Adapting Kriging to the WAAS MOPS Ionospheric Grid

The most productive way to increase the availability of single frequency users of the Wide Area Augmentation System (WAAS) is by decreasing the Grid Ionospheric Vertical Error (GIVE). Currently the GIVE’s are very conservative, since WAAS has to protect against the worst possible case of ionospheric behavior given the measurements. By characterizing more accurately the vertical ionospheric delay model in nominal conditions and by better defining the ‘well sampled’ regions we can be less conservative while maintaining integrity. It has been shown previously that an ionosphere estimation algorithm based on kriging could address these two issues efficiently. In quiet conditions, the kriging method produces at each location an estimate of the Vertical Ionospheric Delay and a confidence bound on the estimate. The confidence bounds obtained seem to be close to the lower limit of what is possible within the thin shell model –before the storm detector is applied. In addition to that, the particular behavior of the kriging variance at the edge of coverage can be used to mitigate the non-stationarity of Total Electron Content (TEC) during storms. To do that, one can either define a ‘well sampled’ region or use it as a metric of coverage. As it has been described, the algorithm would require the user to know at all times the location of all Ionospheric Pierce Points (IPPs). Unfortunately, there is not enough bandwidth for the user to receive all of this information nor would this be efficient. Instead, the user receives a grid of points, the Ionospheric Grid Points (IGPs), which is updated every 5 minutes. For each satellite, the user interpolates both the delay and the GIVEs from the four closest IGPs. However, kriging gives an optimal estimate only at the IGPs, and the kriging variance is only valid at the IGPs. Therefore the delay computed by the user is not optimal, and the confidence bound will necessarily change as we depart from the IGP. In this study, we present a calculation of the GIVE for the kriging method that protects the user at any location. Results of the algorithm will be shown using ionospheric data collected from the WAAS reference stations. We will show that, even after the increase in confidence bound caused by the grid, we can still generate GIVEs below .6 m inland and 1 m in coastal regions. INTRODUCTION The most important attribute of the Wide Area Augmentation System (WAAS) is integrity [1]. Along with the corrections broadcast to the user, WAAS sends strict confidence bounds on those corrections under all conditions. For example, the ionospheric information included in the WAAS message enables the user to correct for the ionospheric delay in each pseudo-range and know accurately the interval in which the true delay lies. Unfortunately, the vast range of ionospheric behavior [2] and the fact that the ionosphere is irregularly sampled have forced these confidence intervals to be very conservative [3]. Now, to reduce the conservativeness, we need to understand better the spatial characteristics of the ionosphere. Within the thin shell approximation [4], each ray path has a corresponding Ionospheric Pierce Point (IPP) and each measurement is converted to an equivalent vertical ionospheric delay. It has been shown that the nominal ionosphere can be well characterized by a planar trend and a random gaussian field with a covariance depending on distance [5]. The minimum mean square estimator corresponding to this structure is called kriging [6]. For each location, kriging provides a confidence bound. In [5], an algorithm for WAAS based on kriging was sketched. This work showed that the confidence bounds where both safe and significantly smaller than the current confidence bounds. This algorithm reused extensively elements of the current WAAS algorithm, in particular the storm detector [7]. A problem with the straight forward implementation of the kriging algorithm is that it supposes the user has all the IPP measurements. In fact, the WAAS user receives the corrections according to the WAAS Minimum Operational Standard (MOPS) which specifies that the ionosphere information be sent in a grid of 5 by 5 degrees in the thin shell at a height of 350 km [7]. At each node of the grid the user receives a vertical ionospheric grid delay (IGD) and a grid ionospheric vertical error (GIVE). The user calculates each of the ionospheric delays corresponding to the satellites in sight as well as the confidence bounds from this grid, according to an algorithm which is also set in the WAAS MOPS. In order to make available the benefits of kriging to WAAS, we need to modify the algorithm presented in [5] such that it can be fit into the ionospheric grid. In the first part, we will review kriging assuming full knowledge of the IPP measurements; we will then show how to compute a GIVE for a kriging algorithm and, finally, we will show the gain in performance that kriging could provide. KRIGING ALGORITHM In this paper, we will skip the discussion about the storm detector. Here, it is sufficient to notice that the storm detector results on an inflation of the confidence bound in the case that the chi-square test passes. More information on the storm detector can be found in [7] and in [9] for a more detailed account of the influence of measurement noise on the chi-square detector. It was shown in [5] that a good model for the vertical ionospheric delay on the thin shell is a planar trend to which a random gaussian field [6], [10] is added, that is: ( ) ( ) ( ) ( ) 0 1 2 east north I x a a x a x r x = + + + In this formula, I(x) is the vertical ionospheric delay at location x, the three coefficients a0, a1, and a2 describe the planar trend and r(x) is the random gaussian field. A nominal ionosphere is such that the field r(x) has a covariance C(x,y) that depends on the distance between x and y: ( ) ( ) ( ) ( ) ( ) , E r x r y C x y C x y = = − Please refer to [5] to see how C was modeled and determined, and the appendix for the analytical expression. Now let us suppose that we have n IPP measurements. Each measurement has a noise pattern which is supposed to be known. ( ) ( ) ( ) ( ) ( ) 0 1 2 east north k k k k k I x a a x a x r x n x = + + + + % Unlike r, n is uncorrelated from one location to another. Kriging gives the best linear unbiased estimate of the field I(x) for each point. The expression: