Extreme Solar Events’ Impact on GPS Positioning Results

: The main objective of the present study is to perform an analysis of the space weather impact on the Latvian CORS (Continuously Operating GNSS (Global Navigation Satellite System) Stations) GPS (Global Positioning System) observations, in situations of geomagnetic storms, sun ﬂares and extreme TEC (Total Electron Content) and ROTI (Rate of change of TEC index) levels, by analyzing the results, i.e., 90-s kinematic post-processing solutions, obtained using Bernese GNSS Software v5.2. To complete this study, the 90-s kinematic time series of all the Latvian CORS for the period from 2007 to 2017 were analyzed, and a correlation between time series outliers (hereinafter referred to as faults) and extreme space weather events was sought. Over 36 million position determination solutions were examined, 0.6% of the solutions appear to be erroneous, 0.13% of the solutions have errors greater than 1 m, 0.05% have errors greater than 10 m, and 0.01% of the solutions show errors greater than 50 m. The correlation between faulty results, TEC and ROTI levels and Bernese GNSS Software v5.2 detected cycle slips was computed. This also includes an analysis of fault distribution depending on the geomagnetic latitude as well as faults distribution simultaneously occurring in some stations, etc. This work is the statistical analysis of the Latvian CORS security, mainly focusing on geomagnetic extreme events and ionospheric scintillations in the region of Latvia, with a latitude around 57 ◦ N.


Introduction
This study uses observation data from the Latvian CORS network, after post processing with the Bernese GNSS Software v5.2 in kinematic mode with a sampling interval of 90 s. These results are used for the study of the Latvian CORS vulnerability control and for the statistical analysis of discrepancies in relation to the TEC and the ROTI levels. The objective of this study is to assess the risks of the CORS reliability for RTK measurements, and the publicly available TEC and ROTI reliability in connection with ionospheric irregularities in the midlatitude region of Latvia.
The term space weather refers to conditions on the sun, solar wind, and Earth's magnetosphere, ionosphere, and thermosphere that can affect the performance and reliability of space-and ground-based technological systems and can endanger human life or health [1]. Improving the understanding and characterization of the effects of space weather phenomena on the Earth and in the space can increase situational awareness, inform decision making, and enable missions to be carried out that depend on technologies and services susceptible to disruption from space weather [2].
Ionospheric disturbances on a small scale can lead to fluctuations in the received satellite signal, so-called signal scintillations. Within GNSS, this reduces the positioning accuracy. Particularly strong events can even lead to a Loss-of-Lock between satellite and receiver, which can delay or completely invalidate a positioning solution. Every GNSS user is affected, especially users with high demands on accuracy, integrity, availability, and continuity [3].
Multiple studies of ionospheric scintillations have been performed. However, the global climate is evolving, and atmosphere irregularities are changing. The use of GNSS positioning is increasing in various applications and the awareness of space weather impact on GNSS observations is increasing.
Spogli et al. [4] discussed the possibility of investigating the dynamics of ionospheric irregularities causing scintillation by combining the information coming from a wide range of latitudes. The authors analyzed the data of ionospheric scintillation from latitudes 44-88 • N during October, November and December 2003.
Similar work was carried out in Belgium by Stankov et al. [5] by studying GPS signal delay during geomagnetic storms of 29 October and 20 November 2003. The anomalous movement of ionosphere walls was studied [5]. Similar ionospheric gradients were found. Instead of the traditional Instrument Landing System (ILS), several prototype airports have used systems for GNSS landings and takeoffs. These prototype airports are in areas in which the occurrence of scintillation is negligible [6][7][8]. Stankov et al. [5] suggest that one important objective is to assess the integrity risk to GBAS/ SBAS services.
Liu et al. [9] studied the variation characteristics of the GPS-based TEC fluctuations over 21 regions of China. They studied the fluctuation intensity at various latitudes, in daytime and nighttime, during winter and summer. The ROTI indices were used to investigate the characteristics of the ionospheric TEC fluctuations during 11-year solar cycle 2002-2012 [9].
To classify the relevant orders of the magnitude and the occurrence rates Hlubek et al. [3] employed a statistical approach and large amounts of measured data were aggregated. The research by Hlubek et al. [3] concluded that a double-peak structure with the greatest scintillation intensity was observed during the spring and autumn equinoxes.
Research on the correlation between GNSS-derived ionospheric spatial de-correlation and space weather intensity for safety-critical differential GNSS systems was carried out by Lee and Lee [8]. Space weather events that occurred in 2015 have been extensively analyzed by the research society around the world. Cherniak et al. [10] investigated the dynamics of the high-latitude ionospheric irregularities during 17 March 2015 (St. Patrick's Day Storm), using ground-based GPS measurements. The St. Patrick's Day geomagnetic storm has been widely considered [10][11][12][13][14][15].
The results reveal interhemispheric differences in the occurrence of ionospheric irregularities. Research on variations of the TEC over the Iberian Peninsula in 2015 was performed by Morozova et al. [11], highlighting the effects of geomagnetic storms, solar flares, and solar eclipses. These authors showed that no definitive conclusions about the dependence of the TEC variation during geomagnetic storms on the season or start time can be made.
At high latitudes, the dynamic behavior of the ionosphere is dominated by the solar wind and electron precipitation (aurora borealis and aurora australis). In mid latitudes, ionospheric dynamics are dominated by the inner magnetosphere and neutral winds, the knowledge of which is incomplete [16].
Advanced studies are carried out by using space-borne exploration techniques like ionosondes, LiDAR, radio waves [17]. The geomagnetic field of the upper atmosphere, the ionospheric plasma and the GPS signal propagation in line of sight from explorer satellite to the GPS vehicle were studied in, for example, the ESA Swarm mission [18].
In high latitudes, Park et al. [19] presented the morphology of GPS TEC "perturbations" with emphasis on the orientation of plasma structures with respect to the line-of-sight direction (CHAMP mission).
Jin et al. and Park et al. [18,19] present the first comprehensive statistical results of high latitude ionospheric plasma irregularities and their dependence on the interplanetary magnetic field (IMF) configurations.
The impact of space weather on the GNSS positioning, navigation, and timing has been recognized as a serious threat [20] to the operational quality of GBAS and SBAS, and for many other positioning and navigation applications as well, such as for remote sensing vehicles, satellites, aviation, cars, trucks, farming, construction, snow removal, etc. Distortion of GNSS signals is of concern for many applications, especially those related to Safety-of-Life. However, despite the fact that the studies of the space weather are developed, not so many research activities are devoted to study the infallibility of the CORS depending on the size of the network and the covered territory.
In our study, performed at the Institute of Geodesy and Geoinformatics, University of Latvia (GGI), the Latvian CORS ground-based GPS observations were collected during the 24th solar cycle. Latvian CORS data is regularly post-processed by the GGI for the Permanent GNSS Network densification of the Regional Reference Frame Sub-Commission for Europe (EUREF), as well as for the EPOS (European Plate Observing System) program [21].
The statistical data of the results of the space weather impact on GPS observations are presented in this study. Conclusions on the security level of the Latvian CORS will be drawn on the basis of these statistics. At the end of this study, Pearson's correlation analysis is performed on the relation characteristics of both the TEC and the ROTI to the impacted GPS positioning discrepancies. The assessment of the TEC and ROTI irregularities will be discussed.

Bernese GNSS Software v5.2 Solutions
The analysis of the kinematic solution results of the 90-s GPS observations was used to approximate the GPS navigation situation. To identify disturbed results caused by extreme solar events of geomagnetic activity and ionospheric scintillations, the Latvian CORS 11-year, selective daily GPS observation data were post-processed in a double-difference (DD) mode using Bernese GNSS Software v5.2 [22]. Information on the ionospheric TEC levels and extreme solar events was obtained from publicly available data sets. The maximum TEC values wereextracted from CODE'S European Ionosphere information INX data files [23]. Data of solar flares and geomagnetic storms were obtained from the auroral and solar activity web page [24]. Bernese GNSS Software v5.2 program RNXSMT (detects cycle slips and outliers on RINEX level using simultaneous code and phase observations from both frequencies to each satellite; code observations are smoothed using the phase measurements) and MAUPRP (automatic phase pre-processing, cycle slip detection and correction, outlier detection, and updating of the Ambiguity List) were used for cycle slip detection [22]. The MAUPRP program was also used to repair cycle slips, with 10 cycles being the minimum size of accepted cycle slip corrections. The outputs from both programs were used to find detected cycle slips for each station and baseline. Daily RINEX observation data (30-s sampling rate) were selected, which included 4-month observation data (with high monthly TEC values) for the full set of Latvian CORS stations for each year from 2007 to 2017. The 90-s sampling interval of kinematic post-processing was chosen. There are 960 kinematic post-processing solutions per 24 h, and 28,800 sessions for each station in 30 days. For the Bernese Software v.5.2 solutions, 4 IGS/EPN (EUREF Permanent GNSS Network) stations were used as reference stations, and the Latvian CORS stations were used as rover stations. The IGS final orbit and clock data, TEC, ocean and atmosphere loading were taken into account. Stochastic ionospheric parameters and CODE's global ionospheric maps are used. The dry Global Mapping Function (GMF) was used to model the tropospheric delay. The solutions were carried out in sets of 4-5 Latvian stations and constantly using the same IGS/ EPN reference stations. The computation of each set of 4-5 Latvian CORS stations, for an observation period of 1 month takes approximately 12-14 h. This type of computation was carried out for all the Latvian CORS stations for 4 to 5 months per year, for 11 years (2007-2017). The main post-processing strategy parameters are listed in Table 1.

Road of Performed Analysis
The post-processed observation data were analyzed by applying software programs developed at the GGI. A total of 30 software programs in Fortran g95 and Python programming language were developed. The faulty solutions were found, and the statistical analysis was performed; the data were prepared for the correlation analysis, and the correlation analysis was performed. The flowchart of the operational functions and data sets are depicted in Figure S1 in Supplementary Materials. The main functions performed were: • CORS observation data were post-processed and 90-s kinematic coordinate solutions were obtained. The Cartesian XYZ coordinates were converted to the national grid coordinates: Northing, Easting, Up (abbreviation denoted in Figure S1-NEh); • The faulty solutions where one of the coordinate components exceeded the 10 cm threshold (SW2, ALL_ERR) were searched; • The cycle slips identified by the Bernese GNSS Software v5.2 were listed (CSLP); • The monthly mean coordinate values were calculated (SW2, MONTH TREND) for each station in each month (ALL_ERR, X4); • The geomagnetic storms over the territory of Latvia, the TEC max values, and solar flares were extracted from the publicly available data sources [23,24] (For_CORR); • The occurrence of the faulty solutions was analyzed, namely: sequences of faulty solutions, simultaneous faulty solutions in numerous stations, count of cycle slips, and faulty solutions for each month and each station were determined (1_z4, Waves, 1_z6, DISCR_4, statistics in Tables S1-S8); • The Pearson's correlation coefficient was computed to find the relation between TEC (set x) and count of cycle slips (set y) and, similarly, between TEC and the count of faulty solutions, as well as TEC and count of cycle slips in faulty solutions, and also between the count of cycle slips and count of faulty solutions (Correlation, R_line).

Latvian CORS Networks
There are two CORS networks included in this study: LatPos, maintained by the Latvian Geospatial Information Agency (LGIA), and EUPOS-RIGA, maintained by the Riga Municipality; and one IGS/ EPN station RIGA, which is operated by the Institute of Astronomy of the University of Latvia. Figure 1 shows the input rate (months in operation) of the Latvian CORS stations with their DOMES names. The maximum rate of input data for a stations/months included in the analysis is 46 months. The map of the Latvian CORS station sites is shown in Figure 2 11-year period were moved to other locations. Therefore, it is more truthful to refer to 46 sites instead of 46 stations. For example, in the city of Kuldiga, the station with a DOMES name KULD was moved to another location two times, correspondingly changing the DOMES names to KUL1 after the first move and to KUL2 after the second move ( Figure 2). Among all the stations included in the analysis, only 8 stations were not moved for 46 months (Figure 2). beginning of 2007, only 23 CORS stations were operational, new stations were gradua created, and in year 2017, the number of operational stations reached 32. Many statio during the 11-year period were moved to other locations. Therefore, it is more truthful refer to 46 sites instead of 46 stations. For example, in the city of Kuldiga, the station w a DOMES name KULD was moved to another location two times, correspondingly chan ing the DOMES names to KUL1 after the first move and to KUL2 after the second mo ( Figure 2). Among all the stations included in the analysis, only 8 stations were not mov for 46 months ( Figure 2). Over 11 years, the total number of months included in the analysis reached 46. F 2015 and 2017, 5-month observation data were analyzed, compared to 4-month observ tion data in each of all the other 9 years.

Monthly Mean Station Coordinates
The knowledge of the correct monthly mean station non-disturbed coordinate valu is the prerequisite for identifying disturbances. Further analysis discovered that 0.6% the whole set of solutions shows disturbed results of great errors. The CORS station co dinates were computed for each month and the corresponding mean monthly coordina were obtained. The values of the monthly mean coordinates were changing during t period of 11 years. To calculate the reliable monthly mean coordinates, in the first attem the outliers exceeding 3σ criteria were excluded. The trend of mean coordinate valu after the data filtration from the first attempt was nearly linear; the time series were ev uated in the second attempt (example in Figure 3). Therefore, it made it easier to appro imate the trend of each stations' coordinates' component changes. The quality control the monthly mean coordinates for the set of filtered solution results becomes possible. beginning of 2007, only 23 CORS stations were operational, new stations were gradually created, and in year 2017, the number of operational stations reached 32. Many station during the 11-year period were moved to other locations. Therefore, it is more truthful to refer to 46 sites instead of 46 stations. For example, in the city of Kuldiga, the station with a DOMES name KULD was moved to another location two times, correspondingly chang ing the DOMES names to KUL1 after the first move and to KUL2 after the second move ( Figure 2). Among all the stations included in the analysis, only 8 stations were not moved for 46 months ( Figure 2). Over 11 years, the total number of months included in the analysis reached 46. Fo 2015 and 2017, 5-month observation data were analyzed, compared to 4-month observa tion data in each of all the other 9 years.

Monthly Mean Station Coordinates
The knowledge of the correct monthly mean station non-disturbed coordinate value is the prerequisite for identifying disturbances. Further analysis discovered that 0.6% o the whole set of solutions shows disturbed results of great errors. The CORS station coor dinates were computed for each month and the corresponding mean monthly coordinate were obtained. The values of the monthly mean coordinates were changing during the period of 11 years. To calculate the reliable monthly mean coordinates, in the first attempt the outliers exceeding 3σ criteria were excluded. The trend of mean coordinate value after the data filtration from the first attempt was nearly linear; the time series were eval uated in the second attempt (example in Figure 3). Therefore, it made it easier to approx imate the trend of each stations' coordinates' component changes. The quality control o the monthly mean coordinates for the set of filtered solution results becomes possible.

Monthly Mean Station Coordinates
The knowledge of the correct monthly mean station non-disturbed coordinate values is the prerequisite for identifying disturbances. Further analysis discovered that 0.6% of the whole set of solutions shows disturbed results of great errors. The CORS station coordinates were computed for each month and the corresponding mean monthly coordinates were obtained. The values of the monthly mean coordinates were changing during the period of 11 years. To calculate the reliable monthly mean coordinates, in the first attempt, the outliers exceeding 3σ criteria were excluded. The trend of mean coordinate values after the data filtration from the first attempt was nearly linear; the time series were evaluated in the second attempt (example in Figure 3). Therefore, it made it easier to approximate the trend of each stations' coordinates' component changes. The quality control of the monthly mean coordinates for the set of filtered solution results becomes possible.

Distribution of the Size of Discrepancies
During the research, the total count of Bernese GNSS Software v5.2 solutions reached 36,728,129, of which 203,981 (i.e., 0.6%) solutions appeared with discrepancies in position greater than 10 cm (3σ). Including the 10 cm threshold, the count reached 204,022. There were 744,689 cycle slips (CSLP) identified by Bernese GNSS Software v5.2. This covers 2% of all Bernese GNSS Software v5.2 solutions. A total of 4849 (i.e., 0.6% of all cycle slips) of these were identified in the subset of disturbed solutions.
The size of the disturbances in coordinates is classified. During the geomagnetic storm, which occurred on 17 March, 2015 (St. Patrick's day) max disturbances in 2 stations (RIGA and VAIV) reached 500 m. The error caused by ionospheric scintillation in 50,430 solutions was greater than 1 meter ( Table 2). This is dangerous in Safety-of-Life critical situations. Classification shows that 75% of disturbances were in the bounds of [0.1; 1.0) meters; 10% of disturbances were in the bounds of [1.0; 5.0) and 4% of disturbances were in the   Table 2). This is dangerous in Safety-of-Life critical situations.

Evil Waves of Disturbances
The term "evil waveform" is used to denote the disturbed information for navigation in some area caused by the GPS clock error [25]. The term "evil waves" in this paper is used to describe the changing distribution of positioning discrepancies over the territory of Latvia in some time period. The movement of "evil wave" is shown in slides of Figure 4a-c and Supplementary Materials Tables S2-S7. The red circles in Figure 4 denote the simultaneously occurring faulty solutions. In each of the (a), (b) and (c) titles in the top row, the period of the "evil wave" is written, in the second row, the beginning of the current 90-s faulty solution is shown.

Evil Waves of Disturbances
The term "evil waveform" is used to denote the disturbed information for navigation in some area caused by the GPS clock error [25]. The term "evil waves" in this paper is used to describe the changing distribution of positioning discrepancies over the territory of Latvia in some time period. The movement of "evil wave" is shown in slides of Figure  4a,b,c and Supplementary Materials Tables S2-S7. The red circles in Figure 4 denote the simultaneously occurring faulty solutions. In each of the (a), (b) and (c) titles in the top row, the period of the "evil wave" is written, in the second row, the beginning of the current 90-s faulty solution is shown. When sorting the disturbances, the occurrence of faulty solutions was found in numerous stations simultaneously. The movement of these disturbances over the territory of Latvia can be described as a "waveform". This could be interpreted as ionospheric scintillations, exposed in a form of table (Table 3, Supplementary Materials Table S4 and Table  S7)) and/or graphs (Figures 4 and 5).   When sorting the disturbances, the occurrence of faulty solutions was found in numerous stations simultaneously. The movement of these disturbances over the territory of Latvia can be described as a "waveform". This could be interpreted as ionospheric scintillations, exposed in a form of table (Table 3, Supplementary Materials Tables S4 and S7) and/or graphs (Figures 4 and 5).  When sorting the disturbances, the occurrence of faulty solutions was found in numerous stations simultaneously. The movement of these disturbances over the territory of Latvia can be described as a "waveform". This could be interpreted as ionospheric scintillations, exposed in a form of table (Table 3, Supplementary Materials Table S4 and Table  S7)) and/or graphs (Figures 4 and 5).   The waves are counted in cases where the groups of at least three simultaneous 90sequences occurred within at least two simultaneous solutions with equal time events Table 3 shows example of two "waves": the first on 29 October 2012, 00:00:00 UT and th second, starting at 0:55: 30   More information on "waves" can be found in Supplementary Materials Tables S3 and S6 and in Table S4 and S7 for December 2014 and March 2015, respectively. Figure 6 depicts the count of "waves" in each analyzed month.   Table S8. The month of December 2009 is at the beginning part of the Solar cycle 24 when the sun activity awakes after a long, calm period.
More information on "waves" can be found in Supplementary Materials Tables S3 and S6  and in Tables S4 and S7 for December 4 5 6 7 9 10 12 13 13 15 16 17 17 17 17 17 18 19 20 22 25 27 29 4 5 6 7 9 10 12 13 13 15 16 17 17 17 17 17 18 19 20 22 25 27 29 Table S4, it ap pears that four stations LUNI, VAIV, KREI and SALP are repeatedly listed in each row meaning that out of five EUPOS-RIGA network stations, four of them on 14 July 2017 were out of normal operation. Consequently, erroneous corrections for GNSS related measure ments were disseminated. Such a search method was adopted for searching Loss-of-Lock of GNSS receivers [26]. Other stations in the city of Riga (OJAR, RIGA, VANG) and in other sites in Latvia (IRBE, TKMS, LIMB and others) are faulty occasionally, but not as often (Table 4). Therefore, there is reason to believe that this is not an effect of jamming.

Loss-of-Lock Situations
The information on sequences of repeatedly occurred 90-s faulty solutions is summa rized in Table 5, where DOY denotes the day of the year.   Table S4, it appears that four stations LUNI, VAIV, KREI and SALP are repeatedly listed in each row, meaning that out of five EUPOS-RIGA network stations, four of them on 14 July 2017 were out of normal operation. Consequently, erroneous corrections for GNSS related measurements were disseminated. Such a search method was adopted for searching Loss-of-Lock of GNSS receivers [26]. Other stations in the city of Riga (OJAR, RIGA, VANG) and in other sites in Latvia (IRBE, TKMS, LIMB and others) are faulty occasionally, but not as often (Table 4). Therefore, there is reason to believe that this is not an effect of jamming. The information on sequences of repeatedly occurred 90-s faulty solutions is summarized in Table 5, where DOY denotes the day of the year.
The detailed analysis of the discrepancies for the two stations LUNI and SALP is shown in Figures 10 and 11.
On other dates, there are similar sequences of repeated discrepancies in other stations of the LatPos network and the IGS/EPN station RIGA. Table 5 gives an example of where the sequences of repeated disturbances occur.  The detailed analysis of the discrepancies for the two stations LUNI and SALP is shown in Figures 10 and 11.   The detailed analysis of the discrepancies for the two stations LUNI and SALP is shown in Figures 10 and 11.  The situation described in Tables 3-5 and shown in Figure 9 can be assumed as a corresponding stations' Loss-of-Lock of receiver. At first, the idea was to remove these sequences of repeated disturbances. However, according to the Figures 10 and 11, the impact of space weather during successive scintillations of the receiver are disturbances of various magnitude, which reflect the strength of the impact. Figure 12 shows the count of frequencies and how often an assumed Loss-of-Lock has occurred (blue). On some days, Loss-of-Lock sequences occurred several times (2-3) per day, e.g., LUNI on 14 July 2017, and 21 July 2017 ( Table 5). The second column (red) in Figure 12 shows the frequency of the days of receivers' Loss-of-Lock occurrence. The maximum number of the count of frequencies of receivers' Loss-of-Lock appears for the IGS/ EPN station RIGA. The receiver of the RIGA station is mounted on a stable basement. Also, the EUPOS-RIGA network stations ANNI, MASK (relocated to VAIV and SALP in 2011, correspondingly), KREI, LUNI, and VANG are covering a small region of the city of Riga. The antennas are mounted on the roofs of buildings with no obstructions. The OJAR station of LatPos network is also located in the city of Riga very close to the station RIGA with the same type of receiver and antenna. However, the occurrence of the positioning disturbances is many times less.
Remote Sens. 2021, 13, 3624 13 of The LatPos network (now 32 stations) covers the entire territory of Latvia. The an ysis discovers that this network is most stable with less Loss-of-Lock situations, exce DAU1 and LIMB stations.    The LatPos network (now 32 stations) covers the entire territory of Latvia. The analysis discovers that this network is most stable with less Loss-of-Lock situations, except DAU1 and LIMB stations.
A summary of 90-s solutions associated with Loss-of-Lock sequences is shown in the histogram (Figure 13), where for each station the count of faulty solutions is displayed. The LatPos network (now 32 stations) covers the entire territory of Latvia. The anal ysis discovers that this network is most stable with less Loss-of-Lock situations, excep DAU1 and LIMB stations.     Figure 13 shows that station VAIV has the largest total count of Loss-of-Lock 90-s faulty solutions. The station VAIV is very close to the seashore. The station LUNI is located in the center of the city of Riga surrounded with a busy traffic environment. Most impacted of the receivers' Loss-of-Lock are the stations of the EUPOS-RIGA network and the single station RIGA.
The stations' DAU1 Loss-of-Lock occasions are very uniform. They are irregular by date, the sequences are not long, and the discrepancies are about 15-20 cm. However, since 2011 there have been 70 sequences in 58 days. The shape of the discrepancy distribution plots is uniform and differs from other stations' discrepancy plots.

Correlation Analysis
The monthly data subsets included the collected daily information of max TEC values over the territory of Latvia, count of cycle slips (CSLP) in all solutions and faulty solutions (CSLP (F)), and count of faulty solutions (>10 cm). A sample of this monthly data subset is presented in Table 6. In Supplementary Materials Table S1 the same data is exposed for the whole 24th solar cycle period 2007-2017. Using the data as in Table 6 the Pearson's correlation coefficient, the covariance coefficient, regression line coefficient, solution's mean square error, both numerator and denominator from Formula (6), R 2 , and value of t-test, were computed and the output was made for each month.
The Pearson's correlation coefficient was computed: The covariance was computed by using the formula Regression line was computed R 2 was computed by formula The Student's distribution t-test was computed by applying the formula t = r xy 1−r xy 2 n−2 A sample of this output is given in Table 7 for the four pairs of data types listed in the explanations after Table 7. This type of computation was carried out in two different versions: the first one with all the data discussed so far, the second version with modified data sets in which the 90-s sequences were removed, which seems to be the GNSS receiver's Loss-of-Lock product. The resulting correlation coefficients are shown in Table 8 and Figure 14.  Table 8. Count of Pearson's correlation coefficients before the removal of the Loss-of-Lock (1st row) and after the removal of the Loss-of-Lock (2nd row). 18  5  0  23  18  4  0  24  25  4  0  17  25  1  2  18  19  5  0  22  16  6  0  24  26  3  0  17  21  0  2  23   Table 8. Count of Pearson's correlation coefficients before the removal of the Loss-of-Lock (1st row) and after the removal of the Loss-of-Lock (2nd row). In Figure 14, the variations of Pearson's correlation coefficient in three cases are depicted: between TEC and count of cycle slips, TEC and count of faulty solutions (f.s.), TEC and faulty solutions with removed Loss-of-Lock sequences (No LoL). The conclusion is that in most situations TEC max, which is defined as a smooth value over the territory of Latvia, is not comparable to the sporadic nature of real time instantaneous spatial distribution of TEC [27].

ROTI Correlation Analysis
The ROTI index is determined from the IGS data of GNSS stations located around the Earth [28]. where: T-type (1-4): 1. TEC and cycle slips; 2.
TEC and cycle slips in faulty solutions; 4.
Cycle slips and faulty solutions.
Corr.c-Pearson's correlation coefficient (Formula (1)); Cov-covariance (Formula (2)); Linear regression line, coefficientâ and coefficientb (Formulas (3)-(5)); S-mean square error; R 2 -coefficient of determination (Formula (6)) and its numerator and denominator values; Student's distribution t-test (Formula (7)). Table 8  In Figure 14, the variations of Pearson's correlation coefficient in three cases are depicted: between TEC and count of cycle slips, TEC and count of faulty solutions (f.s.), TEC and faulty solutions with removed Loss-of-Lock sequences (No LoL). The conclusion is that in most situations TEC max, which is defined as a smooth value over the territory of Latvia, is not comparable to the sporadic nature of real time instantaneous spatial distribution of TEC [27].

ROTI Correlation Analysis
The ROTI index is determined from the IGS data of GNSS stations located around the Earth [28].
Correlation summary of the ROTI is given in Table 9. Unfortunately, the ROTI values are only available starting from year 2010 [28]. In Table S9 (

Estimation of the Relation between the Count of Faulty Solutions and TEC-Max
Geomagnetic storms and solar flares are extreme events. Figure 15 shows the monthly average of the daily maximum TECs and the average numbers of the Latvian CORS networks' faulty 90-s solutions per station/ per month. There is no close correlation between the indices of the mean TEC-max values and disturbance events. The average over a time span of 11 years is compared with sporadic events, and there is no close correlation expected.
Remote Sens. 2021, 13, 3624 17 of Correlation summary of the ROTI is given in Table 9. Unfortunately, the ROTI valu are only available starting from year 2010 [28]. Table 9. Count of Pearson's correlation coefficients between ROTI and faulty solutions. In Table S9 (

Estimation of the Relation between the Count of Faulty Solutions and TEC-Max
Geomagnetic storms and solar flares are extreme events. Figure 15 shows t monthly average of the daily maximum TECs and the average numbers of the Latvi CORS networks' faulty 90-s solutions per station/ per month. There is no close correlati between the indices of the mean TEC-max values and disturbance events. The avera over a time span of 11 years is compared with sporadic events, and there is no close co relation expected.    The count of cycle slips is greater than faulty solutions, the Bernese GNSS Software v5.2 identified most of the affected positions. However, there are still many faulty solutions that Bernese GNSS Software v5.2 does not identify.
A database was created for all collected data and processed data results. The database is stored in the Microsoft Office 365 OneDrive account, provided by the University of Latvia. This database includes also g95 Fortran and Python software programs that were developed as tools for large volume data processing.

Discussion
The impact of space weather on GNSS positioning, navigation and timing has been recognized by many authors as a threat [5,20] to the operational quality of SBAS and GBAS, as well as to many other positioning and navigation applications. The Latvian CORS serves as a basis for the RTK measurements which are used for the land surveying, cadaster and many other branches of engineering, including remote sensing and mapping. So far, no studies have been conducted on the impact of space weather on CORS in Latvia. The researchers of the GGI are working on national geoid improvement and on the application of the digital zenith camera, where short-term GNSS positioning and timing is used [29]. After this study, attention will be paid to the information on space weather and solar activities in the validation of GNSS high quality applications. Faulty solutions in the current study are caused by ionospheric irregularities which are discovered in a specific manner by the application of Bernese GNSS Software v5.2. Other methods, used in the studies of the ionospheric irregularities are reported in most of the references mentioned above.
This study shows that there exists a weak correlation between faulty positioning results and the applied TEC and even ROTI information on ionospheric irregularities caused by solar activity. However, the highest sun activity of the 24th solar cycle occurred in years 2013-2015. The largest positioning disturbances and the frequency of faults appeared in March 2015.
Many research papers are devoted to the studies of the ionospheric irregularities, the TEC fluctuation [19,[30][31][32] and the impact on GNSS and their correlation with GNSS positioning errors [4]. The current research results are, in principle, in agreement with them (Belgium 2002-2012 [30], Northern Europe 2009 [31], China [32]), but the approach of this study is different. However, Norwegian researchers have carried out the positioning tests with 5-minute resolution in a much shorter time span than 11 years, and they concluded The count of cycle slips is greater than faulty solutions, the Bernese GNSS Software v5.2 identified most of the affected positions. However, there are still many faulty solutions that Bernese GNSS Software v5.2 does not identify.
A database was created for all collected data and processed data results. The database is stored in the Microsoft Office 365 OneDrive account, provided by the University of Latvia. This database includes also g95 Fortran and Python software programs that were developed as tools for large volume data processing.

Discussion
The impact of space weather on GNSS positioning, navigation and timing has been recognized by many authors as a threat [5,20] to the operational quality of SBAS and GBAS, as well as to many other positioning and navigation applications. The Latvian CORS serves as a basis for the RTK measurements which are used for the land surveying, cadaster and many other branches of engineering, including remote sensing and mapping. So far, no studies have been conducted on the impact of space weather on CORS in Latvia. The researchers of the GGI are working on national geoid improvement and on the application of the digital zenith camera, where short-term GNSS positioning and timing is used [29]. After this study, attention will be paid to the information on space weather and solar activities in the validation of GNSS high quality applications. Faulty solutions in the current study are caused by ionospheric irregularities which are discovered in a specific manner by the application of Bernese GNSS Software v5.2. Other methods, used in the studies of the ionospheric irregularities are reported in most of the references mentioned above.
This study shows that there exists a weak correlation between faulty positioning results and the applied TEC and even ROTI information on ionospheric irregularities caused by solar activity. However, the highest sun activity of the 24th solar cycle occurred in years 2013-2015. The largest positioning disturbances and the frequency of faults appeared in March 2015.
Many research papers are devoted to the studies of the ionospheric irregularities, the TEC fluctuation [19,[30][31][32] and the impact on GNSS and their correlation with GNSS positioning errors [4]. The current research results are, in principle, in agreement with them (Belgium 2002-2012 [30], Northern Europe 2009 [31], China [32]), but the approach of this study is different. However, Norwegian researchers have carried out the positioning tests with 5-minute resolution in a much shorter time span than 11 years, and they concluded that there is a good correlation of the ROTI and the GNSS positioning errors in the high geographical latitudes of Norway [4]. The results in China show significant regional differences at different latitudes [9]. Liu et al. concluded that "relevant discussions of this phenomenon are still relatively rare, so our results contribute to the development of a more in-depth understanding of irregular ionospheric activities, specifically the characteristics and features that occur over China" [9].

Conclusions
The results show that 0.6% of the solutions appeared with discrepancies in position greater than 10 cm. The largest positioning disturbances and their frequency appeared in March 2015 during the highest sun activity of the 24th solar cycle in years 2013-2015. A very strong geomagnetic storm with Kp index 8 occurred on 6-8 September 2017over the territory of Canada and USA, but this geomagnetic storm did not cover the territory of Latvia. Geomagnetic storm of 17 March 2015 was the only solar activity event that created significant (~500 m) positioning disturbances in the Latvian CORS stations.
The Pearson's correlation coefficients were computed in order to validate the relation between the TEC maximum values over the territory of Latvia. Positioning discrepancies of the Bernese GNSS Software v5.2 solutions discovers that correlation is weak. The ROTI analysis also demonstrated a weak correlation. Even the sum of CSLP and faulty solutions showed a weak correlation between the TEC and the ROTI as well. The performed correlation analysis revealed that the global TEC approximation models are not suitable for the study of the local TEC anomalies. The GPS receiver onboard the ESA Swarm satellite provided the TEC between Swarm and GPS satellite. These electron high-density plasma patches are highly structured with significantly enhanced density fluctuation [18]. This could probably confirm the eventually significant small fluctuation of the TEC that are not included in the global TEC and ROTI models.
The monthly discrepancy diagrams revealed simultaneous discrepancies at numerous individual stations. The output was analyzed, and it was identified that for several stations, the disturbed solutions usually appeared more than 150-200 times. This is assumed to be the Loss-of-Lock of GNSS receivers. The conclusion arises on the dependency between the Loss-of-Lock of GNSS receivers and the GNSS receivers' network geometry and the size of the territorial coverage.
The Loss-of-Lock affected a single operating GNSS receiver the most, which is not included in any network.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/rs13183624/s1, Figure S1: Flowchart of the problem solution functions and related data sets, Table S1: List of geomagnetic storms and sun flares, count of tec-max, identified cycle slips, position discrepancies > 10 cm (faulty solutions) and faulty solution with cycle slips in Latvian CORS 90 s solutions,