## 1 Introduction

Several environmental factors may affect GPS performance, such as electromagnetic interference, multipath, atmospheric delay, and fluctuations in signal amplitude and phase due to the ionosphere, respectively, known as amplitude and phase ionospheric scintillation. These fluctuations are due to diffraction of radio waves caused by kilometer-scale ionospheric plasma density irregularities [*Yeh and Liu*, 1982]. Ionospheric scintillation is responsible for significant deterioration in GPS accuracy that depending on its severity may even lead to a complete system failure [*Basu and Basu*, 1981; *Beach*, 1998]. Such phenomenon is more common in the low-latitude region (between approximately −20° and 20° geomagnetic latitude) and auroral and polar zones (above 55° of latitude) [*Jiao et al.*, 2013; *Mushini et al.*, 2011]. Additionally, scintillation activity that takes place after sunset has a temporal and seasonal dependence and follows the 11 year solar cycle in broad terms [*Kintner et al.*, 2004].

Scintillation affects the performance of GPS receivers notably at the signal tracking loop level. Depending on the scintillation level, there may be an increase in range measurement errors or even losses of lock of the carrier and code loops [*Kintner et al.*, 2001]. In extreme cases, scintillation can result in full disruption of the receiver operation [*Rezende et al.*, 2007].

Recently, *Moraes et al.* [2012, 2014] proposed and validated the use of the *α-μ* model to represent the amplitude and the second-order statistics of the GPS ionospheric scintillation phenomenon. This model was introduced by *Yacoub* [2007] and is based on two parameters, instead of just one, like those adopted in the literature [*Fremouw et al.*, 1980; *Rino*, 2011]. The results obtained by *Moraes et al.* [2012] indicated that the *α-μ* model is the most realistic in describing the amplitude scintillation among the tested distributions, basically due to the additional degree of freedom.

*Conker et al.* [2003] proposed a model for estimating the effects of amplitude scintillation on the availability of GPS receivers. This model assumed the Nakagami-*m* distribution for the statistical characterization of amplitude scintillation. This assumption is based on the work of *Fremouw et al.* [1980], which showed that Nakagami-*m* distribution fitted well amplitude scintillation distributions. The present work, with foundations on the previous work of *Conker et al.* [2003], proposes an extended model for estimating the GPS receiver performance based on the *α-μ* distribution and derives the variances of the carrier and tracking loop errors of the GPS receiver. This extended model is based on a more realistic distribution for amplitude scintillation [*Moraes et al.*, 2012].

Section 2 presents in detail the *α-μ* distribution that will be used by the proposed models. The same section discusses the estimation of these coefficients and their physical interpretation, showing the benefits of *α-μ* distribution for modeling amplitude scintillation. Section 3 revisits the tracking error models for the GPS L1 receiver introduced by *Conker et al.* [2003] in the light of the *α-μ* distribution. The present extended model is explored, showing its mathematical restrictions and its advantages for performance estimations. Based on a large data set from the maximum of solar cycle 23, the typical error values that a user might expect under similar conditions are presented. Section 4 introduces the tracking error models for the GPS L2 aided by L1 receiver and shows the restrictions for the dual-frequency operation. Finally, section 5 presents concluding remarks.