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This paper investigates the inverse scattering problem of recovering a sound-soft obstacle using passive measurements taken from randomly distributed point sources. The randomness introduced by these sources poses significant challenges, leading to the failure of classical direct sampling methods that rely on scattered field measurements. To address this issue, we introduce the Doubly Cross-Correlating Method (DCM), a novel direct imaging scheme that consists of two major steps. Initially, DCM creates a cross-correlation between two passive measurements. This specially designed cross-correlation effectively handles the uncontrollability of incident sources and connects to the active scattering model via the Helmholtz-Kirchhoff identity. Subsequently, this cross-correlation is used to create a correlation-based imaging function that can qualitatively identify the obstacle. The stability and resolution of DCM are theoretically analyzed. Extensive numerical examples, including scenarios with two closely positioned obstacles and multiscale obstacles, demonstrate that DCM is computationally efficient, stable, and fast. |
