185 
A COMPUTER PROGRAM FOR THE VARIANCE-COVARIANCE ESTIMATION 
OF GPS BASELINE VECTORS 
N. Crocetto*, M. Gatti**, C. Materazzo*, L. Toscano’ 
* Second University of Naples, Italy 
** University of Ferrara, Italy 
*** University of Naples "Federico II", Italy 
ISPRS Commission VI, Working Group 3 
KEYWORDS : GPS, baseline vectors, I.G.S./A.S.I., variance factors, software, Fortran, BIQUE. 
ABSTRACT 
We present a computer program in FORTRAN 77S for the least-squares adjustment of geodetic networks 
with GPS measured baseline vectors. The known tridimensional Cartesian geocentric coordinates of some 
GPS points can be assumed as fixed with their variances or as observable. Furthermore, the Baarda test 
can be performed on the normalised weighted least-squares residuals. Finally, the input covariance matrix 
of the GPS measured baseline vectors can be assumed as cofactor matrix, thus the program performs the 
BIQUE (Best Invariant Quadratic Unbiased Estimate) estimates of the two variance components of a 
prefixed stochastic model. At the end we report an example of application of the program. 
1. INTRODUCTION 
The commercial software (i.e. the Trimble and the 
Ashtech) don’t use to estimate the variance factors 
of the a prefixed stochastic model regarding GPS 
data. 
In regard to the least-squares adjustment of geo 
detic networks with GPS measured baseline vector 
components, we have conceived a program written 
in FORTRAN 77S. This program can assume the 
tridimensional Cartesian geocentric coordinates of 
some known GPS points (i.e. I.G.S./A.S.I. perma 
nent GPS network and/or local stations) as fixed 
with their variances or as observable. 
Generally, the input covariance matrix of the GPS 
measured baseline vectors are underestimated by 
the commercial programs of the GPS carrier phase 
data processing. Hence we have implemented, in 
this program, the performance of the BIQUE (Best 
Invariant Quadratic Unbiased Estimate) estimates 
of the two variance components of a prefixed sto 
chastic model. 
Furthermore, the Baarda test can be performed on 
the normalised weighted least-squares residuals, in 
order to localise "gross" errors in the correlated ob 
servations. This test is needed to ensure a correct 
BIQUE estimate of the variance factors. 
The program considers single-baseline vectors and 
hence the weight matrix is diagonal block structured 
with (3,3) full blocks. Hence we remark that the 
program works on block structured matrices in all 
the operations on the matrices and vectors involved 
in all the following procedures. Thus we obtain a 
larger gain in efficiency by the point of view of com 
putational effort. 
In the next sections we present the program struc 
ture, the least-squares adjustment model, the blun 
der detection method, the variance factors BIQUE 
procedure, and finally an application a GPS network 
in Po river Delta (Crocetto el al., 1998b). 
2. PROGRAM STRUCTURE 
The program has a modular structure which con 
sists in two sections: 
- MISPAR.for 
- PARPESI.for 
The options files, mispar.opz and parpesi.opz, of 
both sections contain the network name "rete". 
In the file "misrete.dat" it is possible to choose the 
Trimble output data format, which gives the GPS 
measured baseline vector components and their 
variance-covariance matrices, or, alternatively, the 
Ashtech output data format, which gives the GPS 
measured baseline vector components, their e.q.m. 
and correlation matrices. 
The points are identified with their names in the file 
"misrete.dat", in the files "elenfis.dat" and "xyzfrete. 
dat" (containing the tridimensional Cartesian geo 
centric coordinates with the e.q.m. respectively of 
the I.G.S./A.S.I. and of the known local stations).