## 1. Introduction

[2] Global Navigation Satellite Systems (GNSS) signals traveling from satellite to Earth are subject to a variety of error sources with the most significant being multipath and the effect of the ionosphere. The latter has two components: the background ionosphere introduces both delay and frequency dispersion whereas small scale time-varying irregularities introduce phase scintillation and amplitude fading into the received signal. Whereas correction schemes have been devised for the ionospheric delay using both dual-frequency, differential GPS, wide area augmentation schemes, etc., the effect of scintillation cannot be corrected for, being essentially a quasi-random process. Thus either mitigation and/or increased robustness of GPS receivers to scintillation are the only sensible strategies. GPS (or other GNSS) receivers can suffer from phase lock loss and cycle slip in conditions of severe scintillation, particularly for L2 for which the signal strength is smaller (than L1) and which (for most currently operational GPS satellites) semicodeless or other indirect methods are required for decoding. The phase-locked loop (PLL) minimizes the error between the input phase and its estimated phase output, which is then fed to the receiver processor. The ability of the loop to remain in lock is associated with the magnitude of this error.

[3] The most severe scintillation conditions occur at polar, high (auroral) and low latitudes (±20 of the geomagnetic equator) and would be expected to be more frequent at the maximum of the solar cycle. Considering the approach of the next solar maximum in 2011, it is desirable to be able to forecast the amount of tracking jitter that could occur for GPS receivers in these regions so that the likelihood of cycle slips or phase lock loss for different locations, times and conditions can be determined and the effect on positioning assessed. Further, if the tracking jitter on all simultaneously observed satellites is estimated, these estimates could be used to weight the measurements from each satellite used in the positioning calculation. During a period of occurrence of high phase scintillation, it was found by *Aquino et al.* [2009] that problems related to ambiguity resolution can be reduced by the use of such a mitigation method. The advantage of the present method to such a scheme is that the tracking jitter can be obtained just from time domain calculations and there is no need to continually transform data to the frequency domain to determine the *p* and *T* values required for the tracking jitter calculation.

[4] *Conker et al.* [2003] present models to calculate the variance of this phase tracking error for two common types of PLL (L1 and L1-aided L2, the latter also referred to as L2 semicodeless). including the effects of scintillation on the input phase. They model amplitude scintillation as an increase in thermal noise related to the decrease in received signal power, and phase scintillation as an additive term to the overall phase noise. Required input parameters for their formulae include the spectral parameters *p* (inverse power law of the phase PSD) and *T* (spectral strength of the phase PSD at 1 Hz) as well as S4. A major rational for this was that these spectral parameters were output by the scintillation receivers/monitors they were employing at the time (M. B. El-Arini, private communication 2008) but they are also advantageous in that, unlike scintillation indices, they do not depend intrinsically on the upper and lower limits of the fading frequency spectrum (determined by the lower cutoff and Nyquist frequencies employed). However, this presents a problem for scintillation data for which these spectral parameters are not available or not able to be conveniently determined. An example of such a data set, and one driving force for the present work, is a comprehensive archive of scintillation data gathered by the IESSG, Nottingham University, UK during the high of the last solar cycle containing GPS ionospheric scintillation data gathered simultaneously with four GSV4004 receivers (GPS Silicon Valley) in the UK and Norway, between June 2001 and December 2003, at geographic latitudes varying from 53°N to 71°N. However, this scintillation data contains only the scintillation indices S4 and *σ*_{ϕ} and not the spectral parameters *p* and *T* and thus are insufficient on their own to be used to determine tracking jitter using the formulae of *Conker et al.* [2003]. Although it has been shown by *Aquino et al.* [2007] that it is possible, using high data rate GPS data, to determine *T* if *p* is known, the estimation or prediction of *p* is a difficult problem for all locations and conditions. Thus it is desirable to find a method of transforming scintillation indices into spectral parameters without the recourse of determining PSDs from time series of high sample rate phase data which may not be available and also avoiding the requirement of determining FFTs. Thus, a new method is proposed which uses both scintillation indices (*σ*_{ϕ} and S4) to determine the spectral parameters *p* and *T*. Clearly two equations are required to determine two unknowns. These are formed making use of the known general fading frequency behavior of the PSD spectrum which is different between amplitude and phase scintillation. This difference is exploited, utilizing approximate models of the PSD for both amplitude and phase, to define equations that can be solved for *p* and *T* for any given f_{F}. To do this a relation is established between the standard deviations of the phase and amplitude detrended time series and the integrals of their power spectral densities in the frequency domain. Then it is shown that, by taking account of the range of physically realistic values of f_{F}, the tracking jitter can be determined to a reasonable degree of accuracy, especially for high latitude receiver locations.

[5] In the framework of the theory of weak scintillation and under the assumption of the frozen-in drift, *Yeh and Liu* [1982] have considered the resulting phase S_{ϕ}(f) and amplitude S_{χ}(f) spectra of a received signal. If *V* is a projection of the equivalent drift speed onto the plane perpendicular to the raypath and for an inverse power law spectrum of the electron density fluctuations with 3-D spectral index *p*_{i}, both curves were found to have the same high-frequency behavior, which is proportional to V_{i}^{p−2}f_{i}^{1−p} but in the low frequency limit the curves are different. In this case, the phase spectrum has the same frequency dependence as in the high-frequency case but the amplitude power spectrum tends to a constant value proportional to V_{i}^{p−2}f_{F}^{1−p}_{i}, although it can, for anisotropic irregularities, reduce in amplitude with decreasing frequency below f_{F}, depending on the irregularity aspect ratio, which can also be size dependent [*Wernik et al.*, 1990]. For the method described in this paper it is important to use the relation between experimental estimates of the standard deviation or variances of phase and normalized amplitude of detrended data and the integral of the power spectral densities S_{ϕ}(f) and S_{χ}(f) over their complete frequency range. The lower limit of this integration will be defined by the inverse length of the time series taken for the estimation, or by the cutoff frequency of the high-pass filter used for detrending the data, whichever is higher. The upper limit will generally be determined as one half the sample rate (Nyquist frequency).