187
i = (X|,Y |t Zi) T (3)
. or in the case of a GPS vectors between a
known (I.G.S./A.S.I. or local) station and a un
known point; then the (3,1) observation sub
vector is
Ijk = (^Xjk + Xj , AYj k + Yj , AZj k + Zj) (4)
The A block row and the (3,1) observation sub
vector are respectively:
a, = [0 ... -I ... I ... 0] (5)
lj k ~ (AXj k * AYj k ' AZj k ) (®)
if the GPS vector is between two unknown points.
The A block row and the (3,1) observation sub
vector are respectively:
a, = [0 ... -I ... 0 ... 0] (7)
lj k = (AXj ~ Xk , AYj k — Yk , AZj k - Zk j (8)
when the GPS vector is between a unknown point
and a known (I.G.S./A.S.I. or local) station.
the (3,3) variance-covariance matrix of the GPS
baseline vector between the j-th and k-th points,
where the coordinates of the j-th point are known
and not assumed as observables.
Thus the variance-covariance (n , n) matrix of the
observation is the block diagonal matrix is :
El = diag
(12)
Denoting with
ct = (g?! Gq2) T = (g-i a 2 ) T (13)
the unknown vector (2,1) of the variance factors of
the measured GPS baseline vectors, the stocha
stic model can be put in the following form :
E| = o-| Q-i + a 2 Q2 Q3
(14)
where the known block diagonal cofactor (semi-
definite positive) (n , n) matrices Qj are respecti
vely :
Q 1 formed by the known (3,3) block cofactor
Qjk <
Q 2 by the known (3,3) block cofactor bj k Qj k ,
Q 3 by the known (3,3) block variance Q M .
3.2 Stochastic model
4. BLUNDERS DETECTION
If the Cartesian coordinates of the i-th point are
assumed as observables, then we put: The a P° s f er i° r global variance factor
Qü = Qii
(9)
Sq =(v T Pv)/(n-u)
(15)
where Qj j is the (3,3) known variance matrix of
the coordinates of the i-th point.
Let
Qjk =(<*01 +Go 2 bf k )Qjk
(10)
is computed in order to verify the absence of gross
errors in the observations by the Fisher Global Test.
Since the observations are correlated, in order to
localise one blunder in the set of observations by
the Baarda’s Test, the program performs the com
putation of the normalised weighted LS residuals
the (3,3) variance-covariance matrix of the GPS
baseline vector, whose length is bj k, between the
j-th and k-th points, where the coordinates of the
both points are unknown and none of them is assu
med as observable.
Let
Qjk -
(G§i+G§ 2 bf k )Qjk
+ Qi
(11)
Wj/a Wj = (ej T w)/(ej T S w ej ) (16)
where :
ei is the (n , 1) vector whose i-th com
ponent is 1 and zero everywhere,
w = Pv is the weighted LS residual (n,1)
vector,
E w = P Zy P is the variance-covariance (n , n)
matrix of w.
