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Local Helioseismology as an Inverse Source-Inverse Scattering Problem

The inverse source and inverse scattering problems for a general inhomogeneous medium is investigated within the framework of helioseismic holography. Holographic images, defined by the method of Lindsey and Braun, or via the Porter-Bojarski equation, define a Fredholm integral equation of the first kind in terms of acoustic sources, scatterers, and absorbers. This integral equation is well posed in the sense that it can be inverted by standard constrained inversion methods. The inversion produces an image that does not distinguish between sources, scatterers, and absorbers. The inversion is necessarily approximate because of the null space of the kernel that defines the Fredholm equation. Physically, the null space corresponds to the nonradiating source contribution, as previously shown by Devaney and Porter. Numerical experiments based on a solar model show that the sidelobes in the "imaging point-spread function" are greatly reduced after inversion, such that the image is sharper than the corresponding holographic image.