Evaluation of Upper and Lower Bounds on the Success Probability

Henrik E. Thomsen

Abstract:	Estimation of integer valued phase ambiguities in double differenced phase observations is necessary to obtain high precision in differential GPS. A common approach is to estimate the ambiguities as floats and secondly to search for the integer vector with the small- est distance to the float estimate. The distance is measured by the norm given by the inverse of the covariance matrix of the float estimate. The integer estimates thereby obtained are treated as if they were deterministic and hence can be eliminated from the observations. Thus it is important to know the certainty of these estimates. In order to evaluate the estimates, various ad hoe methods are used, typically by evaluating the distance to the clos- est and second closest integer estimates. However, these methods lack theoretical foundation and therefore no ade- quate probabilistic theory concerning the ambiguities can been presented. Another approach is to consider a reliability measure of the estimation procedure. Recently, the success probability has been used for this. The success probability is the probabil- ity of getting the true integer by a given estimation proce- dure. This probability is hard to compute. To overcome this, Hassibi and Boyd (1998) have published upper and lower bounds on this probability. In addition, Teunissen (1999) has given another lower bound on the success probability. The goal of this article is to evaluate these bounds on the success probability in order to assess which one is to be used. The bounds are evaluated by means of a two-dimensional simulation study as it is easier to evaluate the true success probability by numeric methods in lower dimensions than it is in higher dimensions. The study shows that the upper bound is performing very well. Besides this it is seen that the lower bound given by Teunissen (1999) is also performing very well, and it is performing better than the lower bound given in Hassibi and Boyd (1998).
Published in:	Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000) September 19 - 22, 2000 Salt Palace Convention Center Salt Lake City, UT
Pages:	183 - 188
Cite this article:	Thomsen, Henrik E., "Evaluation of Upper and Lower Bounds on the Success Probability," Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000), Salt Lake City, UT, September 2000, pp. 183-188.
Full Paper:	ION Members/Non-Members: 1 Download Credit Sign In