  In this article we study the inverse problem of thermoacoustic tomography
(TAT) on a medium with attenuation represented by a time- convolution (or
memory) term, and whose consideration is motivated by the modeling of
ultrasound waves in heterogeneous tissue via fractional derivatives with
spatially dependent parameters. Under the assumption of being able to measure
data on the whole boundary, we prove uniqueness and stability, and propose a
convergent reconstruction method for a class of smooth variable sound speeds.
By a suitable modification of the time reversal technique, we obtain a Neumann
series reconstruction formula.
