Results on the Optimal Detection Statistic for Integrity Monitoring

One of the most stringent requirements of positioning systems for aviation is integrity. The provision of integrity is the main function of Satellite-based Augmentation systems (like the Wide Area Augmentation System and EGNOS), Ground Based Augmentation Systems, and Receiver Autonomous Integrity Monitoring (RAIM). All these systems have different threat models, different nominal error models, and different constraints, but they all attempt to do the same thing: maximize the availability while maintaining integrity. Despite this common problem structure, there is a wide variety of integrity algorithms, even within the case of RAIM alone. In this paper, we focus on the choice of the detection statistics, and attempt to determine the optimal statistic given a threat model. There are three contributions in this paper. First, we cast the search of the optimal detection region as a mini-max problem. Second, we use the Neyman-Pearson lemma to limit the search of the detection regions to a class of regions parameterized by a bias. Finally, we have shown that in the case of one threat, even multi-dimensional, the optimal detection statistic is the solution separation, that is, the difference between the all-in-view solution and the solution obtained by assuming that the threat might be present. INTRODUCTION The main task of systems like Augmentation Systems (SBAS, GBAS) or Receiver Autonomous Integrity Monitoring (RAIM) is to provide guaranteed error bounds to the satellite navigation position solution, that is, error bounds with integrity. The error bounds must be small enough so that the service they aim to provide is available. The monitoring algorithms at the core of these systems must therefore attempt to maximize availability while maintaining integrity, or at least meet the required availability while maintaining integrity. For this reason, the design of the monitoring algorithms is in essence an optimization problem (of integrity for example) subject to constraints (the availability). The goal of this paper is to make this optimization with constraints problem explicit and provide some results on its solution. The solution in the case of a fault parameterized by one bias is described in [1], [2], in the context of geodetic networks. Again for the one bias case, this result is presented in [3] as a direct consequence of the NeymanPearson lemma. However, the conditions for the Neyman-Pearson lemma are not met, as the H1 hypothesis (fault) is not well defined. In these papers, the problem is cast in terms of a choice between H0 or H1, which is not the correct question in the integrity context. Although these are not the terms employed in these three papers, the suggested statistic is the square of the solution separation statistic (which coincides with the residual test). Because this result is presented as showing that the optimal statistic is a chi-square statistic, it has sometimes been misinterpreted as meaning that the optimal statistic is the chi-square of the residuals –which is a good detection statistics, but not the optimal as will be seen. In this paper, we formulate the problem as the search of the detection region that minimizes the worst case integrity risk. As a consequence, we will generalize the results included in [1],[2],[3] in two directions. First, we consider threats that can be multidimensional, like simultaneous satellite faults. Second, we include the possibility of having heterogeneous prior probabilities for each fault mode, and attempt to define the detection region given the whole threat model. A possible starting point is given by the Neyman-Pearson lemma, which provides the best test statistic given a known faulted distribution of errors. However, as mentioned earlier, the integrity monitoring problem does not in general fulfill the conditions of this lemma. In particular, in the integrity context, the faulted distribution is typically not completely determined. Instead, it is parameterized by an unknown bias, or vector of biases. The contribution of the fault to the integrity budget is obtained by maximizing over all possible biases. This means that the Neyman Pearson lemma cannot be applied directly. After introducing our notations, the search of the detection region is formulated as a mini-max problem. After some manipulations we apply the Neyman-Pearson lemma, so that the search is reduced to a class of regions for a very wide range of problems. Then, the exact solution is given in the case where the threat model is reduced to one (possibly multidimensional) threat, which is the solution separation statistic. Finally, we show that while the set of solution separation statistics corresponding to each threat is not the optimal statistic in the general case, it might be very close to it. ERROR MODEL AND DEFINITIONS Fault free error model The fault free error model is given by the state equation: other y Gx Hx e = + + (1) G is the geometry matrix, y is the set of measurements, and e is the nominal noise. The vector x is the target of the integrity bounds, and xother is not. For example, xother could be the clock component in the case of RAIM. The vector e is a zero mean multivariate normal random variable. Fault error model In this paper we are interested in the faults that can be described by adding an additional unknown in the state equation: other i y Gx Hx Ab e = + + + (2) Ai and b are a matrix and a vector respectively. Ai is known and b is unknown (it is an additional state). It is assumed that the measurements follow this fault error model with probability pi. The set of fault error models and the fault free error model form a partition, that is, there is only one state at a given time. For notation purposes, we set: 0 0 A = (3) 0 1 1 fault N i i p p = = − ∑ Integrity Risk: Probability of exceeding the error bounds and passing the test The objective of the test is to make sure that the probability of exceeding the error bounds (Alert Limits in the case of RAIM) is below the allowable integrity risk (IR). We label ΩAL the region within which the estimation error must lie with a probability of at least 1 – IR. The region of measurements for which there is no alert is labeled Ω. The test region Ω must be such that: ( ) ˆ , AL P x x y IR − ∉Ω ∈Ω ≤ (4) This expression can be developed as follows: