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.     

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.     

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.     

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.     

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.     

On the feasibility of using satellite gravity observations for detecting large-scale solid mass transfer events

The main focus of this paper is to assess the feasibility of utilizing dedicated satellite gravity missions in order to detect large-scale solid mass transfer events (e.g. landslides). Specifically, a sensitivity analysis of Gravity Recovery and Climate Experiment (GRACE) gravity field solutions in conjunction with simulated case studies is employed to predict gravity changes due to past subaerial and submarine mass transfer events, namely the Agulhas slump in southeastern Africa and the Heart Mountain Landslide in northwestern Wyoming. The detectability of these events is evaluated by taking into account the expected noise level in the GRACE gravity field solutions and simulating their impact on the gravity field through forward modelling of the mass transfer. The spectral content of the estimated gravity changes induced by a simulated large-scale landslide event is estimated for the known spatial resolution of the GRACE observations using wavelet multiresolution analysis. The results indicate that both the Agulhas slump and the Heart Mountain Landslide could have been detected by GRACE, resulting in \({\vert }0.4{\vert }\) and \({\vert }0.18{\vert }\) mGal change on GRACE solutions, respectively. The suggested methodology is further extended to the case studies of the submarine landslide in Tohoku, Japan, and the Grand Banks landslide in Newfoundland, Canada. The detectability of these events using GRACE solutions is assessed through their impact on the gravity field.     

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.     

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.     

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.     

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.     

Performance evaluation of GNSS-TEC estimation techniques at the grid point in middle and low latitudes during different geomagnetic conditions

Global Navigation Satellite Systems (GNSS) have become a powerful tool use in surveying and mapping, air and maritime navigation, ionospheric/space weather research and other applications. However, in some cases, its maximum efficiency could not be attained due to some uncorrelated errors associated with the system measurements, which is caused mainly by the dispersive nature of the ionosphere. Ionosphere has been represented using the total number of electrons along the signal path at a particular height known as Total Electron Content (TEC). However, there are many methods to estimate TEC but the outputs are not uniform, which could be due to the peculiarity in characterizing the biases inside the observables (measurements), and sometimes could be associated to the influence of mapping function. The errors in TEC estimation could lead to wrong conclusion and this could be more critical in case of safety-of-life application. This work investigated the performance of Ciraolo’s and Gopi’s GNSS-TEC calibration techniques, during 5 geomagnetic quiet and disturbed conditions in the month of October 2013, at the grid points located in low and middle latitudes. The data used are obtained from the GNSS ground-based receivers located at Borriana in Spain (40\(^{\circ }\)N, 0\(^{\circ }\)E; mid latitude) and Accra in Ghana (5.50\(^{\circ }\)N, ?0.20\(^{\circ }\)E; low latitude). The results of the calibrated TEC are compared with the TEC obtained from European Geostationary Navigation Overlay System Processing Set (EGNOS PS) TEC algorithm, which is considered as a reference data. The TEC derived from Global Ionospheric Maps (GIM) through International GNSS service (IGS) was also examined at the same grid points. The results obtained in this work showed that Ciraolo’s calibration technique (a calibration technique based on carrier-phase measurements only) estimates TEC better at middle latitude in comparison to Gopi’s technique (a calibration technique based on code and carrier-phase measurements). At the same time, Gopi’s calibration was also found more reliable in low latitude than Ciraolo’s technique. In addition, the TEC derived from IGS GIM seems to be much reliable in middle-latitude than in low-latitude region.     

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.     

Contribution of solar radiation and geomagnetic activity to global structure of 27-day variation of ionosphere

Twenty-seven-day variation caused by solar rotation is one of the main periodic effects of solar radiation influence on the ionosphere, and there have been many studies on this periodicity using peak electron density \(\mathrm{N_{m}F_{2}}\) and solar radio flux index F10.7. In this paper, the global electron content (GEC) and observation of Solar EUV Monitor (SEM) represent the whole ionosphere and solar EUV flux, respectively, to investigate the 27-day variation. The 27-day period components of indices \((\hbox {GEC}_{27}\), \(\hbox {SEM}_{27}\), \(\hbox {F10.7}_{27}\), \(\hbox {Ap}_{27})\) are obtained using Chebyshev band-pass filter. The comparison of regression results indicates that the index SEM has higher coherence than F10.7 with 27-day variation of the ionosphere. The regression coefficients of \(\hbox {SEM}_{27 }\) varied from 0.6 to 1.4 and the coefficients of \(\hbox {Ap}_{27}\) varied from \({-}\)0.6 to 0.3, which suggests that EUV radiation seasonal variations are the primary driver for the 27-day variations of the ionosphere for most periods. TEC map grid points on three meridians where IGS stations are dense are selected for regression, and the results show that the contribution of solar EUV radiation is positive at all geomagnetic latitudes and larger than geomagnetic activity in most latitudes. The contribution of geomagnetic activity is negative at high geomagnetic latitude, increasing with decreasing geomagnetic latitudes, and positive at low geomagnetic latitudes. The global structure of 27-day variation of ionosphere is presented and demonstrates that there are two zonal anomaly regions along with the geomagnetic latitudes lines and two peaks in the north of Southeast Asia and the Middle Pacific where \(\hbox {TEC}_{27}\) magnitude values are notably larger than elsewhere along zonal anomaly regions.     

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.     

Geodetic methods to determine the relativistic redshift at the level of 10$$^{}$$ in the context of international timescales: a review and practical results

The frequency stability and uncertainty of the latest generation of optical atomic clocks is now approaching the one part in \(10^{18}\) level. Comparisons between earthbound clocks at rest must account for the relativistic redshift of the clock frequencies, which is proportional to the corresponding gravity (gravitational plus centrifugal) potential difference. For contributions to international timescales, the relativistic redshift correction must be computed with respect to a conventional zero potential value in order to be consistent with the definition of Terrestrial Time. To benefit fully from the uncertainty of the optical clocks, the gravity potential must be determined with an accuracy of about \(0.1\,\hbox {m}^{2}\,\hbox {s}^{-2}\), equivalent to about 0.01 m in height. This contribution focuses on the static part of the gravity field, assuming that temporal variations are accounted for separately by appropriate reductions. Two geodetic approaches are investigated for the derivation of gravity potential values: geometric levelling and the Global Navigation Satellite Systems (GNSS)/geoid approach. Geometric levelling gives potential differences with millimetre uncertainty over shorter distances (several kilometres), but is susceptible to systematic errors at the decimetre level over large distances. The GNSS/geoid approach gives absolute gravity potential values, but with an uncertainty corresponding to about 2 cm in height. For large distances, the GNSS/geoid approach should therefore be better than geometric levelling. This is demonstrated by the results from practical investigations related to three clock sites in Germany and one in France. The estimated uncertainty for the relativistic redshift correction at each site is about \(2 \times 10^{-18}\).     

Comparison between geodetic and oceanographic approaches to estimate mean dynamic topography for vertical datum unification: evaluation at Australian tide gauges

The direct method of vertical datum unification requires estimates of the ocean’s mean dynamic topography (MDT) at tide gauges, which can be sourced from either geodetic or oceanographic approaches. To assess the suitability of different types of MDT for this purpose, we evaluate 13 physics-based numerical ocean models and six MDTs computed from observed geodetic and/or ocean data at 32 tide gauges around the Australian coast. We focus on the viability of numerical ocean models for vertical datum unification, classifying the 13 ocean models used as either independent (do not contain assimilated geodetic data) or non-independent (do contain assimilated geodetic data). We find that the independent and non-independent ocean models deliver similar results. Maximum differences among ocean models and geodetic MDTs reach >150 mm at several Australian tide gauges and are considered anomalous at the 99% confidence level. These differences appear to be of geodetic origin, but without additional independent information, or formal error estimates for each model, some of these errors remain inseparable. Our results imply that some ocean models have standard deviations of differences with other MDTs (using geodetic and/or ocean observations) at Australian tide gauges, and with levelling between some Australian tide gauges, of \({\sim }\pm 50\,\hbox {mm}\). This indicates that they should be considered as an alternative to geodetic MDTs for the direct unification of vertical datums. They can also be used as diagnostics for errors in geodetic MDT in coastal zones, but the inseparability problem remains, where the error cannot be discriminated between the geoid model or altimeter-derived mean sea surface.     

VMF3/GPT3: refined discrete and empirical troposphere mapping functions

Incorrect modeling of troposphere delays is one of the major error sources for space geodetic techniques such as Global Navigation Satellite Systems (GNSS) or Very Long Baseline Interferometry (VLBI). Over the years, many approaches have been devised which aim at mapping the delay of radio waves from zenith direction down to the observed elevation angle, so-called mapping functions. This paper contains a new approach intended to refine the currently most important discrete mapping function, the Vienna Mapping Functions 1 (VMF1), which is successively referred to as Vienna Mapping Functions 3 (VMF3). It is designed in such a way as to eliminate shortcomings in the empirical coefficients b and c and in the tuning for the specific elevation angle of \(3^{\circ }\). Ray-traced delays of the ray-tracer RADIATE serve as the basis for the calculation of new mapping function coefficients. Comparisons of modeled slant delays demonstrate the ability of VMF3 to approximate the underlying ray-traced delays more accurately than VMF1 does, in particular at low elevation angles. In other words, when requiring highest precision, VMF3 is to be preferable to VMF1. Aside from revising the discrete form of mapping functions, we also present a new empirical model named Global Pressure and Temperature 3 (GPT3) on a \(5^{\circ }\times 5^{\circ }\) as well as a \(1^{\circ }\times 1^{\circ }\) global grid, which is generally based on the same data. Its main components are hydrostatic and wet empirical mapping function coefficients derived from special averaging techniques of the respective (discrete) VMF3 data. In addition, GPT3 also contains a set of meteorological quantities which are adopted as they stand from their predecessor, Global Pressure and Temperature 2 wet. Thus, GPT3 represents a very comprehensive troposphere model which can be used for a series of geodetic as well as meteorological and climatological purposes and is fully consistent with VMF3.     

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.     

Density interface topography recovered by inversion of satellite gravity gradiometry observations

A radial integration of spherical mass elements (i.e. tesseroids) is presented for evaluating the six components of the second-order gravity gradient (i.e. second derivatives of the Newtonian mass integral for the gravitational potential) created by an uneven spherical topography consisting of juxtaposed vertical prisms. The method uses Legendre polynomial series and takes elastic compensation of the topography by the Earth’s surface into account. The speed of computation of the polynomial series increases logically with the observing altitude from the source of anomaly. Such a forward modelling can be easily applied for reduction of observed gravity gradient anomalies by the effects of any spherical interface of density. An iterative least-squares inversion of measured gravity gradient coefficients is also proposed to estimate a regional set of juxtaposed topographic heights. Several tests of recovery have been made by considering simulated gradients created by idealistic conical and irregular Great Meteor seamount topographies, and for varying satellite altitudes and testing different levels of uncertainty. In the case of gravity gradients measured at a GOCE-type altitude of \(\sim \)300 km, the search converges down to a stable but smooth topography after 10–15 iterations, while the final root-mean-square error is \(\sim \)100 m that represents only 2 % of the seamount amplitude. This recovery error decreases with the altitude of the gravity gradient observations by revealing more topographic details in the region of survey.