The high resolution sampling methods for acoustic sources from multi-frequency far field patterns at sparse observation directions

This work is dedicated to novel uniqueness results and high resolution sampling methods for source support from multi-frequency sparse far field patterns. With a single pair of observation directions $\pm\hat{x}$, we prove that the lines $\{z\in\mathbb R^2|\, \hat{x}\cdot z = \hat{x}\cdot y, \,y\in A_{\hat{x}}\}$ can be determined by multi-frequency far field patterns at the directions $\pm\hat{x}$, where $A_{\hat{x}}$ denotes a set containing the corners of the boundary and points whose normal vector to the boundary is parallel to $\hat{x}$. Furthermore, if the source support is composed of polygons and annuluses, then we prove that the support can be determined by multi-frequency far field patterns at sparse directions. Precisely, the lowest number of the observation directions is given in terms of the number of the corners and the annuluses. Inspired by the uniqueness arguments, we introduce two novel indicators to determine the source support. Numerical examples in two dimensions are presented to show the validity and robustness of the two indicators for reconstructing the boundaries of the source support with a high resolution. The second indicator also shows its powerful ability to determine the unknown source function.