  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.
