 |
|
 |
 |
| Figure 1: Superposition of two radial base functions with different positions and shape-parameters |
Figure 2: Spherical harmonic functions of degree n=6 and order m={0,3,6} normalized to values between [-1,1]. The positive and negative areas and the zeros are distributed over the sphere and depend on the degree n and order m |
A good model of the gravity field of the Earth is of great interest for many sciences. This information can be found by satellite observations or by terrestrial measurements. The latter one is mainly used for local approaches, whereas the former data can create a global or regional model of the field. In our days the gravity information is usually generated by the geodetic satellite missions like CHAMP, GRACE and in near future GOCE, where the information is mainly found in the measurement of absolute or relative positions by GPS and the data of the accelerometers (Figure 3). To analyse the satellite information the field is usually approximated in terms of spherical harmonics, which are visualized in Figure 2. On the one hand this system of base functions is easy to implement for all kind of observations and well known in geodesy. On the other hand the spherical harmonics include some restrictions like global support or the smoothing of the data, which avoids regional improvement without changing the whole set of coefficients. Therefore, the idea of residual recovery was born, where the model and the signal are separated into global and residual parts.
The residual signal is created by subtracting a synthetic signal derived from a spherical harmonic model from the observation, to remove the global behaviour (Figure 4). |
 |
 |
| Figure 3: Measurement principle of GRACE according to Rummel et al. 2002 respectively Sharifi 2004. The positions and velocities are derived by GPS observations in high low mode of SST and the distance is measured in the low-low method |
Figure 4: Line-of-sight gradient in the orbit for an arbitrary orbit arc in 2003. The blue line shows the global signal after removing only the ellipsoidal normal field U up to degree 8. The residual line of sight gradient still contains some signal after removing the GGM0s model up to degree N=110 |
The local information can be found by analysing the residual signal by radial base functions. Every base function can be characterized by a scale-factor, a model of the shape-parameter and the position of its centre, where the function has the maximum or minimum value. To analyse an arbitrary signal, a superposition of radial base functions has to be created (Figure 1), so that the difference of observation and approximation is minimized. It is common use in geodesy, to simplify the approximation to a linear problem. In this case, a fixed set of position is chosen, which is often distributed in a regular grid, and a common shape-parameter is derived for example from Kaula's rule. The only free parameters are the scale-factors, and they can be estimated by a linear adjustment. For a high resolution the problem might become overparameterised because of the great number of base functions and this requires a regularisation in the estimation.
To avoid the instability or the regularisation, the idea of the optimized radial base functions was created in our study group. The aim is the minimization of the number of parameters especially by optimising the localisation of a few base functions directly from the data instead of using a fixed grid of positions. The first step is the formulation of the residual signals as in-situ measurement, so that the observations can be considered pointwise without looking of the foretime, while the satellite positions are kept fixed from previous orbit integration. It was shown by closed loop simulations, that the best position of the base functions is connected with some remarkable values in the measurement like the extrema, which produce initial values for the optimisation. A guess for the initial shape-parameter is based on the postulated properties of the base functions, like its convergence or the localisation in space. In the first moment these parameters are usually not the best ones, because of the data distribution, modelling errors, noise and other reasons. In order to optimize the values, a nonlinear problem of heavily dependent parameters has to be solved. This is done by a Levenberg-Marquardt algorithm, which enables us to restrict the estimated values, so that the position are in the area of interest and the shape parameters guarantee the convergence of the functions. In opposite to the classical (linear) least square adjustment, the Levenberg-Marquardt method also minimizes the step size to improve its convergence, which is shown in Figure 5.
The optimization is successfully implemented for the energy balance approach for all missions and the second derivate in flight direction for GRACE, the so called line-of-sight gradiometry. Because of the iterative process it is of great interest to accelerate our algorithms by modelling of the residual signal in terms of radial base functions and their gradients with respect to the parameters in closed formulas, which is successfully done for both kinds of observations. A solution of the line of sight method is demonstrated in Figure 6, with a remarkable correlation of 48% between the approximation and the observation in the orbit. As the total value is still underestimated and biased by an ambiguity, a new range-rate approach is developed. This method is very close to the measurement principle of GRACE, as it deals with the changing of a distance caused by the residual gravity field, but it includes the pastime in the orbit integrations during the optimisation. |
 |
 |
| Figure 5: Solution of a two dimensional minimum problem by Levenberg-Marquardt and least squares adjustment. |
Figure 6: The left picture shows the residual line-of-sight gradient with respect to the GGM02s up to degree N=110 as a reference field. On the right hand side the approximation by the radial base functions is visualised. |
Bibliography
 Antoni M, W Keller and M Weigelt: Regionale Schwerefeldmodellierung durch Slepian- und radiale Basisfunktionen. Zeitschrift für Geodäsie, Geoinformation und Landmanagement 133 (2008) 120-129- Antoni M, W Keller and M Weigelt: Analyse der GRACE-Beobachtungen durch optimierte radiale Basisfunktionen. Geodätische Woche Bremen (30.9.-2.10.)
- Antoni M, W Keller and M Weigelt: Representation of Regional Gravity Fields by Radial Base Functions. IUGG XXIV General Assembly (IUGG 2007), Perugia, Italy, 2.-13.7.2007 (Poster)
- R. Rummel, G. Balmino, J. Johannessen, P. Visser and P. Woodworth: Dedicated gravity field missions-principles and aims. J. Geodynamics, 33:3-20, 2002
- M. A. Sharifi: Satellite gradiometry using a satellite pair. Diploma thesis, Universität Stuttgart, 2004
 Weigelt M, M Antoni and W Keller: Regional gravity recovery from GRACE using position optimized radial base functions. IAG International Symposium on Gravity, Geoid and Earth Observations (GGEO 2008), Chania, Crete, Greece (23.-27.6.)
|
|
|
wm) | © Universität Stuttgart | Legal Notice