Second-order ionospheric correction in precise point positioning

Dual frequency GPS measurements cannot fully compensate higher order ionospheric refraction effects and lead to a residual error in the so-called “ionosphere free” L3 linear phase combination. Due to the dependency on the geomagnetic field, the second-order residual error is highly variable with elevation and azimuth angles and difficult to model. However, based on ionospheric simulations using the Chapman function and a superposed exponential decay for describing the vertical electron density distribution, we developed a correction formula for the second-order residual phase error. Analysis shows that our proposed correction formula limits the second-order propagation effect less than 2 mm residual range error for GNSS users in Germany. Due to the dispersive nature of the ionosphere, the first-order propagation error may fully be corrected by differencing the signal at two spaced frequencies. But whereas the first order effect may be completely removed, the higher order effects are not fully compensated by the difference method. The second-order residual error can vary from a few millimeters to several centimeters depending on the elevation angle, the geographic location and solar-terrestrial relationships. For reasons of simplicity, the ray path bending effect is ignored in the computation of the second-order ionospheric refraction effects i.e. the effects are computed along the line of sight propagation (LoS). This procedure is justified because the second-order effect of another higher order effect will be less than the computational accuracy of the ray-tracing technique and can be neglected. The International Geomagnetic Reference Field (IGRF) model is used to compute the magnetic induction B along the ray path. The IGRF observations are translated into the generalized geocentric XYZ coordinate system to compute Θ, the angle between the wave propagation direction and the geomagnetic field vector. The proposed correction formula can be implemented in real-time applications as it does not require the knowledge of the geomagnetic field or the electron density distribution in the ionosphere along the signal path; only the total electron content (TEC) and geometrical parameters defining the ray path such as azimuth and elevation angles are required. It is expected that the correction will enable a more accurate positioning by using carrier phase measurements