Double-differenced Ambiguities Research Articles

PPP-AR reference satellite selection based on the observation quality factors

Precise point positioning ambiguity resolution (PPP-AR) can effectively improve positioning accuracy and convergence time. In PPP-AR, the double-difference ambiguity between satellite pairs must be fixed. Therefore, it requires the selection of one satellite as a reference to conduct single-difference observations. Usually, the satellite with the highest elevation is selected as the reference satellite, however, once this satellite has a cycle slip or signal interference, its ambiguity will be reinitialized, resulting in the calculated wide-lane and narrow-lane ambiguities are not accurate enough, which further affects all the ambiguities fixed rates and positioning accuracies. In this contribution, we propose a multi-indicators comprehensive evaluation method of the global navigation satellite system (GNSS) based on entropy weight-grey correlation analysis for reference satellite selection of PPP-AR. The comprehensive evaluation includes the observations index selection, the index normalization, the calculation of index entropy weight and the grey correlation analysis. According to the new method, the quality ranking of satellite observations for each epoch can be obtained, and the observation value with the highest ranking is used as the reference satellite during the PPP-AR. One-week observations from 243 multi-GNSS experiment stations are selected to conduct GPS-only, Galileo-only and BDS-3-only kinematic PPP-AR, and the reference satellite selection method using the highest-elevation and the proposed method is applied, respectively. The results show that the PPP performance for the new method can be improved in the positioning accuracies, convergence time and ambiguity fixed rates. The positioning accuracies of three-dimensional directions can be improved by about 5.54%, 8.81% and 6.02% for GPS, BDS-3 and Galileo, respectively. The average improvements of convergence time in the east, north and up directions are 4.67%, 2% and 4% for GPS, BDS-3 and Galileo, respectively. The ambiguity fixed rates are improved by 7.31%, 6.89% and 1.45% for GPS, BDS-3 and Galileo between the 80%-100% range, respectively.

Estimation of BDS-2/3 phase observable-specific signal bias aided by double-differenced model: an exploration of fast BDS-2/3 real-time PPP

GNSS phase observable-specific signal bias (OSB) corrections are essential for widespread application of precise point positioning with ambiguity resolution (PPP-AR) or PPP-RTK. However, subject to the orbital error effects, conventional undifferenced (UD) model-derived BeiDou System (BDS) real-time (RT) OSB products are usually unsatisfactory. In this study, a novel OSB-generating method assisted by the double-differenced (DD) model is proposed. The reliable integer UD ambiguities are obtained by converting DD ambiguities with given ambiguity datums, by which the RT orbit error effects on ambiguity fixing can be reduced during the OSB extraction and PPP-AR process. Validated using data from two regional sparse GNSS reference networks in Shaanxi, China, and Europe, results show that the proposed method-derived OSB products can improve RT PPP-AR performance effectively. In the Shaanxi network, the narrow-lane ambiguity residuals for BDS-3 within ± 0.25 cycles are improved by 23.1% and 33.2% compared to those using the UD model and Centre National d’Etudes Spatiales (CNES)-derived OSB products, respectively, and the corresponding values are 15.2% and 43.1% in the European network. A centimeter- or even millimeter-level positioning accuracy can be achieved for BDS PPP using the poposed OSB products in both networks. In the kinematic PPP-AR test within the Shanxi network, the mean RMS of the BDS-2/3 fixed solutions in the east, north, and up directions is 0.9, 0.7, and 2.3 cm, with a decrease of 57.1%, 53.3%, and 46.5% compared to that using OSB derived by UD model. The median Time-To-First-Fix (TTFF) is also shortened from 23.8 to 7.5 min.

A new inter-system double-difference RTK model applicable to both overlapping and non-overlapping signal frequencies

Aiming at the problem that the traditional inter-system double-difference model is not suitable for non-overlapping signal frequencies, we propose a new inter-system double-difference model with single difference ambiguity estimation, which can be applied for both overlapping and non-overlapping signal frequencies. The single difference ambiguities of all satellites and Differential Inter-System Biases (DISB) are first estimated, and the intra-system double difference ambiguities, which have integer characteristics, are then fixed. After the ambiguities are successfully fixed, high-precision coordinates and DISB can be obtained with a constructed transformation matrix. The model effectively avoids the DISB parameter filtering discontinuity caused by the reference satellite transformation and the low precision of the reference satellite single difference ambiguity calculated with the code. A zero-baseline using multiple types of receivers is selected to verify the stability of the estimated DISB. Three baselines with different lengths are selected to assess the positioning performance of the model. The ionospheric-fixed and ionospheric-float models are used for short and medium-long baselines, respectively. The results show that the Differential Inter-System Code Biases (DISCB) and Differential Inter-System Phase Biases (DISPB) have good stability regardless of the receivers type and the signal frequency used and can be calibrated to enhance the strength of the positioning model. The positioning results with three baselines of different lengths show that the proposed inter-system double-difference model can improve the positioning accuracy by 6–22% compared with the intra-system double-difference model which selects the reference satellite independently for each system. The Time to First Fix (TTFF) of the two medium-long baselines is reduced by 30% and 29%, respectively.

Absolute and relative POD of LEO satellites in formation flying: Undifferenced and uncombined approach

Absolute or relative precise orbit determination (POD) is an essential prerequisite for many low earth orbit (LEO) missions. The POD of LEO satellites typically relays on processing the onboard global navigation satellite system (GNSS) measurements. The absolute POD is usually based on an ionosphere-free (IF) combination, and currently, integer ambiguity resolution (IAR) can be achieved only when external GNSS satellite phase bias (SPB) products are used. The use of these products is not flexible in multi-frequency/multi-constellation scenarios and is difficult to achieve in real-time missions. For relative POD, the double-differenced (DD) with IAR is the most general method. However, the differencing process amplifies observation noise and loses the opportunity to impose dynamic constraints on some eliminated parameters. In this contribution, based on the use of undifferenced and uncombined (UDUC) observations, a new model for both absolute and relative POD is proposed. In this model, the ambiguities of common-view satellites are constructed into DD form, thus IAR can be achieved without any external SPB products. Working with the UDUC observations, multi-frequency scenarios can be easily applied, and residuals can be separated for each frequency. In addition, with precise GNSS satellite clock/orbit products, both the absolute and relative orbits can be derived, which supports absolute and relative LEO POD. Based on onboard GPS observations of T-A and T-B satellites in formation flying, the performance of the UDUC POD model with DD ambiguity was evaluated. With the UDUC algorithm and IAR, the proposed model presented a consistency of 2.8–3.8 cm in 3D with the reference orbits, and the orbit difference was reduced by 16.3% and 10.6% for T-A and T-B compared with the IF-based POD, respectively. In addition, the relative orbit of the two satellites derived from the proposed model showed a consistency of 1.1–1.5 mm, which proved the feasibility of the UDUC POD model with DD ambiguity for formation flying missions.

The Superiority of Multi-GNSS L5/E5a/B2a Frequency Signals in Smartphones: Stochastic Modeling, Ambiguity Resolution, and RTK Positioning

The emerging Internet of Things (IoT) applications, such as intelligent transportation based on vehicular-lane accurate positioning, have a growing demand for precise and reliable positioning with Global Navigation Satellite Systems (GNSSs). It is desirable to use GNSS modules in smartphones to achieve highprecision positioning. The GNSS modules in some brands of smartphones thus far are able to track the new L5 signals of GPS and QZSS, E5a signals of Galileo and B2a signals of BeiDou-3. The L5/E5a/B2a signals have higher quality due to their signal structure, which provides an important potential for highprecision positioning in smartphones. In this paper, we will study the quality of L5/E5a/B2a signals, and their superiorities in integer ambiguity resolution (IAR) and precise positioning with respect to the L1/E1/B1 signals from GPS, QZSS, Galileo and Beidou-2/3 satellites. The signal quality is evaluated in terms of observation precision, multipath, double-differenced ambiguity fractions and ambiguity dilution of precision (ADOP). In addition, we propose a new weighting model that takes into account the variation range of carrier-to-noise density ratio (C/N0). The results indicate that the Beidou-3 B2a signal has comparable quality to L5/E5a signals of other systems, and all of them are better than the L1/E1/B1 signals. However, the ambiguity fractions of B2a signals diverge abruptly in some periods, resulting in the unsuccessful ambiguity fixing. The L5/E5a/B2a signals can generally obtain higher IAR fix-rate and positioning accuracies than the L1/E1/B1 signals. The new weighting model can capture the smartphone noise characteristics better than the traditional weighting model, thus improving the positioning accuracy.

Ambiguity-Fixing in Frequency-Varying Carrier Phase Measurements: Global Navigation Satellite System and Terrestrial Examples

<h3>Abstract</h3> Carrier phase signals are considered among the key observations in global navigation satellite systems (GNSSs) and several other high-precision interferometric measurement systems. However, these ultra-precise measurements are not fully exploited when the integerness of their inherent ambiguities is discarded during the estimation process. Provided that the integer-estimable functions of their phase ambiguities are properly identified, integer ambiguity resolution (IAR) can be utilized to benefit their parameter solutions. For the GNSS code division multiple access systems with transmitters that broadcast carrier phase signals on identical frequencies, these integer-estimable functions have been characterized and are well-known as double differenced ambiguities. However, this is not the case with “frequency-varying” carrier phase signals that are broadcast by GLONASS satellites, Low-Earth-Orbiting communication satellites, or cellular long-term evolution (LTE) transmitters. This study aims to present full-rank models that can be used to identify integer-estimable ambiguity functions, thereby bringing the observation equations of frequency-varying carrier phase measurements into an IAR-applicable form. Our analytical results are supported by several numerical examples, including GNSS and terrestrial-based IAR as well as a new set of “inter-frequency” integer ambiguities that this study discovers in Galileo multi-frequency carrier phase signals.

Massive GNSS Network Analysis Without Baselines: Undifferenced Ambiguity Resolution

AbstractInteger ambiguity resolution (IAR) plays a key role in high‐precision Global Navigation Satellite System (GNSS). While GNSS network analysis is usually achieved through double‐difference (DD) IAR, undifferenced (UD) IAR has also been developed in recent years to achieve network analysis station by station. It is normally acquiesced in that DD‐IAR and UD‐IAR should achieve equivalent results, which is however not the case in a number of studies. We investigated four representatives DD‐IAR and UD‐IAR strategies using 1 year of GPS data from 192 globally distributed stations, where the position repeatability and the root mean squares error (RMSE) against the International GNSS Service weekly solutions were both quantified. We found that the position repeatability and the RMSE of DD‐IAR were both worse than those of UD‐IAR at over 90% of stations for the east component, though the largest deteriorations were only 0.6 and 0.7 mm, respectively. We demonstrate that it is the incorrectly resolved DD ambiguities, though accounting for far less than 1%, that degrade DD‐IAR. Identifying such problematic ambiguities requires sophisticated quality control for DD‐IAR, because the integer offset of an incorrect DD ambiguity might have been absorbed evenly by the cross‐connected mass of DD ambiguities throughout the network, and consequently does not manifest as an outlier in the ambiguity‐fixed solution. We, therefore, recommend UD‐IAR for GNSS network analysis since it is immune to the error propagation resulting from incorrect integer ambiguities at other stations and thus more efficient to achieve ambiguity‐fixed solutions.

Pass-by-Pass Ambiguity Resolution in Single GPS Receiver PPP Using Observations for Two Sequential Days: An Exploratory Study

“Pass-by-pass” or “track-to-track” ambiguity resolution removes Global Navigation Satellite System (GNSS) satellite hardware delays between adjacent undifferenced (UD) ambiguities, which is often applied in precise orbit determination (POD) for Low Earth Orbit (LEO) satellites to improve the accuracy of orbits. In this study, we carried out an exploratory study to use the “pass-by-pass” ambiguity resolution by differencing the undifferenced ambiguity candidates for two adjacent passes in sidereal days for a single Global Positioning System (GPS) receiver static Precise Point Positioning (PPP). Using the GPS observations from 132 globally distributed reference stations of International GPS Service (IGS), we find that 99.08% wide-lane (WL) and 97.83% narrow-lane (NL) double-difference ambiguities formed by the “pass-by-pass” method for all stations can be fixed to their nearest integers within absolute fractional residuals of 0.2 cycles. These proportions are higher than the corresponding values of network solution with multiple receivers with 97.39% and 91.20%, respectively. About 97% to 98% of ambiguities can be fixed finally on average. The comparison of the estimated station coordinates with the IGS weekly solutions reveals that the Root Mean Square (RMS) in East and North directions are 2-4 mm and is about 6 mm in the Up direction. For hourly data, it is found that the mean positioning accuracy improvement can achieve to about 10% after ambiguity resolution. From a dam deformation monitoring application, it shows that the fixing rate of WL and NL ambiguity can be closed to 100% and higher than 90%, respectively. The time series generated by PPP are also in agreement with the short baseline solutions.

Single-Frequency Precise Point Positioning Using Regional Dual-Frequency Observations.

High-precision and low-cost single-frequency precise point positioning (SF-PPP) has been attracting more and more attention in numerous global navigation satellite system (GNSS) applications. To provide the precise ionosphere delay and improve the positioning accuracy of the SF-PPP, the dual-frequency receiver, which receives dual-frequency observations, is used. Based on the serviced precise ionosphere delay, which is generated from the dual-frequency observations, the high-precision SF-PPP is realized. To further improve the accuracy of the SF-PPP and shorten its convergence time, the double-differenced (DD) ambiguity resolutions, which are generated from the DD algorithm, are introduced. This method avoids the estimation of fractional cycle bias (FCB) for the SF-PPP ambiguity. Here, we collected data from six stations of Shanghai China which was processed, and the corresponding results were analyzed. The results of the dual-frequency observations enhanced SF-PPP realize centimeter-level positioning. The difference between the results of two stations estimated with dual-frequency observations enhanced SF-PPP were compared with the relative positioning results computed with the DD algorithm. Experimental results showed that the relative positioning accuracy of the DD algorithm is slightly better than that of the dual-frequency observations enhanced SF-PPP. This could be explained by the effect of the float ambiguity resolutions on the positioning accuracy. The data was processed with the proposed method for the introduction of the DD ambiguity into SF-PPP and the results indicated that this method could improve the positioning accuracy and shorten the convergence time of the SF-PPP. The results could further improve the deformation monitoring ability of SF-PPP.

A Comparative Study of BDS Triple-Frequency Ambiguity Fixing Approaches for RTK Positioning.

Concerning the triple-frequency ambiguity resolution, in principle there are three different realizations. The first one is to fix all the ambiguities of the original frequencies together. However, it is also believed that fixing the combined integer ambiguities with longer wavelength, such as extra-wide-lane (EWL), wide-lane (WL), should be advantageous. Also, it is demonstrated that fixing sequentially EWL, WL and one type of original ambiguities provides better results, as the previously fixed ambiguities increase parameters’ precision for later fixings. In this paper, we undertake a comparative study of the three fixing approaches by means of experimental validation. In order to realize the three fixing approaches from the same information in terms of adjustment, we developed a processing strategy to provide fully consistent normal equations. We first generate the normal equation with the original undifferentiated carrier phase ambiguities, then map it into that with the combined and double-differenced ambiguities required by the individual approach for fixing. Four baselines of 258 m, 22 km, 47 km and 53 km are selected and processed in both static and kinematic mode using the three ambiguity-fixing approaches. Indicators including time of first fixed solution (TFFS), the correct fixing rate, positioning accuracy and RATIO are used to evaluate and investigate results. We also made a preliminary theoretical explanation of the results by looking into the decorrelation procedure of the ambiguity searching algorithm and the intermediate results. As conclusions, integrated searching of original ambiguities or combined ambiguities has almost the same fixing performance, whereas the sequential fixing of EWL, WL and B1 ambiguities overperforms the integrated searching. By the way, the third-frequency data can shorten the TFFS significantly but can hardly improve the positioning.

Tightly combined triple-frequency GPS and BDS for rapid wide-lane RTK positioning with consideration of carrier-phase differential inter-system bias

Multi-frequency signals have been accessible for most Global Navigation Satellite Systems . Existing studies have verified that using multi-frequency extra-wide-lane and wide-lane observations can realize decimeter, sub-decimeter and even centimeter rapid positioning. In this paper, a tightly combined wide-lane real-time kinematic positioning method using triple-frequency GPS and BDS is proposed. The differential inter-system bias is taken into consideration so that an inter-system differencing model is formed. Due to the influence of different frequencies between GPS and BDS, the double-difference wide-lane ambiguity between GPS and BDS reference satellites and the single-difference ambiguity of the BDS reference satellite are estimated jointly with the wide-lane differential inter-system bias. Thus, a full-rank model can be obtained without any external calibration. Using the stability of wide-lane differential inter-system bias in the estimation from epoch to epoch, redundant observations can be introduced, therefore the strength of the positioning model can be enhanced. Positioning performance under simulated obstructed environments is evaluated. The results show that the inter-system model can effectively improve the positioning compared with the conventional intra-system model for the severely obstructed situations.

Real‐time and dynamic time transfer method based on double‐differenced real‐time kinematic mode

In this study, a real-time and dynamic time transfer method based on double-differenced (DD) real-time kinematic (RTK) mode is proposed. First, the method forms the DD observations of Global Navigation Satellite System (GNSS) to fix the DD ambiguity, and then, all of the fixed single-differenced (SD) ambiguities are recovered while defining the SD ambiguity datum; finally, the method uses the SD observations to realise the time transfer by using the fixed SD ambiguities. Five different baseline links were used for the validations. The results demonstrated that the DD-RTK method can be used for both kinematic positioning and precise time transfers, and it provides for much better accuracy and stability than the un-differenced precise point positioning method because it is operated by the broadcast ephemeris, which is more convenient for real-time applications.

A New Method of Real-Time Kinematic Positioning Suitable for Baselines of Different Lengths

This study introduces a new real-time kinematic (RTK) positioning method which is suitable for baselines of different lengths. The method merges carrier-phase wide-lane, and ionosphere-free observation combinations (LWLC) instead of using pseudo-range, and carrier-phase ionosphere-free combination (PCLC), or single-frequency pseudo-range and phase combination (P1L1). In a first step, the double-differenced wide-lane ambiguities were calculated and fixed using the pseudo-range and carrier-phase wide-lane combination observations. Once the double-differenced wide-lane integer ambiguities were known, the wide-lane combined observations were regarded as accurate pseudo-range observations. Subsequently, the carrier-phase wide-lane, and ionosphere-free combined observations were used to fix the double-differenced carrier-phase integer ambiguities, achieving the final RTK positioning. The RTK positioning analysis was performed for short, medium, and long baselines, using the P1L1, PCLC, and LWLC methods, respectively. For a short baseline, the LWLC method demonstrated positioning accuracy similar to the P1L1 method, and performed better than the PCLC method. For medium and long baselines, the positioning accuracy of the LWLC method was slightly higher than those of the PCLC and P1L1 methods. In conclusion, the LWLC method provided high-precision RTK positioning results for baselines with different lengths, as it used high-precision carrier-phase observations with fixed ambiguities instead of low-precision pseudo-range observations.

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.

A multi-system tightly-combined model with consideration of differential inter-systems code biases

Knowledge of differential inter-systems phase/code biases are critical to integrate observations from several different Global Navigation Satellite Systems. Many studies on tightly-combined model have mostly focused on the overlapping frequencies and short baselines, and fewer have focused on non-overlapping frequencies and medium-long baselines. Considering differential inter-systems code biases and inter-frequency code biases, we propose a tightly combined real-time kinematic model by a reparameterization process for non-overlapping frequencies among GPS/GLONASS/BDS systems. Compared with the traditional double-difference ambiguity parameter, we use single-difference ambiguity as the parameter to be estimated, which is not affected by reference satellite switching and wavelength limitation. The paper first analyzes the stability of differential inter-systems code biases and inter-frequency code biases with several zero and ultra-short baselines. Then, the paper tests the positioning performance under occlusion environment. The experiment shows that the tightly-combined model is more robust than the loosely-combined model and the average time to first fix is shortened by approximately 22.4%. At last, the paper compares the positioning performance of tightly-combined model, loosely-combined model, and tightly-combined model after calibration under the medium and long baselines. The experiment shows that the tightly-combined model after calibration is better than the tightly-combined model and loosely-combined model in the ambiguity resolution rate, initialization speed and positioning accuracy, respectively. Compared with intra-system loosely-combined model, the positioning accuracy of tightly-combined model after calibration is improved by 42.1%, 25.5%, and 32.3% in the three-dimensional directions, respectively.

Parameterization-switching GNSS attitude determination considering the success rate of ambiguity resolution

The two key elements of GNSS attitude determination are the correct integer ambiguity resolution (IAR) and the choice of appropriate method. The least-squares ambiguity decorrelation adjustment method is normally used to fix ambiguities using float estimates. However, it is difficult to get all ambiguities fixed in real time. Therefore, a method of partial ambiguity resolution which considers the success rate of IAR and makes full use of the ambiguity information at each epoch is proposed. The ambiguities after decorrelation can be divided into fixed ambiguities and unfixed ambiguities considering the fixing success rate. At the next epoch, the fixed ones are treated as constants, while the float estimates of the unfixed ones, along with the corresponding covariance matrix, are treated as prior information or pseudo-measurements. The parameters of the attitude determination problem can be baseline coordinates (BCs) or attitude angles (AAs). BC parameterization, without considering the baseline constraints, has the merit of resulting in a linear model; however, it involves parameter redundancy, which reduces the model strength. AA parameterization, although avoiding parameter redundancy, brings in nonlinearity and hence linearization errors in the least-squares solution. The linearization errors decrease as the number of fixed ambiguities increases. A switching-parameterization method is proposed; namely, at any epoch, as long as there are at least three fixed double-difference ambiguities, AA parameterization is adopted, otherwise BC is adopted. Even with AA, BC is firstly estimated and then transformed to attitude estimates at which the nonlinear measurement equations are linearized and solved. To verify the performance of the proposed method, a comparative study is implemented with BC and AA methods in the static experiment. The results show that the proposed method can fix ambiguity faster and have better stability and accuracy. In addition, the yaw angle is consistent with the actual running route of the vehicle in the kinematic experiment.

Enhanced orbit determination for formation-flying satellites through integrated single- and double-difference GPS ambiguity resolution

The formation-flying technique is a fundamental concept for earth observing satellite missions, which usually require both absolute and relative orbit accuracies. Their precise orbit determinations are usually exclusively performed based on spaceborne GNSS data, where integer ambiguity resolution (IAR) plays a crucial role in achieving the best orbit accuracy. However, it is found that single-receiver IAR by resolving the single-difference (SD) ambiguities between GNSS satellites at each individual formation-flying satellite cannot achieve the same relative orbit accuracy that is attained by double-difference (DD) IAR between formation-flying satellites. To unravel this problem, 1 year of GPS data collected by the Gravity Recovery and Climate Experiment (GRACE) mission are used and four types of orbits are derived for comparison: (1) orbits where no ambiguities are fixed; (2) orbits where SD IAR is performed for both satellites; (3) orbits where only DD ambiguities between the twin GRACE satellites are resolved; (4) and orbits where SD IAR is carried out on only GRACE A while DD IAR is further accomplished between the twin satellites, namely the integrated SD IAR and DD IAR solutions. They are then evaluated through comparison to the reduced-dynamic orbit generated at the Jet Propulsion Laboratory, residual analysis of satellite laser ranging (SLR), and K-Band ranging (KBR) measurements. As expected, the integrated SD IAR and DD IAR solutions can achieve the highest absolute and relative orbit accuracies simultaneously. Specifically, SLR residuals in case of the integrated IAR are reduced by at least 25% for the kinematic orbit, when compared to the case of DD IAR. KBR residuals in case of the integrated IAR are reduced by 35 and 16% for the dynamic and kinematic orbit, respectively, when compared to those of SD IAR. Importantly, we find that errors in GPS clocks and/or narrow-lane fractional-cycle biases are in part responsible for the deteriorated relative accuracy of SD IAR achieved orbits. Therefore, we suggest that the integrated SD IAR and DD IAR scheme should be implemented for the best orbit solutions of formation-flying missions.

Regularization and particle filtering estimation of phase inter-system biases (ISB) and the lookup table for Galileo E1-GPS L1 phase ISB calibration

The between-receiver phase inter-system bias (ISB) in multi-GNSS integration needs to be calibrated when fixing the inter-system double difference (DD) ambiguities. As a result, the DD ambiguity fixing performance is affected by the accuracy of the ISB calibration; however, those effects have not been investigated until now. Also, the knowledge about the ISB characteristics is not enough which can lead to failure of ambiguity fixing with ISB calibration value such as obtained by the lookup table method. Our study first integrates the regularization method to the existing particle filtering fractional ISB (F-ISB) estimation to derive more precise F-ISB estimates. Afterward, the effects of the F-ISB accuracy on DD ambiguity fixing in single-epoch GPS L1-Galileo E1 integration are investigated. We show that the errors of F-ISB can degrade the success rate of single-epoch ambiguity fixing, and thus, more precise F-ISB values lead to higher empirical success rates. Finally, the characteristics of the F-ISB are investigated based on baselines shorter than 20 km from the GNSS network of International GNSS Services. According to the 132 F-ISB estimates obtained using data collected from day of year 001 2016 to DOY 001, 2019, the F-ISB values for 4 receiver brands including Leica, Septentrio, Trimble and Javad are analyzed. The receiver-type and software-version combinations from the same manufacturer and with similar F-ISB values are regarded as one group and all the receivers can be classified into 6 groups. With F-ISB of Leica receiver group set to zero value, the other groups have F-ISB values of 0.000 m, + 0.055 m and + 0.095 m and the L1-E1 F-ISBs between different groups are provided in a lookup table for practical use.

Improving the initialization speed for long-range NRTK in network solution mode

Initialization speed is one of the most important factors in network real time kinematic (NRTK) performance. Owing to the low correlation among the error sources of reference stations, it is difficult to fix reference station ambiguities of long-range NRTK quickly. In traditional reference stations ambiguity resolution (AR) methods, baselines are usually solved independently which is called baseline solution (BS) mode in this study. Because the correlations among baselines are not taken into consideration in ambiguities estimation, the AR speed is slow. Generally, tens of minutes or longer time is required to initialize. We propose a network solution (NS) mode approach, in which the correlations among the double-difference ambiguities (DDAs) as well as double-difference ionospheric delays (DDIDs) of different baselines are considered in estimating float ambiguity solutions. Experimental results show that the float ambiguity solutions obtained are more accurate with an improved consistency. Thus, initialization speed is significantly increased by 18% in NS mode.

Impact of ambiguity resolution with sequential constraints on real-time precise GPS satellite orbit determination

The real-time precise orbit is an essential prerequisite for a real-time precise point positioning service. Focusing on the impact of ambiguity resolution on real-time orbital precision, we propose an efficient approach for real-time ambiguity resolution, which consists of two modules. The first module resolves the double-differenced ambiguities according to their proximity to the nearest integer. The second module sequentially adds the resolved integer ambiguity constraints to the square-root information filter process, after confirming that the same constraints have not been imposed before the current epoch. To validate our method, GPS data collected from 100 globally distributed stations are used to simulate a real-time precise orbit determination. The convergence performance is analyzed, and the accuracy of the orbit is evaluated. The results show that: (1) Almost 90% of the double-differenced ambiguities are fixed correctly in real time for baselines shorter than 1000 km. (2) After application of the proposed approach, the root-mean-square errors of all the satellite orbits are reduced from (5.9, 3.4, 2.3) cm to (4.7, 2.6, 2.2) cm for the along-track, cross-track and radial directions, respectively, with improvements of about 22% for the along-track and cross-track directions, and 6% for the radial direction. (3) Simulated real-time orbits determined with this method can obtain almost the same accuracy as some ultra-rapid products and, particularly, better accuracy can be achieved for eclipsing satellites.