Satellite to satellite tracking in the space-wise approach

The launch of the champ mission in 2000 has renewed interest in the recovery of the geopotential field from satellite observations which has been a challenging research issue for decades. It was the first dedicated gravity field mission which was followed by the grace spacecrafts. In the grace mission, the high-low (hl-) and the low-low satellite-to-satellitetracking (ll-sst) observations are combined and the resultant observables are expressed in terms of the gravity gradient at the barycenter of two satellites. Each observation at its respective evaluation point can be written in terms of the spherical harmonic coefficients. Consequently, the observations are a sequence of discrete time series which are mathematically related to the unknown coefficients via the corresponding position of the satellites at the evaluation epoch. In this approach, which is called time-wise approach, the determination of unknown coefficients becomes possible after plugging the observations into the mathematical model. Fulfilling the sampling theorem, however, leads to a huge linear system of equations with a large number of unknowns. As an alternative, one can employ the semi-analytical approach which is derived from the timewise approach by imposing some approximations. Observations are still considered as discrete time series on an ideal geometry with a constant radius and/or constant inclination. The coefficients are reordered and then computed via the lumped coefficients or using 2d fft. Another alternative is the space-wise approach in which the observations are mapped on a specific grid on the mean orbital sphere. In this approach, the observation values are predicted on the grid points and the coefficients are derived by implementation of the global spherical harmonic analysis on the gridded observations. Compared to the time-wise approach, the linear system of equations are split into smaller systems which can be solved very easily in ordinary pcs. In this thesis, the ll-sst problem is formulated both in the semi-analytical as well as the space-wise approach. The space-wise approach is then numerically implemented. Despite the spirit of modern geodesy to avoid reduction, the reduction of observations is required both in the semi-analytical and the space-wise approaches. Different formulations are used for downor up-ward continuation of observations on the reference geometry. Optimality of the basis functions and their respective parameters is carefully treated by means of the Genetic Algorithms (ga). Optimizing the approximation methods is carefully investigated using the genetic algorithms. The idea of one-leave-out method or the so-called residual bootstrap approach is successfully used in the definition of the object functions. Compared to the classical error criterion, the modified object function results in a better solution. In order to reduce the linearization and the reduction error, the residual gravity field is recovered. In this study, an adaptive reference orbit is used. Furthermore, the determination of the best fitting reference orbit is expressed as a least squares and an optimization problem. Indeed, mathematical formulation of the gradiometry approach of the cubic order in terms of Taylor series is derived. The contribution of each individual term to the formulation is analyzed and the formulation is simplified accordingly. The relative velocity vector is combined with the high-accuracy ranging observations both in the acceleration difference and the gradiometry approaches. Since it is not directly observed it should numerically be derived from gps observations by means of numerical differentiation. In this regard, a few differentiation algorithms are studied for deriving the relative intersatellite velocity vector. Furthermore, the recovery of the residual field and computing the relative velocity using the reference field are alternatively utilized to bypass the numerical differentiation. Compared to the numerical differentiation, the alternative methods yield more accurate solution. The ranging system observations are more accurate than the gps measurements. Some condition equations are derived for adjusting the low-accuracy observations using the ll-sst measurements. It only improves the cross-track and radial components of the relative position vector whereas the along-track component of the relative velocity benefits from imposing the constraints. Finally, the previously derived formulation is used for recovery of the residual field. Two different iterative approaches are employed for determination of the residual gravity field using the grace non-invariant observable. To sum up, the gradiometry approach using a satellite pair is successfully implemented for the recovery of the residual gravity field in the space-wise approach.