Ionospheric path delay modelling for spacecraft formation flying

In addition to absolute orbit determination information, many current and future space missions require onboard relative navigation solutions between multiple satellites. GNSS is ideal for satisfying such requirements because of its long-term stable accuracy and global availability. It is well known that of the various error sources in the GNSS observations, the ionospheric path delay is one of the largest. Even though most common GNSS errors can be eliminated by double-differencing (DD) the observations, the DD residuals of the ionospheric delay may still be significant when the spacecraft separation is large. An accurate model of the ionospheric delay can improve relative navigation accuracy. This paper describes a method for solving the zero-differenced (ZD) ionospheric delay in dual-frequency measurements. The geometry-free combination of code phase observables is formed after pre-processing the measurements. In the observation model equation, the ionospheric error is transformed into a function of the vertical total electron content (vTEC) using the modified Lear mapping function. Then the differential code biases (DCBs) of all the GNSS satellites and the receivers are estimated using a Least Squares method on the basis that the vTEC is parameterised by the spherical harmonic (SH) function. Once the DCBs are obtained from a long arc of GNSS observations (about 1 hour of data), they are then substituted into the observation equation to calculate the ionospheric delay. The vTEC can also be predicted directly from the ZD and singledifferenced (SD) measurements. Twin satellite data from the Gravity Recovery and Climate Experiment (GRACE) mission are used to test the proposed method.