## 1. Introduction

[2] The ionospheric propagation medium is characterized by a refractive index which is different from that of the free space. The phase refractive index is less than unity resulting in a phase velocity that is greater than the speed of light. However, the group refractive index is greater than unity resulting in a group velocity that is less than the speed of light. Therefore, when the Global Positioning System (GPS) signals propagate through the ionosphere, the carrier experiences a phase advance and the code experiences a group delay. The carrier phase pseudoranges are measured too short and the code pseudoranges are measured too long compared to the geometric range between the satellite and the receiver. Accurate range estimations between a receiver and four or more satellites enable accurate position determination of GNSS users in space and time.

[3] The ionospheric range error is at first order directly proportional to the total number of free electrons along the path of the signal from the satellite to the receiver. It can vary from a few meters to tens of meters at the zenith [*Klobuchar*, 1996]. Since the ionosphere is a dispersive medium, the magnitude of the ionospheric delay depends on the signal frequency and therefore, the first-order effect can be eliminated through a linear combination of dual-frequency observables. However, inhomogeneous plasma distribution and anisotropy cause higher-order nonlinear effects which are not removed in this approach. Mainly the second- and third-order ionospheric terms (in the expansion of the refractive index) and errors due to bending of the signal remain uncorrected. They can be several tens of centimeters of range error at low elevation angles and during high solar activity conditions [*Klobuchar*, 1996].

[4] Early work was done by *Brunner and Gu* [1991] in computing higher-order ionospheric effects and developing correction for them. Since then higher-order ionospheric effects have been studied by different authors during last two decades [e.g., *Bassiri and Hajj*, 1993; *Jakowski et al.*, 1994; *Strangeways and Ioannides*, 2002; *Kedar et al.*, 2003; *Fritsche et al.*, 2005; *Hawarey et al.*, 2005; *Hoque and Jakowski*, 2006, 2007, 2008; *Hernández-Pajares et al.*, 2007; *Datta-Barua et al.*, 2008; *Morton et al.*, 2009]. They found that higher-order ionospheric terms are less than 1% of the first-order term at GPS frequencies; however, they represent millimeter/centimeter level errors in geodetic measurements. *Hernández-Pajares et al.* [2007] found submillimeter level shifting in receiver positions along southward direction for low latitude receivers and northward direction for high latitude receivers applying the second-order term correction. *Fritsche et al.* [2005] found centimeter level correction in GPS positions considering higher-order ionospheric terms. Recently *Petrie et al.* [2010] investigated the potential effects of the bending terms described by *Hoque and Jakowski* [2008] on global GPS network. They found the bending correction for the dual-frequency linear L1-L2 combination to exceed the 3 mm level in the equatorial region. All these studies were conducted to compute higher-order ionospheric effects on GNSS signals for ground-based reception.

[5] Recently *Hoque and Jakowski* [2010] investigated ionospheric impact on GPS occultation signals received onboard Low Earth Orbiting (LEO) CHAMP (Challenging Minisatellite Payload) satellite. LEO satellites have the opportunity to receive signals from occulting GNSS satellites using onboard limb-pointing antennas. The observations have the dual purpose of studying electron densities in the ionosphere as well as temperature and moisture in the neutral atmosphere. A number of satellite and minisatellite missions such as GPS/MET (GPS Meteorology Instrument), SAC-C (Satelite de Aplicaciones Cientificas-C), CHAMP, GRACE (Gravity Recovery And Climate Experiment) and COSMIC satellite network (Constellation Observing System for Meteorology, Ionosphere and Climate, also known as FORMOSAT-3) carry onboard GPS receivers for the GPS radio sounding of the Earth. Although occultation measurements are not usually used for positioning or navigation, it is worthy to know the ionospheric impact on accurate range estimation using these measurements. This paper investigates raypath bending effects in dual-frequency range and total electron content (TEC) estimation and proposes correction for mitigating such effects.

[6] During occultation both transmitter and receiver are out of the Earth's atmosphere. When the transmitted signal approaches the Earth, the closest point of approach to the Earth's surface is known as the tangential point and the altitude corresponding to this point is known as the tangential height. The perpendicular distances of the signal path from the straight line of sight (LOS) propagation are defined as the raypath deviations. Interested readers are referred to Figure 1 of *Hoque and Jakowski* [2010] for an exaggerated view of GPS frequencies geometric paths during radio occultation. Radio wave traverses a long ionospheric limb path and therefore, the refraction effects are the most pronounced during radio occultation. The refraction effects depend on the actual condition of the ionospheric ionization and as well as on the raypath geometry.

[7] Our previous investigation [*Hoque and Jakowski*, 2010] for selected GPS-CHAMP occultation events shows that the straight line propagation assumption errors such as the excess path length of the signal compared to the LOS propagation, raypath deviations and TEC differences along the curved and LOS paths significantly vary with the ionospheric profile shape and raypath geometry. We found the maximum estimates of the excess path length to be about 2.7 m, and the second- and third-order ionospheric terms to be about 13 cm and 2.1 cm, respectively, for the GPS L2 signal for an electron density profile with vertical TEC of about 167 TEC units (1 TEC unit = 10^{16} electrons/m^{2}). We found the separation between the GPS L1 and L2 raypaths to exceed the kilometer level and errors in the GPS dual-frequency range estimation and TEC estimation to exceed the meter and 10 TEC units level, respectively.

[8] In this paper, simulation studies have been done to determine the straight line propagation assumption errors as functions of the signal frequency, different ionospheric parameters such as the maximum ionization and TEC, and geometrical parameters such as the tangential height of the raypath. Based on simulation studies we have proposed correction formulas for computing the excess path length and TEC difference between the signal and LOS paths.