(39)
(40)
bid be
re X',
ad by
vex
WSX
(41)
T2.
215
r4
(42)
F the
1e ei-
(43)
cuated
where
is:
x'? y'? 712
2
Pll=== + ==> + === <K |= P [
2 2
x < K = ] -X(44
1£ 7 0 Sr
X
f:
For the standard ellipsoid (K = 1):
P [Zi 1]= 1-04" 0.199 (45)
which is obtained from distribution with three
degrees of freedom; so the probability that the
point be situated inside the standard ellipsoid
is & 205.
In order to establish confidence regions, we se-
lect the X level and compute the multiplier K.
For X = 0.05 :
2 2
p | 2% x. PIX <7.815 |= 0.95 - (46)
f=3 f:3 $=3
from which it resultsK = 7.815 = 2.7955.
Consequently, the probability that the point be
situated within or on an ellipsoid with the axes
a= 2.7955 VA, b=2.7955 VAZ, c=2.7955 VA31s 95%.
Other typical values are:
a) for P=99%: a=3.368 VA b=3.368 VAZ, 73.398
3
b) for P=99.9%: a=4.037 yA,, b=4.037VA2, c=4.037
For each one-dimensional marginal normal distrita
tion, the probability that each variable X, Y, Z
lies in the region within plus and minus one stan
dard deviation (+ 0x or + Oy or +02), from the
normal distribution function, is 68.27%. By con-
trast, the probability for joint event, which is
falling within the standard ellipsoid, is consi-
derable less, being only æ 20%.
A computer program called EROELIPS was developped
by the author and its formulation is based on the
above principles. The output of the program ERO-
ELIPS provides the eigenvalues, the eigenvectors
of each matrix Z 7) and the parameters of standard
error ellipsoid in three-dimensional space (a,b,
c,W, v, X).
The program also provides the coordinates of the
j-th triangulated point in the shifted coordinate
system X', Y', Z' and the parameters of ellipso-
ids of constant probability for different confi-
dence levels.
5. CONCLUSIONS
By means of error propagation and error ellipso-
ids presented above, it is possible to evaluate
the accuracy of photogrammetric determination of
positions in three-dimensional space and to con-
duct theoretical error studies based upon ficti-
tious photography without resorting to tedious
sampling techniques (e.g. Monte Carlo method).
From such investigations, the potential accuracy
of the multiple station analytical stereotriangu-
lation developped from Bundle Adjustment Method
or from Direct Linear Transformation in various
Situations can be ascertained and also the influ-
ence of various distributions of control can be
determined.
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