Thermoacoustic tomography - consistency conditions and the partial scan problem

Hybrid imaging techniques using either radiofrequency (RF) or near-infrared radiation (NIR) as excitation energy measure averages of tissue RF/NIR absorptivity over spheres centred at ultrasound (US) transducer locations on the bottom of a spherical bowl, where {z < 0}. Inversion formulae for the 'complete data' case where transducers measure all over the bowl weight data from the lower hemisphere more heavily for reconstructions points also satisfying {z < 0}. We derive consistency conditions for the data and use them to compute the unmeasured data corresponding to transducer locations on the top of the bowl. This process is clearly unstable, but somewhat tempered by the reconstruction's 1/r weighting.