ION Technical Content
RAIM with Non-Gaussian Errors
| Abstract: | Papers on RAIM invariably assume that the nominal measurement errors are Gaussian with known means and standard deviations. While this assumption was justified in the 1990s when the dominant errors due to Selective Availability (SA) appeared to be zero-mean Gaussian by design, we continue to make this assumption post-SA apparently to make the problem tractable. We lack a methodology to deal with microscopic probabilities (like probability of missed detection of 10^-7), unless the distributions are Gaussian. But forcing a Gaussian model on the problem exacts a price in terms of conservatism (over-inflated parameters for the over-bounded Gaussian distributions) and the resultant loss of availability of service. The Gaussian error model has several well-known shortcomings. First, empirical data do not fit a Gaussian model: the tails of empirical distributions of pseudorange errors are typically longer and thicker than for a Gaussian distribution with the corresponding standard deviation. Second, true Gaussian errors can get arbitrarily large if observed over a long enough period. Empirical data gathered over 10-plus years suggest that errors under nominal conditions due to orbit and clock mismodeling, atmospheric propagation, and receiver noise do not grow arbitrarily large in the absence of a system fault. The result is that the computed protection level accounts for events we have never seen occur. Third, a Gaussian model is too rigid. Assume values of a mean and standard deviation, and you are locked in. There is no room for sensitivity analysis. After all, the situation doesn’t become hazardous suddenly when position error reaches the alert limit of, say, 20 meters. An error of 19.5 meters may also be cause for concern. But a Gaussian model does not allow for a meaningful sensitivity analysis. We started this investigation thinking there might be a way to dispense with the Gaussian assumption when we have a surfeit of satellites from multiple constellations. Turns out the Gaussian-error assumption appears unnecessary even with GPS alone. We propose an alternate approach to RAIM which dispenses with the Gaussian assumption. If not Gaussian, then what? The answer is we don’t need a full-blown characterization of the probability density function (pdf). After all, ‘small’ pseudorange measurement errors of the kind observed routinely cause no problems and we don’t have to fuss over their precise distribution. Trouble arises only when one or more of the measured pseudoranges have larger than usual errors. What we need, therefore, is a framework for a careful characterization of ‘large’ errors: how large can the errors get in the absence of a system fault, and how often? In other words, what we need is a suitable characterization of the tails of the error distributions, not the entire pdf. This can be done in terms of discrete percentile points associated with the tails, given the vast amount of empirical data collected to date and mathematical models of rare events. We propose to reverse the steps of conventional RAIM which starts out by specifying a Gaussian error model and then determines a protection level consistent with a probability of missed detection associated with a given alert limit. Instead, our two-step implementation focuses on the questions: (i) For a given satellite geometry, how large must the pseudorange errors get before the position error approaches the alert limit? (ii) Given a discrete characterization of the tails of the pdf, what’s the probability that the errors can get this large, or larger? The first question is easy. It is answered with a simple algebraic analysis without any uncertainty. We formulate it as an optimization problem: Given the geometry matrix, what’s the smallest size measurement error vector that can raise the position error to alert limit level? The second question now deals with how to divide this large error among the satellites so as to maximize the probability of its occurrence and the resultant damage, another optimization problem. The value of this approach would depend upon our ability to accommodate in Step 1 above a measure of sensitivity analysis (what if the alert limit were a little lower or higher?), and account in Step 2 for the various modes of system faults, their probabilities and resultant errors. The initial implementations of both steps are presented in the paper along with examples. |
| Published in: |
Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN |
| Pages: | 2664 - 2671 |
| Cite this article: | Misra, P., Rife, J., "RAIM with Non-Gaussian Errors," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 2664-2671. |
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