Journal of Geodesy

 Simplified techniques for high-degree spherical harmonic synthesis are extended to include gravitational potential second derivatives with respect to latitude.

Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaula’s rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newton’s equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.

A new approach for tuning the trajectories of the European remote sensing (ERS) satellites is developed and assessed. Differential dual-pass interferometry is applied to calculate interferograms from the phase difference of synthetic aperture radar (SAR) images acquired by the ERS satellites over the site of the 1992 earthquake in Landers, California. These interferograms contain information about orbital trajectories and geophysical deformation. Beginning with good prior estimates of the orbital trajectories, a radial and an across-track orbital adjustment is estimated at each epoch. The data are the fringe counts along distance and azimuth. Errors in the across-track and radial components of the orbit estimates produce fringes in the interferograms. The spacing between roughly parallel fringes gives the gradients in distance and azimuth coordinates. The approach eliminates these fringes from interferometric pairs spanning relatively short time intervals containing few topographic residuals or atmospheric artefacts. An optimum interferometric path with six SAR acquisitions is selected to study post- and inter-seismic deformation fields. In order to regularize the problem, it is assumed that the radial and across-track adjustments both sum to zero. Applying the adjustment approach to the prior estimates of trajectory from the Delft Institute for Earth-Orientated Space Research (DEOS), root mean squares of 7.3 cm for the across-track correction components and 2.4 cm for the radial ones are found. Assuming 0.1 fringes for the a priori standard deviation of the measurement, the approach yields mean standard deviations of 2.4 cm for the across-track and 4.5 cm for the radial components. The approach allows an ‘interval by interval’ improvement of a set of orbital estimates from which post-fit interferograms of different time intervals spanning a total 3.8-year inter-seismic time interval can be created. The interferograms calculated with the post-fit orbital estimates compare favorably with those corrected with a conventional orbital tuning approach. Using the adjustment approach, it is possible to distinguish between orbital and deformation contributions to interferometric SAR (InSAR) phase gradients. Surface deformation changes over an inter-seismic time interval longer than one year can be measured. This approach is, however, limited to well-correlated interferograms where it is possible to measure the fringe gradient.

The SIGMA-Δ model has been developed for stochastic modelling of global positioning system (GPS) signal diffraction errors in high precision GPS surveys. The basic information used in the SIGMA-Δ model is the measured carrier-to-noise power-density ratio (C/N0). Using the C/N0 data and a template technique, the proper variances are derived for all phase observations. Thus the quality of the measured phase is automatically assessed and if phase observations are suspected to be contaminated by diffraction effects they are weighted down in the least-squares adjustment. The ability of the SIGMA-Δ model to reduce signal diffraction effects is demonstrated on two static GPS surveys as well as on a kinematic high-precision GPS railway survey. In cases of severe signal diffraction the accuracy of the GPS positions is improved by more than 50% compared to standard GPS processing techniques.

Aside from the ionospheric total electron content (TEC) information, root-mean-square (RMS) maps are also provided as the standard deviations of the corresponding TEC errors in global ionospheric maps (GIMs). As the RMS maps are commonly used as the accuracy indicator of GIMs to optimize the stochastic model of precise point positioning algorithms, it is of crucial importance to investigate the reliability of RMS maps involved in GIMs of different Ionospheric Associated Analysis Centers (IAACs) of the International GNSS Service (IGS), i.e., the integrity of GIMs. We indirectly analyzed the reliability of RMS maps by comparing the actual error of the differential STEC (dSTEC) with the RMS of the dSTEC derived from the RMS maps. With this method, the integrity of seven rapid IGS GIMs (UQRG, CORG, JPRG, WHRG, EHRG, EMRG, and IGRG) and six final GIMs (UPCG, CODG, JPLG, WHUG, ESAG and IGSG) was examined under the maximum and minimum solar activity conditions as well as the geomagnetic storm period. The results reveal that the reliability of the RMS maps is significantly different for the GIMs from different IAACs. Among these GIMs, the values in the RMS maps of UQRG are large, which can be used as ionospheric protection level, while the RMS values in EHRG and ESAG are significantly lower than the realistic RMS. The rapid and final GIMs from CODE, JPL and WHU provide quite reasonable RMS maps. The bounding performance of RMS maps can be influenced by the location of the stations, while the influence of solar activity and the geomagnetic storm is not obvious.

The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $$T_{zz})$$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion.