  We consider the inverse source problem of thermo- and photoacoustic
tomography, with data registered on an open surface partially surrounding the
source of acoustic waves. Under the assumption of constant speed of sound we
develop an explicit non-iterative reconstruction procedure that recovers the
Radon transform of the sought source, up to an infinitely smooth additive error
term. The source then can be found by inverting the Radon transform.
  Our analysis is microlocal in nature and does not provide a norm estimate on
the error in the so obtained image. However, numerical simulations show that
this error is quite small in practical terms. We also present an asymptotically
fast implementation of this procedure for the case when the data are given on a
circular arc in 2D.
