We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.
A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.
The Earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0,4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modelling of geoscientifically relevant data on the sphere involving rotational symmetry of the trial functions used for the approximation plays an important role. In this paper we deal with isotropic kernel functions showing local support and (one-dimensional) polynomial structure (briefly called isotropic finite elements) for reconstructing square--integrable functions on the sphere. Essential tool is the concept of multiresolution analysis by virtue of the spherical up function. The main result is a tree algorithm in terms of (low--order) isotropic finite elements.
The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.
Die Grundgleichungen der Physikalischen Geodäsie (in der klassischen Formulierung) werden einer Multiskalenformulierung mittels (sphärisch harmonischer) Wavelets unterzogen. Die Energieverteilung des Störpotentials wird in Auflösung nach Skala und Ort durch Verwendung von Waveletvarianzen beschrieben. Schließlich werden zur Modellierung der zeitlichen Variationen des Schwerefeldes zeit- und ortsgebundene Energiespektren zur Detektion lokaler sowie periodischer/saisonaler Strukturen eingeführt.
In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively.
The following two papers present recent developments in multiscale ocean circulation modeling and multiscale gravitational field modeling that have been presented at the 2nd International GOCE User Workshop 2004 in Frascati. Part A - Multiscale Modeling of Ocean Circulation In this paper the applicability of multiscale methods to oceanography is demonstrated. More precisely, we use convolutions with certain locally supported kernels to approximate the dynamic topography and the geostrophic flow. As data sets the French CLS01 data are used for the mean sea surface topography and are compared to the EGM96 geoid. Since those two data sets have very different levels of spatial resolutions the necessity of an interpolating or approximating tool is evident. Compared to the standard spherical harmonics approach, the strongly space localizing kernels improve the possibilities of local data analysis here. Part B - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, GGM01, UCPH2002_0.5, EGM96 Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, GGM01, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise. In this paper it is the aim of the Geomathematics Group to make these data available to other interested groups. These wavelet coefficients allow not only the reconstruction of the wavelet approximations of the gravitational potential but also of the geoid, of the gravity anomalies and other important functionals of the gravitational field. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CuP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.
The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required both the knowledge of smoothness of the solution function and the overall error behavior. We will show that we can guarantee an asymptotically optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behavior which can mostly obtained out of two input data sets. An application to the determination of the gravitational field out of satellite data will be shown.
Based on the well-known results of classical potential theory, viz. the limit and jump relations for layer integrals, a numerically viable and e±cient multiscale method of approximating the disturbing potential from gravity anomalies is established on regular surfaces, i.e., on telluroids of ellipsoidal or even more structured geometric shape. The essential idea is to use scale dependent regularizations of the layer potentials occurring in the integral formulation of the linearized Molodensky problem to introduce scaling functions and wavelets on the telluroid. As an application of our multiscale approach some numerical examples are presented on an ellipsoidal telluroid.
We introduce splines for the approximation of harmonic functions on a 3-dimensional ball. Those splines are combined with a multiresolution concept. More precisely, at each step of improving the approximation we add more data and, at the same time, reduce the hat-width of the used spline basis functions. Finally, a convergence theorem is proved. One possible application, that is discussed in detail, is the reconstruction of the Earth´s density distribution from gravitational data obtained at a satellite orbit. This is an exponentially ill-posed problem where only the harmonic part of the density can be recovered since its orthogonal complement has the potential 0. Whereas classical approaches use a truncated singular value decomposition (TSVD) with the well-known disadvantages like the non-localizing character of the used spherical harmonics and the bandlimitedness of the solution, modern regularization techniques use wavelets allowing a localized reconstruction via convolutions with kernels that are only essentially large in the region of interest. The essential remaining drawback of a TSVD and the wavelet approaches is that the integrals (i.e. the inner product in case of a TSVD and the convolution in case of wavelets) are calculated on a spherical orbit, which is not given in reality. Thus, simplifying modelling assumptions, that certainly include a modelling error, have to be made. The splines introduced here have the important advantage, that the given data need not be located on a sphere but may be (almost) arbitrarily distributed in the outer space of the Earth. This includes, in particular, the possibility to mix data from different satellite missions (different orbits, different derivatives of the gravitational potential) in the calculation of the Earth´s density distribution. Moreover, the approximating splines can be calculated at varying resolution scales, where the differences for increasing the resolution can be computed with the introduced spline-wavelet technique.