Direct sampling methods for acoustic sources: Advancing to quantitative identification from qualitative probes

This work establishes novel uniqueness theorems and develops high quality sampling methods for reconstructing source supports from multi-frequency sparse far field patterns. For a single pair of observation directions $ \pm\hat{x} $, we demonstrate that the lines $ \{z\in\mathbb R^2|\, \hat{x}\cdot z = \hat{x}\cdot y, \, y\in A_{\hat{x}}\} $ can be uniquely determined using multi-frequency far field patterns at $ \pm\hat{x} $, where $ A_{\hat{x}} $ comprises boundary corners and points where the boundary normal is parallel to $ \hat{x} $. Furthermore, when the source support consists of polygonal and annular components, we prove its complete determination using multi-frequency far field patterns from sparse observation directions. The minimal required number of observation directions is explicitly quantified in terms of the corner and annular count. Building upon these uniqueness proofs, we propose two innovative numerical indicators for source support reconstruction. Two-dimensional numerical experiments validate both indicators' effectiveness and robustness in achieving high quality boundary recovery. Notably, the second indicator demonstrates the capability in simultaneously recovering the unknown source function.