ornia, vol. 2, pp. 484- 
Fenstermaker, L.K, 
Tinney, L.R. 1991, 
. Information System 
nd Research Issues', 
note Sensing, vol. 57, 
/. 1989, 'Observations 
Àn and Treatment of 
Accuracy of Spatial 
X S. Gopal, Taylor & 
f Landsat TM Data to 
ey', in Proceedings of 
ensing | Conference, 
Lins, 4 "1986, "A 
isure of Classification 
ineering & Remote 
n of Systematic and 
he Accuracy of Maps 
Sensed Data", 
mote Sensing, vol. 58, 
3; 1986, 'Accuracy 
ve, Photogrammetric 
|. 52, no. 3, pp. 397- 
sensed data as an 
| information systems 
nternational Journal of 
vol. 5, no. 2, pp. 225- 
for the Map Overlay 
Databases, eds. M.F. 
incis, pp. 3-18. 
ina 1996 
VARIANCE DECOMPOSITION AND ITS APPLICATION IN PHOTOGRAMMETRY 
Mohamed ETTARID, Ph.D. 
Department of Photogrammetry 
Institut Agronomique Hassan II 
B.P. 6202, Rabat 
MOROCCO 
ISPRS-Commission III 
KEY WORDS: Photogrammetry, Calibration, Modeling, Correlation, Identification. 
ABSTRACT: 
The mathematical modeling of dynamic and time-dependant perturbations affecting space sensors is a common practice in 
photogrammetry. This modeling is usually done through the fitting of parametric or interpolative models. However the extension of 
the model and the addition of parameters may lead to unstable solution due to high correlations between parameters. The 
identification of correlated parameters to take corrective measures is often based on the analysis of the correlation matrix. The 
correlation matrix shows however, only correlations pairewise and does not give any indication on functional groupings. 
In this paper, the variance decomposition based on singular value decomposition is presented. In this method the number of small 
singular values indicates the number of near depndencies and parameters involved in these are identified as those that have more 
than 50% of their variances associated with the same small singular value. 
A sase studey based on in-flight camera calibration was conducted with simulated and real data, and showed the efficiency of the 
method in dealing with fuctional groupings of the paramameters. 
1. INTRODUCTION 
The effects of external conditions and errors affecting the 
system constitute a limiting factor on the attainable accuracy in 
computational photogrammetry. The mathematical modeling of 
such phenomena are a common practice so as to take into 
accounts these effects. This modeling is usually done by the 
fitting of: 
- a parameteric model based on the geometrical or 
physical characteristics of the phenomenon. 
- an interpolative model represented by a polynomial. 
Modification of existing models through their extension and 
addition of parameters to account for these perturbation may 
lead to an unstable solution due to the correlation between 
parameters. 
For almost all least square users, the identification of the 
correlated parameters is based on the analysis of the correlation 
matrix. Hence, in the case of additional parameters, the 
decision of rejecting and deleting parameters is essentially 
based on the magnitude of the correlation coefficient. In this 
respect, some authors recommended 0.90 as a rejection standard 
(Grun, 1980), while others suggested 0.85 (Faig and Shih , 
1988). 
However, the alternative of rejecting and deleting parameters on 
the grounds of their significance and stability is not alwys 
justified. In fact, in some applications the physical significance 
of the parameter may be of great importance to the modeling; 
besides this, the rejection decision may not be fully reliable due 
to the fact that hypothesis testing may be rendered inconclusive 
because of the high variances inducued by the ill-conditioning. 
On the other hand, the correlation matrix shows only 
correlations between parameter pairs and does not give any 
indication on fuctional groupings where more than two 
parameters are simultaneously invloved in a correlation. In 
this respect, experience has shown that, it is possible for three 
or more parameters to be correlated when taken together, but no 
International Arc 
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two of these taken in pairs are highly correlated. Moreover, 
when the system is ill-conditioned, high correlation coefficients 
may be indicative of correlated parameters, but the absence of 
high correlation coefficients cannot be considered as evidence 
of no problem. 
To overcome the drawbacks of the correlation matrix 
mentioned above, we present in this paper a method based on 
the singular value decomposition and that deals efficiently with 
multiple correlations or functinal groupings of parameters. 
2. BACKGROUND ON VARIANCE DECOMPOSITION 
2.1 Singular Value Decomposition 
The singular value decomposition is a concept closely related to 
the eigensystem , but that applies directly to the design matrix 
A insted of the normal matrix (ATA) : 
Hence, if A is an (mxn) rectangular matrix, the singular values 
A; of A arethe positive square roots of the eigenvalues of the 
square matrix (ATA) of order n (Lascaux and Theodor, 1986). 
In fact, for any arbitrary (mxn) matrix A, there exists an unitary 
(mxn) matrix U and an unitary (nxn) matrix V such that: 
A = UDV" (2.1) 
with D a diagonal matrix of the form: 
p=/Pu-8 23) 
To "+0 : 
and : 
D4, - diagonal(A, ,À5,.-..,À.,) (2.2a) 
hives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996