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terms were a0 and bO corrections.) This solution illustrates the need for C A to be generated relative to a reasonably 
defined adjustment vector A. 
For the described scenario, the new replacement sensor model error propagation information successfully captured all 
error propagation information associated with the original rigorous sensor model. These results are also indicative for 
all other scenarios analyzed to date, varying in the sensor support data error characteristics, image geometry, and 
number of images. The only exception was a simulated vertical frame camera at a low altitude (4000 m) and with a 
wide field of view (100 degrees). AC, generated relative to the baseline image space adjustment polynomial (6 
adjustment parameters) did not perform well. However, a C, generated relative to a ground space adjustment 
polynomial with the baseline 6 adjustment parameters performed very well. It provided for a replacement solution 
virtually identical to the original solution. This level of performance was also true for all the other scenarios when 
using the baseline ground space adjustment polynomial. 
A final comment concerns the possible refinement of the replacement sensor model when augmented with the new 
error propagation information. It should be straight-forward to refine the adjustable polynomial by solving for 
corrections to the polynomial, i.e., the adjustment vector A becomes non-zero. The a priori value of A (zero) has an a 
priori error covariance C,. This refinement process could involve other replacement sensor models that correspond 
to overlapping images and could emulate the standard triangulation process that refines sensor support data 
parameters. Of course, since the adjustment vector A has no physical relevance, quality assurance checks are 
significantly better when using the standard triangulation. 
6. CONCLUSION 
This paper has presented the background to the use of rational functions and polynomial functions (especially the 
USM) and has determined the use of replacement sensor model techniques which can be used for a wide range of 
imagery. The arguments for and against the use of polynomial functions have been presented and analysed. The 
results from a number of organisations which have tested these have been presented and discussed. 
A critical issue is the lack of complete and rigorous error propagation information. A new method was introduced 
that could alleviate that situation for two of the techniques (RF and USM). It adds increased complexity to the 
techniques, but improved geopositioning performance when using these replacements sensor models may make it well 
worth while. 
The test results reported in this paper show that in the cases reported polynomial functions work well and can be used 
without loss of accuracy compared to rigorous sensor models. However the tests do not report on failures nor on the 
stability of the solutions. In this sense they may be regarded as being limited and further investigation, or some time in 
production use of RFs, is needed. The test carried out with the provision of rigorous error propagation information 
were successful and so a number of the objections to the method are removed. It seems therefore that the use of 
polynomial functions are quite suited to use with many sensors and provide an efficient and accurate way of using the 
data. In addition they provide a means of defining a universal standard for transferring sensor data. However it is 
important that the parameters of rigorous models are transferred wherever possible and that rigour is not sacrificed for 
the sake of expediency. 
We conclude therefore that the USM, and rational functions in general if used with care, are appropriate for use with a 
wide range of sensors. There is no compelling argument that they are suitable for frame cameras, which have none of 
the problems associated with push broom or Radar sensors. Rigorous models for frame cameras should therefore 
continue to be used and work should continue to define a standard for transfer of rigorous sensor models. 
ACKNOWLEDGEMENTS 
This paper has been prepared with the help of a working group set up by ISPRS Council under the chairmanship of the 
first author, to report on rational functions. We acknowledge the help of Mostafa Madani, Manos Baltsavias, Ramon 
Alamus and Vagelis Papapanagiotu who have provided results for inclusion in the paper. We particularly like to thank 
Professor E M Mikhail of Purdue University for his encouragement and helpful comments regarding error 
propagation for replacement sensor models and also Arliss Whiteside for his help and comments. 
REFERENCES 
Baltsavias E P and Stallmann D, 1992. Metric information exraction from SPOT images and the role of polynomial 
mapping functions. International Archives of Photogrammetry and Remote Sensing, 29(B4):358-364. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 265