ZHU Xuefen，LIN Mengying，CHEN Xin，and CHEN Xiyuan

1. Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology of Ministry of Education，School of Instrument Science and Engineering，Southeast University，Nanjing 210096，China; 2. School of Electronic Information and Electrical Engineering，Shanghai Jiao Tong University，Shanghai 200240，China

Abstract: In the equatorial region，deep amplitude fading in global positioning system (GPS) signals frequently occurs during the strong ionospheric scintillation，it can lead to the loss of lock in GPS carrier tracking loops，and result in increased positioning error and even navigation interruption. The relationships between amplitude scintillation indices and detrended carrier frequency are investigated，based on GPS L1 C/A signals during the last peak of the solar cycle at the low latitude site of S?o José dos Campos，Brazil (23.2S，45.9W) from 2013 to 2015.Corresponding mathematic model of the probability distribution function is built for the first time to provide statistical analysis on the above relationships. The results show that the standard carrier frequencies reveal an almost linear relation with the amplitude scintillation indices. Moreover，the frequency widths of detrended frequency are proportional to levels of amplitude scintillation when the value of the peak probability is lower than the corresponding boundary. A conclusion can be drawn that different levels of amplitude scintillation will influence the fluctuation of the carrier frequency. The analysis will provide useful guidance to set the receiver’s bandwidth with respect to the different scintillation levels and design the advanced tracking algorithms to improve the robustness and precision of the GPS receiver.

Keywords: ionospheric scintillation，carrier frequency disturbance，probability distribution，bandwidth.

1. Introduction

Ionospheric scintillation refers to violent vibration of signals，which most frequently occurs in equatorial，auroral，and polar regions which are particularly suffused with free electrons with long duration [1，2]. The radio frequency (RF) signals are vulnerable and characterized by propagation nuisances of amplitude variations，phase shifts，and group delays when propagating through ionospheric plasma of electron density irregularities caused by the Fountain effect [3]. There are numerous factors influencing the incidence frequency，intensity and duration of ionospheric scintillation，including but not limited to solar activity，geomagnetic storm，local electric field，electrical conductivity，wave interaction and so on [4]，which will result in the ionosphere at low and high latitudes particularly susceptible to irregular electron density and rapid fluctuations in signal intensity and phase jittering [5].

In the equatorial region，occurrence of strong ionospheric scintillation will pose a threat to the global navigation satellite system (GNSS) receiver tracking performance，including cycle slips，phase errors and increased carrier Doppler shifts [6]. In severe cases，the receiver measurements where the deep simultaneous signal fading during ionospheric scintillation on all three signal bands rarely occurs，will be plagued with serious destruction，causing deep amplitude fading and a series of errors on the carrier tracking loop，even leading to a degradation in position and navigation solution accuracy，integrity and continuity [7]. The loss of lock on the carrier tracking loop，caused by increased Doppler frequency shift during equatorial ionospheric scintillation，may occur when the total Doppler frequency shift exceeds the preset bandwidth of tracking [8]. However，this kind of interference can rarely be modeled empirically for its sporadic and irregular appearance. Thus，the modeling of ionospheric scintillation effects should be receiver-specific and quantified to estimate the position to mitigate scintillation effects [9，10]. To improve the robust performance of position and navigation，the relationships between ionospheric scintillation indices and carrier frequency on GPS signals will be taken into consideration to reduce the loss of lock on the carrier tracking loop. Meanwhile，the mathematic model of the probability distribution function (PDF)addressed for the first time，will be proposed and calculated to explain the statistical frequency widths in relation to amplitude scintillation.

Since 2012，a GNSS data collection system at S?o José dos Campos，Brazil (23.2S，45.9W) has been collecting segments of raw GNSS intermediate frequency (IF) data at periods of high ionospheric scintillation. The GNSS-dertved total electron content (GTEC) free front-end is exploited to record the GPS IF signals by generating zero IF data streams with 8 bits resolution in-phase (I) and quadrature-phase (Q) samples at a complex sampling rate of 20 MHz. In this paper，raw IF GPS L1 data from 2013 to 2015 are processed to investigate deep amplitude fading across GPS L1 signals. A custom software defined receiver (SDR) is applied to ensure robust tracking of GPS signals throughout strong scintillation events. The highlight of this study is that statistical results have been obtained concerning the relationships between scintillation indices and detrended carrier frequency on GPS L1 signals. The amplitude scintillationS4 index，the histogram of the detrended carrier frequency and the relationships between the carrier frequency andS4 are obtained for each satellite. The histograms of the detrended carrier frequency of all satellites data are then presented with respect to different values ofS4.

The rest of the paper is organized as follows: Section 2 shows the data collecting process and a brief description on ionospheric scintillation indices and filtering procedures. Section 3 presents scintillation results in relation to scintillation indices and carrier frequency for each satellite. Corresponding PDFs are calculated with respect to the amplitude scintillationS4 index and the detrended carrier frequency in Section 4. A conclusion and recommendations for future work are given in Section 5.

2. Ionospheric scintillation indices

2.1 Data collection

The GPS data processed in this paper were collected at S?o José dos Campos，Brazil (23.2S，45.9W) where the phenomenon of ionospheric scintillation is extremely active for its geographical location close to both the South Atlantic magnetic anomaly (SAMA) and the equatorial ionization anomaly (EIA) [11]. Months of data from 2013 to 2015 were recorded based on strong ionospheric scintillation detected on GPS signals during each hour of detection. As a kind of natural phenomenon，the events of ionospheric scintillation are normally unpredictable.Thus，a Septentrio PolaRx ionospheric scintillation monitoring (ISM) receiver was utilized to monitor the scintillation activity. Fig. 1 shows a schematic diagram data collection system driven by ionospheric scintillation events. The wideband antennas are split into several ports connected to commercial ISM receiver Septentrio PolaRx and SDR-based RF front ends. Scintillation-related measurements includingI/Qchannel correlation values as well as carrier phase will be collected incessantly by the ISM receiver，while relevant scintillation indices and event indicators are calculated simultaneously with respect to collected measurements [12]. Furthermore，the indicators will be compared with the threshold values preset before data collecting to trigger the data server to record raw IF samples generated by the SDR front ends[13，14]. It should be mentioned that only the data affected by the natural ionospheric scintillation events will be recorded to economize a large number of memory spaces.These data can be utilized to research on the optimal algorithm with respect to ISM receiver，as well as providing database for the analysis of strong scintillation characteristics. Besides，on the processes of acquisition and tracking，the delay locked loop (DLL) and phase locked loop (PLL) pull-in noise bandwidths are set as 2 Hz and 25 Hz respectively，while the pull-in time is set as 500 ms. Once the signals are locked and the tracking loop achieves stable state，the DLL and PLL pull-in noise bandwidths are reset to 1 Hz and 10 Hz respectively to reduce the influence of noise and other interferences.

Fig. 1 Architecture of the scintillation event-driven data collection system

The GPS L1 C/A signal recorded from the GTEC free front ends which were configured to collect zero frequency data with 8 bits resolution samples at a complex sampling rate of 20 MHz，was selected to carry out the study. The collecting period lasted from March 2013 to February 2015，including the latest peak period of the solar activity. Furthermore，these available data were processed by the SDR receiver advanced by the combination coherent/con-coherent integration acquisition algorithm to enhance the acquisition and tracking performance under strong ionospheric scintillation environment. As the scintillation events normally occur during night-time，the equatorial data were normally collected from universal time coordinated (UTC) 00:00:00 to 03:00:00 and lasted for one hour. The operation days of data used in the following analysis are listed in Table 1. The available pseudo random noise codes (PRNs) used in following research are shown.

Table 1 Hours of GPS L1 C/A available data from the collection system at S?o José dos Campos，Brazil

One hour of data from 2013 to 2015 were selected for analyzing details. During each data collecting period，several satellites were acquired and tracked. However，only approximate four satellites were chosen and available under the condition of elevation above 30° which can mitigate the effects of multipath interference.

2.2 Scintillation indices procedures

There are several relevant observables frequently used to quantitatively analyze characteristics of the GNSS signals during ionospheric scintillation，revealing the severity of scintillation events as indicators. As the most commonly used two observables，the amplitude scintillation indexS4 refers to a magnitude of amplitude scintillation defined as the normalized standard deviation of the signal intensity [15]，while the carrier phase scintillation index is defined as the standard deviation of the detrended carrier phase measurements [16]. The calculations of the above two indices are presented as

whereMrepresents the number of noiseNi，andI/Qsamples are used to measure the wide band power (WBP)and narrow band power (NBP). Furthermore，the difference of N BP and W BP is calculated to explore the connection between N BP ，W BP and the raw signal intensity S Iraw[17].

The raw signal intensity S Irawis basically proportional to the difference of N BP and W BP. It is worth mentioning that the process of detrending refers to remove the fluctuation caused by environmental noise，multipath effect and other factors [18，19]，so that ionospheric scintillation activities can avoid being incorrectly represented by two indices. There are several available detrending methods such as discrete wavelet filtering，continuous wavelet filtering，Butterworth filtering，and polynomial fitting [17，20]. To eliminate the influence of high frequency noise，S Irawmust be normalized. Divided by the detrended signal intensity S Itrendusing a 4th order polynomial fitting on S irawto remove the noise contribution on signal intensity，we can obtain the normalized signal intensity S Inorm=SIraw/SItrend. The SInormwill be not affected by the variable ofMor differents scale factors cancelled out by the numerator and denominator. In this paper，Mis set to 40 for that GPS navigation data rate is 50 Hz，while the product ofMand the coherent integration time should be equal to multiples of 20 ms [17].Meanwhile，numerous experiments with different integration time intervals based on GPS L1 strong scintillation signals have proved that 40 blocks of 1 ms correlator output accommodate sufficient energy to generate measurements with distinct characteristics [21]. Its normalized standard deviation is defined as the amplitude scintillation indexS4raw.

where 〈 ·〉 stands for the average value over the interval.The interval time is set to 10 s and combines with the moving window for 1 s. To remove the contributions from the ambient noise to S Iraw，the normalized standard deviation of noise is defined asrepresents the average value of the noise ratio with the interval time of 10 s.Combining the above equations，the finalS4 can be obtained as

To better reflect the characteristics of equatorial strong ionospheric scintillation，the interval time of (5) is set to 10 s in this paper，and a sliding averaging window is combined to adjust the rate of two indices from 0.1 Hz to 1 Hz.

Another conventional indicator representing the fluctuation of the carrier phase is σφ[22]，which is calculated as

where φ represents the carrier phase. It should be noted that the raw value φ should be detrended before calculation. Different from the low pass filtering process of calculation of the amplitude scintillation index，the high pass filter with a cutoff frequency of 0.1 Hz is utilized to remove the effect of low frequency noise on the phase observable. Besides，the 6th order filter is implemented by three cascaded 2nd order filters in this paper to enhance the stability of filtering response [17]. However，the details of this indicator will not be further presented in remainder analysis for more frequent occurrences of strong scintillation in the polar region than that near the equatorial region.

Moreover，to analyze the relationship between scintillation indices and the carrier frequency，the raw carrier frequency will be processed primarily in the same method as carrier phase to remove interference of other factors. Fig. 2 shows an example of the detrending process of the carrier frequency measurement based on GPS L1 C/A signals for PRN 11 observed from 00:00 to 01:00 on March 26，2013 at S?o José dos Campos，Brazil. During the period，apart from the trend of the carrier frequency shown in the left panel，its corresponding detrended frequency and standard values are also presented in the right panel. The data frequency of the detrended carrier frequency remains consistent with the raw carrier frequency of 1 000 Hz. With the same processing procedure of the amplitude scintillation indexS4，the value of the standard frequency is counted each 10 s and the data rate is 1 Hz. It should be mentioned that the data of filter distortion which occurs in the first 25 s and results in the abnormal phenomenon on the standard deviation，will be ignored and illustrated in the following analysis.

Fig. 2 Detrending process of carrier frequency measurement

3. Relationship on carrier frequency and amplitude scintillation

This section focuses on the analysis of the amplitude scintillation index and its relation with the standard carrier frequency based on the data mentioned in Table 1，to investigate the relationship between amplitude scintillation and the carrier frequency in equatorial regions. Under the condition of the elevation angle mask exceeding 30°，data of the four satellites for each year from 2013 to 2015 will be analyzed as representatives.

To explore the influence extent of the carrier frequency by different intensities of amplitude scintillation initially，F(xiàn)ig. 3(a) and Fig. 3(b) show the fluctuation of the amplitude scintillation indexS4 with collecting time and the standard carrier frequency measurement in relation toS4，respectively. In three panels，signal data on each satellite are influenced by amplitude scintillation in varying intensities，while the standard carrier frequency measurement has an approximately linear relation with correspondingS4 overall. Moreover，comparing PRN25 and PRN29 in Fig. 3(a) as well as PRN11 and PRN31 in Fig. 3(c)，it can be noticed that the divergence of discrete points for each satellite reveals increasing with pronounced fluctuation of amplitude scintillation. In addition，it should be mentioned that there are a small amount of red points in the left three panels and some of red points are located in the anomaly. This phenomenon which performs particularly evident in the right panel of Fig. 3(c) is resulted by the filter distortion mentioned in Section 3.Thus，the first 25 s of data will be discarded in the following analysis to eliminate the influence of filter distortion.

To further exploit the relationship between the detrended frequency and various levels of amplitude scintillation，PRN25 on November 18，2013，PRN31 on February 27，2015 and PRN19 on January 31，2015 are selected to be analyzed and the histograms of the detrended frequency based on 10 levels of amplitude scintillation are presented in Fig. 4. The detrended frequency range is set from -6 Hz to 6 Hz for the reason of lower occurrences on other frequencies with the frequency step of 0.2 Hz.The value ofS4 and the detrended frequency on each second will be defined as an event of occurrence on each segment of one hour data without the first 25 s.

Fig. 3 Examples of the fluctuation of S4 and its relation to standard carrier frequency measurement after detrending

On each level of amplitude scintillation，the trends of occurrences of the detrended frequency are characterized by the Gaussian distribution in the three panels. Meanwhile，comparing with Fig. 4(a) and Fig. 4(c)，it can be observed obviously in Fig. 4(b) that the range of the detrended frequency increases as the rise of amplitude scintillation. This phenomenon reveals that the occurrence of stronger scintillation may cause deeper fluctuation of the carrier frequency. Moreover，the overall trends of the following histograms are fitted to the mathematical model of two terms of Gaussian functions which are plotted by red solid curves，while fitting effects of R-squares are marked on each panel，reaching approximately 99%. The mathematical formal of the fitting process is shown as

wherePstands for the polynomial function of occurrences of various levels of the detrended frequency shown in Fig. 4，frepresents the detrended carrier frequency，a1anda2are the coefficients，μ1，μ2and σ1，σ2stand for the mean value and the standard deviation，respectively. The comparison of Gaussian fitting effects between one term and two terms on three satellites shown in Fig. 4，is listed in Table 2. The coefficient of the determination R-square and the root mean squared error(RMSE) can be combined to evaluate the fitting performance. The value of R-square closer to 1 and the smaller RMSE will generate more corresponding fitting result. It can be obtained from Table 2 that both coefficients perform better on fitting of the three segments of data under the condition on two terms of Gaussian fitting than that on one term. The value of R-square on PRN31 on February 27，2014 is particularly closer to 1 on two terms. Thus，the Gaussian fitting of the two terms model will be utilized in the following statistical analysis.

Fig. 4 Histograms of various levels of amplitude scintillation as functions of detrended carrier frequency

Table 2 Comparison of Gaussian fitting effects

However，in order to avoid loss of lock on the carrier tracking loop caused by the total carrier Doppler frequency exceeding the carrier tracking loop bandwidth，the relationship between scintillation indices and the carrier frequency needs to be quantitative to provide an appropriate frequency for the loop bandwidth.

4. PDF of amplitude scintillation and detrended carrier frequency

In order to obtain conclusions with quantitative and statistical significance，further research will be achieved in this section after the image analysis of each single satellite in the previous section. The data of all satellites will be combined and divided into four groups based on amplitude scintillation to analyze the characteristics of GPS carrier frequency disturbance distribution.

The levels of amplitude scintillation data are divided into three groups in the following four cases in this paper:S4≤0.2，0.2＜S4≤0.5，0.5＜S4≤0.8andS4＞0.8. The frequency step width is set as 0.2 Hz，while the data of distribution of the detrended frequency on each level of amplitude scintillation are superimposed by occurrences of four satellites in each collecting date and the elevation of all satellites on three data collecting periods remain above 30°. The histograms of the detrended carrier frequency on four levels of amplitude scintillation are shown in Fig. 5.

Comparing the three periods of bars above for four levels of amplitude scintillation，the corresponding occurrences of the detrended frequency show diversity on different levels ofS4. To explore further quantitative relationship between the detrended frequency and four levels of amplitude scintillation，the PDF is proposed for the first time to normalize the parameter of occurrence of the detrended frequency so that all groups of data can be analyzed on the same order of magnitude.

Fig. 5 Histograms of detrended carrier frequency versus four levels of amplitude scintillation

The case of four levels of amplitude scintillation has been chosen for further analysis and the plots of probability distribution and the fitting curves corresponding to each segment of data are presented in Fig. 6，while the frequency step width is defined as 0.2 Hz. The fitting PDFs with respect to the above fitting curves are based on (7)mentioned in Section 3. For each PDF，the relevant parameters of fitting effects have been presented in Table 3.

Fig. 6 Probability of detrended carrier frequency disturbance distributions and two terms of Gaussian fitting curves on four levels of amplitude scintillation S4≤0.2，0.20.8

Table 3 Two terms of Gaussian fitting effects of three segments of data on four levels of amplitude scintillation

It can be noted that the trends of fitting curves are almost consistent with the real measured data trends with all the R-squares exceeding 0.9. Thus，the fitting model proposed in Section 3 is further proved reasonable and shows excellent performance. It is worth mentioning that the measured points marked with four different shapes are calculated by the way of dividing occurrences by the sum of themselves of each group of data shown in Fig. 5.

It seems that each pair of measured data and the corresponding fitting curve keeps similar with the others.However，they show apparent differences on data analysis. After the probability of detrended carrier frequency disturbance distribution can be fitted by the fitting model，the frequency width based on the ionospheric scintillation of varying intensity will be calculated accurately so that the receiver bandwidth can be set precisely during the design of the receiver carrier tracking loop. As is shown in each panel，the three boundaries of the peak probability reveal that the corresponding frequency width keep proportional with levels of amplitude scintillation.To take the data on November 18，2013 in the top panel，under the boundary condition of 0.08，the frequency width on level ofS4≤0.2 will be less than that on level of 0.2＜S4≤0.5，which is less than that on level of 0.5＜S4≤0.8 and so on. Meanwhile，it can be roughly observed that the differences will be more evident when the value of the peak probability is smaller.

Table 4 shows the frequency widths on the three segments of data，four levels of the peak probability boundary and four levels of amplitude scintillation，respectively. Varying with the peak-amplitude of probability，the frequency bandwidth may present different relation with the amplitude scintillation. Thus，for each hourly period，four peak probability boundaries are set to compare the variations of frequency bands on four levels of amplitude scintillation. It means that apart from the boundary of the peak probability marked in each panel shown in Fig. 6，other three values of the peak probability of 0.003，0.002 and 0.001 are chosen simultaneously for further analysis.

Table 4 Frequency widths on three segments of data，four levels of peak probability boundaries and amplitude scintillation

It can be obtained from Table 4 that on the condition of certain collected data and value of the probability，frequency widths present incremental progressively as levels of amplitude scintillation. Meanwhile，the smaller the value of the peak probability，the larger the frequency width will be on the same level of amplitude scintillation.Thus，it can be preliminarily inferred that to some extent，the levels of amplitude scintillation show superiority of proportional relation with the frequency bandwidth and provide reasonable basis for setting different bandwidths under different amplitude scintillation indicators.

In addition，the duration of ionospheric scintillation phenomena lasts from normally half an hour to several hours，so the frequency bandwidth as feedback to receiver bandwidth can be updated every fixed period of time such as 10 min or shorter，which can be determinated by various experimental measurements.

5. Conclusions and recommendations

The relationship between the GPS carrier frequency disturbance distribution and the amplitude scintillation index in the environment of strong ionospheric scintillation during the equatorial region is analyzed in details，based on segments of raw GPS intermediate-frequency IF data at periods of high ionospheric scintillation collected at S?o José dos Campos，Brazil (23.2S，45.9W) from 2013 to 2015. The numerous hours of data and three segments of one hour data were collected in 2013，2014 and 2015，respectively，as representatives for analysis. The carrier frequency，as well as its corresponding standard and detrended frequency are the main variables investigated in relation to various levels of amplitude scintillation. Combining with the linear relation between the amplitude scintillation index and the carrier frequency，the fitting model of Gaussian PDF is proposed to fit the probability of carrier frequency disturbance distribution based on various levels of ionospheric scintillation. The results of frequency widths calculated precisely by proposed PDFs can provide scientific basis for research on the carrier tracking algorithm with better performance in the case of strong ionospheric scintillation in low latitude areas to provide an efficient means to adjust the receiver bandwidth during the process of acquisition and tracking，further reaching the purpose of improving the positioning performance of the GPS software receiver. The following conclusions can be substantiated by means of quantitative analysis.

(i) In the equatorial region，strong amplitude scintillation can be detected frequently and corresponding standard carrier frequency presents linear relation with the amplitude scintillation overall，based on removing the abnormal data caused by filter distortion and multipath interference.

(ii) For each single satellite，its data of histogram shows a trend of Gaussian curve，while the two terms of Gaussian fitting show excellent consistence with the fitted data. Comparing with one term of Gaussian fitting，the two terms of fitting contribute to pronounced effects on fitting performance which is especially apparent for some satellites，with the coefficient of determination R-square reaching approximately 0.99.

(iii) On the same data collecting period，all histograms of satellites，which are above 30° superimposed to be fitted to Gaussian function，will result in the frequency width proportional to four levels of amplitude scintillation when the corresponding value of the peak probability is defined appropriately. Moreover，lower the value of the peak probability trends to more apparent relation between frequency widths and levels of amplitude scintillation.

These above conclusions drawn from elementary data analysis，can be of great importance for a better understanding of the relationship between the carrier frequency and amplitude scintillation in the equatorial region. Furthermore，it can also contribute to the acquisition and tracking process and the design of advanced tracking algorithms to improve the robustness and precision of the GPS receiver.

Besides，for the reason that phenomenon of ionospheric scintillation is normally influenced by the time period，season，region and other factors which need to be taken into consideration，the fitting model of Gaussian PDF will be further improved to enhance the fitting effect in various environments. Meanwhile，more mathematical models will be considered to adapt to as various ionospheric scintillation events as possible.

Acknowledgment

The experimental data used in this paper was supported by Prof. Dennis M Akos at Department of Aerospace Engineering Science，University of Colorado，Boulder，USA. Special appreciation is given to Prof. Jade Morton and Prof. Dennis M Akos at Department of Aerospace Engineering Science，University of Colorado，Boulder，USA for their kind technical guidance and assistance.