An SVD in Spherical Surface Wave Tomography
Provides a theoretical foundation and numerical method for an inverse problem in geophysics and medical imaging, though the full-data case is idealized.
The authors derive a singular value decomposition (SVD) for the spherical surface wave tomography problem with full data, and also for the case of arcs with fixed opening angle, generalizing prior work. Numerical tests demonstrate the algorithm's effectiveness.
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.