International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
7
M=|EE..E] (8)
A weight matrix P,, is:
2v-1
F4 70924 (9)
The most probable value of C and its variance — covariance
matrix are:
é-(M'P,M) M'P,d (10)
= an
Q; M' PM
OG is a priori variance factor of unit weight obtained from
Equation (1) or Equation (3). Since the number of targets in
Block B should be small, the postriori variance obtained from
Equation (7) is not employed.
To evaluate whether Block B was deformed, the null hypothesis
Hyg lcm (No deformation occurring)
is tested against the alternative hypothesis
H, :€ — € x (Deformation occurring)
Assuming that the null hypothesis is correct, the tested
statistical quantity is given with unknown oz by:
7 =6 0665) FR. f + fr) (12)
where f, and f, are fin Equation (2) for respective epochs or
degrees of freedom of observations, and 7, = rank ( Q; )
and F(r , f) means F-distribution with degree of freedom (r. f).
if a value of 7 is larger than a value T at the level of
significance q , the null hypothesis can be rejected.
Secondly testing power is considered. If the alternative
hypothesis is correct, the tested statistics with unknown
variance factor od becomes:
^T 5-15 4.2 ’ ; : D
Ja Qs 6C/róg)-Fr,f; - fpi) (13)
- . . . c ^s
where F'is non-central F -distribution, and 6° is a non-
centrality and is expressed as:
Ys AT yh 2
oO 2650 575: (14)
i; . ^ . . "
Because oj; is unknown, it is replaced by a priori variance. A
. 2c. . .
non-centrality à; is determined so that the second kind error
probability at T", equals 1- for the given . Then, it is
sufficient if the following is valid for every target.
07505 (15)
5. RELIABILITY OF NETWORKS CAPABLE OF
DETECTING GROSS ERROR
The data snooping method tests whether there exists gross
errors in the observations for Equation (1) in the object
coordinates that the epoch of each time is tested with. The
symbol of epoch is omitted in the following equations. It is not
impossible to assume that there are multiple gross errors in a
single set of observations. However, here it is assumed that at
most one gross error exists in the observations. The following
null hypothesis
H,:E(e| H)s AX (16)
is tested against the following alternative hypothesis.
H, : E(e| H,) - AX * Z,Vs (17)
where Vs is magnitude of a gross error and a scalar. Z, isa
(m, 1) vector and denotes the location of the gross error. The
k -th element is set to 1 and the others to 0, if testing that there
is a gross error in the k-th (k = 1, ...., m) observation.
With the unknown variance of unit weight c: the following
series of equations hold for the alternative hypothesis (Koch,
1999b).
By specifying rank(4,) r,-n-7.
TzIRIDB/O,/m-r, 1
E mer ché )
6^ e ViQvs/g?
Q- V'PV = e" pO Pe
Q,-Q-R
R=V5'Q'V§=V"PZ QZ PV (18)
Vi-QZIPQ,Be--Q,Z; PF
ri
Q, - (ZI Pg, Pz,)
V = m0); Re
za
-l T T
op -PB -A, (A P.A,) A; J
where VS is the most probable value of Vs, Q; is a co-factor
matrix of residuals, and (2 is a sum of a square of
discrepancies when there is no gross error, R is the sum of a
square of discrepancies accounted for by a gross error when
there is a gross error. €, decreases, if there is a gross error,
and hence 7 increases, then the hypothesis is rejected (it is
interpreted that there was a gross error.)
» 0. . 2 "fo :
The lower limit of non-centrality À, that satisfies the testing
power fjfor a level of significance is evaluated for a gross
error to be detected. After examining the network reliability
against the individual observation / (j 7 1, ...., m), it is judged
sufficient if the following is valid.
6 24 (19)
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