The receiver and external thermal noises.
The work discusses the efficiency of different ionospheric scintillation indices. The new index D2fi based on the GNSS carrier phase observable was introduced. We analyze the accuracy of the phase measurements, in particular its dependence on the GNSS equipment thermal noises, multipath and external noises, and presettings of Phase Lock Loop and Code Delay Discriminator. The performance of DROTI, S4, σφ, and D2fi was considered for the case of high-rate data. The “sensitivity” and reliability of each index differs significantly and depends on the time resolution of the carrier phase data. The new index D2fi advantages are that it is easily derived and has a clear dependence on GNSS hardware and software features. D2fi was proven to be a useful tool to detect the small-scale ionospheric disturbances based on high-rate GPS carrier phase measurements.
- ionospheric scintillation indices
- high-rate GNSS data
GNSS data with high-rate sampling becomes more and more available worldwide [1, 2]. It provides opportunities for the better results in the field of ionospheric scintillation research. Standard ionospheric indices and parameters
It is important to find such an ionospheric scintillation index which is easily derived and has a clear dependence on both the ionospheric turbulence structure and GNSS hardware and software presets. In this work, the second-order derivative of the GPS signal carrier phase based on high-rate carrier phase time series is suggested as a promising means for the ionospheric scintillation detection. No additional complex processing is needed to obtain this new scintillation index.
The work  and the general necessity to define the GPS data time resolution sufficient for the robust scintillation analysis were the motivation for the authors to test the real sensitivity of the ionospheric indices depending on the input data sampling rate. We consider GNSS carrier phase observable to be the most capable of observing the ionospheric disturbances and scintillations. The aims of this study include (a) introduction of the new index that is the second-order derivative of the GPS signal carrier phase (D2fi index) which helps to reveal scintillation events; (b) test of sensitivity of D2fi, DROTI, S4, and σφ indices based on 50 Hz GPS data; and (c) consideration of the benefits and limitations of these indices for scintillation studies. The analysis was performed for the case study and was based on GPS data of the mid-latitude GNSS station located near Irkutsk, Russia, during the intense geomagnetic storm.
2. The carrier phase noise content at the phase lock loop input
Ionospheric phase scintillations are induced with ionospheric irregularities of hundreds of meters to several kilometer size. These irregularities correspond to the Fresnel frequencies from ≈ 0.1 to ≈ 10 Hz [13, 14]. According to [1, 2, 15], it is possible to detect small-scale ionospheric irregularities of hundreds of meters to several kilometer size by observing not only the fast carrier phase variations but also the carrier phase noise variations which were considered earlier as “noise” . This is possible if the data sampling rate is high enough to exclude low-frequency variations and trends from the carrier phase time series. The data sampling rate should contain the sufficient ionospheric information. The authors  showed that the majority of the phase scintillation events can be revealed if data sampling rate between 10 and 40 Hz is used. Therefore, for the analysis of weak ionospheric scintillations, the sampling data rate higher than 10 Hz should be used.
To extract the phase noise variations from the complex carrier phase data, we use the carrier phase derivatives. The second-order derivative works as a high-pass filter and removes the phase ambiguity, all the low-frequency trends (due to the relative motion between satellite and receivers), multipath slow variations, and low-frequency phase variations due to reference oscillator frequency drift. It allows us to extract the phase noise variations from the phase measurements without additional complex processing procedures. The carrier phase noise derivative can be also used as a new parameter in GPS occultation technology .
Let us estimate the values of the main components of the carrier phase noise. They should be small enough to obtain the pure ionospheric phase scintillation based on the D2fi index. In case of the stationary receiver, there are no phase variations and phase measurement noises due to vibration and jerks. Based on this assumption, the noise error in carrier phase measurements depends on two main factors
For an ideal PLL without inner loss, the noise dispersion of phase measurements is determined as follows :
where is the noise bandwidth of the PLL filter (Hz) and is the carrier-to-noise ratio at the PLL unit input (dBW).
Thus, the noise level of the carrier phase measurements is determined by the carrier-to-noise ratio (CNR) at the PLL input. The CNR depends on (1) the level of external noises, (2) the antenna pattern, and (3) the low-noise preamplifier (LPA) gain. In addition to external noise, the inherit receiver thermal noise, the short-term instability of the reference oscillator, the signal sampling, and quantization noise should be considered as well.
According to expression (1), the final accuracy of the carrier phase measurements depends on the filter noise bandwidth. At the same time, the carrier-to-noise ratio at the PLL input depends on the time of accumulation of instantaneous phase measurement samples. Thus, the noise dispersion of phase measurements can be determined more precisely as follows :
where is the dispersion of receiver thermal noise and is the dispersion of noise caused by the Allan deviation.
The noise components of the phase measurements with dispersions and depend on the above factors as follows :
where is the time of accumulation of instantaneous phase measurement samples (ms), is the RMS of the short-term instability of the reference generator frequency (Hz),
The carrier-to-noise ratio at the PLL input is a function of the receiver noise temperature (including the antenna), as well as the environment noise temperature (the Earth noise, the total noise of cosmic radio sources, and the Sun noise). The measurements of noise caused by analog-to-digital signal conversion, as well as signal-to-noise level with regard to filtering, amplification, and antenna gain, can be expressed through the corresponding loss in the resulting carrier-to-noise ratio at the phase detector input. Therefore, the carrier-to-noise ratio at the PLL input can be expressed as follows :
According to formula (5), two factors affecting the carrier-to-noise ratio at the PLL input сan be deduced. The first factor is constant during the measurement and depends on the receiver equipment type. It is defined by the
At the same time, there is a factor that depends not only on the equipment type but also changes randomly. This is the receiver thermal noise
The total noise temperature can be estimated as follows :
Under standard physical conditions, the Earth noise temperature is . The Earth noise component, which is present at the PLL input (
where is the ratio of the angular aperture of a groundward part of the antenna pattern, with respect to the total angular aperture of the antenna pattern.
According to Eq. (8), the higher the ratio, the higher the magnitude of the Earth noise. With regard to the known antenna pattern of typical navigation receiver antennas, the value can be within 0.004–0.01 . Thus, the Earth noise temperature at the PLL input is
Similarly, the Sun noise temperature can be obtained. The total noise temperature of the Sun is . The angular size of the Sun visible from the Earth’s surface is βС = 0,5°. Considering the above mentioned typical antenna pattern, the ratio is about 10−5. When the sunlight falls into the antenna aperture, the Sun noise temperature ТSN = 0,00001 × 6000 ≈ 0,06 К. This corresponds to the Sun noise temperature at the PLL input of about -241 dBW/Hz.
The sky noise temperature (
The inherit antenna noise temperature TA results from the noise of active loss resistance in the antenna :
If the antenna temperature is equal to 300 K and the typical antenna efficiency is between 80 and 90%, the temperature
The noise temperature of the preamplifier is defined as follows :
where ε is the preamplifier noise coefficient and
Table 1 shows the values of noise temperatures and noise spectral power for the above mentioned components of the receiver thermal noise (
|Noise source||Noise temperature, K||Power spectral density dBW/Hz|
|Sky noise (all sources)||100||−208|
Using the information from the Table 1 and formulas (2)–(5), we can estimate the noise level of the phase measurements in a stationary receiver when measuring the phase at different GPS frequencies and satellite elevations. Let us assume that , accumulation time , Allan deviation of the reference generator , the maximum and minimum power levels of the signals (), received at L1, L2 and L5 frequencies are described by curves in Figure 1 [19, 20]. The values of the standard deviation of the phase noise for this case are given in Table 2.
|Frequency, MHz||Minimal value , deg||Maximal value , deg|
|L1 = 1575.42||1.59||7.22|
|L2 = 1227.60||3.35||14.85|
|L5 = 1176.45||1.33||6.06|
The quality of the receiver radio-frequency chain (RFC) and the regular variations in the signal level at the reception point play an important role in the potential accuracy of the signal phase measurements. In particular, the sustainable phase tracking threshold equals 15°  is almost reached under conditions of the worst radio-frequency chain parameters (Table 2) and the minimum signal receiving level at the L2 frequency. Thus, although the phase measurements yield the best accuracy for ionospheric scintillation detection, still the careful presetting of GNSS receiver hardware and the consideration of measurement conditions are needed. To note, under the similar conditions, the best accuracy of the phase measurements is achieved if the signals are used at the L5 GPS frequency. This can be explained by the highest carrier-to-noise ratio in the given measurement channel (Figure 1).
Another important factor for the high accuracy of carrier phase measurements is the correct choice of the PLL settings such as accumulation time (
When using an optimal phase discriminator, the measured parameter (phase) should not be changed during the accumulation time (
The measured parameter is not obligatory constant within
After the determination of the optimal value, the selected PLL noise bandwidth should satisfy Eq. (11). In addition, according to Eq. (1) the noise level of the phase measurement depends on the noise bandwidth . Therefore, the practical choice of the noise bandwidth depends on the expected measurement conditions and usually lies within the range from 10 to 20 Hz. If there is an impact of external electromagnetic interference, the phase tracking stability reduces. Therefore, the choice of the wider noise bandwidth increases the reliability of the phase tracking. Finally, according to expressions (3) and (4), the increase of the noise bandwidth leads to the proportional increase of the average RMS of the receiver equipment thermal noise. On the other hand, as increases, the noise component related to the short-term frequency instability of the reference oscillator decreases. Thus, the noise bandwidth can be reduced without the significant loss of the phase measurement quality by using a better-quality reference oscillator.
The multipath effect is another important source of the carrier phase noises. In general, the phase error due to multipath can be calculated as a difference between the carrier phase of the reflected composite signal and the carrier phase of the direct signal. In the presence of multipath propagation, the composite signal phase shifts randomly from the direct signal phase, and the NCO-generated local carrier locks to the composite carrier phase, resulting in the error of the phase measurement. In the case of one reflected signal, the error of the phase measurement is defined as follows :
where is the autocorrelation function of the PRN code, is the cross-correlation function between the direct GNSS signal and the reflected signal, is the receiver estimate of the incoming signal code delay, is the reflected signal code delay, is the reflection coefficient that corresponds to the Signal to Multipath Ratio (SMR) as , and is the phase of the reflected signal.
If the direct signal has no distortion in the form the PRN code, the autocorrelation function () depends on the front-end bandwidth of the GNSS receiver radio-frequency chain. The PRN codes have one main lobe and several side lobes in the frequency domain. In practice, the signal is band limited, and only the main lobe and one or more side lobes are used for the signal processing. As a result, the sharp correlation peaks are rounded and the ends are trailed-off. It was found earlier that the RFC bandwidth affects the maximum error value significantly [10, 11]. In the case of the unlimited bandwidth, the misalignment in the value computation is zero. In the case of 10 MHz bandwidth, the misalignment is not equal to zero and can vary within ±0.03
The cross-correlation function significantly depends on the early-late correlator spacing (
This equation describes the discriminator output in case if the input tracking error (τe) is within linear part of the discriminator performance. The maximal discriminator output value is limited by the correlator spacing time and depends on the code chip length 
Thus, both the correlator spacing and the PRN code chip length define the maximal code tracking error value and, as a result, the cross-correlation function . Let us consider the particular example of L1 C/A code and the coherent discriminator using a standard correlator with the correlator spacing of
To estimate the possible impact of the PRN code rate on the multipath error, the multipath error envelopes can be used [22, 23]. Table 3 illustrates the GPS PRN code characteristics transmitted at L1 and L5 GPS frequencies. Table 4 was reconstructed based on the results . It illustrates the maximal code multipath error () in relation to the PRN code rate and correlator spacing for the coherent discriminator.
|Frequency/PRN code||Carrier frequency, MHz||Code rate (Mbps)|
|L5 I5, L5 Q5||1176.45||10.23|
|Frequency/PRN code||Maximal code multipath error (), meters||Maximal relative multipath delay (), meters|
|Td = ±0.5||Td = ±0.1||Td = ±0.5||Td = ±0.1|
Table 4 shows that the size of the area, where the multipath effect is significant, depends on the code rate or, to be exact, on the PRN chip length (
The maximum error values of the phase measurement due to multipath are calculated according to Eq. (12). It was supposed that there is only one reflected signal that has the phase shift angle φ1,max, rad and delayed seconds. This angle corresponds to the case when the multipath errors reach the maxima and affects the multipath error envelope which contains all the smaller variations of the ΔΨ values. The angle φ1,max can be found by differentiating Eq. (12) with respect to φ1, putting it to zero and solving it for φ1. It results in the following :
Figure 2 shows the standard deviations of the carrier phase multipath errors with respect to multipath delays for different SMR using the correlator spacing of
According to Figure 2, there is a dependence of the error on SMR. The magnitude of the multipath error (ΔΨ) is proportional to the strength of the multipath signal. Moreover, the multipath error value is independent on the carrier wavelength (Eq. 12), but it is mostly a function of the antenna-reflector distance through the correlation function . If the multipath delay (
3. Experimental results and analysis
3.1 Indices comparison
This section discusses the performance of the “standard” ionospheric scintillation indices and the index D2fi based on high-rate sampling data. The D2fi index and the ionospheric indices/parameters
The 50 Hz
As the de-trended TEC data is used to calculate
The storm period was chosen for the analysis as geomagnetic storms are known to cause ionospheric disturbances including the small-scale disturbances that are of the particular interest for this work. The intense storm of June 22–25, 2015, was under analysis. Figure 3 shows SYM-H index variations during the storm. Main phase (MP) and recovery phase (RP) of the storm are marked with red lines . SYM-H reached its minimum on June 23rd. SYM-H index data was obtained from Data Analysis Center for Geomagnetism and Space Magnetism following the link http://wdc.kugi.kyoto-u.ac.jp/aeasy/index.html (last access: August 2018).
According to , the relationship between
The worst correlation is between the D2fi index and
Let us consider the advantages of the D2fi index in comparison to other indices by the example in Figure 4. First, it marks the sharper and more precise in time response to small-scale turbulences than other indices. Furthermore, only one GPS frequency is needed to obtain the D2fi index. Thus, it avoids the possible impact from the inter-frequency noises and
Now, let us consider the data from high-latitude region, where scintillations are more frequent. Figures 5 and 6 are similar to Figure 4 and show the results derived from the 50-Hz data at stations EDM (53,35° N, 247,02° W) and GJO (68,63° N, 254,15° W). Both stations belong to the Canadian High Arctic Ionospheric Network (CHAIN)  and equipped with the same type SEPTENTRIO PolaRxS GNSS receivers . The station EDM is still within mid-latitudes (however in Canada it strictly depends on current geomagnetic conditions), but the station GJO is in high-latitude region.
It is seen that the weaker scintillation, the weaker the response of D2fi and
The comparison of different indices allows us to reveal the prevalence of phase or amplitude scintillations. In our case (Figure 5) the obvious difference in S4 and
To sum up, Figures 4–6 prove the following: (a) D2fi peaks are caused by scintillation events (as there are also responses in other scintillation indices though less precise) and (b) that the D2fi index shows more sensitivity to phase scintillations.
3.2 Time resolution comparison
The time resolution of input data is very important to detect scintillations. For example, the work  showed the significant sampling rate influence on
The smaller irregularities (from tens of meters to 100–300 m) are mostly considered to provoke the diffractive amplitude and phase variations. To detect them the highest time rate possible is needed (higher that 10 Hz). Diffractive phenomena can cause the phase scintillations that are usually accompanied by the intense amplitude fluctuations. These can be detected by
Several kilometer size irregularities usually cause the refractive scintillations of 0.01–0.1 Hz. When such irregularities dominate,
Usually, the irregularities of different scales are present in the ionosphere simultaneously. It can occur during the volcanic eruptions, powerful explosions, rocket launching, under disturbed geomagnetic conditions, etc. . The ionospheric irregularities can move with the quiet different velocities and in different directions. The 1 Hz or lower time resolution data does not allow us to reveal if the ionospheric event was caused by the diffractive irregularities of hundreds of meters or by the larger refractive irregularities of tens of kilometers.
We suggest that the high data sampling rate such as 10 Hz and higher provides the opportunity to reveal and analyze the weak small-scale ionospheric irregularities. To test this assumption, we compared 1, 10, and 50 Hz time series of the D2fi index for the same events and under the same geomagnetic storm conditions. Figure 7 shows the results of comparison for PRN 04, PRN15, and PRN27 at ISTP station on June 22, 2015, during the main phase of the geomagnetic storm (Figure 3).
The D2fi index obtained from 1 Hz GPS data does not reveal any scintillation event for all three satellites (Figure 7c). In contrast, the time series obtained from 10 Hz data show the clear peaks for the SV PRN 15 and PRN 27 (Figure 7b), but not for the weakest event for SV PRN 04 (Figure 7b, left). The peaks of 50 Hz time series are the most pronounced for all the satellites (Figure 7a). Note that the 1 Hz data shows both the highest noise level and the additional regular trend. The low-frequency trends are mostly removed from the time series of higher sampling rate.
In case of the highest data rate (50 Hz), the background values of D2fi do not exceed 0.4 rad/s*s (Figure 7a). For the lower data rate (10 Hz), the weak regular trend appears, and the background noise increases to 0.6 rad/s*s (Figure 7b). The D2fi variations increase 4–5 times and exceed 2–3 rad/s*s in the last case (1 Hz data, Figure 7c).
Apart from the ionospheric scintillations, one of the common sources of the phase fluctuations is the multipath effect. The majority of the multipath-induced fluctuations are observed at lower elevation angles. It is also not a thorough determination of multipath as it is possible to observe it at the higher elevations as well . Thus, we should test if the scintillation events revealed above are related to multipath and/or blocked signal effects. Usually, the multipath-induced phase variations are caused by the repeating events due to local reflection or diffuse scattering. The picture of such events repeats from day to day at the same location. At the same time, the picture of such “scintillations” has the regular time shift about 16 s from one day to another due to GPS orbits daily motion . This means that to determine whether the scintillation candidate events are caused by repeating local multipath effects, the raw data for the day before and after the scintillation should be analyzed. Figure 8 illustrates such the analysis for 50 Hz data on June 21, June 22, and June 23, 2015, for PRN 04, PRN15, and PRN27.
No significant phase scintillations on the day before (June 21, Figure 8, left column) and/or after (June 23, Figure 8, right column) were observed. In contrast, there were the sharp and rapid variations of the second-order derivative of the carrier phase on June 22, 2015, for all the satellites. This fact proves that the phase scintillation events observed on June 22, 2015, are not related to the multipath effect. Thus, the above mentioned phase scintillation events probably have the ionospheric origin.
The performance of the well-known “standard” ionospheric scintillation indices ROTI, DROTI,
The overall accuracy of the GNSS carrier phase measurements is limited by both thermal and external noises and significantly depends on the GNSS hardware and software presets and architecture. The accuracy of the carrier phase measurements can be improved if the particular specification is used for GNSS equipment suggested for the ionospheric studies. This particular specification means that the narrowband code delay discriminator, the large code rate for the open-access GNSS signals, the expanded front-end bandpass of the RFC, the low-noise preamplifier, and the specific pattern antenna should be specified for the ionospheric study.
In the present study, the new index D2fi is proved to be an effective tool to detect the small-scale ionospheric irregularities. It was shown that the sensitivity of the D2fi index depends on the data sampling rate. The higher the sampling rate, the clearer the peaks of the D2fi index, and the weaker both the noise background and the low-frequency trend. The comparison between the D2fi index and DROTI,
This work was supported by the Russian Federation President Grant No. MK-3265.2019.5 and by grant No. 18-05-00343 from the Russian Foundation for Basic Research. LANCE acknowledges partial support from CONACyT LN-299022, CONACyT PN 2015-173, and CONACyT-AEM Grant 2017-2101-292684. ISTP staton data were recorded by the Angara Multiaccess Center facilities (http://ckp-angara.iszf.irk.ru) at ISTP SB RAS within the base financing of FR program II.16.
Conflict of interest
Vladislav Demyanov is the principal author and the corresponding author of this book chapter. The text and the figures presented in this book chapter were not published anywhere else before.