The second-order Rytov approximation and residual error in dual-frequency satellite navigation systems

The second-order Rytov approximation has been used to determine ionospheric corrections for the phase path up to third order. We show the transition of the derived expressions to previous results obtained within the ray approximation using the second-order approximation of perturbation theory by solving the eikonal equation. The resulting equation for the phase path is used to determine the residual ionospheric first-, second- and third-order errors of a dual-frequency navigation system, with diffraction effects taken into account. Formulas are derived for the biases and variances of these errors, and these formulas are analyzed and modeled for a turbulent ionosphere. The modeling results show that the third-order error that is determined by random irregularities can be dominant in the residual errors. In particular, the role of random irregularities is enhanced for small elevation angles. Furthermore, in the case of small angles the role of diffraction effects increases. It is pointed out that a need to pass on to diffraction formulas arises when the Fresnel radius exceeds the inner scale of turbulence.