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.     

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.     

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.     

Earth orientation parameters estimated from VLBI during the CONT11 campaign

In this paper we investigate the accuracy of the earth orientation parameters (EOP) estimated from the continuous VLBI campaign CONT11. We first estimated EOP with daily resolution and compared these to EOP estimated from GNSS data. We find that the WRMS differences are about 31  $\upmu $ as for polar motion and 7  $\upmu $ s for length of day. This is about the precision we could expect, based on Monte Carlo simulations and the results of the previous CONT campaigns. We also estimated EOP with hourly resolution to study the sub-diurnal variations. The results confirm the results of previous studies, showing that the current IERS model for high-frequency EOP variations does not explain all the sub-diurnal variations seen in the estimated time series. We then compared our results to various empirical high-frequency EOP models. However, we did not find that any of these gave any unambiguous improvement. Several simulations testing the impact of various aspects of, e.g. the observing network were also made. For example, we made simulations assuming that all CONT11 stations were equipped with fast VLBI2010 antennas. We found that the WRMS error decreased by about a factor five compared to the current VLBI system. Furthermore, the simulations showed that it is very important to have a homogenous global distribution of the stations for achieving the highest precision for the EOP.     

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.     

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.     

Non-linear station motions in epoch and multi-year reference frames

In the conventions of the International Earth Rotation and Reference Systems Service (e.g. IERS Conventions 2010), it is recommended that the instantaneous station position, which is fixed to the Earth’s crust, is described by a regularized station position and conventional correction models. Current realizations of the International Terrestrial Reference Frame use a station position at a reference epoch and a constant velocity to describe the motion of the regularized station position in time. An advantage of this parameterization is the possibility to provide station coordinates of high accuracy over a long time span. Various publications have shown that residual non-linear station motions can reach a magnitude of a few centimeters due to not considered loading effects. Consistently estimated parameters like the Earth Orientation Parameters (EOP) may be affected if these non-linear station motions are neglected. In this paper, we investigate a new approach, which is based on a frequent (e.g. weekly) estimation of station positions and EOP from a combination of epoch normal equations of the space geodetic techniques Global Positioning System (GPS), Satellite Laser Ranging (SLR) and Very Long Baseline Interferometry (VLBI). The resulting time series of epoch reference frames are studied in detail and are compared with the conventional secular approach. It is shown that both approaches have specific advantages and disadvantages, which are discussed in the paper. A major advantage of the frequently estimated epoch reference frames is that the non-linear station motions are implicitly taken into account, which is a major limiting factor for the accuracy of the secular frames. Various test computations and comparisons between the epoch and secular approach are performed. The authors found that the consistently estimated EOP are systematically affected by the two different combination approaches. The differences between the epoch and secular frames reach magnitudes of $23.6~\upmu \hbox {as}$ (0.73 mm) and $39.8~\upmu \hbox {as}$ (1.23 mm) for the x-pole and y-pole, respectively, in case of the combined solutions. For the SLR-only solutions, significant differences with amplitudes of $77.3~\upmu \hbox {as}$ (2.39 mm) can be found.     

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.     

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.     

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.     

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.     

EOP and scale from continuous VLBI observing: CONT campaigns to future VGOS networks

Continuous (CONT) VLBI campaigns have been carried out about every 3 years since 2002. The basic idea of these campaigns is to acquire state-of-the-art VLBI data over a continuous time period of about 2 weeks to demonstrate the highest accuracy of which the current VLBI system is capable. In addition, these campaigns support scientific studies such as investigations of high-resolution Earth rotation, reference frame stability, and daily to sub-daily site motions. The size of the CONT networks and the observing data rate have increased steadily since 1994. Performance of these networks based on reference frame scale precision and polar motion/LOD comparison with global navigation satellite system (GNSS) earth orientation parameters (EOP) has been substantially better than the weekly operational R1 and R4 series. The precisions of CONT EOP and scale have improved by more than a factor of two since 2002. Polar motion precision based on the WRMS difference between VLBI and GNSS for the most recent CONT campaigns is at the 30 \(\upmu \)as level, which is comparable to that of GNSS. The CONT campaigns are a natural precursor to the planned future VLBI observing networks, which are expected to observe continuously. We compare the performance of the most recent CONT campaigns in 2011 and 2014 with the expected performance of the future VLBI global observing system network using simulations. These simulations indicate that the expected future precision of scale and EOP will be at least 3 times better than the current CONT precision.     

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.     

International VLBI Service for Geodesy and Astrometry

The International VLBI Service for Geodesy and Astrometry (IVS) regularly produces high-quality Earth orientation parameters from observing sessions employing extensive networks or individual baselines. The master schedule is designed according to the telescope days committed by the stations and by the need for dense sampling of the Earth orientation parameters (EOP). In the pre-2011 era, the network constellations with their number of telescopes participating were limited by the playback and baseline capabilities of the hardware (Mark4) correlators. This limitation was overcome by the advent of software correlators, which can now accommodate many more playback units in a flexible configuration. In this paper, we describe the current operations of the IVS with special emphasis on the quality of the polar motion results since these are the only EOP components which can be validated against independent benchmarks. The polar motion results provided by the IVS have improved continuously over the years, now providing an agreement with IGS results at the level of 20–25 \(\upmu \)as in a WRMS sense. At the end of the paper, an outlook is given for the realization of the VLBI Global Observing System.     

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.     

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.     

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.     

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.     

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.     

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.