Solution to the spectral filter problem of residual terrain modelling (RTM)

In physical geodesy, the residual terrain modelling (RTM) technique is frequently used for high-frequency gravity forward modelling. In the RTM technique, a detailed elevation model is high-pass-filtered in the topography domain, which is not equivalent to filtering in the gravity domain. This in-equivalence, denoted as spectral filter problem of the RTM technique, gives rise to two imperfections (errors). The first imperfection is unwanted low-frequency (LF) gravity signals, and the second imperfection is missing high-frequency (HF) signals in the forward-modelled RTM gravity signal. This paper presents new solutions to the RTM spectral filter problem. Our solutions are based on explicit modelling of the two imperfections via corrections. The HF correction is computed using spectral domain gravity forward modelling that delivers the HF gravity signal generated by the long-wavelength RTM reference topography. The LF correction is obtained from pre-computed global RTM gravity grids that are low-pass-filtered using surface or solid spherical harmonics. A numerical case study reveals maximum absolute signal strengths of \(\sim 44\) mGal (0.5 mGal RMS) for the HF correction and \(\sim 33\) mGal (0.6 mGal RMS) for the LF correction w.r.t. a degree-2160 reference topography within the data coverage of the SRTM topography model (\(56^{\circ }\hbox {S} \le \phi \le 60^{\circ }\hbox {N}\)). Application of the LF and HF corrections to pre-computed global gravity models (here the GGMplus gravity maps) demonstrates the efficiency of the new corrections over topographically rugged terrain. Over Switzerland, consideration of the HF and LF corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 4.41 to 3.27 mGal, which translates into \(\sim 26\)% improvement. Over a second test area (Canada), our corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 5.65 to 5.30 mGal (\(\sim 6\)% improvement). Particularly over Switzerland, geophysical signals (associated, e.g. with valley fillings) were found to stand out more clearly in the RTM-reduced gravity measurements when the HF and LF correction are taken into account. In summary, the new RTM filter corrections can be easily computed and applied to improve the spectral filter characteristics of the popular RTM approach. Benefits are expected, e.g. in the context of the development of future ultra-high-resolution global gravity models, smoothing of observed gravity data in mountainous terrain and geophysical interpretations of RTM-reduced gravity measurements.     

Precise and efficient evaluation of gravimetric quantities at arbitrarily scattered points in space

Gravimetric quantities are commonly represented in terms of high degree surface or solid spherical harmonics. After EGM2008, such expansions routinely extend to spherical harmonic degree 2190, which makes the computation of gravimetric quantities at a large number of arbitrarily scattered points in space using harmonic synthesis, a very computationally demanding process. We present here the development of an algorithm and its associated software for the efficient and precise evaluation of gravimetric quantities, represented in high degree solid spherical harmonics, at arbitrarily scattered points in the space exterior to the surface of the Earth. The new algorithm is based on representation of the quantities of interest in solid ellipsoidal harmonics and application of the tensor product trigonometric needlets. A FORTRAN implementation of this algorithm has been developed and extensively tested. The capabilities of the code are demonstrated using as examples the disturbing potential T, height anomaly \(\zeta \), gravity anomaly \(\Delta g\), gravity disturbance \(\delta g\), north–south deflection of the vertical \(\xi \), east–west deflection of the vertical \(\eta \), and the second radial derivative \(T_{rr}\) of the disturbing potential. After a pre-computational step that takes between 1 and 2 h per quantity, the current version of the software is capable of computing on a standard PC each of these quantities in the range from the surface of the Earth up to 544 km above that surface at speeds between 20,000 and 40,000 point evaluations per second, depending on the gravimetric quantity being evaluated, while the relative error does not exceed \(10^{-6}\) and the memory (RAM) use is 9.3 GB.     

The unexpected signal in GRACE estimates of $$C_{20}$$

For science applications of the gravity recovery and climate experiment (GRACE) monthly solutions, the GRACE estimates of \(C_{20}\) (or \(J_{2}\)) are typically replaced by the value determined from satellite laser ranging (SLR) due to an unexpectedly strong, clearly non-geophysical, variation at a period of \(\sim \)160 days. This signal has sometimes been referred to as a tide-like variation since the period is close to the perturbation period on the GRACE orbits due to the spherical harmonic coefficient pair \(C_{22}/S_{22}\) of S2 ocean tide. Errors in the S2 tide model used in GRACE data processing could produce a significant perturbation to the GRACE orbits, but it cannot contribute to the \(\sim \)160-day signal appearing in \(C_{20}\). Since the dominant contribution to the GRACE estimate of \(C_{20}\) is from the global positioning system tracking data, a time series of 138 monthly solutions up to degree and order 10 (\(10\times 10\)) were derived along with estimates of ocean tide parameters up to degree 6 for eight major tides. The results show that the \(\sim \)160-day signal remains in the \(C_{20}\) time series. Consequently, the anomalous signal in GRACE \(C_{20}\) cannot be attributed to aliasing from the errors in the S2 tide. A preliminary analysis of the cross-track forces acting on GRACE and the cross-track component of the accelerometer data suggests that a temperature-dependent systematic error in the accelerometer data could be a cause. Because a wide variety of science applications relies on the replacement values for \(C_{20}\), it is essential that the SLR estimates are as reliable as possible. An ongoing concern has been the influence of higher degree even zonal terms on the SLR estimates of \(C_{20}\), since only \(C_{20}\) and \(C_{40}\) are currently estimated. To investigate whether a better separation between \(C_{20}\) and the higher-degree terms could be achieved, several combinations of additional SLR satellites were investigated. In addition, a series of monthly gravity field solutions (\(60\times 60\)) were estimated from a combination of GRACE and SLR data. The results indicate that the combination of GRACE and SLR data might benefit the resonant orders in the GRACE-derived gravity fields, but it appears to degrade the recovery of the \(C_{20}\) variations. In fact, the results suggest that the poorer recovery of \(C_{40}\) by GRACE, where the annual variation is significantly underestimated, may be affecting the estimates of \(C_{20}\). Consequently, it appears appropriate to continue using the SLR-based estimates of \(C_{20}\), and possibly also \(C_{40}\), to augment the existing GRACE mission.     

A first modeling of dynamic and static crustal strain field from near-field dilatation measurements: example of the 2013 $$M_w$$ 6.2 Ruisui earthquake,Taiwan

We analyze the high-resolution dilatation data for the October 2013 \(M_w\) 6.2 Ruisui, Taiwan, earthquake, which occurred at a distance of 15–20 km away from a Sacks–Evertson dilatometer network. Based on well-constrained source parameters (\(\hbox {strike}=217^\circ \), \(\hbox {dip}=48^\circ \), \(\hbox {rake}=49^\circ \)), we propose a simple rupture model that explains the permanent static deformation and the dynamic vibrations at short period (\(\sim \)3.5–4.5 s) for most of the four sites with less than 20 % of discrepancies. This study represents a first attempt of modeling simultaneously the dynamic and static crustal strain using dilatation data. The results illustrate the potential for strain recordings of high-frequency seismic waves in the near-field of an earthquake to add constraints on the properties of seismic sources.     

The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates

We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional \({\sim }\)280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at \(1^{\prime \prime }\times 1^{\prime \prime }\) resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to \({\sim }100\) mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.     

Impact of the VLBA on reference frames and earth orientation studies

The geodetic VLBI community began using VLBA antennas in 1989 for geodesy and astrometry. We examine how usage of the VLBA has improved the celestial reference frame, the terrestrial reference frame, and Earth orientation parameters. Without the VLBA, ICRF2 would have had only 1011 sources instead of 3414. ICRF3 will contain at least 4121 sources, with approximately 70 % or more coming exclusively from VLBA astrometry and geodesy sessions. The terrestrial reference frame is also more stable and precise due to VLBA geodesy sessions. Approximately two dozen geodesy stations that have participated in VLBA sessions show average position formal errors that are \(\sim \)13–14 % better in the horizontal components and \(\sim \)5 % better in the vertical component than would be expected solely from the increased number of observations. Also the Earth orientation parameters obtained from the RDV sessions represent the most accurate EOP series of any of the long-term VLBI session types.     

Adding source positions to the IVS combination—First results

The consistent estimation of terrestrial reference frames (TRF), celestial reference frames (CRF) and Earth orientation parameters (EOP) is still an open subject and offers a large field of investigations. Until now, source positions resulting from Very Long Baseline Interferometry (VLBI) observations are not routinely combined on the level of normal equations in the same way as it is a common process for station coordinates and EOPs. The combination of source positions based on VLBI observations is now integrated in the IVS combination process. We present the studies carried out to evaluate the benefit of the combination compared to individual solutions. On the level of source time series, improved statistics regarding weighted root mean square have been found for the combination in comparison with the individual contributions. In total, 67 stations and 907 sources (including 291 ICRF2 defining sources) are included in the consistently generated CRF and TRF covering 30 years of VLBI contributions. The rotation angles \(A_1\), \(A_2\) and \(A_3\) relative to ICRF2 are ?12.7, 51.7 and 1.8 \({\upmu }\) as, the drifts \(D_\alpha \) and \(D_\delta \) are ?67.2 and 19.1 \(\upmu \) as/rad and the bias \(B_\delta \) is 26.1 \(\upmu \) as. The comparison of the TRF solution with the IVS routinely combined quarterly TRF solution shows no significant impact on the TRF, when the CRF is estimated consistently with the TRF. The root mean square value of the post-fit station coordinate residuals is 0.9 cm.     

Evaluation of gravitational curvatures of a tesseroid in spherical integral kernels

Proper understanding of how the Earth’s mass distributions and redistributions influence the Earth’s gravity field-related functionals is crucial for numerous applications in geodesy, geophysics and related geosciences. Calculations of the gravitational curvatures (GC) have been proposed in geodesy in recent years. In view of future satellite missions, the sixth-order developments of the gradients are becoming requisite. In this paper, a set of 3D integral GC formulas of a tesseroid mass body have been provided by spherical integral kernels in the spatial domain. Based on the Taylor series expansion approach, the numerical expressions of the 3D GC formulas are provided up to sixth order. Moreover, numerical experiments demonstrate the correctness of the 3D Taylor series approach for the GC formulas with order as high as sixth order. Analogous to other gravitational effects (e.g., gravitational potential, gravity vector, gravity gradient tensor), numerically it is found that there exist the very-near-area problem and polar singularity problem in the GC east–east–radial, north–north–radial and radial–radial–radial components in spatial domain, and compared to the other gravitational effects, the relative approximation errors of the GC components are larger due to not only the influence of the geocentric distance but also the influence of the latitude. This study shows that the magnitude of each term for the nonzero GC functionals by a grid resolution 15\(^{{\prime } }\,\times \) 15\(^{{\prime }}\) at GOCE satellite height can reach of about 10\(^{-16}\) m\(^{-1}\) s\(^{2}\) for zero order, 10\(^{-24 }\) or 10\(^{-23}\) m\(^{-1}\) s\(^{2}\) for second order, 10\(^{-29}\) m\(^{-1}\) s\(^{2}\) for fourth order and 10\(^{-35}\) or 10\(^{-34}\) m\(^{-1}\) s\(^{2}\) for sixth order, respectively.     

A clear link connecting the troposphere and ionosphere: ionospheric reponses to the 2015 Typhoon Dujuan

The global navigation satellite system (GNSS) total electron content (TEC) sequences were used to capture the arrival time and location of the ionosphere disturbances in response to the 2015 Typhoon Dujuan. After removing the de-trended TEC variation, the clear ionosphere disturbances on the typhoon landing day could be distinguished, and these disturbances disappeared from the TEC sequences before and after the typhoon landing day. The foF2 data observed by Xiamen ionosonde station also show ionosphere disturbances. Based on the advantages of GNSS multi-point observations, the disturbances horizontal velocity in the ionosphere were estimated according to the linear theory for a dispersion relation of acoustic gravity waves (AGWs) in an isothermal atmosphere. The average horizontal velocity (\(\sim \)240 m/s) and the radial velocity (\(\sim \)287 m/s) were used in the two-dimensional grid search for the origin point on the Earth’s surface. The origin area was determined to be on the eastern side of Taiwan. Lastly, a possible physical mechanism is discussed in this study. When typhoons land on Taiwan, the severe convective storms and the drag effect from the Central Mountains create an ideal location for development of AGWs. Topographic conditions, like the high lapse rate, contribute to the formation of AGWs, which then propagates into the ionosphere altitude.     

Joint analysis of celestial pole offset and free core nutation series

Three combined celestial pole offset (CPO) series computed at the Paris Observatory (C04), the United States Naval Observatory (USNO), and the International VLBI Service for Geodesy and Astrometry (IVS), as well as six free core nutation (FCN) models, were compared from different perspectives, such as stochastic and systematic differences, and FCN amplitude and phase variations. The differences between the C04 and IVS CPO series were mostly stochastic, whereas a low-frequency bias at the level of several tens of \(\upmu \)as was found between the C04 and USNO CPO series. The stochastic differences between the C04 and USNO series became considerably smaller when computed at the IVS epochs, which can indicate possible problems with the interpolation of the IVS data at the midnight epochs during the computation of the C04 and USNO series. The comparison of the FCN series showed that the series computed with similar window widths of 1.1–1.2 years were close to one another at a level of 10–20 \(\upmu \)as, whereas the differences between these series and the series computed with a larger window width of 4 and 7 years reached 100 \(\upmu \)as. The dependence of the FCN model on the underlying CPO series was investigated. The RMS differences between the FCN models derived from the C04, USNO, and IVS CPO series were at a level of approximately 15 \(\upmu \)as, which was considerably smaller than the differences among the CPO series. The analysis of the differences between the IVS, C04, and USNO CPO series suggested that the IVS series would be preferable for both precession-nutation and FCN-related studies.     

Calibration for the shear strain of 3-component borehole strainmeters in eastern Taiwan through Earth and ocean tidal waveform modeling

We propose an approach for calibrating the horizontal tidal shear components [(differential extension (\(\gamma _1\)) and engineering shear (\(\gamma _2\))] of two Sacks–Evertson (in Pap Meteorol Geophys 22:195–208, 1971) SES-3 borehole strainmeters installed in the Longitudinal Valley in eastern Taiwan. The method is based on the waveform reconstruction of the Earth and ocean tidal shear signals through linear regressions on strain gauge signals, with variable sensor azimuth. This method allows us to derive the orientation of the sensor without any initial constraints and to calibrate the shear strain components \(\gamma _1\) and \(\gamma _2\) against \(M_2\) tidal constituent. The results illustrate the potential of tensor strainmeters for recording horizontal tidal shear strain.     

A high-precision digital astrogeodetic traverse in an area of steep geoid gradients close to the coast of Perth,Western Australia

We present results from a new vertical deflection (VD) traverse observed in Perth, Western Australia, which is the first of its kind in the Southern Hemisphere. A digital astrogeodetic QDaedalus instrument was deployed to measure VDs with \({\sim }\)0.2\(''\) precision at 39 benchmarks with a \({{\sim }}1~\hbox {km}\) spacing. For the conversion of VDs to quasigeoid height differences, the method of astronomical–topographical levelling was applied, based on topographical information from the Shuttle Radar Topography Mission. The astronomical quasigeoid heights are in 20–30 mm (RMS) agreement with three independent gravimetric quasigeoid models, and the astrogeodetic VDs agree to 0.2–0.3\(''\) (north–south) and 0.6–0.9\(''\) (east–west) RMS. Tilt-like biases of \({\sim }1\,\,\hbox {mm}\) over \({\sim }1\,\,\hbox {km}\) are present for all quasigeoid models within \({\sim }20\,\,\hbox {km}\) of the coastline, suggesting inconsistencies in the coastal zone gravity data. The VD campaign in Perth was designed as a low-cost effort, possibly allowing replication in other Southern Hemisphere countries (e.g., Asia, Africa, South America and Antarctica), where VD data are particularly scarce.     

Contributions to reference systems from Lunar Laser Ranging using the IfE analysis model

Lunar Laser Ranging (LLR) provides various quantities related to reference frames like Earth orientation parameters, coordinates and velocities of ground stations in the Earth-fixed frame and selenocentric coordinates of the lunar retro-reflectors. This paper presents the recent results from LLR data analysis at the Institut für Erdmessung, Leibniz Universität Hannover, based on all LLR data up to the end of 2016. The estimates of long-periodic nutation coefficients with periods between 13.6 days and 18.6 years are obtained with an accuracy in the order of 0.05–0.7 milliarcseconds (mas). Estimations of the Earth rotation phase \(\Delta \)UT are accurate at the level of 0.032 ms if more than 14 normal points per night are included. The tie between the dynamical ephemeris frame to the kinematic celestial frame is estimated from pure LLR observations by two angles and their rates with an accuracy of 0.25 and 0.02 mas per year. The estimated station coordinates and velocities are compared to the ITRF2014 solution and the geometry of the retro-reflector network with the DE430 solution. The given accuracies represent 3 times formal errors of the parameter fit. The accuracy for \(\Delta \)UT is based on the standard deviation of the estimates with respect to the reference C04 solution.     

Slope correction for ocean radar altimetry

We develop a slope correction model to improve the accuracy of mean sea surface topography models as well as marine gravity models. The correction is greatest above ocean trenches and large seamounts where the slope of the geoid exceeds 100  \(\upmu \) rad. In extreme cases, the correction to the mean sea surface height is 40 mm and the correction to the along-track altimeter slope is 1–2  \(\upmu \) rad which maps into a 1–2 mGal gravity error. Both corrections are easily applied using existing grids of sea surface slope from satellite altimetry.     

Seasonal low-degree changes in terrestrial water mass load from global GNSS measurements

Large-scale mass redistribution in the terrestrial water storage (TWS) leads to changes in the low-degree spherical harmonic coefficients of the Earth’s surface mass density field. Studying these low-degree fluctuations is an important task that contributes to our understanding of continental hydrology. In this study, we use global GNSS measurements of vertical and horizontal crustal displacements that we correct for atmospheric and oceanic effects, and use a set of modified basis functions similar to Clarke et al. (Geophys J Int 171:1–10, 2007) to perform an inversion of the corrected measurements in order to recover changes in the coefficients of degree-0 (hydrological mass change), degree-1 (centre of mass shift) and degree-2 (flattening of the Earth) caused by variations in the TWS over the period January 2003–January 2015. We infer from the GNSS-derived degree-0 estimate an annual variation in total continental water mass with an amplitude of \((3.49 \pm 0.19) \times 10^{3}\) Gt and a phase of \(70^{\circ } \pm 3^{\circ }\) (implying a peak in early March), in excellent agreement with corresponding values derived from the Global Land Data Assimilation System (GLDAS) water storage model that amount to \((3.39 \pm 0.10) \times 10^{3}\) Gt and \(71^{\circ } \pm 2^{\circ }\), respectively. The degree-1 coefficients we recover from GNSS predict annual geocentre motion (i.e. the offset change between the centre of common mass and the centre of figure) caused by changes in TWS with amplitudes of \(0.69 \pm 0.07\) mm for GX, \(1.31 \pm 0.08\) mm for GY and \(2.60 \pm 0.13\) mm for GZ. These values agree with GLDAS and estimates obtained from the combination of GRACE and the output of an ocean model using the approach of Swenson et al. (J Geophys Res 113(B8), 2008) at the level of about 0.5, 0.3 and 0.9 mm for GX, GY and GZ, respectively. Corresponding degree-1 coefficients from SLR, however, generally show higher variability and predict larger amplitudes for GX and GZ. The results we obtain for the degree-2 coefficients from GNSS are slightly mixed, and the level of agreement with the other sources heavily depends on the individual coefficient being investigated. The best agreement is observed for \(T_{20}^C\) and \(T_{22}^S\), which contain the most prominent annual signals among the degree-2 coefficients, with amplitudes amounting to \((5.47 \pm 0.44) \times 10^{-3}\) and \((4.52 \pm 0.31) \times 10^{-3}\) m of equivalent water height (EWH), respectively, as inferred from GNSS. Corresponding agreement with values from SLR and GRACE is at the level of or better than \(0.4 \times 10^{-3}\) and \(0.9 \times 10^{-3}\) m of EWH for \(T_{20}^C\) and \(T_{22}^S\), respectively, while for both coefficients, GLDAS predicts smaller amplitudes. Somewhat lower agreement is obtained for the order-1 coefficients, \(T_{21}^C\) and \(T_{21}^S\), while our GNSS inversion seems unable to reliably recover \(T_{22}^C\). For all the coefficients we consider, the GNSS-derived estimates from the modified inversion approach are more consistent with the solutions from the other sources than corresponding estimates obtained from an unconstrained standard inversion.     

The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor

Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components \(V_{xy}\) and \(V_{yz}\) of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of \(96.7^{\circ }\). With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of \(V_{xy}\) and \(V_{yz}\) are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.     

Model improvements and validation of TerraSAR-X precise orbit determination

The radar imaging satellite mission TerraSAR-X requires precisely determined satellite orbits for validating geodetic remote sensing techniques. Since the achieved quality of the operationally derived, reduced-dynamic (RD) orbit solutions limits the capabilities of the synthetic aperture radar (SAR) validation, an effort is made to improve the estimated orbit solutions. This paper discusses the benefits of refined dynamical models on orbit accuracy as well as estimated empirical accelerations and compares different dynamic models in a RD orbit determination. Modeling aspects discussed in the paper include the use of a macro-model for drag and radiation pressure computation, the use of high-quality atmospheric density and wind models as well as the benefit of high-fidelity gravity and ocean tide models. The Sun-synchronous dusk–dawn orbit geometry of TerraSAR-X results in a particular high correlation of solar radiation pressure modeling and estimated normal-direction positions. Furthermore, this mission offers a unique suite of independent sensors for orbit validation. Several parameters serve as quality indicators for the estimated satellite orbit solutions. These include the magnitude of the estimated empirical accelerations, satellite laser ranging (SLR) residuals, and SLR-based orbit corrections. Moreover, the radargrammetric distance measurements of the SAR instrument are selected for assessing the quality of the orbit solutions and compared to the SLR analysis. The use of high-fidelity satellite dynamics models in the RD approach is shown to clearly improve the orbit quality compared to simplified models and loosely constrained empirical accelerations. The estimated empirical accelerations are substantially reduced by 30% in tangential direction when working with the refined dynamical models. Likewise the SLR residuals are reduced from \(-3\,\pm \,17\) to \(2\,\pm \,13\) mm, and the SLR-derived normal-direction position corrections are reduced from 15 to 6 mm, obtained from the 2012–2014 period. The radar range bias is reduced from \(-10.3\) to \(-6.1\) mm with the updated orbit solutions, which coincides with the reduced standard deviation of the SLR residuals. The improvements are mainly driven by the satellite macro-model for the purpose of solar radiation pressure modeling, improved atmospheric density models, and the use of state-of-the-art gravity field models.     

On the consistency of the current conventional EOP series and the celestial and terrestrial reference frames

Precise transformation between the celestial reference frames (CRF) and terrestrial reference frames (TRF) is needed for many purposes in Earth and space sciences. According to the Global Geodetic Observing System (GGOS) recommendations, the accuracy of positions and stability of reference frames should reach 1 mm and 0.1 mm year\(^{-1}\), and thus, the Earth Orientation Parameters (EOP) should be estimated with similar accuracy. Different realizations of TRFs, based on the combination of solutions from four different space geodetic techniques, and CRFs, based on a single technique only (VLBI, Very Long Baseline Interferometry), might cause a slow degradation of the consistency among EOP, CRFs, and TRFs (e.g., because of differences in geometry, orientation and scale) and a misalignment of the current conventional EOP series, IERS 08 C04. We empirically assess the consistency among the conventional reference frames and EOP by analyzing the record of VLBI sessions since 1990 with varied settings to reflect the impact of changing frames or other processing strategies on the EOP estimates. Our tests show that the EOP estimates are insensitive to CRF changes, but sensitive to TRF variations and unmodeled geophysical signals at the GGOS level. The differences between the conventional IERS 08 C04 and other EOP series computed with distinct TRF settings exhibit biases and even non-negligible trends in the cases where no differential rotations should appear, e.g., a drift of about 20 \(\upmu \)as year\(^{-1 }\)in \(y_{\mathrm{pol }}\) when the VLBI-only frame VTRF2008 is used. Likewise, different strategies on station position modeling originate scatters larger than 150 \(\upmu \)as in the terrestrial pole coordinates.     

An a priori solar radiation pressure model for the QZSS Michibiki satellite

It has been noted that the satellite laser ranging (SLR) residuals of the Quasi-Zenith Satellite System (QZSS) Michibiki satellite orbits show very marked dependence on the elevation angle of the Sun above the orbital plane (i.e., the \(\beta \) angle). It is well recognized that the systematic error is caused by mismodeling of the solar radiation pressure (SRP). Although the error can be reduced by the updated ECOM SRP model, the orbit error is still very large when the satellite switches to orbit-normal (ON) orientation. In this study, an a priori SRP model was established for the QZSS Michibiki satellite to enhance the ECOM model. This model is expressed in ECOM’s D, Y, and B axes (DYB) using seven parameters for the yaw-steering (YS) mode, and additional three parameters are used to compensate the remaining modeling deficiencies, particularly the perturbations in the Y axis, based on a redefined DYB for the ON mode. With the proposed a priori model, QZSS Michibiki’s precise orbits over 21 months were determined. SLR validation indicated that the systematic \(\beta \)-angle-dependent error was reduced when the satellite was in the YS mode, and better than an 8-cm root mean square (RMS) was achieved. More importantly, the orbit quality was also improved significantly when the satellite was in the ON mode. Relative to ECOM and adjustable box-wing model, the proposed SRP model showed the best performance in the ON mode, and the RMS of the SLR residuals was better than 15 cm, which was a two times improvement over the ECOM without a priori model used, but was still two times worse than the YS mode.     

Design and validation of broadcast ephemeris for low Earth orbit satellites

Low Earth orbit (LEO) constellations have potentialities to augment global navigation satellite systems for better service performance. The prerequisite is to provide the broadcast ephemerides that meet the accuracy requirement for navigation and positioning. In this study, the Kepler ephemeris model is chosen as the basis of LEO broadcast ephemeris design for backward compatibility and simplicity. To eliminate the singularity caused by the smaller eccentricity of LEO satellites compared to MEO satellites, non-singular elements are introduced for curve fitting of parameters and then transformed to Kepler elements to assure the algorithm of ephemeris computation remains unchanged for the user. We analyze the variation characteristics of LEO orbital elements and establish suitable broadcast ephemeris models considering fit accuracy, number of parameters, fit interval, and orbital altitude. The results of the fit accuracy for different fit intervals and orbital altitudes suggest that the optimal parameter selections are \((Crs3,Crc3)\), \((Crs3,Crc3, \, \dot{a},\dot{n})\) and \(\left( {Crs3,Crc3, \, \dot{a},\dot{n}, \, \ddot{i},\ddot{a}} \right)\), i.e., adding two, four or six parameters to the GPS 16-parameter ephemeris. When adding four parameters, the fit accuracy can be improved by about one order of magnitude compared to the GPS 16-parameter ephemeris model, and fit errors of less than 10 cm can be achieved with 20-min fit interval for a 400–1400 km orbital altitude. In addition, the effects of the number of parameters, fit interval, and orbit altitude on fit accuracy are discussed in detail. The validation with four LEO satellites in orbit also confirms the effectiveness of proposed models.