Uncalibrated Phase Delay Research Articles

Improving the Performance of Galileo Uncombined Precise Point Positioning Ambiguity Resolution Using Triple-Frequency Observations

Compared with the traditional ionospheric-free linear combination precise point positioning (PPP) model, the un-differenced and uncombined (UDUC) PPP model using original observations can keep all the information of the observations and be easily extended to any number of frequencies. However, the current studies about the multi-frequency UDUC-PPP ambiguity resolution (AR) were mainly based on the triple-frequency BeiDou navigation satellite system (BDS) observations or simulated data. Limited by many factors, for example the accuracy of BDS precise orbit and clock products, the advantages of triple-frequency signals to UDUC-PPP AR were not fully exploited. As Galileo constellations have been upgraded by increasing the number of 19 useable satellites, it makes using Galileo satellites to further study the triple-frequency UDUC-PPP ambiguity resolution (AR) possible. In this contribution, we proposed the method of multi-frequency step-by-step ambiguity resolution based on the UDUC-PPP model and gave the reason why the performance of PPP AR can be improved using triple-frequency observations. We used triple-frequency Galileo observations on day of year (DOY) 201, 2018 provided by 166 Multi-GNSS Experiment (MGEX) stations to estimate original uncalibrated phase delays (UPD) on each frequency and to conduct both dual- and triple-frequency UDUC-PPP AR. The performance of UDUC-PPP AR based on post-processing mode was assessed in terms of the time-to-first-fix (TTFF) as well as positioning accuracy with 2-h observations. It was found that triple-frequency observations were helpful to reduce TTFF and improve the positioning accuracy. The current statistic results showed that triple-frequency PPP-AR reduced the averaged TTFF by 19.6% and also improved the positioning accuracy by 40.9%, 31.2% and 23.6% in the east, north and up directions respectively, compared with dual-frequency PPP-AR. With an increasing number of Galileo satellites, it is expected that the robustness and accuracy of the triple-frequency UCUD-PPP AR can be improved further.

Triple-frequency PPP ambiguity resolution with multi-constellation GNSS: BDS and Galileo

Multi-constellation GNSS (multi-GNSS) and multi-frequency signals open new prospects for fast ambiguity resolution (AR) of precise point positioning (PPP). Currently, all the BDS and Galileo satellites are capable of transmitting signals on three or more frequencies. In this contribution, we investigate the triple-frequency PPP ambiguity resolution with B1, B2 and B3 observations from BDS satellites and E1, E5a and E5b observations from Galileo satellites and evaluate the contribution of BDS + Galileo combination to triple-frequency PPP AR. The uncalibrated phase delay (UPD) products are estimated based on triple-frequency observations, and the temporal characteristic as well as the residual distributions are analyzed. Our results show that the extra-wide-lane (EWL) and wide-lane (WL) UPDs for BDS and Galileo satellites are both stable during the 30 days and the daily narrow-lane (NL) UPD series are also steady with no obvious fluctuation. The Galileo UPDs exhibit better performance than BDS UPDs due to the high-quality observations. It is also interesting to find that the EWL UPD corrections for all Galileo satellites are very close to the zero. With the precise UPD products, the triple-frequency PPP AR with BDS and Galileo observations was implemented in both static and kinematic modes. Compared to the ambiguity-float solution, the performance can be significantly improved by triple-frequency PPP AR with the positioning accuracy improved by 30–70% in both static and kinematic modes. Moreover, the triple-frequency PPP fixed solutions also present better performance than the dual-frequency PPP fixed solutions in terms of time to the first fix and positioning accuracy, especially for the Galileo-only and BDS + Galileo solutions. And the fusion of multi-GNSS (BDS and Galileo) can further improve the position estimations compared to the single system with more satellites and better spatial geometry.

Regional BDS satellite clock estimation with triple-frequency ambiguity resolution based on undifferenced observation

Real-time precise satellite clock estimation is fundamental for real-time precise point positioning. To take full advantage of the triple-frequency (TF) signal of the BeiDou Navigation Satellite System (BDS), we used undifferenced observations directly in the TF data processing. In addition, TF ambiguity resolution (AR) was performed in a sequential manner, in which the wide-lane (WL) and extra wide-lane (EWL) ambiguities were fixed based on the Hatch---Melbourne---W?bbena (HMW) combination of B1/B2 and B1/B3, and then we followed with narrow-lane (NL) ambiguity fixing. By applying the ambiguity integer constraints, an ambiguity-fixed BDS satellite clock was obtained. To evaluate the contribution of TF AR in a satellite clock solution when a regional station network is involved, experiments were carried out based on the observations collected by 11 stations from the crustal movement observation network of China (CMONOC) during day of year (DOY) 153---188, 2016. First, the temporal behavior of the BDS EWL and WL uncalibrated phase delays (UPDs) were analyzed. The results suggested that they were quite stable over several days with standard deviations (STDs) of 0.004 cycles and 0.015 cycles for EWL and WL, respectively. Then, the ambiguity-fixed solution of the satellite clock was obtained with average success rates of about 99%, 90%, and 63% for EWL, WL, and NL AR, respectively. Attributed to the efficient AR, not only was the initialization time shortened but also the precision was improved. The overall averaged improvements of STD for geostationary orbits (GEO), inclined geostationary orbits (IGSO), and medium-altitude Earth orbit (MEO) satellites were 48.7%, 40.8%, and 34.4%, respectively, for the EWL and WL ambiguity-fixed solution and 72.6%, 71.0%, and 62.8%, respectively, for the NL ambiguity-fixed solution. As a comparison, a dual-frequency (DF) ambiguity-fixed satellite clock was also generated. Although the precision of the NL ambiguity-fixed clock was roughly the same, a larger root mean square (RMS) by a factor of 52% suggested that the DF ambiguity-fixed solution may have been biased with incorrect integer ambiguity fixing, whereas the TF ambiguity-fixed solution was more reliable because of the third frequency observation. In addition, precise point positioning (PPP) was carried out to assess the performance of these clock products. Compared with the PPP based on the ambiguity-float clock solution, improvements of 32.0% and 42.9% for horizontal and vertical, respectively, were obtained with the TF NL ambiguity-fixed clock solution.

Capturing coseismic displacement in real time with mixed single- and dual-frequency receivers: application to the 2018 Mw7.9 Alaska earthquake

High-density GPS monitoring networks have been built in seismically active areas to obtain high-precision real-time seismic displacements with high spatial resolution, which is necessary for rapid source inversion and decreasing the blind zone for earthquake early warning. To reduce the costs for large regions such as China, single-frequency (SF) receivers are preferred to densify the current sparse dual-frequency (DF) GPS network. We developed a new approach for SF precise point positioning ambiguity resolution (PPP-AR) with mixed SF and DF receivers. The ionospheric corrections, the main issue in processing SF data, were derived directly from geometry-free carrier-phase observations with fixed ambiguities, and then interpolated to the SF stations using the Kriging interpolator. The uncalibrated phase-delay (UPD) at L1 frequency was also estimated based on a regional network to enable PPP-AR. A dataset of 30 days from SAPOS network with different station densities was utilized to validate the performance of SF PPP-AR. The experimental results show that the accuracy of original ionospheric estimates is about 0.016 m, which is better than 0.035 m obtained from interpolation for an average station separation of less than 56 km. The L1 UPDs can be estimated hourly due to its high temporal stability; all of the phase residuals are within 0.3 cycle. Compared with the post-processed daily solutions, the positioning accuracies of SF kinematic PPP-AR can reach 1–2 cm and 2–3.5 cm for the horizontal and vertical components, respectively. Convergence time can be achieved within 10 min when the station spacing is less than 52 km, but it will increase to more than 20 min when the station spacing is larger than 99 km since most ambiguities cannot be fixed correctly. Finally, we applied this approach to the 2018 Mw7.9 Alaska earthquake and successfully retrieved the seismic displacement waveforms at a comparable accuracy level when compared to the ionospheric-free (IF) PPP solutions. The accuracy of the coseismic offset estimates relative to the final solutions based on the combination of GPS and accelerometer daily solutions, and is about 5–7 mm which demonstrates the capability for capturing the coseismic displacements.

Spatial–temporal characteristic of BDS phase delays and PPP ambiguity resolution with GEO/IGSO/MEO satellites

GPS precise point positioning (PPP) ambiguity resolution (AR) can improve the positioning accuracy and shorten the convergence time. However, for the BeiDou Satellite Navigation System (BDS), the problems of satellite-induced code bias, imperfections in the error models and the inadequate accuracy of orbit products limit the applications of the BDS PPP AR system, which requires more than 6 h to achieve the first ambiguity-fixed solution. In this study, the accuracy of a wide-lane (WL) uncalibrated phase delay (UPD) is improved after careful consideration of the code bias and multipath. Meanwhile, the accuracy of the BDS float ambiguity is also improved by multi-GNSS fusion and improved precise orbit and clock products, which are critical for high-quality narrow-lane (NL) UPD estimations. With three tracking networks of different scales, including Hong Kong, the Crustal Movement Observation Network of China (CMONOC) and the multi-GNSS experiment (MGEX) networks, the spatial–temporal characteristics of WL and NL UPDs for BDS GEO/IGSO/MEO satellites are analyzed, and the PPP AR is performed. Numerous results show that WL and NL UPDs with a standard deviation (STD) of less than 0.15 cycles can be achieved for BDS GEO satellites, while a STD of less than 0.1 cycles can be obtained for IGSO and MEO satellites. With the precise UPD estimation, for the first time, the BDS PPP rapid ambiguity resolution for GEO/IGSO/MEO satellites is achieved. We found that the average time to first fix (TTFF) of the BDS PPP AR is shortened significantly, to approximately 40 min for Hong Kong and the CMONOC, while the TTFF was 57.4 min for the MGEX networks. With ambiguity resolution, the accuracy of the daily BDS PPP in the east, north and vertical directions improves from 1.74 cm, 1.08 cm, and 5.52 cm to 0.72 cm, 0.54 cm, and 3.21 cm for the Hong Kong network, 2.24 cm, 2.31 cm, and 5.64 cm to 1.18 cm, 0.79 cm, and 3.30 cm for the CMONOC, and 2.71 cm, 1.80 cm, and 6.00 cm to 1.58 cm, 1.15 cm, and 4.33 cm for the MGEX networks. Significant improvement is also achieved for kinematic PPP, with improvements of 40.41%, 34.33% and 37.17% in the east, north and vertical directions for the MGEX networks, respectively.

Accelerating real-time PPP ambiguity resolution by incorporating multi-GNSS observations

The performance of Real-Time (RT) Precise Point Positioning (PPP) ambiguity resolution can be further improved by incorporating multiple Global Navigation Satellite System (GNSS) observations. RT PPP with ambiguity resolution based on raw GNSS observations, aided by the satellite clock offsets and GPS un-calibrated phase delays (UPDs), is realized to evaluate the effect of GNSS observations. Compared with previous studies, Galileo observations are integrated in this analysis and the effect of new BeiDou/Galileo systems in RT PPP ambiguity resolution is analyzed for global users. Comparing with the GPS-only and GPS/GLONASS solution, the time to first fix (TTFF) is 18.8% and 7.9% improved for those users in the Asia-Pacific regions, while it is 9.6% and 3.8% improved for other users by adding the BeiDou/Galileo observations. The improvement of the fixing rate and positioning accuracy is similar for users in different regions. The fixing rate is 2.1% and 0.5% improved when adding BeiDou/Galileo observations on top of GPS and GPS/GLONASS respectively. The accuracy of the north, east and up components are 0.4, 0.0 and 0.5 mm improved when adding on top of GPS, while 0.2, 0.1 and 0.4 mm when adding on top of GPS/GLONASS. By utilizing observations of four systems, an accuracy of 9.0, 9.1 and 28.3 mm is achieved for the fixed kinematic PPP solution after a mean initialization time of 797.9 s.

Calibration of GLONASS Inter-Frequency Code Bias for PPP Ambiguity Resolution with Heterogeneous Rover Receivers

Integer ambiguity resolution (IAR) is important for rapid initialization of precise point positioning (PPP). Whereas many studies have been limited to Global Positioning System (GPS) alone, there is a strong need to add Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) to the PPP-IAR solution. However, the frequency-division multiplexing of GLONASS signals causes inter-frequency code bias (IFCB) in the receiving equipment. The IFCB causes GLONASS wide-lane uncalibrated phase delay (UPD) estimation with heterogeneous receiver types to fail, so GLONASS ambiguity is therefore traditionally estimated as float values in PPP. A two-step method of calibrating GLONASS IFCB is proposed in this paper, such that GLONASS PPP-IAR can be performed with heterogeneous receivers. Experimental results demonstrate that with the proposed method, GLONASS PPP ambiguity resolution can be achieved across a variety of receiver types. For kinematic PPP with mixed receiver types, the fixing percentage within 10 min is only 33.5% for GPS-only. Upon adding GLONASS, the percentage improves substantially, to 84.9%.

BDS code bias periodical mitigation by low-pass filtering and its applications in precise positioning

The code-phase divergences, which are minimal for GPS, GLONASS, and Galileo satellites, are commonly found in BeiDou Navigation Satellite System (BDS) Geostationary Orbit (GEO), Inclined GeoSynchronous Orbit (IGSO) and Medium Earth Orbit (MEO) satellites. Several precise positioning applications which use code observations are severely affected by these code biases. We present an analysis of code bias based on multipath (MP) combination observations. To mitigate the effect of BDS code bias on precise positioning, we proposed a periodical correction method using a low-pass filter for BDS GEO, IGSO and MEO satellites. The auto-correlation of MP series over long periods is analyzed to obtain the periods of the dominant repeating components for three types of BDS satellites. The periods of the dominant daily repeating components are close to 86,160 s for BDS GEO and IGSO satellites while 603,120 s for MEO satellites. The zero phase-shift low-pass filter was used to extract the low-frequency components of MP series and then low-frequency components are applied to mitigate the code bias periodically. The developed correction methods can make a more remarkable improvement for the accuracy of MP series, compared to the current elevation-dependent correction models. Data sets collected at 50 Global Navigation Satellite System (GNSS) ground stations including 15 of the International GNSS Monitoring and Assessment System (iGMAS) and 35 of the Multi-GNSS Experiment (MGEX) stations are employed for this study. To analyze the influence of code bias on precise positioning and validate the effectiveness of the correction methods, some applications such as single point positioning (SPP), wide-lane (WL) ambiguity analysis and Uncalibrated Phase Delays (UPDs) estimation are conducted. After applying the proposal correction method to the code observations, SPP solutions outperform the uncorrected ones in term of positioning accuracy. The positioning accuracy decreased by 0.28 and 0.1 m in the north and east components and the improvements are more significant for the U components decreased by 0.42 m. In addition, the systematic variations of Melbourne-Wübbena (MW) combination are greatly removed and the convergence time of the MW series are decreased. Moreover, significant improvement is also achieved in terms of the usage rate and residuals of UPDs estimation.

Three-frequency BDS precise point positioning ambiguity resolution based on raw observables

All BeiDou navigation satellite system (BDS) satellites are transmitting signals on three frequencies, which brings new opportunity and challenges for high-accuracy precise point positioning (PPP) with ambiguity resolution (AR). This paper proposes an effective uncalibrated phase delay (UPD) estimation and AR strategy which is based on a raw PPP model. First, triple-frequency raw PPP models are developed. The observation model and stochastic model are designed and extended to accommodate the third frequency. Then, the UPD is parameterized in raw frequency form while estimating with the high-precision and low-noise integer linear combination of float ambiguity which are derived by ambiguity decorrelation. Third, with UPD corrected, the LAMBDA method is used for resolving full or partial ambiguities which can be fixed. This method can be easily and flexibly extended for dual-, triple- or even more frequency. To verify the effectiveness and performance of triple-frequency PPP AR, tests with real BDS data from 90 stations lasting for 21 days were performed in static mode. Data were processed with three strategies: BDS triple-frequency ambiguity-float PPP, BDS triple-frequency PPP with dual-frequency (B1/B2) and three-frequency AR, respectively. Numerous experiment results showed that compared with the ambiguity-float solution, the performance in terms of convergence time and positioning biases can be significantly improved by AR. Among three groups of solutions, the triple-frequency PPP AR achieved the best performance. Compared with dual-frequency AR, additional the third frequency could apparently improve the position estimations during the initialization phase and under constraint environments when the dual-frequency PPP AR is limited by few satellite numbers.

A comprehensive analysis of satellite-induced code bias for BDS-3 satellites and signals

Five BDS-3 satellites were launched successfully in 2015 and 2016, including two new-generation BDS-3 IGSO satellites I1-S and I2-S, and three new-generation BDS-3 Medium Earth Orbit (MEO) satellites M1-S, M2-S and M3-S. The BDS-3 satellites are able to transmit several new navigation frequencies including B1C at 1575.42 MHz, B2a at 1176.45 MHz, and B2b at 1207.14 MHz, in addition to B1 (1561.098 MHz) and B3 (1268.520 MHz) frequencies transmitted by BDS-2 satellites. Since the satellite-induced and elevation-dependent code biases were observed for the 14 older BDS-2 satellites (C01-C14), a comprehensive analysis and characterization of the code observations for the BDS-3 satellites were conducted based on observation data collected at 14 stations including 13 of the International GNSS Monitoring and Assessment System (IGMAS) and one station named SGG0 operated at the School of Geodesy and Geomatics (SGG) of Wuhan University. The Multipath (MP) combination, Melbourne-Wübbena (MW) combination and wide-lane uncalibrated phase delays (UPDs) are calculated to verify the existence and magnitude of the satellite-induced code bias for the BDS-3 satellites. The elevation-dependent polynomial models of MP combination for BDS-3 satellites were also established in order to analyze the characteristic of satellite-induced bias observed at different receiver types or different frequencies. For comparison, the MP combination of a newly launched BDS-2 satellite I6 was also calculated. The results indicate that the satellite-induced code bias is negligible for the BDS-3 satellites while newly launched BDS-2 satellite I6 has similar elevation-dependent code bias as the 14 older BDS-2 satellites.

Multi-GNSS phase delay estimation and PPP ambiguity resolution: GPS, BDS, GLONASS, Galileo

This paper focuses on the precise point positioning (PPP) ambiguity resolution (AR) using the observations acquired from four systems: GPS, BDS, GLONASS, and Galileo (GCRE). A GCRE four-system uncalibrated phase delay (UPD) estimation model and multi-GNSS undifferenced PPP AR method were developed in order to utilize the observations from all systems. For UPD estimation, the GCRE-combined PPP solutions of the globally distributed MGEX and IGS stations are performed to obtain four-system float ambiguities and then UPDs of GCRE satellites can be precisely estimated from these ambiguities. The quality of UPD products in terms of temporal stability and residual distributions is investigated for GPS, BDS, GLONASS, and Galileo satellites, respectively. The BDS satellite-induced code biases were corrected for GEO, IGSO, and MEO satellites before the UPD estimation. The UPD results of global and regional networks were also evaluated for Galileo and BDS, respectively. As a result of the frequency-division multiple-access strategy of GLONASS, the UPD estimation was performed using a network of homogeneous receivers including three commonly used GNSS receivers (TRIMBLE NETR9, JAVAD TRE_G3TH DELTA, and LEICA). Data recorded from 140 MGEX and IGS stations for a 30-day period in January in 2017 were used to validate the proposed GCRE UPD estimation and multi-GNSS dual-frequency PPP AR. Our results show that GCRE four-system PPP AR enables the fastest time to first fix (TTFF) solutions and the highest accuracy for all three coordinate components compared to the single and dual system. An average TTFF of 9.21 min with $$7{^{\circ }}$$ cutoff elevation angle can be achieved for GCRE PPP AR, which is much shorter than that of GPS (18.07 min), GR (12.10 min), GE (15.36 min) and GC (13.21 min). With observations length of 10 min, the positioning accuracy of the GCRE fixed solution is 1.84, 1.11, and 1.53 cm, while the GPS-only result is 2.25, 1.29, and 9.73 cm for the east, north, and vertical components, respectively. When the cutoff elevation angle is increased to $$30{^{\circ }}$$ , the GPS-only PPP AR results are very unreliable, while 13.44 min of TTFF is still achievable for GCRE four-system solutions.

An improved method for multi-GNSS baseline processing using single difference

The rapid development of multi-GNSS (Global Navigation Satellite System) constellations has provided numerous benefits for common PNT (Positioning, Navigation and Timing) services. Generally, DD (Double Difference) observations are used in GNSS baseline processing. However, in traditional DD data processing methods, DD observations are normally not formed between different GNSS systems or the inter-system DD ambiguities are rather difficult to be resolved due to the different wavelength. As a result, DD observations are normally formed in each system or their inter-system ambiguities are left unresolved, which could not be regarded as the optimal baseline processing methods for multi-GNSS. This paper presents an improved algorithm for multi-GNSS short baseline processing where SD (Single Difference) rather than DD observations are formed between two receivers and this method is found to be equivalent to the ideal DD method with inter-system DD observations and resolved inter-system DD ambiguities. Furthermore, an average approach and linear model are proposed to isolate the receiver-dependent Uncalibrated Phase Delays (UPDs) from the SD ambiguities for CDMA (Code Division Multiple Access) and FDMA (Frequency Division Multiple Access) GNSS systems, respectively, to address ambiguity resolution. Experiments show that after removing the UPDs derived by the proposed methods, the fractional parts of the residuals for almost all ambiguities are less than 0.1 cycles for the GPS, BDS, GLONASS and Galileo systems, which confirms the validity of our UPD calibration methods. Experiments also show that the mean differences between the daily solutions derived by GAMIT and the 4-h solutions derived by the proposed SD method are approximately −0.12, −0.33, 1.04 and 0.31cm in the north, east, up and length components, respectively. We also find that the baseline repeatability of the new SD method outperforms that of the DD method with only DD observations formed in each GNSS system by 36% and 26% in the north and up directions, respectively. These improvements are more significant in environments with poor observation conditions. An improvement of 40% and 30% in the north and up directions are, respectively, found when only 5GPS and 4 BDS satellites are available. Therefore, more reliable and precise baselines can be derived with our new baseline processing method in multi-GNSS cases.

Performance analysis of PPP ambiguity resolution with UPD products estimated from different scales of reference station networks

Integer ambiguity fixing with uncalibrated phase delay (UPD) products can significantly shorten the initialization time and improve the accuracy of precise point positioning (PPP). Since the tracking arcs of satellites and the behavior of atmospheric biases can be very different for the reference networks with different scales, the qualities of corresponding UPD products may be also various. The purpose of this paper is to comparatively investigate the influence of different scales of reference station networks on UPD estimation and user ambiguity resolution. Three reference station networks with global, wide-area and local scales are used to compute the UPD products and analyze their impact on the PPP-AR. The time-to-first-fix, the unfix rate and the incorrect fix rate of PPP-AR are analyzed. Moreover, in order to further shorten the convergence time for obtaining precise positioning, a modified partial ambiguity resolution (PAR) and corresponding validation strategy are presented. In this PAR method, the ambiguity subset is determined by removing the ambiguity one by one in the order of ascending elevations. Besides, for static positioning mode, a coordinate validation strategy is employed to enhance the reliability of the fixed coordinate. The experiment results show that UPD products computed by smaller station network are more accurate and lead to a better coordinate solution; the PAR method used in this paper can shorten the convergence time and the coordinate validation strategy can improve the availability of high precision positioning.

Property Analysis of the Real-Time Uncalibrated Phase Delay Product Generated by Regional Reference Stations and Its Influence on Precise Point Positioning Ambiguity Resolution

The real-time estimation of the wide-lane and narrow-lane Uncalibrated Phase Delay (UPD) of satellites is realized by real-time data received from regional reference station networks; The properties of the real-time UPD product and its influence on real-time precise point positioning ambiguity resolution (RTPPP-AR) are experimentally analyzed according to real-time data obtained from the regional Continuously Operating Reference Stations (CORS) network located in Tianjin, Shanghai, Hong Kong, etc. The results show that the real-time wide-lane and narrow-lane UPD products differ significantly from each other in time-domain characteristics; the wide-lane UPDs have daily stability, with a change rate of less than 0.1 cycle/day, while the narrow-lane UPDs have short-term stability, with significant change in one day. The UPD products generated by different regional networks have obvious spatial characteristics, thus significantly influencing RTPPP-AR: the adoption of real-time UPD products employing the sparse stations in the regional network for estimation is favorable for improving the regional RTPPP-AR up to 99%; the real-time UPD products of different regional networks slightly influence PPP-AR positioning accuracy. After ambiguities are successfully fixed, the real-time dynamic RTPPP-AR positioning accuracy is better than 3 cm in the plane and 8 cm in the upward direction.

Formal Uncertainty and Dispersion of Single and Double Difference Models for GNSS-Based Attitude Determination.

With multi-antenna synchronized global navigation satellite system (GNSS) receivers, the single difference (SD) between two antennas is able to eliminate both satellite and receiver clock error, thus it becomes necessary to reconsider the equivalency problem between the SD and double difference (DD) models. In this paper, we quantitatively compared the formal uncertainties and dispersions between multiple SD models and the DD model, and also carried out static and kinematic short baseline experiments. The theoretical and experimental results show that under a non-common clock scheme the SD and DD model are equivalent. Under a common clock scheme, if we estimate stochastic uncalibrated phase delay (UPD) parameters every epoch, this SD model is still equivalent to the DD model, but if we estimate only one UPD parameter for all epochs or take it as a known constant, the SD (here called SD2) and DD models are no longer equivalent. For the vertical component of baseline solutions, the formal uncertainties of the SD2 model are two times smaller than those of the DD model, and the dispersions of the SD2 model are even more than twice smaller than those of the DD model. In addition, to obtain baseline solutions, the SD2 model requires a minimum of three satellites, while the DD model requires a minimum of four satellites, which makes the SD2 more advantageous in attitude determination under sheltered environments.

Ambiguity Resolution In Precise Point Positioning Technique: A Case Study

Abstract Because of the dynamics of the GPS technique used in different domains like geodesy, near real-time GPS meteorology, geodynamics, the precise point positioning (PPP) becomes more than a powerful method for determining the position, or the delay caused by the atmosphere. The main idea of this method is that we need only one receiver – preferably that have dual frequencies pseudorange and carrier-phase capabilities – to obtain the position. Because we are using only one receiver the majority of the residuals that are eliminated in double differencing method, we have to estimate them in PPP. The development of the PPP method allows us, to use precise satellite clock estimates, and precise orbits, resulting in a much more efficient way to deal with the disadvantages of this technique, like slow convergence time, or ambiguity resolution. Because this two problem are correlated, to achieve fast convergence we need to resolve the problem of ambiguity resolution. But the accuracy of the PPP results are directly influenced by presence of the uncalibrated phase delays (UPD) originating in the receivers and satellites. In this article we present the GPS errors and biases, the zenith wet delay and the necessary time for obtaining the convergence. The necessary correction are downloaded by using the IGS service.

Ionospheric effects in uncalibrated phase delay estimation and ambiguity-fixed PPP based on raw observable model

Zero-difference (ZD) ambiguity resolution (AR) reveals the potential to further improve the performance of precise point positioning (PPP). Traditionally, PPP AR is achieved by Melbourne–Wubbena and ionosphere-free combinations in which the ionosphere effect are removed. To exploit the ionosphere characteristics, PPP AR with L1 and L2 raw observable has also been developed recently. In this study, we apply this new approach in uncalibrated phase delay (UPD) generation and ZD AR and compare it with the traditional model. The raw observable processing strategy treats each ionosphere delay as an unknown parameter. In this manner, both a priori ionosphere correction model and its spatio-temporal correlation can be employed as constraints to improve the ambiguity resolution. However, theoretical analysis indicates that for the wide-lane (WL) UPD retrieved from L1/L2 ambiguities to benefit from this raw observable approach, high precision ionosphere correction of better than 0.7 total electron content unit (TECU) is essential. This conclusion is then confirmed with over 1 year data collected at about 360 stations. Firstly, both global and regional ionosphere model were generated and evaluated, the results of which demonstrated that, for large-scale ionosphere modeling, only an accuracy of 3.9 TECU can be achieved on average for the vertical delays, and this accuracy can be improved to about 0.64 TECU when dense network is involved. Based on these ionosphere products, WL/narrow-lane (NL) UPDs are then extracted with the raw observable model. The NL ambiguity reveals a better stability and consistency compared to traditional approach. Nonetheless, the WL ambiguity can be hardly improved even constrained with the high spatio-temporal resolution ionospheric corrections. By applying both these approaches in PPP-RTK, it is interesting to find that the traditional model is more efficient in AR as evidenced by the shorter time to first fix, while the three-dimensional positioning accuracy of the RAW model outperforms the combination model by about \(7.9\,\%\). This reveals that, with the current ionosphere models, there is actually no optimal strategy for the dual-frequency ZD ambiguity resolution, and the combination approach and raw approach each has merits and demerits.

A new differential code bias (C1–P1) estimation method and its performance evaluation

The current satellite clock products are computed using the ionosphere-free phase (L1/L2) and code (P1/P2) observations. Thus, if users conduct undifferenced positioning using these clock products together with C1 and P2 observations, the differential code bias (DCB) (C1---P1) should be properly compensated. The influence of DCB (C1---P1) on the undifferenced ambiguity solutions is investigated. Based on the investigation, we propose a new DCB (C1---P1) estimation method. Using it, the satellite DCB (C1---P1) can be computed. A 30-day (DOY 205---234, 2012) dual-frequency GPS data set is processed to estimate the DCB (C1---P1). Comparing the estimated results with that of IGS DCB products, the accuracy is better than 0.13 m. The performances of DCB (C1---P1) in the code-based single-point positioning, precise point positioning (PPP) convergence and wide-lane uncalibrated phase delay (UPD) estimation are investigated using the estimated DCB (C1---P1). The results of the code-based single-point positioning show that the influence of DCB (C1---P1) on the up direction is more evident than on the horizontal directions. The accuracy is improved by 50 % and reaches to decimeter level with DCB (C1---P1) application. The performance of DCB (C1---P1) in PPP shows that it can accelerate PPP convergence through improving the accuracy of the code observation. The computed UPD values show that influence of DCB (C1---P1) on UPD of each satellite is different, and some values are larger than 0.3 cycles.

Near-field surface displacement and permanent deformation induced by the Alaska Mw 7.5 earthquake determined by high-rate real-time ambiguity-fixed PPP solutions

High-rate GPS data from the United States continuously operating reference stations in the Alaska region are processed using the recently developed precise point positioning (PPP) technique. The traditional PPP technique does not fix ambiguities into their integers because these ambiguities do not have an integer nature when data from a single receiver, as well as precise orbit and clock corrections, are used. Additional corrections, i.e., uncalibrated phase delay (UPD), are needed to fix integer ambiguities and consequently improve positioning accuracy. This study proposes a methodology to compute for wide-lane and L2 (the second L-band frequency) UPDs using the geometry-based model and subsequently applies these parameters to the computation for ambiguity-fixed solutions. The instantaneous displacements of near-field sites, as well as the permanent deformations after the earthquake, are therefore obtained for the January 5, 2013, Alaska earthquake. The real-time performance of PPP solutions are assessed by considering realistic data latency and data interval of corrections. Ambiguity-fixed solutions are compared with ambiguity-float ones. The comparison shows that the positioning accuracy can be improved significantly when the ambiguities are fixed correctly. The solutions using the real-time corrections are also compared with those using post-processing corrections, i.e., Center for Orbit Determination in Europe final orbit and clock. Although the accuracy is somehow degraded because of the data latency and data interval, the real-time results are satisfactory for use in monitoring the small-scale deformation (1–2 cm) caused by the Alaska earthquake. In addition, the kinematic ambiguity-fixed PPP solutions for 7 days around the earthquake are calculated to obtain permanent pre- and post-earthquake deformations. The deformations computed from real-time and post-processing corrections do not appear to be significantly different.

An improved approach to model regional ionosphere and accelerate convergence for precise point positioning

Given the severe effects of the ionosphere on global navigation satellite system (GNSS) signals, single-frequency (SF) precise point positioning (PPP) users can only achieve decimeter-level positioning results. Ionosphere-free combinations can eliminate the majority of ionospheric delay, but increase observation noise and slow down dual-frequency (DF) PPP convergence. In this paper, we develop a regional ionosphere modeling and rapid convergence approach to improve SF PPP (SFPPP) accuracy and accelerate DF PPP (DFPPP) convergence speed. Instead of area model, ionospheric delay is modeled for each satellite to be used as a priori correction. With the ionospheric, wide-lane uncalibrated phase delay (UPD) and residuals satellite DCBs product, the wide-lane observations for DF users change to be high-precision pseudorange observations. The validation of a continuously operating reference station (CORS) network was analyzed. The experimental results confirm that the approach considerably improves the accuracy of SFPPP. For DF users, convergence time is substantially reduced.