## 1. Introduction

[2] It is well known that GNSS satellite systems operating in L band are subject to ionospheric amplitude and phase scintillation effects and, in the equatorial region, (±20° geomagnetic latitude), these are generally most prominent for a few hours after local sunset. In the high latitude (polar and auroral) regions, phase scintillation is more dominant than amplitude scintillation and can occur at any time during the day especially during geomagnetic storms. In midlatitude regions, both amplitude and phase scintillation are negligible. Amplitude scintillation is more dominant at low latitudes and phase scintillation at high latitudes. The scintillation arises from ionospheric irregularities which can be embedded in mesoscale structures such as polar patches or low latitude plasma bubbles. Such scintillation has an adverse effect on GPS range estimation and on positioning by introducing tracking jitter variance in receiver PLLs, which can lead to cycle slips and, for sufficiently strong scintillation conditions, even loss of carrier phase lock. It is therefore desirable to be able to mitigate the scintillation effect. This can be attempted via receiver hardware modifications (e.g., to make the phase tracking more robust) or by software means. The latter can involve either leaving out the satellites in the positioning calculation whose paths to the receiver have been severely affected by scintillation [*Beniguel et al.*, 2004] or weighting all the satellites in the positioning calculation inversely according to the scintillation present on the respective satellite to receiver paths. This can be done by utilizing the estimated tracking error variances in the receiver PLL for each satellite receiver path to weight the respective ranges in the positioning calculation [*Aquino et al.*, 2009]. These variances can be determined for GPS receivers from the formulae given by *Conker et al.* [2003] if the spectral parameters *p* (slope of the phase spectrum plotted on log-log axes) and T (phase power spectral density at 1Hz) are known. We will follow a similar procedure here but instead of determining the spectral parameters *p* and *T* from the phase spectrum (which requires high rate raw data of carrier phase and continuous FFTs to be performed on this data for all the satellites) or from estimating *p* just based on the prevalent conditions which is rather difficult [*Aquino et al.*, 2007], we will instead obtain them from the scintillation indices using the method of *Strangeways* [2009, hereinafter S09].

[3] In the S09 method, the phase parameters are determined directly from the phase and amplitude scintillation indices making use of approximate models of the amplitude and phase spectra and an approximate value for the Fresnel frequency for the path. The Fresnel frequency is an important feature of the amplitude PSD and is given by the velocity of the scintillation-inducing irregularities perpendicular to the satellite to receiver path divided by the Fresnel scale. The phase and amplitude spectra are modeled as shown in Figure 1 (except that the sloping parts of the spectra are assumed to coincide whereas they are shown a little separated in Figure 1 for clarity). *Rino*'s [1979] representation of the phase scintillation PSD is given by:

where *f*_{0} is the outer scale, *p* is the spectral slope of the phase PSD and *T* is its spectral strength at 1 Hz. If *f* ≫ *f*_{0} then we can write *S*_{ϕ} = *Tf*^{−p}. Then the variance of the phase of the detrended GPS phase data (equivalent to the scintillation index *σ*_{ϕ}^{2}) is equal to twice the area under the curve S_{ϕ} (f) (also determined from the PDF of the detrended phase data) and thus corresponding to the same range of fading frequency between a lower cutoff frequency f_{c} (generally set by the detrending filter) and an upper cutoff frequency f_{u} (generally given by half the sampling frequency). A similar relationship will exist between the (normalized) amplitude scintillation index *σ*_{χ} and the area under the curve of the PSD for amplitude. Both these relations follow from Parseval's theorem for discrete signals. Then the difference in the squares of the scintillation indices must be equal to twice the difference in these areas yielding [*Strangeways*, 2009]:

where *σ*_{χ}, the (normalized) amplitude scintillation index, is equivalent to S2 and *S*4 ≈ 2*σ*_{χ}, (providing that, for the distribution of amplitude, the variation from the mean is much less than the mean; e.g., see *Yakovlev* [2002] who takes S2 = 0.52S4). *f*_{c} is the lower cutoff of the detrended data generally given by high-pass filter cutoff used for detrending it, *f*_{u} is upper cutoff frequency (generally given by half the sampling frequency) and *r* = 1 − *p*. Then, utilizing a known or estimated value of the Fresnel frequency (*f*_{F}), we can find the value of the slope of the phase spectrum *p* that will result in given values of *σ*_{ϕ} sand *σ*_{χ} by finding the zero of the function:

[4] Here *σ*_{ϕ} and *σ*_{χ} are (for GPS observations) the scintillation indices determined from the detrended data. The precision of the value of the upper cutoff in equation (3) is not so important as long as it is above the Fresnel frequency. However, the value of the lower cutoff can have a significant effect on the value of *σ*_{ϕ} since there is a power law increase is the PSD (resulting from scintillation) with decreasing frequency. Thus it is important that equation (3) is solved for the correct value, which should in any case be known, as otherwise the value of *σ*_{ϕ} itself is of limited usefulness. Equation (3) is solved in MATLAB by an algorithm that uses a combination of bisection, secant, and inverse quadratic interpolation methods. This equation can be modified to include the effect of filter roll-off, an irregularity outer scale or other factors that modify the fading spectrum as explained in section 7 of *Strangeways* [2009]. When *p* has been found, *T* can then be determined from the relation between *p*, *T* and *σ*_{ϕ}^{2} given above.

[5] Since no experimental verification of the method was given by *Strangeways* [2009], before utilizing it to obtain tracking jitter values to use in weighting individual satellite links, or for other purposes, we will first establish the reliability of the method from experimental data. This will be accomplished using 50 Hz amplitude and phase data together with the corresponding scintillation indices (S4 and *σ*_{ϕ}) from high latitude receiving stations at Tromso (69.67°N, 18.97°E) and Longyearbyen (78.169°N, 15.992°E). Once the spectral parameters are known then the tracking jitter can be determined using the formulae given by *Conker et al.* [2003] who introduced the model of tracking error variance at the output of the L1 carrier PLL as:

*σ*_{ϕs}^{2}, *σ*_{ϕT}^{2} and *σ*_{ϕosc}^{2} are the phase scintillation, the thermal noise and the oscillator noise components of the tracking error variance respectively. Amplitude scintillation is modeled as:

where *B*_{n} is the L1 third-order PLL one-sided bandwidth (∼10 Hz); (*c*/*n*_{0})_{L1−C/A} is the SNR and *η* is the predetection integration time (0.02s for GPS and 0.002s for WAAS). The formula is valid for S4(L1) < 0.707 and so does not apply to the strongest amplitude scintillation conditions.

[6] The representation of phase scintillation of *Rino* [1979] was used to calculate the phase error at the input of the PLL by:

where *τ* is a system parameter relating to the phase stability time of the receiver. Then the phase scintillation component of the tracking error variance at the output of PLL is given by [*Conker et al.*, 2003]:

where *k* is the order of the PLL, *f*_{n} is the loop natural frequency. Thus the formula gives the tracking jitter in terms of the characteristics of the phase lock loop, the phase stability of he receiver and its oscillator phase noise, the CNR, the parameters *p* and *T* of the phase spectrum of the detrended signal and the amplitude scintillation index S4. For the calculations of tracking jitter presented below *k* was taken as 3, *η* as 0.02, f_{n} as 1.91 Hz, *B*_{n} as 10 Hz, *σ*_{ϕOSC} as 0.1 rad. and (c/n_{0})_{L1−C/A} was obtained from the receiver each minute.