Eliminating Obliquity Error from the Estimation of Ionospheric Delay in a Satellite-Based Augmentation System

Current satellite-based augmentation systems estimate ionospheric delay using algorithms that assume the electron density of the ionosphere is non-negligible only in a thin shell located near the peak of the actual profile. In its initial operating capability, for example, the Wide Area Augmentation System incorporated the thin shell model into an estimation algorithm that calculates vertical delay using a planar fit. Under disturbed conditions or at low latitude where ionospheric structure is complex, however, the thin shell approximation can serve as a significant source of estimation error. A recent upgrade of the system replaced the planar fit algorithm with an algorithm based upon kriging. The upgrade owes its success, in part, to the ability of kriging to mitigate the error due to this approximation. Previously, alternative delay estimation algorithms have been proposed that eliminate the need for invoking the thin shell model altogether. Prior analyses have compared the accuracy achieved by these methods to the accuracy achieved by the planar fit algorithm. This paper extends these analyses to include a comparison with the accuracy achieved by kriging. It concludes by examining how a satellite-based augmentation system might be implemented without recourse to the thin shell approximation. INTRODUCTION Signals broadcast by global navigation satellite systems (GNSS) experience delay when they propagate through the ionosphere. For airline navigation, satellite-based augmentation systems (SBAS) have been implemented to guarantee the accuracy and integrity of user position estimates based upon GNSS signals. These systems provide accurate calibration of ionospheric delays. Systems such as the United States’ Wide Area Augmentation System (WAAS) and the European Geostationary Navigation Overlay Service (EGNOS) employ the thin shell model of the ionosphere when estimating ionospheric delay, a crude model where the electron density is assumed to be non-negligible only in an infinitesimally thin layer located at a shell height representative of the altitude where the electron density attains its peak value. Such a crude approximation necessarily introduces error into delay estimation. In its initial operational capability, WAAS performed vertical delay estimation by incorporating the thin shell model into a planar fit algorithm. A recent upgrade of WAAS replaced the planar fit algorithm with an algorithm based upon a geo-statistical technique known as kriging [1]. The success of kriging is partly due to its ability to mitigate the error due to the thin shell approximation. Previous papers [2][3], however, have proposed alternative methods of delay estimation that eliminate the need for adopting the thin shell approximation altogether. These papers have compared the accuracy achieved to the accuracy attained by the planar fit algorithm. This paper extends these analyses by comparing the accuracy achieved to that attained by kriging, and it examines the implications of these results for SBAS. The primary reason for adopting the thin shell approximation is that it greatly simplifies the process of ionospheric delay estimation. In particular, the thin shell model serves two very useful purposes which account for its popularity in ionospheric analyses: (1) it associates each signal raypath with an ionospheric pierce point (IPP) where the raypath intersects the ionospheric shell, and (2) it provides a means of converting a slant delay measurement to a vertical delay estimate at the IPP using a simple geometrical conversion factor. These benefits are not achieved cost-free, however. It is well known that the thin shell approximation can also serve as a significant source of estimation error [4]. This error is designated obliquity error. The ionosphere at low latitude poses a particularly critical challenge for any SBAS that relies upon the thin shell model. Here the structure of the ionosphere is complex. Slant delay values display large magnitudes. Nearly every day, for example, large gradients in electron density occur as a manifestation of the presence of the equatorial (Appleton) anomaly. Previous studies [5] of slant delay residual errors based upon a planar fit have, in fact, concluded that the equatorial ionosphere exhibits such irregular structure under nominally quiet conditions that the occurrence of a major magnetic storm does not dramatically alter the statistical nature of the fit residuals. To base an SBAS at low latitude upon the thin shell model is clearly problematic. In prior papers [2][3] that address delay estimation, alternative methods have been proposed that eliminate the need for adopting the thin shell approximation, thereby removing obliquity error as a source of delay estimation error. The conical domain model [2] avoids the thin shell approximation by restricting each fit of GNSS measurements to a spatial domain encompassing signals from only a single satellite, that is, the fit domain is a cone with a satellite at the vertex. An SBAS based upon the conical domain model would broadcast fit parameters associated with each satellite. A user would then determine her line-of-sight delay to a given satellite by evaluating a model fit centered on her location, using the parameters associated with the cone of the satellite in question. In the context of SBAS, the conical domain model suffers from a major limitation: each fit is comprised of relatively few measurements, far fewer than are used in the corresponding standard planar fit algorithm. Each planar fit incorporates measurements from multiple satellites. Given a spatial distribution of receivers similar to that of WAAS, there are, in practice, typically thirty or more vertical delay observations processed per fit. In contrast, for a conical domain fit, the number of measurements available from a single satellite is typically only four or five. Since the reliability of the fit is dependent upon the number of observations fit, a small number represents a threat to the integrity of the delay values estimated. The multi-cone model [3] addresses this limitation by adopting a two-phase estimation scheme. In the first phase, a set of locations on the surface of the earth is identified, and the conical domain approach is used to estimate delays for raypaths that connect each of these earth points to each of the GNSS satellites that are locally visible. The signals that follow these raypaths are designated pseudo-observations. In the second phase of the multi-cone algorithm, the conical domain model is inverted; the spatial domain for each estimation is restricted to pseudo-observations whose raypaths intersect the earth at a single point, that is, the vertex of the conical fit domain now resides on the surface of the earth. A fit of pseudo-observations is henceforth designated a pseudo-fit. A given delay estimate will thus depend upon all the measurements used to define each of the pseudo-observations in the pseudofit, a number generally comparable to the number of observations used in the planar fit algorithm. In this fashion the multi-cone algorithm combines the accuracy of the conical domain model with the integrity of the planar fit algorithm. In the sections that follow, we begin by reviewing the planar fit algorithm for estimating ionospheric delay. In particular, we focus on how the thin shell approximation is used and how estimation errors arise as a consequence. Next we provide an overview of the manner in which the multi-cone algorithm eliminates the need to invoke the thin shell approximation. Subsequently, we describe briefly how estimation based upon kriging serves to mitigate obliquity error. Next, we report the results of computations that compare the accuracy of the planar fit, kriging, and the multi-cone algorithms. We assess accuracy in terms of the residual differences between the equivalent vertical delay of actual measurements and that of estimates based upon model fits. Using a Mexican data set from a day when the ionosphere becomes highly disturbed, we find that the multi-cone model reduces the RMS error of the planar fit algorithm on average by a factor of 1.9 and that of the kriging algorithm by a factor of 1.7. We conclude with a discussion of alternative means for implementing the multi-cone algorithm in an SBAS.