An Inverse Diffraction Technique for Scaling ... - Boston College

An Inverse Diffraction Technique for Scaling Measurements of Ionospheric Scintillations on the GPS L1, L2, and L5 Carriers to Other Frequencies Charles S. Carrano, Keith M. Groves, Susan H. Delay, and Patricia H. Doherty Institute for Scientific Research, Boston College, Boston, MA

has led ionospheric research in support of SatelliteBased Augmentation Systems (SBAS), of which the FAA’s Wide Area Augmentation System (WAAS) is an example. She has served in numerous volunteer offices in the Institute including the ION’s Executive Committee and as program and general chair of the ION GNSS meeting. She is currently executive vice president of the Institute.

BIOGRAPHY Charles S. Carrano is a senior research physicist and Principal Investigator at Boston College’s Institute for Scientific Research. He has authored and coauthored more than 40 technical papers concerning the impacts of ionospheric scintillation on radar, satellite communications, and Global Navigation Satellite Systems (GNSS). He is an Associate Editor of the AGU journal Radio Science, and serves on the Editorial Board of GPS Solutions. He has a B.S. degree from Cornell University and M.S. and Ph.D. degrees from The Pennsylvania State University.

ABSTRACT To predict the impact of ionospheric scintillations on communications, navigation, and space radar systems it can be useful to simulate the RF conditions under which these systems must operate. GNSS satellites broadcast radio signals that can be monitored by receivers on the ground to specify scintillations at multiple frequencies in the L band range. In this paper, we describe an inverse diffraction technique for mapping the deterministic structure of GPS scintillations measured at the L1, L2, and L5 frequencies to other frequencies. In the case of weak scatter, a statistical theory that describes how the S4 index scales from one frequency to another is well established (Rino, Radio Sci., 14, 1135, 1979). However, this statistical scaling breaks down when the S4 index at either frequency saturates due to multiple scatter effects. The inverse diffraction technique can be used under both weak and strong scatter conditions, and it works as follows. The complex signal fluctuations on the ground are backpropagated up to ionospheric altitudes to infer an equivalent phase screen. Forward-propagation through a wavelength-scaled version of this screen provides a realization of the complex fluctuations at the desired operating frequency. Using 50 Hz

Keith Groves works as a Senior Research Scientist at Boston College where his research interests include radio wave scintillations, high power HF ionospheric modification, wave-particle interactions, and space weather impacts on communication, navigation and surveillance systems. He has authored and coauthored more than 60 papers and is an internationally recognized expert in the field of ionospheric scintillations. He has a Ph.D. in Space Physics from MIT and a B.S. in Physics from Andrews University. Susan Delay is a Senior Research Analyst with the Boston College ISR. Her work includes data and numerical analysis, programming and graphics development. Patricia Doherty is Director of the Institute for Scientific Research at Boston College. She has been an active researcher in the area of radio wave propagation for the past 20 years, focusing on ionospheric effects on satellite-based navigation. She

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measurements of C/No and phase fluctuations on the GPS L1, L2, and L5 carrier signals from Brazil, we validate the technique by predicting scintillations on each GPS carrier signal from the others. We quantify the accuracy of the results by calculating the Euclidean norm of the difference between predicted and measured signals. We find the results to be most accurate for equatorial scintillations when the relative velocity between the penetration point and zonal drift is directed across the magnetic field lines. The reason for this is that the propagation calculations are restricted to the plane of the satellite scan, and when this scan is across highly field-aligned irregularities all structure with spectral content at the Fresnel scale is resolved.

validate this technique using observations of GPS scintillations on L1, L2C, and L5. The availability of civil (open) codes on L2 and L5 allows robust measurement of scintillations on these frequencies (Carrano et al., 2012; Carrano et al., 2013). Sokolovskiy et al. (2013) applied a similar inverse diffraction technique to radio occultation data; their aim was to reduce ionospheric correction errors by back-propagation of the 50-Hz sampled L1 and L2 signals. Back-propagation of the L1 signal has also been applied to radio occultation data to localize ionospheric irregularities along the radio propagation path (Gorbunov et al. 2002; Sokolovskiy et al. 2002). In this paper, we employ back-propagation to scale scintillation observations from one frequency to another.

1. INTRODUCTION To predict the impact of ionospheric scintillation on radio-based systems, including GNSS, one often wishes to scale scintillation measurements from one frequency (at which observations are made) to another (the frequency at which the system of interest operates). As examples, one might use GNSS scintillation observations to predict the structure of scintillations along the propagation path of a UHF satellite communications link or a space-radar operating at X band.

To demonstrate a practical application of the inverse diffraction technique, we predict the intensity correlation between GPS L1 and L2 using observations on L1 alone. As discussed by Carrano et al. (2012), this enables use of L1 only observations to infer the correlations between L1, L2, and L5 during the previous solar maximum (before L2C and L5 signals were actually broadcast). The previous solar cycle was much more intense than the present, and it is very likely that impacts on the L2C and L5 signals would have been greater than those we observe now. The intensity correlation between carrier pairs during scintillation is of interest because this dictates the extent to which frequency diversity may be leveraged to mitigate scintillation impacts on navigation accuracy.

There are two ways to perform this frequency scaling, statistical and deterministic. In the statistical scaling approach, one generally invokes the weak scatter theory (Rino, 1979; Franke, et al., 1984) which predicts a simple power-law dependence of the scintillation index on frequency, i.e. S4  f ( p 3)/4 .

2. MULTI-FREQUENCY GPS SCINTILLATION OBSERVATIONS

Here, f is the radio frequency and p is the phase spectral index associated with path integration through random irregularities in the electron density distribution within the ionosphere. To make predictions using the statistical power-law scaling model, the spectral index must be known, and neither the observation nor target frequency can be in saturation.

With funding and support from the Federal Aviation Administration (FAA), Boston College and National Institute for Space Research (INPE) in Brazil have collaborated to collect GPS scintillation observations on the L1, L2, and L5 carrier signals since April 2012. A Septentrio PolaRxS Pro GNSS receiver reports 50 Hz samples of post-correlator in-phase (I) and quadrature (Q) data, along with carrier phase measurements at the L1, L2, and L5 frequencies. This receiver is located at the INPE headquarters in São José dos Campos, Brazil (23.2S, 45.9W, 17.5S dip

In this paper, we explore an alternative to statistical scaling—an inverse diffraction technique that does not require a-priori information about the irregularity structure, and provides both deterministic and statistical scaling of scintillation observations. We

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latitude). São José is located near the southern crest of the equatorial anomaly, where the strongest scintillations tend to occur globally. The raw I&Q samples and carrier-phase measurements are postprocessed to obtain the intensity and phase fluctuations due to ionospheric scintillation, using the methodology described in Carrano et al. (2012). For this study, we selected data from 26 evenings with strong scintillation events (S4>0.6) between October 2012 through February 2013, unless stated otherwise.

deviation 2.04. This relatively large standard deviation suggests that there is significant variability in the statistical structure of the irregularities. As we shall see, this natural variability limits the accuracy with which we can use the statistical power-law scaling method to make frequency scaling predictions (since we must assume some value of p in order to perform the frequency scaling). This variability was, in fact, one of our primary motivations for developing of the inverse diffraction technique which makes no assumptions regarding the structure of the irregularities—instead it infers this information from the scintillation data itself.

3. STATISTICAL FREQUENCY SCALING As discussed by Franke et al. (1984), and previous authors, the theory of scintillations caused by a weak and thin phase changing screen suggests that the scintillation index S4 exhibits a power-law dependence on frequency S4  f  n

(1)

where the frequency spectral index n is related to the phase spectral index p of the irregularities according to (2) n  ( p  3) / 4 Figure 1. Log-log plot of GPS L2 S4 vs L1 S4 for 26 evenings with strong scintillation between Oct 2012 Feb 2013 at São José dos Campos, Brazil.

The assumptions upon which the above relations depend include the following: 1) the irregularities can be described by a single-component power-law, 2) the receiver is sufficiently close to the irregularities that the influence of an outer scale in the turbulence is not perceived, 3) the scintillations are sufficiently weak that multiple-scatter effects can be neglected.

To see how this variability degrades our predictive capability, we now assume a fixed spectral index p=2.5 and use (1) and (2) to predict the S4 at L2 from the measured S4 at L1 and vice-versa. Figure 2 shows the results of these predictions plotted against the measured S4 at L1 and L2. The dotted lines with unit slope represent a perfect prediction. The RMS deviation from this line for this 26 evening dataset is 0.061 for L1 and 0.087 for L2, suggesting slightly larger errors in predicting L2 from L1 than the converse. Note that the accuracy of both predictions increasingly degrades as the S4 increases. This is due to the breakdown of the weak scatter theory when the scattering is strong. We will show that the inverse diffraction method, under certain conditions (to be described), produces more accurate predictions for all levels of scattering strength than does this simple power law scaling.

These assumptions are not overly-restrictive for the case of GNSS scintillations at L band frequencies— this we demonstrate by plotting the S4 measured at L2 against the S4 measured at L1, shown in Figure 1. On a log-log plot, equation (1) implies a linear relationship between with unit slope and intercept equal to n(log f1 - log f2). Using the method of leastsquares, we fit a line with unit slope to this data between L2 S4 of 0.1 (below which the contribution from receiver noise is problematic) and 0.8 (above which multiple-scatter and saturation effects cause a departure from linearity). The least-squares fit to this data produced an estimate of n=1.39 with standard deviation 0.24. From (2), this implies a phase spectral index of p=2.55, in expectation, with standard

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between Oct 2012 - Feb 2013. We also applied the method to the data from day 52 of 2013, since on this evening there was a scintillation event on PRN01 which transmits signals on all three frequencies (L1, L2, and L5) simultaneously. In what follows, we will describe two examples of the method’s application, in detail, followed by a summary of the method’s accuracy for the dataset overall. Figure 3 shows the intensity fluctuations due to scintillation on L1, L2, and L5 for GPS PRN01 observed from São José dos Campos on day 52 of 2013. The detrending interval used was 60 seconds. The magnitude of the intensity fluctuations increases with the carrier wavelength, as expected. Figure 4 shows the phase fluctuations due to scintillation on L1, L2, and L5 for GPS PRN01 observed from São José dos Campos on day 52 of 2013. The detrending interval used for the phase was also 60 seconds. The magnitude of the phase fluctuations also increases with the wavelength of the carrier.

Figure 2. Frequency scaling predictions produced by simple power-law scaling: (left) predicted L1 S4 using L2 versus measured L1 S4, (right) predicted L2 S4 using L1 versus measured L2 S4.

3. THE INVERSE DIFFRACTION METHOD The inverse diffraction method was introduced in (Carrano et al., 2012) and it works as follows. The raw I&Q samples and carrier-phase measurements from the receiver are post-processed to obtain the complex field on the ground due to ionospheric scintillation. The complex field is then backpropagated up to ionospheric altitudes until the amplitude fluctuations are minimized. The field at this altitude serves an equivalent amplitude- and phase-changing screen. We note that this equivalent screen is necessarily one-dimensional, since satellite scintillation data provides only a one-dimensional scan through the two-dimensional diffraction pattern on the ground. As a metric of back-propagation effectiveness, we define the “S4 reduction” as ratio of the S4 index of the field at the altitude of the screen to the S4 index of the field on the ground. The amplitude variations in the equivalent screen are then discarded, to obtain a purely phase-changing screen. This phasechanging screen is then scaled by the ratio of the desired output wavelength to the observation wavelength. Finally, a new wave at the desired output wavelength is passed through the wavelength-scaled phase-screen down to the receiver on the ground. The result is a realization of the complex fluctuations on the ground due to scintillation at the desired frequency. The full details of this procedure are given in Section 3 of (Carrano et al., 2012).

Figure 3. Detrended intensity fluctuations on L1 (top), L2 (middle), and L5 (bottom) for PRN01 observed from São José dos Campos on day 52 of 2013.

We applied the inverse diffraction method to the data from all 26 evenings with strong scintillations

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predicted (red) and measured (black) intensity fluctuations are compared and the relative amplitude errors of the predictions are shown for each of the L1, L2, and L5 carriers. The relative amplitude errors in predicting L1 from L2, predicting L2 from L1, and predicting L5 from L1 were 7.68%, 10.10%, and 10.76%, respectively. Generally speaking, we find the relative error to be slightly larger when predicting the lower frequency carriers (L2 and L5) from the higher frequency carrier (L1). A possible explanation for this is as follows. Since the scintillations tend to be weaker on the high frequency carrier (L1), the relative contribution from receiver noise (e.g. both amplitude noise and phase noise) to the complex fluctuations used for back-propagation will be larger. In other words, more accurate predictions can be made using the lower frequency carriers as input because the scintillations are stronger and the contribution from receiver noise is less significant.

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Figure 4. Detrended phase fluctuations on L1 (top), L2 (middle), and L5 (bottom) for PRN01 observed from São José dos Campos on day 52 of 2013.

Black – measured, Red - Predicted PRN 01

Figure 5 shows the results of applying the inverse diffraction method to a four minute sub-segment of the data shown in Figures 3 and 4. We chose a four minute sub-segment of data for the back- and forward-propagation steps, since this will generally include a few outer scale lengths of the turbulence for GPS satellite scans through equatorial irregularities. The results were very similar for all other subsegments during this hour, and thus we show only one example here. For the purpose of validation, we predicted the intensity fluctuations on each carrier from one of the others. More specifically, the L1 fluctuations were predicted from L2 measurements, while the L2 and L5 fluctuations were predicted from L1 measurements. We quantify the accuracy of the results by calculating the Euclidean norm of the difference between predicted and measured signals. For example, the relative amplitude error for each carrier signal was calculated according to E  AM (t )  AP (t ) / AM (t )

Predicted from L2 PRN 01

Predicted from L1 PRN 01

Predicted from L1

Figure 5. Measured (black) and predicted (red) intensity fluctuations for L1, L2, and L5. The L1 fluctuations were predicted from L2 measurements. The L2 and L5 fluctuations were predicted from L1 measurements.

(3) Next we demonstrate the inverse diffraction technique using data acquired during a period of stronger scintillations. Figure 6 shows the intensity fluctuations due to scintillations on L1, L2, and L5 for GPS PRN25 observed from São José dos Campos on day 1 of 2013. Figure 7 shows the corresponding

where AM(t) represents the measured time series of amplitude fluctuations, AP(t) represents the predicted time series of amplitude fluctuations, and  represents the Euclidean norm. In Figure 5, both the

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phase fluctuations on L1, L2, and L5. Figure 8 shows the results of applying the inverse diffraction method to a four minute sub-segment of the data shown in Figures 6 and 7. The results were very similar for all other sub-segments during this hour, so we show only one example. The predicted and measured intensity fluctuations are compared and the relative amplitude errors of the predictions are shown for each of the L1, L2, and L5 carriers. The relative amplitude errors in predicting L1 from L2, predicting L2 from L1, and predicting L5 from L1 were 13.30%, 16.83%, and 17.57%, respectively. As in the previous example, the relative error is slightly larger when predicting the lower frequency carriers (L2 and L5) from the higher frequency carrier (L1). While the accuracy of the technique is excellent in the two cases shown, the results are not always this accurate, as we shall discuss in Section 4. There are two basic motivations for this paper, 1) to establish that measurements of the complex signal fluctuations at one frequency can be used to accurately predict the deterministic structure of scintillations at other frequencies, and 2) to precisely establish the conditions under which a one-dimensional satellite scan through the equatorial ionosphere is adequate to perform this scaling.

Figure 7. Detrended phase fluctuations on L1 (top), L2 (middle), and L5 (bottom) for PRN01 observed from São José dos Campos on day 1 of 2013.

Black – measured, Red - Predicted PRN 25

Predicted from L2 PRN 25

Predicted from L1 PRN 25

Predicted from L1

Figure 8. Measured (black) and predicted (red) intensity fluctuations for L1, L2, and L5. The L1 fluctuations were predicted from L2 measurements. The L2 and L5 fluctuations were predicted from L1 measurements.

Figure 6. Detrended intensity fluctuations on L1 (top), L2 (middle), and L5 (bottom) for PRN25 observed from São José dos Campos on day 1 of 2013.

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This effective scan azimuth reduces to the ordinary scan azimuth for normal propagation through a horizontal magnetic field. The effective scan azimuth is 90 when the satellite scan is directed across fieldaligned irregularities, and 0 when directed along the irregularities for any propagation direction and any local magnetic field orientation.

4. PERFORMANCE OF THE METHOD The inverse diffraction method may be applied without any knowledge of the statistical structure of the irregularities responsible for the scintillation. Nevertheless, it is helpful at this point to introduce a model for the irregularities in order to interpret the accuracy of the results. Rino (1979) proposed a model for the irregularities that are described statistically in terms of a three-dimensional powerlaw spectrum. A coordinate stretching transformation is used to elongate the irregularities along the direction of the local magnetic field. The irregularities in this model are permitted to translate with the bulk plasma motion, but are otherwise considered frozen (i.e. evolution of the structure is forbidden) during the time interval over which field statistics are computed.

Figure 9 shows the S4 reduction (shown as a percentile) as a function of effective scan azimuth angle. Predictions of L1 (from L2) are shown with circles, while predictions of L2 (from L1) are shown with plus signs. The symbols are colored according to the corresponding S4 measurement (L1 or L2). While the spread of the data is considerable, the S4 reduction tends to increase as the effective scan azimuth increases toward 90, i.e. as the satellite scan is oriented across field-aligned irregularities. The S4 reduction does not exhibit a significant dependence on the measured S4, which corroborates our claim that the inverse diffraction method is applicable to both weak and strong scatter conditions.

Rino (1979) showed that for a one-dimensional satellite scan through a two-dimensional diffraction pattern due to scintillation, the effective scan velocity, Veff, is the appropriate translation from temporal fluctuation scales to spatial scales. Mathematically, Veff, is the rate at which the line of sight scans across isocontours of phase correlation and is given by: 1/2

Veff

 CVsx2  BVsxVsy  AVsy2    AC  B 2 / 4  

(4)

where A, B, and C are coefficients of a coordinate rotation and stretching operation that relate the propagation line of sight to the irregularity axes. The components (Vsx, Vsy) represent the horizontal velocity of the drifting irregularities with respect to the ionospheric penetration point of the satellite (which is in motion). Evaluation of (Vsx, Vsy) therefore requires knowledge of the bulk plasma drift velocity. For the analysis in this paper, we used spaced UHF antenna measurements of the zonal drift from nearby Cuiaba, Brazil.

Figure 9. S4 reduction versus effective scan azimuth angle for 26 evenings with strong scintillation between Oct 2012 - Feb 2013 at São José dos Campos, Brazil.

Figure 10 shows the relative amplitude error (shown as a percentile) as a function of effective scan azimuth angle. Predictions of L1 (from L2) are shown with circles, while predictions of L2 (from L1) are shown with plus signs. The symbols are colored according to the corresponding S4 measurement. The relative error also tends to be smallest when the effective scan azimuth is large, i.e. when the satellite scan is directed across field-aligned irregularities. There is also a notable decrease in accuracy as the

We define an effective scan azimuth angle as



 eff  sin 1 Veff / Vsx2  Vsy2

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(5)

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measured S4 increases; this effect becomes more pronounced as the effective scan azimuth decreases. We will propose an explanation for this effect shortly. Note that for a purely transverse scan (eff ~ 90), the accuracy of the inverse diffraction technique is virtually independent of the scintillation strength.

Figure 11. Frequency scaling predictions produced by inverse diffraction method: (left) predicted L1 S4 using L2 versus measured L1 S4, (right) predicted L2 S4 using L1 versus measured L2 S4. Figure 12 shows the same data as Figure 11, except that only samples for which the S4 reduction was at least 60% are shown while the other samples have been discarded. In this case, the RMS deviations from the perfect-prediction line are only 0.033 for L1 and 0.041 for L2. These values are twice as small as the RMS deviations using the simple power-law scaling method (Figure 2). Moreover, unlike the simple power-law scaling method, the accuracy of the inverse diffraction predictions (when the S4 reduction is at least 60%) shows very little dependence on the strength of scatter.

Figure 10. Relative amplitude error versus effective scan azimuth angle for 26 evenings with strong scintillation between Oct 2012 - Feb 2013 at São José dos Campos, Brazil.

5. FREQUENCY SCALING COMPARISON In this section, we compare the accuracy of frequency scaling predictions made using the inverse diffraction method with the simple power law statistical scaling method described in Section 2.

Inverse diffraction with > 60% S4 reduction

Figure 11 shows the results of the inverse diffraction method when predicting L1 from L2 and when predicting L2 from L1. The predictions are plotted against the actual S4 measured at L1 and L2, for reference. Once again, the dotted lines with unit slope represent a perfect prediction. Note that while there is a heavy concentration of samples along the perfect prediction line, there are many results which significantly under-predict the measured values. As a result, the RMS deviations from the perfectprediction line for this 26 evening dataset is large, namely 0.162 for L1 and 0.194 for L2. These values are 2.7 times and 2.2 times larger, respectively, than the corresponding RMS deviations for the simple power-law scaling method (Figure 2).

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Inverse diffraction with > 60% S4 reduction

Figure 12. Frequency scaling predictions produced by inverse diffraction method, keeping only results for which the S4 reduction is at least 60%: (left) predicted L1 S4 using L2 versus measured L1 S4, (right) predicted L2 S4 using L1 versus measured L2 S4.

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when the effective satellite scan is directed along the irregularities rather than across them.

6. L1 / L2 INTENSITY CORRELATION In this section, we demonstrate a practical application of the inverse diffraction method, namely to predict the intensity correlation between GPS L1 and L2 using observations only on L1. This enables use of L1 only observations to infer the correlation between L1, L2, and L5 during the previous solar maximum, which was much more intense than the present. The intensity correlation between carrier pairs during scintillation is of interest because this dictates the extent to which frequency diversity may be leveraged to mitigate scintillation impacts on navigation accuracy (Carrano et al., 2012). Figure 13 shows the correlation between L1 and L2 intensity fluctuations versus the measured S4 at L1. These measured correlations (top panel) were calculated from 60 second sub-segments of the scintillation intensity data. The predicted correlations (bottom panel) were computed by correlating L1 with the predicted L2 obtained from L1 using the inverse diffraction method. The samples are colored by the effective scan azimuth angle. The solid curve is a theoretical result for the expected value of the intensity correlation, corresponding to an assumed phase spectral index p=3.0 and calculated using the approach described in the Appendix of (Carrano et al., 2012). When the scintillations are weak (S4