SVD solutions to inverse source problems in the time domain: Application to complex point sources

Abstract

The inverse source problem is formulated in terms of the singular value decomposition (SVD) for scalar sources that radiate time-domain wavefields. Both spatial and temporal constraints are imposed on the sources. The temporal constraint takes the form of a preselected bandlimit that is imposed through the use of sinc-interpolation. To make the inverse source problem numerically tractable, both the field exterior and interior to the source can be specified. A time-domain spherical-harmonics truncation formula is derived for the complex point source with time dependence given by analytic δ−pulses. This pulsed complex point source is then realized using an array of real point sources that are evenly distributed over a smooth array surface. The array surface and the inner surface are bodies of revolution with cross sections in the shape of superellipses. Accurate well-behaved array solutions are obtained in a straightforward manner by appropriate truncation of the singular-value expansion.