  The attenuation of ultrasound waves in photoacoustic and thermoacoustic
imaging presents an important drawback in the applicability of these
modalities. This issue has been addressed previously in the applied and
theoretical literature, and some advances have been made on the topic. In
particular, stability inequalities have been proposed for the inverse problem
of initial source recovery with partial observations under the assumption of
unique determination of the initial pressure. The main goal of this work is to
fill this gap, this is, we prove the uniqueness property for the inverse
problem and establish the associated stability estimates as well. The problem
of reconstructing the initial condition of acoustic waves in the complete-data
setting is revisited and a new Neumann series reconstruction formula is
obtained for the case of partial observations in a semi-bounded geometry. A
numerical simulation is also included to test the method.
