Monitoring and Imaging Ionospheric Total Electron Content Without the Thin-Shell Approximation

The thin-shell model of the ionosphere relies on the coarse approximation that ionospheric electron density is non-negligible only in the vicinity of a specified reference height (typically the peak of the F-layer). The utility of this approximation resides primarily in the ease with which measurements of slant total electron content (TEC) may be converted into estimates of vertical TEC: if we identify the ionospheric pierce point (IPP) where a signal raypath intersects the shell height, then the vertical TEC at this IPP is estimated by scaling the TEC measured along the raypath by a simple geometric factor that depends upon the elevation angle of the signal. Developed to ensure the accuracy and integrity of user position estimates based upon global navigation satellite system (GNSS) measurements, all satellite-based augmentation systems (SBAS) to date, such as the United States' Wide Area Augmentation System (WAAS), the European Geostationary Navigation Overlay Service (EGNOS), and the Multi-Functional Satellite Augmentation System (MSAS) in Japan, use the thin-shell model as the basis for estimating vertical ionospheric delay at a specified set of regularly-spaced intervals in latitude and longitude, i.e., ionospheric grid points (IGPs). The vertical delay estimate at each IGP is calculated from a planar fit of neighboring slant delay measurements projected to vertical using the standard thin-shell obliquity factor. For an estimate of vertical TEC based upon the thin-shell approximation to be valid, two conditions must generally be satisfied: (1) the ionospheric electron density must be azimuthally symmetric with respect to the IPP; and (2) the choice of shell height must be appropriate. The successful operation of WAAS over the past fives years is a testament to the fact that these conditions are roughly satisfied under nominal ionospheric conditions at mid-latitudes. In the presence of significant horizontal electron density gradients, however, distinct measurements, which share a common IPP, can result in inconsistent estimates of the vertical TEC at the IPP. This is particularly true under storm conditions and at low latitudes, in regions characterized by complex ionospheric structure, high TEC values, and steep electron density gradients. The resulting error, often designated obliquity error, can seriously degrade the accuracy of the estimate. We present a two-stage approach for estimating TEC that does not use the thin-shell approximation and thereby eliminates obliquity error from the estimate. In the first stage, pseudo-measurements are calculated for a set of raypaths connecting earth grid points (EGPs) and satellites, where each fit of GNSS measurements used to define a pseudo-measurement is restricted to a spatial domain encompassing signals from only one satellite. In the second stage, we perform a fit of pseudo-measurements at each EGP. The resulting fit parameters may then be interpolated to provide an estimate of slant TEC along any arbitrary raypath. This method is designated the multi-cone model, since the spatial domain of each fit is a cone with a satellite (stage one) or an EGP (stage two) at the vertex of the cone. We assess the improvement in fit accuracy by comparing results achieved with the multi-cone model to those of the thin-shell model using data sets from networks of dual-frequency GPS receivers in Mexico and the United States, under both quiet conditions and disturbed conditions representative of those likely to be encountered at the next solar maximum.