Determination of the sound speed and an initial source in photoacoustic tomography

In thermoacoustic and photoacoustic tomography an electromagnetic wave is sent through a medium, heating it and therefore generating an elastic expansion that in turns generates an acoustic wave that is measured outside the medium. The general problem is to recover both the inhomogeneous sound speed and the initial pressure from the boundary measurements of the solution of the acoustic wave equation with a single measurement of the pressure. We show that one can recover both the sound speed and the initial pressure assuming a monoticitiy condition on the sound speed that includes, under suitable assumptions, the physically interesting case of some class of piecewise constant sound speeds. We also establish a link between this problem and the corresponding transmission eigenvalues problem.