An SVD in Spherical Surface Wave Tomography

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.

References in corpus (1)

• On the Determination of Star Bodies from Their Half-Sections

Cited by in corpus (7)

• The cone-beam transform and spherical convolution operators
• Sliced Optimal Transport on the Sphere
• Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group
• Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion
• Approximation properties of the double Fourier sphere method
• A Frame Decomposition of the Funk-Radon Transform
• A double Fourier sphere method for $d$-dimensional manifolds