Ionosphere-free Linear Combination Research Articles

Phase Residuals Analysis in Kinematic Orbit Determination of GRACE-FO

The GRACE-FO based scientific research will benefit from the improvement of the kinematic orbit quality. This study investigated the errors that could potentially impact the accuracy of orbit determination and applied error corrections to the observations, including phase center variation (PCV) and multipath error, with a specific focus on enhancing the performance of GRACE-FO kinematic precise orbit determination based on the zero-difference method. When the observation model for precise orbit determination is highly consistent with the actual situation, the phase residuals should only consist of observation noise. However, the phase residuals encompass both modeled errors and unmodeled errors, such as PCV and multipath errors. The study utilized the GPS observation data of the GRACE-FO satellites in January 2022 to derive the fixed ambiguity solutions based on the zero-difference ionosphere-free linear combination, thereby obtaining the phase residuals. Subsequently, the antenna PCV was estimated based on the residual method, resulting in a substantial reduction in carrier phase residuals through PCV correction. A comparison with the reference orbits revealed that the root mean square (RMS) of the fixed solutions with PCV correction improved by 2.8 to 4.3 mm in three directions. Furthermore, the wavelet decomposition was performed on the phase residuals to extract the multipath errors and reconstruct the carrier phase observations. The RMS of the kinematic orbits improved by 4.3 to 5.9 mm with the incorporation of both the PCV and multipath error corrections. The RMS of the satellite laser ranging (SLR) residuals of the two GRACE-FO (GRACE-C and GRACE-D) satellites were 1.43 cm and 1.48 cm, respectively. The results indicate that these methods have the potential to enhance the kinematic orbit accuracy of the GRACE-FO satellites, bringing it closer to the a posteriori scientific orbit accuracy.

Extension of the BDS IGSO and MEO satellite antenna patterns with FengYun-3C and LuTan-1 onboard data

The International Global Navigation Satellite System (GNSS) Service (IGS) has recommended calibrating the extended satellite antenna Phase Center Variation (PCV) model for all navigation satellites to support low-Earth-Orbit (LEO) satellite applications. Owing to the absence of ground-calibrated PCVs for the BeiDou Navigation Satellite System (BDS), tracking data from global ground stations were used for in-orbit estimations. However, limited by the observed maximum nadir angle, PCVs based on ground data cannot cover the entire range of nadir angles observed using by LEO satellites. In this study, the FengYun-3C and LuTan-1 missions were used to extend the PCVs of the BDS Inclined GeoSynchronous orbit (IGSO) and medium-Earth-orbit (MEO) satellites. The PCVs corresponding to the ionosphere-free linear combinations of B1I/B2I, B1I/B3I, and B1C/B2a were calibrated in-orbit by extending the maximum nadir angle from 9° to 10° for the IGSO and from 13° to 15° for the MEO. There was good consistency among the estimated PCVs of satellites of the same type. In addition, these estimated PCVs were close to the corresponding satellite block-specific values estimated based on ground tracking data. The impact of the extended PCVs of the BDS satellite antenna was investigated by comparing them with the unextended PCVs model in the receiver antenna PCVs pattern estimation and precise orbit determination of the FengYun-3C and LuTan-1 satellites. The results showed that the extended BDS satellite PCVs effectively reduced the unmodeled error of the unextended BDS satellite PCVs in the low-elevation region of the LEO receiver antenna PCV patterns. In addition, compared with the introduction of phase center offset alone, the introduction of ground-based PCVs can improve the overlapping orbit differences (OODs) of LEO satellites, and the extended BDS satellite PCVs can further improve the OODs. Taking B1C/B2a as an example, the improvement ratio of OODs in the 3D direction of Lutan-1A increased from 5.2 % to 8.0 %, and that of Lutan-1B can rise from 1.7 % to 4.2 %.

Comprehensive Analysis on GPS Carrier Phase under Various Cutoff Elevation Angles and Its Impact on Station Coordinates’ Repeatability

When processing the carrier phase, the global navigation satellite system (GNSS) grants the highest precision for geodetic measurements. The analysis centers (ACs) from the International GNSS Service (IGS) provide different data such as precise clock data, precise orbits, reference frame, ionosphere and troposphere data, as well as other geodetic products. Each individual AC has its own strategy for delivering the abovementioned products, with one of the key elements being the cutoff elevation angle. Typically, this angle is arbitrarily chosen using generic values without studying the impact of this choice on the obtained results, in particular when very precise positions are considered. This article addresses this issue. To this end, the article has two key sections, and the first is to evaluate the impact of using the two different cutoff elevation angles that are most widely used: (a) 3 degrees cutoff and (b) 10 degrees cutoff elevation angle. This analysis is completed in two major parts: (i) the analysis of the root mean square (RMS) for the carrier phase and (ii) the analysis of the station position in terms of repeatability. The second key section of the paper is a comprehensive carrier phase analysis conducted by adopting a new approach using a mean of the 25-point average RMS (A-RMS) and the single-point RMS and using an ionosphere-free linear combination. By using the ratio between the 25-point average RMS and the single-point RMS we can define the type of scatter that dominates the phase solution. The analyzed data span a one-year period. The tested GNSS stations belong to the EUREF Permanent Network (EPN) and the International GNSS Service (IGS). These comprise 55 GNSS stations, of which only 23 GNSS stations had more than 95% data availability for the entire year. The RMS and A-RMS are analyzed in conjunction with the precipitable water vapor (PWV), which shows clear signs of temporal correlation. Of the 23 GNSS stations, three stations show an increase of around 50% of the phase RMS when using a 3° cutoff elevation angle, and only four stations have a difference of 5% between the phase RMS when using both cutoff elevation angles. When using the A-RMS, there is an average improvement of 37% of the phase scatter for the 10° cutoff elevation angle, whereas for the 3° cutoff elevation angle, the improvement is around 33%. Based on studying this ratio, four stations indicate that the scatter is dominated by the stronger-than-usual dominance of long-period variations, whereas the others show short-term noise. In terms of station position repeatability, the weighted root mean square (WRMS) is used as an indicator, and the results between the differences of using a 3° and 10° cutoff elevation angle strategy show a difference of −0.16 mm for the North component, −0.21 mm for the East component and a value of −0.75 mm for the Up component, indicating the importance of using optimal cutoff angles.

Consistency of Galileo satellite antenna phase center offsets

Information about satellite antenna phase center offsets (PCOs) is indispensable for high-precision applications of global navigation satellite systems. Pre-flight manufacturer calibrations of the PCOs are available for all individual Galileo satellites and each frequency. So far, geodetic usage of Galileo is focused on the ionosphere-free linear combination of the E1 and E5a signals. In view of the growing number of E5b- and E6-capable receivers and upcoming multi-frequency applications, the consistency of the PCO values for different frequencies becomes a topic of increasing importance. Galileo satellite antenna PCOs have been estimated from the ionosphere-free linear combinations of E1/E5a, E1/E5b, and E1/E6. The mean horizontal PCOs of the different frequencies agree on the few millimeter level. The X-PCOs show a bias of about 1 cm compared to the manufacturer calibrations, whereas the Y-PCOs are free of such a bias. The Z-PCOs have a systematic offset of -11 to {-15} cm w.r.t. the manufacturer calibrations due to scale inconsistencies of the current version of the International Terrestrial Reference Frame (ITRF2020) and the manufacturer calibrations. The maximum Z-PCO difference between the various linear combinations amounts to 4 cm and provides a measure of the presently achieved consistency of ground and space antenna calibrations across different frequencies. This inconsistency would translate into height differences of about 1.6 mm and associated scale differences of the terrestrial reference frame of 0.25 ppb, when adjusting station coordinates with manufacturer calibrated Galileo PCOs for different frequency pairs.

Analysis of the Influence of Flood on the L4 Combination Observation of GPS and GLONASS Satellites

With global warming, extreme weather such as floods and waterlogging occurs more frequently and seriously in recent years. During the flood, the surrounding environment of the GNSS (Global Navigation Satellite System) station will change as the volume of water increases. Considering the multipath error is directly relevant to the observation environment, thus, the influence of flood on the L4 combination observation (a geometry-free ionosphere-free linear combination of carrier phase) which is related to the multipath error of GPS (Global Positioning System) and GLONASS satellites is investigated in depth. In addition, the ground track repetition periods of GPS and GLONASS satellites are analyzed in the sky plot to illustrate the rationality of chosen reference day. Based on the results of the satellite sky plot, one and eight days are adopted to demonstrate the influence of flood on L4 combination observation for GPS and GLONASS satellites, respectively. Real data sets collected at the ZHNZ GNSS observation station during the flood from DOY (Day of Year) 193 to DOY 204, 2021 are used. Experimental results show that the flood has a significant impact on the L4 combination observation of GPS and GLONASS satellites, and the fluctuation of L4 under flood performs much larger than that of without flood. For GPS satellites, the maximum RMS (root mean square) increase rate of L4 under flood is approximately 186.67% on the G31 satellite. Even for the minimum RMS increase rate, it can reach approximately 23.52%, which is the G02 satellite. Moreover, the average RMS increase rate of GPS and GLONASS satellites can reach approximately 109.53% and 43.65%, respectively. In addition, the influence of rainfall and hardware device are also investigated, which can further demonstrate that the fluctuation of L4 is mainly caused by the flood but not by the rainfall and hardware device elements. Thus, based on the above results, the influence of flood on L4 observation should be taken into account during the applications of L4 used, such as the retrieval of soil moisture and vegetation water content based on GNSS L4 combination observations

Performance of Galileo satellite products determined from multi-frequency measurements

Each Galileo satellite provides coherent navigation signals in four distinct frequency bands. International GNSS Service (IGS) analysis centers (ACs) typically determine Galileo satellite products based on the E1/E5a dual-frequency measurements due to the software limitation and the limited tracking capability of other signals in the early time. The goal of this contribution is to evaluate the quality of Galileo satellite products determined by using different dual-frequency (E1/E5a, E1/E5b, E1/E5, E1/E6) and multi-frequency (E1/E5a/E5b/E5/E6) measurements based on different sizes of ground networks. The performance of signal noise, the consistency of frequency-specific satellite phase center offsets and the stability of satellite phase biases are assessed in advance to confirm preconditions for multi-frequency processing. Orbit results from different dual-frequency measurements show that orbit precision determined from E1/E6 is clearly worse (about 35%) than that from other dual-frequency solutions. In view of a similar E1, E5a, E5b and E6 measurement quality, the degraded E1/E6 orbit performance is mainly attributed to the unfavorable noise amplification in the respective ionosphere-free linear combination. The advantage of using multi-frequency measurements over dual-frequency for precise orbit determination is clearly visible when using small networks. For instance, the ambiguity fixing rate is 80% for the multi-frequency solution while it is less than 40% for the dual-frequency solution if 150 s data sampling is employed in a 15-station network. Higher fixing rates result in better (more than 30%) satellite orbits and more robust satellite clock and phase bias products. In general, satellite phase bias products determined from a 20-station (or more) network are precise enough to conduct precise point positioning with ambiguity resolution (PPP-AR) applications. Multi-frequency kinematic PPP-AR solutions always show 5–10% precision improvement compared to those computed from dual-frequency observations.

Time transfer with BDS-3 signals: CV, PPP and IPPP

The third phase of BeiDou navigation satellite system (BDS-3) was officially commissioned on 31 July 2020. In this study, we make a comprehensive evaluation of BDS-3 time transfer with the B1I, B3I, B1C and B2a measurements. The multi-path (MP) errors and noises of different BDS-3 ranging signals are analyzed to illustrate characteristic of the code observations firstly. Then dual-frequency ionosphere-free linear combinations of BDS-3 B1I&B3I and B1C&B2a measurements are used to achieve time transfer. Using Multi-GNSS Experiment station observations, we evaluate the performance of BDS-3 time transfer with common-view (CV), precise point positioning (PPP) and integer ambiguity PPP (IPPP) techniques. Analysis results show that BDS-3 B1C&B2a CV time transfer links show a better performance than GPS L1P&L2P links, whereas BDS-3 B1I&B3I links are worse than GPS links. For the PPP time transfer, GPS links show the best performance, followed by BDS-3 B1C&B2a links and B1I&B3I links. Furthermore, frequency stability of BDS-3 IPPP time transfer is more stable than PPP solutions at the long average interval time. And the long-term frequency stability of BDS-3 IPPP is comparable with GPS IPPP.

On the potential contribution of BeiDou-3 to the realization of the terrestrial reference frame scale

Since the release of the phase center calibrations for both the receivers and the satellites, the BeiDou Navigation Satellite System (BDS) became a new potential contributor to the realization of the terrestrial reference frame (TRF) scale of future International Terrestrial Reference Frame releases. This study focuses on the evaluation of the potential usage of the BDS-3 Medium Earth Orbit (MEO) constellation to the definition of the TRF scale. To that aim, we used ground calibrated BDS-3 satellite PCOs provided by the China Satellite Navigation Office and multi-GNSS robot calibrations for the ground antennas conducted by Geo++. Two ionosphere-free linear combinations of signals, namely B1I/B3I and B1C/B2a, have been investigated to find out whether using different frequencies may lead to different TRF scale definitions. Differences between the z components of the satellite phase offsets as given by manufacturer calibrations and those estimated based on IGS14 scale amount to 6.55 ± 12.56 cm and − 0.32 ± 10.99 cm for B1I/B3I and B1C/B2a frequency pairs, respectively. On the one hand, the substantial deviation from the mean reflects the disparities in the quality of calibrations for the individual spacecraft, especially those manufactured by the Shanghai Engineering Center for Microsatellites (SECM). On the other hand, the difference between the two frequency pairs arises to a great extent from the doubtful quality of the SECM PCO calibrations, which certainly do not reflect the frequency dependence of the PCOs. Eventually, the mean scale bias with respect to IGS14 equals + 0.546 ± 0.085 ppb and + 0.026 ± 0.085 ppb for B1I/B3I and B1C/B2a solutions, respectively, when using all 24 BDS-3 MEO satellites.

A Novel Clock Parameterization and Its Implications for Precise Point Positioning and Ionosphere Estimation.

By convention, IGS precise clock products are computed using the ionosphere-free linear combination. Due to the broad use of IGS products, this convention is exploited in PPP-RTK models not using such a linear combination. So, in different carrier phase combinations, the code hardware biases are contained in different combinations, thus making the problem of separating biases from integer ambiguities more complicated. In this paper, we proposed a novel clock parameterization which allows facilitating this problem. Based on the proposed parameterization, we derived a dual-frequency PPP-RTK model for the undifferenced measurements and assessed this model for the static positioning case in terms of positioning accuracy, convergence, and ambiguity resolution performance. The results showed that a cm-level accuracy level is achievable with the derived models with nearly instant convergence and almost 100% successfully resolved ambiguities. We demonstrated the use of this parameterization for slant ionosphere estimation. We derived the analog of the equation linking the wide-lane, geometry-free, and ionosphere-free biases from the Fast-PPP system and used it to retrieve slant ionosphere information. Our TEC estimates showed some evidence of capability to reach an agreement of 1–2 TECU and the standard deviation of 3–4 TECU with GIM TEC values.

Ambiguity Resolution for Long Baseline in a Network with BDS-3 Quad-Frequency Ionosphere-Weighted Model

For long baseline in a network, the traditional combined ionosphere-free (IF) + wide-lane (WL) strategy is commonly used, but the residual tropospheric delays and larger noise hamper the basic ambiguity resolution (AR). With the completion of the BeiDou global navigation satellite system (BDS-3) and the quad-frequency signals provided by BDS-3 satellites, we can construct more combinations that are conducive to ambiguity resolution. Compared with ionosphere-free linear combinations, we estimated ionospheric delay using three independent WL observations, and formed an ionosphere-weighted model using uncombined code and phase observations, which proved to be quite effective. Based on the real quad-frequency BDS-3 observations of two CORS (Continuously Operating Reference Stations) and two user stations, we processed eight days of data to study the formal and empirical ambiguity success rates and user positioning errors. The rounding success rate of WL ambiguity was significantly improved with ionospheric correction. The success rate of the basic ambiguity increased from 94.4 and 96.1% to 98.0% using the quad-frequency ionosphere-weighted (QFIW) model compared with the double-frequency ionosphere-free (DFIF) model and the triple-frequency geometry-based (TFGB) model. Furthermore, the user E/N/U positioning accuracy improved by 20.6/31.5/13.1% and 6.3/22.9/5.8%, respectively.

The refractive and diffractive contributions to GPS signal scintillation at high latitudes during the geomagnetic storm on 7–8 September 2017

Different indices have been used to reflect, or monitor the ionospheric scintillation, e.g. the detrended carrier phase,σφ,S4,the rate of change of the vertical total electron content index (vROTI), as well as the ionosphere‐free linear combination (IFLC) of two carrier phases. However, few studies have been performed to investigate the refractive and diffractive contributions to these indices, especially during geomagnetic storms. In this study, we analyze the high-resolution (50 Hz) phase and amplitude measurements from four high-latitude stations in Svalbard, Norway during the geomagnetic storm on 7–8 September 2017. Our results show that at high latitudes, the high-pass filter with a standard cutoff frequency of 0.1 Hz sometimes cannot effectively remove the refraction-driven phase variations, especially during the geomagnetic storm, leading to a remaining refraction contribution to the detrended carrier phase andσφwhen scintillation happens. In the meanwhile, asvROTI is sensitive to the TEC gradients, regardless of small- or large-scale ionospheric structures, both refraction and diffraction effects can cause visible fluctuations ofvROTI. For most of the scintillation events, the phase indices (including detrended carrier phase,σφ, andvROTI), IFLC,andS4show consistent fluctuations, indicating that diffraction usually occurs simultaneously with refraction during scintillation. One interesting feature is that although the IFLC andS4are thought to be both related to the diffraction effect, they do not always show simultaneous correspondence during scintillations. The IFLC is enhanced during the geomagnetic storm, while such a feature is not seen inS4. We suggest that the enhanced IFLC during the geomagnetic storm is caused by the increased high-frequency phase power, which should be related to the enhanced density of small-scale irregularities during storm periods.

Estimation and Analysis of the Observable-Specific Code Biases Estimated Using Multi-GNSS Observations and Global Ionospheric Maps

Observable-specific bias (OSB) parameterization allows observation biases belonging to various signal types to be flexibly addressed in the estimation of ionosphere and global navigation satellite system (GNSS) clock products. In this contribution, multi-GNSS OSBs are generated by two different methods. With regard to the first method, geometry-free (GF) linear combinations of the pseudorange and carrier-phase observations of a global multi-GNSS receiver network are formed for the extraction of OSB observables, and global ionospheric maps (GIMs) are employed to correct ionospheric path delays. Concerning the second method, satellite and receiver OSBs are converted directly from external differential code bias (DCB) products. Two assumptions are employed in the two methods to distinguish satellite- and receiver-specific OSB parameters. The first assumption is a zero-mean condition for each satellite OSB type and GNSS signal. The second assumption involves ionosphere-free (IF) linear combination signal constraints for satellites and receivers between two signals, which are compatible with the International GNSS Service (IGS) clock product. Agreement between the multi-GNSS satellite OSBs estimated by the two methods and those from the Chinese Academy of Sciences (CAS) is shown at levels of 0.15 ns and 0.1 ns, respectively. The results from observations spanning 6 months show that the multi-GNSS OSB estimates for signals in the same frequency bands may have very similar code bias characteristics, and the receiver OSB estimates present larger standard deviations (STDs) than the satellite OSB estimates. Additionally, the variations in the receiver OSB estimates are shown to be related to the types of receivers and antennas and the firmware version. The results also indicate that the root mean square (RMS) of the differences between the OSBs estimated based on the CAS- and German Aerospace Center (DLR)-provided DCB products are 0.32 ns for the global positioning system (GPS), 0.45 ns for the BeiDou navigation satellite system (BDS), 0.39 ns for GLONASS and 0.22 ns for Galileo.

Estimating ambiguity fixed satellite orbit, integer clock and daily bias products for GPS L1/L2, L1/L5 and Galileo E1/E5a, E1/E5b signals

Ambiguity resolution of a single receiver is becoming more and more popular for precise GNSS (Global Navigation Satellite System) applications. To serve such an approach, dedicated satellite orbit, clock and bias products are needed. However, we need to be sure whether products based on specific frequencies and signals can be used when processing measurements of other frequencies and signals. For instance, for Galileo E5a frequency, some receivers track only the pilot signal (C5Q) while some track only the pilot-data signal (C5X). We cannot compute the differences between C5Q and C5X directly since these two signals are not tracked concurrently by any common receiver. As code measurements contribute equally as phase in the Melbourne-Wuebbena (MelWub) linear combination it is important to investigate whether C5Q and C5X can be mixed in a network to compute a common satellite MelWub bias product. By forming two network clusters tracking Q and X signals, respectively, we confirm that GPS C5Q and C5X signals cannot be mixed together. Because the bias differences between GPS C5Q and C5X can be more than half of one wide-lane cycle. Whereas, mixing of C5Q and C5X signals for Galileo satellites is possible. The RMS of satellite MelWub bias differences between Q and X cluster is about 0.01 wide-lane cycles for both E1/E5a and E1/E5b frequencies. Furthermore, we develop procedures to compute satellite integer clock and narrow-lane bias products using individual dual-frequency types. Same as the finding from previous studies, GPS satellite clock differences between L1/L2 and L1/L5 estimates exist and show a periodical behavior, with a peak-to-peak amplitude of 0.7 ns after removing the daily mean difference of each satellite. For Galileo satellites, the maximum clock difference between E1/E5a and E1/E5b estimates after removing the mean value is 0.04 ns and the mean RMS of differences is 0.015 ns. This is at the same level as the noise of the carrier phase measurement in the ionosphere-free linear combination. Finally, we introduce all the estimated GPS and Galileo satellite products into PPP-AR (precise point positioning, ambiguity resolution) and Sentinel-3A satellite orbit determination. Ambiguity fixed solutions show clear improvement over float solutions. The repeatability of five ground-station coordinates show an improvement of more than 30% in the east direction when using both GPS and Galileo products. The Sentinel-3A satellite tracks only GPS L1/L2 measurements. The standard deviation (STD) of satellite laser ranging (SLR) residuals is reduced by about 10% when fixing ambiguity parameters to integer values.

Применение теории решетчатых упаковок в задаче высокоточного абсолютного местоопределения по ионосферосвободным измерениям в ГНСС с кодовы м разделением

The paper considers the use of the lattice packing theory for Integer Precise Point Positioning (Integer PPP) with the errors usually not exceeding 1–3 cm based on GNSS signals with code division multiple access (CDMA). Positioning is carried out by processing ionosphere-free linear combinations of code and phase measurements with ambiguity resolution employing satellite corrections. The main issue of PPP algorithms is overcoming the rank deficiency problem of the linear equation system obtained by linearization of nonlinear mathematical models of measurements. Nowadays Float PPP is quite well developed, where rank deficiency is tackled by combining systematic biases in measurement models with integer carrier phase ambiguities. As a result, the number of unknowns is reduced to the rank of design matrix, which allows unambiguous estimation of precise user coordinates and values of new variables generated by the performed combinations. However, under such conditions the information about integer nature of carrier phase ambiguities is lost, and this leads to a significant increase in convergence time to obtain user coordinates estimates with the errors of 1–3 cm. It is possible to involve the information on the integer nature of phase ambiguities into processing by applying ambiguity resolution algorithms. Though, as a result of the conducted combinations, the integer nature is destroyed, which makes it impossible to apply these algorithms. In Integer PPP rank deficiency is overcome by projecting the state space of the initial linear equation system onto a so-called S-space, whose dimension is equal to the rank of this system. The orientation of the S-space and the direction of projecting are chosen so that the variables of the initial system corresponding to user coordinates are not changed during the projecting and the projections of integer variables remain integer. This makes it possible to estimate precise user coordinates involving information on the integer nature of phase ambiguities. In the literature on Integer PPP based on CDMA GNSS signals processing the description of the S-space orientation with the desired properties is given, but there is no description of the method to determine this orientation. This paper based on the notions of the lattice packing theory considers an algorithm for determining the S-space with the desired properties. It is shown that there exists an infinite set of such S-spaces connected by unimodular transformations, and a technique is proposed to enable selection from this set the S-space, which requires minimal computational cost. The use of the lattice packing theory to the Integer PPP network solution with CDMA GNSS signals will be considered in the following publication of the authors.

An SBAS Integrity Model to Overbound Residuals of Higher-Order Ionospheric Effects in the Ionosphere-Free Linear Combination

The next generation of satellite-based augmentation systems (SBAS) will support aviation receivers that take advantage of the ionosphere-free dual-frequency combination. By combining signals of the L1 and L5 bands, about 99% of the ionospheric refraction effects on the GNSS (Global Navigation Satellite Systems) signals can be removed in the user receivers without additional SBAS corrections. Nevertheless, even if most of the negative impacts on GNSS signals are removed by the ionospheric-free combination, some residuals remain and have to be taken into account by overbounding models in the integrity computation conducted by safety-of-live (SoL) receivers in airplanes. Such models have to overbound residuals as well, which result from the most rare extreme ionospheric events, e.g., such as the famous “Halloween Storm”, and should thus include the tails of the error distribution. Their application shall lead to safe error bounds on the user position and allow the computation of protection levels for the horizontal and vertical position errors. Here, we propose and justify such an overbounding model for residual ionospheric delays that remain after the application of the ionospheric-free linear combination. The model takes into account second- and third-order ionospheric refraction effects, excess path due to ray bending, and increased ionospheric total electron content (TEC) along the signal path due to ray bending.

GPS triple-frequency undifferenced and uncombined precise orbit determination with the consideration of receiver time-variant bias

As the development of the global navigation satellite system (GNSS), the navigation satellite transmitting multi-frequency signals becomes a prevailing trend. Currently, the usual GNSS precise orbit determination (POD) strategy uses dual-frequency (DF) ionosphere-free (IF) linear combination to build up the observation equation. In the area of precise positioning, it has been validated that the third frequency can decrease the convergence time of ambiguity resolution and has a subtle improvement on the positioning accuracy. However, there is no report about the benefits of the third frequency on GNSS POD. Although the precision of dual-frequency POD can meet requirements for generating different types of products, an obvious advantage of multi-frequency observations is that more robust results can be derived. Besides, with the third frequency observations, the improvements of orbits and clocks need to be validated. Thus, a triple-frequency (TF) uncombined (UC) POD method of GNSS satellites is developed. The hardware delay of carrier phase is divided by time-invariant and -variant components. Then the UC observation model is given by re-parameterizing the unknown parameters. The datum of satellite clocks is aligned to IGS products. The step-by-step ambiguity fixing method, i.e. the extra-wide-lane, wide-lane and narrow-lane ambiguities being fixed in sequence, is deduced by using double-differenced ambiguities in a network. We select 74 ground stations and 12 GPS BLOCK IIF satellites to process POD with a length of 20 days. Results of TF-UC POD show that about 10% improvements on orbits and clocks are received compared to L1/L2 DF-IF POD. The time-variant characteristic of phase biases from satellites and stations at the third frequency is analyzed, which shows that the bias at the station has a slight influence on the derived products. These results imply that the precision of orbits and clocks can be improved with observations from the third frequency added.

Precision point positioning with additional baseline vector constraint

Abstract Traditional precise point positioning (PPP) based on undifferenced ionosphere-free linear combination of observations has many advantages such as high accuracy and easy operation. PPP usually uses the Kalman Filter (KF) to estimate state vector. However, the positioning performance depends on the accuracy of the kinematic model and initial value. The inaccurate kinematic model or initial value will lead to filter performance degradation or even divergence. To overcome this problem, this paper proposes a PPP method with an additional baseline vector constraint, which uses the direction information and length information of the baseline to correct the estimated position of the receiver. By reducing the error covariance matrix of the float solution, the algorithm improves the accuracy of the float solution. By using the collected real GPS static and kinematic data, the performance of the traditional model and the proposed model in this paper are compared. It is shown that the additional baseline vector constraint improves the PPP solution effectively in comparison with that of traditional PPP model. Additionally, the contribution of the additional constraint is up to the accuracy of the prior information.

Precise point positioning with decimetre accuracy using wide-lane ambiguities and triple-frequency GNSS data

Abstract This paper proposes precise point positioning (PPP) methods that offer an accuracy of a few decimetres (dm) with triple frequency GNSS data. Firstly, an enhanced triple frequency linear combination is presented for rapid fixing of the extra wide-lane (EWL) and wide-lane (WL) ambiguities for GPS, Beidou-2 and Galileo. This has improved performance compared to the Melbourne-Wübbena (MW) linear combination, and has 6.7 % lower measurement noise for the GPS L1/L2 signals, 12.7 % for L1/L5 and 0.7 % for L2/L5. Analysis with tested data showed a 5–6 % reduction in time required to fix the N 21 {N_{21}} and N 51 {N_{51}} ambiguities. Once the EWL/WL ambiguities are fixed with the proposed linear combinations, three methods are presented that aim to provide positioning accuracy of a few dm. In the first approach, the three EWL/WL ambiguities in their respective phase equations are used to derive a low-noise ionosphere-free (IF) linear combination. The second method uses a low noise IF combination with two carrier-phase EWL/WL equations and a single pseudorange measurement. The third method uses a low noise IF combination with a single carrier phase EWL equation and two pseudorange measurements. These proposed methods can provide dm level positioning accuracy if carrier phase measurements with mm precision is tracked by the receiver. When comparing these combinations with a combination proposed in [22], it is found that superior performance is achieved with the third method when carrier phase noise is >5–6 mm for GPS and Beidou-2 and >2–3 mm for Galileo. This model only requires the EWL ambiguity to be fixed which typically takes just one epoch of data. Thus, the user achieves instant decimetre level PPP accuracy.

Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation.

This contribution presents and assesses the methodology aiming at the characterization of the structural vibrations with high-rate GNSS measurements. As commonly employed precise point positioning (PPP) based on ionosphere-free linear combination of undifferenced signals may not meet the high requirements in terms of displacement precision, a modified processing strategy has been proposed. The algorithms were implemented in the own-developed GNSS processing software and validated using the designed experiment. For this purpose, we have set up a field experiment taking advantage of the prototype shake-table, which simulated the dynamic horizontal displacements of the GNSS antenna. The device ensured a periodic motion of the antenna with modifiable characteristics, namely amplitude and frequency. In this experiment, we have set the amplitudes from 1.5 to 9 mm and the frequency to 3.80 Hz. As a dataset, we have used 100 Hz GPS, Galileo, and BDS measurements. The results confirmed a high applicability of the enhanced PPP processing strategy for precise displacement detection. Specifically, it was feasible to obtain the dynamic displacements with precision at the level of millimeters. The differences between the PPP-derived amplitude and the true amplitude of the simulated displacements were in the range of 0.5–1.3 mm, whereas the difference between the detected and benchmark frequency did not exceed 0.026 Hz. Hence, the proposed methodology allows meeting the specific demands of structural displacement monitoring.

Monitoring Zenithal Total Delays over the three different climatic zones from IGS GPS final products: A comparison between the use of the VMF1 and GMF mapping functions

The International GNSS Service (IGS) final products (ephemeris and clocks-correction) have made the GNSS an indispensable low-cost tool for scientific research, for example sub-daily atmospheric water vapor monitoring. In this study, we investigate if there is a systematic difference coming from the choice between the Vienna Mapping Function 1 (VMF1) and the Global Mapping Function (GMF) for the modeling of Zenith Total Delay (ZTD) estimates, as well as the Integrated Precipitable Water Vapor (IPWV) estimates that are deduced from them. As ZTD estimates cannot be fully separated from coordinate estimates, we also investigated the coordinate repeatability between subsequent measurements. For this purpose, we monitored twelve GNSS stations on a global scale, for each of the three climatic zones (polar, mid-latitudes and tropical), with four stations on each zone. We used an automated processing based on the Bernese GNSS Software Version 5.2 by applying the Precise Point Positioning (PPP) approach, L3 Ionosphere-free linear combination, 7 ̊ cutoff elevation angle and 2 h sampling. We noticed an excellent agreement with the ZTD estimates and coordinate repeatability for all the stations w.r.t to CODE (the Center for Orbit Determination in Europe) and USNO (US Naval Observatory) products, except for the Antarctic station (Davis) which shows systematic biases for the GMF related results. As a final step, we investigated the effect of using two mapping functions (VMF1 and GMF) to estimate the IPWV, w.r.t the IPWV estimates provided by the Integrated Global Radiosonde Archive (IGRA). The GPS-derived IPWV estimates are very close to the radiosonde-derived IPWV estimates, except for one station in the tropics (Tahiti).