Carl Christian Tscherning

Determination of semi‐diurnal ocean tide loading constituents using GPS in Alaska

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.

Geoid Determination by 3D Least-Squares Collocation

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.

Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCE Gravimetry

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.

Calibration and Validation of GOCE Gravity Gradients

GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.

Generalizing the Harmonic Reduction Procedure in Residual Topographic Modeling

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.

Upward continuation of Dome-C airborne gravity and comparison with GOCE gradients at orbit altitude in east Antarctica

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.

Using ground gravity to improve ice mass change estimation from GOCE gravity gradients in mid-west Greenland

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 Isbræ, 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.

Crustal deformations at permanent GPS sites in Denmark

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.