Determining scintillation effects on GPS receivers



[1] Although formulae are available to determine tracking jitter (variance of the phase tracking error at the output of the Global Navigation Satellite Systems receiver phase-locked loop) resulting from scintillation for GPS/SBAS C/A code processing and semicodeless GPS L1 and L2 Y-code, these require input of the spectral parameters p (inverse power law of the phase power spectral density (PSD)) and T (spectral strength of the phase PSD at 1 Hz) which will not generally be available. It would certainly be more convenient if tracking jitter could be determined just from scintillation indices (S4 and σϕ) enabling determination when spectral parameters are not readily available and permitting tracking jitter for all the simultaneously observed satellites to be easily determined and used in a scintillation mitigation scheme. The main difficulty is that the Fresnel frequency, fF, which is an important feature of the amplitude PSD, should be known. Here a method is proposed which uses both scintillation indices (σϕ and S4) to give an additional relation to find both p and T. This makes 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 fF. Even when fF is not known, it is shown that by taking account of the range of physically realistic values of fF, the tracking jitter can generally be determined to a reasonable degree of accuracy.

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 σϕ 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 (σϕ 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 fF. 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 fF, 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 Sϕ(f) and amplitude Sχ(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 pi, both curves were found to have the same high-frequency behavior, which is proportional to Vip−2fi1−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 Vip−2fF1−pi, although it can, for anisotropic irregularities, reduce in amplitude with decreasing frequency below fF, 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 Sϕ(f) and Sχ(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).

2. Model of Amplitude and Phase Spectra

[6] The phase spectrum is modeled to have a single slope spectrum (slope p) between its upper and lower limits on log-log axes. The amplitude spectrum is modeled similarly to the phase spectrum above the Fresnel frequency. Below the Fresnel frequency the amplitude spectrum is taken to be constant (although a variation resulting from anisotropy of the irregularities could also be modeled providing no additional variable were introduced; see section 7). σϕ2 is taken to be equal to twice the area under the PSD of the phase variation [Rino, 1979] and a similar relation is assumed for the normalized σχ2 with respect to the PSD of amplitude variation. The curves representing the spectra above the Fresnel frequency can be considered to have the same slope, as mentioned above. They are shown a little apart for clarity in Figure 1 but it will be assumed that they are coincident except for strong scintillation conditions. In reality the PSD of amplitude would not of course be independent of the absolute signal amplitude but this is not a problem here since a normalized parameter is employed. In practice the exact GPS signal amplitude is seldom known and the scintillation index S4 does not depend on it as it is normalized to near unity. Thus the amplitude spectra shown is for a similar normalization to the amplitude scintillation index σχ which would be equivalent to the S2 index. Now, providing strong scintillation conditions do not occur, the areas under the two spectral curves are quite closely similar below the Fresnel frequency. Figure 2 gives an example of the variation of the area under the spectral curves with lower cutoff frequency, the upper cutoff being half the sample rate. This simulation has been performed using the LSP (Leeds St. Petersburg) transionospheric simulation simulator [Gherm et al., 2005] for a transionospheric path from a GPS satellite at L2 frequency. Although for a specific propagation path (azimuth 180° and elevation 50° with a magnetic dip of 45°), particular background and stochastic ionosphere parameters (an east–west frozen-in drift velocity of 200 m/s, an outer scale of 30 km, an aspect ratio of 5, a variance of the electron density fluctuations of 10−4 with an inverse power law anisotropic spatial spectrum governed by pi = 3.7), a similar result of the close coincidence of the spectral curves above the Fresnel frequency (and equal areas measured from close to this frequency to the Nyquist) results for a wide range of propagation geometries and ionospheric parameters. However, as would be expected, for strong scintillation conditions the high frequency parts of the spectra do separate and the amplitude also shows an asymmetric distribution about its maxima. Such a separation can, however, also be modeled as shown in section 7.

Figure 1.

Model of phase and log amplitude spectra.

Figure 2.

Area under the curves of spectral densities of amplitude and phase above the frequency shown on the x axis to the cutoff frequency.

3. Relation Between Phase and Amplitude Scintillation Indices and the Corresponding Power Spectral Densities

[7] Rino [1979] presented the relation between the phase scintillation index and phase PSD as:

equation image

[8] This relation can be considered to follow from Parseval's theorem which equates the energies in the time and frequency domains. By the same reasoning, a similar relationship will exist between the amplitude scintillation index σχ and the area under the curve of the PSD for amplitude. However, since the time series for amplitude will not, at least for conventional GPS detrending, have zero mean, this relation would not be true if the integration were performed right down to zero frequency because of the “dc term” that would result in the Fourier transform. This is not a problem for the method proposed here in practice as only time domain measurements are used (providing the Fresnel frequency is estimated without determining the amplitude spectrum). The relation between the normalized σχ index (equivalent to S2) and S4 depends on the distribution but if the variation is much less than the mean S4 ≈ 2σχ. Because of the way that the amplitude detrending is performed for GPS signals, this normalization is actually performed by the standard detrending procedure. The S4 index is generally derived from detrended signal intensity (received signal power). Because the S4 index is normalized, the receiver's absolute gain is not important, as long as it is relatively constant during the detrending period. It is also important that the intensity measurement be linear with respect to the signal power over its entire range, including deep scintillation fades [van Dierendonck, 1999]. To obtain the S4 index, the received power is normally detrended by first filtering the intensity measurements in a (sixth-order Butterworth) low path filter and then the detrending is accomplished by dividing the raw data by the output of the low-pass filter which results in a signal which fluctuates around a value of unity [Van Dierendonck et al., 1993]; thus proving the normalization required. The σχ index is introduced to quantify the difference between the amplitude and phase spectra as a means of setting up the equations to determine the spectral parameters from the scintillation indices which, as will be shown in the next section, is accomplished by deriving relations between the amplitude and phase scintillation indices and the spectral parameters T and p given by Rino [1979].

4. Determination of p and T From σϕ and S4 Using the Spectral Models

[9] Rino's [1979] representation of the phase scintillation PSD is given by:

equation image

where fo is the outer scale frequency, T is the spectral strength of the phase noise at 1 Hz and p is the spectral slope of the phase PSD. (Note p = pi −1) If ffo then we can write Sϕ = Tf –p. Then:

equation image

where r = 1 − p, and using the approximate form of the amplitude spectrum introduced in section 2 the variance of the normalized amplitude

equation image

[10] The area between the phase and log-amplitude PSDs will then be:

equation image

[11] We can substitute T = 0.5σϕ2/[(fur − fcr)/r] from (1). Then we can find the value of the slope of the phase spectrum p that will result in given values of σϕ and σχ by finding the zero of the function:

equation image

[12] This is done in MATLAB by an algorithm that uses a combination of bisection, secant, and inverse quadratic interpolation methods.

5. Determining Tracking Jitter

[13] Conker et al. [2003] introduced models which can be used to calculate the variance of the PLL phase tracking error (tracking jitter) for two common types of PLL (a 3rd order L1 carrier PLL and a 2nd order L1-aided L2 carrier PLL (L2 semicodeless). The model for the L1 carrier PLL accounts for the effects of scintillation on the input phase and computes the tracking error variance at the output of the PLL, as

equation image

where σϕs2, σϕT2 and σϕosc2 are respectively the phase scintillation, thermal noise and the receiver/satellite oscillator noise components of the tracking error variance. This assumes no correlation between the amplitude and phase scintillations; an assumption that may break down for strong scintillation conditions [Gherm et al., 2003]. Amplitude scintillation is modeled as an increase in the thermal noise, related to the decrease in the received signal power

equation image

Bn is the L1 third-order PLL one-sided bandwidth, (∼10 Hz); (c/n0)L1-C/A is the SNR and η is the predetection integration time (0.02 s for GPS and 0.002 s for WAAS). The formula is valid for S4(L1) < 0.707.

[14] Phase scintillation jitter is determined starting with

equation image

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 is given by:

equation image

where k is the order of the PLL and fn is the loop natural frequency. For the calculations of tracking jitter presented below k was taken as 3, (c/n0)L1-C/A as 30 dB, η as 0.02, fn as 1.91 Hz, Bn as 10 Hz and σϕ,osc was taken to be 0.1 rad.

6. Investigation of Tracking Jitter Variation

[15] Using this method the variation of tracking jitter that can occur for given values of the phase and amplitude scintillation indices can be explored. The maximum variation is likely to occur when both scintillation indices are large. A low latitude example with σϕ = 0.5 and S4 = 0.624 is first considered. For Fresnel frequencies in the range 0.15 to 2.5, we solve in each case for the values of p and T that result in the correct scintillation index values. With the above mentioned constraints, increasing values of fF result in increasing values of T and decreasing values of p. The Fresnel frequency is given by

equation image

where vrel is the relative drift (depending on both the satellite motion vsat and the irregularity drift vd) and z is the distance from the equivalent diffraction screen to the receiver. At L1 for a diffraction screen at 350 km, dF = 360 m. From the results of p, T and tracking jitter versus Fresnel frequency, as shown in Figure 3, the advantage of choosing the Fresnel frequency as a control parameter can be appreciated. Although the range of tracking jitter for given values of σϕ and S4 is large for the total possible Fresnel frequency range, for the range of realistic values of the Fresnel frequency at low latitudes from about 0.2 to 0.45 Hz [Forte and Radicella, 2002], it is a lot smaller; from 9.7 to 11.7° (2°). This Fresnel frequency range is based on a typical GPS satellite velocity of 26–37 m/s (mapped to the assumed 350 km height of the IPP (ionospheric pierce point) at an elevation of around 40° from a low latitude receiver and irregularity drift velocities in the range 70–170 m/s. Thus, in this way, the error in estimating the tracking jitter at low latitudes, when only scintillation indices are available, can be very considerably decreased.

Figure 3.

(left) Slope p, (center) T, and (right) standard deviation of tracking jitter versus Fresnel frequency for scintillation indices σϕ = 0.5 and S4 = 0.624.

[16] Next the tracking jitter is determined as a function of the Fresnel frequency for a typical high latitude case. Here we consider σϕ > S4, using σϕ = 0.6 and S4 = 0.2 (first example) and also σϕ = 0.6 and S4 = 0.1 (second example). As before, we determine p, T and the standard deviation of the tracking jitter (Figure 4) corresponding to these scintillation indices as a function of the Fresnel frequency. Based on a typical GPS satellite velocity at the IPP of 26–37 m/s at 350 km height (at an angle of elevation of around 40°) and an irregularity drift velocity of 300 m/s, the Fresnel frequency would be about 0.8 Hz whereas, for a more extreme value of 1000 m/s, it would be about 2.75 Hz [Forte and Radicella, 2002]. The total range of the standard deviation of the tracking jitter for σϕ = 0.6 and S4 = 0.2 is much smaller than for the low latitude case (8.2° to 9.5°); only 1.3°. From the graphs it can be seen that typical values of p (2 to 3) correspond to higher Fresnel frequencies (>0.5 Hz) as expected at higher latitudes. (Note that higher values of p than 7 which correspond to near zero Fresnel frequencies are strictly beyond the validity of the equation for the phase scintillation component of the tracking error variance.) When S4 is reduced to 0.1, the range of tracking jitter for the whole range of Fresnel frequency from 0.15 to 2.5 Hz (considered to be the range of physically reasonable values at high latitudes [Forte and Radicella, 2002]) is 8.18° to 8.52°; only 0.34° as seen in Figure 5 (left). Thus the tracking jitter can be estimated fairly accurately by finding it for physically realistic values of the Fresnel frequency. Figure 5 (right) shows by how much the estimate of the tracking jitter can vary if not constrained by physically realistic values of Fresnel frequency. The tracking jitter has been plotted (see Figure 5, right) for the same values of σϕ = 0.6 and S4 = 0.1 as the plot shown in Figure 5 (left) but for the total range of Fresnel frequency that is possible based on just the values of the two scintillation indices. Then the range of tracking jitter is found to be much larger (8.2° to 49.6°). However, all the larger values (>9°) correspond to unrealistically large Fresnel frequencies (>1 km/s up to 22 km/s irregularity velocities). This is in complete contrast to employing the restraint of physically acceptable values of p. For the last example (σϕ = 0.6 and S4 = 0.1) allowing p in the range 1 to 4 gives no constraint at all on the jitter which is found to vary between 8.2° and 49.6° = 41.4° range, whereas, using physically realistic values of the Fresnel frequency limits the jitter to the range of only 0.34° (8.18° to 8.52°).

Figure 4.

(left) Slope p, (center) T, and (right) standard deviation of tracking jitter versus Fresnel frequency for scintillation indices σϕ = 0.6 and S4 = 0.2.

Figure 5.

Tracking jitter (standard deviation) versus Fresnel frequency for σϕ = 0.6 and S4 = 0.1 for Fresnel frequencies (left) up to 2.5 Hz and (right) up to 62 Hz.

7. Refining the Spectra Models

[17] In the work so far presented the phase spectra has been taken to have the approximate form Sϕ = Tf –p. This has the advantage that the integral over frequency can be obtained analytically. However, this does not include the effect of an outer scale (which will cause the phase spectrum to flatten out at the lowest frequencies) and thus will only be strictly valid when fc ≫ fo where fo is the frequency corresponding to the maximum irregularity size in the ionosphere and fc is the lower cutoff. (Note that if the reciprocal of the data interval or integration time τc (whichever is smaller) is less than fo−1, then fo should be replaced by τc−1 in the formula for the spectral density [Rino, 1979].) If the formula

equation image

is employed, then to find p, the zero needs to be found of the function

equation image

[18] The integrals were found using the Matlab function fzero which uses an adaptive Simpson quadrature algorithm. Figure 6 shows how the tracking jitter varies with p for an outer scale of 0.05 Hz compared with no outer scale. The outer scale of 0.05 Hz can be seen to make little difference to the determined tracking jitter. When fo is increased the determined value of p is found to increase also. This is because the outer scale has the effect, for fixed scintillation indices and Fresnel frequency, of decreasing the slope of the phase PSD so that an increased value of p is needed to compensate for the same difference between the scintillation indices.

Figure 6.

Variation of tracking jitter with p (blue line with no outer scale) and green line (fo = 0.05 Hz).

[19] Determining the integrals of the phase and log amplitude spectra using a numerical method opens the way for further enhancements of the method to include: (1) modified spectra for strong scintillation (tails of amplitude and phase spectra do not coalesce at the highest frequencies and amplitude spectrum broadens) (2) the decrease in the PSD for amplitude with decreasing frequency below fF due to anisotropic irregularities, and (3) modeling of the effect of the detrending filter on the spectra. Considering the aforementioned effect 2, although, in principle, the modeled amplitude spectrum could be modified by including a positive slope, pb, with increasing frequency below the Fresnel frequency, (pb depending on the irregularities anisotropy), in practice the axial ratio of the irregularities is not likely to be known although it could estimated on the basis of the geographical location, level of geomagnetic activity, etc. However, although on a commonly used plot, showing the amplitude and phase spectra in dB, the change in area between the amplitude and phase PSDs below fF might appear appreciable, this is a significant overestimate from the proper relation for area as would be seen on a linear plot. For example, take the case of a Fresnel frequency of 1 Hz and lower cutoff frequency of 0.1 Hz together with a slope for both spectra above fF of p = 3 (these are all considered to be typical values). Then assuming a positive slope of the amplitude spectra below fF given by a power law of pb = 2 (this is probably a fairly extreme value), the difference in the area between the two spectra below fF (which is the important factor in determining the spectral values T and p) only increases by 1.3% compared with the flat spectrum for amplitude below fF.

8. Problems Concerning Detrending to Determine σϕ and S4 Indices

[20] Application of the method to σϕ and S4 indices received from orbiting GPS satellites involves additional issues. Clearly the scintillation indices input to the phase jitter finding program must properly correspond be ionospheric scintillation and not to noise, multipath or others causes and should be well detrended. Significant problems have been discovered in recent years with detrending both phase and amplitude/intensity time series data, particularly for GPS signals. Forte and Radicella [2002] have pointed out perils of the phase scintillation index and it is well known that because, unlike S4, it is influenced by spatial-scale structures much larger than the Fresnel radius, this index can be very sensitive to the lower cutoff frequency of the data. Forte [2008] has also shown how erroneous data detrending can be responsible for high phase scintillation with low intensity scintillation events which take place when a fixed filtering window, used to detrend raw GPS signal components, is not appropriate to the actual plasma dynamics. He saw a large decrease in the phase scintillation index when, for a particular set of data, the lower cutoff frequency was increased from 0.1 to 0.3 Hz. This problem of the sensitivity of the phase scintillation index to large-scale structures is not a problem for the method proposed here (providing the spectrum slope remains constant) as the lower cutoff frequency is taken into account in the calculations.

[21] Detrending problems with the S4 index have been investigated by Materassi and Mitchell [2007]. They point out that the S4 index is strictly only valid when the signal is stationary for the normal 60 s period used to compute it. Wavelet techniques have been used to resolve individual events when nonstationary conditions on this timescale exist and also possible differences introduced by the different detrending methods have been investigated [Materassi et al., 2009]. As far as the current method is concerned it is thought that there is not a problem with the method finding T and p which, for the nonstationary case, will correspond to some “averaged” spectra over the time interval which is normally 60 s.

[22] It is not necessary for the method to operate correctly that the scintillation should be just due to diffractive scattering from the irregularities. It is well known that fast TEC variations can contribute to the phase scintillation index (and sometimes even S4) at low fading frequencies [e.g., Mitchell et al., 2005; van de Kamp and Cannon, 2009]: this will not introduce error to the method providing the phase spectrum is still single slope. The presence of such a contribution to the phase scintillation index could be checked by examining the derivative of rate of change of TEC (DROTI) [Bhattacharyya et al., 2000]. Further although the spectral model could be modified for strong scintillation conditions and thus p and T determined if there is still just a single slope spectrum, the equations given by Conker et al. [2003] for the tracking jitter are not valid for S4(L1) > 0.707. These equations, like the spectral models used in the method proposed here, assume a single slope spectra whereas, particularly for strong scintillation conditions, the spectra may be dual slope. Dual slop spectra have been seen at both high latitudes [e.g., Forte, 2008] due to large scale irregularities and low latitudes [e.g., Vijayakumar and Pasricha, 1997] due to (large scale) plasma bubbles. For refractive scattering, Forte [2008] has described a lowering and spreading of the intensity spectrum toward both high and low spectral frequencies together with a roll-off of the intensity spectrum that is more Gaussian (thus departing from a single slope characteristic). Such strong scintillation conditions are, however, less likely to be encountered for the higher frequency GPS signals (>1 GHz) than for the 150 and 400 MHz measurements of Forte or the 244 MHz measurements of Vijayakumar and Pasricha.

9. Conclusions

[23] The problem of determining tracking jitter using only scintillation indices has been considered, requiring conversion between the scintillation indices and the spectral parameters T and p. To facilitate this, analytical models of phase and amplitude PSDs have been constructed. Using these models, an equation has been found, based on the difference between the scintillation indices, which can be used to determine p and T for a presumed value of the Fresnel frequency. This has then been used to determine the range of p, T and Fresnel frequency for values of these scintillation indices for both low and high latitude conditions. The results show that, particularly at high latitudes, a good approximate value of the standard deviation of the tracking jitter can be obtained by using the constraint of the range of physically realistic values of the Fresnel frequency. The method could also be used with a model that can estimate the Fresnel frequency. For previously acquired scintillation indices this could, for example for high northern latitude receiver sites, employ use of CUTLASS radar data. It is also shown how the method can be refined to include more complex spectral models. In this way the effect of the outer scale or anisotropic irregularities can be accounted for.

[24] The relations and method described above rely on S4 and σϕ being true measures of the amplitude and phase scintillation. For accurate results care should be taken to correct S4 for thermal noise and detrend to remove multipath if present. Phase noise in the receiver can dominate phase scintillation and it is important to employ receivers with low phase noise to make reliable scintillation measurements. Van Dierendonck [2008] gives a detailed account of how GPS receivers measure (or should measure) ionospheric scintillation and quotes a thermal noise limitation of about 0.1 rad at 30 dB-Hz and phase noise typically better than 0.05 radians achievable with SC-cut Oven Controlled oscillators.

[25] Thus the method advocated above can be used to obtain the spectral parameters p and T when these are otherwise unavailable or would be too time intensive or too computationally extensive to determine, for example for real time applications such as scintillation mitigation. Determining the tracking jitter for all the GPS satellites seen by a receiver at any one time could be the basis of a mitigation scheme to reduce the effect of scintillation on the positioning either by not using the satellites that display a high tracking jitter [Beniguel et al., 2004] or by weighting the measurements from each satellite in the positioning calculation inversely according to the tracking jitter present. The advantage of the latter in reducing problems related to ambiguity resolution has recently been demonstrated by Aquino et al. [2009] For such an application it would be the accuracy of the estimate of tracking jitter rather than those of the spectral parameters p and T that would be the most important.