MULTIDIMENSIONAL TESTS FOR MODEL ERRORS AND THEIR RELIABILITY MEASURES. 
Theodore Bouloucos 
International Institute for Aerospace Survey and Earth Sciences (ITC) 
Enschede, The Netherlands 
ABSTRACT 
The modelling of gross and systematic errors or deformations of terrain 
points, as composite alternative hypotheses, leads to multidimensional 
statistical tests. In this paper, test quantities are derived and 
reliability measures are formulated for such hypotheses. 
1. INTRODUCTION 
Statistical hypotheses formulated about the mean values of the observa- 
tions, are of importance in geodesy and photogrammetry. The initial 
assumption, i.e. the null hypothesis H on which the adjustment of 
geodetic networks or photogrammetric blocks is based, E{x'}=R", can be 
tested against alternative hypotheses H. formulated as 
E(x )-x* + cl vk kil =1, ...5m 
Such hypotheses cover the possible presence of model errors in the func- 
tional model, i.e. gross errors, systematic deformations or deformations 
of terrain points. 
The structure of the matrix with elements ci in the alternative 
hypothesis depends upon the type of the model error. When a number of 
observations is suspected of gross errors, then the matrix ci consists 
of unit vectors with the units in appropriate positions. In the case of 
systematic deformations, the matrix of contains the coefficient of a 
linear or linearized systematic deformation model. Similarly, deforma- 
tions of terrain points can be expressed by filling-up the matrix ct 
appropriately. The testing procedure developed in this paper, is essen- 
tially a generalization of Baarda's testing theory /2/, which deals with 
the alternative hypotheses generated by a vector c! times a scalar V. 
Apart of the test quantities developed in section 3, internal and exter- 
nal reliability measures are formulated in section 5. These measures can 
be used in the design stage of a point determination system. The former, 
for the design of a network or block so that the test for the specific 
model errors becomes more sensitive, and the latter for the design of 
-: 93 -