This model preserves the high accuracy of the conical domain model while providing superior integrity.

The multi-cone model is a computational model for estimating ionospheric delays of Global Positioning System (GPS) signals. It is a direct descendant of the conical-domain model, which was described in “Conical-Domain Model for Estimating GPS Ionospheric Delays” (NPO-40930), Software Tech Briefs, special supplement to NASA Tech Briefs, September 2009, page 18. A primary motivation for the development of this model is the need to find alternatives for modeling slant delays at low latitudes, where ionospheric behavior poses an acute challenge for GPS signal-delay estimates based upon the thin-shell model of the ionosphere.

Figure 1. A Conical Domain having a vertex at a given IGP is defined, and a set of pseudo-measurements for ray paths that intersect at the IGP is fit to this conical domain.
Since ionospheric signal delay contributes error to GPS position and time measurements, it is necessary to estimate the delay to correct and bound this error. Several national and international systems, denoted generally as satellite-based augmentation systems (SBASs), are under development worldwide to enhance the integrity and accuracy of GPS measurements for airline navigation.

Figure 2. The Slant Delay From a Satellite of Interest to a user airplane is estimated by interpolating among slant delays for ray paths between (a) the user airplane and (b) each of the four adjacent IGPs for which pseudo-measurements have been fit to conical domains as depicted in Figure 1.
A prominent example is the Wide Area Augmentation System (WAAS) of the United States, in which slant ionospheric delay errors and confidence bounds are derived from estimates of vertical ionospheric delay modeled on a grid at regularly spaced intervals of latitude and longitude. The estimate of vertical delay at each ionospheric grid point (IGP) is calculated from a planar fit of neighboring slant delay measurements, projected to vertical using a standard thin-shell model of the ionosphere.

Interpolation on the WAAS grid enables estimation of the vertical delay at the ionospheric pierce point (IPP) of any arbitrary user’s measurement. (The IPP of a given user’s measurement is the point where the ray path of the measured GPS signal intersects a reference ionospheric height.) The product of the interpolated value and the user’s thin-shell obliquity factor provides an estimate of the user’s ionospheric slant delay.

Two types of error restrict the accuracy of delay estimates based upon the thin-shell model: (1) error arising from the implicit assumption that, at the IPP, the electron density is independent of the azimuthal angle, and (2) error due to an invalid obliquity factor (e.g., error due to a suboptimal choice of shell height). Under nominal conditions at mid-latitudes, the magnitude of the error incurred from these sources is small. However, at low latitudes or at mid-latitudes under disturbed conditions, the error grows due to the presence of enhanced ionization, complex ionospheric structure, and large electron- density gradients. In the conical-domain model, these sources of error are mitigated by eliminating the use of both the thin-shell model and the vertical delay grid. Instead, a user’s slant delay to a given satellite is calculated directly by fitting measured slant delays for nearby ray paths to the same satellite.

The conical domain model is so named because the receiver and satellite positions define a cone with the satellite position at the vertex. In an SBAS based upon the conical domain model, fits of delay on a grid of IGPs are replaced with fits inside cones, each having a GPS satellite at its vertex. A user (e.g., an airplane in flight) within a given cone evaluates the delay to the satellite directly, using (1) the IPP coordinates of the line of sight to the satellite and (2) broadcast fit parameters associated with the cone.


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