Journal ArticleDOI
Constructive approximation and numerical methods in geodetic research today : an attempt at a categorization based on an uncertainty principle
Willi Freeden,Volker Michel +1 more
TLDR
In this article, the authors present a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets, as well as the uncertainty principle for the quantification of space and momentum localization of trial functions.Abstract:
Current activities and recent progress on constructive approximation and numerical analysis in physical geodesy are reported upon Two major topics of interest are focused upon, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantification of space and momentum localization of trial functions The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and waveletsread more
Citations
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Hilbert Space Methods in Signal Processing
TL;DR: In this article, the authors introduce the concept of Hilbert spaces and introduce operators, including bounded operators, compact operators, and integral operators, for signal concentration and joint spatio-spectral analysis on 2-sphere.
Journal ArticleDOI
Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity
Frederik J. Simons,Ignace Loris,Guust Nolet,Ingrid Daubechies,Sergey Voronin,J. S. Judd,P. Vetter,J. Charléty,Cédric Vonesch +8 more
TL;DR: In this article, a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data is proposed, which is numerically efficient and can be implemented with parallel computing.
Journal ArticleDOI
Spatiospectral localization of global geopotential fields from the Gravity Recovery and Climate Experiment (GRACE) reveals the coseismic gravity change owing to the 2004 Sumatra‐Andaman earthquake
TL;DR: In this paper, a significant gravity change after the 2004 Sumatra-Andaman earthquake is found in the time series of the GRACE coefficients after localization with a single band-limited window centered near the north of the island of Sumatra.
Journal ArticleDOI
Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth's gravitational field at satellite height
TL;DR: In this article, a multiscale regularization technique using scaling functions and wavelets constructed for corresponding integro-differential equations is introduced and its numerical applications are discussed, where the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential model.
Journal ArticleDOI
Thin plate spline interpolation
TL;DR: In this paper, the concept of the thin plate spline from the plane to the sphere is carried over to the spherical TPS, where the idea of an elastic shell is attributed with the bending energy and the external energy.
References
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Spline models for observational data
TL;DR: In this paper, a theory and practice for the estimation of functions from noisy data on functionals is developed, where convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework.
The lifting scheme: A construction of second generation wavelets
TL;DR: The lifting scheme is presented, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function, and can be adapted to intervals, domains, surfaces, weights, and irregular samples.
Journal ArticleDOI
The lifting scheme: a construction of second generation wavelets
TL;DR: The lifting wavelet as discussed by the authors is a simple construction of second generation wavelets that can be adapted to intervals, domains, surfaces, weights, and irregular samples, and it leads to a faster, in-place calculation of the wavelet transform.
Journal ArticleDOI
Wavelet and multiscale methods for operator equations
TL;DR: Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile.
Book
Constructive Approximation on the Sphere: With Applications to Geomathematics
TL;DR: In this article, the authors present a set of tensorial approximations of spherical harmonic functions for tensor and tensorial approximation methods, respectively, using the Gabor and Toeplitz Transform and Discrete Wavelet Transform.