On the determination of sets supporting unknown sources for the wave equation from radiated fields

TL;DR

This paper develops a method to identify minimal space-time regions containing unknown acoustic sources from wave measurements, leveraging wave propagation properties and support restrictions to address non-uniqueness in inverse source problems.

Contribution

It introduces a novel approach that combines wave propagation characteristics with support restrictions to determine small domains containing unknown sources from measurements.

Findings

Method effectively identifies small support regions for sources.

Supports theoretical uniqueness under certain restrictions.

Addresses non-uniqueness issues in inverse wave problems.

Abstract

Given near or far field wave measurements generated by some unknown time- and space-dependent acoustic source, we seek to rapidly determine a domain in space-time, as small as possible, that contains the support of a source radiating these measurements. As for any inverse source problem, this task is challenging without further restrictions on the source, particularly due to the infinite-dimensional space of silent sources radiating zero measurements. This first causes non-uniqueness, that is, the source in general cannot be determined uniquely, and second prevents the computation of, e. g., a largest set that must contain the support of the source. To determine small domains containing the support of some source that radiates given measurements, we exploit that solutions to the wave equation propagate along characteristics. We further indicate restrictions on the support of a source that allow to theoretically characterize this support uniquely from near or far field measurements.