ing 
rix 
for 
the 
the 
ing 
tic 
rid 
the 
the 
tor 
This can give some trouble in the fixing of the 
weights of the different elements. However by 
repeating the integrated geodesy approach 
adjustment, the uncertainty about the weight 
ratios can be eliminated, and suitable values for 
the weights can be established. 
Moreover all the data are supposed outlier free; 
however because outliers occur in the data, due to 
gross errors and/or unmodelled effects, a suitable 
strategy combining robustness and efficiency has 
to be used. Indeed robust estimators are useful 
for the identification of suspected outliers, 
while the least squares are very powerful for 
testing about acceptance or rejection. 
The system of observation equations is now 
rewritten as: 
with S containing both stochastic and 
non-stochastic parameters s -[x's ] and the design 
matrix B defined as B - [A B], expressing both the 
chosen functional and stochastic modelling., The 
observations « are related to the estimates s of s 
by the same linearized model: 
Be - n-a=0 (A. 1) 
The covariance matrix C for the newly defined 
signal s contains four Blocks, two diagonal blocks 
containing the covariance matrices of the 
stochastic and non-stochastic part of the signal, 
and two zero off-diagonal blocks: 
The covariance matrix of the stochastic parameters 
is determined by one or more auto and 
crosscovariance functions, which can be estimated 
empirically with the results of preceding separate 
adjustments. The covariance matrix of the non 
stochastic parameters is a diagonal matrix, the 
elements of which have to be chosen in balance 
with the variances of the stochastic parameters: 
in such a way that the solution is not contrained 
too much to either type of parameters. The general 
variance of the noise o^, which also has to be 
known a priori, can "be assumed equal to the 
estimated variance factor c of the last separate 
preceding adjustment. 
The least squares criterion can now be used to 
RE A LZ. 
minimize contemporaneously the norm sS C ^s and the 
norm of the residuals of the observation equations 
n Pn/o : 
ut au s uude. 68 
zs n*] Se + A‘(Bs - n - a ) = min 
0 P/o*||n 
n 
with P the weight matrix of the observations and À 
a vector of Lagrange multipliers. According to 
this criterion, the estimates for the signal and 
the noise become (taking into account expression 
1.1): 
14-1 © 
:€-c mium B«oP!j!a (A.2) 
ss ss n 
nec pin pec pP'!ylese- Bs (A. 3) 
n ss n 
The computation of expressions (A.2) and (A.3) 
requires the solution of a system with dimension 
m, equal to the number of observations. It would 
however be more convenient to have analogous 
expressions, which require the solution of a 
system with dimension n « m, equal to the number 
of parameters. A further requirement would be the 
absence of inverse matrices which contain inverse 
matrices. Both can be achieved by the application 
of the two theorems of linear algebra, which are 
stated below: 
(0 * nS)" gs 9g 'n (s! s To !m r9! (4.4) 
-145-1 1 
gig !sr sg! «(9 € oso)" (A.B) 
Precisely, applying first two times theorem (A.4) 
and then theorem (A.5), one obtains: 
(c B. ply: 
ss n 
& P/o?- PR-(o" C^! + BPB) BPC = 
n n ss n 
P/o* - PB (B*PB) ! B'P/o* + 
+ 
PB(B'PB) 'I[C, * c^(B'PB) ^ 1  (B'PB) "B'P = 
P/o* - PB(B'PB) !B'p7o? + 
n 
+ PB(B“PBC__B“PB + o^ p'rp) 'B'P 
The estimate for the noise can now be rewritten 
as: 
a - prc pp) = o^(B'PBC B'PB + o^B'PB) !]B'Pa^ 
5 
M 
«a - Bs (A.6) 
Il 
Taking into account expression (A.6) the estimate 
for the signal becomes: 
s = (B*PB) 'B'Pa’+ 
_ c? (p'PBc B'PB « o? B'PB) !B'Po^ (A. 7) 
n ss n 
387