nd 
6) 
che 
the 
7) 
Moreover, indicating with the symbol e the error 
in the estimate of the predicted signal, i.e. the 
difference between its theoretic and its estimated 
value: e = t - t, by applying the law of variance 
propagation, and taking into account expression 
(A.17), the covariance matrix C becomes: 
C er Ct. C = (A.18) 
2 
n 
= C: +C. B'PBAB'PRC B'PB + © 
t ts ss 
B'PB) !p'PBC 
t st 
Unfortunately, expressions (A.17) and (A.18) are 
not very convenient in computation, and it is not 
possible to find others more suitable. Therefore 
their computation is usually omitted. 
Some statistic properties of the mentioned 
estimates are now considered. The estimate for the 
filtered and predicted signal is consistent and 
unbiased under the hypothesis that the expected 
value of the observables is zero. The estimate of 
the filtered and predicted signal and the estimate 
of its error are efficient, i.e. their variance is 
smaller than the a priori variance of the signal. 
The estimate for the residual noise is efficient, 
i.e. its variance is smaller than the a priori 
variance of the observations. The estimate of the 
filtered and predicted signal has minimal variance 
of all linear estimates. 
The variance of the noise can also be estimated a 
posteriori. Imposing its estimate to be unbiased, 
ke? = k Elo?) = E(n‘Pn) = Tr(PC--) (A. 19) 
n n nn 
one obtains: 
kem-nt Trío^ P/ 7B (B'PBC  B'PB + 
. : 
* o gp Bp? (A. 20) 
n 
where m is the number of observations and n the 
number of parameters. Therefore the a posteriori 
estimate of the variance of the noise becomes: 
of = (n Pn)/k (4.21) 
n 
This estimate is also consistent under the 
hypothesis that the observations are normally 
distributed. Formula (A.19) can be used for the a 
posteriori estimate of variances and therefore 
also of weights of a priori defined groups of 
observations. 
The a posteriori estimate of the covariance 
function of the signal requires fairly 
sophisticated procedures, which are often 
computationally heavy and do not always produce 
reliable results. 
With respect to the computability some 
considerations are made concerning the 
applications of theorems (A.4) and (A.B). As was 
already said before, a suitable application of 
these theorems provides systems of dimension n «m, 
without inverse matrices which contain other 
inverse matrices. The expressions (A.7) and (A. 12) 
contain the expressions: 
389 
1 
(B*PB) ! BPa? ; (B PB). (A 22) 
The solution of this system and the computation of 
the inverse matrix are standard procedures in any 
least squares problem and, are computable with 
direct solution algorithms, which are capable to 
work with sparse matrices. Expressions (A.7) and 
(1.12) also contain the expressions: 
(B“PBC__B“PB + o? BPR) !p'po? 
1 
(B*PBC. B'PB + c^ B*PB)' (423) 
The normal matrix B'PB was already obtained 
before. The covariance matrix: C = C * S.» of 
the properly called stochastic *Parameéters is a 
sparse matrix when constructed by multiplying, 
according to Hadamard, the proper covariance 
matrix C by a suitable finite covariance matrix 
Sas” Its”Sparseness depends on the "persistence of 
c8Prelation" of the finite covariance functions. 
Its dispersion however is influenced by the 
(re)numbering of the points. The product of three 
sparse matrices (B*PB)C* (B PB) is a sparse matrix 
itself. The solution ofthe corresponding system 
therefore can be computed with iterative solution 
algorithms for sparse matrices. 
Finally starting from the use of the Hadamard 
product to obtain a sparse covariance matrix, an 
acceptable approximation of its inverse matrix can 
be obtained by multiplying, according to Hadamard, 
once more the inverse of the covariance matrix 
(C!* S )' by the previous defined finite cova- 
riance "matrix S In such a way the matrix 
(C5 e S s * S is Sparse too and the expressions: 
(o? toit s 25 "et B'PB) ! Ep a* 
i (A.24) 
(o (c * s )! ^9 Brey 
n ss ss ss 
could be preferred to the expressions (A.23) dn 
term of a greater sparseness of the matrices and a 
better numerical conditioning of the systems. This 
new approach, which could be called "approximated 
integrated geodesy", has not been tested very well 
yet, but should be applied in the next future. 
REFERENCES 
Ammannati, F., Benciolini, B., Mussio, L., Sanso, 
F. (1983). An Experiment of Collocation Applied to 
Digital Height Model Analysis. Proceedings of the 
Int. Colloquium on Mathematical Aspects of Digital 
Elevation Models, K. Torlegard (Ed), Dept. of 
Photogrammetry, Royal Institute of Technology, 
Stockholm, 19-20 April, 1983. 
Barzaghi, R., Crippa, B., Forlani, G., Mussio, L. 
(1988). Digital Modelling by Using the Integrated 
Geodesy Approach. Int. Archives of Photogrammetry 
and Remote Sensing, vol. 27, part B3, Kyoto, 1-10 
July, 1938. 
Barzaghi, R., Crippa, B., (1880). 3-D Collocation 
Filtering. Int. Archives of Photogrammetry and 
Remote Sensing, vol. 28, part 5/2, Zurich, 3-7 
September, 1990. 
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