## 65D07 Splines

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#### Dokumenttyp

- Preprint (4)
- Diplomarbeit (1)
- Dissertation (1)

#### Schlagworte

- Inverses Problem (3)
- Spline (3)
- Gravimetrie (2)
- Harmonische Spline-Funktion (2)
- Mehrskalenanalyse (2)
- reproducing kernel (2)
- reproduzierender Kern (2)
- Approximation (1)
- CHAMP <Satellitenmission> (1)
- Cauchy-Navier equation (1)

This paper provides a brief overview of two linear inverse problems concerned with the determination of the Earth’s interior: inverse gravimetry and normal mode tomography. Moreover, a vector spline method is proposed for a combined solution of both problems. This method uses localised basis functions, which are based on reproducing kernels, and is related to approaches which have been successfully applied to the inverse gravimetric problem and the seismic traveltime tomography separately.

The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.

We show the numerical applicability of a multiresolution method based on harmonic splines on the 3-dimensional ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, e.g. different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, e.g. determined out of the splitting function of free oscillations, to an approximation of the whole mass density function. The applicability of the presented tool is demonstrated by several test calculations based on simulated gravity values derived from EGM96. The method yields a multiresolution in the sense that the localization of the constructed spline basis functions can be increased which yields in combination with more data a higher resolution of the resulting spline. Moreover, we show that a locally improved data situation allows a highly resolved recovery in this particular area in combination with a coarse approximation elsewhere which is an essential advantage of this method, e.g. compared to polynomial approximation.

In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves.

Satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG), respectively, are two measurement principles in modern satellite geodesy which yield knowledge of the first and second order radial derivative of the earth's gravitational potential at satellite altitude, respectively. A numerical method to compute the gravitational potential on the earth's surface from those observations should be capable of processing huge amounts of observational data. Moreover, it should yield a reconstruction of the gravitational potential at different levels of detail, and it should be possible to reconstruct the gravitational potential from only locally given data. SST and SGG are modeled as ill-posed linear pseudodifferential operator equations with an injective but non-surjective compact operator, which operates between Sobolev spaces of harmonic functions and such ones consisting of their first and second order radial derivatives, respectively. An immediate discretization of the operator equation is obtained by replacing the signal on its right-hand-side either by an interpolating or a smoothing spline which approximates the observational data. Here the noise level and the spatial distribution of the data determine whether spline-interpolation or spline-smoothing is appropriate. The large full linear equation system with positive definite matrix which occurs in the spline-interplation and spline-smoothing problem, respectively, is efficiently solved with the help of the Schwarz alternating algorithm, a domain decomposition method which allows it to split the large linear equation system into several smaller ones which are then solved alernatingly in an iterative procedure. Strongly space-localizing regularization scaling functions and wavelets are used to obtain a multiscale reconstruction of the gravitational potential on the earth's surface. In a numerical experiment the advocated method is successfully applied to reconstruct the earth's gravitational potential from simulated 'exact' and 'error-affected' SGG data on a spherical orbit, using Tikhonov regularization. The applicability of the numerical method is, however, not restricted to data given on a closed orbit but it can also cope with realistic satellite data.

SST (satellite-to-satellite tracking) and SGG (satellite gravity gradiometry) provide data that allows the determination of the first and second order radial derivative of the earth's gravitational potential on the satellite orbit, respectively. The modeling of the gravitational potential from such data is an exponentially ill-posed problem that demands regularization. In this paper, we present the numerical studies of an approach, investigated in [24] and [25], that reconstructs the potential with spline smoothing. In this case, spline smoothing is not just an approximation procedure but it solves the underlying compact operator equation of the SST-problem and the SGG-problem. The numerical studies in this paper are performed for a simplified geometrical scenario with simulated data, but the approach is designed to handle first or second order radial derivative data on a real satellite orbit.