Radboud Koop

Simulation of gravity gradients: a comparison study

At different European institutes software has been developed for evaluation of the gravitational potential of the Earth using high degree spherical harmonic expansions. In this report the results of a comparison of a number of these software packages are presented. We compared the results for the second order derivatives (gravity gradients). It appeared that one of the most critical points in these computations is the definition of the coordinates, which should be as accurate as possible. Machine dependency and algorithm setup were of less importance, the former being only reflected in CPU timing results.

Calibration and Error Assessment of GOCE Data

The GOCE level lb data products consist, among others, of time lines of calibrated and corrected gravity gradient data and SST data. The calibration for GOCE can be described as a multi-step procedure consisting of a pre-flight, on-ground calibration, an in-flight internal calibration and a so-called external (or absolute) calibration step. The latter consists of checking and correcting the data by comparison with external data, models or other external gravity field knowledge in order to improve absolute calibration parameters determined earlier and to correct for known or unknown errors in the data which have not been corrected for before. Obvious choices for external calibration are to compare the GOCE data with ground based gravity data, with existing global gravity field models or to intercompare different GOCE data products, like the accelerometer common mode observations with the GPS data. The latter two methods are studied in more detail here. It appears that for the gradiometer observations, the use of existing global gravity field models may help the calibration for the low frequencies below the measurement bandwidth. The calibration of the common mode observations using GPS data depends heavily on the quality of prior conservative force models. In addition, we consider here the problem of in flight performance monitoring of the GOCE measurements using a collinear track technique in analogy to satellite altimetry. We discuss the results of a new algorithm designed for a sun-synchronous frozen repeat orbit and treat two extreme scenarios. For the best case scenario our conclusion is that it is possible to monitor the stability of the instrument within the measurement band to around (MATH) whereas for our worst case scenario this is a factor of 10 less accurate.

Validation of fast pre-mission error analysis of the GOCE gradiometry mission by a full gravity field recovery simulation

Abstract Spherical harmonic error analysis fully relies on the validity of the a priori observational and stochastic models. In this paper we validate error analysis results of the gradiometer mission GOCE by a full-fledged spherical harmonic coefficient recovery. Both methods (least squares error analysis and full recovery) are based on a semi-analytical approach. The results compare very well in spectral and spatial domains. Thus, it is demonstrated that, besides being fast, the least squares error analysis is a reliable premission error assessment tool.

Simulation of the GOCE Gravity Field Mission

GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) is one of the four selected ESA Earth Explorer Missions. The main objective of GOCE is the determination of the Earth’s gravity field with high spatial resolution and with high homogeneous accuracy. For this purpose, two observation concepts will be realised. Satellite-to-Satellite Tracking (SST) in high-low mode will be used for the orbit determination and for the retrieval of the long-wavelength part of the gravity field. Satellite Gravity Gradiometry (SGG) will be employed for the derivation of the medium/short-wavelength parts of the gravity field..

Data analysis for the GOCE mission

We investigate the time-wise approach to the data anlaysis for the GOCE mission. The number of observations collected during the mission, the number of potential coefficients to be estimated, and the complexity of the mathematical model for the time-wise approach require a special strategy, which has to reduce the CPU-time and storage requirements considerably. Our approach is based on (1) the iterative solution of the normal equations using a Richardson-iteration scheme and (2) the approximation of the design matrix in order to assemble the right-hand side in each iteration step efficiently. We demonstrate the performance of the approach for white noise and coloured noise observations along a simulated GOCE orbit up to degree and order 180. We provide error estimates anal show that the solution is unbiased. We also prove that the method does not converge to the solution of the normal equations. However, the approximation error can be neglected in our simulations.

On the information contents and regularisation of lunar gravity field solutions

Till the present day the recovery of the lunar gravity field from satellite tracking data depends in a crucial way on the level and method of regularisation. With Earth-based tracking only, the spatial data coverage is limited to only slightly more than 50% and the inverse problem remains severely ill-posed. The development of global gravity models suitable for precise orbit modeling as well as geophysical studies therefore requires a significant level of regularisation, limiting the solution power over the far-side where no gravity information is available. Unconstrained solutions, within the framework of global harmonic base functions, are only possible for very low degrees (< 10). Any significant change to this situation is only to be expected when global satellite-to-satellite tracking data of high quality becomes available early in the next decade. Yet, a rigorous analysis of the impact of the chosen method and level of regularisation is lacking. Most gravity models employ a Kaula-type signal smoothness constraint of 15 × 10−5 /l2, which allows a good overall data fit as well as a smooth field over the far-side. Furthermore, a geographical type of constraint has been suggested, where surface accelerations have been introduced in areas of no data coverage. Modern numerical methods, on the other hand, offer direct tools and search mechanisms for the optimal level of regularisation. This paper presents a study of Tikhonov-type regularisation of lunar gravity solutions, with emphasis on the so-called L-curve and quasi-optimality methods for regularisation parameter estimation. Furthermore, new quality measures of lunar gravity solutions are presented, which account for the bias introduced by the regularisation.

Quality Improvement of Global Gravity Field Models by Combining Satellite Gradiometry and Airborne Gravimetry

The expected high resolution and precision of a global gravity field model derived from satellite gradiometric observations is unprecedented compared to nowadays satellite-only models. However, a dedicated gravity field mission will most certainly fly in a non-polar (sun-synchronous) orbit, such that small polar regions will not be covered with observations. The resulting inhomogeneous global data coverage, together with the downward continuation problem, leads to unstable global solutions and regularization is necessary. Regularization gives rise to a bias in the solution, mainly in the polar areas although in other regions as well.

Determination of the Gravity Field from Satellite Gradiometry

For the analysis of the yield of a satellite gradiometry mission, different methods can be employed, see e.g. (Koop, 1993) or (Rummel et al., 1993). Most studies carried out so far rely on covariance propagation. In the present study full simulations are used to evaluate future missions like step. This gives the opportunity to investigate several effects in more detail than before.

POTENTIAL COEFFICIENT RECOVERY FROM THE CSR SET OF SIMULATED SATELLITE GRADIOMETRY OBSERVATIONS

An investigation was carried out to demonstrate the possibility of recovering spherical harmonic potential coefficients up to high degree and order from gradiometric measurements. Observations were taken from one month simulated GRM gradient data computed at the Center of Space Research at the University of Texas. For least squares adjustment a linear observation model was chosen from which the radial orbit uncertainties were eliminated. A solution up to degree and order 180 proved to be very promising, whereas a 360 solution showed degraded results due to the too short mission and the non-homogeneous data coverage.