Modern Improvements of GNSS Technologies: New Opportunities in Exploration of the Earth’s Ionosphere

1. Introduction

The modern Global Navigation Satellite Systems (GNSS) and networks of GNSS receivers covered the globe and became a unique and high-precision tool for both navigation and geosciences. Today, it is possible to observe 30–40 satellites transmitting signals at three frequency bands simultaneously. Besides, most of the GNSS signals consist of both the in-phase and quadrature signal components carrying different pseudo-random codes with very different correlation properties. Manufacturers produce multisystem GNSS receivers that are able to get and record code/carrier measurements of all GNSS signal components with temporal resolution up to 100 Hz. Such rapid advances in GNSS technologies open unprecedented opportunities to maintain an effective remote probing of the Earth’s ionosphere using a great number of GNSS signals simultaneously and globally.

On the other hand, modern improvements in GNSS go far in advance compared to research technologies involving GNSS signals and equipment to monitor the atmosphere and ionosphere. These research technologies totally rely on GNSS signals, receivers, and processing methods of the GNSS data, considering these elements as a “black box” with constant and a priori known features. Until recently, the mentioned “black box” has not been examined deeply as a possible impactful factor degrading the results of the ionosphere remote probing. Typically, the known set of pre-adjusted parameters of a GNSS receiver, as well as the parameters of processing procedures of GNSS code/carrier time series, were proposed long ago [1, 2], and they have not been revised seriously since that time.

Note that the novel capabilities of GNSS allow monitoring more ionospheric events, including weak turbulences at the boundary of the noises [3]. However, recent studies demonstrated that ionospheric total electron content (TEC) and TEC-based indices can differ significantly from each other when derived from GNSS receivers of different types [4]. Yang and Liu [5] observed the difference in TEC and ROTI calculated from the data of L2P(Y) and L2C GPS observations. McCaffrey et al. [6] demonstrated that the independently tracked carrier phase dynamics are significantly more accurate than the L1-aided observables to detect ionospheric scintillations. Bolla and Borre [7] defined an optimal ionosphere-free combination for dual-frequency L1, L2C, and L5 GPS signals in terms of the best sensitivity and lowest observation noise. All these issues emerge with the evolution of GNSS technologies today. Thus, a geophysicist needs to take into account the receiver characteristics and the new GNSS signal properties in order to be able to define accurate pre-adjustments of the data processing procedures to achieve better results in the study of the ionosphere. It is time to not consider a GNSS receiver or a typical data processing methodology as just a “black box”.

In this chapter, we present a review of new research results, methods, and ideas that allow utilizing the modern improvements of GNSS technologies for deep and highly effective studies of the ionosphere. We discuss the following items here:

1. GNSS receiver sensitivity as an instrument of the ionosphere remote probing;

2. The noise component of Total Electron Content depends on the GNSS receiver type;

3. The performance of the ionospheric scintillation indices depending on the data temporal resolution and the pre-adjustments of de-trending and filtering procedures.

2. Factors defining the sensitivity of a GNSS receiver as an instrument of the ionosphere remote probing

There are two main parameters measured from GNSS signals propagating throughout the ionosphere: signal carrier and code delay. Both of these parameters and their combinations are utilized to reconstruct TEC along the satellite line of sight (LOS). Besides these parameters, measured at particular frequencies of the L1, L2, or L5 bands are involved to compute some ionospheric indices [2, 8, 9, 10]. The main question here is about the sensitivity of such measurements and their combinations to detect as weak ionospheric turbulences as possible.

Let us consider the content of the code and carrier measurements to point out the components defining the measurement sensitivity. The code range, defined as the code delay multiplied by the speed of light (c), is [11, 12]:

Where k is a satellite vehicle (SV) number; τk is the time of the signal traveling between a receiver and SV; drk is a geometric distance between the satellite and the receiving antennas; Ik is a group ionospheric ranging error; Tk is a tropospheric ranging error; с∆tt is a receiver timescale deviation error; с∆tkt−τk is a satellite timescale deviation error; сdtt is a signal delay in the receiver; сdtkt−τk is a signal delay in the satellite; dmk is a multipath error; dsk is an error induced by ionospheric scintillations; ek is noise error.

Similarly, the content of the carrier phase, defined as a phase range (in meters), is as follows:

The Eq. (2) contains the same main components as (1), but there are some features as follows: the ionospheric ranging error has an opposite sign; there are additional terms corresponding to inequality to zero of the initial signal phases (λ∙…) and the phase ambiguity (λ∙Nk); the multipath error (δmk) and noise (Ek) are much lower compared to the ones in (1); ionospheric scintillations of the carrier phase (δsk) mostly appear as a result of the refractive events, but not as a result of the amplitude diffraction (as it is for the code delay scintillations—dsk).

When the carrier-to-noise ratio (CNR) is stable and there are no ionospheric scintillations and multipath, the sensitivity of the remote probing of the ionosphere is defined by the thermal noise of a GNSS receiver. It is illustrated in Figure 1a, which shows the log spectrum of TEC disturbances of different spatial and temporal scales.

In reality, the code/carrier measurements significantly degrade by the errors due to multipath (dmk, δmk), ionospheric scintillations (dsk,δsk), and noises (ek, Ek). Hence, to define the real sensitivity of the ionosphere remote probing the sensitivity threshold should be introduced. We can use the threshold called “deviation frequency” (fd) as an idea. This term was introduced first in Ref. [6] as a border between two parts of the spectrum of amplitude or carrier phase variations. The left part of the spectrum describes variations of amplitude or carrier phase, but the right one presents uncorrelated noises. Using fd value, we can define the real sensitivity of the ionosphere remote probing as it is illustrated in Figure 1b. Comparing Figure 1a and b, we see that the impact of multipath, scintillations, and thermal noise may significantly change the sensitivity. Besides, the sensitivity of the ionosphere remote probing strictly depends on the time resolution of the GNSS data. Lower time resolution limits the observable TEC disturbances by the Nyquist frequency (FNQ and Figure 1).

The impact of multipath, scintillations, and noise strictly depends on the current user’s location, GNSS receiver characteristics, signal properties, and the CNR at the input of a code and phase tracking loops. The main advantages provided by modern GNSS are follows: (1) increasing the power of some GNSS signals; (2) launching new signals with better correlation properties at three frequency bands; (3) improving GNSS receivers by means of reducing thermal noises and Allan variance of reference oscillators; (4) improvements of signal tracking techniques inside a GNSS receiver.

Tables 1 and 2 demonstrate features of the modern GNSS signals and constellations in detail [13, 14, 15, 16, 17, 18].

One can see that there are plenty of signal components available for tracking in a modern GNSS receiver simultaneously. The signal parameters, such as CNR, PRN properties, and modulation type, differ from one signal component to the other very significantly. All of the mentioned may yield very different results of the ionosphere remote probing made by different GNSS signals and components. Let us consider how this GNSS modernization can change the quality of the ionosphere remote probing.

Standard deviation of the filtering error of the code delay or carrier phase depends on pseudo-random code (PRN) parameters and on the carrier-to-noise ratio as follows:

where β=1/τPRN is the signal spectrum width, Hz; τPRN is the PRN code chip length, s.

The CNR depends on both the PRN length (NPRN) and τ_PRN as follows [19]:

where S2f∆F is the averaged signal power within the signal frequency band; ∆F=1/(NPRN∙τPRN) is the signal frequency band; N02 is the noise power within the signal frequency band.

Besides CNR, value depends on the thermal noises of a GNSS receiver, including environmental thermal noises, as follows [19, 20]:

where Prec is the signal level at the receiving antenna (dBW); GA is the antenna gain (dB); NT,E is the spectral density of the receiver thermal noise including the environmental noises (dBW); Ltr is the total power loss during filtering, frequency conversion, and the signal attenuation in the cable (dB); and Ldg is the signal power loss due to its analog-to-digital conversion (dB).

GNSS receiver architecture and its pre-adjustments are also important factors for the sensitivity of the remote probing of the ionosphere. For a stationary receiver in the absence of vibration-induced phase jitter, the standard deviation of the error of the code delay (σΔρ) and carrier phase (σφ) evaluation can be found from the following equations [19]:

Where k and m are the parameters of a code delay discriminator and a correlator, correspondingly; ΔFDLL, ΔFPLL are the noise bandwidth of the code delay loop (DLL) and the carrier phase lock loops; ΔT is an integration time; dτ is the correlator spacing between early, prompt, and late; σFτ is the Allan variance-induced oscillator jitter; and n is the phase lock loop (PLL) parameter.

The signal modulation type (BPSK, BOC, AltBOC) defines the PRN code autocorrelation function (Rτ). The PRN parameters, such as NPRN and τPRN, as well as the function Rτ, define the multipath error figure for both code and carrier phase. Indeed, one reflected signal is delayed with t1 mixes with the direct signal and produces the correspondent multipath error in the code delay computation as follows [21]:

Where α1 is the reflection coefficient; t1 is the delay in propagation of a reflected signal relative to the direct signal, sec.

In the same case (one reflected signal), the multipath error of the carrier phase evaluation (δmi) it is also dependent on the PRN correlation properties as follows [21]:

where Rτ̂ is the PRN autocorrelation function; Rτ̂−t1 is the cross-correlation function between the direct and the reflected GNSS signals; τ̂=τ0±Δτ is the estimated code delay; ϕ1 is the phase of a reflected signal; τ0 is the pure code delay; Δτ is an error in the code delay calculation.

Interference between the direct and the reflected signals can cause significant degradation in the PRN autocorrelation function and cause fading of the signal power at the reception point [21].

These multipath-induced fluctuations of the Prec may cause corresponding fading and rapid fluctuations of the CNR (5) at the PLL/DLL unit and produce proportional errors in code/carrier estimates.

The ionospheric scintillations (dsi,δsi) may cause rapid fluctuations in both code delay and carrier phase estimates. The author [22] offered to set these effects of the ionospheric scintillation by the S4 index. Based on it, the Eqs. (6) and (7) can be rewritten as follows:

The effects of the ionospheric scintillations mainly depend on the signal frequency as follows [22]:

Eq. (13) demonstrates that scintillation studies may yield much more results by involving more GNSS frequencies and signal components. Review of the Eqs. (3)–(13) allows us to identify the main factors defining the sensitivity of a GNSS receiver as an instrument of ionosphere remote probing. Here there are

• carrier frequency;

• PRN code characteristics and its correlation properties;

• satellite signal power at the reception point;

• thermal noises of a radio frequency chain of a GNSS receiver;

• pre-adjustments of PLL and DLL units;

• available temporal resolution of the code/carrier recordings;

• code/phase tracking technique inside a GNSS receiver.

Now let us consider some important results, demonstrating opportunities to use the modern improvements of GNSS technologies to achieve better results in deep studies of the ionosphere.

3. The multipath and noise component of TEC depending on the GNSS signal and model of a receiver

Ionospheric total electron content is the most popular parameter utilized to explore the ionosphere by means of GNSS signals. TEC value along the satellite’s LOS can be reconstructed from the ionosphere-free linear combinations of code or carrier measurements obtained at two frequencies. The ionosphere-free linear combination of two carrier phases at different frequencies, fi and fj, allows for the calculation of the slant TEC along the LOS between the satellite and receiver as follows [12]:

where λi, λj, Li, and Lj are the wavelengths and carrier phase counts (including integer and fractional parts of the phase cycles) at fi and fj frequencies; const is the phase ambiguity; MPPH is the multipath term combined from multipath errors δmi and δmj (2) at fi and fj frequencies; and nφ is the sum of the phase noise components from the Li and Lj linear combination.

Until recently, the TEC reconstructed from the carrier phase measurements has been preferred as the most accurate and containing much lower noise compared to the TEC computed from the code measurements. The situation changed when Galileo and BeiDou systems started to exploit modern, extra-wideband E5 and B2 AltBOC signals [23]. The authors Padokhin et al. [24] and Chen et al. [25] demonstrated that the novel Galileo E5 and BeiDou B2 AltBOC signals have great potential in TEC estimation from both code and carrier measurements. These signals provide a comparable level of noise for the dual-frequency phase TEC and for the single-frequency phase-code TEC estimates. It has become possible thanks to both more sophisticated correlation properties of the new signals and increased CNR of these signals.

TEC utilizing combinations of signal code (P) and phase (L) measurements at the i-th frequency is computed from the corresponding linear combinations as follows [12, 26]:

where fi and fj are GNSS operating frequencies (depending on the system); DCB is the sum of differential code biases (including the satellite and the receiver components); MPPG and MPGG are the multipath term combined from multipath errors dmk and δmk from Eqs. (1) and (2) correspondingly.

Looking at Eqs. (14)–(16), we have to note that the quality of TEC computation obviously depends on the following factors:

1. Thermal noises of a satellite and a receiver radio chain, which define the terms nφ, nE, and nε;

2. Signal tracking technique defining the correlation between thermal noises at fi and fj frequencies;

3. Signal modulation type, which defines the multipath term according to Eqs. (8)–(9).

Figure 2 demonstrates that signal modulation of the new GNSS signals, such as Galileo E5 and BeiDou B2, may provide significant improvement in multipath mitigation. It is especially true for the AltBOC modulated signals compared to traditional BPSK modulation [23, 27]. Figure 2 demonstrates it clearly, presenting the multipath-induced code-range noise figure versus CNR when different types of modulation are involved. AltBOC signal outperforms both BPSK and QPSK signals with code-range noise below 5 cm when CNR is greater than 35 dB-Hz. The estimates presented in Figure 2 were obtained with coding parameters, such as integers (m, n) and (n) marked in brackets. These integers are the multipliers for subcarrier frequency, utilized as follows: fs=m×f0 and chip rate frequency fchip=n×f0, with f0 = 1.023 MHz [25].

According to Figure 2 and Eqs. (14)–(16), different signal combinations utilized to reconstruct TEC have to yield very different TEC values for the same observational conditions. Figure 3 proves this and demonstrates TEC vs. time dependence obtained from different combinations of Galileo and BeiDou E5 (a + b) and BeiDou B2 (a + b) signals. The TEC estimations were obtained on February 28, 2024, at the ACRG receiver (5.603°N, 0.187°W) by the authors Chen et al. [25]. The left panel of Figure 3 shows TEC obtained from the signal components of BeiDou satellite C24. The right panel of Figure 3 displays TEC reconstructed from the signals of the Galileo E05 satellite. Both of the satellites were observed at the elevation angles from 10° to 80° during ∼8 hours.

We can see that the dual-frequency code-range combination (4) constructed with C2 and C5 observables demonstrates the largest noise among all the considered combinations. Single-frequency code/phase combination (3) of the signals utilizing QPSK modulation is slightly less affected by noise. The lowest noise was observed for the dual-frequency phase combination (1). However, the single-frequency code/phase combination (2), utilizing sophisticated AltBOC modulation, also demonstrates a low level of noise, which is comparable with the noise observed for the dual-frequency phase combination (1).

Thermal phase noise of the radio frequency chain of a receiver (Ek, Eq. 2) differs significantly from one receiver type to the other. The authors Demyanov et al. [28] demonstrated it by comparing the carrier phase noises obtained from the Leica, Trimble, JAVAD, and Septentrio. Figure 4 summarizes these results and proves that the phase noise term may differ by about an order from one receiver type to the other: ∼0.01 rad for JAVAD compared to ∼0.001 for the Septentrio receiver at the satellite elevation of 80°.

Besides this receiver feature, some types of GNSS receivers use the L1-aided technique to track signals at different frequencies. Such a technique may degrade the results of remote probing of the ionosphere by different GNSS signals. Usually, it is not known whether the L2 or L5 signal is tracked utilizing the L1-aided algorithm or not. For the ionosphere-free linear combinations (14) and (16), the standard deviation of the TEC noise component (nφ and nε) depends on the covariance between the noises at fi and fj frequencies as follows:

where σLi,σLj are r.m.s. of the phase noises at fi and fj frequencies; RLi,Lj is the correlation coefficient between the noises at these two frequencies.

The authors Demyanov et al. [28] introduced a model to evaluate the summarized noise of the phase TEC (nφ, Eq. (14)) depending on the correlation between the noises of carrier phase at fi and fj frequencies.

where σn is defined by Eq. (17) and obtained from the experiment; n0, k and a are the model parameters; and γ is the satellite elevation angle.

The difference between the correlated and uncorrelated phase noises at fi and fj frequencies can be computed by the model (18) as follows:

where nϕ1MODEL and nϕ0MODEL are the model TEC noise calculated by the model (18) assuming RL1,L2≠0 and RL1,L2=0, respectively.

Figure 5 presents the difference in TEC computation taking into account the correlation between the phase noises at frequencies L1 and L2 GPS for different types of GNSS receivers.

One can see that different receiver types demonstrate different impacts in TEC calculation due to the correlation between the noises at two frequencies. The inter-channel correlation of the phase noises does not have a significant impact on the TEC noise component in the case of elevation angles >20°. The largest value Δnφ≈0.01 of TECU was observed for the JAVAD receiver at an elevation angle of 20°. In contrast, the lowest value of Δnφ≈0.0075 TECU, was found for the Septentrio receiver at the same elevation angle.

4. Ionospheric scintillation indices vs. GNSS data temporal resolution, detrending and filtering procedures

Temporal resolution of GNSS data defines the Nyquist frequency in the spectra of carrier phase variations of the trans-ionospheric signals (FNQ, Figure 1). Thus, the temporal resolution directly defines an opportunity to observe smaller-scale weak ionospheric turbulences. Figure 6 demonstrates the effect of reduction of the temporal resolution when the same ionospheric turbulence is observed from the 50 Hz (panel C), 10 Hz (panel B), and 1 Hz data (panel A). The data relates to GPS PRN 27 recorded from the JAVAD receiver at the ISTP site (52° N, 104° W) on June 22, 2015, under the strong geomagnetic storm (see details in Ref. [29]).

One can see that higher temporal resolution allows observing higher frequency phase variations in the spectra (see the left panel of the figure). The small-scale ionospheric disturbance corresponding to the scintillation event is reliably obtained from 50 Hz data (panel C). By reducing the temporal resolution, we are losing an opportunity to detect the same ionospheric disturbance reliably (panel B). Finally, the lowest time resolution leads to the fault in the observation of the mentioned ionospheric disturbance (panel A).

In practice, it may be required to detect small-scale ionospheric turbulences but not to evaluate their characteristics. The authors Demyanov et al. [3] offered to use the term “deviation frequency” (fd) as a promising indicator to detect small-scale and weak ionospheric turbulences, including ones that are below the noise level. It was found that fd value shifts toward the higher frequencies in the spectra of the phase variations when the ionospheric turbulences appear. As expected, the reaction of fd on the same ionospheric turbulence significantly differs depending on the signal component. Figure 7 illustrates it and demonstrates the histograms of fd values under geomagnetically quiet conditions on September 2, 2022, and under the geomagnetic storm on September 4, 2022. The data were obtained from 100 Hz carrier phase recordings of Galileo signals registered from all satellites in view during 24 hours at the ISTP site on both September 2 and 4, 2022. The histogram related to the particular Galileo signals (L1C, L6C and L7Q) is marked on the corresponding panels of Figure 7.

Figure 7 demonstrates the apparent response of fd on the geomagnetic storm. The fd shifts toward the higher frequencies for all Galileo signals up to almost the Nyquist frequency (50 Hz). In contrast, the fd is quite stable under the undisturbed conditions. It varies within 41–42 Hz for all Galileo signals on September 2, 2022 (Figure 7a–e). Reaction of fd values differs depending on the signal component under the geomagnetic storm on September 4, 2022. The L1C and L5Q signals demonstrate the highest variations in fd value, which reaches 48–49 Hz (Figure 7f and j). In contrast, L6C, L7Q, and L8Q components react less and the corresponding fd values do not exceed 45–46 Hz under the same conditions (Figure 7g–i).

Taking into account the higher availability of GNSS data with high temporal resolution, it is time to revise some typical methodologies of the ionospheric indices computation. Until recently, this issue has not been studied deeply as a significant factor affecting the accuracy of the remote probing of the ionosphere by GNSS signals. On the other hand, the higher temporal resolution replaces the focus of the data analysis close to the background noise, defining the accuracy and sensitivity of the remote probing methods (see Figure 1). Let us provide a short overview of new opportunities in the ionosphere indices computation based on the GNSS data with high temporal resolution. All the details of this full research can be found in Ref. [30].

The most popular ionospheric index providing a measure of the ionospheric disturbances isROTI [2]. The index is computed using the averaging of the de-trended TEC time series (dTEC) as follows:

From (20) it is clear that the accuracy and sensitivity of ROTI as an indicator of the ionospheric turbulences totally defined by the averaging time and the data temporal resolution. Figure 8 demonstrates ROTI indices obtained from the detrended phase TEC (14) utilizing different averaging intervals of 0.1, 1.0, and 10 sec. TEC data was emulated and recorded with a 100 Hz registration frequency [30].

One can see that the shorter averaging time gives higher noise and higher peaks in the ROTI value, but it becomes more difficult to detect the weak ionospheric turbulence that gets buried in the noise. Increased averaging time results in better noise suppression. It provides more reliable detection of the ionospheric turbulence by the ROTI peak.

The next very popular ionosphere index is the phase scintillation index (σφ) which is calculated from the detrended phase measurements φDTR as follows [1]:

Looking at Eq. (21), one can conclude that the result of σφ computation depends on both the detrending procedure and on the averaging time. Typically, it is recommended to use the sixth-order Butterworth filter with a cutoff frequency of 0.1 Hz as a detrending procedure. Figure 9 demonstrates the results of the σφ computations when using the Butterworth filter with a cutoff frequency of 0.1 Hz (left panel) and 1 Hz (right panel). The carrier phase data was emulated and recorded with 100 Hz registration frequency [30].

One can see that employing the Butterworth filter with different cutoff frequencies does not lead to suppression of measurement noise. However, the increased averaging time suppresses the noise and causes a significant increase in the calculated index σφ—from 0.023 (averaging time is 0.1 s—curve 1) to 0.04 when the averaging time is 10 s (curve 3, Figure 9b). Thus, the incorrect averaging time can result in the overestimated σφ value.

The authors Demyanov et al. [29] proposed the second-order derivative (d2fi index) to extract noise and ionospheric scintillations directly from the time series of carrier phase registered with high temporal resolution. Sensitivity of the d2fi index, as a detector of weak ionospheric turbulences, crucially depends on the background noise of a receiver. Figure 10 presents the results of the d2fi index calculation from the emulated carrier phase time series recorded at 100 Hz frequency. Different phase recordings consist of the same phase scintillations (marked with the black arrow) but different receiver noise [30]. It allows us to compare the results of the d2fi computation at different “scintillation-to-noise” ratio and define the index sensitivity threshold. Figure 10a demonstrates the d2fi computed at “scintillation-to-noise” ratio σφ∼N/σd2fi=1.0. Figure 10b and c demonstrate the same but at the “scintillation-to-noise” ratio of 0.5 and 0.1, correspondingly.

Figure 10 clearly demonstrates that the procedure of the d2fi index computation doubles the noise. The weak scintillations cannot be detected when σφ∼N/σd2fi≥0.5 (panels a and b). Therefore, the d2fi index is recommended to detect weak ionospheric phase scintillations when the low-noise receiver is employed and the condition σφ∼N/σd2fi<0.5 is satisfied.

5. Conclusions

The rocketing development of GNSS, including new-generation satellites, signals, and receivers, presents a wealth of new research opportunities in the study of the ionosphere. However, the full potential of these technologies for studying the atmosphere, ionosphere, and other interdisciplinary fields has not been fully realized yet. To realize all the abilities of modern GNSS as a global and high-precision tool for ionosphere remote probing, one needs to work out the corresponding novel methodologies, methods, and procedures of GNSS data analysis. Such a sophisticated analysis has to take into account the following features of the modern GNSS technologies:

• improved correlation properties of new PRN codes and types of a signal modulation;

• increased signal power transmitted by the new generations of satellites;

• real thermal noise of different types of GNSS receivers;

• adoptive pre-adjustments PLL and DLL units according to the remote probing requirements;

• high temporal resolution of the code/carrier recordings;

• code/phase tracking technique inside a GNSS receiver.

A GNSS receiver must not be considered as just a ‘black box’ in a process of geophysical studies anymore. We demonstrated that the very GNSS-based methods of the explorations of the ionosphere have to become the subject of research and modernization. It will bring better opportunities in ionosphere explorations thanks to the most effective utilization of the novel technological capabilities of GNSS.

To prove the aforementioned statements, the procedures of the ionospheric indices TEC, ROTI, σϕ, d2fi, and fd calculation were examined, taking into account the novel features in GNSS. We clearly demonstrated that the results of the ionospheric indices computation obviously depend on the following items:

1. time resolution of the data when to compute ROTI index;

2. cutoff frequency of the Butterworth filter and the averaging time when to compute σφ index;

3. the scintillation-to-noise rate when to compute d2fi index;

4. the type of GNSS signal components when to compute fd index.

The presented review demonstrates the efficiency and usability of the GNSS data processing procedures can be improved significantly by taking into account the novel advantages of GNSS.

Acknowledgments

The authors sincerely appreciate Dr. A. M. Padokhin from Moscow State University, who provided original figures and materials we used in Figures 2 and 3 of this manuscript.

The work is financially supported the by Russian Science Foundation (project № 23-17-00157, https://rscf.ru/pro-ject/23-17-00157/).

Conflict of interest

The authors declare no conflict of interest.