Comparison of three techniques for modeling the Earth's gravity field on the basis of a satellite orbit

At present, there are three techniques for the computation of the Earth's gravity field from a satellite orbit: (i) the "classical" approach based on the integration of variational equations (IVEA); (ii) the energy balance approach (EBA); (iii) the acceleration approach (AA), which directly relates the satellite accelerations to the gravity field in accordance with Newton's second law. Most of the results have been obtained so far with the IVEA and EBA. The AA is believed to be inferior because the double differentiation needed to convert the satellite orbit into the satellite accelerations amplifies data noise dramatically. We show that that a poor performance of the AA is a myth. One can easily prove that the solution of an inverse problem is invariant with respect to the linear transformation of the data vector of the kind d' = B d (where d is the original data vector, d' is the transformed data vector, and B is the transformation matrix) provided that the matrix B is square and invertible. The only pre-requisite is that the optimal estimation procedure is followed, including the usage of the properly transformed covariance matrix: C_d' = B C_d B^T. In other words, such data vectors d' and d are equivalent. It is easy to show that the satellite positions and satellite accelerations are two nearly equivalent data sets (in order to reach a strict equivalence, the latter can be supplied, e.g., with the initial state vector). Therefore, these data sets may result in nearly the same gravity field model. A decision which technique is preferable should be made on the basis of practical considerations, e.g. the numerical efficiency. According to our experience, the AA leads to a much faster computational scheme than the IVEA. Furthermore, we have considered the EBA. It is easy to show that a set of kinetic energy measurements is nearly equivalent to a set of along-track satellite accelerations. The other two components of the acceleration vectors are ignored by the EBA. Therefore, one can expect that this technique is about √ {3} times less accurate than the IVEA and AA (provided that all 3 acceleration components are equally informative and accurate). This conclusion can also be justified from the physical point of view. The cross-track and the radial components of the of the gravitational forces, which are responsible for corresponding accelerations, are always directed normally to the elementary path. Therefore, they do no work and are not perceptible in terms of the energy balance. Our theoretical findings are supported by a numerical example. A 10-day drag-free repeat satellite orbit of 246-km altitude with 1-s sampling is considered; the orbit corresponds to the EGM96 gravity field model truncated at degree and order 80. The satellite positions are artificially contaminated with 1-cm white noise, after which satellite accelerations are derived. Then, the gravity field model is computed with the AA and EBA. The accuracy of the results obtained, expressed in terms of average geoid height errors (in the latitudinal band +/- 80^o), turns out to be the following: AA (all three acceleration components are considered): 29.7 cm; AA (only the along-track component is considered): 50.3 cm; EBA: 52.1 cm. In order to compare the AA and the IVEA, we have considered a similar data set but with 15-s sampling. The models obtained are characterized by the following accuracy: IVEA: 111 cm; AA: 114 cm. These results agree very well with the theoretical expectations.