## 1. Introduction

[2] The ionosphere significantly affects observations of the code phase delay and the carrier phase advance of Global Positioning System (GPS) signals at the two GPS frequencies currently in use (1227.60 and 1575.42 MHz) [e.g., *Brunner and Gu*, 1991; *Bassiri and Hajj*, 1993]. The ionospheric errors are dominated by the electron density of the ionospheric plasma along the signal propagation path and result in a ranging error on the order of tens of meters [e.g., *Klobuchar*, 1996]. Additional ionospheric errors are introduced (although to a lesser extent) by the presence of the Earth's magnetic field [e.g., *Brunner and Gu*, 1991; *Bassiri and Hajj*, 1993; *Datta-Barua et al.*, 2006, 2008]. When a predominantly right-hand circularly polarized (RCP) GPS signal propagates through the ionosphere, which is an anisotropic medium, it may propagate as the linear combination of two different modes: the extraordinary (or X) mode and the ordinary (or O) mode, depending on the angle between the GPS wave normal and the Earth's magnetic field. These two modes correspond to two different magneto-ionic polarizations and two different refractive indices, affecting the second-order ionospheric error introduced to GPS ranging calculations. The calculations provided herein demonstrate that accounting for the magneto-ionic polarization of the GPS signal produces a second-order error that is asymmetric about the Earth's geomagnetic equator. The specifics of the asymmetry are location dependent. For instance, we will show that near the geomagnetic equator, signals arriving from the north propagate with O mode polarization, and the second-order ionosphere carrier phase error is positive, implying that the first-order ionosphere error is an underestimation of the total signal delay. On the other hand, GPS signals arriving from the south propagate with X mode polarization, and the second-order ionosphere carrier phase error is negative, implying that the first-order ionosphere error is an overestimation of the total delay.

[3] The difference between the magneto-ionic modes of propagation was overlooked in previous studies. For example, *Bassiri and Hajj* [1993] state that “the (−) and (+) signs in equation (11) correspond to the ordinary and extraordinary waves, respectively. Ignoring the LCP component of the GPS signal which has less than 0.35% and 2.5% of the total power for L1 and L2, respectively, only the (+) sign will be of relevance to us in the subsequent analysis.” In the paper, equation (11) is the expression that computes the second-order error. The paper mistakenly relates the left-hand circularly polarized (LCP) component of the GPS signal with the ordinary wave propagation mode and failed to recognize that a RCP signal can propagate in both ordinary and extraordinary modes. *Bassiri and Hajj* [1993] were not alone in this misconception: *Brunner and Gu* [1991] provide a very similar description in their Appendix A. This misconception persists today, due at least in part to the propagation of the concept from previous works. As recently as 2008, *Hoque and Jakowski* [2008] state “Since the anisotropic plasma is doubly refracting (indicated by the ±sign in refractive index equation (14)) there are actually two waves. The wave with the upper (+) sign is usually called the ordinary wave, whereas the lower (−) sign is related to the extraordinary wave. The ordinary mode is left-hand circularly polarized, while the extraordinary mode is right-hand circularly polarized [*Hartmann and Leitinger*, 1984]. However, since GPS signals are transmitted in the right-hand polarization [*Parkinson and Gilbert*, 1983], only the results of extraordinary mode are considered here.” Additionally, the several articles [e.g., *Hartmann and Leitinger*, 1984; *Datta-Barua et al.*, 2006, 2008; *Hoque and Jakowski*, 2007, 2008] that have critically analyzed second-order ionospheric effects have employed the second-order error terms provided by *Bassiri and Hajj* [1993] and by *Brunner and Gu* [1991], which are inconsistent. We note that several recent articles [e.g., *Kedar et al.*, 2003; *Fritsche et al.*, 2005; *Hoque and Jakowski*, 2008; *Datta-Barua et al.*, 2008; *Petrie et al.*, 2010] employ the same formula as that derived herein to calculate second-order GPS errors, and these analyses are therefore consistent with the work presented here. None of these other works deviate from the concept that GPS signals propagate solely as an X mode polarized signal, however, and none have provided a complete mathematical justification for using this formula.

[4] This paper provides a rigorous Taylor-series expansion of the refractive index and the group refractive index for GPS signals propagating through the ionosphere, which we take to be a cold, collisionless, and magnetized electron plasma. It is demonstrated that the magneto-ionic polarization of the predominantly right-hand circularly polarized GPS signal depends on the direction of the GPS signal *k* vector with respect to the Earth's magnetic field. To illustrate the effect of the GPS signal magneto-ionic polarization on GPS positioning accuracy, we interpret second-order GPS carrier phase ranging errors for three geographically distinct receiver locations. This dependence is shown to produce an asymmetry in the second-order GPS ranging error about the geomagnetic equator.