1oted by solid line in
k and fatter tails, the
igure 3) also deviates
on. These departures
from making statis-
n distributions men-
statistical inference
o based sampling dis-
tion. This statistical
'omputation which is
mphasized that such
hich are problem de-
ks
nt design tool used to
ivity prior to execut-
veral reliability mea-
random errors for the
da, 1968). Although
ability measures may
e not been completed
of such derivations,
evaluate network re-
liability is examined
tected under any cir-
external reliability is
nonunique estimator
ral reliability is to de-
vation must be before
sh examination of the
ept is modified such
rroneous observation
ally or otherwise. To
n under this scenario,
1 conjunction with a
igle observation is re-
ts impact on blunder
is presented in Fig-
tions. After 100,000
1g distribution of each
ms in Figure 2. No-
luctuate in height and
uctuations is network
0 m is identical for all
i. The L2 norm resid-
riences these fluctua-
k geometry. However,
| the trilateration net-
simulations are com-
tion changes substan-
la 1996
Figure 4: 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
0.2 0.2 Il 0.2 |
Lise Ex zz o E Lir
A 0 1 23 0 1 A 0 1
0.8 0.8 0.8
0.6 0.6 0.6
4 5 6
0.4 Ji 0.4 0.4
0.2 | 0.2 0.2
Ls étre elite
g À 0 1 = Oo 1 = 0 1
=
5
os 0.8 0.8
ce
5 0.6 0.6 0.6
= 7 8 9
50.4 0.4 0.4
o2 nmn o2 0.2
$ o d o | o
x zZ 0 1 21 0 1 =1 0 1
œ
0.8 0.8 0.8
0.6 0.6 0.6
10 11 12
0.4 0.4 0.4
0.2 0.2 0.2
x D cc ne ge o
24 o 1 = o 1 e 0 1
0.8 0.8
0.6 0.6
13 14
0.4 0.4
0.2 0.2
i Dit o !
-1 o 1
Residual Magnitude (meters)
tially as shown in Figure 4. The most significant change
occurred in observation 9 which is illustrated by a “spike-
only? distribution, indicating that residual 9 was equal to
zero for all 100,000 simulations. Consequently, a blunder
in observation 9 will never be found using L; norm estima-
tion given the existing network geometry and the stated 14
observations. To remedy this deficiency, the network could
be redesigned to incorporate new observations and/or new
network stations into the network. Although the L1 norm
fails to identify an erroneous observation under this low
redundancy scenario (redundancy = 14 - 9 = 5), the L;
norm holds more promise at blunder detection given ample
redundancy.
5.3 External Reliability
External reliability refers to the effect an erroneous obser-
vation has on the parameter estimates and has been stud-
ied by many authors (Baarda, 1968), (Mackenzie, 1985).
Experimentation with external reliability and the L1 norm
indicates that 1n some instances an erroneous observation
may have no effect on the L4 parameter estimates at all.
This unusual circumstance arises because Li norm min-
imization fails to provide unique parameter estimates in
some cases. As an example, the simple trilateration net-
work in Figure 5 was evaluated to illustrate the region
where the Z1 norm has an infinite number of correct solu-
tions. These solutions lie on the plane defined by ABCD in
Figure 6. When examples such as these arise, the network
could be redesigned to incorporate additional observations
and/or stations which will reduce or eliminate the effects
of the network deficiencies.
6 Statistical Evaluation of Residuals
Removing observations from a network must be well jus-
tified, especially given the extensive time, cost and labor
involved in acquiring high quality observations. There-
fore the purpose of this section is to determine whether an
observation should be removed from a network based on
41
Figure 5: Trilateration Network II
^ (1300, 1400»
zt
À
su
92
"p
300.04 m 300.04 m
(Observed) (Observed)
A^ > A
«1000, 1000) (1600, 1000)
3
£ v
>
£6
eoo
Su
“a
Legend
A Fixed Station e
«1300, 600)
o Unknown Station
(xy? Coordinate Definition Not to scole
Figure 6: L1 Objective Function Corresponding to Figure
5
40
35
30
Objective Function, Tivl (unitless)
<>
NAN
NS OZ
2
SSZS
= — ==
LL
DE
2S
Soe
SZ S>ZS>
Nea e
NEED AR
NE SZ
1000.05 1300.1
1300.05
999.95 1299.95
Y-coordinate (meters) X-coordinate (meters)
the empirical sampling distribution presented in Section
4. The statistical test used to identify erroneous observa-
tions is based on Monte Carlo generated distributions for
the null hypothesis.
6.1 Tests on Individual Residuals
In this section statistical inference is based on a technique
of simple exponential curve fitting to the empirical sam-
pling distribution of each residual. Using the area under
the exponential curve, a straightforward computation of
critical values results. The residual sampling distribution
displayed in Figure 3 is symmetric and contains a sharp
change in grade at the apex of the sampling distribution.
Since the sampling distribution is symmetric, the curve
fitting process will take place in the positive quadrant of
the residual sampling distribution. The exponential equa-
tion used to fit the sampling distribution (with the spike
removed) is,
bx
"s zz 60:
$2.2, db TR (9)
where
a, b, c are unknown parameters
yi is the relative frequency in the i'^ case
x; is the residual magnitude in the it? case.
In addition to accounting for the exponentially shaped por-
tion of the sampling distribution, the sampling distribu-
tion must include the zero residuals (spike) in the basic
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
