(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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