The Disparity of Parity, Determining Position Confidence Bounds in the Presence of Biases

Tom Arthur

Abstract:	With all navigation systems it is desirable to obtain observability into positional errors. While direct observability is preferred, indirect observability is often the only available method. One method of indirect observability used in GPS is that of Parity, also called Null Space. Parity, a measurement of consistency, is a key component of the Receiver Autonomous Integrity Monitoring system (RAIM). Because of the importance of Parity to RAIM and other navigation systems, this paper will introduce some important insights into Parity. Specifically it will explore the observability and non-observability of positional errors, due to both variances and biases, while using the Parity concept as the primary observable. This paper will initially explore the concepts of Parity in a one-dimension system. Principles from the one-dimensional view will be further applied to three dimensional systems for application to GPS RAIM in double-differenced systems. New concepts of Disparity and Gamma-space will be introduced to facilitate an intuitive understanding of the underlying mechanisms. Typical confidence bounds of a random variable in a directly observable system are formed by using ? ? k + , where ? is the process mean, k is a constant to give the desired bounds, and ? is the standard deviation of the observed process. However, when using indirect observability into a process, there may be an additional scale factor for the mean, which will ensure the desired error bounds. This paper will determine this mean, or bias, scale factor as a function of the number of dimensions in position space and the number of redundant measurements. Assuming that the probability of a bias on a particular measurement is equally likely, the Probability Distribution Function (PDF) and Cumulative Distribution Function (CDF) of positional errors due to measurement biases will be determined. Unlike previous methods to contend with biases, this method does not assume that only one measurement contains a bias. The goal of this paper is, that with a given parity statistic, to determine the conditional probabilities of positional errors. Additionally, objective methods for integrating variances and biases into a combined position estimate will be derived.
Published in:	Proceedings of the 18th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2005) September 13 - 16, 2005 Long Beach Convention Center Long Beach, CA
Pages:	1841 - 1853
Cite this article:	Arthur, Tom, "The Disparity of Parity, Determining Position Confidence Bounds in the Presence of Biases," Proceedings of the 18th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2005), Long Beach, CA, September 2005, pp. 1841-1853.
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