Experimental Estimation of Deviation Frequency within the Spectrum of Scintillations of the Carrier Phase of GNSS Signals

: The term deviation frequency (fd) denotes the boundary between the variable part of the amplitude and phase scintillation spectrum and the part of uninformative noises. We suggested the concept of the “characteristic deviation frequency” during the observation period. The characteristic deviation frequency is deﬁned as the most probable value of the deviation frequency under current local conditions. Our case study involved GPS, GLONASS, Galileo and SBAS data under quiet and weakly disturbed geomagnetic conditions (geomagnetic storm on 16 April 2021, Kp max = 5, SYM-H min = − 57 nT) at the mid-latitude GNSS station. Our results demonstrated that the deviation frequency for all signal components of GPS, GLONASS and Galileo varies within 15–22 Hz. The characteristic deviation frequency was 20 Hz for the mentioned GNSS signals. The SBAS differs from other systems: deviation frequency varies within 13–20 Hz. The characteristic deviation frequency is lower and equal to 18 Hz. We suggest the characteristic deviation frequency to determine the optimal sampling rate of the GNSS carrier phase data for the ionospheric studies. In turn, the deviation frequency can be considered as a promising index to estimate the boundary of non-variability of the ionosphere.


Introduction
Global Navigation Satellite Systems (GNSS) form part of a technological basis for different applications [1]. Their data is widely used for fundamental research tasks in different fields, for example in geodynamics [2], radio propagation environment including the GNSS remote sensing (GNSS-RO) [3] and GNSS Reflectometry of Earth surface (GNSS-R) [4]. In particular, one of the important geophysical studies that was carried out on the basis of GNSS signal processing was the study of the Earth's ionosphere and upper atmosphere [5][6][7] and their impact on different levels of applications [8][9][10][11][12].
In many cases the main measured parameter of satellite vehicle (SV) signals is the carrier phase that is characterized by the lowest multipath noise level and the highest measurement accuracy. Various stochastic techniques report normally distributed carrier phase noise of 2 mm and code pseudo range noise of 0.5-0.8 m [13]. Such precise measurements allow us to detect the effects of rather weak geophysical events and eventually reconstruct the structure of the ionosphere.
Determination of optimum sensitivity of the carrier phase lock loop (PLL) is an important task for the remote sensing of the ionosphere. The PLL sensitivity depends on both the internal noises and the sampling rate of carrier phase measurements [14]. The PLL Remote Sens. 2021, 13, 5017 2 of 12 sensitivity is considered as an optimal if the probability of detection of a weak event at the background of the uninformative noise within the phase variation spectrum is the highest at a given noise level and a given sampling rate.
On the other hand, the higher boundary in the spectrum of the phase variations (at which weak disturbances can still be detected) may be considered as the boundary of the regular ionosphere. Studies that are focused on estimation of this boundary and its dependence on observation conditions are of particular importance for geophysical research advances.
McCaffrey and Jayachandran [15] suggested the "deviation frequency" term to denote the boundary between the variable part of the amplitude and phase scintillation spectrum and the part of uninformative noise in the spectrum. Developing this idea, we suggest introducing the concept of the "characteristic deviation frequency" during the observation period. The characteristic deviation frequency is defined as the most probable value of the deviation frequency (fd) under current local conditions. We assume that the characteristic deviation frequency is an estimate of the optimal sensitivity of the PLL under the current conditions.
It is worth pointing out that the spectral slope of the phase variations is mostly defined with large-scale ionospheric structures such as manifestations of the acoustic-gravity waves in the form of large-scale travelling ionospheric disturbances. Such disturbances can be caused by auroral oval border pulsations (during geomagnetic storms and substorms), earthquakes of high magnitude, tsunamis, cyclones and other power processes in the lower atmosphere. Thus, the deviation frequency as a break point between the phase variations spectra (with the spectral slope < 0) and the noise part of the spectra (with the spectral slope~0) varies depending on presence or absence of the large-scale ionospheric disturbances. This allows us to assume that the deviation frequency can be considered as a promising index to estimate the boundary of non-variability of the ionosphere.
The aim of this work is to estimate the characteristic deviation frequency under particular observation conditions. The research tasks are to reveal the features of variations of the characteristic deviation frequency for GPS, GLONASS, Galileo and SBAS satellites, types of satellite signals, time of day and the level of geomagnetic activity.

Data and Processing Method
The measurements of GNSS signal carrier phase were performed during 13 and 16 April 2021 with use of the multi-system multi-band navigation receiver Javad Delta-G3T connected to the RingAnt-G3T antenna [16]. The equipment was installed at the ISTP station (geographic coordinates 52.24 • N, 104.26 • E; geomagnetic coordinates 42.70 • N, 177.43 • E). The station belongs to the SibNet GNSS receiver network [17]. Figure 1 shows variations of Kp and SYM-H geomagnetic indices during the considered period.
The intensity of the geomagnetic storm on 16 April 2021 was weak. SYM-H reached its minimum value of −57 nT at 20:02 UT. The Kp index reached the value equal to 5 during the period of 18:00-21:00 UT. The main phase of the storm occurred in the local midnight sector. This means the minimal background electron concentration level. Such conditions imply the appearance of weak ionospheric disturbances registered at the border of the stationary of the ionosphere. Therefore, we chose the period of this storm for our analysis.
The level of carrier phase measurement noises significantly differs for signals of different systems and for different signal components [18][19][20]. This issue is important for estimation of the characteristic deviation frequency in the spectrum of phase variations and scintillations. Hence, the carrier phase noise for the signal components at L1, L2 and L5 frequencies of GPS, GLONASS, Galileo and SBAS satellites were studied first ( Table 1). The measurements were performed with a 50 Hz sampling rate. The signal components description mentioned in Table 1 can be found in the data format description available at [21]. Remote Sens. 2021, 13, x FOR PEER REVIEW 3 of 14 The level of carrier phase measurement noises significantly differs for signals of different systems and for different signal components [18][19][20]. This issue is important for estimation of the characteristic deviation frequency in the spectrum of phase variations and scintillations. Hence, the carrier phase noise for the signal components at L1, L2 and L5 frequencies of GPS, GLONASS, Galileo and SBAS satellites were studied first ( Table 1). The measurements were performed with a 50 Hz sampling rate. The signal components description mentioned in Table 1 can be found in the data format description available at [21].

Navigation System Signal Components
We applied the following procedure for deviation frequency estimation.
(1) The second order phase derivative was used to detect cycle slips and abnormal data in the carrier phase records. Cycle slip and anomalies in data yield sharp peaks of the phase derivative. All the carrier phase records that contain one or more events defined by Equation (1) were excluded from our analysis where dMax is the threshold defined empirically as dMax = 10d2Lj−1; d2Lj−1 is the second order derivative of the carrier phase at the previous step of calculation; and Li is the carrier phase at i-th time point (in cycles).
We applied the following procedure for deviation frequency estimation.
(1) The second order phase derivative was used to detect cycle slips and abnormal data in the carrier phase records. Cycle slip and anomalies in data yield sharp peaks of the phase derivative. All the carrier phase records that contain one or more events defined by Equation (1) were excluded from our analysis where dMax is the threshold defined empirically as dMax = 10d2L j−1 ; d2L j−1 is the second order derivative of the carrier phase at the previous step of calculation; and L i is the carrier phase at i-th time point (in cycles).
(2) Phase ambiguity was resolved by means of polynomial filtering. We used a 2-step algorithm to reduce the phase ambiguity. First, the main trend of the carrier phase was removed by means of linear interpolation as follows Further, the second-order trends of the carrier phase were removed by means of the 5-th order polynomial interpolation: Remote Sens. 2021, 13, 5017 4 of 12 (3) The final de-trending of the carrier phase was performed by means of moving averaging window filter as follows The parameter of the filter N defines the window width. Considering the results reported in [14], we chose the 5 min averaging window. According to [14], phase fluctuations caused by small-scale irregularities usually have a period of several seconds. Indeed, Pi et al. [7] reported a period of~(2-13) s and Forte and Radicella [22] reported~(0.4-5) s.
After the data preprocessing, a fast Fourier transform was applied to the de-trended phase data series LS (4). We did not apply the Hanning window as in work [15], because of uncertainty of filter parameter choice under the unknown spread and behavior of deviation frequencies. The deviation frequency was determined at the logarithmic spectrum of phase variations as a "break point" at which the maximum decrease in the slope of the spectrum passes into near-zero decrease in a given frequency range. Figure 2 illustrates the example of deviation frequency identification from the logarithmic spectrum of phase variations for line-of-site to GPS satellite PRN07 on 13 April (panel a) and 16 April (panel b).
Further, the second-order trends of the carrier phase were removed by means of the 5th order polynomial interpolation: (3) The final de-trending of the carrier phase was performed by means of moving averaging window filter as follows The parameter of the filter N defines the window width. Considering the results reported in [14], we chose the 5 min averaging window. According to [14], phase fluctuations caused by small-scale irregularities usually have a period of several seconds. Indeed, Pi et al. [7] reported a period of ~(2-13) s and Forte and Radicella [22] reported ~(0.4-5) s.
After the data preprocessing, a fast Fourier transform was applied to the de-trended phase data series LS (4). We did not apply the Hanning window as in work [15], because of uncertainty of filter parameter choice under the unknown spread and behavior of deviation frequencies. The deviation frequency was determined at the logarithmic spectrum of phase variations as a "break point" at which the maximum decrease in the slope of the spectrum passes into near-zero decrease in a given frequency range. Figure   When performing this analysis, the frequency range from 1 to 25 Hz in the spectrum was studied in which the deviation frequency was expected to vary. The "break point" defines the part of the phase variations spectrum where the expected power law character is shallowing towards white noise. Therefore, the carrier phase noise is a key issue to define the precise value of deviation frequency. That is why it is important to analyze the level of phase measurement noise for signals of different navigation systems, different frequencies and components.
The second-order derivative of the signal carrier phase was used as the noise magnitude (Please, see the methodology description in [14]). Table 2 provides root-mean-square (RMS) estimates for the noise of phase measurements for signals from different systems and components.
According to Table 2, the level of the carrier phase noise for the similar types of components and signal frequencies of GPS, GLONASS and Galileo does not differ much. At the same time, RMS of the phase measurements for SBAS signals is 1.5-3 times higher, Remote Sens. 2021, 13, 5017 5 of 12 which is in accordance with the known results by [23]. The lowest noise level among the considered signal frequencies and components was detected for L5 frequency.

Discussion of Results
Our results are presented in the histogram form. To construct each histogram we evaluated the number of fd values for each particular case: one fd value per one spectrum of the phase variations. Each spectra was obtained by applying the Fast Fourier Transformation (FFT) from the 1-h carrier phase record for the particular satellite and signal component with 50-Hz sampling. Thus, the general statistics of this study includes~2900 particular spectra of the phase variations (for instance, two days × 24 h × (6-8 GPS satellites in view × 5 GPS signal components + (4-6) GLONASS satellites × 4 GLONASS signal components + (2-4) GALILEO satellites × 2 GALILEO signal components) + SBAS data). Figure 3 shows the histograms of distribution of the deviation frequency (fd) for signal components L1X and L5X of GALILEO satellites during 24 h. The deviation frequency value varied within 15-22 Hz during both days. The most probable deviation frequency in either day of two and for all signal components was 20 Hz. The character of distribution shown for L1X and L5X components under the same conditions has no fundamental difference. At the same time, the carrier phase noise for the L5X component is significantly lower than for the L1X component (Table 2). Moreover, the histograms constructed for the control day of 13 April and the geomagnetically disturbed day of 16 April differ essentially. Histograms for the quiet day are characterized by more gradual rise at their left part. In contrast, the most probable deviation frequency is pronounced more clearly at the histograms for the disturbed day.  Figure 5a,c,e,g,i). It is also worth noting the rather uniform appearance of the deviation frequency distribution in the histograms for GPS with the only exception of the 20 Hz peak during geomagnetically disturbed day (Figure 4 right panels). same time, the carrier phase noise for the L5X component is significantly lower than for the L1X component (Table 2). Moreover, the histograms constructed for the control day of 13 April and the geomagnetically disturbed day of 16 April differ essentially. Histograms for the quiet day are characterized by more gradual rise at their left part. In contrast, the most probable deviation frequency is pronounced more clearly at the histograms for the disturbed day.   4a,c,e,g and 5a,c,e,g,i). It is also worth noting the rather uniform appearance of the deviation frequency distribution in the histograms for GPS with the only exception of the 20 Hz peak during geomagnetically disturbed day (Figure 4 right panels). It is worth noting that regardless of the significant difference in the carrier phase noise (see Table 2) the results for GPS, GLONASS and GALILEO signals are quite similar. The difference in the phase noise did not affect the characteristic deviation frequency. Figure 6 shows the distribution of deviation frequency of carrier phase for L1X and L5I components of SBAS satellite signals during 24 h. The measurements were obtained for three SBAS satellites whose angular characteristics are given in Table 3.

Deviation Frequency Estimates for Different GNSS Signal Components
According to Table 3, all observed SBAS satellites were concentrated in one narrow southwest sector and at low elevation angles. Consequently, spatial variability of the ionosphere due to the line-of-sight movement through different ionospheric/atmospheric regions may not be taken into account.
The histogram form in Figure 6 differs significantly from the histograms shown in Figures 3-5. The SBAS deviation frequency varied within 13-20 Hz, which is smaller than one of GPS, GLONASS or Galileo. The characteristic deviation frequency can be determined with confidence only in one case of four in Figure 6 (in panel b). For example, the characteristic deviation frequency in Figure 6a is 18 Hz with probability of 0.25, and it is 15 or 16 Hz with the probability of 0.22. In contrast, in Figure 6b the characteristic deviation frequency is pronounced much clearer at 18 Hz. Other two histograms in Figure 6 show no clear maximum. On the quiet day of 13 April we recorded smaller deviation frequencies as compared to 16 April. Under disturbed conditions the maximum is more pronounced and varied within 18-19 Hz (Figure 6b,d).

Deviation Frequency Dependence on Geomagnetic Conditions
The previous section showed high stability of characteristic deviation frequency fd = 20 Hz under particular conditions in this case study. This allows us to suggest that this characteristic frequency is primarily associated with the boundary of the regular ionosphere under current conditions. To test this assumption we analyzed diurnal variations in the most probable deviation frequency. Remote Sens. 2021, 13, x FOR PEER REVIEW 8 of 14 Figure 4. The same as in Figure 3 but for the GLONASS signal components (Table 1). Deviation frequency distribution on 13 April 2021 (a,c,e,g) and 16 April 2021 (b,d,f,h) for GLONASS signals. Figure 4. The same as in Figure 3 but for the GLONASS signal components (Table 1). Deviation frequency distribution on 13 April 2021 (a,c,e,g) and 16 April 2021 (b,d,f,h) for GLONASS signals.  (Table 1). Deviation frequency distribution on 13 April 2021 (a,c,e,g,i) and 16 April 2021 (b,d,f,h,j) for GPS signals.  According to Table 3, all observed SBAS satellites were concentrated in one narrow southwest sector and at low elevation angles. Consequently, spatial variability of the ionosphere due to the line-of-sight movement through different ionospheric/atmospheric regions may not be taken into account.
The histogram form in Figure 6 differs significantly from the histograms shown in Figures 3-5. The SBAS deviation frequency varied within 13-20 Hz, which is smaller than   Figure 7 (upper panels) shows diurnal variations in the most probable deviation frequency for all signal components and for all navigation systems (excluding SBAS). We compute one fd value per one hour. The local noon and the night period indicated by the red arrow and blue rectangle, respectively. The left panels show the results for the quiet day, and the right panels show those for the disturbed day.
The lower panels of Figure 7 present the diurnal variations of the relative number of cases when fd = 20 Hz, calculated hourly as follows where N 20 is the number of cases when fd = 20 Hz during one hour; N TOT is the total number of all fd values derived from all phase variations spectra for all satellites in view for all signal components and for all navigation systems (excluding SBAS) during one hour. Remote Sens. 2021, 13, x FOR PEER REVIEW 12 of   Figure 7 (lower panels) shows that the characteristic deviation frequency of 20 H not always dominant during the 24 h. This means that the deviation frequency can other values during most of the day. Unfolding of how the particular deviation freq value is related to certain processes in the ionosphere is the issue for future research. I case study no clear tendency in the character of NCFd variations was revealed. Neverth the observed NCFd value at the particular moments was higher under quiet geoma conditions (NCFd ≤ 0.9-1.0) than under disturbed conditions (NCFd ≤ 0.8).

Conclusions
We suggested the concept of the characteristic deviation frequency to determin optimal sampling rate of the GNSS carrier phase data for the ionosphere studies. Cu research considers that the characteristic deviation frequency is optimal if its fu increase does not provide more information on the small-scale structure of the ionosph Our case study showed that the characteristic deviation frequency tends to be during the short observation intervals (1 h). However, its value can vary significantly d 24 h (within [15][16][17][18][19][20][21]. This agrees with the current knowledge on the physical ionosp structure [24][25][26][27][28].   Figure 7 (lower panels) shows that the characteristic deviation frequency of 20 Hz was not always dominant during the 24 h. This means that the deviation frequency can take other values during most of the day. Unfolding of how the particular deviation frequency value is related to certain processes in the ionosphere is the issue for future research. In this case study no clear tendency in the character of N CFd variations was revealed. Nevertheless, the observed N CFd value at the particular moments was higher under quiet geomagnetic conditions (N CFd ≤ 0.9-1.0) than under disturbed conditions (N CFd ≤ 0.8).

Conclusions
We suggested the concept of the characteristic deviation frequency to determine the optimal sampling rate of the GNSS carrier phase data for the ionosphere studies. Current research considers that the characteristic deviation frequency is optimal if its further increase does not provide more information on the small-scale structure of the ionosphere.
The maximal probability of the characteristic deviation frequency of 20 Hz at some moments did not exceed 0.8 under geomagnetically disturbed conditions and reached 0.9-1.0 under quiet conditions. The increase in this probability can indicate the increase in the spatial-temporal stability of the ionosphere under quiet conditions. Correspondingly, its decrease implies the presence of the disturbance effects that change the phase spectrum slope. This allows us to assume that the deviation frequency can be considered as a promising index to estimate the boundary of non-variability of the ionosphere at the presence or absence of large-scale ionospheric disturbances.
The probability distribution of the characteristic deviation frequency obtained from SBAS data differs from the probability distribution of the other GNSS signals significantly. We showed that the carrier phase noise for SBAS signals is notably higher than for the signals of other GNSS. This is in accordance with the results of [23]. It is probable that the plasmasphere could impact the characteristic deviation frequency for SBAS signals. This hypothesis should be tested with further experiments and analysis of data. Moreover, the signals of geostationary SBAS satellites passed from the same narrow angular sector. This means that spatial variability of the ionosphere due to the line-of-site movement through different ionospheric/atmospheric regions did not impact on the deviation frequency variations significantly. This circumstance is absent for the radio propagation of signals from medium-orbit (and lower orbital) GNSS constellations. The data of geostationary BeiDou satellites which is characterized by the same phase noise level as GPS/GLONASS may be used for checking this hypothesis further.