Spherical fast multiscale approximation by locally compact orthogonal wavelets

Abstract

Using a stereographical projection to the plane we construct an O(N log(N )) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB R ⃝implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.

Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion

“…This despite there being a wealth of available constructions on the sphere. [16][17][18][19][20][21][22][23][24][25][26][27][28] However, the cited studies are mostly concerned with surfaces, not volumes, and many of them require custom design and special algorithms. For the application to global seismic tomography that we envisage, we take advantage of the ease and flexibility of existing Cartesian constructions by implementing standard algorithms on a spherical-to-Cartesian map proposed by Ronchi et al (1996), a grid that has long since proven its utility in the geosciences and beyond.…”

Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity

“…This despite, or perhaps because, there being a wealth of available constructions relevant for global geophysics, in other words: on the sphere (e.g. Schröder & Sweldens 1995; Narcowich & Ward 1996; Antoine et al 2002; Holschneider et al 2003; Freeden & Michel 2004a; Fernández & Prestin 2006; Hemmat et al 2005; Schmidt et al 2006; Starck et al 2006; McEwen et al 2007; Wiaux et al 2007; Lessig & Fiume 2008; Bauer & Gutting 2011), if not on the ball. Indeed, inasmuch as they involve the analysis of cosmological data, satellite observations or computer‐generated images, the above studies are mostly concerned with surfaces, not volumes.…”