ALBERT

Description: Of the three satellite geodetic techniques contributing to the International Terrestrial Reference Frame (ITRF), Satellite Laser Ranging (SLR) is generally held to provide the most reliable time series of geocentre coordinates and exclusively defines the ITRF origin. Traditionally, only observations to the two LAser GEOdynamics Satellite (LAGEOS) and Etalon pairs of satellites have been used for the definition of the ITRF origin. Previous simulation studies using evenly sampled LAGEOS-like data have shown that only the Z component of geocentre motion suffers minor collinearity issues, which may explain its lower quality compared to the equatorial components. Using collinearity diagnosis, this study provides insight into the actual capability of SLR to sense geocentre motion using the existing geographically unbalanced ground network and real observations to eight spherical geodetic satellites. We find that, under certain parameterisations, observations to the low Earth orbiters (LEOs) Starlette, Stella, Ajisai and LAser RElativity Satellite are able to improve the observability of the geocentre coordinates in multi-satellite solutions compared to LAGEOS-only solutions. The higher sensitivity of LEOs to geocentre motion and the larger number of observations are primarily responsible for the improved observability. Errors in the modelling of Starlette, Stella and Ajisai orbits may contaminate the geocentre motion estimates, but do not disprove the intrinsic strength of LEO tracking data. The sporadically observed Etalon satellites fail to make a significant beneficial contribution to the observability of the geocentre coordinates derived via the network shift approach and can be safely omitted from SLR data analyses for TRF determination.

Description: The European Space Agency Gravity field and steady-state Ocean Circular Explorer (GOCE) carries a gradiometer consisting of three pairs of accelerometers in an orthogonal triad. Precise GOCE science orbit solutions (PSO), which are based on satellite-to-satellite tracking observations by the Global Positioning System and which are claimed to be at the few cm precision level, can be used to calibrate and validate the observations taken by the accelerometers. This has been done for each individual accelerometer by a dynamic orbit fit of the time series of position co-ordinates from the PSOs, where the accelerometer observations represent the non-gravitational accelerations. Since the accelerometers do not coincide with the center of mass of the GOCE satellite, the observations have to be corrected for rotational and gravity gradient terms. This is not required when using the so-called common-mode accelerometer observations, provided the center of the gradiometer coincides with the GOCE center of mass. Dynamic orbit fits based on these common-mode accelerations therefore served as reference. It is shown that for all individual accelerometers, similar dynamic orbit fits can be obtained provided the above-mentioned corrections are made. In addition, accelerometer bias estimates are obtained that are consistent with offsets in the gravity gradients that are derived from the GOCE gradiometer observations.

Description: This paper reports on different sources of errors that occur in the calibration process of a superconducting gravimeter (SG), determined by comparison with a ballistic absolute gravimeter (AG); some of them have never been discussed in the literature. We then provide methods to mitigate the impact of those errors, to achieve a robust calibration estimate at the level. We demonstrate that a standard deviation at the level of can be reached within 48 h by measuring at spring tides and by increasing the AG sampling rate. This is much shorter than what is classically reported in previous empirical studies. Measuring more than 5 days around a tidal extreme does not improve the precision in the calibration factor significantly, as the variation in the error as a function of \(1/\sqrt{N} \) does not apply, considering the decrease in signal amplitude due to the tidal modulation. However, we investigate the precision improvement up to 120 days, which can be useful if an AG is run continuously: at mid-latitude it would require 21 days to ensure a calibration factor at the level with a 99.7 % confidence interval. We also show that restricting the AG measurement periods to tidal extrema can reduce instrument demand, while this does not affect the precision on the calibration factor significantly. Then, we quantify the effect of high microseismic noise causing aliasing in the AG time series. We eventually discuss the attenuation bias that might be induced by noisy time series of the SG. When experiments are performed at the level, 7 are needed to ensure that the error in the calibration estimate will be at the 1 per mille level with a 99 % confidence.

Description: We review the measurement of the mean dynamic topography (MDT) of the Mediterranean using ellipsoidal heights of sea level at discrete tide gauge locations, and across the entire basin using satellite altimetry, subtracting estimates of the geoid obtained from recent models. This ‘geodetic approach’ to the determination of the MDT can be compared to the independent ‘ocean approach’ that involves the use of in situ oceanographic measurements and ocean modelling. We demonstrate that with modern geoid and ocean models there is an encouraging level of consistency between the two sets of MDTs. In addition, we show how important geodetic MDT information can be in judging between existing global ocean circulation models, and in providing insight for the development of new ones. The review makes clear the major limitations in Mediterranean data sets that prevent a more complete validation, including the need for improved geoid models of high spatial resolution and accuracy. Suggestions are made on how a greater amount of reliable geo-located tide gauge information can be obtained in the future.

Description: The long wavelength part of the Earth’s gravity field can be determined, with varying accuracy, from satellite laser ranging (SLR). In this study, we investigate the combination of up to ten geodetic SLR satellites using iterative variance component estimation. SLR observations to different satellites are combined in order to identify the impact of each satellite on the estimated Stokes coefficients. The combination of satellite-specific weekly or monthly arcs allows to reduce parameter correlations of the single-satellite solutions and leads to alternative estimates of the second-degree Stokes coefficients. This alternative time series might be helpful for assessing the uncertainty in the impact of the low-degree Stokes coefficients on geophysical investigations. In order to validate the obtained time series of second-degree Stokes coefficients, a comparison with the SLR RL05 time series of the Center of Space Research (CSR) is done. This investigation shows that all time series are comparable to the CSR time series. The precision of the weekly/monthly \(C_{21}\) and \(S_{21}\) coefficients is analyzed by comparing mass-related equatorial excitation functions \(\chi ^{\text {mass}}_{1,2}\) with geophysical model results and reduced geodetic excitation functions. In case of \(\chi ^{\text {mass}}_{1}\) , the annual amplitude and phase of the DGFI solution agrees better with three of four geophysical model combinations than other time series. In case of \(\chi ^{\text {mass}}_{2}\) , all time series agree very well to each other. The impact of \(C_{20}\) on the ice mass trend estimates for Antarctica are compared based on CSR GRACE RL05 solutions, in which different monthly \(C_{20}\) time series are used for replacing. We found differences in the long-term Antarctic ice loss of \(12.3\)  Gt/year between the GRACE solutions induced by the different \(C_{20}\) SLR time series of CSR and DGFI, which is about 13 % of the total ice loss of Antarctica. This result shows that Antarctic ice mass loss quantifications must be carefully interpreted.

Description: The Empirical CODE Orbit Model (ECOM) of the Center for Orbit Determination in Europe (CODE), which was developed in the early 1990s, is widely used in the International GNSS Service (IGS) community. For a rather long time, spurious spectral lines are known to exist in geophysical parameters, in particular in the Earth Rotation Parameters (ERPs) and in the estimated geocenter coordinates, which could recently be attributed to the ECOM. These effects grew creepingly with the increasing influence of the GLONASS system in recent years in the CODE analysis, which is based on a rigorous combination of GPS and GLONASS since May 2003. In a first step we show that the problems associated with the ECOM are to the largest extent caused by the GLONASS, which was reaching full deployment by the end of 2011. GPS-only, GLONASS-only, and combined GPS/GLONASS solutions using the observations in the years 2009–2011 of a global network of 92 combined GPS/GLONASS receivers were analyzed for this purpose. In a second step we review direct solar radiation pressure (SRP) models for GNSS satellites. We demonstrate that only even-order short-period harmonic perturbations acting along the direction Sun-satellite occur for GPS and GLONASS satellites, and only odd-order perturbations acting along the direction perpendicular to both, the vector Sun-satellite and the spacecraft’s solar panel axis. Based on this insight we assess in the third step the performance of four candidate orbit models for the future ECOM. The geocenter coordinates, the ERP differences w. r. t. the IERS 08 C04 series of ERPs, the misclosures for the midnight epochs of the daily orbital arcs, and scale parameters of Helmert transformations for station coordinates serve as quality criteria. The old and updated ECOM are validated in addition with satellite laser ranging (SLR) observations and by comparing the orbits to those of the IGS and other analysis centers. Based on all tests, we present a new extended ECOM which substantially reduces the spurious signals in the geocenter coordinate \(z\) (by about a factor of 2–6), reduces the orbit misclosures at the day boundaries by about 10 %, slightly improves the consistency of the estimated ERPs with those of the IERS 08 C04 Earth rotation series, and substantially reduces the systematics in the SLR validation of the GNSS orbits.

Description: We investigate the effects of quasar structure on geodetic very long baseline interferometry (VLBI) measurements. We create catalogues of simulated and real quasars with a range of structure indices, and use these to generate synthetic CONT11 observations with the Vienna VLBI Software simulator tool. We systematically investigate the effects of quasars with different amounts of source structure, and find that source structure can affect station positions at the one-millimetre level. This effect is stronger for isolated stations. Overall, source structure is found to contribute to about 10 % of the troposphere and clock effects. Our simulations confirm analytical predictions that source structure mitigation strategies must be developed in order to achieve millimetre-level VLBI position accuracy.

Description: The rapid development of the Chinese BeiDou Navigation Satellite System (BDS) brings a promising prospect for the real-time retrieval of zenith tropospheric delays (ZTD) and precipitable water vapor (PWV), which is of great benefit for supporting the time-critical meteorological applications such as nowcasting or severe weather event monitoring. In this study, we develop a real-time ZTD/PWV processing method based on Global Positioning System (GPS) and BDS observations. The performance of ZTD and PWV derived from BDS observations using real-time precise point positioning (PPP) technique is carefully investigated. The contribution of combining BDS and GPS for ZTD/PWV retrieving is evaluated as well. GPS and BDS observations of a half-year period for 40 globally distributed stations from the International GNSS Service Multi-GNSS Experiment and BeiDou Experiment Tracking Network are processed. The results show that the real-time BDS-only ZTD series agree well with the GPS-only ZTD series in general: the RMS values are about 11–16 mm (about 2–3 mm in PWV). Furthermore, the real-time ZTD derived from GPS-only, BDS-only, and GPS/BDS combined solutions are compared with those derived from the Very Long Baseline Interferometry. The comparisons show that the BDS can contribute to real-time meteorological applications, slightly less accurately than GPS. More accurate and reliable water vapor estimates, about 1.3–1.8 mm in PWV, can be obtained if the BDS observations are combined with the GPS observations in the real-time PPP data processing. The PWV comparisons with radiosondes further confirm the performance of BDS-derived real-time PWV and the benefit of adding BDS to standard GPS processing.

Description: Satellite laser ranging (SLR) to the satellites of the global navigation satellite systems (GNSS) provides substantial and valuable information about the accuracy and quality of GNSS orbits and allows for the SLR-GNSS co-location in space. In the framework of the NAVSTAR-SLR experiment two GPS satellites of Block-IIA were equipped with laser retroreflector arrays (LRAs), whereas all satellites of the GLONASS system are equipped with LRAs in an operational mode. We summarize the outcome of the NAVSTAR-SLR experiment by processing 20 years of SLR observations to GPS and 12 years of SLR observations to GLONASS satellites using the reprocessed microwave orbits provided by the center for orbit determination in Europe (CODE). The dependency of the SLR residuals on the size, shape, and number of corner cubes in LRAs is studied. We show that the mean SLR residuals and the RMS of residuals depend on the coating of the LRAs and the block or type of GNSS satellites. The SLR mean residuals are also a function of the equipment used at SLR stations including the single-photon and multi-photon detection modes. We also show that the SLR observations to GNSS satellites are important to validate GNSS orbits and to assess deficiencies in the solar radiation pressure models. We found that the satellite signature effect, which is defined as a spread of optical pulse signals due to reflection from multiple reflectors, causes the variations of mean SLR residuals of up to 15 mm between the observations at nadir angles of 0 \(^{\circ }\) and 14 \(^{\circ }\) . in case of multi-photon SLR stations. For single-photon SLR stations this effect does not exceed 1 mm. When using the new empirical CODE orbit model (ECOM), the SLR mean residual falls into the range 0.1–1.8 mm for high-performing single-photon SLR stations observing GLONASS-M satellites with uncoated corner cubes. For best-performing multi-photon stations the mean SLR residuals are between \(-12.2\) and \(-25.6\)  mm due to the satellite signature effect.

Description: The key issue to enable precise point positioning with ambiguity resolution (PPP-AR) is to estimate fractional-cycle biases (FCBs), which mainly relate to receiver and satellite hardware biases, over a network of reference stations. While this has been well achieved for GPS, FCB estimation for GLONASS is difficult because (1) satellites do not share the same frequencies as a result of Frequency Division Multiple Access (FDMA) signals; (2) and even worse, pseudorange hardware biases of receivers vary in an irregular manner with manufacturers, antennas, domes, firmware, etc., which especially complicates GLONASS PPP-AR over inhomogeneous receivers. We propose a general approach where external ionosphere products are introduced into GLONASS PPP to estimate precise FCBs that are less impaired by pseudorange hardware biases of diverse receivers to enable PPP-AR. One month of GLONASS data at about 550 European stations were processed. From an exemplary network of 51 inhomogeneous receivers, including four receiver types with various antennas and spanning about 800 km in both longitudinal and latitudinal directions, we found that 92.4 % of all fractional parts of GLONASS wide-lane ambiguities agree well within \(\pm \) 0.15 cycles with a standard deviation of 0.09 cycles if global ionosphere maps (GIMs) are introduced, compared to only 51.7 % within \(\pm \) 0.15 cycles and a larger standard deviation of 0.22 cycles otherwise. Hourly static GLONASS PPP-AR at 40 test stations can reach position estimates of about 1 and 2 cm in RMS from ground truth for the horizontal and vertical components, respectively, which is comparable to hourly GPS PPP-AR. Integrated GLONASS and GPS PPP-AR can further achieve an RMS of about 0.5 cm in horizontal and 1–2 cm in vertical components. We stress that the performance of GLONASS PPP-AR across inhomogeneous receivers depends on the accuracy of ionosphere products. GIMs have a modest accuracy of only 2–8 TECU (Total Electron Content Unit) in vertical which confines PPP-AR to an approximately \(800\times 800\)  km area in Europe. We expect that a regional ionosphere map with a better than 1 TECU accuracy is likely to improve the GLONASS PPP-AR efficiency.

Description: On 17 April 2011, all analysis centers (ACs) of the International GNSS Service (IGS) adopted the reference frame realization IGS08 and the corresponding absolute antenna phase center model igs08.atx for their routine analyses. The latter consists of an updated set of receiver and satellite antenna phase center offsets and variations (PCOs and PCVs). An update of the model was necessary due to the difference of about 1 ppb in the terrestrial scale between two consecutive realizations of the International Terrestrial Reference Frame (ITRF2008 vs. ITRF2005), as that parameter is highly correlated with the GNSS satellite antenna PCO components in the radial direction. For the receiver antennas, more individual calibrations could be considered and GLONASS-specific correction values were added. For the satellite antennas, all correction values except for the GPS PCVs were newly estimated considering more data than for the former model. Satellite-specific PCOs for all GPS satellites active since 1994 could be derived from reprocessed solutions of five ACs generated within the scope of the first IGS reprocessing campaign. Two ACs separately derived a full set of corrections for all GLONASS satellites active since 2003. Ignoring scale-related biases, the accuracy of the satellite antenna PCOs is on the level of a few cm. With the new phase center model, orbit discontinuities at day boundaries can be reduced, and the consistency between GPS and GLONASS results is improved. To support the analysis of low Earth orbiter (LEO) data, igs08.atx was extended with LEO-derived PCV estimates for big nadir angles in June 2013.

Description: Homogeneously reprocessed combined GPS/GLONASS 1- and 3-day solutions from 1994 to 2013, generated by the Center for Orbit Determination in Europe (CODE) in the frame of the second reprocessing campaign REPRO-2 of the International GNSS Service, as well as GPS- and GLONASS-only 1- and 3-day solutions for the years 2009 to 2011 are analyzed to assess the impact of the arc length on the estimated Earth Orientation Parameters (EOP, namely polar motion and length of day), on the geocenter, and on the orbits. The conventional CODE 3-day solutions assume continuity of orbits, polar motion components, and of other parameters at the day boundaries. An experimental 3-day solution, which assumes continuity of the orbits, but independence from day to day for all other parameters, as well as a non-overlapping 3-day solution, is included into our analysis. The time series of EOPs, geocenter coordinates, and orbit misclosures, are analyzed. The long-arc solutions were found to be superior to the 1-day solutions: the RMS values of EOP and geocenter series are typically reduced between 10 and 40 %, except for the polar motion rates, where RMS reductions by factors of 2–3 with respect to the 1-day solutions are achieved for the overlapping and the non-overlapping 3-day solutions. In the low-frequency part of the spectrum, the reduction is even more important. The better performance of the orbits of 3-day solutions with respect to 1-day solutions is also confirmed by the validation with satellite laser ranging.

Description: At present, the BeiDou system (BDS) enables the practical application of triple-frequency observable in the Asia-Pacific region, of many possible benefits from the additional signal; this study focuses on exploiting the contribution of zero difference (ZD) ambiguity resolution (AR) to the precise point positioning (PPP). A general modeling strategy for multi-frequency PPP AR is presented, in which, the least squares ambiguity decorrelation adjustment (LAMBDA) method is employed in ambiguity fixing based on the full variance-covariance ambiguity matrix generated from the raw data processing model. Because of the reliable fixing of BDS L1 ambiguity faces more difficulty, the LAMBDA method with partial ambiguity fixing is proposed to enable the independent and instantaneous resolution of extra wide-lane (EWL) and wide-lane (WL). This mechanism of sequential ambiguity fixing is demonstrated for resolving ZD satellite phase bias and performing triple-frequency PPP AR with two reference station networks with a typical baseline of up to 400  and 800 km, respectively. Tests show that about \(90\,\%\) of the EWL and WL phase bias of BDS has a consistency of better than 0.1 cycle, and this value decreases to \(〈\) 80 % for L1 phase bias for Experiment I, while all the solutions of Experiment II have a similar RMS of about 0.12 cycles. In addition, the repeatability of the daily mean phase bias agree to 0.093 cycles and 0.095 cycles for EWL and WL on average, which is much smaller than 0.20 cycles of L1. To assess the improvement of fixed PPP brought by applying the third frequency signal as well as the above phase bias, various ambiguity fixing strategy are considered in the numerical demonstration. It is shown that the impact of the additional signal is almost negligible when only float solution involved. It is also shown that by fixing EWL and WL together, as opposed to the single ambiguity fixing, will leads to an improvement in PPP accuracy by about \(20.6\,\%\) on average. Attributed to the efficient resolution of EWL  \(+\)  WL within about 2 min in Experiment I, the 0.5 m level positioning can be achieved in 10 min for both horizontal and vertical, compared to 50 min for horizontal and 30 min for vertical by the NONE/EWL/WL fixed solution. While, for Experiment II, the improvement in the convergence can only be seen for the horizontal as the TTFF takes about 40 min for EWL and WL to be resolved.

Description: Previous work has shown, that strapdown airborne gravimeters can have a comparable or even superior performance in the higher frequency domain (resolution of few kilometres), compared to classical stable-platform air gravimeters using springs, such as the LaCoste and Romberg (LCR) S-gravimeter. However, the longer wavelengths (tens of kilometres and more) usually suffer from drifts of the accelerometers of the strapdown inertial measurement unit (IMU). In this paper, we analyse the drift characteristics of the QA2000 accelerometers, which are the most widely used navigation-grade IMU accelerometers. A large portion of these drifts is shown to come from thermal effects. A lab calibration procedure is used to derive a thermal correction, which is then applied to data from 18 out of 19 flights from an airborne gravity campaign carried out in Chile in October 2013. The IMU-derived gravity closure error can be reduced by 91 % on average, from 3.72 mGal/h to only 0.33 mGal/h (RMS), which is an excellent long-term performance for strapdown gravimetry. Also, the IMU results are compared to the LCR S-gravimeter, which is known to have an excellent long-term stability. Again, the thermal correction yields a significant reduction of errors, with IMU and LCR aerogravity results being consistent at the 2 mGal level.

Description: The current high-degree global geopotential models EGM2008 and EIGEN-6C4 resolve gravity field structures to \(\sim \) 10 km spatial scales over most parts of the of Earth’s surface. However, a notable exception is continental Antarctica, where the gravity information in these and other recent models is based on satellite gravimetry observations only, and thus limited to about \(\sim \) 80–120 km spatial scales. Here, we present a new degree-2190 global gravity model (GGM) that for the first time improves the spatial resolution of the gravity field over the whole of continental Antarctica to \(\sim \) 10 km spatial scales. The new model called SatGravRET2014 is a combination of recent Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite gravimetry with gravitational signals derived from the 2013 Bedmap2 topography/ice thickness/bedrock model with gravity forward modelling in ellipsoidal approximation. Bedmap2 is a significantly improved description of the topographic mass distribution over the Antarctic region based on a multitude of topographic surveys, and a well-suited source for modelling short-scale gravity signals as we show in our study. We describe the development of SatGravRET2014 which entirely relies on spherical harmonic modelling techniques. Details are provided on the least-squares combination procedures and on the conversion of topography to implied gravitational potential. The main outcome of our work is the SatGravRET2014 spherical harmonic series expansion to degree 2190, and derived high-resolution grids of 3D-synthesized gravity and quasigeoid effects over the whole of Antarctica. For validation, six data sets from the IAG Subcommission 2.4f “Gravity and Geoid in Antarctica” (AntGG) database were used comprising a total of 1,092,981 airborne gravimetric observations. All subsets consistently show that the Bedmap2-based short-scale gravity modelling improves the agreement over satellite-only data considerably (improvement rates ranging between 9 and 75 % with standard deviations from residuals between SatGravRET2014 and AntGG gravity ranging between 8 and 25 mGal). For comparison purposes, a degree-2190 GGM was generated based on the year-2001 Bedmap1 (using the ETOPO1 topography) instead of 2013 Bedmap2 topography product. Comparison of both GGMs against AntGG consistently reveals a closer fit over all test areas when Bedmap2 is used. This experiment provides evidence for clear improvements in Bedmap2 topographic information over Bedmap1 at spatial scales of \(\sim \) 80–10 km, obtained from independent gravity data used as validation tool. As a general conclusion, our modelling effort fills—in approximation—some gaps in short-scale gravity knowledge over Antarctica and demonstrates the value of the Bedmap2 topography data for short-scale gravity refinement in GGMs. SatGravRET2014 can be used, e.g. as a reference model for future gravity modelling efforts over Antarctica, e.g. as foundation for a combination with the AntGG data set to obtain further improved gravity information.

Description: This contribution summarizes the strategy used by Wuhan University (WHU) to determine precise orbit and clock products for Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS). In particular, the satellite attitude, phase center corrections, solar radiation pressure model developed and used for BDS satellites are addressed. In addition, this contribution analyzes the orbit and clock quality of the quad-constellation products from MGEX Analysis Centers (ACs) for a common time period of 1 year (2014). With IGS final GPS and GLONASS products as the reference, Multi-GNSS products of WHU (indicated by WUM) show the best agreement among these products from all MGEX ACs in both accuracy and stability. 3D Day Boundary Discontinuities (DBDs) range from 8 to 27 cm for Galileo-IOV satellites among all ACs’ products, whereas WUM ones are the largest (about 26.2 cm). Among three types of BDS satellites, MEOs show the smallest DBDs from 10 to 27 cm, whereas the DBDs for all ACs products are at decimeter to meter level for GEOs and one to three decimeter for IGSOs, respectively. As to the satellite laser ranging (SLR) validation for Galileo-IOV satellites, the accuracy evaluated by SLR residuals is at the one decimeter level with the well-known systematic bias of about \(-5\)  cm for all ACs. For BDS satellites, the accuracy could reach decimeter level, one decimeter level, and centimeter level for GEOs, IGSOs, and MEOs, respectively. However, there is a noticeable bias in GEO SLR residuals. In addition, systematic errors dependent on orbit angle related to mismodeled solar radiation pressure (SRP) are present for BDS GEOs and IGSOs. The results of Multi-GNSS combined kinematic PPP demonstrate that the best accuracy of position and fastest convergence speed have been achieved using WUM products, particularly in the Up direction. Furthermore, the accuracy of static BDS only PPP degrades when the BDS IGSO and MEO satellites switches to orbit-normal orientation, particularly for COM products, whereas the WUM show the slightest degradation.

Description: The FormoSat-3/ Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) has been proven a successful mission on profiling ionospheric electron density \(( {N_e })\) using the radio occultation (RO) technique. A follow-on program (called FS7/COSMIC2) is now in progress. The FS3/COSMIC follow-on mission will have six 24 \(^{\circ }\) -inclination and 550-km low Earth orbiting (LEO) satellites and six 72 \(^{\circ }\) -inclination and 750-km LEO satellites to receive Tri-G (GPS, GLONASS, and Galileo) satellite signals. FS7/COSMIC2 RO observations were simulated in this study by calculating limb-viewing GNSS-to-LEO TEC values separately through two independent ionospheric models (the TWIM and NeQuick models). We propose a compensatory Abel-inversion scheme to improve vertical \(N_e \) profiling and three-dimensional (3D) \(N_e \) modeling in this FS7/COSMIC2 simulation study with future real observations. In this FS7/COSMIC2 feasibility study the number of RO observations will increase of around 10 times compared with FS3/COSMIC, and the windowing day number to collect \(N_e \) profiles and to derive every half-hour 3D \(N_e \) model could be decreased from 30 to 3 days. The results show that the root-mean-square (RMS) foF2 and hmF2 difference improvements are 46 % (32 %) and 21 % (4.6 %), respectively, in relative percentage over the standard Abel inversion at the TWIM-background (NeQuick-background) simulation experiment. The RMS modeling errors are about one order less than those from FS3/COSMIC simulations.

Description: This paper proposes an enhanced algorithm to estimate the differential code biases (DCB) on three frequencies of the BeiDou Navigation Satellite System (BDS) satellites. By forming ionospheric observables derived from uncombined precise point positioning and geometry-free linear combination of phase-smoothed range, satellite DCBs are determined together with ionospheric delay that is modeled at each individual station. Specifically, the DCB and ionospheric delay are estimated in a weighted least-squares estimator by considering the precision of ionospheric observables, and a misclosure constraint for different types of satellite DCBs is introduced. This algorithm was tested by GNSS data collected in November and December 2013 from 29 stations of Multi-GNSS Experiment (MGEX) and BeiDou Experimental Tracking Stations. Results show that the proposed algorithm is able to precisely estimate BDS satellite DCBs, where the mean value of day-to-day scattering is about 0.19 ns and the RMS of the difference with respect to MGEX DCB products is about 0.24 ns. In order to make comparison, an existing algorithm based on IGG: Institute of Geodesy and Geophysics, China (IGGDCB), is also used to process the same dataset. Results show that, the DCB difference between results from the enhanced algorithm and the DCB products from Center for Orbit Determination in Europe (CODE) and MGEX is reduced in average by 46 % for GPS satellites and 14 % for BDS satellites, when compared with DCB difference between the results of IGGDCB algorithm and the DCB products from CODE and MGEX. In addition, we find the day-to-day scattering of BDS IGSO satellites is obviously lower than that of GEO and MEO satellites, and a significant bias exists in daily DCB values of GEO satellites comparing with MGEX DCB product. This proposed algorithm also provides a new approach to estimate the satellite DCBs of multiple GNSS systems.

Description: Tracking L-band signals of GNSS satellites by radio telescopes became a new observation type in recent years and will be used to improve reference system realizations and links between Earth- and space-fixed frames. First successful test observations were done, with the drawback of being single-frequency only. In order to correct the ionospheric delay by using GNSS phase observations from co-located receivers, the L4R approach was developed. Based on residuals derived by a least-squares processing of the GNSS geometry-free linear combination corresponding corrections could be derived. As a first validation step L4R corrections were applied to GNSS \(L_1\) data analysis. Station coordinate repeatibilities at the 1-cm level were obtained for baselines of a few thousand kilometers. Comparing the derived delay corrections to VLBI ionospheric delays for quasars located in same directions, differences with a standard deviation of 2.2 TECU could be achieved.

Description: The primary purpose of the International very long baseline interferometry (VLBI) Service for Geodesy and Astrometry Intensive sessions is the rapid estimation of UT1-TAI. Improving the robustness and the precision of the UT1 estimates from the Intensives is an important goal. The INT01 series, which usually uses the Kokee–Wettzell baseline and runs on weekdays, is the most regular IVS Intensive series. The United States Naval Observatory which schedules these sessions traditionally used a small list of strong sources. In 2009, the authors requested and received the use of nine IVS R&D sessions for the evaluation of a new strategy which draws on all sources mutually visible on the Kokee–Wettzell baseline. Analysis of these sessions was sufficiently promising that in July 2010, USNO began to alternate the use of the original and the new strategy in scheduling the INT01 sessions to assess the operational effectiveness of the proposed strategy. In this paper, we summarize our analysis of the R&D sessions, and we also analyze 2 years of operational INT01 sessions. Considered in toto, the new strategy performs as well as, or better than, the original strategy in terms of several measures of robustness and precision. Furthermore, the RMS difference of the UT1 estimates from the 1 h operational INTO1 sessions and concurrently run 24 h VLBI sessions is 21.0 \(\upmu \) s, compared to 30.7 \(\upmu \) s using the standard strategy, indicating that the new strategy is, on average, 30 % more accurate.

Description: The time variable Earth’s gravity field contains information about the mass transport within the system Earth, i.e., the relationship between mass variations in the atmosphere, oceans, land hydrology, and ice sheets. For many years, satellite laser ranging (SLR) observations to geodetic satellites have provided valuable information of the low-degree coefficients of the Earth’s gravity field. Today, the Gravity Recovery and Climate Experiment (GRACE) mission is the major source of information for the time variable field of a high spatial resolution. We recover the low-degree coefficients of the time variable Earth’s gravity field using SLR observations up to nine geodetic satellites: LAGEOS-1, LAGEOS-2, Starlette, Stella, AJISAI, LARES, Larets, BLITS, and Beacon-C. We estimate monthly gravity field coefficients up to degree and order 10/10 for the time span 2003–2013 and we compare the results with the GRACE-derived gravity field coefficients. We show that not only degree-2 gravity field coefficients can be well determined from SLR, but also other coefficients up to degree 10 using the combination of short 1-day arcs for low orbiting satellites and 10-day arcs for LAGEOS-1/2. In this way, LAGEOS-1/2 allow recovering zonal terms, which are associated with long-term satellite orbit perturbations, whereas the tesseral and sectorial terms benefit most from low orbiting satellites, whose orbit modeling deficiencies are minimized due to short 1-day arcs. The amplitudes of the annual signal in the low-degree gravity field coefficients derived from SLR agree with GRACE K-band results at a level of 77 %. This implies that SLR has a great potential to fill the gap between the current GRACE and the future GRACE Follow-On mission for recovering of the seasonal variations and secular trends of the longest wavelengths in gravity field, which are associated with the large-scale mass transport in the system Earth.

Description: Spatial leakage is a major limitation for quantitative interpretation of satellite gravity measurements from the gravity recovery and climate experiment (GRACE). Using synthetic data to simulate ice mass changes in the Amundsen Sea Embayment and Antarctic Peninsula, we analyze quantitatively the effects of a limited range of spherical harmonics (SH) coefficients and additional filtering, which in combination can significantly attenuate signal amplitudes. We present details of a forward modeling algorithm and show that it is capable of removing these biases from GRACE estimates. Examples show how to implement the method by constraining locations of presumed mass changes, or leaving these locations unspecified within a continental region. Our analysis indicates that leakage effects from far-field mass signals (e.g., terrestrial water storage change and glacial melting over other continents) on Antarctic mass rate estimates appear to be negligible. However, leakage from long-term ocean bottom pressure change in the surrounding Antarctic Circumpolar Current regions may bias Antarctic mass rate estimates by up to 20 Gigatonne per year (Gt/year). Experiments based on proxy GRACE measurement noise indicate that the effects of GRACE spatial noise on estimated Antarctic mass rates via constrained and unconstrained forward modelings are \(\sim \) 5 and 15 Gt/year, respectively.

Description: Classical degree variance models (such as Kaula’s rule or the Tscherning-Rapp model) often rely on low-resolution gravity data and so are subject to extrapolation when used to describe the decay of the gravity field at short spatial scales. This paper presents a new degree variance model based on the recently published GGMplus near-global land areas 220 m resolution gravity maps (Geophys Res Lett 40(16):4279–4283, 2013 ). We investigate and use a 2D-DFT (discrete Fourier transform) approach to transform GGMplus gravity grids into degree variances. The method is described in detail and its approximation errors are studied using closed-loop experiments. Focus is placed on tiling, azimuth averaging, and windowing effects in the 2D-DFT method and on analytical fitting of degree variances. Approximation errors of the 2D-DFT procedure on the (spherical harmonic) degree variance are found to be at the 10–20 % level. The importance of the reference surface (sphere, ellipsoid or topography) of the gravity data for correct interpretation of degree variance spectra is highlighted. The effect of the underlying mass arrangement (spherical or ellipsoidal approximation) on the degree variances is found to be crucial at short spatial scales. A rule-of-thumb for transformation of spectra between spherical and ellipsoidal approximation is derived. Application of the 2D-DFT on GGMplus gravity maps yields a new degree variance model to degree 90,000. The model is supported by GRACE, GOCE, EGM2008 and forward-modelled gravity at 3 billion land points over all land areas within the SRTM data coverage and provides gravity signal variances at the surface of the topography. The model yields omission errors of \(\sim \) 9 mGal for gravity ( \(\sim \) 1.5 cm for geoid effects) at scales of 10 km, \(\sim \) 4 mGal ( \(\sim \) 1 mm) at 2-km scales, and \(\sim \) 2 mGal ( \(\sim \) 0.2 mm) at 1-km scales.

Description: The developments in GNSS receiver and antenna technologies, especially the increased sampling rate up to 100 sps, open up the possibility to measure high-rate earthquake ground motions with GNSS. In this paper we focus on the GPS errors in the frequency band above 1 Hz. The dominant error sources are mainly the carrier phase jitter caused by thermal noise and the stress error caused by the dynamics, e.g. antenna motions. To generate a large set of different motions, we used a single-axis shake table, where a GNSS antenna and a strong motion seismometer were mounted with a well-known ground truth. The generated motions were recorded with three different GNSS receivers with sampling rates up to 100 sps and different receiver baseband parameters. The baseband parameters directly dictate the carrier phase jitter and the correlations between subsequent epochs. A narrow loop filter bandwidth keeps the carrier phase jitter on a low level, but has an extreme impact on the receiver response for motions above 1 Hz. The amplitudes above 3 Hz are overestimated up to 50 % or reduced by well over half. The corresponding phase errors are between 30 and 90 degrees. Compared to the GNSS receiver response, the strong motion seismometer measurements do not show any amplitude or phase variations for the frequency range from 1 to 20 Hz. Due to the large errors for dynamic GNSS measurements, it is essential to account for the baseband parameters of the GNSS receivers if high-rate GNSS is to become a valuable tool for seismic displacement measurements above 1 Hz. Fortunately, the receiver response can be corrected by an inverse filter if the baseband parameters are known.

Description: The Fresnel–Fizeau effect is a special relativistic effect that makes the speed of light dependent on the velocity of a transparent, moving medium. We present a theoretical formalism for discussing propagation of electromagnetic signals through the moving Earth atmosphere taking into account the Fresnel–Fizeau effect. It provides the rigorous relativistic derivation of the atmospheric time delay equation in the consensus model of geodetic VLBI observations which has never been published before. The paper confirms the atmospheric time delay of the consensus VLBI model used in IERS standards and provides a firm theoretical basis for calculation of even more subtle relativistic corrections.

Description: Integral transforms of the disturbing gravitational potential derived from satellite altimetry onto satellite gradiometric data are formulated, investigated and applied in this article. First, corresponding differential operators, that relate the disturbing gravitational potential to the six components of the disturbing gradiometric tensor in the spherical local north-oriented frame, are applied to the spherical Abel-Poisson integral equation. This yields six new integral equations for which respective kernel functions are given in both spectral and spatial forms. Second, truncation error formulas for each of the integral transforms are provided in the spectral form. Also expressions for the corresponding truncation error coefficients are derived. Third, practical estimators for evaluation of the disturbing gravitational gradients are formulated, and their correctness and expected accuracy are investigated. Finally, the practical estimators are applied for validation of a sample of the gradiometric data provided by the GOCE satellite mission. Obtained results demonstrate applicability of the new apparatus as an alternative validation method of the satellite gravitational gradients.

Description: An analytical approach to single-frequency precise point positioning (PPP) is discussed in this paper. To obtain highest precision results, all biases must be eliminated or modelled to centimetre level. The use of the GRAPHIC ionosphere-free linear combination that is based on single-frequency phase and code observations eliminates the ionosphere bias; however, the rank deficient Gauss–Markov model is obtained. We explicitly determine rank deficiency of a Gauss–Markov model as a number of all ambiguity clusters, each of them defined as a set of all ambiguities overlapping in time. On the basis of S-transformation we prove that the single-frequency PPP represents an unbiased estimator for station coordinates and troposphere parameters, while it presents a biased estimator for ambiguities and receiver-clock error parameters. Additionally we describe the estimable parameters in each ambiguity cluster as the differences between ambiguity parameters and the sum of receiver-clock parameters with one of the ambiguities. We also show that any other particular solution on the basis of S-transformation is obtained only when the common least-squares estimation in single step is applied. The recursive least-squares estimation with parameter pre-elimination only determines the vector of unknowns as possible to transform through S-transformation, whereas the same does not hold for the cofactor matrix of unknowns. For a case study, we present our method on GPS data from 19 permanent stations (14 IGS and 5 EPN) in Europe, for 89 consecutive days in the beginning of 2013. The static case study revealed the precision of daily coordinates as 7.6, 11.7 and 19.6 mm for \(N\) , \(E\) and \(U\) , respectively. The accuracies of the \(N\) , \(E\) and \(U\) components were determined as 6.9, 13.5 and 31.4 mm, respectively, and were calculated using the Helmert transformation of weighted-mean daily single-frequency PPP and IGb08 coordinates. The estimated convergence times were relatively diverse, expanding from 1.75 h (CAGL) to 5.25 h (GRAZ) for the horizontal position with the 10-cm precision threshold, and from 1.00 h (GRAS) to 3.25 h (BZRG) for the height component with a 20-cm precision threshold. The convergence times were shown to be strongly correlated to the remaining unmodelled biases in the GRAPHIC linear combination, primarily with multipath, where the correlation coefficient for the horizontal position was determined as \(\rho _P\) \(=\) 0.68 and for height as \(\rho _U\) \(=\) 0.85. The comparison to the model where raw observations are used ( \(C\) , \(L\) ) and where the ionosphere bias is mitigated with global ionosphere models (GIM) revealed the supremacy of the proposed single-frequency PPP method based on the GRAPHIC linear combination in both the static and the semi-kinematic case study. In the static case study, the proposed single-frequency PPP model was superior both in terms of precision and accuracy. In the semi-kinematic case study, the usage of raw observations with GIM would improve results only when multipath and noise of code observations would prevail over the remaining ionosphere bias, i.e. after applying GIM.

Description: A surface spherical harmonic expansion of gravity anomalies with respect to a geodetic reference ellipsoid can be used to model the global gravity field and reveal its spectral properties. In this paper, a direct and rigorous transformation between solid spherical harmonic coefficients of the Earth’s disturbing potential and surface spherical harmonic coefficients of gravity anomalies in ellipsoidal approximation with respect to a reference ellipsoid is derived. This transformation cannot rigorously be achieved by the Hotine–Jekeli transformation between spherical and ellipsoidal harmonic coefficients. The method derived here is used to create a surface spherical harmonic model of gravity anomalies with respect to the GRS80 ellipsoid from the EGM2008 global gravity model. Internal validation of the model shows a global RMS precision of \(〈\) 1 nGal. This is significantly more precise than previous solutions based on spherical approximation or approximations to order \(e^2\) or \(e^3\) , which are shown to be insufficient for the generation of surface spherical harmonic coefficients with respect to a geodetic reference ellipsoid. Numerical results of two applications of the new method (the computation of ellipsoidal corrections to gravimetric geoid computation, and area means of gravity anomalies in ellipsoidal approximation) are provided.

Description: The increasing accuracy and growing time span of Very Long Baseline Interferometry (VLBI) observations allow the determination of seasonal signals in station positions which still remain unmodelled in conventional analysis approaches. In this study we focus on the impact of the neglected seasonal signals in the station displacement on the celestial reference frame and Earth orientation parameters. We estimate empirical harmonic models for selected stations within a global solution of all suitable VLBI sessions and create mean annual models by stacking yearly time series of station positions which are then entered a priori in the analysis of VLBI observations. Our results reveal that there is no systematic propagation of the seasonal signal into the orientation of celestial reference frame but position changes occur for radio sources observed non-evenly over the year. On the other hand, the omitted seasonal harmonic signal in horizontal station coordinates propagates directly into the Earth rotation parameters causing differences of several tens of microarcseconds.

Description: Precise knowledge and consistent modeling of the yaw-attitude of GNSS satellites are essential for high-precision data processing and applications. As the exact attitude control mechanism for the satellites of the BeiDou Satellite Navigation System (BDS) is not yet released, the reverse kinematic precise point positioning (PPP) method was applied in our study. However, we confirm that the recent precise orbit determination (POD) processing for GPS satellites could not provide suitable products for estimating BDS attitude using the reverse PPP because of the special attitude control switching between the nominal and the orbit-normal mode. In our study, we propose a modified processing schema for studying the attitude behavior of the BDS satellites. In this approach, the observations of the satellites during and after attitude switch are excluded in the POD processing, so that the estimates, which are needed in the reverse PPP, are not contaminated by the inaccurate initial attitude mode. The modified process is validated by experimental data sets and the attitude yaw-angles of the BDS IGSO and MEO satellites are estimated with an accuracy of better than \(9^{\circ }\) . Furthermore, the results confirm that the switch is executed when the Sun elevation is about \(4^{\circ }\) and the actual orientation is very close to its target one. Based on the estimated yaw-angles, a preliminary attitude switch model was established and reintroduced into the POD, yielding to a substantial improvement in the orbit overlap RMS.

Description: Daily Very Long Baseline Interferometry (VLBI) intensive measurements make an important contribution to the regular monitoring of Earth rotation variations. Since these variations are quite rapid, their knowledge is important for navigation with global navigation satellite system and for investigations in Earth sciences. Unfortunately, the precision of VLBI intensive observations is 2–3 times worse than the precision of regular 24h-VLBI measurements with networks of 5–10 radio telescopes. The major advancement of research in this paper is the improvement of VLBI intensive results by (a) using twin telescopes instead of single telescopes and (b) applying an entirely new scheduling concept for the individual observations. Preparatory investigations of standardintensive sessions suggest that the impact factors of the observations are well suited for the identification of the most influential observations which are needed for the determination of certain parameters within the entire design of a VLBI session. Based on this experience, the scheduling method is designed for optimizing the observations’ geometry for a given network of radio telescopes and a predefined set of parameters to be estimated. The configuration of at least two twin telescopes, or one twin and two single telescopes, offers the possibility of building pairwise sub-nets that observe two different sources simultaneously. In addition to an optimized observing plan, a special parametrization for twin telescopes leads to an improved determination of Earth rotation variations, as it is shown by simulated observations. In general, an improvement of about 50 % in the formal errors can be realized using twin radio telescopes. This result is only due to geometric improvements as higher slew rates of the twin telescopes are not taken into account.

Description: GNSS integer ambiguity validation is considered to be a challenge task for decades. Several kinds of validation tests are developed and widely used in these years, but theoretical basis is their weakness. Ambiguity validation theoretically is an issue of hypothesis test. In the frame of Bayesian hypothesis testing, posterior probability is the canonical standard that statistical decision should be based on. In this contribution, (i) we derive the posterior probability of the fixed ambiguity based on the Bayesian principle and modify it for practice ambiguity validation. (ii) The optimal property of the posterior probability test is proved based on an extended Neyman–Pearson lemma. Since validation failure rate is the issue users most concerned about, (iii) we derive the failure rate upper bound of the posterior probability test, so the user can use the posterior probability test either in the fixed posterior probability or in the fixed failure rate way. Simulated as well as real observed data are used for experimental validations. The results show that (i) the posterior probability test is the most effective within the R-ratio test, difference test, ellipsoidal integer aperture test and posterior probability test, (ii) the posterior probability test is computational efficient and (iii) the failure rate estimation for posterior probability test is useful.

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

Description: The concept of reliability was introduced into geodesy by Baarda (A testing procedure for use in geodetic networks. Publications on Geodesy, vol. 2. Netherlands Geodetic Commission, Delft, 1968 ). It gives a measure for the ability of a parameter estimation to detect outliers and leads in case of one outlier to the MDB, the minimal detectable bias or outlier. The MDB depends on the non-centrality parameter of the \(\chi ^2\) -distribution, as the variance factor of the linear model is assumed to be known, on the size of the outlier test of an individual observation which is set to 0.001 and on the power of the test which is generally chosen to be 0.80. Starting from an estimated variance factor, the \(F\) -distribution is applied here. Furthermore, the size of the test of the individual observation is a function of the number of outliers to keep the size of the test of all observations constant, say 0.05. The power of the test is set to 0.80. The MDBs for multiple outliers are derived here under these assumptions. The method is applied to the reconstruction of a bell-shaped surface measured by a laser scanner. The MDBs are introduced as outliers for the alternative hypotheses of the outlier tests. A Monte Carlo method reveals that due to the way of introducing the outliers, the false null hypotheses cannot be rejected on the average with a power of 0.80 if the MDBs are not enlarged by a factor.

Description: Although the analytical solutions for total least-squares with multiple linear and single quadratic constraints were developed quite recently in different geodetic publications, these methods are restricted in number and type of constraints, and currently their computational efficiency and applications are mostly unknown. In this contribution, it is shown how the weighted total least-squares (WTLS) problem with arbitrary applicable constraints can be solved based on a Newton type methodology. This iterative process with quadratic convergence is expanded upon to become a compact solution for the WTLS with or without constraints. This compact solution is then further interpreted as a universal formula for the symmetrical adjustment of the errors-in-variables model which represents affine, similarity and rigid transformations in two- and three-dimensional space. Furthermore, statistical analysis of the constrained WTLS including the first-order approximation of precision and the bias was investigated. In order to substantiate our proposed method’s applicability, it was used to solve the affine, similarity and rigid transformation problem in two- and three-dimensional cases, where the structure of the coefficient matrix and multiple constraints were taken into account simultaneously.

Description: Multiple-aperture SAR interferometry (MAI) has demonstrated outstanding measurement accuracy of along-track displacement when compared to pixel-offset-tracking methods; however, measuring slow-moving (cm/year) surface displacement remains a challenge. Stacking of multi-temporal observations is a potential approach to reducing noise and increasing measurement accuracy, but it is difficult to achieve a significant improvement by applying traditional stacking methods to multi-temporal MAI interferograms. This paper proposes an efficient MAI stacking method, where multi-temporal forward- and backward-looking residual interferograms are individually stacked before the MAI interferogram is generated. We tested the performance of this method using ENVISAT data from Kīlauea Volcano, Hawai‘i, where displacement on the order of several centimeters per year is common. By comparing results from the proposed stacking methods with displacements from GPS data, we documented measurement accuracies of about 1.03 and 1.07 cm/year for the descending and ascending tracks, respectively—an improvement of about a factor of two when compared with that from the conventional stacking approach. Three-dimensional surface-displacement maps can be constructed by combining stacked InSAR and MAI observations, which will contribute to a better understanding of a variety of geological phenomena.

Description: In the high-precision application of Global Navigation Satellite System (GNSS), integer ambiguity resolution is the key step to realize precise positioning and attitude determination. As the necessary part of quality control, integer aperture (IA) ambiguity resolution provides the theoretical and practical foundation for ambiguity validation. It is mainly realized by acceptance testing. Due to the constraint of correlation between ambiguities, it is impossible to realize the controlling of failure rate according to analytical formula. Hence, the fixed failure rate approach is implemented by Monte Carlo sampling. However, due to the characteristics of Monte Carlo sampling and look-up table, we have to face the problem of a large amount of time consumption if sufficient GNSS scenarios are included in the creation of look-up table. This restricts the fixed failure rate approach to be a post process approach if a look-up table is not available. Furthermore, if not enough GNSS scenarios are considered, the table may only be valid for a specific scenario or application. Besides this, the method of creating look-up table or look-up function still needs to be designed for each specific acceptance test. To overcome these problems in determination of critical values, this contribution will propose an instantaneous and CONtrollable (iCON) IA ambiguity resolution approach for the first time. The iCON approach has the following advantages: (a) critical value of acceptance test is independently determined based on the required failure rate and GNSS model without resorting to external information such as look-up table; (b) it can be realized instantaneously for most of IA estimators which have analytical probability formulas. The stronger GNSS model, the less time consumption; (c) it provides a new viewpoint to improve the research about IA estimation. To verify these conclusions, multi-frequency and multi-GNSS simulation experiments are implemented. Those results show that IA estimators based on iCON approach can realize controllable ambiguity resolution. Besides this, compared with ratio test IA based on look-up table, difference test IA and IA least square based on the iCON approach most of times have higher success rates and better controllability to failure rates.

Description: Two gimbal-mounted GNSS antennas were installed on each side of the radome-enclosed 20 m VLBI radio telescope at the Onsala Space Observatory. GPS data with a 1 Hz sampling rate were recorded for five semi-kinematic and four kinematic observing campaigns. These GPS data were analysed together with data from the IGS station ONSA with an in-house Matlab-based GPS software package, using the double-difference analysis strategy. The coordinates of the GNSS antennas on the telescope were estimated for different observation angles of the telescope, at specific epochs, and used to calculate the geodetic reference point of the telescope. The local tie vector between the VLBI and the ONSA GNSS reference points in a geocentric reference frame was hence obtained. The two different types of observing campaigns gave consistent results of the estimated local tie vector and the axis offset of the telescope. The estimated local tie vector obtained from all nine campaigns gave standard deviations of 1.5, 1.0, and 2.9 mm for the geocentric X, Y, and Z components, respectively. The result of the estimated axis offset of the VLBI telescope shows a difference of 0.3 mm, with a standard deviation of 1.9 mm, with respect to a reference value obtained by two local surveys carried out in 2002 and 2008. Our results show that the presented method can be used as a complement to the more accurate but more labour intensive classical geodetic surveys to continuously monitor the local tie at co-location stations with an accuracy of a few millimetres.

Description: The concept of minimal detectable bias (MDB) as initiated by Baarda (Publ Geod New Ser 2(5), 1968 ) and later developed by Wang and Chen (Acta Geodaet et Cartograph Sin Engl Edn 42–51, 1994 ), Schaffrin (J Eng Surv 123:126–137, 1997 ), Teunissen (IEEE Aerosp Electron Syst Mag 5(7):35–41, 1990 , J Geod 72:236–244 1998 , Testing theory: an introduction. Delft University Press, Delft, 2000 ) and others, refers to the issue of outlier detectability. A supplementation of the concept is proposed for the case of correlated observations contaminated with a single gross error. The supplementation consists mainly of an outlier identifiability index assigned to each individual observation in a network and a mis-identifiability index being the maximum probability of identifying a wrong observation. To those indices there can also be added the MDB multiplying factor to increase the identifiability index to a satisfactory level. As auxiliary measures there are indices of partial identifiability concerning pairs of observations. The indices were derived assuming the generalized outlier identification procedure as in Knight et al. (J Geod. doi: 10.1007/s00190-010-0392-4 , 2010 ), which with one outlier case being assumed is similar to Baarda’s w -test (Baarda in Publ Geod New Ser 2(5), 1968 ). The following two options of identifiability indices and partial identifiability indices are distinguished: I. the indices related to identification of a contaminated observation within a set of observations suspected of containing a gross error (identifiability), II. the indices related to identification of a contaminated observation within a whole set of observations (pseudo-identifiability). To characterize the proposed approach in the context of the existing solutions of similar topic being the separability testing, the properties of both types of identifiability indices are discussed with reference to the concept of Minimal Separable Bias (Wang and Knight in J Glob Position Syst 11(1):46–57, 2012 ) and a general approach in Yang et al. (J Geod 87(6):591–604, 2013 ). Numerical examples are provided to verify the proposed approach.

Description: A new methodology is proposed to estimate changes in the Earth’s dynamic oblateness ( \(\Delta {J_{2}}\) or equivalently, \(-\sqrt{5}\Delta {C_{20}}\) ) on a monthly basis. The algorithm uses monthly Gravity Recovery and Climate Experiment (GRACE) gravity solutions, an ocean bottom pressure model and a glacial isostatic adjustment (GIA) model. The resulting time series agree remarkably well with a solution based on satellite laser ranging (SLR) data. Seasonal variations of the obtained time series show little sensitivity to the choice of GRACE solutions. Reducing signal leakage in coastal areas when dealing with GRACE data and accounting for self-attraction and loading effects when dealing with water redistribution in the ocean is crucial in achieving close agreement with the SLR-based solution in terms of de-trended solutions. The obtained trend estimates, on the other hand, may be less accurate due to their dependence on the GIA models, which still carry large uncertainties.

Description: The geoid-to-quasigeoid separation is often computed only approximately as a function of the simple planar Bouguer gravity anomaly and the height of the computation point while disregarding the contributions of terrain geometry and anomalous topographic density as well as the sub-geoid masses. In this study we demonstrate that these contributions are significant and, therefore, should be taken into consideration when investigating the relation between the normal and orthometric heights particularly in the mountainous, polar and geologically complex regions. These contributions are evaluated by applying the spectral expressions for gravimetric forward modelling and using the EIGEN-6C4 gravity model, the Earth2014 datasets of terrain, ice thickness and inland bathymetry and the CRUST1.0 sediment and (consolidated) crustal density data. Since the global crustal density models currently available (e.g. CRUST1.0) have a limited accuracy and resolution, the comparison of individual density contributions is—for consistency—realized with a limited spectral resolution up to a spherical harmonic degree 360 (or 180). The results reveal that the topographic contribution globally varies between \(-\) 0.33 and 0.57 m, with maxima in Himalaya and Tibet. The contribution of ice considerably modifies the geoid-to-quasigeoid separation over large parts of Antarctica and Greenland, where it reaches \(\sim \) 0.2 m. The contributions of sediments and bedrock are less pronounced, with the values typically varying only within a few centimetres. These results, however, have still possibly large uncertainties due to the lack of information on the actual sediment and bedrock density. The contribution of lakes is mostly negligible; its maxima over the Laurentian Great Lakes and the Baikal Lake reach only several millimetres. The contribution of the sub-geoid masses is significant. It is everywhere negative and reaches extreme values of \(-\) 4.43 m. According to our estimates, the geoid-to-quasigeoid separation globally varies within \(-\) 4.19 and 0.26 m while the corresponding values computed according to a classical definition are only negative and reach extreme values of \(-\) 3.5 m. A comparison of these results reveals that inaccuracies caused by disregarding the terrain geometry and mass density heterogeneities distributed within the topography and below the geoid surface can reach \(\pm \) 2 m or more in the mountainous regions.

Description: Two levelling-based vertical datums have been used in North America, namely CGVD28 in Canada and NAVD88 in the USA and Mexico. Although the two datums will be replaced by a common and continent-wide vertical datum in a few years, their connection and unification are of great interest to the scientific and user communities. In this paper, the geodetic boundary value problem (GBVP) approach is studied as a rigorous method for connecting two or more vertical datums through computed datum offsets from a global equipotential surface defined by a GOCE-based geoid. The so-called indirect bias term, the effect of the GOCE geoid omission error, the effect of the systematic levelling datum errors and distortions, and the effect of the geodetic data errors on the datum unification are four important factors affecting the practical implementation of this approach. These factors are investigated numerically using the GNSS-levelling and tide gauge stations in Canada, the USA, Alaska, and Mexico. The results show that the indirect bias term can be omitted if a GOCE-based global geopotential model is used in gravimetric geoid computations. The omission of the indirect bias term simplifies the linear system of equations for the estimation of the datum offset(s). Because of the existing systematic levelling errors and distortions in the Canadian and US levelling networks, the datum offsets are investigated in eight smaller regions along the Canadian and US coastal areas. Using GNSS-levelling stations in the US coastal regions, the mean datum offset can be estimated with a 1 cm standard deviation if the GOCE geoid omission error is taken into account by means of the local gravity and topographic information. In the Canadian Atlantic and Pacific regions, the datum offsets can be estimated with 2.3 and 3.5 cm standard deviation, respectively, using GNSS-levelling stations. However, due to the low number of tide gauge stations, the standard deviation of the CGVD28 and NAVD88 datum offsets can reach one decimetre in the Pacific regions. With the available GNSS-levelling stations in Alaska and Mexico, the NAVD88 datum offset can be estimated with a standard deviation below 3 cm. The numerical investigations of this study provide, for the first time, the datum offsets between North American vertical datums and their associated standard deviations with which the offsets can be estimated. The results of this study demonstrate the importance of the aforementioned four factors in the practical implementation of the GBVP approach for the unification of the levelling-based vertical datums.

Description: A critical point in the analysis of ground displacement time series, as those recorded by space geodetic techniques, is the development of data-driven methods that allow the different sources of deformation to be discerned and characterized in the space and time domains. Multivariate statistic includes several approaches that can be considered as a part of data-driven methods. A widely used technique is the principal component analysis (PCA), which allows us to reduce the dimensionality of the data space while maintaining most of the variance of the dataset explained. However, PCA does not perform well in finding the solution to the so-called blind source separation (BSS) problem, i.e., in recovering and separating the original sources that generate the observed data. This is mainly due to the fact that PCA minimizes the misfit calculated using an \(L_{2}\) norm ( \(\chi ^{2})\) , looking for a new Euclidean space where the projected data are uncorrelated. The independent component analysis (ICA) is a popular technique adopted to approach the BSS problem. However, the independence condition is not easy to impose, and it is often necessary to introduce some approximations. To work around this problem, we test the use of a modified variational Bayesian ICA (vbICA) method to recover the multiple sources of ground deformation even in the presence of missing data. The vbICA method models the probability density function (pdf) of each source signal using a mix of Gaussian distributions, allowing for more flexibility in the description of the pdf of the sources with respect to standard ICA, and giving a more reliable estimate of them. Here we present its application to synthetic global positioning system (GPS) position time series, generated by simulating deformation near an active fault, including inter-seismic, co-seismic, and post-seismic signals, plus seasonal signals and noise, and an additional time-dependent volcanic source. We evaluate the ability of the PCA and ICA decomposition techniques in explaining the data and in recovering the original (known) sources. Using the same number of components, we find that the vbICA method fits the data almost as well as a PCA method, since the \(\chi ^{2}\) increase is less than 10 % the value calculated using a PCA decomposition. Unlike PCA, the vbICA algorithm is found to correctly separate the sources if the correlation of the dataset is low ( \(〈\) 0.67) and the geodetic network is sufficiently dense (ten continuous GPS stations within a box of side equal to two times the locking depth of a fault where an earthquake of \(M_\mathrm{{w}} 〉6\) occurred). We also provide a cookbook for the use of the vbICA algorithm in analyses of position time series for tectonic and non-tectonic applications.

Description: Differential code biases (DCBs) are important parameters that must be estimated accurately and reliably for high-precision GNSS applications. For optimal operational service performance of the Beidou navigation system (BDS), continuous monitoring and constant quality assessment of the BDS satellite DCBs are crucial. In this study, a global ionospheric model was constructed based on a dual system BDS/GPS combination. Daily BDS DCBs were estimated together with the total electron content from 23 months’ multi-GNSS observations. The stability of the resulting BDS DCB estimates was analyzed in detail. It was found that over a long period, the standard deviations (STDs) for all satellite B1–B2 DCBs were within 0.3 ns (average: 0.19 ns) and for all satellite B1–B3 DCBs, the STDs were within 0.36 ns (average: 0.22 ns). For BDS receivers, the STDs were greater than for the satellites, with most values \(〈\) 2 ns. The DCBs of different receiver families are different. Comparison of the statistics of the short-term stability of satellite DCBs over different time intervals revealed that the difference in STD between 28- and 7-day intervals was small, with a maximum not exceeding 0.06 ns. In almost all cases, the difference in BDS satellite DCBs between two consecutive days was \(〈\) 0.8 ns. The main conclusion is that because of the stability of the BDS DCBs, they only require occasional estimation or calibration. Furthermore, the 30-day averaged satellite DCBs can be used reliably for the most demanding BDS applications.

Description: Precise orbit determination is an essential part of the most scientific satellite missions. Highly accurate knowledge of the satellite position is used to geolocate measurements of the onboard sensors. For applications in the field of gravity field research, the position itself can be used as observation. In this context, kinematic orbits of low earth orbiters (LEO) are widely used, because they do not include a priori information about the gravity field. The limiting factor for the achievable accuracy of the gravity field through LEO positions is the orbit accuracy. We make use of raw global positioning system (GPS) observations to estimate the kinematic satellite positions. The method is based on the principles of precise point positioning. Systematic influences are reduced by modeling and correcting for all known error sources. Remaining effects such as the ionospheric influence on the signal propagation are either unknown or not known to a sufficient level of accuracy. These effects are modeled as unknown parameters in the estimation process. The redundancy in the adjustment is reduced; however, an improvement in orbit accuracy leads to a better gravity field estimation. This paper describes our orbit determination approach and its mathematical background. Some examples of real data applications highlight the feasibility of the orbit determination method based on raw GPS measurements. Its suitability for gravity field estimation is presented in a second step.

Description: The 2010, (Mw 8.8) Maule, Chile, earthquake produced large co-seismic displacements and non-secular, post-seismic deformation, within latitudes 28 \(^{\circ }\) S–40 \(^{\circ }\) S extending from the Pacific to the Atlantic oceans. Although these effects are easily resolvable by fitting geodetic extended trajectory models (ETM) to continuous GPS (CGPS) time series, the co- and post-seismic deformation cannot be determined at locations without CGPS (e.g., on passive geodetic benchmarks). To estimate the trajectories of passive geodetic benchmarks, we used CGPS time series to fit an ETM that includes the secular South American plate motion and plate boundary deformation, the co-seismic discontinuity, and the non-secular, logarithmic post-seismic transient produced by the earthquake in the Posiciones Geodésicas Argentinas 2007 (POSGAR07) reference frame (RF). We then used least squares collocation (LSC) to model both the background secular inter-seismic and the non-secular post-seismic components of the ETM at the locations without CGPS. We tested the LSC modeled trajectories using campaign and CGPS data that was not used to generate the model and found standard deviations (95 % confidence level) for position estimates for the north and east components of 3.8 and 5.5 mm, respectively, indicating that the model predicts the post-seismic deformation field very well. Finally, we added the co-seismic displacement field, estimated using an elastic finite element model. The final, trajectory model allows accessing the POSGAR07 RF using post-Maule earthquake coordinates within 5 cm for \(\sim \) 91 % of the passive test benchmarks.

Description: In this contribution, we present a GPS+GLONASS+BeiDou+Galileo four-system model to fully exploit the observations of all these four navigation satellite systems for real-time precise orbit determination, clock estimation and positioning. A rigorous multi-GNSS analysis is performed to achieve the best possible consistency by processing the observations from different GNSS together in one common parameter estimation procedure. Meanwhile, an efficient multi-GNSS real-time precise positioning service system is designed and demonstrated by using the multi-GNSS Experiment, BeiDou Experimental Tracking Network, and International GNSS Service networks including stations all over the world. The statistical analysis of the 6-h predicted orbits show that the radial and cross root mean square (RMS) values are smaller than 10 cm for BeiDou and Galileo, and smaller than 5 cm for both GLONASS and GPS satellites, respectively. The RMS values of the clock differences between real-time and batch-processed solutions for GPS satellites are about 0.10 ns, while the RMS values for BeiDou, Galileo and GLONASS are 0.13, 0.13 and 0.14 ns, respectively. The addition of the BeiDou, Galileo and GLONASS systems to the standard GPS-only processing, reduces the convergence time almost by 70 %, while the positioning accuracy is improved by about 25 %. Some outliers in the GPS-only solutions vanish when multi-GNSS observations are processed simultaneous. The availability and reliability of GPS precise positioning decrease dramatically as the elevation cutoff increases. However, the accuracy of multi-GNSS precise point positioning (PPP) is hardly decreased and few centimeter are still achievable in the horizontal components even with 40 \(^{\circ }\) elevation cutoff. At 30 \(^{\circ }\) and 40 \(^{\circ }\) elevation cutoffs, the availability rates of GPS-only solution drop significantly to only around 70 and 40 %, respectively. However, multi-GNSS PPP can provide precise position estimates continuously (availability rate is more than 99.5 %) even up to 40 \(^{\circ }\) elevation cutoff (e.g., in urban canyons).

Description: In this note, the 3D similarity datum transformation problem with Gauss–Helmert model, also known as the 3D symmetric Helmert transformation, is studied. The closed-form least-squares solution, i.e., without iteration, to this problem is derived. It is found that the rotation parameters in this solution are the same to that for the transformation with Gauss–Markov model, while the scale and translation parameters differ from each other.

Description: Some estimates of GPS velocity uncertainties are very low, \(〈\) 0.1 mm/year with 10 years of data. Yet, residual velocities relative to rigid plate models in nominally stable plate interiors can be an order of magnitude larger. This discrepancy could be caused by underestimating low-frequency time-dependent noise in position time series, such as random walk. We show that traditional estimators, based on individual time series, are insensitive to low-amplitude random walk, yet such noise significantly increases GPS velocity uncertainties. Here, we develop a method for determining representative noise parameters in GPS position time series, by analyzing an entire network simultaneously, which we refer to as the network noise estimator (NNE). We analyze data from the aseismic central-eastern USA, assuming that residual motions relative to North America, corrected for glacial isostatic adjustment (GIA), represent noise. The position time series are decomposed into signal (plate rotation and GIA) and noise components. NNE simultaneously processes multiple stations with a Kalman filter and solves for average noise components for the network by maximum likelihood estimation. Synthetic tests show that NNE correctly estimates even low-level random walk, thus providing better estimates of velocity uncertainties than conventional, single station methods. To test NNE on actual data, we analyze a heterogeneous 15 station GPS network from the central-eastern USA, assuming the noise is a sum of random walk, flicker and white noise. For the horizontal time series, NNE finds higher average random walk than the standard individual station-based method, leading to velocity uncertainties a factor of 2 higher than traditional methods.

Description: Carrier-phase integer ambiguity resolution (IAR) is the key to highly precise, fast positioning and attitude determination with Global Navigation Satellite System (GNSS). It can be seen as the process of estimating the unknown cycle ambiguities of the carrier-phase observations as integers. Once the ambiguities are fixed, carrier phase data will act as the very precise range data. Integer aperture (IA) ambiguity resolution is the combination of acceptance testing and integer ambiguity resolution, which can realize better quality control of IAR. Difference test (DT) is one of the most popular acceptance tests. This contribution will give a detailed analysis about the following properties of IA ambiguity resolution based on DT: The sharpest and loose upper bounds of DT are derived from the perspective of geometry. These bounds are very simple and easy to be computed, which give the range for the critical values of DT. The definition of DT integer aperture bootstrapping (IAB) estimator is firstly given. The relationships between DT-IAB and DT-IA are deeply investigated, which also firstly give a new perspective to review the IAB and IA least square (IALS) estimators. Based on the properties of the second best integer candidates in integer least square and integer bootstrapping estimators, the definition of DT-IA is given from another perspective, which is mathematically equivalent to its original definition. The analytical expressions of the success rate lower bound and upper bound of DT-IA estimator are firstly derived. Then, the quality measure for DT-IA estimator can be completely calculated as integer estimator without measurements. Both sharp and loose bounds of DT-IA success rate are given so that the success rates are easily evaluated, which also can provide reasonable approximation for DT-IA estimator. All these conclusions are verified based on the single and combination GNSS simulation experiments. The experiment results indicate the correctness of these conclusions. These properties demonstrate the special properties of DT-IA estimator, and also provide the research frame to investigate other IA estimators. They are helpful to realize better use of IA estimators in quality control and precise positioning in future.

Description: Gravity field parameters are usually determined from observations of the GRACE satellite mission together with arc-specific parameters in a generalized orbit determination process. When separating the estimation of gravity field parameters from the determination of the satellites’ orbits, correlations between orbit parameters and gravity field coefficients are ignored and the latter parameters are biased towards the a priori force model. We are thus confronted with a kind of hidden regularization. To decipher the underlying mechanisms, the Celestial Mechanics Approach is complemented by tools to modify the impact of the pseudo-stochastic arc-specific parameters on the normal equations level and to efficiently generate ensembles of solutions. By introducing a time variable a priori model and solving for hourly pseudo-stochastic accelerations, a significant reduction of noisy striping in the monthly solutions can be achieved. Setting up more frequent pseudo-stochastic parameters results in a further reduction of the noise, but also in a notable damping of the observed geophysical signals. To quantify the effect of the a priori model on the monthly solutions, the process of fixing the orbit parameters is replaced by an equivalent introduction of special pseudo-observations, i.e., by explicit regularization. The contribution of the thereby introduced a priori information is determined by a contribution analysis. The presented mechanism is valid universally. It may be used to separate any subset of parameters by pseudo-observations of a special design and to quantify the damage imposed on the solution.

Description: The article describes the estimation of a priori error associated with heterogeneous, non-correlated noise within one dataset. The errors are estimated by restricted maximum likelihood (REML). The solution is composed of a cross-validation technique named leave-one-out (LOO) and REML estimation of a priori noise for different groups obtained by LOO. A numerical test is the main part of this case study and it presents two options. In the first one, the whole data is split into two subsets using LOO, by finding potentially outlying data. Then a priori errors are estimated in groups for the better and worse subset, where the latter includes the mentioned outlying data. The second option was to select data from the neighborhood of each point and estimate two a priori errors by REML, one for the selected point and one for the surrounding group of data. Both ideas have been validated with the use of LOO performed only in points of the better subset from the first kind of test. The use of homogeneous noise in the two example sets leads to LOO standard deviations equal 1.83 and 1.54 mGal, respectively. The group estimation generates only small improvement at the level of 0.1 mGal, which can also be reached after the removal of worse points. The pointwise REML solution, however, provides LOO standard deviations that are at least 20 % smaller than statistics from the homogeneous noise application.

Description: Global Positioning System (GPS) has become a cost-effective tool to determine troposphere zenith total delay (ZTD) with accuracy comparable to other atmospheric sensors such as the radiosonde, the water vapor radiometer, the radio occultation and so on. However, the high accuracy of GPS troposphere ZTD estimates relies on the precise satellite orbit and clock products available with various latencies. Although the International GNSS Service (IGS) can provide predicted orbit and clock products for real-time applications, the predicted clock accuracy of 3 ns cannot always guarantee the high accuracy of troposphere ZTD estimates. Such limitations could be overcome by the use of the newly launched IGS real-time service which provides \(\sim \) 5 cm orbit and 0.2–1.0 ns (an equivalent range error of 6–30 cm) clock products in real time. Considering the relatively larger magnitude of the clock error than that of the orbit error, this paper investigates the effect of real-time satellite clock errors on the GPS precise point positioning (PPP)-based troposphere ZTD estimation. Meanwhile, how the real-time satellite clock errors impact the GPS PPP-based troposphere ZTD estimation has also been studied to obtain the most precise ZTD solutions. First, two types of real-time satellite clock products are assessed with respect to the IGS final clock product in terms of accuracy and precision. Second, the real-time GPS PPP-based troposphere ZTD estimation is conducted using data from 34 selected IGS stations over three independent weeks in April, July and October, 2013. Numerical results demonstrate that the precision, rather than the accuracy, of the real-time satellite clock products impacts the real-time PPP-based ZTD solutions more significantly. In other words, the real-time satellite clock product with better precision leads to more precise real-time PPP-based troposphere ZTD solutions. Therefore, it is suggested that users should select and apply real-time satellite products with better clock precision to obtain more consistent real-time PPP-based ZTD solutions.

Description: Non-gravitational accelerations acting on two closely co-orbiting satellites are highly correlated, and one satellite’s non-gravitational accelerations, sensing by accelerometer, can be transferred for the other satellite. NASA/DLR GRACE mission has been suffering from intermittent single accelerometer situations due to power limitation beyond its design life time. To overcome this situation, three estimation methods to predict one satellite’s non-gravitational acceleration using a weighted moving average of another satellite’s data are proposed Differential non-gravitational acceleration projection along velocity direction is used to evaluate the three methods with the GRACE flight data. If no bias adjustment is performed during preprocessing, one of the new methods shows an accuracy improvement over the time-shift method, which utilizes a single epoch data from the other satellite. With a bias adjustment, the time-shift method is preferred for its simplicity. The annual variations of the differential acceleration projection and estimation errors are analyzed using long-term GRACE flight data. The differential acceleration magnitude is closely related to solar activity, and therefore large estimation errors occur in the geomagnetic equator around noon, local time.

Description: This article first clearly figures out the relationship between parameters of timing group delay (TGD) and differential code bias (DCB) for BDS, and demonstrates the equivalence of TGD and DCB correction models combining theory with practice. The TGD/DCB correction models have been extended to various occasions for BDS positioning, and such models have been evaluated by real triple-frequency datasets. To test the effectiveness of broadcast TGDs in the navigation message and DCBs provided by the Multi-GNSS Experiment (MGEX), both standard point positioning (SPP) and precise point positioning (PPP) tests are carried out for BDS signals with different schemes. Furthermore, the influence of differential code biases on BDS positioning estimates such as coordinates, receiver clock biases, tropospheric delays and carrier phase ambiguities is investigated comprehensively. Comparative analysis show that the unmodeled differential code biases degrade the performance of BDS SPP by a factor of two or more, whereas the estimates of PPP are subject to varying degrees of influences. For SPP, the accuracy of dual-frequency combinations is slightly worse than that of single-frequency, and they are much more sensitive to the differential code biases, particularly for the B2B3 combination. For PPP, the uncorrected differential code biases are mostly absorbed into the receiver clock bias and carrier phase ambiguities and thus resulting in a much longer convergence time. Even though the influence of the differential code biases could be mitigated over time and comparable positioning accuracy could be achieved after convergence, it is suggested to properly handle with the differential code biases since it is vital for PPP convergence and integer ambiguity resolution.

Description: A drift mode accelerometer is a precision instrument for spacecraft that overcomes much of the acceleration noise and readout dynamic range limitations of traditional electrostatic accelerometers. It has the potential of achieving acceleration noise performance similar to that of drag-free systems over a restricted frequency band without the need for external drag-free control or continuous spacecraft propulsion. Like traditional accelerometers, the drift mode accelerometer contains a high-density test mass surrounded by an electrode housing, which can control and sense all six degrees of freedom of the test mass. Unlike traditional accelerometers, the suspension system is operated with a low duty cycle so that the limiting suspension force noise only acts over brief, known time intervals, which can be neglected in the data analysis. The readout is performed using a laser interferometer which is immune to the dynamic range limitations of even the best voltage references typically used to determine the inertial acceleration of electrostatic accelerometers. The drift mode accelerometer is a novel offshoot of the like-named operational mode of the LISA Pathfinder spacecraft, in which its test mass suspension system is cycled on and off to estimate the acceleration noise associated with the front-end electronics. This paper presents the concept of a drift mode accelerometer, describes the operation of such a device, develops models for its performance with respect to non-drag-free satellite geodesy and gravitational wave missions, and discusses plans for testing the performance of a prototype sensor in the laboratory using torsion pendula.

Description: Because of its geophysical interpretation, Earth’s polar motion excitation is generally decomposed into prograde (counter-clockwise) and retrograde (clockwise) circular terms at fixed frequency. Yet, these later are commonly considered as specific to the frequency and to the underlying geophysical process, and no study has raised the possibility that they could share features independent from frequency. Complex Fourier Transform permits to determine retrograde and prograde circular terms of the observed excitation and of its atmospheric, oceanic and hydrological counterparts. The total prograde and retrograde parts of these excitations are reconstructed in time domain. Then, complex linear correlation between retrograde and conjugate prograde parts is observed for both the geodetic excitation and the matter term of the hydro-atmospheric excitation. In frequency domain, the ratio of the retrograde circular terms with their corresponding conjugate prograde terms favours specific values: the amplitude ratio follows a probabilistic gamma distribution centred around 1.5 (maximum for 1), and the argument ratio obeys a distribution close to a normal law centred around \(2 \alpha = 160^{\circ }\) . These frequency and time domain characteristics mean an elliptical polarisation towards \(\alpha ={\sim } 80^{\circ }\) East with an ellipticity of 0.8, mostly resulting from the matter term of the hydro-atmospheric excitation. Whatsoever the frequency band above 0.4 cpd, the hydro-atmospheric matter term tends to be maximal in the geographic areas surrounding the great meridian circle of longitude \({\sim }80^{\circ }\) or \({\sim } 260^{\circ }\) East. The favoured retrograde/prograde amplitude ratio around 1.5 or equivalently the ellipticity of 0.8 can result from the amplification of pressure waves propagating towards the west by the normal atmospheric mode \(\Psi _3^1\) around 10 days.

Description: In this article, the realization of a global terrestrial reference system (TRS) based on a consistent combination of Global Navigation Satellite System (GNSS) and Satellite Laser Ranging (SLR) is studied. Our input data consists of normal equation systems from 17 years (1994–2010) of homogeneously reprocessed GPS, GLONASS and SLR data. This effort used common state of the art reduction models and the same processing software (Bernese GNSS Software) to ensure the highest consistency when combining GNSS and SLR. Residual surface load deformations are modeled with a spherical harmonic approach. The estimated degree-1 surface load coefficients have a strong annual signal for which the GNSS- and SLR-only solutions show very similar results. A combination including these coefficients reduces systematic uncertainties in comparison to the single-technique solution. In particular, uncertainties due to solar radiation pressure modeling in the coefficient time series can be reduced up to 50 % in the GNSS+SLR solution compared to the GNSS-only solution. In contrast to the ITRF2008 realization, no local ties are used to combine the different geodetic techniques. We combine the pole coordinates as global ties and apply minimum constraints to define the geodetic datum. We show that a common origin, scale and orientation can be reliably realized from our combination strategy in comparison to the ITRF2008.

Description: In 1884, the International Meridian Conference recommended that the prime meridian “to be employed as a common zero of longitude and standard of time-reckoning throughout the globe” pass through the “centre of the transit instrument at the Observatory of Greenwich”. Today, tourists visiting its meridian line must walk east approximately 102 m before their satellite-navigation receivers indicate zero longitude. This offset can be accounted for by the difference between astronomical and geodetic coordinates—deflection of the vertical—in the east–west direction at Greenwich, and the imposed condition of continuity in astronomical time. The coordinates of satellite-navigation receivers are provided in reference frames that are related to the geocentric reference frame introduced by the Bureau International de l’Heure (BIH) in 1984. This BIH Terrestrial System provided the basis for orientation of subsequent geocentric reference frames, including all realizations of the World Geodetic System 1984 and the International Terrestrial Reference Frame. Despite the lateral offset of the original and current zero-longitude lines at Greenwich, the orientation of the meridian plane used to measure Universal Time has remained essentially unchanged.

Description: This paper presents a fast and accurate algorithm for high-frequency trans-ionospheric path length determination. The algorithm is merely based on the solution of the Eikonal equation that is solved using the conformal theory of refraction. The main advantages of the algorithm are summarized as follows. First, the algorithm can determine the optical path length without iteratively adjusting both elevation and azimuth angles and, hence, the computational time can be reduced. Second, for the same elevation and azimuth angles, the algorithm can simultaneously determine the phase and group of both ordinary and extra-ordinary optical path lengths for different frequencies. Results from numerical simulations show that the computational time required by the proposed algorithm to accurately determine 8 different optical path lengths is almost 17 times faster than that required by a 3D ionospheric ray-tracing algorithm. It is found that the computational time to determine multiple optical path lengths is the same with that for determining a single optical path length. It is also found that the proposed algorithm is capable of determining the optical path lengths with millimeter level of accuracies, if the magnitude of the squared ratio of the plasma frequency to the transmitted frequency is less than \(1.33\times 10^{-3}\) , and hence the proposed algorithm is applicable for geodetic applications.

Description: In order to better understand the differential code biases (DCBs) of global navigation satellite system, the IGGDCB method is extended to estimate the intra- and inter-frequency biases of the global positioning system (GPS), GLONASS, BeiDou navigation satellite system (BDS), and Galileo based on observations collected by the multi-GNSS experiment (MGEX) of the international GNSS service (IGS). In the approach of IGGDCB, the local ionospheric total electronic content is modeled with generalized triangular series (GTS) function rather than using a global ionosphere model or a priori ionospheric information. The DCB estimated by the IGGDCB method is compared with the DCB products from the Center for Orbit Determination in Europe (CODE) and German Aerospace Center (DLR), as well as the broadcast timing group delay (TGD) parameters over a 2-year span (2013 and 2014). The results indicate that GPS and GLONASS intra-frequency biases obtained in this work show the same precision levels as those estimated by DLR (about 0.1 and 0.2–0.4 ns for the two constellations, respectively, with respect to the products of CODE). The precision levels of IGGDCB-based inter-frequency biases estimated over the 24-month period are about 0.29 ns for GPS, 0.56 ns for GLONASS, 0.36 ns for BDS, and 0.24 ns for Galileo, respectively. Here, the accuracies of GPS and GLONASS biases are assessed relative to the products of CODE, while those of BDS and Galileo are compared with the estimates of DLR. In addition, the monthly stability indices of IGGDCB-based DCBs are 0.11 (GPS), 0.18 (GLONASS), 0.17 (BDS), and 0.14 (Galileo) ns for the individual constellation.

Description: The Rosborough approach was developed in the 1980s for the modeling of orbit perturbations of altimeter satellites. It is a formalism that is rooted in the so-called time-wise approach, in which gravitational functionals are described as an along-orbit time series. Nevertheless, through a transformation of the orbital variables, the along-orbit functional can be mapped back onto the sphere. As such, the Rosborough formulation is a so-called space-wise approach at the same time. Both the conventional time-wise and the space-wise approaches have been improved and optimized over the past decade. When we explore the utility of the Rosborough approach in this contribution, we do not expect improved solutions for this particular GOCE-based case study. However, we aim to show the special characteristics of the Rosborough approach for processing the GOCE gradiometry data. In particular, we show that this approach can successfully deal with the problems that come with real data like bandwidth limitation and mispointing. Based on the first 71 days of the GOCE gravity gradients, we obtain solutions up till spherical harmonic degree 200. Compared to a high-quality gradiometric-only time-wise model, our solution shows a similar performance with just 8 cm geoid RMS difference in the relevant bandwidth. Moreover, relative contributions from the individual components are provided for the geographically mean gravity gradient components \(T_{xx}\) , \(T_{yy}\) and \(T_{zz}\) , and for the geographically variable gravity gradient components \(T_{xx}\) and \(T_{yy}\) . It is shown that the spatially variable components provide a direct access to understanding the mapping of time-variable error effects on the sphere. For instance, the known geomagnetic equator effect comes out clearly in the variable components of gravity gradients, as do track-specific errors. In conclusion, it is demonstrated that the Rosborough approach is a complementary method to the conventional approaches for GOCE data processing.

Description: Satellite altimeter sea surface height observations include the geocentric displacements caused by the pole tide, namely the response of the solid Earth and oceans to polar motion. Most users of these data remove these effects using a model that was developed more than 20 years ago. We describe two improvements to the pole tide model for satellite altimeter measurements. Firstly, we recommend an approach that improves the model for the response of the oceans by including the effects of self-gravitation, loading, and mass conservation. Our recommended approach also specifically includes the previously ignored displacement of the solid Earth due to the load of the ocean response, and includes the effects of geocenter motion. Altogether, this improvement amplifies the modeled geocentric pole tide by 15 %, or up to 2 mm of sea surface height displacement. We validate this improvement using two decades of satellite altimeter measurements. Secondly, we recommend that the altimetry pole tide model exclude geocentric sea surface displacements resulting from the long-term drift in polar motion. The response to this particular component of polar motion requires a more rigorous approach than is used by conventional models. We show that erroneously including the response to this component of polar motion in the pole tide model impacts interpretation of regional sea level rise by \(\pm \) 0.25 mm/year.

Description: Moving-base gravity gradiometry has proven to be a convenient method to determine the Earth’s gravity field. The ESA mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) has enabled to map the Earth gravity field and its gradients with a resolution of 80 km, leading to significant advances in physical oceanography and solid Earth physics. At smaller scales, airborne gravity gradiometry has been increasingly used during the past decade in mineral and hydrocarbon exploration. In both cases the sensitivity of gradiometers to the short wavelengths of the gravity field is of crucial interest. Here, we quantify and characterize the error on the gravity gradients estimated from measurements performed with a new instrument concept, called GREMLIT, for typical airborne conditions. GREMLIT is an ultra-sensitive planar gravitational gradiometer which consists in a planar acceleration gradiometer together with 3 gyroscopes. To conduct this error analysis, a simulation of a realistic airborne survey with GREMLIT is carried out. We first simulate realistic GREMLIT synthetic data, taking into account the acceleration gradiometer and gyroscope noises and biases and the variation of orientation of the measurement reference frame. Then, we estimate the gravity gradients from these data. Special attention is paid to the processing of the gyroscopes measurements whose accuracy is not commensurate with the ultra-sensitive gradiometer. We propose a method to calibrate the gyroscopes biases with a precision of the order \(10^{-8}\)  rad/s. In order to transform the tensor from the measurement frame to the local geodetic frame, we estimate the error induced when replacing the non-measured elements of the gravity gradient tensor by an a priori model. With the appropriate smoothing, we show that it is possible to achieve a precision better than 2E for an along-track spatial resolution of 2 km.

Description: GLONASS could hardly reach the positioning performance of GPS, especially for fast and real-time precise positioning. One of the reasons is the phase inter-frequency bias (IFB) at the receiver end prevents its integer ambiguity resolution. A number of studies were carried out to achieve the integer ambiguity resolution for GLONASS. Based on some of the revealed IFB characteristics, for instance IFB is a linear function of the received carrier frequency and L1 and L2 have the same IFB in unit of length, most of recent methods recommend estimating the IFB rate together with ambiguities. However, since the two sets of parameters are highly correlated, as demonstrated in previous studies, observations over several hours up to 1 day are needed even with simultaneous GPS observations to obtain a reasonable solution. Obviously, these approaches cannot be applied for real-time positioning. Actually, it can be demonstrated that GLONASS ambiguity resolution should also be available even for a single epoch if the IFB rate is precisely known. In addition, the closer the IFB rate value is to its true value, the larger the fixing RATIO will be. Based on this fact, in this paper, a new approach is developed to estimate the IFB rate by means of particle filtering with the likelihood function derived from RATIO. This approach is evaluated with several sets of experimental data. For both static and kinematic cases, the results show that IFB rates could be estimated precisely just with GLONASS data of a few epochs depending on the baseline length. The time cost with a normal PC can be controlled around 1 s and can be further reduced. With the estimated IFB rate, integer ambiguity resolution is available immediately and as a consequence, the positioning accuracy is improved significantly to the level of GPS fixed solution. Thus the new approach enables real-time precise applications of GLONASS.

Description: A new approach to determine a multi-point deformation of the earth’s surface or objects upon it, represented by point fields measured in two epochs, is presented. The problem of determining, which points have been deformed, is not approached by testing point-by-point, but by formulating alternative hypotheses that test if one, two or more subsets of points have been deformed, each subset in its own way. The method is based on the least squares connection adjustment, defines alternative hypotheses and searches the best one by testing a large amount of them. If the best hypothesis is found, a least squares estimation of the deformations is provided. The test results of the presented method are invariant under changes of the S-systems in which the point coordinates are defined. The results of a numerical test of the method applied to a simulated network are given. In designing a geodetic deformation network minimal detectable deformations can be computed, belonging to likely deformation patterns. The proposed method leads to a reconsideration of the duality of reference and object points. A comparison with the method of testing confidence ellipsoids is made. The relevance of the difference between geometric and physical interpretations of deformations and the consequences of the presented method for future developments are discussed.

Description: PPP-RTK extends the PPP concept by providing single-receiver users, next to orbits and clocks, also information about the satellite phase and code biases, thus enabling single-receiver ambiguity resolution. It is the goal of the present contribution to provide an analytical study of the quality of the PPP-RTK corrections as well as of their impact on the user ambiguity resolution performance. We consider the geometry-free and the geometry-based network derived corrections, as well as the impact of network ambiguity resolution on these corrections. Next to the insight that is provided by the analytical solutions, the closed form expressions of the variance matrices also demonstrate how the corrections depend on network parameters such as number of epochs, number of stations, number of satellites, and number of frequencies. As a result we are able to describe in a qualitative sense how the user ambiguity resolution performance is driven by the data from the different network scenarios.

Description: The characterization of the accuracy of ionospheric models currently used in global navigation satellite systems (GNSSs) is a long-standing issue. The characterization remains a challenging problem owing to the lack of sufficiently accurate slant ionospheric determinations to be used as a reference. The present study proposes a methodology based on the comparison of the predictions of any ionospheric model with actual unambiguous carrier-phase measurements from a global distribution of permanent receivers. The differences are separated as hardware delays (a receiver constant plus a satellite constant) per day. The present study was conducted for the entire year of 2014, i.e. during the last solar cycle maximum. The ionospheric models assessed are the operational models broadcast by the global positioning system (GPS) and Galileo constellations, the satellite-based augmentation system (SBAS) (i.e. European Geostationary Navigation Overlay System (EGNOS) and wide area augmentation system (WAAS)), a number of post-process global ionospheric maps (GIMs) from different International GNSS Service (IGS) analysis centres (ACs) and, finally, a more sophisticated GIM computed by the research group of Astronomy and GEomatics (gAGE). Ionospheric models based on GNSS data and represented on a grid (IGS GIMs or SBAS) correct about 85 % of the total slant ionospheric delay, whereas the models broadcasted in the navigation messages of GPS and Galileo only account for about 70 %. Our gAGE GIM is shown to correct 95 % of the delay. The proposed methodology appears to be a useful tool to improve current ionospheric models.

Description: With the advent of modernized GPS, triple-frequency phase measurements (L1, L2, and L5) are available for civil use. The successful ambiguity resolution of the integer ambiguities of the phase measurements will be the key to centimeter-level positioning. In order to achieve ambiguity resolution over long baselines, code measurements (pseudorange) are regularly incorporated with the phase measurements in the observation model. However, code multipath affects ambiguity resolution and thus completely eliminating the influence is an important issue. Therefore, the present study proposes an approach that uses only the phase measurements in the observation model. The proposed approach has three steps and focuses on resolving the integer ambiguities of the triple-frequency phase measurements. Simulation baseline data were processed by the proposed approach and the results show that the integer ambiguities of the phase measurements can be successfully resolved and that satellite geometry is an important factor for the phase-only ambiguity resolution performance. Real triple-frequency GPS data from currently available Block IIF satellites were also processed to demonstrate the feasibility of the proposed approach.

Description: The concept of integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) relies on appropriate network information for the parameters that are common between the single-receiver user that applies and the network that provides this information. Most of the current methods for PPP-RTK are based on forming the ionosphere-free combination using dual-frequency Global Navigation Satellite System (GNSS) observations. These methods are therefore restrictive in the light of the development of new multi-frequency GNSS constellations, as well as from the point of view that the PPP-RTK user requires ionospheric corrections to obtain integer ambiguity resolution results based on short observation time spans. The method for PPP-RTK that is presented in this article does not have above limitations as it is based on the undifferenced, uncombined GNSS observation equations, thereby keeping all parameters in the model. Working with the undifferenced observation equations implies that the models are rank-deficient; not all parameters are unbiasedly estimable, but only combinations of them. By application of \(\mathcal {S}\) -system theory the model is made of full rank by constraining a minimum set of parameters, or S-basis. The choice of this S-basis determines the estimability and the interpretation of the parameters that are transmitted to the PPP-RTK users. As this choice is not unique, one has to be very careful when comparing network solutions in different \(\mathcal {S}\) -systems; in that case the S-transformation, which is provided by the \(\mathcal {S}\) -system method, should be used to make the comparison. Knowing the estimability and interpretation of the parameters estimated by the network is shown to be crucial for a correct interpretation of the estimable PPP-RTK user parameters, among others the essential ambiguity parameters, which have the integer property which is clearly following from the interpretation of satellite phase biases from the network. The flexibility of the \(\mathcal {S}\) -system method is furthermore demonstrated by the fact that all models in this article are derived in multi-epoch mode, allowing to incorporate dynamic model constraints on all or subsets of parameters.

Description: Multipath is one major error source in high-accuracy GNSS positioning. Various hardware and software approaches are developed to mitigate the multipath effect. Among them the MHM (multipath hemispherical map) and sidereal filtering (SF)/advanced SF (ASF) approaches utilize the spatiotemporal repeatability of multipath effect under static environment, hence they can be implemented to generate multipath correction model for real-time GNSS data processing. We focus on the spatial–temporal repeatability-based MHM and SF/ASF approaches and compare their performances for multipath reduction. Comparisons indicate that both MHM and ASF approaches perform well with residual variance reduction (50 %) for short span (next 5 days) and maintains roughly 45 % reduction level for longer span (next 6–25 days). The ASF model is more suitable for high frequency multipath reduction, such as high-rate GNSS applications. The MHM model is easier to implement for real-time multipath mitigation when the overall multipath regime is medium to low frequency.

Description: In the geometrical optics approximation, the ionospheric part of error in measuring phase and code delays of the satellite signal may be represented as a rapidly decreasing series in inverse power of frequency. Such a simple frequency dependence allows us to use multi-frequency measurements for eliminating the error in such multi-frequency Global Navigation Satellite Systems as GPS, GLONASS, BeiDou, and Galileo. However, the elimination of errors is handicapped by diffraction effects during signal propagation through turbulent ionospheric plasma. The numerical simulation has shown that when using the spatial processing in the form of Fresnel inversion the transition from dual-frequency to triple-frequency measurements reduces the average error of measurement. Yet fluctuations of the error diminish only if the inner scale exceeds the Fresnel radius. In the opposite case of excess of the Fresnel radius over the inner scale, the random component of the residual error is growing during the transition to triple-frequency measurements. The numerical simulation results also suggest that the Fresnel spatial processing in dual-frequency measurements at the optimal distance to the virtual screen can reduce the average error from centimeter to submillimeter level, which renders the transition to triple-frequency measurements unnecessary. The study of the residual error dependence on the distance from the virtual screen to the observer has revealed that the optimum value of this distance may be found from the minimum condition of amplitude scintillation index of the processed signal. The signal thus processed may be utilized both in geodetic precise measurements and in diagnostics of the lower atmosphere.

Description: Low-latency high-rate ( \(〉\) 1 Hz) precise real-time kinematic (RTK) can be applied in high-speed scenarios such as aircraft automatic landing, precise agriculture and intelligent vehicle. The classic synchronous RTK (SRTK) precise differential GNSS (DGNSS) positioning technology, however, is not able to obtain a low-latency high-rate output for the rover receiver because of long data link transmission time delays (DLTTD) from the reference receiver. To overcome the long DLTTD, this paper proposes an asynchronous real-time kinematic (ARTK) method using asynchronous observations from two receivers. The asynchronous observation model (AOM) is developed based on undifferenced carrier phase observation equations of the two receivers at different epochs with short baseline. The ephemeris error and atmosphere delay are the possible main error sources on positioning accuracy in this model, and they are analyzed theoretically. In a short DLTTD and during a period of quiet ionosphere activity, the main error sources decreasing positioning accuracy are satellite orbital errors: the “inverted ephemeris error” and the integration of satellite velocity error which increase linearly along with DLTTD. The cycle slip of asynchronous double-differencing carrier phase is detected by TurboEdit method and repaired by the additional ambiguity parameter method. The AOM can deal with synchronous observation model (SOM) and achieve precise positioning solution with synchronous observations as well, since the SOM is only a specific case of AOM. The proposed method not only can reduce the cost of data collection and transmission, but can also support the mobile phone network data link transfer mode for the data of the reference receiver. This method can avoid data synchronizing process besides ambiguity initialization step, which is very convenient for real-time navigation of vehicles. The static and kinematic experiment results show that this method achieves 20 Hz or even higher rate output in real time. The ARTK positioning accuracy is better and more robust than the combination of phase difference over time (PDOT) and SRTK method at a high rate. The ARTK positioning accuracy is equivalent to SRTK solution when the DLTTD is 0.5 s, and centimeter level accuracy can be achieved even when DLTTD is 15 s.

Description: In the next generation VLBI network, the VLBI global observing system (VGOS), there will be several twin telescopes, i.e. stations equipped with a pair of VLBI telescopes with identical design. In this work we test the possibility of combining the tropospheric parameters of these two telescopes within the VLBI data analysis. This is done through simulations of a possible future VGOS network containing one twin telescope. We simulate the tropospheric delays with the help of a turbulence model, approximately taking into account the distance between the antennas. The results show that the combination of tropospheric delays can improve the station position repeatability by about 15 % as long as the distance is smaller than 1 km. The main improvement is in the repeatability of the baseline vector between the antennas. However, the results are strongly dependent on how the observations are scheduled for the twin telescope. The simulation results are confirmed by an analysis of the CONT14 campaign, where the tropospheric parameters of the two Hobart antennas are combined. Furthermore, we also discuss the study of combining other parameters for the twin telescope, i.e. the clocks and/or the station positions.

Description: The growing desire for better spatial and also temporal distribution of radio occultation data is a motivation for extensive researches considering either number of GNSS/receiver satellites or better optimization tools resulting in better distributions. This paper addresses the problem of designing a global positioning system-only radio occultation mission with the optimal performance in Asia Pacific region. Constellation Patterns are discussed and 2D-lattice and 3D-lattice flower constellations are adopted to develop a system with circular and elliptical orbits, respectively. A perturbed orbit propagation model leading to significantly more accurate pre-analysis is used. Emphasizing on the spatial and also temporal distribution of radio occultation events for the first time, distribution norm is provided as a volumetric distribution measure using Voronoi diagram concept in a 3D space consisting temporal and spatial intervals. Optimizations are performed using genetic algorithm to determine optimal constellation design parameters by the suitable fitness function and constraints devised. The resulted constellation has been evaluated by a regional comparison to the globally distributed FORMOSAT-3/COSMIC in terms of the distribution norm, number of radio occultation events and also coverage as an additional point-to-point distribution measure. Although it is demonstrated that the optimal 3D-lattice enjoys better performance than FORMOSAT-3, the design approach results in a 2D-lattice flower constellation which is superior to other constellations in regional emphasis of radio occultation events. Its global performance is discussed and it is demonstrated that using multi-GNSS receiver to increase satellites may not guarantee a good distribution of radio occultation data in some aspects.

Description: A new synthetic model of the time-variable global gravity field is now available based on realistic mass variability in atmosphere, oceans, terrestrial water storage, continental ice-sheets, and the solid Earth. The updated ESA Earth System Model is provided in Stokes coefficients up to degree and order 180 with a temporal resolution of 6 h covering the time period 1995–2006, and can be readily applied as a source model in future gravity mission simulation studies. The model contains plausible variability and trends in both low-degree coefficients and the global mean eustatic sea level. It depicts reasonable mass variability all over the globe at a wide range of frequencies including multi-year trends, year-to-year variability, and seasonal variability even at very fine spatial scales, which is important for a realistic representation of spatial aliasing and leakage. In particular on these small spatial scales between 50 and 250 km, the model contains a range of signals that have not been reliably observed yet by satellite gravimetry. In addition, the updated Earth System Model provides substantial high-frequency variability at periods down to a few hours only, thereby allowing to critically test strategies for the minimization of temporal aliasing.

Description: Star cameras (SCs) on board the GRACE satellites provide information about the attitudes of the spacecrafts. This information is needed to reduce the K-band ranging data to the centre of mass of the satellites. In this paper, we analyse GRACE SC errors using two months of real data of the primary and secondary SCs. We show that the errors consist of a harmonic component, which is highly correlated with the satellite’s true anomaly, and a stochastic component. We built models of both error components, and use these models for error propagation studies. Firstly, we analyse the propagation of SC errors into inter-satellite accelerations. A spectral analysis reveals that the stochastic component exceeds the harmonic component, except in the 3–10 mHz frequency band. In this band, which contains most of the geophysically relevant signal, the harmonic error component is larger than the random component. Secondly, we propagate SC errors into optimally filtered monthly mass anomaly maps and compare them with the total error. We found that SC errors account for about 18 % of the total error. Moreover, gaps in the SC data series amplify the effect of SC errors by a factor of \(5\) . Finally, an analysis of inter-satellite pointing angles for GRACE data between 2003 and 2010 reveals that inter-satellite ranging errors were exceptionally large during the period February 2003 till May 2003. During these months, SC noise is amplified by a factor of 3 and is a considerable source of errors in monthly GRACE mass anomaly maps. In the context of future satellite gravity missions, the noise models developed in this paper may be valuable for mission performance studies.

Description: We use a series of simulated scenarios to characterize the observability of geocenter location with GPS tracking data. We examine in particular the improvement realized when a GPS receiver in low Earth orbit (LEO) augments the ground network. Various orbital configurations for the LEO are considered and the observability of geocenter location based on GPS tracking is compared to that based on satellite laser ranging (SLR). The distance between a satellite and a ground tracking-site is the primary measurement, and Earth rotation plays important role in determining the geocenter location. Compared to SLR, which directly and unambiguously measures this distance, terrestrial GPS observations provide a weaker (relative) measurement for geocenter location determination. The estimation of GPS transmitter and receiver clock errors, which is equivalent to double differencing four simultaneous range measurements, removes much of this absolute distance information. We show that when ground GPS tracking data are augmented with precise measurements from a GPS receiver onboard a LEO satellite, the sensitivity of the data to geocenter location increases by more than a factor of two for Z-component. The geometric diversity underlying the varying baselines between the LEO and ground stations promotes improved global observability, and renders the GPS technique comparable to SLR in terms of information content for geocenter location determination. We assess a variety of LEO orbital configurations, including the proposed orbit for the geodetic reference antenna in space mission concept. The results suggest that a retrograde LEO with altitude near 3,000 km is favorable for geocenter determination.