Acoustic range error is the dominant error affecting the positioning accuracy of GNSS-Acoustics (GNSS-A). To reduce the impact, the sound speed profile (SSP) measurement data are typically used. However, the SSP measurement data may not be obtained when an unmanned observation platform is used for positioning GNSS-A. To address the issue of lacking SSP measurements, a new method that does not require SSP has been proposed. The method uses an extended cubic spline function to model the temporal variation of sound speed, employs cubic polynomials to model the spatial variation of sound speed, utilizes a squared function to model sound ray bending error, and uses least squares estimation to determine the coordinates of seafloor transponder. The observations of seafloor stations TOS2 and MYGI from 2011 to 2019 were used to test the method. The results were compared with the coordinates of the seafloor transponder provided by the GARPOS software. The results show that (1) the average root mean square (RMS) of the GNSS-A positioning residuals for TOS2 and MYGI are 6.84 cm and 8.59 cm, respectively, with the residuals exhibiting a normal distribution; (2) the centimeter-level positioning of the seafloor transponder can be realized by using the method of SSP-free GNSS-A positioning. (3) the approach using the extended cubic spline function improves the position accuracy by approximately 9.25% for TOS2 and 24.94% for MYGI, compared to the method using the cubic spline function; (4) the average difference between the baselines obtained from GARPOS software and those calculated by the method using the extended cubic spline function is 0.23 cm for TOS2 and 0.03 cm for MYGI; (5) the accuracy of the baseline derived from the method using the extended cubic spline function improves by about 17.05% for TOS2 and 13.53% for MYGI when compared to the GARPOS software results. Since no sound speed profile observation is required, the SSP-free GNSS-A positioning method is particularly suitable for unmanned GNSS-A observation systems where in-field SSP measurements are difficult to conduct. However, the SSP-free GNSS-A positioning method with the extended cubic spline function is only applicable for the positioning of seafloor stations equipped with multiple transponders within a limited sea area.

In recent years, the flex power of the global positioning system (GPS) Block IIR-M and Block IIF satellites has frequently been activated and deactivated. Ground-based GPS flex power monitoring faces several challenges, including uneven receiver distribution, multipath error arising from complex environments, and significant fluctuations in low-elevation carrier-to-noise ratio data. To overcome these challenges, we propose a novel GPS flex power detection method by introducing a detection metric, CN0el. Spaceborne receiver data from 15 low earth orbit (LEO) satellites and historical data from February 2022 to December 2023 are utilized to train an extreme gradient boosting (XGBoost) machine learning model. The detection model achieves an accuracy exceeding 99

The BeiDou global satellite navigation system (BDS) is an important tool for precise remote time and frequency transfer. The current day-boundary discontinuity in the data process of the BDS-3 carrier phase (CP) time transfer limits the exploitation of the full potential of the BeiDou system, particularly over extended periods (e.g. >1 day). This study focused on the daily discontinuities in BDS-3 ground time and frequency transfer induced by the current satellite orbit and clock products from the individual Multi-GNSS experiment (MGEX) analysis center. An interpolation approach that bridges the daily jump of satellite products is proposed to improve the continuity of BDS-3 time and frequency transfer when using the CP technique. Only third-order polynomial is required to smooth and bridge the impact of the daily jump for medium earth orbit (MEO) and inclined geosynchronous satellite orbit (IGSO) satellite products. Eight MGEX network stations equipped with various types of receivers and antennas with dual-frequency BDS-3 signals were used to establish four time transfer links (i.e. BRUX – PTBB, AMC4–HOB2, LCK4–USUD, and ONS1–LCK3) and evaluate their effectiveness. These results reveal that the current BDS-3 satellite product discontinuity from individual analysis center easily caused the drift of phase residuals near the day-boundary epoch, with mean amplitudes of 0.081 m and 0.434 m for MEO and IGSO satellite products, respectively. The proposed approach can contribute to effectively eliminating this drift in time and frequency transfer. Furthermore, it achieved a noticeable reduction in the percentage of negative day-boundary discontinuities in BDS-3 ground time and frequency transfer results. The average values of the jump were 0.143 ns, 0.367 ns, 0.138 ns, and 0.131 ns for the BRUX – PTBB, AMC4–HOB2, LCK4–USUD, and ONS1–LCK3 time links, respectively, representing improvements of 11.4%, 40.5%, 4.5%, and 12.0%, respectively, compared with the traditional approach.

Real-Time Precise Point Positioning (RTPPP) technology is one of the most important means of positioning and navigation, whereas this type of service requires Internet access. As an alternative, satellite-based RTPPP can be utilized, especially for the PPP-B2b service of the BeiDou Navigation Satellite System (BDS). However, there is no systematic study on how to fully employ all satellite-based augmented services such as B2b, B2a, and B1C services in BDS. To ensure the continuity and availability of the PPP-B2b service, an Integrated Real-Time PPP (InRPPP) system is proposed which consists of B2b, B2a, B1C, and broadcast ephemeris messages. Specifically, the satellite orbit and clock errors are corrected by the B2b and B2a services, and the ionospheric delays by the B1C services. That is, the B2b, B2a, B1C, and broadcast ephemeris messages are resiliently utilized in the InRPPP system. Both static and kinematic experiments under complex conditions are carried out. The results indicate that compared to the PPP based on B2b, B2a, or B1C services, the InRPPP system performs the best in terms of visible satellite numbers and Position Dilution of Precision (PDOP) values. Also, in the static experiment, the InRPPP mode achieves the highest accuracy and the shortest convergence time compared to the PPP based on B2b, B2a, or B1C services. The Three-Dimensional (3D) Root-Mean-Square (RMS) value of the InRPPP mode is 0.49 m, improving 59.6% on average. For the convergence time, 65.9% improvements can be obtained on average. In the kinematic experiment, for the InRPPP solutions the jumps are suppressed and the availability and stability are increased, which exhibiting average 34.3% improvements in terms of the 3D RMS value. In conclusion, the InRPPP system with B2b/B2a/B1C services presents superior performance and has a great potential in real applications.

Satellite-based positioning navigation and timing (PNT) system is the most fundamental and convenient application system in PNT fields now and in the future, thus the satellite-based PNT service system should be more and more robust, reliable and easy to be applied. Aiming at the weak service performance of the existing BeiDou constellation in the Antarctic and Arctic regions and the possible discontinuous problems in the autonomous orbit determination using BeiDou terminals on board of low earth orbit (LEO) satellites, we propose a scheme to optimize the BeiDou constellation configuration. A relatively optimized deep space PNT service constellation is designed in order to overcome the weak service capability of the BeiDou navigation constellation in deep space. We find that more satellites in the same orbital plane have better positioning geometry than increasing the number of constellation orbital planes. Furthermore, the idea of resilient satellite-based PNT service mode combining deep, high, medium, and low earth orbit navigation constellations is proposed, in order to improve the service capability of multi navigation constellations. To strengthen the satellite PNT service ability, the resilient strategies of on-board phased array antenna, on-board atomic clocks and satellite signal generator are proposed, in which the adaptively possible adjustment of signal power, signal frequency, signal modulation waveform and information format are discussed. Some examples to demonstrate the resilient satellite-based PNT application modes are given.

The strap-down inertial navigation system (SINS) and ultrashort baseline (USBL) (SINS/USBL) integrated system are the highly promising tool for navigating the autonomous underwater vehicles (AUVs). The measurement in the deep sea often contains unknown, time-varying noise and outliers. The traditional Kalman filter (KF) may face challenges in achieving high-precision underwater navigation due to its limited robustness and adaptivity. Although the robust KF has been developed and can effectively handle non-Gaussian noises in most cases, it may still suffer a significant loss in accuracy under nonstationary noise conditions. This study presents an adaptive robust KF that integrates the maximum correntropy criterion (MCC) with the variational Bayesian (VB) method to effectively mitigate the effects of complex noise. The proposed method achieves adaptivity by employing the VB method to estimate measurement noise covariance while enhancing robustness by mitigating outliers using the variable kernel bandwidth MCC strategy. According to simulation and offshore experiments, the proposed method provides superior estimation accuracy compared to related adaptive and robust algorithms.

Large constellations composed of a great amount of Low Earth Orbit (LEO) satellites are widely applied in satellite communication, remote sensing, augmented satellite navigation, environment monitoring, and so on. The satellite Orbit Determination (OD) is critical for the various function realization of the large constellation. Three different stepwise autonomous OD strategies for the large constellation of LEO satellites are proposed based on the spaceborne Global Navigation Satellite System (GNSS) observations and Inter-Satellite Link (ISL) range measurements, including the stepwise OD with both GNSS and ISL range measurements, stepwise OD with ISL range constraints, and an adaptive stepwise OD with both kinds of measurements. All of the three proposed stepwise autonomous OD approaches first estimate the initial orbit parameters for each satellite utilizing the spaceborne GNSS observations based on either a kinematic or dynamic OD strategy. The correction vector for the orbit parameters of each satellite is then individually calculated using the partial ISL range observations or ISL range constraints. The difference of the adaptive stepwise OD algorithm is that the covariance matrix of the predicted orbit parameters based on the dynamic model is modified by an adaptive factor. The LEO satellite parameters estimated with the stepwise OD strategies are equivalent to those obtained with the related integrated OD strategies. The main advantages of the proposed stepwise OD estimators are: (1) the orbit parameters of each satellite can be estimated in parallel, reducing the OD computational load for a large LEO constellation; (2) the spaceborne GNSS observations and the ILS range measurements in the three proposed approaches can be separately joined the OD procedures, making the parameter estimation flexible; (3) the adaptive stepwise OD mode with an adaptive factor acting on the covariance matrix of the predicted orbit parameters can effectively control the effects of the abnormal dynamic model information on the orbit parameter estimates. The simulation results for different OD strategies are analyzed. It is shown that the Root Mean Square Error (RMSE) of the estimated positions of the LEO satellites using the kinematic OD method is 60.527 cm, assuming the GNSS pseudorange noise of 30 cm. In contrast, the RMSE for the stepwise strategy, which considers only four adjacent ISL range measurements with an accuracy of 5 cm, is 18.287 cm. When the dynamic models for the LEO satellites are adopted, the RMSE of the estimated satellite positions using the stepwise orbit determination is further reduced to 11.340 cm. If the ISI ranging accuracy is better than 5 cm, the results remain nearly the same disregarding the ISL ranges are employed as observations or as constraints in the stepwise OD approaches. If the dynamic model information contains a few outliers, the adaptive stepwise OD can effectively control their effects on the orbit parameter estimates.

Spatio-temporal variation of sound speed, in seafloor geodetic precise positioning, can always be attributed to the time error. Firstly, this paper analyzes the existing error compensation model, i.e., the time ratio model, which is expressed by the recorded time multiplying a ratio coefficient. And then a time split model is proposed by expressing the acoustic ray traveling time as the recorded time pluses a perturbation time error. The theoretical differences between the proposed time bias compensation model and the time ratio model are analyzed. Under the new framework, sound speed perturbation models with optimal single-layer spatial gradient and multi-layer spatial gradients are developed to compensate for sound speed error in the complex cases. Numerical computation shows that the simple time split model keeps the same accuracy as some complicated models while considering the distribution of random error. Furthermore, multi-layer model can improve the positioning accuracy without putting the pressure on parametrization.

The seafloor hybrid constellation, composed of fixed and moored stations equipped with acoustic beacons, serves as a crucial infrastructure and holds promising prospects for possible applications in ocean submesoscale current monitoring and acoustic navigation when compared with traditionally unalloyed seafloor constellations. However, most of the acoustic positioning models are designed to handle fixed seafloor stations and do not match the actual motion characteristics of moored stations in a hybrid constellation, which may degrade the accuracy of beacon position estimation. To address this gap, a novel GNSS-acoustic (GNSS-A) positioning model is proposed in this contribution. First, the critical factor of acoustic measurements, namely, observation error of sound speed, is processed by error modeling based on the geometric angle of acoustic rays. Second, the smooth variation characteristic of physical marine signal processing is taken into consideration to estimate parameters related to time-delay error. Furthermore, the motion depiction of moored beacons is established and introduced into the observation equation system to obtain more reasonable positioning results of seafloor beacons. Finally, the proposed model is validated through tests on a sea-trial experimental dataset, along with an analysis of seafloor baseline measurements. Results and analysis show that, compared with those of traditional methods, the motion of moored beacons can be tracked in detail, and the trajectories of the four beacons maintain an overall consistency, which is expected to aid in deriving the possible ocean submesoscale currents.

In recent years, the Global Positioning System has commonly activated flex power, including global flex power and regional flex power. The activation and deactivation periods of satellite flex power are distinguished, and the impact of flex power on the estimation of the differential code biases (DCBs), ionospheric delays and positioning are analyzed. The results indicate that the DCBs are more stable when taking flex power into account. The impact of flex power on the inter-frequency DCBs is 0.19 ns, and on ionosphere estimation is 0.68 TECU. The impact of flex power on the intra-frequency DCBs for L1 and L2 is 0.17 ns and 0.25 ns, respectively. Single point positioning is affected by flex power in the E, N and U components by 0.04 m, 0.12 m and 0.17 m, respectively. (c) 2024 Published by Elsevier B.V. on behalf of COSPAR.

Least-squares collocation is a method for determining the anomalous gravitational field by a combination of geodetic measurements of different kinds. Its reliability is based on a precise covariance function and a suitable observation distribution as well as the smoothness of the stochastic collocation field. In this paper, three kinds of robustized collocation formulae are derived under the assumptions that both of the observations and the stochastic collocation field are contaminated by outliers, or one of them is contaminated respectively. The procedure in robustifying the covariance function and the calculations of the robust collocation is discussed. The influence functions of an observation error to the estimators are given.

A Low Earth Orbit (LEO) constellation augmenting satellite navigation is important in the future development of Global Navigation Satellite System (GNSS). GNSS augmented by LEO constellations can improve not only the accuracy of Positioning, Navigation, and Timing (PNT), but also the consistency and reliability of secure PNT system. This paper mainly analyzes the diverse demands of different PNT users for LEO augmented GNSS, including the precision demand in real-time, the availability demand in special areas, the navigation signal enhancement demand in complex electromagnetic environments, and the integrity demand with high security. Correspondingly, the possible contributions of LEO constellations to PNT performance are analyzed from multiple aspects. A particular attention is paid to the special PNT user requirements that cannot be fulfilled with existing GNSS, such as the PNT service demand in the polar regions and the onboard GNSS orbit determination demand of some LEO satellites. The key technologies to be considered in the constellation design, function realization, and payload development of the LEO-augmented navigation system are summarized.

The carrier phase (CP) technique based on the BeiDou Global Satellite Navigation System (BDS-3) has proven to be a crucial spatial tool for remote time and frequency transfer. The current CP technique models the receiver clock offset as a white noise stochastic process and easily absorbs some unmodeled errors, thus compromising the time and frequency transfer performance. To further improve the performance of time and frequency transfer, a new BDS-3 receiver clock estimation algorithm based on the epoch difference (ED) model is presented, and the mathematical principles and applied modes are discussed. The algorithm makes full use of both observations of the current epoch and practical variations of the receiver clock offset, further improving the performance of time and frequency transfer. Five Multi-Global Navigation Satellite System Experiment network stations equipped with various types of receivers and antennas with dual-frequency BDS-3 signals were used to establish four time transfer links (i.e., AMC4-PTBB, BRUX-PTBB, OP71-PTBB, and WTZS-PTBB) to evaluate their effectiveness. The ED model improves all the four time links in terms of noise level, with improvements of 17.0%, 18.3%, 20.3%, and 5.9%, respectively, when compared with the results from a non-ED model. The ED model outputs were better than the raw solutions in terms of frequency stability at all time links, particularly for average time intervals (tau) < 1000 s. The mean improvement was 8.1% for AMC4-PTBB, 16.1% for BRUX-PTBB, 10.0% for OP71-PTBB, and 18.6% for WTZS-PTBB when the average time (tau) was less than 1000 s.

The accuracy of underwater acoustic navigation is significantly influenced by geometry configuration and systematic errors coming from geophysical environment. Long baseline (LBL) systems exhibit superior positioning precision than other acoustic navigation systems, primarily attributable to the better Geometric Dilution of Precision (GDOP). However, the geometry configuration tends to degrade when the underwater vehicle is far away from the seafloor beacons. Factors such as acoustic ray bending error, Doppler effect and Earth curvature may seriously affect the navigation accuracy. To address these issues, we present a robust Kalman filter with systematic error compensation for long-range LBL systems. Firstly, a reversed acoustic ray tracking method is proposed for a unique phenomenon in acoustic wave propagation. Following that, we construct an acoustic positioning model with a time bias parameter, to weaken the impact of Doppler effect on acoustic ranging. This additional parameter was estimated with the state of the user vehicle in real time. At last, an observation equation of pressure gauge considering Earth curvature is presented for depth constraint. Long-range LBL experiments conducted in the South China Sea is employed to validate the performance of the proposed methods, taking the positioning results of inverted ultra-short baseline (iUSBL) system as reference. The results show that the proposed method outperform traditional method with a decrease of about 60 % in the three-dimensional root mean square (3DRMS) of navigation errors. With mentioned three improvements, the 3DRMS of the long-range LBL system with the longest distance of 20 km from the beacon center is less than 14 m. These findings can provide theoretical basis for other long-range LBL systems.

Measurement trajectory error, geometric error of the measurement trajectory, and systematic error attributable to the velocity of sound are the main sources of the errors that affect the positioning accuracy of seafloor geodetic stations. Here, we propose a systematic error difference constraint model designed to control the influence of systematic errors in seafloor geodetic station positioning. An error equation is constructed using system error parameters and the position parameters of the seafloor geodetic station to be estimated. On the basis of the subtle positional change characteristics of an adjacent epoch, the constraint condition of the adjacent epoch difference is added. Finally, for the case of a single seafloor transponder, analysis of the results of a simulation and an actual experiment confirm that the proposed model with a systematic error difference constraint can effectively control the influence of systematic errors on seafloor positioning and overcome the ill-posed model issue, thereby enabling positioning results that are more accurate.

Global navigation satellite system-acoustic (GNSS-A) combined underwater positioning technique is widely applied in seafloor displacement monitoring and offshore exploration. The conventional GNSS-A positioning strategy is under the assumption of equal-precision sea-surface transducer's positions determined by GNSS positioning, which weakens the positioning accuracy of single seafloor transponder-equipped station. In this article, the extended joint adjustment (JA) of the measurements of the sea-surface transducers to seafloor transponders and ranging measurements of the transponder to transponder is proposed. First, we refine the transducer-to-transponder timing observation equation system by acoustic ray-tracing strategy to reduce the sound-speed-related errors. Second, we establish the mathematical model for extended JA with interstation ranging measurements for seafloor geodetic network (SGN) positioning. Finally, the efficacy of the proposed method is demonstrated both in simulations and in real measurement datasets. The experimental results show that the proposed method outperforms the traditional positioning methods, especially in the horizontal components.

Addressing the challenge of real-time estimation of instantaneous sound speed profiles (SSP for short). We proposes a SSP inversion method based on the travel time of vertically launched sound waves from seafloor stations to the sea surface. The basic principles of the inversion method are introduced, followed by an investigation into the calculation of instantaneous SSP area considering tidal and wave effects. The proposed method is validated by using semi-experimental data, the results indicate that the accuracy of predictions is poor if the correcting for the tidal and wave influences are not taken. However, after applying the correction equation proposed in this study to adjust for travel time, the relative error in the prediction of SSP area difference is less than 4%.

Robust estimation with independent observations has been investigated in the field of geodesy. However, robust estimation in the dependent situation has not widely been studied. Robust estimation models for correlated observations, based on the principles of M-estimation and equivalent weights, are established in this paper. The general linear expressions of solutions and their corresponding influence functions are derived. Based on iterative calculation procedures, six computation schemes are performed and compared. A new equivalent weight function for correlated observations is proposed.

The ocean sound speed changes drastically with time and space. It is almost impossible to achieve centimeter-level-precision Global Navigation Satellite System (GNSS)-Acoustic (GNSS-A) positioning without regarding spatio-temporal variations of sound speed. Inspired by the atmospheric delay estimation, we define zenith acoustic delay (ZAD) as the zenith-direction ranging error sourced by the sound speed variations, and then it is decomposed into one temporal component and two horizontal components for the sound speed variations in time and space, respectively. We propose a network solution with the three ZAD components of each seafloor geodetic point as common parameters to be estimated, and then present a piece-wise estimation to characterize their time-varying nature. To remedy the ray bending effect this study is implemented in the context of the ray tracing algorithm based on a reference sound speed profile (SSP). Experimental tests show that the proposed network solution improves ZAD estimation and results in a better position determination as compared to the single-point positioning (SPP) solution. The proposed network solution for a single campaign can achieve a horizontal positioning precision better than 5 cm (1-sigma) for the seafloor geodetic array centroid. The temporal variation of ZAD is up to one meter while the horizontal variations vary within a few decimeters in the time domain and show an azimuthal asymmetry in space.

The accuracy of underwater acoustic positioning is significantly influenced by observing geometry configuration and sound speed variation. The geometric configuration includes the sailing track of the surface vessel as well as the geometry of the seafloor array, all of which are relevant to data processing and error analysis. In this paper, we study how to reduce the influence of sound speed error on underwater positioning based on geometric configuration. For the single point positioning model, an integrated sound speed compensation approach considering the vessel track is proposed to improve positioning accuracy. For the seafloor geodetic network, a network positioning model with virtual seafloor baseline constraints is proposed to improve positioning accuracy. Acoustic positioning experiments conducted in the South China Sea and Japanese open data verify the effectiveness of presented methods. The results show that there is significant diurnal sound speed variation in the South China Sea. The discrepancies between positioning results of different data sets are reduced by sound speed compensation, and the standard deviations of horizontal components are better than 10 cm. The positioning accuracy of repeated surveys in Japanese geodetic network is improved with virtual baseline constraints. Compared with solutions with no virtual baseline constraints, the standard deviation of array center coordinate series is reduced by 8.7%, 21.3%, and 57.1% for the east, north, and up components when dealing with long-term GNSS-A data.