Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise

Satellite data that are used to model the global gravity field of the Earth are typically corrupted by correlated noise, which can be related to a frequency dependence of the data accuracy. We show an opportunity to take such noise into account by using a proper noise covariance matrix in the estimation procedure. If the dependence of noise on frequency is not known a priori, it can be estimated on the basis of a posteriori residuals. The methodology can be applied to data with gaps. Non-stationarity of noise can also be dealt with, provided that the necessary a priori information exists. The proposed methodology is illustrated with CHAllenging Mini-satellite Payload (CHAMP) data processing. It is shown, in particular, that the usage of a proper noise model can make the measurements of non-gravitational satellite accelerations unnecessarily. This opens the door for high-quality modeling of the Earth’s gravity field on the basis of observed orbits of non-dedicated satellites (i.e., satellites without an on-board accelerometer). Furthermore, the processing of data from dedicated satellite missions – GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Explorer) – may also benefit from the proposed methodology.     

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [-89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for the SHA program SHA by Heck and Seitz.     

Deformation measurements using SAR interferometry: potential and limitations

Most applications of Synthetic Aperture Radar (SAR) make only use of the amplitude information in just one image. Interferometric SAR (InSAR) makes use mainly of the phase measurements in two or more SAR images of the same scene, acquired at two different moments and/or at two slightly different locations. By interference of the two images, very small slant-range changes of the same surface can be inferred. These slant-range changes can be related to topography and/or surface deformations. InSAR thus has the potential of mapping centimeter-scale ground displacements over a region many tens of kilometers in size at a resolution of a few meters making it one of the most promising space-geodetic techniques for monitoring Earth's surface deformations. The goal of this paper is to discuss some of the potential new applications of InSAR for the monitoring of deformations, and to show its major limitations. Some potential new applications of InSAR related to surface-change detection including earthquake and crustal studies, the monitoring of volcanoes and anthropogenic effects, and the monitoring of glaciers and ice sheets are presented. The discussion on the limitations of InSAR for surface-change detection focuses on atmospheric perturbations and the problem of temporal decorrelation.     

Impact of accounting for coloured noise in radar altimetry data on a regional quasi-geoid model

We study the impact of an accurate computation and incorporation of coloured noise in radar altimeter data when computing a regional quasi-geoid model using least-squares techniques. Our test area comprises the Southern North Sea including the Netherlands, Belgium, and parts of France, Germany, and the UK. We perform the study by modelling the disturbing potential with spherical radial base functions. To that end, we use the traditional remove-compute-restore procedure with a recent GRACE/GOCE static gravity field model. Apart from radar altimeter data, we use terrestrial, airborne, and shipboard gravity data. Radar altimeter sea surface heights are corrected for the instantaneous dynamic topography and used in the form of along-track quasi-geoid height differences. Noise in these data are estimated using repeat-track and post-fit residual analysis techniques and then modelled as an auto regressive moving average process. Quasi-geoid models are computed with and without taking the modelled coloured noise into account. The difference between them is used as a measure of the impact of coloured noise in radar altimeter along-track quasi-geoid height differences on the estimated quasi-geoid model. The impact strongly depends on the availability of shipboard gravity data. If no such data are available, the impact may attain values exceeding 10 centimetres in particular areas. In case shipboard gravity data are used, the impact is reduced, though it still attains values of several centimetres. We use geometric quasi-geoid heights from GPS/levelling data at height markers as control data to analyse the quality of the quasi-geoid models. The quasi-geoid model computed using a model of the coloured noise in radar altimeter along-track quasi-geoid height differences shows in some areas a significant improvement over a model that assumes white noise in these data. However, the interpretation in other areas remains a challenge due to the limited quality of the control data.     

Earth’s gravity field modelling based on satellite accelerations derived from onboard GPS phase measurements

GPS data collected by satellite gravity missions can be used for extracting the long-wavelength part of the Earth’s gravity field. We propose a new data processing method which makes use of the ‘average acceleration’ approach to gravity field modelling. In this method, satellite accelerations are directly derived from GPS carrier phase measurements with an epoch-differenced scheme. As a result, no ambiguity solutions are needed and the systematic errors that do not change much from epoch to epoch are largely eliminated. The GPS data collected by the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission are used to demonstrate the added value of the proposed method. An analysis of the residual accelerations shows that accelerations derived in this way are more precise, with noise being reduced by about 20 and 5% at the cross-track component and the other two components, respectively, as compared to those based on kinematic orbits. The accelerations obtained in this way allow the recovery of the gravity field to a slightly higher maximum degree compared to the solution based on kinematic orbits. Furthermore, the gravity field solution has an overall better performance. Errors in spherical harmonic coefficients are smaller, especially at low degrees. The cumulative geoid height error is reduced by about 15 and 5% up to degree 50 and 150, respectively. An analysis in the spatial domain shows that large errors along the geomagnetic equator, which are caused by a high electron density coupled with large short-term variations, are substantially reduced. Finally, the new method allows for a better observation of mass transport signals. In particular, sufficiently realistic signatures of regional mass anomalies in North America and south-west Africa are obtained.     

Efficient gravity field recovery from GOCE gravity gradient observations

 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000     

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.     

Data requirements for a 5-mm quasi-geoid in the Netherlands

We assess the surface gravity data requirements for a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf in terms of omission and commission errors. The omission error critically depends on the roughness of the topography and bathymetry. For the Netherlands continental shelf, Central and Northern Netherlands, the omission error is well described by the model 0.32d mm, where d is the data spacing in km. For the more hilly Southern Netherlands, the omission error model is 0.92d mm. The commission error depends on the kernel modification, the data spacing, and the data accuracy. When using the spheroidal Stokes kernel, it is well described by 0.277 d σ_Δg mm, where σ_Δg is the noise standard deviation of surface gravity data in mGal. An upper bound of the commission error of the state-of-the-art satellite-only gravity model GOCO05S over the Netherlands is e^{0.03676L–11.419} mm, where L is the maximum degree up to which this model is used. Only if this model is truncated at a sufficiently low degree, e.g., at degree 100, its contribution to the total commission error can be neglected. We determine the total error as the sum of commission and omission errors. Hence, to realize a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf, a data spacing of 3.5 km is needed when assuming a noise standard deviation of 1.5 mGal for surface gravity data. The currently available land-based gravity data fulfill this requirement. This does not apply to the situation at sea, where the density of the shipboard gravity data and the accuracy of the radar altimeter-derived data do not allow the realization of a 5-mm quasi-geoid model.     

Numerical solution of geodetic boundary value problems using a global reference field

One of the most serious practical limitations of boundary element methods for gravity field determination is that they cannot make efficient use of existing satellite geopotential models. Three basic approaches to solving the problem are developed: (1) alternative representation formulas; (2) modified kernel functions of classical representation formulas; and (3) modified trial and test spaces. The three methods are tested and compared for the altimetry–gravimetry II boundary value problem. It is shown that there is in fact a significant improvement when compared to the pure boundary element solution. Most promising is the method of multiscale trial and test spaces which, in addition, yields sparse system matrices. Received: 7 September 1998 / Accepted: 16 June 1999     

A Kalman Filter Approach to Realize the Lowest Astronomical Tide Surface

In this paper , we present a novel Kalman filter approach to combine a hydrodynamic model-derived lowest astronomical tide (LAT) surface with tide gauge record-derived LAT values. In the approach, tidal water levels are assimilated into the model. As such, the combination is guided by the model physics. When validating the obtained “Kalman-filtered LAT realization” at all tide gauges, we obtained an overall root-mean-square (RMS) difference of 15.1 cm. At the tide gauges not used in the data assimilation, the RMS is 17.9 cm. We found that the assimilation reduces the overall RMS difference by ～ 31% and ～ 22%, respectively. In the Dutch North Sea and Wadden Sea, the RMS differences are 6.6 and 14.8 cm (all tide gauges), respectively. Furthermore, we address the problem of LAT realization in intertidal waters where LAT is not defined. We propose to replace LAT by pseudo-LAT, which we suggest to realize similarly as LAT except that all water level boundary conditions and assimilated tidal water levels have to be enlarged by a constant value that is removed afterward. Using this approach, we obtained a smooth reference surface for the Dutch Wadden Sea that fits LAT at the North Sea boundary within a few centimeters.