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Vertical gravity gradient anomalies from the Gravity and steady-state Ocean Circulation Explorer (GOCE) DIR-3 model have been used to determine gravity anomalies in mid-west Greenland by using Least-Squares Collocation (LSC) and the Reduced Point Mass (RPM) method. The two methods give nearly identical results. However, compared to LSC, the RPM method needs less computational time as the number of equations to be solved in LSC equals the number of observations. The advantage of the LSC, however, is the acquired error estimates. The observation periods are winter 2009 and summer 2012. In order to enhance the accuracy of the calculated gravity anomalies, ground gravity data from West Greenland is used over locations where the gravity change resulting from ice mass changes is negligible, i.e. over solid rock. In the period considered, the gravity anomaly change due to changes in ice mass varies from −5 mGal to 4 mGal. It is negative over the outlet glacier Jacobshavn Isbrae, where the mass loss corresponds to a gravity change of approximately −4 mGal. When using only GOCE vertical gravity gradients, the error estimates range from 5 mGal at the coast to 17 mGal over the ice sheet. Introducing the ground gravity data from West Greenland in the prediction reduces the errors to range from 2 to 10 mGal.

An airborne gravity campaign was carried out at the Dome-C survey area in East Antarctica between the 17th and 22nd of January 2013, in order to provide data for an experiment to validate GOCE satellite gravity gradients. After typical filtering for airborne gravity data, the cross-over error statistics for the few crossing points are 11.3 mGal root mean square (rms) error, corresponding to an rms line error of 8.0 mGal. This number is relatively large due to the rough flight conditions, short lines and field handling procedures used. Comparison of the airborne gravity data with GOCE RL4 spherical harmonic models confirmed the quality of the airborne data and that they contain more high-frequency signal than the global models. First, the airborne gravity data were upward continued to GOCE altitude to predict gravity gradients in the local North-East-Up reference frame. In this step, the least squares collocation using the ITGGRACE2010S field to degree and order 90 as reference field, which is subtracted from both the airborne gravity and GOCE gravity gradients, was applied. Then, the predicted gradients were rotated to the gradiometer reference frame using level 1 attitude quaternion data. The validation with the airborne gravity data was limited to the accurate gradient anomalies (TXX, TYY, TZZ and TXZ) where the long-wavelength information of the GOCE gradients has been replaced with GOCO03s signal to avoid contamination with GOCE gradient errors at these wavelengths. The comparison shows standard deviations between the predicted and GOCE gradient anomalies TXX, TYY, TZZ and TXZ of 9.9, 11.5, 11.6 and 10.4 mE, respectively. A more precise airborne gravity survey of the southern polar gap which is not observed by GOCE would thus provide gradient predictions at a better accuracy, complementing the GOCE coverage in this region.

We combine the complementary characteristics of laser altimeter data and stereoscopic digital elevation models (DEMs) to construct high-resolution (∼100 m) maps of surface elevations and elevation changes over rapidly changing outlet glaciers in Greenland. Measurements from spaceborne and airborne laser altimeters have relatively low errors but are spatially limited to the ground tracks, while DEMs have larger errors but provide spatially continuous surfaces. The principle of our method is to fit the DEM surface to the altimeter point clouds in time and space to minimize the DEM errors and use that surface to extrapolate elevations away from altimeter flight lines. This reduces the DEM registration errors and fills the gap between the altimeter paths. We use data from ICESat and ATM as well as SPOT 5 DEMs from 2007 and 2008 and apply them to the outlet glaciers Jakobshavn Isbræ (JI) and Kangerdlugssuaq (KL). We find that the main trunks of JI and KL lowered at rates of 30–35 and 7–20 m a−1,respectively. The rates decreased inland. The corresponding errors were 0.3–5.2 m a−1for JI and 0.3–5.1 m a−1for KL, with errors increasing proportionally with distance from the altimeter paths.

This paper deals with the analysis of gravity anomaly and precise levelling in conjunction with GPS-Levelling data for the computation of a gravimetric geoid and an estimate of the height system bias parameter No for the vertical datum in Pakistan by means of least squares collocation technique. The long term objective is to obtain a regional geoid (or quasi-geoid) modeling using a combination of local data with a high degree and order Earth gravity model (EGM) and to determine a bias (if there is one) with respect to a global mean sea surface. An application of collocation with the optimal covariance parameters has facilitated to achieve gravimetric height anomalies in a global geocentric datum. Residual terrain modeling (RTM) technique has been used in combination with the EGM96 for the reduction and smoothing of the gravity data. A value for the bias parameter No has been estimated with reference to the local GPS-Levelling datum that appears to be 0.705 m with 0.07 m mean square error. The gravimetric height anomalies were compared with height anomalies obtained from GPS-Levelling stations using least square collocation with and without bias adjustment. The bias adjustment minimizes the difference between the gravimetric height anomalies with respect to residual GPS-Levelling data and the standard deviation of the differences drops from 35 cm to 2.6 cm. The results of this study suggest that No adjustment may be a good alternative for the fitting of the final gravimetric geoid as is generally done when using FFT methods.

During the past years, the accuracy of relative positioning using differential GPS (DGPS) has been improved significantly. The present accuracy of DGPS allows us to directly estimate the differential amplitudes and Greenwich phase lags of the main semi-diurnal ocean tide loading constituents (S2, K2, M2 and N2). For this purpose a test is carried out using two GPS stations in Alaska. One station, Chi3, is located on an island in the Gulf of Alaska, while the second station, Fair, is located far away from the coastal areas. Processing hourly GPS solutions for the baseline between Fair and Chi3 during 49 days gives differential amplitudes of 23.21 mm and 4.71 mm for M2 and N2, respectively, while the theoretically differential amplitudes of M2 and N2 are 20.90 mm and 4.21 mm, respectively (using the GOT99.2 ocean tide model). The diurnal ocean tide loading constituents are not considered, because unmodeled troposphere effects increase the noise level near the diurnal frequency band and prevent us from obtaining significant results.

The Gravity field and steady state Ocean Circulation Explorer (GOCE) maps variations in the gravity field by observing second order derivatives (gradients) of the Earth gravitational potential. Flying in the low altitude of 255 km and having a spatially dense data distribution of short wavelengths of the gravity field, GOCE may be used to enhance the time varying gravity signal coming fromthe GRACE satellites. The GOCE gradients may potentially be used for the determination of residual masses in local regions. This can be done using Least-Squares Collocation (LSC) or the Reduced Point Mass (RPM) method. In this study, different gravity field solutions are calculated by the use of RPM, LSC and GOCE gradients, respectively. Gravity field time series are created and presented for the six consecutive months of GOCE gradient observations, data being acquired between November 2009 and June 2010. Corresponding gravity anomaly results are used for the calculation of ice mass changes by the use of theRPMmethod. The results are then compared with the computed topographic effect of the ice by the use of a modified topographic correction and the Gravsoft TC program. The maximal gravity changes at the ground predicted from GOCE gradients are between 2 and 4 mGal for the period considered. The gravity anomaly estimation error arising from the GOCE gradient data using only Tzz with an associated error of 20 mE is 11 mGal. This analysis shows the potential of using GOCE data for observations of ice mass changes although the GOCE dataset is limited to only six months. We expect four years of GOCE gradient observations to be available by mid-2014. This will increase the accuracy and spatial resolution of the GOCE measurements, which may lead to an accuracy necessary for observing ice mass changes.

The use of 3D Least-Squares Collocation (LSC) for the determination of a regional or local approximation to the anomalous (gravity) potential as implemented by the GRAVSOFT Fortran programs is described. The software also implements the remove-restore method so that gravity variations outside the region of computation are accounted for by subtracting the contribution of an Earth Gravity Model (EGM) and so that statistical homogenisation is achieved by removing the contribution of topographic short wavelength features. It is also described how LSC may be used to determine parameters like a height datum off-set or to detect possible gross errors. Examples using data from New Mexico, USA, illustrates the use of the method.

The GOCE gradients, having a spatially dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model. The aim of this study is to develop a methodology to improve the use of GOCE gradients and to determine the Earth’s gravity field with better accuracy than by using global models, which have been truncated at a specific harmonic degree and order. The method makes use of all available GOCE gradient data in addition to the global models and aims at improving the determination of Earth’s gravitational field in regional areas. Subsequently, the calculated geoid is used together with measurements of sea surface height in a calculation of the Mean Dynamic Topography.In regional geoid recovery from GOCE gradients, two methods are used, one of them being Least-Squares Collocation (LSC). The second method is developed as a part of this study, and it is based on the Reduced Point Mass (RPM) response. The results show that the RPM method and LSC method give very similar results when using the same data, i.e. the difference is insignificant when compared to the EGM2008 results. However, when all of the available GOCE gradient data are used with the RPM method, an improvement in the gravitational field determination is achieved. The enhanced geoid by the RPM method is then used for the improvement of the MDT in the North Atlantic region.

In gravity field modeling measurements are usually located on or above the terrain. However, when using the residual topographic modeling (RTM) method, measurements may end up inside the masses after adding the mean topography. These values do not correspond to values evaluated using a harmonic function. A so-called harmonic correction has been applied to gravity anomalies to solve this problem. However, for height anomalies no correction has been applied. To generalize the correction to e.g. height anomalies we interprete that the vertical gravity gradient inside the masses multiplied by height equals the correction. In principle the procedure is applicable to all gravity field functionals. We have tested this generalization of the procedure which consist in determining equivalent quantities in points Q on the mean surface if this surface is in free air. The procedure has as data the reduced values in P inside the masses but considered as being located at the mean surface. Numerical tests with height anomaly data from New Mexico and Norway as control data show that for gravity anomalies the general procedure is better than using the original harmonic correction procedure.

The GRAVSOFT suite of Fortran programs enables gravity field modeling using 3D or 2D Least-Squares Collocation and Fourier techniques, the computation of topographic effects, the evaluation of high-degree spherical harmonic series and several other functions. It has been developed since the early 1970s with a line-oriented DOS-interface. Sponsored by the Geodetic Survey of Malaysia a modern graphical interface has been designed using Python (www.python.org) and the widget toolkit Tk, following the Apple Design Guidelines.

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10 cm in terms of geoid heights, and 3 mGal in terms of gravity anomalies, globally at a resolution of 100 km (degree/order 200).

The impact of the GOCE satellite mission on the recovery of the gravity field is analysed for two simulated cases. In the first case the GOCE Level 2 product is used where the gravity field is approximated by spherical harmonic coefficients up to degree and order 200. In the second case synthetic GOCE Level 1B data are used directly in a gravity field determination using least squares collocation. In case two the full spectrum geoid error was improved from 31 cm to 15 cm and the resolution was doubled.

From energy considerations the relationship in inertial space between the gravitational potential and the state vector of the satellite can be found. The state vector from the rapid science orbit of CHAMP has been used after the velocities have been converted to the inertial system.

[1] The atmospheric effects represent one of the major limits of SAR interferometry as a quantitative technique to monitor subsidences. In this work, which focuses on subsidences of small spatial extent, a procedure to reduce these effects is described. The atmospheric component of the interferometric phase is estimated adopting a filtering and prediction procedure, which exploits the phase over stable areas identified in the vicinity of the analysed subsidence area. The procedure was validated on a suitable test site located in North-eastern Spain. Furthermore, it was employed in the study of a small-scale subsidence, where the interferometric results were validated using precise geodetic observations.

One of the products derived from the gravity field and steady-state ocean circulation explorer (GOCE) observations are the gravity gradients. These gravity gradients are provided in the gradiometer reference frame (GRF) and are calibrated in-flight using satellite shaking and star sensor data. To use these gravity gradients for application in Earth scienes and gravity field analysis, additional preprocessing needs to be done, including corrections for temporal gravity field signals to isolate the static gravity field part, screening for outliers, calibration by comparison with existing external gravity field information and error assessment. The temporal gravity gradient corrections consist of tidal and nontidal corrections. These are all generally below the gravity gradient error level, which is predicted to show a 1/f behaviour for low frequencies. In the outlier detection, the 1/f error is compensated for by subtracting a local median from the data, while the data error is assessed using the median absolute deviation. The local median acts as a high-pass filter and it is robust as is the median absolute deviation. Three different methods have been implemented for the calibration of the gravity gradients. All three methods use a high-pass filter to compensate for the 1/f gravity gradient error. The baseline method uses state-of-the-art global gravity field models and the most accurate results are obtained if star sensor misalignments are estimated along with the calibration parameters. A second calibration method uses GOCE GPS data to estimate a low-degree gravity field model as well as gravity gradient scale factors. Both methods allow to estimate gravity gradient scale factors down to the 10?3 level. The third calibration method uses high accurate terrestrial gravity data in selected regions to validate the gravity gradient scale factors, focussing on the measurement band. Gravity gradient scale factors may be estimated down to the 10?2 level with this method.

The National Survey and Cadastre (KMS) is responsible for the geodetic definition of the reference network in Denmark. Permanent GPS stations play an important role in the monitoring and maintenance of the geodetic network. During 1998 and 1999 KMS established three permanent GPS station in Denmark, SMID, SULD and BUDP. Using almost 4.5 years of continuous data from the Danish station and the Swedish station, ONSA, we analyse the daily GPS solution due to crustal deformation caused by glacial isostatic adjustment (GIA). Although, displacements due to GIA are only 1–3 mm/year at the Danish GPS sites, the current precision of positioning using GPS allows us to observe these effects. The modelled horizontal GIA velocities and the observed horizontal residuals obtained from GPS show almost the same direction for all station. However, the observed velocity residuals are larger than the modelled GIA velocities. Furthermore, the difference of the observed vertical velocity residual rate between ONSA and SMID, and between ONSA and SULD seems to agree with the expected rate according to the post-glacial model by Milne. However, the vertical velocity rate of BUDP seems to disagree with the model. Moreover, the analysis presented here uses a relative short period of data. The error of the vertical velocities (including reference frame drift) is order of ±1.1 mm/yr.

The objective of this study is to develop methodology to use GOCE gravity gradients for enhanced geoid modelling and ocean circulation modelling. In specific regions with a rough gravity field, the resolution of the geoid may be enhanced substantially if GOCE gradiometer data are used in addition of the GOCE spherical harmonic coefficient model (EGMs) since in such areas the GOCE gradients contain more information than the EGM itself. Hence, the use of gradients may lead to improve the resolution of e.g. the marine geoid which in turn will improve the estimation of the ocean circulation. This is tested using GOCE gravity gradient data, the GEOCOL program (GRAVSOFT) and Reduced Point Mass (RPM) program. Tests are carried out in the GOCINA region and in the Mediterranean basin. Furthermore, the effect of the decreasing height of the GOCE satellite on gravity gradients and associated MDT is investigated.

GOCE TRF (terrestrial reference frame) vertical anomalous gradients (Tzz) from two periods have been used to determine gravity anomalies changes in mid-west Greenland, where a large mass-loss has been detected using GRACE (Fig. 1). As additional data were used the GOCE DIR-3 model and ground gravity at the coast on solid rock, where no mass loss is expected. The methods of Least-Squares Collocation (LSC) and the Reduced Point Mass (RPM) methods have been used, however only LSC included the ground data.