Discovery About GPS Positioning Regular Daily Ionospheric Scintillation Disturbances

1. Introduction

Global Navigation Satellite Systems (GNSS) have become so widely used that it's hard to imagine an industry in the national economy where they are not used. However, those people who use GNSS for obtaining high-precision results in positioning, navigation and timing (PNT) understand that high precision requires taking into account various nuances, including the impact of space weather. In this article, we will touch upon several topics related to GNSS positioning and navigation in the introduction, and in more detail about the daily regular ionospheric scintillation waves of interplanetary origin, which cause positioning accuracy errors.

Recently, some Low Earth Orbiter (LEO) systems have been rapidly developing. He, B. et al [1] explain that GNSS precise point positioning (PPP) technique has been widely used due to its high accuracy, low cost, and flexible operation. Nevertheless, its convergence time usually takes tens of minutes, which limits its applications, especially in the navigation field. There are two main reasons for this slow convergence. On the one hand, GNSS satellites’ spatial geometry change is too slow due to higher GNSS orbit altitude. On the other hand, atmospheric delay parameters need to be estimated due to a lack of accurate atmospheric delay corrections, leading to over-parameterization and simultaneously increasing the parameters’ correlation. Although a dual-frequency ionosphere-free combination can be applied to remove most effects of the ionospheric delay instead of direct estimation as unknowns, it sacrifices the number of usable observations. Fortunately, the emergence of Low Earth Orbit (LEO) satellites provides an opportunity to speed up the PPP convergence. The orbit altitudes of the LEO satellites are typically between 300 and 2000 km, which facilitates the quick variation of the satellites’ geometric distribution. Also, LEO satellites can increase the redundancy of observation data and improve the integrity of the navigation system. In addition, regional GNSS tracking networks have become commonly available, which can be utilized to generate precise atmospheric delay information at the user station. Both the LEO- and atmosphere-augmented information contribute to speeding up the PPP convergence. Recently, some LEO systems have been rapidly developing, like the Iridium system, SpaceX, OneWeb, Hongyun, and Hongyan systems. In addition, some research institutions have attempted to launch LEO satellites, such as Luojia-1 from Wuhan University, China [1]. LEO-augmented multi-GNSS PPP has been widely investigated in recent years, performing the GPS/LEO combined PPP and concluded that LEO satellites can accelerate the PPP convergence and improve its accuracy. They also compared the GPS/LEO combination with the GPS/GLONASS combination, and the results demonstrate that the LEO satellites contribute more to improving the GPS PPP performance than the GLONASS satellites [1].

He, B. et al. [1] processed the GNSS observations and simulated LEO observations under a Walker/polar mixed constellation that used to test the double augmentation PPP sample solutions. Improved convergence time is achieved by over 73% compared to GNSS constellation scenarios and over 83% compared to the LEO only. The double augmentation PPP observation equations are presented as the integrated atmosphere-augmented LEO and the atmosphere-augmented GNSS observations. The LEO constellation design is discussed. The double-augmentation PPP data simulation and solution validation is summarized in conclusion [1].

Sun X. et al [2] evaluated the International GNSS Service (IGS) 10 day continuous observations in 8 combinations of BeiDou (BDS-3) and Galileo frequencies. The dual frequency ionospheric free (DFIF) Precise Point Positioning (PPP) in static and kinematic mode were used by applying the open source Nett_Diff software produced by GNSS Analysis Center of Shanghai Astronomical Observatory [2]. The observations were selected from 60 stations of IGS Multi-GNSS Pilot Project MGEX that were equipped with receivers produced by different GNSS production companies. The satellite orbit and clock bias products obtained from Deutsche Geo-Forschungs Zentrum (GFZ) data base, the daily multi-GNSS differential code biases (DCB) from Chinese Academy of Science. For troposphere delay the Saastamoinen model used and GPT2_6w and VMF1 models applied. The PPP solution mathematics is explained. The computed results are compared to IGS Solution Independent Exchange (SINEX) daily solutions [2]. The comparison of Root Mean Square Error (RMSE ) of computed positioning components Northing, Easting,UP (NEU) are used as the main tool for evealuation of various frequency combination results. Sun et al [2] concluded that BDS-3 is competitive with Galileo. The best results gained with equal accuracy of BDS-3 only, B1C/B3I and Galileo only, E1/E5 [2].

GNSS provides users with all-weather continous high precision PNT services. However, the influence of the space weather is a factor that must be considered [3]. During geomagnetic storms a series of changes in the Earth’s magnetosphere, ionosphere and upper atmosphere affect GNSS’s positioning performance. Xing S. [3] selected three geometric storm events that occurred from September to December 2023. Utilizing the global positioning system GPS/Beidou navigation satellite system (BDS) dual system kinematic PPP experiments were conducted and the raw observational data from 100 stations worldwide was analyzed. The experimental results show that the positioning accuracy of some stations in high-latitude areas decreases significantly when using the conventional Geometry Free (GF) cycle-slip detectionthreshold during geomagnetric storms which means that GF is no longer applicable to high precision services [3].

To start talking about the potential use of GNSS on the Moon, it is necessary to mention the successful 2024 expedition of Chinese science mission Chong’e -6 to the far side of the Moon, bringing back about 2 kg of soil samples to Earth [4].

The lunar PNT is very important issue for lunar exploration currently carried out by many countries and private companies.The article [5] mainly is devoted to LuGRE mission that aims to receive GPS and Galileo signals at the Moon.

Authors characterize the lunar GNSS signal environment for PNT estimation and analize collected data for support of development of GNSS receivers specific to lunar use. This contribution investigated a case of code-based differential GNSS for two lunar orbiters as it has never been done before. In particular, it has been assessed the potential of Inter-Spacecraft Range (ISR) estimation both through theoretical simulations and Hardware-in-the-Loop (HWiL) tests using the “soon-to-be-flying NaviMoon Galileo-GPS receiver” [5]. It has been shown that commonly used terrestrial algorithms for kinematic code-based differential GNSS ranging techniques are not applicable for a scenario of GNSS receiver in lunar orbit. Using the standard code-based DGNSS algorithm, a significant bias is introduced. It’s magnitude increases with the angular separation between the two users with respect to the GNSS constellation. The results are obtained and experimentally,making use of GNSS observables generated by a GNSS receiver specifically designed for a Moon mission, through a Hardware-in-the-Loop (HWiL) set-up. The Moon receiver is called the NaviMoon and will be hosted by the Lunar Pathfinder space mission [5]. The space scenario considered for the present analysis, the architecture of the theoretical Monte Carlo simulations as well as the functional blocks of the test bench designed to test the presented techniques on the NaviMoon receiver [5]. This work shows that modelling assumptions used for linearization of the relationship between the inter-user vector and the pseudorange single and double differences taken for granted in terrestrial and LEO applications are violated in the case of lunar GNSS differential scenarios [5]. Updating the modelling assumption significantly improved the performances with respect to the terrestrial standard algorithm by reducing the bias introduced by the increasing baseline. The potential performances of inter-spacecraft GNSS ranging in space, showing that the various techniques presented seem to be lower bounded by taking the difference of the individual Single Point Position (SPP) [5]. This is physically grounded in the fact that sensor-specific errors are typically independent, enabling the more effective decoupling of instrumental biases from true atmospheric signals [6]. Space weather events such as the May 2024 Mother’s Day Superstorm, induce hazardous perturbations in the coupled thermosphere–ionosphere–magnetosphere system [7]. Filjar et al. [8] demonstrate that in the case of the short-term fast-developing geomagnetic storm, a machine learning-based environment-aware GNSS ionospheric correction model for sub-equatorial regions may provide a substantial improvement over the global mode Ning Huang et al [9] presented the analysis of the GNSS precipitable water vapor (PWV) for evaluation of the accuracy of MERRA-2 and ERA5 water vapor products under extreme rainfall conditions. The relationship between the evolution of heavy rainfall and changes in water vapor are examined. The method of GNSS PWV retrieval described and the accuracy analysis of MERRA-2 and ERA5 water vapor products under different rainfall levels is presented and compared them with ground-based precipitation time series.

If up to now this introduction has discussed the problems of GNSS data processing, then further on we will talk a little about the effects of ionospheric scintillation and their studies. Ionospheric scintillation, caused by irregularities in the ionosphere, can lead to rapid changes in signal amplitude and phase, particularly affecting GPS signals at high latitudes and during space weather events [10,11,12,13,14]. The presence of moderate to strong scintillation can double positioning errors and induce clustering effects on positioning solutions, complicating navigation tasks [11]. The positioning performance of GNSS is notably affected during severe geomagnetic storms, with studies indicating that the accuracy of precise point positioning (PPP) can be significantly reduced due to frequent cycle slips [12]. The correlation between geomagnetic storm indices and GNSS signal loss highlights the need for effective monitoring and forecasting to mitigate the impacts of space weather on navigation systems [13]. Ionospheric scintillation, can lead to rapid changes in signal amplitude and phase, particularly affecting GPS signals during space weather events [14,15,16,17]. The positioning accuracy of some stations in high-latitude areas decreases significantly when using the conventional Geometry Free (GF) cycle-slip detection threshold during geomagnetic storms, which means that the GF is no longer applicable to high-precision positioning services [18]. The need for effective monitoring and forecasting to mitigate the impacts of space weather on navigation and positioning systems is of continuous concern of many users [19,20,21,22].

Space weather events such as the May 2024 Mother’s Day Superstorm, caused by a series of interplanetary coronal mass ejections (ICMEs) and high-speed streams, induce hazardous perturbations in the coupled thermosphere–ionosphere–magnetosphere system [23,24]. Analyzing such unique events offers valuable insight into the evolution of extreme geospace dynamics [23,24,25]. The Mother’s Day storm event originated from the solar active region classified as AR13664 [26], which produced at least four M-class flares and one X1.0 flare on 8 May 2024, followed by further strong solar flares (including six X-class flares) between 9 and 11 May, launching a series of fast ICMEs. For a detailed analysis of this storm, including a list of the ICMEs and their associated flares, see Spogli et al. [12,24]. Several authors have studied the ionospheric response and the impact on GPS positioning accuracy during the tropical cyclons [27,28] and volcano eruption [29].

By comparing solar X-ray observations (RHESSI) with in situ electron data from STEREO/SEPT [30], the researchers identified patterns in electron acceleration and transport. X-ray observations provide crucial insights into solar system dynamics, including planetary auroras, solar wind interactions, and interplanetary electrons [31,32]. High-energy electrons from solar flares follow complex propagation paths, affecting their detection in space. X-ray and radio studies help track energetic electrons from the Sun into interplanetary space [33]. Interplanetary X-rays are high-energy electromagnetic emissions that originate from interactions within the interplanetary medium, including the solar wind, planetary atmospheres, and cosmic radiation. These X-rays can be produced by several mechanisms, such as solar Wind Charge Exchange (SWCX), scattering and fluorescence, high-energy particle interactions of electrons and ions from the Sun or cosmic rays producing X-ray emissions [34]. Recent studies have identified various celestial bodies, including planets and moons, as X-ray emitters, with emissions primarily resulting from solar X-ray scattering and charge exchange processes [30,31,34].

An extended dataset of Loss of Lock (LoL) events recorded by the Swarm constellation from December 2013 to December 2020, the longest ever used, and discusses the corresponding occurrence as a function of latitude, local time, season, and solar activity [35]. In addition, the analyses aim was at finding a relation between the LoL occurrence and defined values of the following two ionospheric indices: the rate of change of electron density index (RODI) and the rate of change of total electron content (TEC) index (ROTI). This kind of research was done both to characterize the background conditions of the ionosphere for such events, and to start understanding whether these events can be som The large-scale ducting of Pc1 pulsations observed by Swarm satellites and multiple ground magnetometer networks on 25 June and September 2015 [36].

Pc1 waves (Pc1 pulsations) are ultra-low-frequency (ULF) electromagnetic waves in the range of 0.2–5 Hz.They are part of the geomagnetic pulsation spectrum and are closely associated with Electromagnetic Ion Cyclotron (EMIC) waves, which play a key role in magnetosphere-ionosphere interactions [37]. Pilipenko et al [38] compare the magnetosphere-ionosphere current system with a circuit analogy, where the nonsteady field-aligned currents interact with the ionosphere in a different way depending on the ratio between the driver time scale ԏ and the Alfven field line eigenperiod TA [39]. The theoretical predictions with observational results from conjugate high latitude stations in Antarctica and Greenland during summer (7 June 2014) and winter (7 January 2014) periods performed. A disturbance is a sudden commencement (SC) pulse caused by the impact of an interplanetary shock (IS) on the magnetosphere. At high latitude when the duration of the local field line eigenperiod TA is ~5-10 min which can be longer than an SC impulse with ԏ ~ 1-2 min [38]. ehow forecast, which would be very important for Space Weather effects mitigation purposes [35].

It is still a challenge to determine the apparent relationship between ionospheric conditions and Pc1 wave occurrence [37]. Pc1 waves originate in the Earth's magnetosphere, often near the plasmapause or in regions of high plasma density [40]. They are usually triggered by solar wind pressure variations or wave-particle interactions with energetic protons. Fransia et al [40] examined a Pc1 event observed at a high-latitude station during geomagnetic quiet conditions, with the aim at looking for possible correspondences with the ionospheric response. The results can be summarized as follows: 1) correspondence is found between the Pc1 pulsation activity and the occurrence of ionospheric irregularities, as clearly showed by the analysis of ROT fluctuations; 2) the study clearly shows that signatures of EMIC waves, driven by increased solar wind pressure, can be observed also at polar latitudes, simultaneously in both hemispheres, due to their propagation in the ionospheric waveguide. Such waves, generated just inside the magnetopause, propagate through the magnetosphere and transmit as Alfven waves along geomagnetic field lines up to the auroral ionosphere. They can produce the precipitation of magnetospheric energetic electrons into the atmosphere, causing electron/ion density variations as measured by TEC fluctuations. Fransia et al believe [40] that the observations of this work, although present a very interesting result, should be substantiated by further investigations, based on a number of events and additional data (satellite and/or riometer data) to be fully explained. 3) These waves propagate along geomagnetic field lines and can reach the Earth’s ionosphere, where they influence ionospheric conductivity and electron density. Pc1 waves can modulate ionospheric Total Electron Content (TEC), leading to irregularities and affecting radio signals [40]. Geomagnetic measurements from ground and space represent the main ingredient in modelling and understanding the Earth’s magnetic field [41].

The issues of space weather related to PNT are relevant worldwide. [42,43]. The significant research centers for geomagnetism, space physics, and space weather in Europe are ESA space weather network and GFZ Helmholtz Centre for Geosciences in Potsdam, Germany. There are also studies in Latvia on the impact of space weather on GNSS measurements 44].

The second section of this study provides a brief overview of the study's raw data, describing those data from March 2015 that clearly illustrate the presence of regular daily ionospheric scintillation waves. Further analysis datasets and their reduction formulas are presented. The third section presents the results of the data analysis and reflects the characteristics of the obtained results. Additional information about the resu and compares them with geomagnetic measurements of the Pc1 waves mentioned lts is provided in Appendix A and Appendix B. The fourth section discusses the results obtained and comoare them with a Pc1 observations repoorted in the literature.

2. Materials and Methods

In this study, the hypothesis regarding the possible regular daily ionospheric scintillation disturbances affecting GPS positioning observations was tested in 46 selected month of 24^th solar activity cycle (Years 2007-2017).

GPS observation data of Latvian Continuously Operating GPS Station (CORS) network were used with an elevation cut-off angle of 15˚ for 90 second (sampling data 30 sec) intervals of kinematic post-processing. The FES2004 ocean tidal model was used, along with correction of the solid Earth tide effect. The Dry Global Mapping Function (DRY-GMF) was used for the tropospheric delay modelling. The maximum size of accepted cycle slip corrections was 10. The results of Bernese 5.2 post-processed kinematic solution data were used for furter analysis by using the authors made software programs [45].

2.1. Ionospheric Scintillation Waves

The regular daily disturbances were observed in Latvian CORS stations in March 2015 [42]. The graphical image of the information subset of the ionospheric scintillation wave is depicted in Figure 1.

Starting from first row is a No. of recorded information of scintillation event, date and time and list of Latvian CORS station DOMEs where simultaneously occurred ionospheric scintillation causing positioning 3D discrepancy ≥10 cm. In next row is information on the next event occurred in the next 90 seconds. The various subsets of stations imagine the covered area of station network that is impacted by various size of electron clouds. Therefore, the information subset of each row we call a cloud and whole sequence of uninterrupted 90-sec events by ionospheric scintillation wave (evil wave [43]). During one day there may occurre various count of waves depending of the solar activity.

Subset of the any wave $w_k$ is described in Formula (1) as a union of subsets of scintillation clouds with their date and time and GPS position discrepancy data 3. Results

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

$$w_k:={\displaystyle \cup }\left\{\text{‹}{k}_i,D_k,t_{k_i},s_{k_{ij}},d_{k_{i}j},r_{k_{i}j},w_k^{\text{'}} \text{›} : j \in \text{‹} D_k,t_{k_i}\text{›}, n_{k_i}\in \text{‹}{D}_k,t_{k_i}\text{›}, 1\le \mathrm{i}\le n_k,1\le \mathrm{j}\le \left|S_{k_i}\right|\right\}$$

(1)

where $k_i$ is No. of recorded scintillation’s wave in date $D_k$and $k_i$-cloud’s time $t_{k_i}$; $s_{k_{i}j}$- subset of station DOME names in each row’s subsets $S_{k_i}$with cardinality $\left|S_{k_i}\right|$ where simultaneously fixed scintillations. $S_{k_i}$ subsets will be named a clouds, similarly to electron clouds in space impacted area of station subset of cardinality j. Positioning 3D discrepamcies subset $d_{k_{i}j}$ and subset of Rate of Total Electron Content Index ROTI denoted by $r_{k_{i}j}$, correspondingly,. Further in this article will be described the tuple $w_k^{\text{'}}$ of Modified Julian Days (MJD) denoted by $\mathrm{formula} \left(2\right)$ where day and time converted to MJD. Cardinality of subset $w_k$ is $n_k \mathrm{and}$the count of CORS statins in each $k_i-$cloud is $\left|S_{k_i}\right| , \mathrm{correspondingly}.$ The number of waves per day varies, and the number of clouds in the wave also varies. Therefore, based on a data-driven approach, the actual spatiotemporally matched structure of the electron cloud can be inferred from these patterns on ground [6].

2.2. Graphical Representation of the Registered Ionospheric Wave Sets Over the Month

In Figure 2 depicted the occurrence of monthly ionospheric scintillation waves in March 2015. There are visible waves of type of Figure 1, that clearly repeating daily. The biggest peak is geomagnetic storm of St Patric’s day in March 17.

Table 1. Monthly ionospheric impact waves with minimum 2 cloudsand clouds.

#	Year	Month	Clouds	Waves	#	Year	Month	Clouds	Waves
1	2007	FEB	1569	120	24	2012	OCT	837	88
2	2007	MAY	4359	308	25	2013	MAY	1587	201
3	2007	JUN	3501	272	26	2013	OCT	3772	152
4	2007	AUG	5830	491	27	2013	NOV	935	114
5	2008	MAR	726	69	28	2013	DEC	1155	136
6	2008	JUN	1600	140	29	2014	FEB	1268	99
7	2008	SEP	1986	193	30	2014	JUN	3393	295
8	2008	OCT	1328	107	31	2014	OCT	1241	117
9	2009	JUL	2473	216	32	2014	DEC	1837	126
10	2009	AUG	1413	126	33	2015	MAR	1584	119
11	2009	OCT	1304	107	34	2015	MAY	1749	170
12	2009	DEC	2056	115	35	2015	JUN	2499	154
13	2010	JAN	255	20	36	2015	OCT	880	118
14	2010	FEB	1123	66	37	2015	DEC	988	119
15	2010	APR	449	59	38	2016	FEB	1166	104
16	2010	NOV	1263	45	39	2016	APR	1401	105
17	2011	MAR	747	48	40	2016	MAY	2541	175
18	2011	AUG	1943	248	41	2016	JUL	3445	339
19	2011	SEP	1477	134	42	2017	APR	1980	120
20	2011	NOV	523	77	43	2017	MAY	2894	136
21	2012	JAN	366	35	44	2017	JUL	6442	214
22	2012	MAR	461	56	45	2017	SEP	1181	126
23	2012	JUL	2155	227	46	2017	OCT	1145	112

Table 1. Monthly ionospheric impact waves with minimum 2 cloudsand clouds.

#	Year	Month	Clouds	Waves	#	Year	Month	Clouds	Waves
1	2007	FEB	1569	120	24	2012	OCT	837	88
2	2007	MAY	4359	308	25	2013	MAY	1587	201
3	2007	JUN	3501	272	26	2013	OCT	3772	152
4	2007	AUG	5830	491	27	2013	NOV	935	114
5	2008	MAR	726	69	28	2013	DEC	1155	136
6	2008	JUN	1600	140	29	2014	FEB	1268	99
7	2008	SEP	1986	193	30	2014	JUN	3393	295
8	2008	OCT	1328	107	31	2014	OCT	1241	117
9	2009	JUL	2473	216	32	2014	DEC	1837	126
10	2009	AUG	1413	126	33	2015	MAR	1584	119
11	2009	OCT	1304	107	34	2015	MAY	1749	170
12	2009	DEC	2056	115	35	2015	JUN	2499	154
13	2010	JAN	255	20	36	2015	OCT	880	118
14	2010	FEB	1123	66	37	2015	DEC	988	119
15	2010	APR	449	59	38	2016	FEB	1166	104
16	2010	NOV	1263	45	39	2016	APR	1401	105
17	2011	MAR	747	48	40	2016	MAY	2541	175
18	2011	AUG	1943	248	41	2016	JUL	3445	339
19	2011	SEP	1477	134	42	2017	APR	1980	120
20	2011	NOV	523	77	43	2017	MAY	2894	136
21	2012	JAN	366	35	44	2017	JUL	6442	214
22	2012	MAR	461	56	45	2017	SEP	1181	126
23	2012	JUL	2155	227	46	2017	OCT	1145	112

More complicated is Figure 3 where scintillation events on August 2007 are depicted.

In Figure 4 depicted scintillation events in July 2016. On the surface, it is hard to say whether scintillation waves have been regularly recurring on a daily basis this month.

The total number of waves is 6718 and 84827 clouds. It is necessary to choose some algorithm to detect the presence of regular waves in these monthly subsets of data. Based on the example of the month of March 2015 daily regular waves are repeated with a 4.5-minute lag on the time scale graduated in increments of 1.5 minutes. For better transparency the count of clouds and waves depicted in Figure 5.

2.3. The Classification of the Characteristic Elements of the Waves Of Ionospheric Scintillation - Clouds

Figure 6 once again shows some scintillation k-wave and sketches the details of that wave used in the subsequent analysis process.

Additionally, to the elements of each wave with index k the information subset of the tuples of Modified Julian Days (MJD) is introduced, where MJD for peak cloud is denoted by $p_k$, beginning cloud $b_k$ , end cloud $e_k$, middle cloud $g_k$, wave length $l_k$, station count in peak cloud $\left|S_{k_i}\right|$ , k is No. of wave in month, $c_1$ is record No.of first wave’s cloud and $n_k is$count of clouds in a wave. Initially was planned to use the differences between element values of adjacent waves (denited by o) but in practice, its use turned out to be insignificant.

$$W^{\prime }:=\underset{k=1}{\overset{\left|W^{\prime }\right|}{{\displaystyle \mathrm{U}}}} \left\{\left(k,c_1,b_k,e_k,p_k,g_k,l_k,o_k^b,o_k^e o_k^p,o_k^g,n_k,max\left|S_{k_i}\right|\right) \in w_k,n_{k_i}\in w_k,1\le \mathrm{i}\le n_k\right\}$$

(2)

An example from such a subset of tuples is shown in Table 2 for March 11-13 2015.

2.4. Characteristic of regular daily wave elements in March 2015

The duration of the wave will be named in this article a length of wave. The length of the waves will be of various size in dependance of annual season and solar activity level. In March 2015 the regularity of waves is clearly visible (Figure 2). But it is quite a difficult to imagine regular daily waves in Figure 3 or in Figure 4.

With a 4.5-minute lag in adjacent days of March 2015 each of the 1.5-minute (90 sec) clouds shifts. However. the time lag for each of these proper clouds (peak, med and others) varies due to the changes of daily wave configuration. It turned out that the times of regular daily repetition are variable. Table 2 shows how the parameters of daily waves are changing. The Shift time is given in minutes for peak cloud. Uncertainties dpeak in time minutes (min) are obtained in result of linear approximation of daily peak times. Similarly. linear approximation performed for median (dmed) and beginning (dbeg). Length of wave in minutes mean the duration of the GPS positioning discrepansies. Time differences for the peak minus beginning (peak-beg). median minus peak (med-peak) and end-peak (end-peak) characterizes the changing shape of the waves.

Table 2. Regular daily wave characterization parameters (min) in March 2015.

Table 2. Regular daily wave characterization parameters (min) in March 2015.

The root mean square error (RMS) of linear approximation is 1.12 min for time of occurrence of the peak cloud (peak), 7.75 min for cloud at the middle of time sequence (further median - med) and 6.15 min for time sequence beginning of wave (further beginning – beg). The linear approximation errors of the daily time are depicted in Figure 7. The peak’s daily repetition is most regular. The monthly mean value of the length of wave (Length) is 30.0 min. mean value without outlier is 24.4 min (Figure 7). Mean time difference from beginning to peak is 9.05 min. peak to middle of wave 5.95 min and from peak to the end of wave 20.95 min. This means that in average the peak of the ionospheric wave is closer to the beginning of the wave, but the wave part from the peak to the end is 2 times more stretched. Daily regular waves repeated with a -4.11 minute lag on the time for peak, -3.76 min for wave median and -3.78 min for the beginning of wave.

Variation of the length of regular waves depicted in Figure 8. It will be proven later in the article that the wavelength changes depending on solar activity. This is also shown in this image, where the longest wavelength occurs on the day of the geomagnetic storm, March 17. The geomagnetic storm appears in the Latvian CORS observations from 14:27 UT to 18:18 UT, while the regular daily scintillation wave is from 21:43:30 UT to 23:58:30 UT.

Since the most stable regularity is indicated by the time lag of the peak, it is useful to carry out the search for the next regular daily wave in relation to the forecast of the next day of the peak time. . In order to search for the peak of the next day it was necessary to take into account both the fluctuation of the lag time and the discrepancy between the gradation of the measurement time 1.5 min (see Figure 1) and the regular predicted daily time lag -4.11 min.

When the study began, there were only thoughts that there might be regular waves of GPS interference in a few other months as well. However, the example from March 2015 sparked interest in exploring the regularity of positioning disorders in other months.

2.5. Search of Regular Daily Waves

The monthly set of regular daily scintillation waves is

$$W :={\displaystyle \cup } \left\{w_k:\exists p_k \exists p_k^{\text{'}} \mathrm{abs}\left(p_k-p_k^{\text{'}}\right)\le c^{\prime }, p_k, p_{k }^{\text{'}}\in w_k , 1 \le \mathrm{k} \le 31 \right\}$$

(3)

$$p_k^{\text{'}}=p_{k-1}+\mathrm{const}$$

(4)

The $c^{\prime }$ is a threshold for uncertainty of the constant for peak clooud $p_k$occurrence time daily regularity. For the initial search test, sets of CORS GPS measurement erroneous results of the October and December 2014 were selected. A cloud of the largest cardinality of the first wave of the first day of the month was chosen as the starting point. The search process confirmed the daily varying difference in time between the wave beginnings, peaks, and midpoints.The results of experimentation will be reflected in the next section of article.

The final test was based on the assumption that the predicted regular wave of the next day partially overlaps the regular wave of the current day in terms of time in minutes and seconds, unless one of the regular wavelengths of the comparable adjacent days is less than a certain time limit.

$$W_{tested} :={\displaystyle \cup } \left\{\left(w_i,w_j\right): \left(\left(t_{j_1}^{\text{'}}<t_{i_l}\right)\wedge \left(t_{j_l}^{\text{'}}>t_{i_1}\right) \right),t_{i1} ,t_{il}\in D_i , w_j, t_{j_1}^{\text{'}} ,t_{j_l}^{\text{'}}\in D_{i+1} ,1 \le \mathrm{i} \le \left|w_i\left| , 1 \le \mathrm{j}\le \right|w_j\right|\right\}$$

(5)

$$t_j^{\text{'}}=t_i+\mathrm{const}$$

(6)

where $t_{i_1}$ means time of the first cloud of the wave, $t_{i_l}$ - the last cloud of the wave, $\left|w_i\right|$is cardinality - count of the clouds in current day. The $t_{j_1}^{\text{'}}$and $t_{j_l}^{\text{'}}$belong to the wave found for the next day., correspondingly.. Const means a daily lag in time changes of the regular waves occurred in adjacent days. The value of const will be discussed in section of Discussion.

2.6. ROTI@ground and Positioning Errors

In conclusion, in a few months, the average values of ROTI@ground and the affected CORS station average positioning errors will be calculated to gain insight into what impact these regular daily scintillation waves have on positioning accuracy.

$\mathrm{In} \mathrm{Table} 4$is shown the sample o+f discrepancies in GPS positioning for all the stations of one cloud: Northing dN, Easting dE, height dh, 3D error, azimuth Az abd Rate of Total electron content Index ROTI with time resolution 8 min for ROTI for each station. Both the results of ROTI@ ground and average positioning error is computed for each daily regular wave. Results will be discussed in next sections, but the sample of input data for each station on fixed time is shown in Table 3.

3. Results

By analogy with the month of March 2015, initially the search for regular waves in other month began from the first wave of the first day of the month. The results were disappointing. Then the search began with the largest wave of the first day, then with the highest peak of the first day. It happened that the amount of waves on the first day of a few months was unexpectedly large, and it was necessary to change the initial search parameters in software program. Several search parameters were improved step by step and as a result, the search was carried out 6 times with both different and matching results.

Doubts and disbelief about the regularity of scintillation led to its existence being tested again and again. The search conditions were not fundamentally different, The differences were only the change in the choice of the starting point cloud or in the choice of regular daily repetition time shift constant (Const). By changing the Const value, the number of identified waves per month changed. In the search process with a different Const value there was a peak cloud as the starting point always. But in seach of subsequent day's wave, a different cloud of the same wave entered instead of the peak cloud because of it’s fluctuating time. In months with shorter wave lengths, the search process collapsed, as the accepted time Const shifted beyond the limits of the next wave. In Figure 9 depicted the fluctuation of the time of peak cloud and subsequent variation of Const value. Dashed line shows the mean Const that is close to the length of siderial day.

The results of the experiments were summarized. Result is shown in Figure 10 and example of summary in Table 4. The column St mens number of stations in cloud, Stations/cloud in 6 columns expose the search results in 6 experiments. For control purposes numbers in Štations/cloud” mean count of stations in cloud that coinside with data in column St. “Adj.days time lag” show the difference in minutes (min) between identified cloud of regular waves in adjacent days (current minus previous). Information in lines 31-33 are about clouds in one wave, lines 50-54 other regular wave, lines 114-115 other very small wave, lines 127-130 one more wave. The complete information of this data subset for June 2014 is in Table A1 where the summarized search results of selected month are shpwn.

November of year 2011 in a 24^th solar activity cycle is a month of low solar activity. The article will demonstrate that the wavelength of ionospheric scintillation regular waves is dependent on solar activity. In the example of Table 4, the regular waves are very short. According to Formula (5) the predicted regular wave of the next day partially overlaps the regular wave of the current day in terms of time in minutes and seconds, unless one of the regular wavelengths of the comparable adjacent days is less than a certain time limit.$\mathrm{For} \mathrm{examle}, \mathrm{in} \mathrm{Table} 4 \mathrm{in} \mathrm{NOV} 2 \mathrm{and} 3 \mathrm{the} \mathrm{time}$1:42:00 < 1:49:30 and 1:51:00 >1:46:29. But wave of NOV 6 is not covering the adjacent day NOOV 7 in sense of minutes and seconds.

3.1. Linear Approximation

Due to the fact that in some months it was not possible to find regular ionospheric scintillation waves on all days, a linear approximation of the time series of the detected waves was performed for both peak clouds, median clouds, and initial (beginning) clouds. The approximation for initial clouds showed the greatest errors, making its use for forecasts invalid. The number of input data for peak and median clouds is shown in Figure 10.

In the linear approximation algorithm, approximation precision criteria were incorporated. For example, the criterion for the precision of peak time series approximation was 3 minutes. To achieve this precision, it was allowed to discard times with the largest discrepancy during iteration, but keeping the number of remaining times no less than 4 in the solution. In the case of the median, there were 2 solutions. One with the same conditions as for peak cloud times, but the other with different precision conditions: a precision of 17 minutes and a minimum of 6 time counts.

Based on the results of the linear approximation, it was checked whether such scintillation waves were present in the CORS dataset. The search results are shown in Figure 11.

As a result of linear approximation, the peak cloud times of the unidentified waves were also predicted. The search results are shown in Table 5. The information in columns as follows: “Pred” – number of predicted wave’s peak time; “Not”- number of waves that are not found in search procedures; “found”- found in final search procedures based on linear approximation results; “Wlen”- mean wave lrngth (min); “For-mean”- numbrr of waves without outliers of length;”Mean”- mean wave lrngth (min) without outliers; “Outliers”- number of outliers; “Thr”- threshold for outliers 60 (min).

The results demonstrated in Table 5 and Figure 11 confirm that the method of finding waves predicted by linear transformation is not applicable in months of low solar activity, when the scintillation wave length is less than 10 minutes. It is quite successfully used in months of elevated solar activity.

3.2. Solar Activity and Length of Regular Ionospheric Scintillation Waves

The owerview of the percentage of distribution of wave’s length of the identified regular waves is shown in Table 6 for various month and years of 24^th solar cycle in mid-latitude country Jatvia.

The period was selected from the first to the last MJD day in which the regular disturbance was identified over the course of a month. For better visibility, Figures 12-17 show the distributions of wavelengths mentioned in Table 6 by individual years. These graphs llustrate the dependence of wavelength distribution on solar geomagnetic activity. In the years 2007 and 2008, solar activity decreased, but it is still relatively high.In period of low solar activity in November 2011 up to 80% of wavelength is very short [0;5) minutes (Table 6, Figure 13). However, 20 % of waves are lasting more than 1 hour.

Figure 12-17. The percentage distribution of wave lengths in the months analyzed of 2007 and 2008 (Fig 12), 2011 (Fig 13),2014 (Fig 14), 2015 (Fig 15), 2016 (Fig 16) and 2017 (Fig 17).

Figure 12-17. The percentage distribution of wave lengths in the months analyzed of 2007 and 2008 (Fig 12), 2011 (Fig 13),2014 (Fig 14), 2015 (Fig 15), 2016 (Fig 16) and 2017 (Fig 17).

In period of high solar activity June 2014 in 36.8% of occurrances the wavelength was [[15,20] minutes of time, in October 2014 45.1%, in December 2014 67.7% of occurrances of wave length [20;25) and 22.6% of more than 1 hour waves (Figure 14). Figure 15 shows how wave lengths change depending on changes in solar activity. In March, May, and June 2015, solar activity is very high, while in October and December it is significantly lower. In March, there is a very uniform distribution starting from 15 minutes and more (42% [20;25)). In October, 50%, and in December, in 67% of cases, the wavelength does not exceed 5 minutes. The Figure 16 shows the wave length distribution in low solar activity year 2016. Solar geomagnetic activity has begun to increase since April 2017, and shaply decreased in October. In September 2017 very strong geomagnetic storm was observed in USA and Canada, but it doesn’t appears in Latvia. This is also reflected in the distribution of wavelengths in Figure 17.

3.3. The Impact of Daily Regular Ionospheric Waves on Positioning Accuracy

To assess the impact of regular scintillation wave on positioning accuracy, error data and ROTI data were randomly selected from the Latvia CORS dataset defined in formula (1). In Table 7 shown example of ROTI@ground and mean positioning 3D discrepancies for all clouds of regular wave in March 23, 2015. In column St is a number of staations in cloud.

In Table 8 depicyed the ROTI@ground and mean discrepancies for individual clouds in situation example when solar activity was low.

In March 2015, there was increased solar activity. Table 9 shows the impact of regular wave ROTI@ground and the average positioning error of each day. It is not for individual clouds like in Table 7 and Table 8. There is computed the meam values of impact of daily regular scincillation waves. Table 9 data shows that daily regular scintillation waves cause serious positioning errors.

Table 9 data shows that daily regular scintillation waves cause serious positioning errors.

3.4. Time Lag of Daily Regular Ionospheric Scintillation Waves

The period was selected from the first to the last MJD day in which the regular disturbance was identified over the course of a year. The average daily time lag for regular ionospheric scintillations was calculated (Table 10). The value of daily time lag is close to the length of siderial day. This confirms the hypothesis that the initial source of the radiation causing the scintillation is not the Sun. It is enhanced by solar activity, but origin is from interplanetary media, similarly like Pc1 pulsation waves. 4. Time lag of daily regular ionospheric scintillation waves.

Researchers conclude [37] that Pc1 waves appear at midnight just before dawn. Our studies demonstrate the likelihood of the regular cyclical occurrence of daily regular scintillation waves at any time of day (Table B1). Of course, if they are indeed the scintillation waves referred to similarity as Pc1 waves.

Pc1 waves enhanced by solar activity [37,38,39], It is seen in Figures 12-17 that wavelength of regular wavws is dependent from the solar activity.enhancement.

Regular waves repetition shift is 0.997156 days (or -4. 1 minutes), that is approximate siderial day different from solar UT day. Consequently, the origin of waves is supposed to be interplanetary media.

4. Discussion

It is revealed that the regularity of daily disturbances is time-shifted, close to the length of a sidereal day. The duration (length) of daily scintillation waves) is influenced by the olar activity. This daily regular ionospheric scintillation wave phenomenon is similar to the Pc1 interplanetary wave phenomenon, but unlike Pc1 observations, they are recorded at any time of the day (Table B1). Only during periods of low solar activity the regular scintillation waves not appear. Table 11 provides the dates of the observed Pc1 waves [37] and a comparison of the lengths of these waves with the number and length of waves observed by GPS in Latvia on those dates. The Pc1 dates were published in a preprint of publication [37], unfortunately, without the indication of the observation time.

In two other studies, the dates and times of the Pc1 wave magnetometric measurements have been mentioned.

The theoretical predictions with observational results from conjugate high latitude stations in Antarctica and Greenland performed by Pilipenko et al [38] during summer (7 June 2014) and wintwr (7 January 2014) periods. A disturbance is a sudden commencement (SC) pulse caused by the impact of an interplanetary shock (IS) on the magnetosphere. At high latitude when the duration of the local field line eigenperiod T_A is ~5-10 min which can be longer than an SC impulse with 1-2 min [38]. In 7 June 2014 accordingtothe SuperDARNTEC maps, for the 7 June 2014 event (1650–1655). In Figure 18 shown daily regular scintillaion event registered by Latvian CORS on 7 June 2014

Francia et al [40] studied the correspondence between Pc1 activity and ionospheric irregularities at polar latitudes. They performed magnetometric observations in 22 february 2007 in MST station in Antarctica. between 0200 and 1400 UT.

Latvian CORS GPS observations confirmed that the month of February 2007 had low solar activity. However, on February 22, two cases of ionospheric scintillation were 872

Figure 19. Ionospheric scintillaion 2 event registered by Latvian CORS on 22 Feb 2007.

Figure 19. Ionospheric scintillaion 2 event registered by Latvian CORS on 22 Feb 2007.

It is still a challenge to determine the apparent relationship between ionospheric conditions and Pc1 wave occurrence [37].

It cannot be claimed that the comparison of daily regular ionospheric scintillation cases over time with the mentioned Pc1 wave cases indicates any interrelation. But it can be asserted that, judging by the periodic shift of the day, the scintillation recurrence impulse comes from the interplanetary environment.with some mutual relationship.