# Measuring phase scintillation at different frequencies with conventional GNSS receivers operating at 1 Hz

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## Abstract

Ionospheric scintillation causes rapid fluctuations of measurements from Global Navigation Satellite Systems (GNSSs), thus threatening space-based communication and geolocation services. The phenomenon is most intense in equatorial regions, around the equinoxes and in maximum solar cycle conditions. Currently, ionospheric scintillation monitoring receivers (ISMRs) measure scintillation with high-pass filter algorithms involving high sampling rates, e.g. 50 Hz, and highly stable clocks, e.g. an ultra-low-noise Oven-Controlled Crystal Oscillator. The present paper evolves phase scintillation indices implemented in conventional geodetic receivers with sampling rates of 1 Hz and rapidly fluctuating clocks. The method is capable to mitigate ISMR artefacts that contaminate the readings of the state-of-the-art phase scintillation index. Our results agree in more than 99.9% within ± 0.05 rad (2 mm) of the ISMRs, with a data set of 8 days which include periods of moderate and strong scintillation. The discrepancies are clearly identified, being associated with data gaps and to cycle-slips in the carrier-phase tracking of ISMR that occur simultaneously with ionospheric scintillation. The technique opens the door to use huge databases available from the International GNSS Service and other centres for scintillation studies. This involves GNSS measurements from hundreds of worldwide-distributed geodetic receivers over more than one Solar Cycle. This overcomes the current limitations of scintillation studies using ISMRs, as only a few tens of ISMRs are available and their data are provided just for short periods of time.

## Keywords

Phase scintillation index Ionospheric scintillation Global Navigation Satellite System (GNSS) Ionospheric scintillation monitoring receiver (ISMR) Geodetic receiver Cycle-slip detection## 1 Introduction

The Earth ionosphere is defined as the upper part of the atmosphere (at an altitude comprised between 60 and 2000 km), where ions and free electrons are present in quantities sufficient to affect the propagation of radio waves (Institute of Electrical and Electronics Engineers Standard 211 1997). Ionospheric scintillation occurs when Global Navigation Satellite System (GNSS) signals experience fast fluctuations, when they are refracted or diffracted by irregularities of the electron distribution along their propagation paths (Kintner et al. 2007). These irregularities are present at equatorial and high latitudes, predominantly in the F layer at altitudes comprised from 250 to 400 km, but also in the E layer at high latitudes (Prikryl et al. 2016) with altitudes ranging from 90 to 120 km (Aarons 1982). Ionospheric perturbations affecting GNSS are associated with space weather events (such as geomagnetic storms) at high latitudes, and associated with plasma bubbles after the sunset at low latitudes (Juan et al. 2018a).

*L*

_{f}can be decomposed as (Sanz Subirana et al. 2013):

\( I_{f}^{r} \) is the refractive ionospheric effect at frequency \( f \), which can be eliminated up to 99.9% with the dual-frequency ionosphere-free (IF) combination, which is commonly used in the precise point positioning (PPP) method (Zumberge et al. 1997).

\( I_{f}^{d} \) is the diffractive ionospheric effect at frequency \( f \). In low-latitude regions, ionospheric irregularities with a size close to the Fresnel length for GNSS frequencies, which is 400 m, can scatter the signal into multiple paths producing signal diffraction (Kintner and Humphreys 2009). The diffractive effects can be observed as rapid fluctuations in both carrier-phase and signal amplitude, losses of lock, and frequent cycle-slips (Carrano et al. 2013). Unlike the ionospheric refraction, the diffraction is not proportional to the inverse squared frequency. Thus, diffractive effects cannot be eliminated with the IF combination and degrade the accuracy of highly accurate GNSS positioning under severe scintillation conditions (Béniguel et al. 2009).

In order to measure the scintillation of GNSS signals, it is common to use a special type of equipment termed ionospheric scintillation monitoring receiver (ISMR). Thanks to the high sampling rate (SR), typically 50 Hz, ISMRs are able to track signals experiencing rapid phase variations due to scintillation. Moreover, ISMRs are equipped with ultra-low-noise Oven-Controlled Crystal Oscillators that are more precise and stable than the internal clocks equipped in conventional geodetic receivers, such as those used in the International GNSS Service (IGS) network (Beutler et al. 2009).

ISMRs provide two types of scintillation indices. The first one is the amplitude scintillation index, denoted as \( S_{4} \), defined as the standard deviation of the signal intensity normalized by its mean (Briggs and Parkin 1963). In the current work, we focus on the second one, which is the phase scintillation index, denoted as \( \sigma_{{\varphi_{f} }} \).

In order to compute \( \sigma_{{\varphi_{f} }} \), the first step consists in detrending \( L_{f} \) into \( \varphi_{f} \). That is, to apply a high-pass filter (HPF) to \( L_{f} \), typically a sixth-order Butterworth (Van Dierendonck and Arbesser-Ratsburg 2004), with a cut-off frequency of \( f_{\text{c}} = 0.1\,{\text{Hz}} \). The HPF cancels out all low-frequency components caused by the variation of receiver–satellite geometry \( \rho \) and the tropospheric delay \( {\text{Tr}} \) or even variations in the hardware delays associated with temperature (Zhang et al. 2017). Therefore, the HPF isolates high-frequency effects such as carrier-phase fluctuations associated with ionospheric scintillation in \( I_{f} \).

This contamination is continuous in time, and it is the reason why, up to now, conventional receivers are not used to compute the phase scintillation index \( \sigma_{{\hat{\emptyset }_{f} }} \).

The second source of contamination of \( \sigma_{\varphi f} \) and \( \sigma_{{\hat{\emptyset }_{f} }} \) in ISMR and conventional receivers occurs during scintillation. The carrier-phase tracked by the receiver may experience variations on the ambiguity \( N_{f} \) present in the carrier-phase measurements, named cycle-slips (Takasu and Yasuda 2008; Liu et al. 2018; Juan et al. 2018b). These changes are not necessarily discontinuities, see, for instance, Fig. 6 in Juan et al. (2018b). Indeed, transition between integer cycles can last several seconds, so they are difficult to detect. If cycle-slips are not detected, the HPF of the ISMR cannot filter out high-frequency parts caused by cycle-slips. As a result, erroneous values of \( \sigma_{{\varphi_{f} }} \) can be calculated. Unlike the clock fluctuations, the effect of cycle-slips remains as a challenge for ISMRs and conventional receivers.

Thus, ROTI is calculated as the standard deviation of \( \dot{L}_{\text{GF}}\), i.e. \( {\text{ROTI}} = \sigma_{\tau } \left( {\dot{L}_{\text{GF}} } \right) \), for a moving window of \( \tau \) samples. A typical value of \( \tau \) is 300 s when the SR is 30 s.

Notice that all frequency-independent terms are eliminated in (4), including the tropospheric effect, the receiver clock \( \delta t_{\text{rec}} \) and satellite clock \( \delta t^{\text{sat}} \). In this way, one can have a straightforward sampling of scintillation without requiring a stable receiver clock. However, ROTI presents some drawbacks with respect to \( \sigma_{{\varphi_{f} }} \).

First, unlike \( \sigma_{{\varphi_{f} }} \), ROTI measures the scintillation effect in the GF combination of \( L_{1} \) and \( L_{2} \). But, as it is shown in (Bhattacharyya et al. 2000) and (Juan et al. 2017), when diffractive scintillation is present the scintillation effects on \( L_{1} \) and \( L_{2} \) frequencies are not proportional. Then, with ROTI, one cannot extract the scintillation on each individual frequency.

Second, miss-detected cycle-slips may cause a high value of ROTI not associated with any ionospheric fluctuation but to receiver artefacts (Juan et al. 2017). These cycle-slips are more frequent at \( L_{2} \), so large values of ROTI in low latitude can be associated with miss-detected cycle-slips. On the contrary, if cycle-slips are detected, the transitions can last several seconds and this period should be excluded from the ROTI computation, thus reducing the availability of ROTI values under scintillation conditions. This reduction would not occur, if one could isolate the ionospheric effect in \( L_{1} \), which is less affected by cycle-slips, as it is the case of \( \sigma_{{\varphi_{1} }} \).

A new scintillation index termed \( \sigma_{\text{IF}} \) was introduced in (Juan et al. 2017), computed as the standard deviation of the residuals in the IF combination of carrier-phase measurements. Because the refractive effect of scintillation is cancelled in the IF combination, \( \sigma_{\text{IF}} \) measures the diffractive effect, which is relevant to the accuracy of PPP. One of the key innovations was the estimation of the receiver clock to remove the influence of its fluctuation on the \( \sigma_{\text{IF}} \) indicator.

The current paper proposes an evolution of the technique described in Juan et al. (2017). The main advantage of the method presented in this contribution is that the scintillation effect can be studied on each frequency individually. This is a clear benefit with respect to indicators using the GF combination (e.g. ROTI) and the IF combination (e.g. \( \sigma_{\text{IF}} \)). The proposed evolution also takes benefit of the receiver-clock removal introduced in (Juan et al. 2017), which is explained with a great level of detail in this paper. Hence, the method can exploit data from conventional geodetic receivers operating at 1 Hz without the requirement of a high stable clock.

The second contribution of this study addresses the cycle-slip problem. Not only the cycle-slips are detected as in (Juan et al. 2017), but also the carrier-phases are corrected in real time, obtaining continuous measurements. Thus, the phase scintillation index can be computed despite the cycle-slip occurrence. The third contribution is the extension to multiple frequencies of the comparisons regarding the phase scintillation index values obtained with conventional receivers with respect to those readings of co-located ISMRs introduced in Juan et al. (2018b).

The paper is organized as follows. Section 2 describes the methodology. Then, we detail the data set of 8 days and the experiment design in Sect. 3. The results of the phase scintillation index using our method are compared to those of the traditional ISMR in Sect. 4. Section 5 discusses the effects of cycle-slips and satellite clock fluctuation on the computation. The summary and conclusions of the work are presented in the last section.

## 2 Methodology

### 2.1 Geodetic detrending

In this regard, the detrending with geodetic models at centimetre level of accuracy eliminates most of the effects except the receiver-clock offset \(\delta t_{\text{rec}} \), phase ambiguity \( \left( {B_{f} + \lambda_{f} N_{f} } \right) \), ionospheric effects \( I_{f} \) and measurement noise \( \epsilon_{f} \). The first two terms are addressed in Sects. 2.2 and 2.3.

### 2.2 Receiver-clock estimation

Therefore, the evolved approach can sample refractive \( I_{f}^{r} \) and diffractive \( I_{f}^{d} \) scintillation in the uncombined carrier-phase measurements.

### 2.3 Cycle-slips detector–corrector

The third step of the proposed method is to detect and correct cycle-slips occurring in the GNSS carrier-phase measurements. Cycle-slips are variations of integers, \( \Delta N_{f} \), that cause unalignment in \( r_{{L_{f} }}^{ *} \) (and hence in \( r_{{L_{\text{IF}} }}^{*} \)) in the form of jumps. Those discontinuities are proportional to the wavelength \( \lambda_{\text{f}}^{\text{IF}} \) or \( \lambda_{f} \) (recall that \( \lambda_{1}^{IF} = 48.4 \) cm and \( \lambda_{1} = 19.03 \) cm), several times greater than the fluctuation attributable to the diffractive scintillation. Indeed, \( I_{\text{IF}}^{d} \) is typically less than 20 cm and during conditions of strong phase scintillation to 1 rad, which corresponds to \( I_{1}^{d} \) of 3 cm. Thus, cycle-slips can be isolated from scintillation. Conversely, undetected cycle-slips would contaminate the scintillation measurements. The cycle-slip detection–correction approach is described hereafter.

#### 2.3.1 Cycle-slip detection

The cycle-slip occurrence is detected in the IF combination, exploiting the fact that the detrended \( r_{{L_{\text{IF}} }}^{*} \) should be flat. A predicted value of \( r_{{L_{\text{IF}} }}^{*} \) at epoch \( k \), denoted as \( \tilde{r}_{{L_{\text{IF}} }} \left( k \right) \), is computed averaging the previous \( r_{{L_{\text{IF}} }}^{*} \) during an interval of 6 s. When the difference between the actual value and the prediction, defined as \( \xi_{\text{IF}} = r_{{L_{\text{IF}} }}^{*} \left( k \right) - \tilde{r}_{{L_{\text{IF}} }} \left( k \right) \), is greater than a threshold \( \theta_{\text{IF}} = 20\,{\text{cm}} \), a cycle-slip is declared.

#### 2.3.2 Cycle-slip identification

Following the detection of one cycle-slip, we target to identify on which frequency (or frequencies) the variation of cycles \( \Delta N_{1} \) and/or \( \Delta N_{2} \) occurred. If the computation is conclusive, the cycle-slip can be corrected and the uncombined signal \( r_{{L_{f} }}^{ *} \) repaired. Otherwise, a new computation arc starts.

*i”*denotes the “

*i*th” integer candidate per frequency and ranges from \( \widehat{\Delta N}_{f}^{1} \,{\text{to}}\,\widehat{\Delta N}_{f}^{9} \). As we have two frequencies, the complete search space accounts for a total of 81 possible pairs of \( \widehat{\Delta N}_{1}^{i} \) and \( \widehat{\Delta N}_{2}^{j} \) being “

*i*” and “j” the indices for candidates at frequencies \( f_{1} \) and \( f_{2} \), respectively.

*i*,

*j*” pair, we compute the residual at epoch \( k \) subtracting the candidate integer values \( \widehat{\Delta N}_{1}^{i} \left( k \right) \) and \( \widehat{\Delta N}_{2}^{j} \left( k \right) \) to the combined \( r_{{L_{IF} }}^{*} \):

*i*,

*j*” pair that provides the minimum jump with respect to the previous six \( r_{{L_{\text{IF}} }}^{*} \) samples, i.e. before the cycle-slip was detected. For this purpose, we use the predicted \( \tilde{r}_{{L_{\text{IF}} }} \left( k \right) \) in the “

*i*,

*j*” integer search with the following criteria:

\( \left| {r_{{L_{\text{IF}} }}^{i,j} \left( k \right) - \tilde{r}_{{L_{\text{IF}} }} \left( k \right)} \right| \) is minimized;

\( \left| {r_{{L_{\text{IF}} }}^{i,j} \left( k \right) - \tilde{r}_{{L_{\text{IF}} }} \left( k \right)} \right| \le \theta_{\text{IF}} \).

The last condition guarantees that the selected pair (\( \widehat{\Delta N}_{1}^{{\min} } ,\widehat{\Delta N}_{2}^{{\min} } ) \) aligns with the previous samples within the cycle-slip tolerance previously defined. This protection is necessary, as we only evaluate ± 4 cycles from the rough initial estimation \( \Delta N_{f}^{0} \), whereas the number of integers \( \Delta N_{f} \), occurred by the cycle-slip, might fall out of the search space.

In case that a cycle-slip is detected, but no candidate pair fulfils simultaneously the previous two conditions, the identification is inconclusive. Then, a new computing arc is started with the new value of \( N_{f} \).

#### 2.3.3 Cycle-slip correction

### 2.4 Phase scintillation index

## 3 Data and experimental design

Details of the experimental campaign: selected stations, location, dates and IGS product availability

Station | Coordinates | Sunset at 18 h LT | Receiver | Tropospheric correction | Data campaign | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Geographic | Geomagnetic | UT (s) | Manufacturer | Clock | Type | SR (Hz) | Year | Day of Year | Max. \( \sigma_{{\varphi_{1} }} \)(rad) | |||

Latitude | Longitude | Latitude | ||||||||||

GLPS | − 0.7 | − 90.3 | 8.4 | 0 | Ashtech UZ-12 | Noisy | IGS | 1 | IGS-ZPD | 2014 | 083 | N/A |

FAA1 | − 17.5 | − 149.6 | − 15.11 | 14,400 | Septentrio PolaRx4 | Noisy | IGS | 1 | IGS-ZPD | 081 082 083 084 086 | 0.93 0.97 1.31 0.24 0.60 | |

FAAS | Septentrio PolaRxS | Stable | ISMR | 50 | IGS-ZPD | |||||||

JNAV | 21.0 | 105.8 | 11.4 | 39,600 | Trimble NetR9 | Noisy | IGS | 1 | gLAB | 2017 | 251 260 263 | 0.38 0.70 0.67 |

TQBS | Septentrio PolaRxS | Stable | ISMR | 50 | gLAB |

Although the geodetic detrending proposed in Sect. 2 can sample any kind of scintillation in the carrier-phase measurements, we have focused on low-latitude receivers because the equatorial scintillation affects differently each GNSS frequency, see, for instance, Jiao and Morton (2015) or Juan et al. (2017). Thus, studying this particular type of scintillation requires isolating the ionospheric effects on different frequencies as the proposed geodetic detrending does. In contrast, the effect of scintillation at high-latitude is usually proportional at different frequencies, and therefore, it can be isolated by building the GF combination of carrier-phase measurements as in (4), which is a more straightforward manner for detrending the carrier-phase measurements than the geodetic detrending proposed in Sect. 2.

The geodetic detrending has been performed using the GNSS-Lab Tool (gLAB) (Ibáñez et al. 2018). The precise satellite orbits and clocks were obtained from the final products of IGS every 900 s and 30 s, respectively. In order to crosscheck results, we have used also satellite clocks every 5 s computed by the Center for Orbit Determination in Europe (CODE), obtaining similar results.

It is assumed that close stations have common tropospheric and ionospheric effects. Tropospheric Zenith Path Delay (ZPD) data from the IGS are available for FAA1 and GLPS. ZPD data of FAA1 are also used for FAAS. The tropospheric delays of JNAV and TQBS are modelled with a centimetre-level accuracy using the nominal tropospheric delay prediction from Black and Eisner (1984) and the mapping of Niell (1996). Equivalently, the ionospheric scintillation indices \( \sigma_{{\varphi_{f} }} \) from ISMRs of FAAS and TQBS are representative of the collocated IGS receivers FAA1 and JNAV, respectively.

Table 1 lists the dates with high values of \( \sigma_{{\varphi_{f} }} \) selected in the experiment, which include moderate and strong scintillation. The high \( \sigma_{{\varphi_{f} }} \) associated with scintillation have been found in JNAV/TQBS on days 251, 260 and 263 of 2017 and in FAA1/FAAS on days from day 081 to 084 and 086 of 2014. In order to facilitate the correspondence from local time (LT) to universal time (UT), Table 1 indicates the UT at which sunset occurs, assuming 18 h LT.

\( \left. {\sigma_{{\varphi_{f} }} } \right|_{{50{\text{Hz}}}} \): Output of ISMRs, calculated as in (2) from data with SR of 50 Hz. This can be considered as the reference value;

\( \left. {\sigma_{{\hat{\emptyset }_{f} }} } \right|_{{1{\text{Hz}}}} \): Index calculated by the conventional HPF method as in (3), from RINEX data with SR of 1 Hz;

\( \left. {\sigma_{{\emptyset_{f} }} } \right|_{{1{\text{Hz}}}} \): Index by the proposed methodology described in Sect. 2 as in (20), calculated from RINEX data with SR of 1 Hz.

## 4 Results

This section presents the results of applying the procedure described in Methodology section. In order to illustrate how the process works as clearly as possible, each step of the calculus is applied to a selected subset of the data presented in Table 1. Then, we compare the capability of the final \( \left. {\sigma_{{\emptyset_{f} }} } \right|_{{1{\text{Hz}}}} \) to measure phase scintillation with respect to the state-of-the-art HPF method as \( \left. {\sigma_{{\hat{\emptyset }_{f} }} } \right|_{{1{\text{Hz}}}} \) calculated from 1 Hz RINEX data and with respect to \( \left. {\sigma_{{\varphi_{f} }} } \right|_{{50{\text{Hz}}}} \) provided directly by the ISMRs at 50 Hz.

### 4.1 Estimation of receiver-clock fluctuation

### 4.2 Correction of cycle-slips

Once the receiver-clock fluctuation \( \widehat{\delta t}_{\text{rec}} \) is estimated, we can apply (13) in Sect. 2.2 to obtain a receiver-clock-free residual in the IF combination, \( r_{{L_{\text{IF}} }}^{ *} \). After this operation, two effects remain: carrier-phase ambiguities and diffractive ionospheric effect. If the geodetic detrending is accurate enough, cycle-slips can be identified as jumps larger than the noise of the remaining diffractive effect (Juan et al. 2017).

### 4.3 Calculation of phase scintillation index

It can be observed that the fluctuations and jumps in the receiver-clock offset, labelled as \( \widehat{\delta t}_{\text{rec}} \) in the estimation depicted in the bottom subplots of Fig. 5 cause high values and spikes in the \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\hat{\emptyset }_{2} }} } \right|_{{1{\text{Hz}}}} \), as shown in the top panels. In such cases, the ionospheric phase scintillation \( r_{{L_{1} }} \) and \( r_{{L_{2} }} \) of GPS17 cannot be properly sampled, due to the contamination of the high-noise receiver clock. On the contrary, our proposed method based on the geodetic detrending, the receiver-clock estimation, and the cycle-slip correction is capable of sampling the scintillation, as confirmed in the values of \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\emptyset_{2} }} } \right|_{{1{\text{Hz}}}} \) observed in the top panels of Fig. 5. In this manner, the proposed approach opens the door to perform climatological studies in the long term (e.g. an entire Solar Cycle) with hundreds of receivers that will contribute to a better understanding of scintillation phenomena.

### 4.4 The capability of the proposed method in comparison with the state of the art

In every panel, the black dots depict the \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) readings output by the ISMR. Those values are considered as the reference values. In Tahiti (top row), phase scintillation values up to 0.385 rad are recorded in epoch 27,420 s in day 81 of 2014, whereas in Hanoi scintillation up to 0.382 rad can be seen in epoch 45,480 s in day 251 of 2017.

We start examining the capability to sample scintillation of the phase scintillation index \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{1{\text{Hz}}}} \). That is, to apply directly (3) to carrier-phase \( \hat{\emptyset }_{1} \) detrended by the HPF without correcting the receiver clock, nor detecting cycle-slips, neither applying the geodetic detrending. In both ISMRs, Fig. 6a (FAAS) and Fig. 6c (TQBS) depict equivalent results obtained, \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \). This occurs thanks to the stable clock oscillator embedded in the ISMRs, confirmed in the bottom subplots depicting \( \widehat{\delta t}_{\text{rec}} . \) In contrast, the results of \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{1{\text{Hz}}}} \) obtained from data of conventional geodetic receivers clearly fail to detect scintillation. Figure 6b (FAA1) and Fig. 6d (JNAV) depict continuous \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{1{\text{Hz}}}} \) values of 1.3 rad in FAA1 and 0.3 rad in JNAV. As in the previous case of GLPS depicted in Fig. 5, fluctuations of the receiver clocks, \( \widehat{\delta t}_{\text{rec}} \), mask the ionospheric scintillation.

The results obtained in the geodetic receivers by the proposed method based on the receiver-clock removal are examined hereafter. It can be observed how \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) produces significantly different values in scintillation and non-scintillation periods in the geodetic receivers Fig. 6b (FAA1) and Fig. 6d (JNAV). Thus, \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) can correctly identify the scintillation, even using a geodetic receiver with an unstable clock. In Tahiti, Fig. 6b shows an excellent agreement of \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) in FAA1 with the \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of its collocated ISMR FAAS. In Hanoi, Fig. 6d shows that in non-scintillation periods, the \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) of the proposed method in JNAV is approximately 0.03 rad (1 mm) larger than the \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of its collocated ISMR of TQBS. Other few non-negligible differences between \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) of the proposed method and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) from ISMRs have also been identified. We discuss these differences of the two methods with a statistical analysis next Sect. 4.5.

### 4.5 Statistics of the difference between \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \)

Differences between the phase scintillation indices \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \): percentiles, epochs computed and outliers found

Receiver pair | Total number of epochs | Differences | |||
---|---|---|---|---|---|

Phase scintillation index | Percentile 68th | Percentile 95th | Outliers \( \left| {\sigma_{{\emptyset_{1} }} - \sigma_{{\varphi_{1} }} } \right| > 0.05 \,{\text{rad}} \) | ||

FAA1 and FAAS | 253,440 | \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}}^{\text{FAAS}} - \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}}^{\text{FAAS}} \) | 0.008 rad (0.24 mm) | 0.013 rad (0.39 mm) | 7 (0.003%) |

213,118 | \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}}^{\text{FAA1}} - \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}}^{\text{FAAS}} \) | 0.006 rad (0.18 mm) | 0.014 rad (0.42 mm) | 4 (0.002%) | |

JNAV and TQBS | 146,879 | \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}}^{\text{TQBS}} - \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}}^{\text{TQBS}} \) | 0.007 rad (0.21 mm) | 0.015 rad (0.45 mm) | 66 (0.045%) |

185,757 | \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}}^{\text{JNAV}} - \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}}^{\text{TQBS}} \) | 0.020 rad (0.60 mm) | 0.040 rad (1.20 mm) | 295 (0.159%) |

We start the comparison at the same ISMR: Fig. 8a for FAAS and in Fig. 8c for TQBS, which corresponds to the second and fourth columns in Table 2. Over 99% of the \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) values computed by the proposed method agree within 0.03 rad (1 mm) of the reference \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) values. A total of 11 outliers (less than 0.1%) have been found. The reason of this observed disagreement is full- or half-cycle-slips occurring in the 50 Hz data that appear as data gaps in the 1 Hz RINEX file of the ISMR. These cases are discussed in Sect. 5.1 with more details.

We continue the comparison by looking at the values of the conventional geodetic receivers and their co-located ISMR. Figure 8b depicts an excellent agreement between \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{1Hz} \) values of FAA1 and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of FAAS for scintillation and non-scintillation conditions. The fourth row of Table 2 confirms this finding, with a difference smaller than 0.015 rad (0.45 mm) at the 95th percentile.

At the other geodetic/ISMR pair, Table 2 reads that the 95th percentile of the difference between the \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) of JNAV and the \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of TQBS is 0.04 rad (1.20 mm). This larger discrepancy is attributable to the difference between the measurement noise of the receiver of JNAV and that of TQBS. Indeed, according to Table 1, JNAV and TBQS are equipped with a Trimble and a Septentrio receiver, respectively. In contrast, FAAS and FAA1 are equipped with Septentrio receivers.

The analysis finishes with observing the \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) values higher than 0.3 rad in Fig. 8c, d that correspond to satellites GPS28 of day 260 (red triangles) and GPS20 of day 263 (green squares). Both indices \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of these particular days are influenced by fast satellite clock fluctuations rather than by ionospheric scintillation. The black pentagons indicate \( \sigma_{{\emptyset_{1} }} \) values from other satellites in view that are contaminated by these rapidly fluctuating satellites. For instance, Fig. 8d shows that the difference of \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) of JNAV and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) of TQBS increases up to 0.1 rad for \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) and \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) values smaller than 0.2 rad, that is, in the absence of scintillation conditions. The effect of unmodelled satellite clock is discussed in detail in Sect. 5.3.

From the previous analysis, one can conclude that \( \sigma_{{\emptyset_{1} }} \), computed with the new method, and \( \sigma_{{\varphi_{1} }} \), provided by the ISMR, are nearly equivalent, being the 95th percentile of the differences below 0.04 rad and the largest differences are due to outliers which represent less than 0.02% of the comparisons. We will analyse these outliers in the next section.

## 5 Discussion

This section analyses the discrepancies previously observed in the results of the phase scintillation indices. In particular, it assesses the effect of cycle-slips observed in the ISMR readings, which are considered as the reference values. Second, it analyses the effect of different SR in the calculation of the indices. The section ends with a discussion of the contamination of phase scintillation indices in the presence of satellite clock fluctuations that affect ISMRs and geodetic receivers using the proposed approach.

### 5.1 Effect of uncorrected cycle-slips in the index \( \sigma_{{\varphi_{1} }} \) of ISMR

This subsection discusses the first origin of the outliers found in Fig. 8. That is, how undetected cycle-slips contaminate the phase index \( \sigma_{\varphi } \) output by ISMR that we consider as a reference value. In order to ease the explanation of the phenomenon, we use the previous case depicted in Fig. 4, where several cycle-slips are detected in \( L_{5} \) for satellite GPS24 in the ISMR FAAS on day 81 of 2014.

### 5.2 Effect of half-cycle-slips in the 50 Hz data

The uncorrected half-cycle jump affects the value of \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) at epochs 21,060 s and 21,120 s. Note that although the jump occurred at epoch 21,055 s, the effect lasts until epoch 21,120 s due to the HPF. In order to confirm this finding, the index \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{50{\text{Hz}}}} \) of the conventional HPF method as in (3) has been computed with the 50 Hz data that contains two half-cycle jumps. The results show that both the computed \( \left. {\sigma_{{\hat{\emptyset }_{1} }} } \right|_{{50{\text{Hz}}}} \) and the ISMR reference \( \left. {\sigma_{{\varphi_{1} }} } \right|_{{50{\text{Hz}}}} \) with the half-cycle-slips present values up to 0.4 rad higher than those of \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) computed as in (20) from 1 Hz data.

### 5.3 Effect of satellite clock fluctuation

This phenomenon also contaminates the estimation of the receiver clock in (12). The reason is that ROTIM calculus in (9) cancels the satellite clock, thus the measurements with mismodelled satellite clock are not down-weighted in (11). Consequently, this erroneous receiver-clock estimate is propagated and contaminates the estimated ionospheric fluctuation \( r_{{L_{1} }} \) of all other satellites in view and their scintillation measurements. An example of this contamination is depicted in right panel of Fig. 11, where the \( \left. {\sigma_{{\emptyset_{1} }} } \right|_{{1{\text{Hz}}}} \) for satellite GPS19 is artificially increased by the fluctuations of GPS28 during the aforementioned 11 min.

A possible protection against erroneous readings of the phase scintillation index due to high-frequency effects on other satellites is to use a geometry-and-clock-free index, e.g. ROTIM, as an extra indicator. This is the reason to include the ROTIM values in the two upper panels in Fig. 11. In both cases, low ROTIM values can be used to identify false scintillation due to mismodelled fluctuation of non-dispersive effects such as the satellite clock. In this way, when a particular satellite presents high values of \( \sigma_{{\emptyset_{f} }} \) simultaneously with low values of ROTIM, the satellite should be thus discarded to avoid the contamination of the estimation of the receiver-clock fluctuation and subsequently \( \sigma_{\emptyset } \) of all satellites in view.

## 6 Conclusions

This paper contributes to a recently introduced approach to sense the ionospheric phase scintillation with GNSS signals collected by conventional multi-frequency geodetic receivers, operating at 1 Hz. The technique is based on an accurate geodetic modelling of carrier measurements, at the centimetre level. The method can be applied in geodetic receivers, without the requirement of high SR nor the stability of receiver clock as in ISMRs. Thanks to the GNSS growth, hundreds of geodetic receivers are available worldwide and capable of adopting the proposed approach. This fact constitutes an unprecedented frame to improve radio-navigation and ionospheric-sounding techniques, especially in Southeast Asia, the only region where all global and regional constellations of navigation satellite can be tracked.

Up to now, scintillation studies could be performed using combinations of carrier-phase measurements such as the GF in ROTI or the IF in \( \sigma_{\text{IF}} \). The proposed evolution overcomes some of the problems associated with those indicators. First, it is capable to estimate ionospheric fluctuations on each individual frequency rather than in a combination of signals. This turns very adequate when studying diffractive scintillation at low latitude, in which effects are not proportional between frequencies. Second, the accurate modelling of the carrier-phase measurements allows identifying and correcting cycle-slips which are due to receiver artefacts. Miss-detected cycle-slips contaminate the readings \( \sigma_{{\varphi_{f} }} \) provided by ISMRs or ROTIs provided by geodetic receivers.

The results of the phase scintillation index \( \sigma_{{\emptyset_{f} }} \), obtained with our evolved method, agree with the \( \sigma_{{\varphi_{f} }} \) provided by ISMRs at different frequencies. We have found some cases where mismodelled satellite clock fluctuations contaminate the phase scintillation indices measured by ISMRs and by geodetic receivers. However, using a GF index such as ROTIM, it is possible to detect and counteract mismodelled satellite clock fluctuations at a higher frequency than the cadence of precise satellite clock determinations used within the geodetic detrending.

## Notes

### Acknowledgements

This work was supported in part by the Spanish Ministry of Science, Innovation and Universities Project RTI2018-094295-B-I00, in part by the H2020 Projects BELS-PLUS and NAVSCIN, and in part by Vietnamese Government-funded project No DTDL.CN-54/16. Authors acknowledge the use of data from the International GNSS Service and project SCIONAV, ESA-ITT 1-8214/15/NL/LvH of the European Space Agency.

### Author contributions

V. K. Nguyen and J. M. Juan performed the research and wrote the paper. A. Rovira-Garcia analysed the data and wrote the paper. J. Sanz and G. González-Casado designed the research. La T. V. and T. H. Ta reviewed and improved the manuscript.

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