Journal of Geodesy

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Methods for coherent optical Doppler orbitography

Fri, 06/05/2020 - 00:00
Abstract

Doppler orbitography uses the Doppler shift in a transmitted signal to determine the orbital parameters of satellites including range and range rate (or radial velocity). We describe two techniques for atmospheric-limited optical Doppler orbitography measurements of range rate. The unstabilised technique determines the Doppler shift directly from a heterodyne measurement of the returned optical signal. The stabilised technique aims to improve the precision of the first by suppressing atmospheric phase noise imprinted on the transmitted optical signal. We demonstrate the performance of each technique over a \(2.2\,\hbox {km}\) horizontal link with a simulated in-line velocity Doppler shift at the far end. A horizontal link of this length has been estimated to exhibit nearly half the total integrated atmospheric turbulence of a vertical link to space. Without stabilisation of the atmospheric effects, we obtained an estimated range rate precision of \(17\,{\upmu }\hbox {m}\,\hbox {s}^{-1}\) at \(1\,\hbox {s}\) of integration. With active suppression of atmospheric phase noise, this is improved by three orders of magnitude to an estimated range rate precision of \(9.0\,\hbox {nm}\,\hbox {s}^{-1}\) at \(1\,\hbox {s}\) of integration, and \(1.1\,\hbox {nm}\,\hbox {s}^{-1}\) when integrated over \(60\,\hbox {s}\). This represents four orders of magnitude improvement over the typical performance of operational ground to space X-Band systems in terms of range rate precision at the same integration time. The performance of this system is a promising proof of concept for coherent optical Doppler orbitography. There are many additional challenges associated with performing these techniques from ground to space that were not captured within the preliminary experiments presented here. In the future, we aim to progress towards a \(10\,\hbox {km}\) horizontal link to replicate the expected atmospheric turbulence for a ground to space link.

Correction to: Dynamic harmonic regression modeling for monthly mean sea levels at tide gauges within the Arabian Gulf

Mon, 05/25/2020 - 00:00

The editors alerted the author to the editorial policy that ocean and sea areas should be named following agreed international standards.

IAG Newsletter

Fri, 05/22/2020 - 00:00

Towards an International Height Reference System: insights from the Colorado geoid experiment using AUSGeoid computation methods

Mon, 05/11/2020 - 00:00
Abstract

We apply the AUSGeoid data processing and computation methodologies to data provided for the International Height Reference System (IHRS) Colorado experiment as part of the International Association of Geodesy Joint Working Groups 0.1.2 and 2.2.2. This experiment is undertaken to test a range of different geoid computation methods from international research groups with a view to standardising these methods to form a set of conventions that can be established as an IHRS. The IHRS can realise an International Height Reference Frame to be used to study physical changes on and within the Earth. The Colorado experiment study site is much more mountainous (maximum height 4401 m) than the mostly flat Australian continent (maximum height 2228 m), and the available data over Colorado are different from Australian data (e.g. much more extensive airborne gravity coverage). Hence, we have tested and applied several modifications to the AUSGeoid approach, which had been tailored to the Australian situation. This includes different methods for the computation of terrain corrections, the gridding of terrestrial gravity data, the treatment of long-wavelength errors in the gravity anomaly grid and the combination of terrestrial and airborne data. A new method that has not previously been tested is the application of a spherical harmonic high-pass filter to residual anomalies. The results indicate that the AUSGeoid methods can successfully be used to compute a high accuracy geoid in challenging mountainous conditions. Modifications to the AUSGeoid approach lead to root-mean-square differences between geoid models up to ~ 0.028 m and agreement with GNSS-levelling data to ~ 0.044 m, but the benefits of these modifications cannot be rigorously assessed due to the limitation of the GNSS-levelling accuracy over the computation area.

Self-tuning robust adjustment within multivariate regression time series models with vector-autoregressive random errors

Sun, 05/10/2020 - 00:00
Abstract

The iteratively reweighted least-squares approach to self-tuning robust adjustment of parameters in linear regression models with autoregressive (AR) and t-distributed random errors, previously established in Kargoll et al. (in J Geod 92(3):271–297, 2018. https://doi.org/10.1007/s00190-017-1062-6), is extended to multivariate approaches. Multivariate models are used to describe the behavior of multiple observables measured contemporaneously. The proposed approaches allow for the modeling of both auto- and cross-correlations through a vector-autoregressive (VAR) process, where the components of the white-noise input vector are modeled at every time instance either as stochastically independent t-distributed (herein called “stochastic model A”) or as multivariate t-distributed random variables (herein called “stochastic model B”). Both stochastic models are complementary in the sense that the former allows for group-specific degrees of freedom (df) of the t-distributions (thus, sensor-component-specific tail or outlier characteristics) but not for correlations within each white-noise vector, whereas the latter allows for such correlations but not for different dfs. Within the observation equations, nonlinear (differentiable) regression models are generally allowed for. Two different generalized expectation maximization (GEM) algorithms are derived to estimate the regression model parameters jointly with the VAR coefficients, the variance components (in case of stochastic model A) or the cofactor matrix (for stochastic model B), and the df(s). To enable the validation of the fitted VAR model and the selection of the best model order, the multivariate portmanteau test and Akaike’s information criterion are applied. The performance of the algorithms and of the white noise test is evaluated by means of Monte Carlo simulations. Furthermore, the suitability of one of the proposed models and the corresponding GEM algorithm is investigated within a case study involving the multivariate modeling and adjustment of time-series data at four GPS stations in the EUREF Permanent Network (EPN).

Marine gravity determined from multi-satellite GM/ERM altimeter data over the South China Sea: SCSGA V1.0

Wed, 05/06/2020 - 00:00
Abstract

High-precision and high-resolution gravity fields can be derived from multi-source satellite altimeter data. A gravity anomaly model around the South China Sea (SCSGA) V1.0 on a 1′ × 1′ grid is established from sea surface heights (SSHs) of several geodetic missions (GMs) and exact repeat missions. Gridded deflections of the vertical are first calculated from SSHs by the least squares collocation (LSC) method and then used to derive gravity anomalies by the inverse Vening Meinesz formula. In gravity derivation processing, we establish an approximate relationship among the precision of altimetric gravity, precision of geoid gradients, and density of geoid gradients (the average number of geoid gradients per 1′ × 1′ region). The weights of geoid gradients from the Ka-band altimeter for the LSC are innovatively determined by an iterative method. Finally, SCSGA V1.0 is assessed by ship-borne gravity anomalies and marine gravity models. The performance of GMs in gravity derivation is evaluated. In general, the altimetric gravity precision in regions with many islands and reefs increases more obviously than those in other regions when the geoid gradient density increases. The standard deviation of SCSGA V1.0 is 2.78 mGal, which is slightly better than those of four recognized global marine gravity models around the SCS. CryoSat-2 is the most important dataset for SCSGA V1.0. Ka-band SARAL/AltiKa plays a major role in gravity derivation, and the contribution of Haiyang-2A is greater than those of other Ku-band satellites, except CryoSat-2. SCSGA V1.0 is concluded to reach an international advanced level for marine gravity from altimeter data around the SCS.

IAG Newsletter

Thu, 04/30/2020 - 00:00

Tiltmeter data inversion to characterize a strain tensor source at depth: application to reservoir monitoring

Thu, 04/30/2020 - 00:00
Abstract

Surface deformation measured by geodetic data is the sum of single-strain sources deforming at depth. A combination of volume changes from several analytical models (e.g. a point source or dislocation along a plane) can be used to model the different sources. However, solving for the best fit of volume variations, dislocations, position and orientation parameters of all sources is a nonlinear problem, and its solution is generally non-unique. This problem can be converted into a linear one by assimilating the sum of sources to a simplified model formed by three orthogonal planes of dislocations at fixed position and orientation. This strain source model is equivalent to having all neighbouring deformation sources contained in a small size volume. The determination of the strain tensor components can be performed by inverting geodetic data. Because of their high resolution, tiltmeters are well adapted to survey shallow deformation of volcanoes and geological reservoirs. However, they are known to display unknown long-term drift. We propose an approach to jointly estimate the temporal evolution of the strain source and time-dependent instrumental parameters. We verify the approach using synthetic data, giving confidence intervals for each component of the strain tensor. Finally, we link geological information to the internal deformation by interpreting the strain tensor as principal directions of deformation. This approach seems promising for the identification of fracture onset and fault reactivation in geothermal, hydrocarbon exploitations or volcanic systems.

Fusing adjacent-track InSAR datasets to densify the temporal resolution of time-series 3-D displacement estimation over mining areas with a prior deformation model and a generalized weighting least-squares method

Thu, 04/23/2020 - 00:00
Abstract

Interferometric synthetic aperture radar (InSAR) technology can be used to observe high spatial resolution one-dimensional (1-D) deformation along the line-of-sight direction from a single-track synthetic aperture radar (SAR) dataset. With the aid of multi-track InSAR data or a prior model, InSAR can be extended to infer 3-D deformation information, but the temporal resolution is generally limited. This paper presents an InSAR-based method to retrieve high spatio-temporal resolution 3-D displacements over mining areas (hereafter referred to as the MTI-based method). The core idea of the proposed method is to enhance the temporal resolution of the time-series 3-D displacement estimates by fusing multi-track InSAR observations and a prior model. Firstly, we retrieve high spatial resolution 3-D mining displacements from single-track InSAR 1-D deformation observations, with the assistance of the prior deformation model. By applying this approach to multi-track InSAR data over the same area, we obtain much denser 3-D mining displacement samples in time than those derived from a single-track InSAR dataset. Secondly, we propose a generalized weighted least-squares method to integrate the denser 3-D displacement samples, to solve the high temporal resolution 3-D mining displacements, in which the rank deficiency needs to be tackled. Finally, time-series 3-D mining displacements at the chronological dates of all the available multi-track SAR images are estimated. The Yungang coal mining area of China was selected to test the proposed method using two adjacent-track ALOS PALSAR-1 datasets. Compared with the single-track InSAR-derived results, the proposed method not only significantly improves the temporal resolution of the monitoring results by 42.6%, obtaining more detailed 3-D displacements, but it also provides important data support for understanding and modeling the distinctive kinematics of mining deformation and assessing mining-related geohazards. What is more, the core idea of the proposed method will be beneficial to high spatio-temporal resolution 3-D deformation estimation in other geophysical processes.

BDS-3 differential code bias estimation with undifferenced uncombined model based on triple-frequency observation

Sat, 04/04/2020 - 00:00
Abstract

Since 2015, the new generation global BDS system, i.e., BDS-3, has started its development with five experimental satellites demonstration system and has announced its initial global service officially on December 27, 2018. Among the various characteristics to be analyzed for the new generation BDS satellites, the differential code bias (DCB) is of special attention since that it has a direct dependence on the new signals, i.e., B1C and B2a, and it is one of the most intricacy problems in the ionosphere sensing and positioning with multi-GNSS and multi-frequency observations. To take the full capability of the triple-frequency BDS signals, this paper proposed a new method for the DCB estimation in which the undifferenced uncombined observations are processed in PPP mode. In addition, with the intention to estimate all the unknowns, including the DCB, in a single filter, the DESIGN (deterministic plus stochastic ionosphere model for GNSS) method is applied for the ionospheric delay constrains in this method. In the formula derivation, special attention is given to the DCB and clock parameters due to different frequencies for B1I/B1C, etc. Then, the efficiency of the new method is assessed with observations of 23 iGMAS stations capable for BDS triple-frequency tracking and 21 IGS stations capable for GPS triple-frequency tracking during DOY 001 to 090, 2019. Moreover, the traditional DCB estimation method by employing the geometry-free (GF) combination with the ionospheric delay removed by global ionosphere map product is also performed for comparison purpose. The experimental results suggest that by using the undifferenced uncombined solution, rather than the GF combination, the BDS-2 DCB on B1IB2I and B1CB3I can be improved, especially for the MEO satellites. Regarding to the DLR products, the undifferenced uncombined DCB solution presents a RMS of 0.32 ns and 0.27 ns for B1IB2I and B1CB3I, respectively. Concerning the BDS-3 satellites DCB, it is GF combination that performs better by a factor of \(12.7\%\) and \(15.2\%\) for B1CB2a and B1CB3I, respectively. This is mainly due to the fact that the undifferenced uncombined DCB solution is sensitive to the limited precision of the BDS-3 orbit and clock. This conclusion is further confirmed by the improvement in the GPS DCB solution with the new method. Compared with the GF combination solution, the STD for daily repeatability improves from 0.088 to 0.061 ns and 0.119 to 0.090 ns for satellite on C1WC2W and C1WC5X, respectively, by using the undifferenced uncombined model.

Dynamic harmonic regression modeling for monthly mean sea levels at tide gauges within the Arabian Gulf

Sat, 04/04/2020 - 00:00
Abstract

Time series with strong periodicity and non-stationary character in terms of magnitude and frequency which are changing over measurement period can be decomposed into time-varying trend, seasonal and cyclical components by dynamic harmonic regression (DHR) modeling in the state-space framework. The time-variable parameters of the components are first associated with a generalized random walk process, and then, state parameters are estimated by the recursive Kalman filtering and fixed interval smoothing algorithms. Missed points are filled by the DHR interpolation, and change points are detected by both visual inspection and the Pettitt test. Sea level series at tide gauges, which consist of accelerated trend and strong periodicity, are therefore suitable for DHR modeling. Time-varying trend, seasonal and cyclical components are extracted by the DHR modeling from the monthly mean sea level series longer than 15 years at seven stations within the Arabian Gulf. The DHR model accounts for 90–96% of variation of the monthly series, while the seasonal and cyclical components account for 64–85% and 2–7%, respectively. Average relative sea level rate (RSLR) and absolute sea level rate (ASLR) over the Arabian Gulf are found \(1.67 \pm 0.05\) mm/year and 1.93 ± 0.05 mm/year, respectively.

IAG Newsletter

Sat, 03/28/2020 - 00:00

Extended forward and inverse modeling of radiation pressure accelerations for LEO satellites

Thu, 03/26/2020 - 00:00
Abstract

For low Earth orbit (LEO) satellites, activities such as precise orbit determination, gravity field retrieval, and thermospheric density estimation from accelerometry require modeled accelerations due to radiation pressure. To overcome inconsistencies and better understand the propagation of modeling errors into estimates, we here suggest to extend the standard analytical LEO radiation pressure model with emphasis on removing systematic errors in time-dependent radiation data products for the Sun and the Earth. Our extended unified model of Earth radiation pressure accelerations is based on hourly CERES SYN1deg data of the Earth’s outgoing radiation combined with angular distribution models. We apply this approach to the GRACE (Gravity Recovery and Climate Experiment) data. Validations with 1 year of calibrated accelerometer measurements suggest that the proposed model extension reduces RMS fits between 5 and 27%, depending on how measurements were calibrated. In contrast, we find little changes when implementing, e.g., thermal reradiation or anisotropic reflection at the satellite’s surface. The refined model can be adopted to any satellite, but insufficient knowledge of geometry and in particular surface properties remains a limitation. In an inverse approach, we therefore parametrize various combinations of possible systematic errors to investigate estimability and understand correlations of remaining inconsistencies. Using GRACE-A accelerometry data, we solve for corrections of material coefficients and CERES fluxes separately over ocean and land. These results are encouraging and suggest that certain physical radiation pressure model parameters could indeed be determined from satellite accelerometry data.

Comparative analysis of different atmospheric surface pressure models and their impacts on daily ITRF2014 GNSS residual time series

Fri, 03/20/2020 - 00:00
Abstract

To remove atmospheric pressure loading (ATML) effect from GNSS coordinate time series, surface pressure (SP) models are required to predict the displacements. In this paper, we modeled the 3D ATML surface displacements using the latest MERRA-2 SP grids, together with four other products (NCEP-R-1, NCEP-R-2, ERA-Interim and MERRA) for 596 globally distributed GNSS stations, and compared them with ITRF2014 residual time series. The five sets of ATML displacements are highly consistent with each other, particularly for those stations far away from coasts, of which the lowest correlations in the Up component for all the four models w.r.t MERRA-2 become larger than 0.91. ERA-Interim-derived ATML displacement performs best in reducing scatter of the GNSS height for 90.3% of the stations (89.3% for NCEP-R-1, 89.1% for NCEP-R-2, 86.4% for MERRA and 85.1% for MERRA-2). We think that this may be possibly due to the 4D variational data assimilation method applied. Considering inland stations only, more than 96% exhibit WRMS reduction in the Up direction for all five models, with an average improvement of 3–4% compared with the original ITRF2014 residual time series before ATML correction. Most stations (> 67%) also exhibit horizontal WRMS reductions based on the five models, but of small magnitudes, with most improvements (> 76%) less than 5%. In particular, most stations in South America, South Africa, Oceania and the Southern Oceans show larger WRMS reductions with MERRA-2, while all other four SP datasets lead to larger WRMS reduction for the Up component than MERRA-2 in Europe. Through comparison of the daily pressure variation from the five SP models, we conclude that the bigger model differences in the SP-induced surface displacements and their impacts on the ITRF2014 residuals for coastal/island stations are mainly due to the IB correction based on the different land–sea masks. A unique high spatial resolution land–sea mask should be applied in the future, so that model differences would come from only SP grids. Further research is also required to compare the ATML effect in ice-covered and high mountainous regions, for example the Qinghai–Tibet Plateau in China, the Andes in South America, etc., where larger pressure differences between models tend to occur.

Impact of temperature stabilization on the strapdown airborne gravimetry: a case study in Central Turkey

Tue, 03/17/2020 - 00:00
Abstract

Airborne gravimetry with strapdown inertial sensors has been a valuable tool for many years to fill in the gravity data gaps on the areas not accessible by land. Accuracies of 1 mGal level with off-the-shelf navigation-grade inertial measurement units (IMU) can only be achieved provided that the accelerometer drifts mainly caused by the temperature variations inside the IMU housing are separated from the gravity signal. Although there are several strategies proposed in the literature to deal with this inseparability problem, we use a thermal stabilization system (iTempStab) added on an iNAT-RQH navigation-grade IMU and investigate its performance over a test region in central Turkey with moderate topography and highly qualified ground truth gravity data. Two test flights were performed in 2017 and 2018 with and without iTempStab add-on following almost the same flight trajectories. During the first flight in 2017 with iNAT-RQH only, which lasted almost 5.5 h, there were considerable temperature variations inside the IMU housing from 39.1 to 46.0 °C. A simple thermal correction based on a laboratory calibration done before the flight was applied to the vertical Z-accelerometer in the pre-processing stage. However, temperature changes were within 0.1 °C during the second test flight in 2018 with TempStab add-on. The temperature stabilization gained by the iTempStab add-on produced better cross-over statistics. While the RMSE of the non-adjusted cross-over residuals was about 2.6 mGal, it reduced by 50% with iTempStab add-on. The adjusted cross-over differences of the 2018 flight yielded an RMSE of about 0.5 mGal, which is a remarkable precision for the strapdown gravimetry. The comparison with upward continued ground gravity data at flight altitudes suggests that the thermal stabilization system shows also remarkable improvements in the residual statistics. The range of the residuals decreases from ± 10 to ± 5 mGal, the standard deviation decreases from 2.19 to 0.94 mGal, and the RMSE decreases from 2.24 to 1.48 mGal, respectively, with the iTempStab add-on. It can be concluded that the thermal stabilization system significantly improves the accelerometer stability and therefore the precision and accuracy of the strapdown airborne gravity estimates.

Correction to: “Methodology and consistency of slant and vertical assessments for ionospheric electron content models” and to “Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle”

Mon, 03/16/2020 - 00:00

In the original publication of the articles, “Methodology and consistency of slant and vertical assessments for ionospheric electron content models” and “Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle”, a common typo affecting the text only (not the computations) has been recently noticed. It compromised the definition of the scaling factor from Global Navigation Satellite Systems ionospheric delay to electron content is clarified in this erratum.

Simulation of tracking scenarios to LAGEOS and Etalon satellites

Mon, 03/16/2020 - 00:00
Abstract

The International Laser Ranging Service (ILRS) provides weekly solutions for coordinates of Satellite Laser Ranging (SLR) stations coordinates, geocenter coordinates, as well as Earth rotation parameters with a daily resolution. The ILRS standard solution is an important contribution to the International Terrestrial Reference Frame (ITRF). As of today, it is derived from SLR observations to the pairs of LAGEOS and Etalon satellites exclusively. In this paper, the effect of altering the tracking strategy for the LAGEOS and Etalon satellites on the weekly ILRS standard solution is studied. This is done by simulating various tracking scenarios and by comparing the parameters of the solutions for each scenario. In particular, the focus lies on redistributing observation time between the LAGEOS and Etalon satellites as a possible optimization for the tracking scheduling. By this, the current tracking capability of each station is taken into account with no change of the overall tracking activity to LAGEOS and Etalon. It is shown that the quality of the solution for the ITRF-relevant parameters is not significantly degraded when reducing the number of observations to LAGEOS by up to 20% with respect to the number of available normal points in 2016. The vacant tracking capability obtained from the reduction of LAGEOS observations may be used to increase the number of measurements to Etalon by a factor of three. This leads to nearly 10% improvement of the recovery of Earth rotation parameters within the combined LAGEOS–Etalon solution. With our study, we contribute to the ongoing discussions regarding tracking strategies for SLR stations within the ILRS. In particular, the stations could adjust their individual tracking priorities according to these results in the future without major investments or the need for new infrastructure.

Relationship between free core nutation and geomagnetic jerks

Wed, 03/11/2020 - 00:00
Abstract

Recent studies have indicated a correlation between Earth’s free core nutation (FCN) and geomagnetic jerks (GMJs). However, some uncertainties still need to be resolved before their relationship can be confirmed. The variations in the amplitude and phase of the FCN result from the comprehensive influence of the surface fluid layer and core–mantle couplings, which makes its correlation with GMJs difficult to verify. The FCN period mainly depends on the inertia coupling and the dissipative couplings (such as viscous, electromagnetic and topographic couplings) at the core–mantle boundary according to the theory of Earth rotation. Whatever the GMJ mechanism, it is most likely to affect the FCN by changing the core–mantle couplings. This study was conducted to effectively determine variations in the FCN period by considering atmospheric and oceanic effects, investigate any correlation between the two phenomena, and analyze how the FCN relates to GMJs. Using the normal time–frequency transform, we extracted signals in the nutation band from the atmospheric and oceanic angular momentum functions. We used the broadband Liouville equations to estimate the atmospheric and oceanic effects on nutation terms. Using a sliding window of 2 years, we fitted five nutation terms most affected by FCN resonance from the celestial pole offsets with FCN model removed. The FCN period variation was estimated by using weighted least square method. The results indicated a correlation between the FCN and GMJ. Further, we analyzed the relationship between the geomagnetic fluctuations and FCN based on both the core–mantle couplings and the possible GMJ mechanism.

A single-receiver geometry-free approach to stochastic modeling of multi-frequency GNSS observables

Mon, 03/09/2020 - 00:00
Abstract

The proper choice of stochastic model is of great importance to global navigation satellite system (GNSS) data processing. Whereas extensive investigations into stochastic modeling are mainly based on the relative (or differential) method employing zero and/or short baselines, this work proposes an absolute method that relies upon a stand-alone receiver and works by applying the least-squares variance component estimation to the geometry-free functional model, thus facilitating the characterization of stochastic properties of multi-frequency GNSS observables at the undifferenced level. In developing the absolute method, special care has been taken of the code multipath effects by introducing ambiguity-like parameters to the code observation equations. By means of both the relative and absolute methods, we characterize the precision, cross and time correlation of the code and phase observables of two newly emerging constellations, namely the Chinese BDS and the European Galileo, collected by a variety of receivers of different types at multiple frequencies. Our first finding is that so far as the precision is concerned, the absolute method yields nearly the same numerical values as those derived by the zero-baseline-based relative method. However, the two methods give contradictory results with regard to the cross correlation, which is found (not) to occur between BDS phase observables when use has been made of the relative (absolute) method. Our explanation to this discrepancy is that the cross correlation found in the relative method originates from the parts (antenna, cable, low noise amplifier) shared by two receivers creating a zero baseline. The time correlation is only of significance when the multipath effects are present, as is the case with the short-baseline-based relative method; this correlation turns out to be largely weaker (or ideally absent) in the absolute (or zero-baseline-based relative) method. Moreover, with the absolute method, the stochastic properties determined for two receivers of the same type but subject to different multipath effects are virtually the same. We take this as a convincing evidence that the absolute method is robust against multipath effects. Hence, the absolute method proposed in the present work represents a promising complement to the relative method and appears to be particularly beneficial to GNSS positioning, navigation and timing technologies based on the undifferenced observables, typically the precise point positioning.

Research on empirical correction models of GPS Block IIF and BDS satellite inter-frequency clock bias

Fri, 03/06/2020 - 00:00
Abstract

Triple-frequency observations will introduce an inter-frequency clock bias (IFCB) between the new frequency and the original dual-frequency observations. It has been verified that satellite IFCB can reach dozens of centimeters and several centimeters for GPS Block IIF satellite and BDS satellite, respectively. The existence of satellite IFCB will significantly affect undifferenced triple-frequency data processing. Based on 4-year data collected from 80 globally distributed stations, the long-term characteristics of IFCB coefficients obtained by using harmonic analysis have been studied. The results demonstrate that the coefficients of IFCB periodic model cannot be well fitted only by using sun elevation angle. Also, coefficients have obvious periodic characteristics and their periods differ among different satellites. Thus, a new linear-plus-periodic model is proposed to fit the long-term coefficients. Then, IFCB empirical correction models for 12 GPS Block IIF satellites and BDS GEO and IGSO satellites are built. In order to validate the correction model, IFCB standard deviation (STD), triple-frequency precise point positioning (PPP) and undifferenced extra-wide-lane (EWL) ambiguity resolution are employed. The results based on more than 4-year observations show that, with correction model applied, the average IFCB STD decreases by about 65.5% and 45.5% for GPS and BDS satellites, respectively. Compared to triple-frequency PPP without IFCB correction, triple-frequency PPP results with IFCB correction show that Up, North and East components accuracy are improved by 12.3%, 16.0% and 13.2%, respectively. Besides, IFCB correction will greatly improve the consistence of EWL fractional cycle bias among different stations and improve the success rate of EWL ambiguity resolution.

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