Ellipsoidal Wavelet Representation for Gravity Field

1- The determination and the representation of the gravity field of the earth are some of the most important topics of physical geodesy. Traditionally in satellite gravity recovery problems the global gravity field of the earth is modeled as a series expansion in terms of spherical harmonics. Since the Earth’s gravity field shows heterogeneous structures over the globe, a multi-resolution representation is an appropriate candidate for an alternative spatial modeling.2- Spherical Harmonics are mostly used in global geodetic application, because they are simple and the surface of Earth is a nearly a sphere. However, an ellipsoid of rotation, i.e., a spheroid, means a better approximation of the Earth’s shape. Consequently, ellipsoidal harmonics are more appropriate than spherical harmonics to model the gravity field of the Earth. However, the computation of the coefficients of a series expansion for the geopotential in terms of both, spherical or ellipsoidal harmonics, requires preferably homogeneous distributed global data sets.3- Applying scaling and wavelet functions as spherical base functions a multi-resolution representation can be established. Scaling and wavelet functions are characterized by the ability to localize both in the spatial and in the frequency domain. Thus, regional or even local structures of the Gravity field can be modeled by means of an appropriate wavelet expansion. To be more specific, the application of the wavelet transform allows the decomposition of a given data set into a certain number of frequency – dependent detail signals.we treat in this report the ellipsoidal wavelet theory to model the Earth’s geopotential.4- Modern satellite Gravity missions such as the Gravity recovery And Climate Experiment (GRACE) allow the determination of spatio-temporal, I.e., four-dimensional Gravity fields.In the last part of this report we outline regional spatio-temporal ellipsoidal modeling. To be more specific, we represent the time-dependent part of our ellipsoidal (spatial) wavelet model by series expansions in terms of one-dimensional B-spline functions. Thus, our concept allows to establish a four-dimensional multi-resolution representation of the Gravity field by applying the tensor product technique.