Radio Science

Perils of the GPS phase scintillation index (σϕ)

Authors


Abstract

[1] The phase scintillation index (σϕ), equal to the standard deviation of measured phase, is often used to characterize Global Positioning System (GPS) observations in ionospheric environments that may be scintillated. Since σϕ is dominated by large-scale fluctuations, questions of usage and interpretation exist as will be illustrated here. Beyond traditional concerns with detrending, multipath and receiver phase noise, there are at least two issues to be considered. The first is the marginal suitability of σϕ to characterize a power law phase screen with a poorly defined low-frequency component (e.g., outer scale). Second, observed σϕ parameters may not be relevant to GPS receiver tracking impacts. These arguments are outlined here in greater detail and are illustrated with simple one-dimensional phase screen propagation modeling results. The conclusion is that GPS σϕ values depend critically on the circumstances of measurement and are difficult to compare among observations without additional knowledge, particularly of relative ionospheric drift and irregularity orientation, that may not be available from an isolated GPS receiver. The development of suitable alternative measures requires careful consideration of the elements of GPS scintillation and its impacts. The broader GPS scintillation community should take an active role in developing suitable replacement measures for σϕ.

1. Introduction

[2] The phase scintillation index (σϕ), defined as the standard deviation of measured phase, has historically been used to characterize the phase component of ionospheric scintillation [Fremouw et al., 1978; Yeh and Liu, 1982]. With the advent of Global Positioning System (GPS) scintillation monitoring, the research community has tended to apply traditional measures like σϕ to the newer observations. Previous analysis [Forte and Radicella, 2002] demonstrated some of the inadequacies of using the traditional phase scintillation index with GPS. A key problem is unknown relative motion between the ionosphere and the GPS line of sight (LOS) coupled with the fact that the bulk of phase fluctuation power due to ionospheric irregularities resides at the larger spatial scales. The goal of this paper is to expand on some of those concerns and to suggest reappraisal of the continued use of σϕ as a quantitative measure, particularly in GPS studies.

[3] That phase fluctuations have fundamental importance to ionospheric scintillation is not in dispute. Indeed, perturbations in phase at the ionosphere are the source of the developed diffraction pattern in the phase screen model. At issue, rather, is the utility of characterizing ionospheric phase by a single parameter that masks crucial information such as fluctuation power at particular scale sizes. The discussion of utility will be informed by several perspectives: (1) applicability to propagation studies, (2) observational limitations and (3) relevance to receiver impacts. In order to keep the arguments focused on the most fundamental concepts they will be illustrated with one-dimensional (1-D) phase screen examples, although the discussion will mention two-dimensional (2-D) effects where appropriate. The upshot will be that suitable phase fluctuation measures must account for spatial scales and velocity effects pertinent to the reasons chosen to quantify phase fluctuations in the first place.

[4] Many, but not all, of these considerations stem from the fact that the magnitude of relative ionospheric motion of the GPS LOS typically falls into an intermediate regime compared to other satellite beacon measurements. It is not possible to assume with GPS that the ionosphere is essentially fixed during the satellite pass as is the case with a low Earth orbit (LEO) beacon; neither is it possible to assume that the satellite link is fixed and all motion is due to the ionosphere as with a geostationary beacon. There may be cases of observations from moving receivers where some of these simplifying assumptions could apply, for example GPS measurements from LEO satellites, but in general they do not.

[5] Finally, note that some of the observational factors to be discussed may apply to the amplitude scintillation index (S4), the root-mean-square deviation of normalized signal power, as well as σϕ. Fortunately, spatial-scale structures much larger than the Fresnel radius do not influence S4 as strongly as they do σϕ unless a significant refractive component to the scintillation exists. Discussion of observational and operational concerns regarding S4 will be considered beyond the scope of the present text.

2. Definitions

[6] In mathematical form, the phase scintillation index is

equation image

where σϕ is phase and the angled brackets 〈…〉 denote expected value, usually replaced by temporal averaging in practice. Already this definition incorporates some deliberate haziness to make a point; namely, which phase is to be used? In principle, the phase variance of interest to theoretical studies has been that of the phase at the phase screen [Salpeter, 1967], sometimes denoted ϕ02 or by other notation. In practice, the variance of measured phase at the ground is used as a proxy since the equivalent phase screen representing the ionospheric state is not available. This paper will adopt the convention that σϕ refers to the phase at the screen, while the standard deviation of the phase of the propagated wavefront is σϕp.

[7] Figure 1 illustrates the geometry for a 1-D phase screen and the numerical difference between equation image and σϕp for a particular phase screen realization, an error of about 20% in this case. The expression for 1-D phase screen propagation is

equation image

where eiωt time dependence is implied and constant phase factors in front of the integral have been omitted. The quantities Ap(x) and σp(x) are, respectively, the amplitude and phase of the propagated wave while rF = equation image is the Fresnel radius, where λ is the radio wavelength and z is the perpendicular distance away from the phase screen that the wavefront has propagated. Note that the phase σp(x) has been “unwrapped” for purposes of presentation and of computing σϕp. Also, spatial averaging is used to estimate the expected values in equation (1). Inspection of Figure 1 clearly shows that larger-scale structures dominate in their contribution to σϕ and σϕp for this realization, a power law phase screen example with ℓ0 = 8.0 km to be detailed in the next section on propagation studies. A further upcoming section will discuss effects related to temporal averaging of the received phase and its velocity dependence.

Figure 1.

Phase screen geometry.

3. Propagation Studies

[8] Rino [1982, p. 860] notes that “most theoreticians have used” a parameter equivalent to σϕ as a “strength parameter” but that this usage “can be misleading” because it is possible to construct screens with different power spectra and a broad range of σp values that give essentially identical signal strength patterns. Rather than delve into the theoretical details of his brief argument we shall examine the effect through concrete examples.

[9] Consider two realizations of 1-D power law phase screens depicted in Figure 2, where ϕ(x) is the phase at the screen as in equation (2) above. The comparison visually demonstrates the qualitative differences between power law realizations having different outer scales; specifically, the bulk of the fluctuation power appears at larger scale sizes for the realization with a greater outer scale, ℓ0. These phase screens began with identical realizations of Gaussian-distributed, band-limited white noise at 0.19-m sample spacing (=λ1, the wavelength of the GPS L1 carrier) that were then spectrally shaped. The mathematical form of the power spectral density (PSD) used is

equation image

where the exponent p is the spectral index, q is the spatial frequency expressed in radians per meter and C is a scale factor. The values of the spectral index and outer scale are listed in the PSD plots (top plots). Note that, as intended, the spectra match above the q corresponding to ℓ0, with the exception of a slight tendency toward developing a noise floor in the ℓ0 = 8 km case. In fact, this tendency, coupled with the 50-km total length of the screen, precludes the use of an outer scale much larger than about 10 km here. The total number of points in the screen, 218, is a compromise between computer memory constraints and allowing sufficient length to vary ℓ0, given the fundamental 0.19-m point spacing.

Figure 2.

Power law (top) spectra and (bottom) phase realizations.

[10] Figure 3 compares the results for signal power, P(x) = ∣Ap(x)∣2, for the two phase-screen realizations obtained from FFT-based computations [Buckley, 1975] of the 1-D propagation integral (2). The power plots are qualitatively very similar; many common features between the two can be identified. Quantitatively, in both cases S4 = 0.800. Other calculation details include the assumption of a zenith phase screen at a distance z = 300 km, giving a Fresnel scale rF = equation image = 240 m. Thus the outer scales are greater than the Fresnel scale for both phase screens presented.

Figure 3.

Comparative received power plots (enlargement, bottom curve offset by −20 dB).

[11] The immediate conclusion is that σϕ, which is dominated by the larger scale variations in phase, does not bear close relation to amplitude scintillation levels if the phase variance in a band about the Fresnel wave number is relatively constant. It is possible to tailor σϕ almost at will for a given S4 by altering the low-frequency portion of the PSD, to include spectral breaks as well as a relatively arbitrary outer scale. Beach and Kintner [1999] model and discuss the sensitivity of the dependence of the relationship of S4 to σϕ on ℓ0 for a PSD similar to Equation (3) in the weak scatter approximation. (Jokipii [1970] proves that the weak scatter approximation is valid up to higher S4 values than is perhaps commonly appreciated; one requirement is that the PSD fall off sufficiently rapidly.)

[12] Again, what is measured as “σϕ” by a GPS receiver is in fact related to σϕp, not σp. This approximation is perhaps not too far wrong in the examples above. One can compute the standard deviation of propagated field phase directly, after unwrapping, to find σϕp = 12.5 rad in the ℓ0 = 8.0 km example (cf. Figure 1) and σϕp = 4.5 rad, an error of a factor of 3, in the ℓ0 = 0.8 km case. Another observational consideration is that GPS phase is often derived from dual-frequency total electron content (TEC) estimates, which contain errors due to the frequency dependence of phase scintillation [e.g., Bhattacharyya et al., 2000].

4. Temporal Sampling

[13] Greater concern develops with the conversion of spatial diffraction patterns into the time series that GPS receivers actually record. A recent series of papers [Kintner et al., 2001, 2004; Ledvina et al., 2004] describes some of these factors for the equatorial region and illustrates them with observational examples. Factors for a stationary receiver include LOS geometry relative to the magnetic field, ionospheric drift, satellite motion and the temporal evolution of the irregularities. At auroral or polar cap latitudes the situation becomes even more complicated because of varying 2-D configurations of ionospheric structure [e.g., Livingston et al., 1982]. This discussion will consider only the effects of net ionospheric drift speed, and then in a rather simplistic fashion, to make some illustrative points.

[14] Forte and Radicella [2002] treat the effects of a fixed (temporal) frequency filter cutoff in the phase detrending process for variable relative drift speeds. They demonstrate in a semiquantitative fashion that the drift, usually unknown for a single-receiver setup, can have a significant impact on temporally averaged σϕp. Conceptually, this result should be quite evident. Changes in the relative drift cause the phase fluctuation spectrum to shift left or right. In the mean time, the detrending process will remove a portion of the low-frequency power from the fluctuation spectrum; the power removed depends critically on where the detrending cutoff falls relative to the shifted spectrum. Since low-frequency power dominates phase's standard deviation in a power law environment, observed σϕp will vary even if the phase fluctuation spectrum remains constant. If the low-frequency phase spectrum varies under different ionospheric conditions (due to, for example, storm time effects [Forte and Radicella, 2002]), additional variability in measured σϕp will be introduced even as the amplitude fluctuations remain substantially the same, as in the above demonstration.

[15] An even more basic effect not considered by Forte and Radicella [2002] is implicit high-pass filtering due to the period over which σϕp is computed. Figure 4 shows sequences of temporally averaged σϕp estimates obtained from the received phase of the ℓ0 = 8 km case above using two different assumed relative drift speeds, vdrift, and a fixed averaging period of 1 min. Here we treat the LOS as fixed and consider the ionosphere to be moving horizontally with the speed vdrift. For simplicity, the irregularities do not evolve with time; that is, we assume “frozen in” irregularities. In the left-hand case of Figure 4, the standard deviation of phase is computed directly over the simulated 1-min data segments (i.e., number of data points = nearest integer to vdrift · 60 s/Δx, where Δx is the grid spacing, 0.19 m). Recall that the value computed above by spatially averaging over the entire propagated wavefront is σϕp = 12.5 rad. In the right-hand case, a linear trend has been subtracted from each 1-min data segment before computing σϕp for that segment. Note that the 50-km-wide phase screen contains fewer total minutes of data as the relative drift increases.

Figure 4.

Effects of relative drift on the temporally averaged phase scintillation index.

[16] The upshot is, the slower the relative drift of this simple 1-D ionosphere, the lower the temporally averaged σϕp estimate and linear detrending reduces the value further. The range of drift speeds used above is consistent with typical equatorial cases and that range can be extended further in many cases, for example, postmidnight zonal drift slowdown or reversal for equatorial irregularities [Kil et al., 2000] or “fade stretching” due to satellite motion [Kintner et al., 2001]. It should be noted that perfectly “frozen in” irregularities have been assumed and that the zenith geometry is particularly simple. In a realistic case, with irregularities evolving in time as they drift past, an unknown outer scale (indeed, an unknown lower-frequency portion of the irregularity spectrum) and LOS motion relative to complex and variable multidimensional structures, the quantitative meaning of temporally averaged σϕp becomes quite obscured. Thus the use of temporally averaged GPS σϕp by itself provides a highly inadequate characterization of the irregular ionosphere.

5. Receiver Performance

[17] The discussion of the implicit filtering in σϕp due to a fixed measurement period above leads naturally to a discussion of receiver impacts. Intuitively, it does not seem likely that the portion of phase variance due to fluctuations at frequencies lower than a few tenths of a Hz (i.e., periods of tens of seconds) will have much influence on the ability of a typical GPS receiver to track. Nevertheless, the σϕp estimates over 1-min periods include a significant amount of fluctuation power at these longer timescales; if it were available, the spatially averaged σϕp would contain even more power for the reasons given above. Careful reading of the work of Conker et al. [2003] shows that they incorporate a tracking-loop transfer function with a roll-off at low frequencies as a weighting factor in the integral of the phase PSD that forms their phase scintillation term, equation image. Unfortunately, they call this term the “phase scintillation”. A casual reading might then give the false impression that the variance thus symbolized is the square of the usual phase scintillation index. It should be strongly emphasized that any phase variance measure of scintillation used in receiver tracking analysis must be appropriately frequency weighted.

[18] Another way to examine this effect is to consider a hypothetically stationary GPS satellite and a constant and motionless 1-D ionosphere, in this case defined by the ℓ0 = 8.0 km example above, with receivers that are moving relative to it. Figure 5 depicts the received phase (ϕp) over 10-s segments at two different receiver speeds. Clearly, the 300-m/s (583-knot) case appears that it could potentially be more stressful to receiver tracking, particularly around the 9-s mark, than the 75-m/s (146-knot) case. Yet the spatially averaged σϕ (at the ionosphere) or σϕp (at the receiver altitude) is identical between the two cases. Not only is there potential for “fade stretching” at low relative drifts to degrade receiver performance but also “phase compression” at high relative motion. Critically, the phase scintillation index alone cannot be used to characterize this effect; characterization requires detailed knowledge of the spatial phase fluctuation spectrum and relative motion.

Figure 5.

Effect of receiver motion against a stationary ionospheric scintillation pattern.

[19] An extreme example of such “phase compression” may account for observations of space shuttle GPS velocity errors in the equatorial regions [Kramer and Goodman, 2001; Goodman, 2002]. Here the GPS receiver moves at about 7 km/s relative to the ionosphere. The shuttle typically orbits at bottomside altitudes where amplitude scintillation will not yet be developed in response to phase perturbations from an irregular ionosphere overhead. Nevertheless, the large relative motion undoubtedly leads to additional phase tracking stress as discussed above. It is conceivable that this effect may even cause tracking loop stress in cases where the associated phase perturbations might be regarded as inconsequential in stationary ground observations. Unfortunately, no correlative studies have been performed and, since the observations have not appeared in the ionospheric literature or at geophysics conferences, there may not even be wide awareness of them in the ionospheric GPS community.

6. Conclusions

[20] The GPS phase scintillation index has many problems as a quantitative measure of ionospheric phase fluctuation—primarily due to the dominance of the low-frequency component of the phase power spectrum in its makeup. First, σϕ can be nearly independent of the level of amplitude scintillation as it appears on the ground. Furthermore, the same ionosphere can produce different observed phase variance levels under different relative drift conditions due to implicit high-pass filtering. Second, phase variance must be appropriately frequency weighted to parameterize receiver impact. When computed under ideal observing circumstances, the usual phase scintillation index incorporates no frequency weighting. This presentation has illustrated each of these problems with concrete, if simplistic, examples and adding realistic complications does not improve the prognosis. Finally, GPS σϕ has many additional observational concerns not treated here: multipath, receiver phase noise and oscillator stability, cycle slips under scintillation conditions, large phase trends due to Doppler shifts, use of TEC as a proxy for phase, etc. For these reasons it is recommended that the use of σϕ be seriously reconsidered by the GPS scintillation community.

[21] Nevertheless, GPS phase, or related TEC, fluctuations should not be overlooked since they can indicate physical structuring in the ionosphere even in the absence of significant GPS amplitude scintillation. The question is one of a suitable measure. Alternative measures to σϕ must account for the spectral characteristics of the phase fluctuation as appropriate to its intended use. For relating phase fluctuations to amplitude scintillation, a parameter like the rate-of-TEC index (ROTI) defined by Pi et al. [1997] has improved stability under typical equatorial conditions [Beach and Kintner, 1999], although there has been some inconsistency in defining the ROTI averaging interval (A. Coster, private communication, 2005). A recently suggested alternative may be to store subsets of phase samples and later use them to estimate S4 from a “sub-sampled phase screen” [Beach et al., 2004], although the practical details of this technique have not yet been fully explored. For receiver-tracking impacts, the frequency-weighted phase variance of Conker et al. [2003] is a better parameter, although it must be recomputed if the relative drift changes and for different receivers with different tracking loop transfer functions. In fact, in some cases the best alternative may be to have representative snapshots of the phase time series or PSD available together with estimates of the net drift and spatial structuring. Ultimately, the GPS scintillation community should carefully investigate, recommend and standardize appropriate phase fluctuation measures for particular observing situations and applications.

Acknowledgments

[22] The author thanks A. Coster and P. Doherty for suggesting that the author develop the original talk upon which this paper was based. This work was partially supported under AFOSR task 2311AS.

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