B2: Multi-Scale Representation of the Gravity Field

Usually the gravitational potential of the Earth is modeled in a series expansion in terms of spherical harmonics. Whereas the current and future gravity field satellite missions like Champ, Grace and Goce provide almost global data sets of high resolution and accuracy, the terrestrial data sets are non-uniformly distributed. Among other things this item gives reason why the gravity field should be modeled by means of a multi-scale approach.

A multi-scale representation can be achieved with the help of the wavelet transform. In comparison to spherical harmonics spherical wavelet functions are characterized by their ability to localize. Hence, regional or even local structures of the gravity field can be modeled appropriately using a spherical wavelet approach.

The basic idea of the multi-scale representation is the decomposition of the signal to be analyzed into a certain number of so-called detail signals. Each detail signal is related to a specific resolution level or frequency-band. Hence, the smoothed signals of two successive resolution levels differ from each other by a detail signal.

Fig. 1 Multi-scale representation of the disturbing potential computed from Champ data. The sum of the detail signals means an approximation (reconstruction) of the disturbing potential.
Fig. 2 (right) a) Regional high-frequency part of EGM 96 gravity anomalies b) Reconstruction with wavelet functions. Due to the localizing property of the wavelet function there are only very small deviations (c) between the reconstruction and the input signal.

The future work in this project will be concentrated mainly (1) on the problem of regularization, (2) on the application of parameter estimation methods and (3) on the consideration of the distribution of the scattered observation points.

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