GPS Phase Observations Research Articles

Bandwidth correction of Swarm GPS carrier phase observations for improved orbit and gravity field determination

Gravity fields derived from GPS tracking of the three Swarm satellites have shown artifacts near the geomagnetic equator, where the carrier phase tracking on the L2 frequency is unable to follow rapid ionospheric path delay changes due to a limited tracking loop bandwidth of only 0.25 Hz in the early years of the mission. Based on the knowledge of the loop filter design, an analytical approach is developed to recover the original L2 signal from the observed carrier phase through inversion of the loop transfer function. Precise orbit determination and gravity field solutions are used to assess the quality of the correction. We show that the a posteriori RMS of the ionosphere-free GPS phase observations for a reduced-dynamic orbit determination can be reduced from 3 to 2 mm while keeping up to 7% more data in the outlier screening compared to uncorrected observations. We also show that artifacts in the kinematic orbit and gravity field solution near the geomagnetic equator can be substantially reduced. The analytical correction is able to mitigate the equatorial artifacts. However, the analytical correction is not as successful compared to the down-weighting of problematic GPS data used in earlier studies. In contrast to the weighting approaches, up to 9–10% more kinematic positions can be retained for the heavily disturbed month March 2015 and also stronger signals for gravity field estimation in the equatorial regions are obtained, as can be seen in the reduced error degree variances of the gravity field estimation. The presented approach may also be applied to other low earth orbit missions, provided that the GPS receivers offer a sufficiently high data rate compared to the tracking loop bandwidth, and provided that the basic loop-filter parameters are known.

Integrated processing of ground- and space-based GPS observations: improving GPS satellite orbits observed with sparse ground networks

The precise orbit determination (POD) of Global Navigation Satellite System (GNSS) satellites and low Earth orbiters (LEOs) are usually performed independently. It is a potential way to improve the GNSS orbits by integrating LEOs onboard observations into the processing, especially for the developing GNSS, e.g., Galileo with a sparse sensor station network and Beidou with a regional distributed operating network. In recent years, few studies combined the processing of ground- and space-based GNSS observations. The integrated POD of GPS satellites and seven LEOs, including GRACE-A/B, OSTM/Jason-2, Jason-3 and, Swarm-A/B/C, is discussed in this study. GPS code and phase observations obtained by onboard GPS receivers of LEOs and ground-based receivers of the International GNSS Service (IGS) tracking network are used together in one least-squares adjustment. The POD solutions of the integrated processing with different subsets of LEOs and ground stations are analyzed in detail. The derived GPS satellite orbits are validated by comparing with the official IGS products and internal comparison based on the differences of overlapping orbits and satellite positions at the day-boundary epoch. The differences between the GPS satellite orbits derived based on a 26-station network and the official IGS products decrease from 37.5 to 23.9 mm (34% improvement) in 1D-mean RMS when adding seven LEOs. Both the number of the space-based observations and the LEO orbit geometry affect the GPS satellite orbits derived in the integrated processing. In this study, the latter one is proved to be more critical. By including three LEOs in three different orbital planes, the GPS satellite orbits improve more than from adding seven well-selected additional stations to the network. Experiments with a ten-station and regional network show an improvement of the GPS satellite orbits from about 25 cm to less than five centimeters in 1D-mean RMS after integrating the seven LEOs.

Fully populated VCM or the hidden parameter

Abstract Least-squares estimates are trustworthy with minimal variance if the correct stochastic model is used. Due to computational burden, diagonal models that neglect correlations are preferred to describe the elevation dependency of the variance of GPS observations. In this contribution, an improved stochastic model based on a parametric function to take correlations between GPS phase observations into account is presented. Built on an adapted and flexible Mátern function accounting for spatiotemporal variabilities, its parameters can be fixed thanks to Maximum Likelihood Estimation or chosen apriori to model turbulent tropospheric refractivity fluctuations. In this contribution, we will show in which cases and under which conditions corresponding fully populated variance covariance matrices (VCM) replace the estimation of a tropospheric parameter. For this equivalence “augmented functional versus augmented stochastic model” to hold, the VCM should be made sufficiently largewhich corresponds to computing small batches of observations. A case study with observations from a medium baseline of 80 km divided into batches of 600 s shows improvement of up to 100 mm for the 3Drms when fully populated VCM are used compared with an elevation dependent diagonal model. It confirms the strong potential of such matrices to improve the least-squares solution, particularly when ambiguities are let float.

Utilization PPP method in aircraft positioning in post-processing mode

PurposeThe purpose of this paper is to study the implementation of GNSS technique in aviation for recovery of aircraft’s position using Precise Point Positioning (PPP) method.Design/methodology/approachThe aircraft’s coordinates in ellipsoidal frame were obtained based on GPS code and phase observations for PPP method. The numerical computations were executed in post-processing mode in the CSRS-PPP and magicPPP online services. The mathematical scheme of PPP method was development using indifference equations of Ionosphere-Free linear combination. In the experiment, airborne test using Cessna 172 aircraft on June 01, 2010 in the military airport in Deblin was realized. The aircraft’s position was determined using data from GNSS receiver (Topcon HiperPro with interval of 1 s).FindingsIn this paper, the accuracy of aircraft’s position is better than 0.07 m for CSRS-PPP service and better than 0.27 m for magicPPP service. In case of the Mean Radial Spherical Error parameter, the average value for CSRS-PPP service equals to 0.01 m, whereas for magicPPP, it is about 0.38 m. The values of vertical coordinate of Cessna 172 aircraft were also checked with results of Real Time Kinematic–On The Fly technique.Research limitations/implicationsIn this paper, the analysis of aircraft positioning is focused on the application of the PPP method in post-processing mode. In near real time, the PPP method still has limitations, especially in the area of ambiguity resolution and also instrumental biases (e.g. Narrow Lane Hardware Delays).Practical implicationsThe PPP method can be applied in aviation in post-processing mode for verification of true aircraft coordinates and elimination of blunder errors from adjustment processing of GNSS observations. The Zenith Wet Delay term as a product of troposphere delay and receiver clock bias as a product of precise time transfer can be obtained in the PPP method.Originality/valueThe paper presents that the PPP method is an alternative solution for the recovery of aircraft’s position in aviation, and this method can be also applied in the positioning of aircraft based on GLONASS or GPS/GLONASS data.

On modelling GPS phase correlations: a parametric model

Least-squares estimates are unbiased with minimal variance if the correct stochastic model is used. However, due to computational burden, diagonal variance covariance matrices (VCM) are often preferred where only the elevation dependency of the variance of GPS observations is described. This simplification that neglects correlations between measurements leads to a less efficient least-squares solution. In this contribution, an improved stochastic model based on a simple parametric function to model correlations between GPS phase observations is presented. Built on an adapted and flexible Matern function accounting for spatiotemporal variabilities, its parameters are fixed thanks to maximum likelihood estimation. Consecutively, fully populated VCM can be computed that both model the correlations of one satellite with itself as well as the correlations between one satellite and other ones. The whitening of the observations thanks to such matrices is particularly effective, allowing a more homogeneous Fourier amplitude spectrum with respect to the one obtained by using diagonal VCM. Wrong Matern parameters—as for instance too long correlation or too low smoothness—are shown to skew the least-squares solution impacting principally results of test statistics such as the apriori cofactor matrix of the estimates or the aposteriori variance factor. The effects at the estimates level are minimal as long as the correlation structure is not strongly wrongly estimated. Thus, taking correlations into account in least-squares adjustment for positioning leads to a more realistic precision and better distributed test statistics such as the overall model test and should not be neglected. Our simple proposal shows an improvement in that direction with respect to often empirical used model.

Taking correlations into account: a diagonal correlation model

The true covariance matrix of the GPS phase observations is unknown and has to be assumed or estimated. The variance of the least-squares residuals was empirically shown to have an elevation dependency and is often expressed as a sum of a constant and an exponential function. Disregarding correlations that are for instance due to atmospheric effects, the variance covariance matrices are diagonal. This simplification leads to errors in the estimates, including the float ambiguity vector, as well as to an overoptimistic precision. Thus, results of test statistics such as the outlier or the overall model test are impacted. For the particular case of GPS positioning, an innovative proposal was made to take correlations into account easily, condensed in an equivalent diagonal matrix. However, the a posteriori variance factor obtained with this simplification is strongly underestimated and in most cases the inversion of fully populated matrices has anyway to be carried out. In this contribution, we propose an alternative diagonal correlation model based on a simple exponential function to approximate the developed equivalent model. This way, correlations can be included in a diagonal variance covariance matrix without computation burden. A case study with an 80-km baseline where the ambiguities are estimated together with the coordinates in the least-squares adjustment demonstrates the potential of the model. It leads to a proposal based on the autocorrelation coefficient for fixing its parameters.

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

In this contribution, using the example of the Matern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schon model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Matern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Matern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Matern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).

1 × 10−16 frequency transfer by GPS PPP with integer ambiguity resolution

For many years, the time community has been using the precise point positioning (PPP) technique which uses GPS phase and code observations to compute time and frequency links. However, progress in atomic clocks implies that the performance of PPP frequency comparisons is a limiting factor in comparing the best frequency standards. We show that a PPP technique where the integer nature of phase ambiguities is preserved consitutes significant improvement of the classical use of floating ambiguities. We demonstrate that this integer-PPP technique allows frequency comparisons with 1 × 10−16 accuracy in a few days and can be readily operated with existing products.

A Realistic and Easy-to-Implement Weighting Model for GPS Phase Observations

Observation weighting is an essential component of GPS stochastic modeling and plays a key role in reliable outlier detection and parameter estimation. Nowadays, satellite elevation angle and SNR are used as quality indicators for GPS phase measurements in high-accuracy geodetic applications. In comparison with elevation-dependent models, SNR-based weighting schemes represent the reality better, but usually require greater implementation efforts. Relying upon a representative analysis of empirical SNR-based weights, this paper proposes the elevation-dependent exponential weighting function EXPZ, which benefits from realistic SNR-based weights and enables easy software implementation. To process GPS data from a regional network, this advanced weighting scheme is implemented in the Bernese GPS Software 5.0 and is compared with the conventional elevation-dependent COSZ model in terms of phase ambiguity resolution, troposphere parameter (TRP) estimation, and site coordinate determination. The results show that the proposed EXPZ model significantly attenuates the downweighting effects on low-elevation observations and improves the success rates of ambiguity resolution by about 10%, the standard deviations of site-specific TRPs by about 40%, and the repeatability of daily coordinate estimates by up to 2.3 mm (50%).

GPS phase difference variation statistics: A comparison between phase scintillation index and proxy indices

Scintillated GPS phase observations are traditionally characterized by the phase scintillation index, derived from specialized GPS receivers usually tracking at 50Hz. Geodetic quality GPS receivers, on the other hand, are normally tracking at frequencies up to 1Hz. However, availability of continuously operating geodetic receivers both in time and geographical location are superior to scintillation receiver’s coverage in many parts of the world. This motivates scintillation studies using regional and global geodetic GPS networks. Previous studies have shown the usefulness of GPS estimated total electron content variations for detecting ionospheric irregularities. In this paper, collocated geodetic and scintillation receivers are employed to compare proxy indices derived from geodetic receivers with the phase scintillation index during quiet and moderately disturbed ionospheric conditions. Sensitivity of the phase scintillation indices at high latitude stations to geomagnetic activity is discussed. Global mapping of ionospheric disturbances using proxy indices from real-time 1Hz GPS stations are also presented.

A case study of using Raman lidar measurements in high-accuracy GPS applications

This paper investigates the impact of rapid small-scale water vapor fluctuations on GPS height determination. Water vapor measurements from a Raman lidar are used for documenting the water vapor heterogeneities and correcting GPS signal propagation delays in clear sky conditions. We use data from four short observing sessions (6 h) during the VAPIC experiment (15 May–15 June 2004). The retrieval of wet delays from our Raman lidar is shown to agree well with radiosonde retrievals (bias and standard deviation (SD) were smaller than 1 and 2.8 mm, respectively) and microwave radiometers (from two different instruments, bias was 6.0/−6.6 mm and SD 1.3/3.8 mm). A standard GPS data analysis is shown to fail in accurately reproducing fast zenith wet delay (ZWD) variations. The ZWD estimates could be improved when mean post-fit phase residuals were removed. Several methodologies for integrating zenith lidar observations into the GPS data processing are also presented. The final method consists in using lidar wet delays for correcting a priori the GPS phase observations and estimating a scale factor for the lidar wet delays jointly with the GPS station position. The estimation of this scale factor allows correcting for a mis-calibration in the lidar data and provides in the same way an estimate of the Raman lidar instrument constant. The agreement of this constant with an independent determination using radiosonde data is at the level of 1–4%. The lidar wet delays were derived by ray-tracing from zenith pointing measurements: further improvement in GPS positioning is expected from slant path lidar measurements that would properly account for water vapor anisotropy.

Kilauea slow slip events: Identification, source inversions, and relation to seismicity

Several slow slip events beneath the south flank of Kilauea Volcano, Hawaii, have been inferred from transient displacements in daily GPS positions. To search for smaller events that may be close to the noise level in the GPS time series, we compare displacement fields on Kilauea's south flank with displacement patterns in previously identified slow slip events. Matching displacement patterns are found for several new candidate events, although displacements are much smaller than previously identified events. One of the candidates, 29 May 2000, is coincident with a microearthquake swarm, as are all of the previously identified slow slip events. The microearthquakes follow the onset of slow slip, implying that they are triggered by stress changes during slip. The new slow slip event brings the total number of events on Kilauea, between 1997 and 2007, to eight, the smallest having MW = 5.3, and the largest having MW = 6.0. While the recurrence time between the four largest events is 2.11 ± 0.01 years, the repeat time for all eight events is 0.9 ± 0.6 years. We invert for the fault geometry and distribution of slip during the slow slip events. The optimal source depths of 5 km, assuming uniform slip dislocations in an elastic half‐space, are considerably shallower than the accompanying swarm earthquakes (6.5–8.5 km), which would place the earthquakes in a zone of decreased Coulomb stress. Inversions including the effects of topography and layered elastic structure in the forward models favor depths comparable to microearthquake depths, such that the earthquakes are located in a region of increased Coulomb stress. We also invert for time‐dependent fault slip directly from the 30 s GPS phase observations, constraining the source to the optimal uniform slip geometry. On the basis of these inversions, the larger events last between 1.5–2.2 days. The data are unable to resolve migration of slip along the fault. The temporal pattern of accompanying microearthquakes is consistent with the fault slip history assuming a seismicity rate theory based on rate and state‐friction, making the swarm earthquakes coshocks and aftershocks of the slow slip events.

A proposal for modelling physical correlations of GPS phase observations

Physical correlations are generally not considered in the stochastic model of GPS observations. Consequently, the resulting estimated parameters (such as station coordinates) are affected and their variances are too optimistic and therefore not appropriate to statistically test the significance of the estimates. Considering atmospheric physics, turbulent irregularities in the lower atmosphere cause physical correlations between GPS signals propagating through them. Based on turbulence theory, a variance–covariance model for double-differenced (DD) GPS phase observations is developed in this paper, termed SIGMA-C model. The resulting fully populated variance–covariance matrix depends not only on the satellite-station geometry but also on the prevailing atmospheric conditions. It is shown that especially the wind velocity and its direction play a key role for the adequate modelling of physical correlations. The comparison of the correlations predicted by the SIGMA-C model and the empirical auto- and cross-correlation functions of DD from a specially designed GPS test network shows good agreement between data and model. Consequently, the SIGMA-C model seems to be suited for the proper description of physical correlations in GPS phase data. Furthermore, the application of the SIGMA-C model now attenuates the unbounded decrease of formal coordinate variances when the number of observations is increased. Hence for small GPS networks, the new model yields more realistic formal coordinate variances—even for high sampling rates and longer observation sessions.

Water vapour tomography using GPS phase observations: Results from the ESCOMPTE experiment

Water vapour tomography using GPS phase observations: Results from the ESCOMPTE experiment

Affine distortion of small GPS networks with large height differences

GPS is a promising tool for real-time monitoring of deformations of slopes or large structures. However, remaining systematic effects in GPS phase observations after double differencing and application of a priori models affect the resulting coordinates. They complicate the proper separation of the actual deformations from pseudo-deformations induced by the systematic effects. This paper shows that for small monitoring networks (baseline lengths <5 km) only affine distortions of the network geometry are generated by the remaining distance dependent systematic effects, e.g. unmodelled tropospheric and ionospheric propagation effects, or satellite orbit errors. Hence, a generic correction model is given by a three-dimensional affine transformation involving a maximum of 12 transformation parameters. For the determination of these parameters, four high quality GPS stations are necessary which are not affected by the actual deformations to be monitored. Based on the analysis of network geometries of synthetic GPS networks with large height differences and considering the physics of the GPS observations it is shown, however, that less than 12 parameters are sufficient for the computation of the corrections. The proposed 8 parameter model was applied to the GPS monitoring network of the Gradenbach landslide. For this small network with large height differences, it was shown that the distortions can be reduced by about 75%.

EXPERIMENTAL GPS/INS/PSEUDOLITE SYSTEM FOR KINEMATIC POSITIONING

The requirements for accuracy and reliability of kinematic positioning have dramatically increased in the past few years. Stand-alone GPS can, in general, provide high accuracy and stability over time, however, may still fail in adverse conditions (jamming, high multipath, high ionospheric activity, etc.) or in the environments with obstructions or locations with weak satellite constellation (high altitudes). One example of rather demanding positioning accuracy requirements is in mobile mapping technology that has significantly expanded its market during the last decade of the twentieth century. In order to meet the stringent accuracy requirements and to provide a continuous trajectory, the most common means of positioning used in mobile mapping is an integrated GPS/INS (inertial navigation system) system, used to provide positioning and attitude information for land-based or airborne imaging sensors. GPS contributes high accuracy and long-term stability (under no losses of lock and properly resolved ambiguities), providing means of error estimation of the INS sensors. A GPS-calibrated inertial navigation system provides reliable bridging during GPS outages and supports the ambiguity resolution after the GPS lock is reestablished. Such a system works well if GPS gaps are not too frequent and not excessively long. However, in case of navigation in urban canyons or indoor mapping, there is usually very limited or no GPS signal. Consequently, a pseudolite (PL) array may be used to supplement the satellite signal.In this paper, an experimental GSP/INS/PL system is presented, with a special emphasis on the error spectrum and the navigation performance analysis, based on a medium-accuracy and high-reliability INS, limited GPS constellation, and varying number of pseudolites. Double difference GPS and PL phase observations are formed and used as a measurement update in the extended Kalman filter, facilitating tight integration of the navigation sensors. The simulated data and the actual test results in typical noisy environments are presented, including the geometry and the vehicle dynamics analysis, and their impact on the navigation accuracy.

EXPERIMENTAL GPS/INS/PSEUDOLITE SYSTEM FOR KINEMATIC POSITIONING

The requirements for accuracy and reliability of kinematic positioning have dramatically increased in the past few years. Stand-alone GPS can, in general, provide high accuracy and stability over time, however, may still fail in adverse conditions (jamming, high multipath, high ionospheric activity, etc.) or in the environments with obstructions or locations with weak satellite constellation (high altitudes). One example of rather demanding positioning accuracy requirements is in mobile mapping technology that has significantly expanded its market during the last decade of the twentieth century. In order to meet the stringent accuracy requirements and to provide a continuous trajectory, the most common means of positioning used in mobile mapping is an integrated GPS/INS (inertial navigation system) system, used to provide positioning and attitude information for land-based or airborne imaging sensors. GPS contributes high accuracy and long-term stability (under no losses of lock and properly resolved ambiguities), providing means of error estimation of the INS sensors. A GPS-calibrated inertial navigation system provides reliable bridging during GPS outages and supports the ambiguity resolution after the GPS lock is reestablished. Such a system works well if GPS gaps are not too frequent and not excessively long. However, in case of navigation in urban canyons or indoor mapping, there is usually very limited or no GPS signal. Consequently, a pseudolite (PL) array may be used to supplement the satellite signal.In this paper, an experimental GSP/INS/PL system is presented, with a special emphasis on the error spectrum and the navigation performance analysis, based on a medium-accuracy and high-reliability INS, limited GPS constellation, and varying number of pseudolites. Double difference GPS and PL phase observations are formed and used as a measurement update in the extended Kalman filter, facilitating tight integration of the navigation sensors. The simulated data and the actual test results in typical noisy environments are presented, including the geometry and the vehicle dynamics analysis, and their impact on the navigation accuracy.

Time-Relative Positioning with a Single Civil GPS Receiver

Time-relative positioning is a recent method for processing GPS phase observations. The operational method undertaken in this paper consists of the following steps: first, recording phase observations at a station of known coordinates; second, moving the GPS receiver to an unknown station (which can be located up to a few hundred meters away, dependint on what type of transportation – e. g., walking, motorcycle – is available) while continuously observing carrier phases; and, third, recording phase observations at a second station of unknown coordinates with a single GPS receiver. To obtain the position of the unknown station relative to the first (known) station, the processing method uses combined observations taken at two different epochs and two different stations with the same receiver. For this reason, the errors that vary between two epochs must be taken into account in an appropriate way, especially errors in satellite clock corrections and ephemerides, and errors related to tropospheric and ionospheric delays. Ionospheric modeling using IONEX files (the ionospheric maps calculated by the International GPS Service) was also tested to correct L1 phase observations. This method has been used to calculate short vectors with an accuracy of a few centimeters (for a processing interval of 30 s) using a single civil GPS receiver. © 2001 John Wiley & Sons, Inc.

Accuracy of GPS Phase and Code Observations in Practice

In order to examine the accuracy of GPS observations, both precision and biases have to be taken into account. The precision of a GPS observation reflects its stochastic variability and is described by the standard deviation. Biases are non-random effects in the GPS observations (like for instance multipath).

An extended weight model for GPS phase observations

Low elevation data may be included in the processing of small scale engineering GPS applications, in order to improve satellite geometry and reduce the required observation time. Careful weighting of the phase observations is needed then, because of increased noise and systematic errors. The measured carrier-to-noise-power-density ratio (C/N0) has proven to be an excellent tool for the estimation of the random errors of the phase observables. In this paper an extended weight model using robust estimation is presented, in particular the Danish Method. This model combines the information inherent in C/N0 and the residuals of the double differenced phase observations in order to model random errors and signal distortion effects.