Measurement Noise versus Process Noise in Ionosphere Estimation for WAAS

One of the parameters driving the performance of the Wide Area Augmentation System (WAAS) is the Grid Ionospheric Vertical Error (GIVE). The GIVE bounds the estimation error of the ionospheric delay at each Ionospheric Grid Point (IGP). The GIVE is generated such that a user interpolating both the vertical ionospheric delays at the IGP and the GIVE, is protected. The GIVE is a function of several parameters: the geometry of the measurements, the measurement noise, and the state of the ionosphere, which yields the process noise. It is very important to distinguish carefully between measurement noise and process noise, as they have a different behavior in the generation of the confidence bound. The measurement noise contribution to the confidence bound tends to zero as the number of measurements increases. The process noise, characterized in the current WAAS algorithm by a standard deviation around the planar trend in nominal conditions, indicates a lower bound for the confidence bound. For each satellite, the measurement noise is well characterized as a function of the elevation and tracking time. However, the process noise, which has a large variability due to the possibility of storms or mild irregularities, is measured or tested only through observations that have measurement noise. For this reason, the observability of the state of the ionosphere is impaired by the measurement noise. Although this is not a problem for the current level of service for WAAS, it could become an issue as we try to increase availability by decreasing the conservatism of the nominal state of the ionosphere. In this study, we develop a set of formulas to evaluate the Probability of Hazardously Misleading Information (PHMI) for two possible algorithms. This analysis takes into account the loss of observability of the state of the ionosphere (which determines the process noise) due to the measurement noise. It can be applied to any vertical ionospheric delay model that has a deterministic trend and a random gaussian component. We will also see in what cases it is essential to distinguish between measurement noise and process noise. This analysis will help maintain integrity for further improvements to the WAAS ionospheric algorithms. INTRODUCTION It is well known that the large ionosphere variability over time and space together with the –necessarilyirregular sampling of the ionosphere has caused the WAAS ionosphere confidence bounds or Grid Ionospheric Vertical Errors (GIVE) to be very large [1], [2]. The GIVE calculation needs to take into account the geometry of the measurements, the noise affecting each of those measurements –the measurement noiseand the state of the ionosphere –which determines the process noise. While the measurement noise is well known at all times [3], the process noise is unknown in real time, but can be inferred from the measurements (which are affected by measurement noise). The goal of this paper is to understand the effect of the measurement noise on the uncertainty of the ionosphere state, and, as a consequence, on the probability of hazardously misleading information (PHMI) [4]. Despite its variability, the ionosphere can be well described by a very simple model. This model states that, locally, the ionosphere follows a planar trend [1]. Once the trend is removed, the residuals can be modeled by a random gaussian field –the process noise. One can either assume a constant covariance, or more accurately, a covariance that depends on distance [5]. The most common covariance structure is called the nominal ionosphere. Because the ionosphere does not always follow the nominal model, a chi-square test statistic is computed on the available measurements to check that they are compatible with the assumed nominal model [1]. Even if the measurements pass the test, the confidence bound needs to be inflated by a factor labeled Rirreg, to take into account the possibility that noise is impeding our ability to detect disturbed ionospheric conditions. The statistic can be used in several ways. In this work we will focus on two possible ways of computing this inflation. The first one is currently used in the WAAS ionospheric correction algorithm. In this option, the chi-square statistic is only used to check whether the measurements are compatible with the model. The second one, a proposed enhancement of the current algorithm, uses the chi-square statistic explicitly to correct the confidence bound [6]. This algorithm is called Real Time Rirreg. In both cases we need to compute the PHMI in order to evaluate the integrity of the system. This problem can only be approached using models for the random processes and for the ionospheric structure –both in quiet and storm conditions. In the first part, we will introduce the models and the assumptions we need to make. Then we will derive formulas to evaluate the PHMI for the two algorithms mentioned above. For both algorithms, we will study the dependency of the PHMI on measurement noise. DESCRIPTION OF THE PROBLEM It has been shown that within the thin shell model, the ionosphere is well described by a planar trend and a random gaussian field: ( ) ( ) ( ) ( ) 1 2 0 1 2 I x a a x a x r x = + + + where x is the location. The covariance function of r is called C. However, this covariance structure is usually unknown. At each time frame we have m ionospheric measurements, collected at the reference stations with dual frequency receivers. These measurements are corrupted by the measurement noise n, whose mean should be zero (the biases are removed using an off line least square process): ( ) ( ) ( ) ( ) ( ) 1 2 0 1 2 I x a a x a x r x n x = + + + + % We call N the covariance matrix of n(x). It is usually a diagonal matrix since measurement noise is uncorrelated from one receiver to another. In the two algorithms considered here the user computes the vertical delay correction for each of the satellites in view by forming a linear estimate of the measurements [5]: ( ) ( ) 1 ˆ m i i i I x I x λ = = ∑ % In this work it is not relevant how the weights are computed. The estimation variance is, for each of the delays: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2