On an inverse source problem connected with photo-acoustic and thermo-acoustic tomographies

An inverse source problem which aims to determine the source density p0(x) taking place in the wave equation Δp(x,t)−(1/c2)∂2p(x,t)/∂t2=−p0(x)δ′(t) is considered. One assumes that p0(x) is a function of bounded support while p(x,t) can be measured on the boundary SS of a convex domain D during a certain finite   time interval [0,T]. An explicit expression of the solution is given in terms of the surface integral of the data on SS. Two illustrative examples show the applicability as well as the effectiveness of the method. In one of these examples SS consists of a spheroid while in the other it consists of a half of the spheroid and a disc. The problem is motivated by photo-acoustic and thermo-acoustic applications.

► An explicit expression for the source density taking place in the inverse problem is given.
► An effective numerical algorithm for practical applications is proposed.
► Two illustrative examples show the applicability as well as the accuracy of the theory.