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# Conical-Domain Model for Estimating GPS Ionospheric Delays

- Friday, 04 September 2009

### Sources of error in a standard ionospheric delay model are eliminated.

The conical-domain model is a computational model, now undergoing development, for estimating ionospheric delays of Global Positioning System (GPS) signals. Relative to the standard ionospheric delay model described below, the conical-domain model offers improved accuracy.

In the absence of selective availability, the ionosphere is the largest source of error for single-frequency users of GPS. Because ionospheric signal delays contribute to errors in GPS position and time measurements, satellite-based augmentation systems (SBASs) have been designed to estimate these delays and broadcast corrections. Several national and international SBASs are currently in various stages of development to enhance the integrity and accuracy of GPS measurements for airline navigation.

In the Wide Area Augmentation System (WAAS) of the United States, 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) corresponding to any arbitrary measurement of a user. (The IPP of a given user’s measurement is the point where the GPS signal ray path intersects a reference ionospheric height.) The product of the interpolated value and the user’s thinshell obliquity factor provides an estimate of the user’s ionospheric slant delay.

Two types of error that restrict the accuracy of the thin-shell model are absent in the conical domain model: (1) error due to the implicit assumption that the electron density is independent of the azimuthal angle at the IPP and (2) error arising from the slant-to-vertical conversion. At low latitudes or at mid-latitudes under disturbed conditions, the accuracy of SBAS systems based upon the thin-shell model suffers due to the presence of complex ionospheric structure, high delay values, and large electron density gradients. Interpolation on the vertical delay grid serves as an additional source of delay error.

The conical-domain model permits direct computation of the user’s slant delay estimate without the intervening use of a vertical delay grid. The key is to restrict each fit of GPS measurements to a spatial domain encompassing signals from only one satellite. The conical domain model is so named because each fit involves a group of GPS receivers that all receive signals from the same GPS satellite (see figure); the receiver and satellite positions define a cone, the satellite position being the vertex. A user 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.

The conical-domain model partly resembles the thin-shell model in that both models reduce an inherently four-dimensional problem to two dimensions. However, unlike the thin-shell model, the conical domain model does not involve any potentially erroneous simplifying assumptions about the structure of the ionosphere. In the conical domain model, the initially four-dimensional problem becomes truly two-dimensional in the sense that once a satellite location has been specified, any signal path emanating from a satellite can be identified by only two coordinates; for example, the IPP coordinates. As a consequence, a user’s slant-delay estimate converges to the correct value in the limit that the receivers converge to the user’s location (or, equivalently, in the limit that the measurement IPPs converge to the user’s IPP).

*This work was done by Lawrence Sparks,
Attila Komjathy, and Anthony Mannucci of
Caltech for NASA’s Jet Propulsion
Laboratory.*

*The software used in this innovation is
available for commercial licensing. Please
contact Karina Edmonds of the California
Institute of Technology at (626) 395-2322.
Refer to NPO-40930.*

### This Brief includes a Technical Support Package (TSP).

** Conical-Domain Model for Estimating GPS Ionospheric Delays** (reference NPO-40930) is currently available for download from the TSP library.

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