## 1. Introduction

[2] The ionosphere is a significant error source of the GNSS error budget. Ionosphere-induced range errors vary from a few meters to tens of meters at the zenith [*Klobuchar*, 1996]. Since the ionosphere is a dispersive medium, the most part of the ionospheric delay can be eliminated through linear combinations of dual-frequency observables; therefore, only the higher order ionospheric errors will remain in the range estimation. However these errors can reach several tens of centimeters at low elevation angles and during extreme space weather conditions (i.e., ionospheric storms), which represent large errors in geodetic measurements [*Klobuchar*, 1996]. Satellite positions can also be improved to the centimeter level when higher order ionospheric corrections are applied [*Fritsche et al.*, 2005].

[3] Early work is done by *Brunner and Gu* [1991] in developing correction models for higher order ionospheric errors, especially to correct the second-order term and raypath bending error. Their approach requires the knowledge of the actual ionosphere and the average value of the longitudinal component of the Earth's magnetic field along raypaths. In practical cases, these parameters are not easy to estimate accurately.

[4] Later *Bassiri and Hajj* [1993] have done similar work assuming an Earth-centered tilted dipole approximation for the geomagnetic field. They consider the ionosphere as a single (thin) layer at a certain altitude (300 km) and compute the magnetic field vector at the ionospheric pierce point (IPP). However investigation by *Hawarey et al.* [2005] for Very Long Baseline Interferometry (VLBI) shows that further improvements for the second-order ionospheric term can be achieved using a more realistic magnetic field model such as the International Geomagnetic Reference Field (IGRF) instead of a dipole model.

[5] *Strangeways and Ioannides* [2002] presented a method for the range correction to less than 1 cm accuracy by employing three factors: one representing the ratio of the slant TECs experienced by the two frequencies and the other two representing the ratio of their geometric path lengths with respect to the true range. However, these parameters are not easy to estimate accurately in practical cases.

[6] In our previous work [*Hoque and Jakowski*, 2006, 2007], we proposed models for the second-order ionospheric error correction as a function of geographic latitude and longitude and geometrical parameters such as elevation and azimuth angles for GNSS users in Germany and Europe (latitude range 30–65°N, longitude range 15°W–45°E).

[7] In the present work, we have derived approximation formulas to estimate and correct dual-frequency range errors due to excess path length of the signal in addition to the free space path length, TEC difference at two GNSS frequencies, and third-order ionospheric term.