  A numerical method is developed for recovering both the source locations and
the obstacle from the scattered Cauchy data of the time-harmonic acoustic
field. First of all, the incident and scattered components are decomposed from
the coupled Cauchy data by the representation of the single-layer potentials
and the solution to the resulting linear integral system. As a consequence of
this decomposition, the original problem of joint inversion is reformulated
into two decoupled subproblems: an inverse source problem and an inverse
obstacle scattering problem. Then, two sampling-type schemes are proposed to
recover the shape of the obstacle and the source locations, respectively. The
sampling methods rely on the specific indicator functions defined on
target-oriented probing domains of circular shape. The error estimates of the
decoupling procedure are established and the asymptotic behaviors of the
indicator functions are analyzed. Extensive numerical experiments are also
conducted to verify the performance of the sampling schemes.
