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.     

Validation of Galileo orbits using SLR with a focus on satellites launched into incorrect orbital planes

The space segment of the European Global Navigation Satellite System (GNSS) Galileo consists of In-Orbit Validation (IOV) and Full Operational Capability (FOC) spacecraft. The first pair of FOC satellites was launched into an incorrect, highly eccentric orbital plane with a lower than nominal inclination angle. All Galileo satellites are equipped with satellite laser ranging (SLR) retroreflectors which allow, for example, for the assessment of the orbit quality or for the SLR–GNSS co-location in space. The number of SLR observations to Galileo satellites has been continuously increasing thanks to a series of intensive campaigns devoted to SLR tracking of GNSS satellites initiated by the International Laser Ranging Service. This paper assesses systematic effects and quality of Galileo orbits using SLR data with a main focus on Galileo satellites launched into incorrect orbits. We compare the SLR observations with respect to microwave-based Galileo orbits generated by the Center for Orbit Determination in Europe (CODE) in the framework of the International GNSS Service Multi-GNSS Experiment for the period 2014.0–2016.5. We analyze the SLR signature effect, which is characterized by the dependency of SLR residuals with respect to various incidence angles of laser beams for stations equipped with single-photon and multi-photon detectors. Surprisingly, the CODE orbit quality of satellites in the incorrect orbital planes is not worse than that of nominal FOC and IOV orbits. The RMS of SLR residuals is even lower by 5.0 and 1.5 mm for satellites in the incorrect orbital planes than for FOC and IOV satellites, respectively. The mean SLR offsets equal \(-44.9, -35.0\), and \(-22.4\) mm for IOV, FOC, and satellites in the incorrect orbital plane. Finally, we found that the empirical orbit models, which were originally designed for precise orbit determination of GNSS satellites in circular orbits, provide fully appropriate results also for highly eccentric orbits with variable linear and angular velocities.     

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.     

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.     

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.     

Homogeneous reprocessing of GPS,GLONASS and SLR observations

The International GNSS Service (IGS) provides operational products for the GPS and GLONASS constellation. Homogeneously processed time series of parameters from the IGS are only available for GPS. Reprocessed GLONASS series are provided only by individual Analysis Centers (i. e. CODE and ESA), making it difficult to fully include the GLONASS system into a rigorous GNSS analysis. In view of the increasing number of active GLONASS satellites and a steadily growing number of GPS+GLONASS-tracking stations available over the past few years, Technische Universität Dresden, Technische Universität München, Universität Bern and Eidgenössische Technische Hochschule Zürich performed a combined reprocessing of GPS and GLONASS observations. Also, SLR observations to GPS and GLONASS are included in this reprocessing effort. Here, we show only SLR results from a GNSS orbit validation. In total, 18 years of data (1994–2011) have been processed from altogether 340 GNSS and 70 SLR stations. The use of GLONASS observations in addition to GPS has no impact on the estimated linear terrestrial reference frame parameters. However, daily station positions show an RMS reduction of 0.3 mm on average for the height component when additional GLONASS observations can be used for the time series determination. Analyzing satellite orbit overlaps, the rigorous combination of GPS and GLONASS neither improves nor degrades the GPS orbit precision. For GLONASS, however, the quality of the microwave-derived GLONASS orbits improves due to the combination. These findings are confirmed using independent SLR observations for a GNSS orbit validation. In comparison to previous studies, mean SLR biases for satellites GPS-35 and GPS-36 could be reduced in magnitude from \(-35\) and \(-38\)  mm to \(-12\) and \(-13\)  mm, respectively. Our results show that remaining SLR biases depend on the satellite type and the use of coated or uncoated retro-reflectors. For Earth rotation parameters, the increasing number of GLONASS satellites and tracking stations over the past few years leads to differences between GPS-only and GPS+GLONASS combined solutions which are most pronounced in the pole rate estimates with maximum 0.2 mas/day in magnitude. At the same time, the difference between GLONASS-only and combined solutions decreases. Derived GNSS orbits are used to estimate combined GPS+GLONASS satellite clocks, with first results presented in this paper. Phase observation residuals from a precise point positioning are at the level of 2 mm and particularly reveal poorly modeled yaw maneuver periods.     

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.     

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.     

Contribution of Starlette,Stella, and AJISAI to the SLR-derived global reference frame

The contribution of Starlette, Stella, and AJISAI is currently neglected when defining the International Terrestrial Reference Frame, despite a long time series of precise SLR observations and a huge amount of available data. The inferior accuracy of the orbits of low orbiting geodetic satellites is the main reason for this neglect. The Analysis Centers of the International Laser Ranging Service (ILRS ACs) do, however, consider including low orbiting geodetic satellites for deriving the standard ILRS products based on LAGEOS and Etalon satellites, instead of the sparsely observed, and thus, virtually negligible Etalons. We process ten years of SLR observations to Starlette, Stella, AJISAI, and LAGEOS and we assess the impact of these Low Earth Orbiting (LEO) SLR satellites on the SLR-derived parameters. We study different orbit parameterizations, in particular different arc lengths and the impact of pseudo-stochastic pulses and dynamical orbit parameters on the quality of the solutions. We found that the repeatability of the East and North components of station coordinates, the quality of polar coordinates, and the scale estimates of the reference are improved when combining LAGEOS with low orbiting SLR satellites. In the multi-SLR solutions, the scale and the \(Z\) component of geocenter coordinates are less affected by deficiencies in solar radiation pressure modeling than in the LAGEOS-1/2 solutions, due to substantially reduced correlations between the \(Z\) geocenter coordinate and empirical orbit parameters. Eventually, we found that the standard values of Center-of-mass corrections (CoM) for geodetic LEO satellites are not valid for the currently operating SLR systems. The variations of station-dependent differential range biases reach 52 and 25 mm for AJISAI and Starlette/Stella, respectively, which is why estimating station-dependent range biases or using station-dependent CoM, instead of one value for all SLR stations, is strongly recommended. This clearly indicates that the ILRS effort to produce CoM corrections for each satellite, which are site-specific and depend on the system characteristics at the time of tracking, is very important and needs to be implemented in the SLR data analysis.     

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.     

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.     

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.     

A neural network model for predicting weighted mean temperature

Water vapor is an important element of the Earth’s atmosphere, and most of it concentrates at the bottom of the troposphere. Knowledge of the water vapor measured by Global Navigation Satellite Systems (GNSS) is an important direction of GNSS research. In particular, when the zenith wet delay is converted to precipitable water vapor, the weighted mean temperature \(T_\mathrm{m}\) is a variable parameter to be determined in this conversion. The purpose of the study is getting a more accurate \(T_\mathrm{m}\) model for global users by a combination of two different characteristics of \(T_\mathrm{m}\) (i.e., the \(T_\mathrm{m}\) seasonal variations and the relationships between \(T_\mathrm{m}\) and surface meteorological elements). The modeling process was carried out by using the neural network technology. A multilayer feedforward neural network model (the NN) was established. The NN model is used with measurements of only surface temperature \(T_\mathrm{S}\). The NN was validated and compared with four other published global \(T_\mathrm{m}\) models. The results show that the NN performed better than any of the four compared models on the global scale.     

Consistent realization of Celestial and Terrestrial Reference Frames

The Celestial Reference System (CRS) is currently realized only by Very Long Baseline Interferometry (VLBI) because it is the space geodetic technique that enables observations in that frame. In contrast, the Terrestrial Reference System (TRS) is realized by means of the combination of four space geodetic techniques: Global Navigation Satellite System (GNSS), VLBI, Satellite Laser Ranging (SLR), and Doppler Orbitography and Radiopositioning Integrated by Satellite. The Earth orientation parameters (EOP) are the link between the two types of systems, CRS and TRS. The EOP series of the International Earth Rotation and Reference Systems Service were combined of specifically selected series from various analysis centers. Other EOP series were generated by a simultaneous estimation together with the TRF while the CRF was fixed. Those computation approaches entail inherent inconsistencies between TRF, EOP, and CRF, also because the input data sets are different. A combined normal equation (NEQ) system, which consists of all the parameters, i.e., TRF, EOP, and CRF, would overcome such an inconsistency. In this paper, we simultaneously estimate TRF, EOP, and CRF from an inter-technique combined NEQ using the latest GNSS, VLBI, and SLR data (2005–2015). The results show that the selection of local ties is most critical to the TRF. The combination of pole coordinates is beneficial for the CRF, whereas the combination of \(\varDelta \hbox {UT1}\) results in clear rotations of the estimated CRF. However, the standard deviations of the EOP and the CRF improve by the inter-technique combination which indicates the benefits of a common estimation of all parameters. It became evident that the common determination of TRF, EOP, and CRF systematically influences future ICRF computations at the level of several \(\upmu \)as. Moreover, the CRF is influenced by up to \(50~\upmu \)as if the station coordinates and EOP are dominated by the satellite techniques.     

Self-consistent treatment of tidal variations in the geocenter for precise orbit determination

We show that the current levels of accuracy being achieved for the precise orbit determination (POD) of low-Earth orbiters demonstrate the need for the self-consistent treatment of tidal variations in the geocenter. Our study uses as an example the POD of the OSTM/Jason-2 satellite altimeter mission based upon Global Positioning System (GPS) tracking data. Current GPS-based POD solutions are demonstrating root-mean-square (RMS) radial orbit accuracy and precision of \({<}1\)  cm and 1 mm, respectively. Meanwhile, we show that the RMS of three-dimensional tidal geocenter variations is \({<}6\)  mm, but can be as large as 15 mm, with the largest component along the Earth’s spin axis. Our results demonstrate that GPS-based POD of Earth orbiters is best performed using GPS satellite orbit positions that are defined in a reference frame whose origin is at the center of mass of the entire Earth system, including the ocean tides. Errors in the GPS-based POD solutions for OSTM/Jason-2 of \({<}4\)  mm (3D RMS) and \({<}2\)  mm (radial RMS) are introduced when tidal geocenter variations are not treated consistently. Nevertheless, inconsistent treatment is measurable in the OSTM/Jason-2 POD solutions and manifests through degraded post-fit tracking data residuals, orbit precision, and relative orbit accuracy. For the latter metric, sea surface height crossover variance is higher by \(6~\hbox {mm}^{2}\) when tidal geocenter variations are treated inconsistently.     

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.     

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.     

Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by \(90^{\circ }\) such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are \(4 \pi \) fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271–285, 2012a). As an example, we obtained \(2190\times 2190\) coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.     

Impact of ambient temperature on spring-based relative gravimeter measurements

In this paper, we investigate the impact of ambient temperature changes on the gravity reading of spring-based relative gravimeters. Controlled heating experiments using two Scintrex CG5 gravimeters allowed us to determine a linear correlation (R \(^{2}>\) 0.9) between ambient temperature and gravity variations. The relation is stable and constant for the two CG5 we used: ?5 nm/s\(^{2}/^\circ \)C. A linear relation is also seen between gravity and residual sensor temperature variations (R \(^{2}>\) 0.75), but contrary to ambient temperature, this relation is neither constant over time nor similar between the two instruments. The linear correction of ambient temperature on the controlled heating time series reduced the standard deviation at least by a factor of 2, to less than 10 nm/s\(^{2}\). The laboratory results allowed for reprocessing the data gathered on a field survey that originally aimed to characterize local hydrological heterogeneities on a karstic area. The correction of two years of monthly CG5 measurements from ambient temperature variations halved the standard deviation (from 62 to 32 nm/s\(^{2}\)) and led us to a better hydrological interpretation. Although the origin of this effect is uncertain, we suggest that an imperfect control of the sensor temperature may be involved, as well as a change of the properties of an electronic component.     

Gravity field error analysis for pendulum formations by a semi-analytical approach

Many geoscience disciplines push for ever higher requirements on accuracy, homogeneity and time- and space-resolution of the Earth’s gravity field. Apart from better instruments or new observables, alternative satellite formations could improve the signal and error structure compared to Grace. One possibility to increase the sensitivity and isotropy by adding cross-track information is a pair of satellites flying in a pendulum formation. This formation contains two satellites which have different ascending nodes and arguments of latitude, but have the same orbital height and inclination. In this study, the semi-analytical approach for efficient pre-mission error assessment is presented, and the transfer coefficients of range, range-rate and range-acceleration gravitational perturbations are derived analytically for the pendulum formation considering a set of opening angles. The new challenge is the time variations of the opening angle and the range, leading to temporally variable transfer coefficients. This is solved by Fourier expansion of the sine/cosine of the opening angle and the central angle. The transfer coefficients are further applied to assess the error patterns which are caused by different orbital parameters. The simulation results indicate that a significant improvement in accuracy and isotropy is obtained for small and medium initial opening angles of single polar pendulums, compared to Grace. The optimal initial opening angles are \(45^\circ \) and \(15^\circ \) for accuracy and isotropy, respectively. For a Bender configuration, which is constituted by a polar Grace and an inclined pendulum in this paper, the behaviour of results is dependent on the inclination (prograde vs. retrograde) and on the relative baseline orientation (left or right leading). The simulation for a sun-synchronous orbit shows better results for the left leading case.