  ThermoAcoustic Tomography (TAT) is a promising, non invasive, medical imaging
technique whose inverse problem can be formulated as an initial condition
reconstruction. In this paper, we introduce a new algorithm originally designed
to correct the state of an evolution model, the \emph{back and forth nudging}
(BFN), for the TAT inverse problem. We show that the flexibility of this
algorithm enables to consider a quite general framework for TAT. The backward
nudging algorithm is studied and a proof of the geometrical convergence rate of
the BFN is given. A method based on Conjugate Gradient (CG) is also introduced.
Finally, numerical experiments validate the theoretical results with a better
BFN convergence rate for more realistic setups and a comparison is established
between BFN, CG and a usual inversion method.
