An SVD in Spherical Surface Wave Tomography
Ralf Hielscher, Daniel Potts, Michael Quellmalz
TL;DR
This paper develops a singular value decomposition for spherical surface wave tomography, extending the Funk-Radon transform, and provides numerical algorithms for inversion with full and partial data.
Contribution
It introduces an SVD framework for surface wave tomography on the sphere, including cases with limited arc data, and generalizes previous results for specific arc configurations.
Findings
Derived an SVD for full data tomography on the sphere.
Extended SVD to arcs with fixed opening angles, ensuring injectivity.
Developed a numerical algorithm based on the SVD and demonstrated its effectiveness.
Abstract
In spherical surface wave tomography, one measures the integrals of a function defined on the sphere along great circle arcs. This forms a generalization of the Funk--Radon transform, which assigns to a function its integrals along full great circles. We show a singular value decomposition (SVD) for the surface wave tomography provided we have full data. Since the inversion problem is overdetermined, we consider some special cases in which we only know the integrals along certain arcs. For the case of great circle arcs with fixed opening angle, we also obtain an SVD that implies the injectivity, generalizing a previous result for half circles in [Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math., 126(2):117--124, 1998]. Furthermore, we derive a numerical algorithm based on the SVD and illustrate its merchantability by numerical tests.
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