GPS Solutions

Low-cost sensor navigation has risen in the past decade with the onset of many modern applications that demand decimeter-level accuracy using mass-market sensors. The key advantage of the precise pointing positioning (PPP) technique over real-time kinematic (RTK) is the non-requirement of local infrastructure and still being able to attain decimeter to sub-meter level accuracy while using mass-market low-cost sensors. Achieving decimeter to sub-meter-level accuracy is a challenge in urban environments. Therefore, adaptive filtering for low-cost sensors is necessary along with motion-based constraining and atmosphere constraints. The traditional robust adaptive Kalman filter (RAKF) uses empirical limits that are derived by analyzing the GNSS receiver data learning statistics based on confidence intervals beforehand to determine when the adaptive factor needs to be applied. In this research, a new technique is proposed to determine the adaptive factor computation based on the detection of an increase in the number of satellite signals after a partial outage. The proposed method provides 6–46% better accuracy than the traditional RAKF and 11–55% better accuracy performance when compared to a tightly coupled solution without enhancements when multiple datasets were tested. The results prove to be a significant improvement for the next generation of applications, such as low-autonomous and intelligent transportation systems.

Multipath (MP) reception has been among the main issues for accurate and reliable positioning in urban environments. It has been shown to introduce positioning errors of up to tens of meters for conventional two-step (2SP) receivers. The direct position estimation (DPE) has been introduced as a more robust positioning algorithm compared to the conventional two-step (2SP) receivers in terms of MP mitigation. However, its high computational load prevents DPE from real-time positioning for commercial receivers. Thus, we present a novel grid-based maximum likelihood estimation (MLE) algorithm based on DPE by making use of pseudorange measurements to obtain the correlogram on a predefined searching space. Unlike DPE, which performs correlations at the intermediate frequency (IF) level, correlations are done by directly comparing the code phase of each candidate position, velocity, and timing with the incoming pseudorange. This way, the proposed method retains MP mitigation properties from DPE through the use of MLE from DPE and allows for a significantly reduced computational load compared to DPE. The proposed method was tested with both open-sourced datasets collected in urban environments as well as IF simulation data, and its performance is evaluated against a 2SP receiver. Results show that the proposed method manages to acquire the MP mitigation capability of DPE and outperforms 2SP by up to around 23% in the tested urban datasets and 91% in the simulation data, at a much-reduced computational time. The resilience of our proposed method against MP and NLOS could even potentially offer applications in geodetic networks, where robust estimators are traditionally employed to counteract outliers. 

With the increasing number of low-cost GNSS antennas available on the market, there is a lack of comprehensive analysis and intercomparison of their performance. Moreover, multi-GNSS observation noises are not well recognized for low-cost receivers. This study characterizes the quality of GNSS signals acquired by low-cost GNSS receivers equipped with eight types of antennas in terms of signal acquisition, multipath error and receiver noise. The differences between various types of low-cost antennas are non-negligible, with helical antennas underperforming in every respect. Compared with a geodetic-grade station, GPS and Galileo signals acquired by low-cost receivers are typically weaker by 3–9 dB-Hz. While the L1, E1 and E5b signals are well-tracked, only 72% and 86% of L2 signals are acquired for GPS and GLONASS, respectively. The signal noise for pseudoranges varies from 0.12 m for Galileo E5b to over 0.30 m for GLONASS L1 and L2, whereas for carrier-phase observations it oscillates around 1 mm for both GPS and Galileo frequencies, but exceeds 3 mm for both GLONASS frequencies. Antenna phase center offsets (PCOs) vary significantly between frequencies and constellations, and do not agree between two antennas of the same type by up to 25 mm in the vertical component. After a field calibration a of low-cost antenna and consistent application of PCOs, the horizontal and vertical accuracy is improved to a few millimeter and a few centimeter level for the multi-GNSS processing with double-differenced and undifferenced approach, respectively. Last but not least, we demonstrate that PPP-AR is possible also with low-cost GNSS receivers and antennas, and improves the precision and convergence time. The results prove that selection of low-cost antenna for a low-cost GNSS receiver is of great importance in precise positioning applications.

The Global Positioning System (GPS) signals-based roll angle measurement methods are key technologies for spinning vehicles with side-mounted single patch antenna. Most existing research focuses on attitude measurement at low spin rates, and it is difficult to achieve accurate roll angle measurement at a time-varying situation with high spin rates, and the attitude measurement errors are not modeled and analyzed. A multi-channel roll angle measurement method based on a third-order phase-locked loop is proposed to improve the attitude measurement accuracy and stability. The effects of phase-locked loop (PLL) tracking errors, errors induced by antenna gain, and antenna phase center offset errors are analyzed and verified by simulation. The intermediate frequency (IF) GPS signals received by a single patch antenna are sampled on a testbed to verify the roll angle measurement method. The experiment results show that the root-mean-square error (RMSE) of the roll angle measurement is 5° ± 1° at spin rates ranging from 50 to 200 r/s.

The thermal expansion effect exhibited by the GNSS monument and their nearby bedrock due to surface temperature variations is an important factor affecting the nonlinear variation of GNSS height. However, current thermal expansion models either consider only the above-surface GNSS monuments or only non-seasonal temperature variations of the subsurface bedrock, and lack a comprehensive thermal expansion model. Furthermore, previous studies on the contribution of thermal expansion effects to nonlinear variations in GNSS height have been analyzed only against one set of GNSS time series products per single research, but in fact the GNSS time series provided by each agency varied considerably. In this study, we use a refined comprehensive thermal expansion model (TEVDFSD) to evaluate its contribution to the nonlinear variations from several GNSS height products (SOPAC, JPL, and COMBINED) obtained using different data processing strategies. The results show that the most GNSS stations (about 95%) using the TEVDFSD model exhibit an annual amplitude increase and phase lag, with an amplitude increase of up to 0.5 mm and phase lag of up to 13° compared with the finite element model, especially for inland and those with deeper GNSS monument stations. This phase lag improves its correlation with the GNSS height, which reduces the GNSS height value to improve the geophysical interpretation. The TEVDFSD model estimates an annual amplitude of up to 7.5 mm, explaining at most 13.6% of the nonlinear variation in the COMBINED height. The COMBINED product exhibits a further WRMS reduction of up to 20% and 18.7% compared with the SOPAC and JPL products, respectively, which are likely due to the higher accuracy of the combined GNSS solution than of the independent data processing strategy. Our work indicates that differences in data processing strategies for GNSS height time series products significantly affect the interpretability of thermal expansion effects to nonlinear variations.

Seasonal signals in satellite geodesy time series are mainly derived from a number of loading sources, such as atmospheric pressure and hydrological loading. The most common method for modeling the seasonal signal with quasi-period is to use the sine and cosine functions with the constant amplitude for approximation. However, due to the complexity of environmental changes, the time-varying period part is very difficult to model by the geometric or physical method. We present a machine learning method with Gaussian process to capture the quasi-periodic signals in the geodetic time series and optimize the estimation of model parameters by means of maximum likelihood estimation. We test the performance of the method using the synthetic time series by simulating the time-varying and quasi-periodic signals. The results show that the fitting residuals of the new model show a better random fluctuation, while the traditional models still leave the clear periodic systematics signals without being fully modeled. The new model illustrates a higher reliability of linear trend estimation, and a lower uncertainty and model fitting RMSE, even in time series with shorter time span. On the other hand,  it shows a strong capacity to restore the missing data and predict the future changes in time series. The method is successfully applied to modeling the real coordinate time series of the GNSS site (BJFS) from IGS network, and the equivalent water height (EWH) time series in North China obtained from gravity satellites. Therefore,  it is recommended as an alternative for precise model reconstruction and signals extraction of satellite geodesy time series, especially in modeling the complex time-varying signals, estimating the secular motion velocity, and recovering the large missing data.

The International Reference Ionosphere (IRI) model is a widely used empirical model to describe ionospheric climatology. However, IRI represents the monthly averages of the ionospheric parameters, which makes it difficult to capture the local and short-term ionospheric variations. To overcome this limitation, we propose a data ingestion method using a combination of ground-based and space-borne observations. The ionospheric parameters from ground-based Global Navigation Satellite System (GNSS), ionosondes, space-borne GNSS radio occultation and satellite altimetry observations are ingested into the IRI-2020 model to improve its accuracy. The outputs of the ingested IRI (IRIinge) are assessed by case study and statistical analysis, with reference to independent ionosonde observations and global ionospheric maps. The case study shows that IRIinge expresses the diurnal and local variations of the ionosphere better than the standard IRI (IRIstan) in both high and low solar activity periods. The relative error of ionospheric electron density profiles from IRIinge is generally less than 10%, and the vertical total electron content from IRIinge has an accuracy improvement of 39.0% compared to that from IRIstan. The statistical analysis shows that IRIinge performs more stable than IRIstan, and its output generally has smaller REs and root-mean-square errors, especially in daytime and storm time. The proposed method significantly improves IRI-2020 on the accuracy of the output parameters and the ability to present the short-term variations of the ionosphere.

The massive number of global navigation satellite system (GNSS) users and frequent positioning demands in cities, as well as the complexity of urban scenarios, pose many challenges for the accuracy and reliability of precise positioning. Since urban environments tend to suffer from GNSS non-line-of-sight (NLOS) signal conditions, leading to large ranging errors, NLOS signal identification and processing are of great importance. Usually, a visual camera can reflect real occlusion, and machine learning is efficient and accurate in processing multiple types of features. Therefore, an algorithm is proposed that combines the advantages of both methods. First, NLOS labels are generated using a combination of an inertial navigation system (INS) and a fisheye camera, and a total of nine features, namely, the elevation angle as well as the signal-to-noise ratios (SNRs), SNR fluctuation magnitudes, pseudorange consistencies, and pseudorange multipath errors at two frequencies, are extracted. Then, to improve efficiency and avoid overfitting, the nine original features are aggregated into three common factors via factor analysis, and these three factors can be well interpreted. Finally, a NLOS signal identification model based on the random forest (RF) algorithm is designed. In addition, to improve the precise point positioning (PPP) performance, a weighting scheme based on the elevation angle and SNR is optimized in accordance with the probability of NLOS occurrence. In an experiment, the RF model is trained using on-board dynamic multi-GNSS dual-frequency data collected by a low-cost UBLOX F9P receiver in Wuhan, and then validation is performed using data collected in Wuhan and Zhengzhou. The experimental results show that compared with the gradient boosted decision tree (GBDT), support vector machine (SVM), naive Bayes (NB), and convolutional neural network (CNN) algorithms, the RF model shows superior performance. While achieving 87.5% and 72.5% accuracy on the local and remote test datasets, respectively, the RF model costs only 12.2 ms for LOS/NLOS classification per epoch. Moreover, through factor analysis, the computational efficiency is improved by 29.5% for all five algorithms. Additionally, the accuracy and stability of uncombined PPP are improved using the proposed weighting strategy.

Dual-frequency multi-constellation (DFMC) satellite-based augmentation system (SBAS) is a new SBAS standard for aeronautical navigation systems. It supports aircraft navigation from the enroute to approach phases via the L1 and L5 frequencies (1575.42 and 1176.45 MHz). Although the ionosphere-free (IF) combination in the DFMC SBAS operation removes the first-order ionospheric delays in the pseudorange measurement, remaining terms including the satellite-clock offset errors and higher-order ionospheric (HOI) delays are still unaccounted for. The DFMC SBAS accuracy and integrity can be affected by the HOI effects, especially during severe ionospheric disturbances. In this work, we present the local DFMC SBAS corrections with and without the mitigation of HOI delays. We first estimate the HOI delay terms using the received pseudorange followed by separate satellite and receiver bias estimations based on the minimum sum-variance technique. The integrity terms can then be obtained. The performances of DFMC SBAS using the global navigation satellite system (GNSS) data including GPS, Galileo, and QZSS are evaluated using obtained GNSS data at stations in Thailand on the ionospheric quiet and disturbed days. The results show that with the HOI mitigation, the vertical positioning errors (VPE) on the quiet and disturbed days can be improved by 12% and 9%, whereas the vertical protection levels (VPL) are improved by 16% and 21%, respectively. In addition, we perform a preliminary assessment of DFMC SBAS based on the International Civil Aviation Organization (ICAO) requirements of two categories: Localizer Performance with Vertical guidance (LPV-200) and Category I precision approach (CAT-I) showing promising results.

In land surface remote sensing using Global Navigation Satellite System reflectometry (GNSS-R) signals, it is common to observe signal coupling between the reflections from the soil surface and vegetation. But most recent research focuses on either bare soil or single vegetation. The vegetation significantly reduces the amplitude of the GNSS signal and increases the standard deviation (STD) of the carrier phase and pseudorange calculations. This study proposes a solution that uses GNSS transmission reflectometry (GNSS-TR) and wavelet transform to decouple signals reflected off vegetation-covered and bare soil surfaces. This coupling persists despite the ability of wavelet transform to initially separate the signal-to-noise ratio (SNR) sequences of GNSS reflected signals into different frequency components. This study uses the power of GNSS transmission signals, which carry almost exclusively vegetation information, as a priori information input to calibrate the influence of vegetation on the reflected signal from the soil surface to maximize the decoupling of the two reflected signals. Furthermore, signals from above- and below-vegetation GNSS antennas were simultaneously collected using a low-cost, dual-channel GNSS chipset, which can increase GNSS signal processing channels while reducing equipment costs. The results show that the solution proposed in this study can reach a correlation coefficient of 0.96 between the retrieval and in situ soil moisture content (SMC), and the root mean square error (RMSE) is reduced to 0.013 cm3/cm3. Moreover, the transmitted signal power, pseudorange STD, and carrier phase STD showed a clear trend with the vegetation growth status (VGS).

The atmospheric weighted mean temperature (Tm) is a key parameter in determining the precipitable water vapor (PWV). Conventional meteorological parameter empirical models have a lower spatial resolution and poor regional applicability, resulting in lower accuracy in obtaining the Tm values in global navigation satellite system (GNSS) PWV retrieval. We discuss a long short-term memory-based ERA5 temperature (LSTM-ERATM) model and evaluate the accuracy of calculating the Tm. Considering Tm’s annual, semi-annual, and daily cycle characteristics, an ERATM model was developed based on the ERA5 data from 2017 to 2020 provided by the European Center for Mesoscale Weather Forecasts (ECMWF). Then, the LSTM model was used to train the differences between the Tm values obtained by discrete integration of the ERA5 data and Tm values calculated by the ERATM model to enhance the accuracy of the ERATM model. We use the ERA5 and sounding data from 2021 to 2022 to analyze the calculation effect of the LSTM-ERATM, ERATM, GPT3, UNB3, and Bevis models. The results show that the ERATM model has broad regional applicability and can provide high-accuracy Tm. Compared with the UNB3, GPT3, and Bevis models, the mean root-mean-square (RMS) values of the ERATM model is reduced by 43.4%, 3.4%, and 11.7% respectively when using the ERA5 data as the reference values, and reduced by 22.9%, 13.9%, and 0.2% respectively when using the sounding data as the reference values. Moreover, the accuracy of the LSTM-ERATM is generally better than ERATM at different time points and regions, which shows that the LSTM model effectively improves the accuracy of the ERATM model in calculating Tm. For example, the mean RMS values of LSTM-ERATM were reduced by 50.8%, 37.4%, 26.2%, and 18.9% in the next time points of 6:00, 12:00, 18:00, and 24:00 respectively when using the ERA5 data as the reference values, and reduced by 31.3%, 27.2%, 35.9%, and 8.6% respectively when using the sounding data as the reference values. The LSTM-ERATM model in this study provides a powerful tool to improve the accuracy of calculating Tm, which can provide more reliable data for meteorology and climate research.

High-precision satellite laser ranging measurements to Galileo retroreflector panels are analyzed to determine the angle of incidence of the laser beam based on specific orientations of the panel with respect to the observing station. During the measurements, the panel aligns with respect to the observing station in such a way that multiple retroreflectors appear at the same range, forming regions of increased data density—separated by a few millimeters. First, measurements to a spare IOV-type retroreflector mounted on an astronomical mount at a remote location 32 km away from the Graz laser ranging station are performed. In addition, more than 100 symmetry passes to Galileo satellites in orbit have been measured. Two novel techniques are described to form laser ranging normal points with improved precision compared to traditional methods. An individual normal point can be formed for each set of retroreflectors at a constant range. The central normal point was shown to be up to 4 mm more accurate when compared with a precise orbit solution. Similar offsets are determined by applying a pattern correlation technique comparing simulated with measured data, and the first method is verified. Irregular reflection patterns of Galileo FOC panels indicate accumulated far-field diffraction patterns resulting from non-uniform retroreflector distributions.

Orbit prediction (OP) recently tends to be a very crucial step for supporting real-time GNSS orbit services due to the dynamic stability of navigation satellite orbits. The OP performance depends on the length of the predicted orbits and the accuracy of precise orbit determination (POD) as basis. Considering this, a new automatic processing engine is established for improving the multiple global navigation satellite systems (multi-GNSS) constellation OP performance. From the architecture-oriented high-performance parallel processing perspective, the multi-node and multi-core computer sources are fully exploited to implement the hourly update of the current multi-GNSS POD. For MEO-type satellites (e.g., Galileo satellites), the accuracy of predicted orbits is improved from 3.8 cm, 6.5 cm, and 12.3 cm to 3.5 cm, 4.3 cm, and 6.3 cm, in the radial, cross, and along directions, respectively, compared to the three-hour POD update. Despite the shortened OP length, the OP performance of regional navigation satellite system (RNSS) satellites is still limited due to their regional observability. The BDS-IGSO and QZSS-IGSO satellitesexhibit radial directional orbital errors of up to 36.9 cm and 28.9 cm, respectively. Therefore, an orbit fitting (OF) processing method with orbit reconstruction is implemented into the processing engine. By utilizing this method, the radial orbital errors for BDS-IGSO and QZSS-IGSO satellites can be reduced to 7.0 cm and 10.4 cm, respectively. The mean real-time positioning errors are thus reduced from 28.3 to 18.4 cm and from 24.4 to 18.2 cm in the horizontal and vertical components, respectively.

The Galileo high accuracy service (HAS) is a free-of-charge service designed to deliver decimeter-level accuracy in real-time precise point positioning (PPP) applications using global navigation satellite systems (GNSS). With the intention of facilitating the use of HAS corrections with open access tools, we present the open-source library named HASlib and its integration with another open-source library named RTKLIB. HASlib decodes the Reed–Solomon encoded Galileo E6 navigation data pages and outputs the corrections in commonly used formats. This enables the utilization of HAS with conventional GNSS receivers and PPP engines that lack native support for HAS formats. For instance, the outputs from HASlib enable the use of HAS corrections in RTKLIB. In order to validate this integration, we demonstrate that HAS can allow decimeter-level accuracy using only free-of-charge services and tools. We have obtained a 3D root mean square error below 20 cm (1 sigma) after a convergence time of 10–90 min in Finland. This accuracy has overcome classical real-time solutions with broadcast and satellite-based augmentation system (SBAS) data by one order of magnitude. Compared to post-processed multi-GNSS PPP, HAS corrections required longer convergence times, given the real-time nature. Furthermore, our assessment revealed that the longer convergence time, compared to prior literature, was attributed to RTKLIB filtering procedure and geometry deficiencies in high latitudes. Nevertheless, once convergence was attained, a sub-decimeter level of accuracy was observed in both horizontal and vertical components. These findings highlight the effectiveness of Galileo HAS, HASlib, and RTKLIB as powerful tools for providing open-access to real-time PPP solutions.

The International GNSS Service (IGS) provides combined satellite and station clock products, which are generated from the individual clock solutions produced by the analysis centers (ACs). Combinations for GPS and GLONASS are currently available, but there is still a lack of combined products for the new constellations such as Galileo, BeiDou, and QZSS. This study presents a combination framework based on least squares variance component estimation using the ACs’ aligned clock solutions. We present the various alignments required to harmonize the solutions from the ACs, namely the radial correction derived from the differences of the associated orbits, the alignment of the AC clocks to compensate for different reference clocks within each AC solution, and the inter-system bias (ISB) alignment to correct for different AC ISB definitions when multiple constellations are used. The combination scheme is tested with IGS MGEX and repro3 products. The RMS computed between the combined product and the aligned ACs’ solutions differ for each constellation, where the lowest values are obtained for Galileo and GPS with on average below 45 psec (13 mm) and reaching more than 150 psec (45 mm) for QZSS. The same behavior is repeated when the process is performed with the repro3 products. A clock and orbit combination validation is done using precise point positioning (PPP) that shows ionosphere-free phase residuals below 10 mm for all constellations, comparable with the AC solutions that are in the same level.

Ionospheric indices give information about ionospheric perturbations, which may cause absorption, diffraction, refraction, and scattering of radio signals, including those from global navigation satellite systems (GNSS). Therefore, there may be a relationship between index values and GNSS positioning results. A thorough understanding of ionospheric indices and their relationship to positioning results can help monitor and forecast the reliability and accuracy of GNSS positioning and support the precision and safety of life applications. In this study, we present the relationship between three indices: Gradient Ionosphere indeX (GIX), Sudden Ionospheric Disturbance indeX (SIDX), and Rate of Total electron content Index (ROTI) in relation to precise positioning results. We used two approaches: precise point positioning (PPP) and relative positioning for long baselines. We focus on GNSS stations located in Europe for two selected geomagnetic storms: March 17, 2015, and May 22, 2015. Our results show that in the case of PPP, positioning degradation occurred mainly at high latitudes and was mostly caused by rapid small-scale changes in ionospheric electron content represented by SIDX and ROTI. We also showed a significant correlation between cycle slips of GNSS signals and ROTI (0.88). The most significant degradations for relative positioning for low and medium latitudes were associated with large spatial gradients reflected by the GIX.