Improved Ionospheric Correction for Dual Frequency and Differential GPS Positioning Methods

Abstract

A new three dimensional (3-D) electron density model has been developed that can give a very accurate description of the real ionospheric variation; latitudinally,longitudinally and altitudinally. An important advantage of this model is that the electron density and all its spatial derivatives are continuous as required for high accuracy in a ray tracing program. This model was then incorporated in the Jones 3-D ray-tracing program in order to determine the effect of ionospheric horizontal gradient on Global Positioning System (GPS) ray paths. This was used to investigate the accurate mapping of slant to vertical in the presence of horizontal gradient. In differential GPS (DGPS), the effect of the ionosphere is assumed to be the same for two closely spaced Earth receiving stations (e. g. 10km baseline distance). However, the presence of ionospheric horizontal gradient, especially for receivers in the equatorial and polar regions and the difference in elevation angle at these two spaced receivers, that are observing the same satellite, do introduce some range errors. These range errors need to be considered when the ionospheric error is corrected in the single difference approach. Some mathematical expressions have been developed to show the significance of these errors in final DGPS user positioning by performing ray tracing calculations between a reference station and a user station using a simple block ionospheric model incorporating a linear horizontal gradient. The baseline distance between the stations was 10km. Then, these models have been used to show the improvement in DGPS positioning by taking into account the error due to the effect of an ionospheric horizontal gradient and the difference in the elevation angle at the reference and user receivers observing the same satellite. Final positioning improvement of about 10cm has been obtained. Additionally, methods have been proposed to determine the magnitude of the ionospheric gradient from real data (e. g. from GPS satellites). It has been found that the Rutherford Appleton Laboratory United Kingdom's (RAL UK's) Total Electron Content (TEC) online map (updated at every 10 minutes) can give a gradient magnitude and direction which can be applied in DGPS horizontal gradient correction. This determined gradient, which is 1.6853/rad (in all direction), was then used in the ray tracing program (linear gradient approach) to show the improvement possible in the user positioning. Further, since it has been found that the RAL UK's TEC map correlates very well with the vertical TEC from International Reference Ionosphere (IRI), the gradient was then obtained from IRI for GPS receiving stations in the equatorial region, such as Malaysia. 15cm of improvement in the user positioning was then obtained showing the importance of correcting for the effect of the horizontal gradient for GPS stations in the region like Malaysia. The amount of improvement was also investigated for different Geometrical Dilution of Precision(GDOP) factors to see for what satellite configurations there would be the most positional improvement. In addition, the component of the gradient in the satellite direction was approximated by using a simple mathematical relation taking account of the elevation angle and the azimuths of the satellite from the navigation data. Then, these `corrections' were applied to the carrier phase measurements of GPS observation data to show positional improvement at the user in DGPS using GPSurvey. The resolution of the ambiguity 3 minutes and 30 seconds earlier than for the case before corrections, shows the improvement in the user positional when the magnitude and direction of the gradients was taken into account. The dual frequency correction scheme uses the dispersive nature of the ionosphere to eliminate the ionospheric range error. Nevertheless, the dual frequency model cannot totally remove the effect of the ionosphere as it does introduce some approximations. For an example, it does not take into account the presence of the higher order terms in the phase refractive index equation's expansion (in the power of X-1). Though the higher order terms are about two orders of magnitude lower than the first order term at L-band frequencies, in applications such as geophysics and surveying which require millimetre level positioning accuracy, these terms need to be considered. In this work, these higher order terms have been obtained by using a program based on an analytical perturbation method (which required as input the azimuth and elevation angle of the satellite and an approximate electron density profile), which is much less numerically intensive than using the numerical ray tracing method based on the Haselgrove equations. Then, about 4cm of improvement in the final positioning using a dual frequency receiver has been shown to be possible by correcting for these higher order terms