Geometrical interpretation of acoustic holography: Adaptation of the propagator and minimum quality guaranteed in the presence of errors

A large number of inverse problems in acoustics consist of a reverse propagation of the acoustic pressure measured with an array of microphones. The goal is usually to identify the acoustic source location and strength or the surface velocity of a vibrating structure. The quality of the results obtained depends on the propagation model, on the accuracy of the pressure measurements and, finally, on the inverse problem conditioning. How to quantify this quality is the issue addressed in this paper. For this purpose, a geometrical interpretation of the inverse acoustic problem is proposed. The main application will, eventually, be near-field acoustic holography (NAH), but it is expected that the proposed approach will also apply to other types of inverse acoustic problems. First, the geometrical representation of the inverse problem is proposed. The inverse problem is stated from a direct linear problem in the frequency domain. For each frequency, an overdetermined system of linear complex algebraic equations must be inverted. The concept of quality is discussed and a quality index is proposed based upon the residue of the inverse problem, solved in a mean square sense. Then, a simple one-dimensional (plane wave) acoustic example consisting of a source and two pressure measurements is used to illustrate the proposed geometrical representation of the inverse problem and the quality criterion inspired by it. In the simple example, the propagation model can be improved by searching for a reflection coefficient at the origin of the simulated hologram. This reflection coefficient is used to simulate the presence of a hidden source placed behind the source. An artificial attenuation is introduced to simulate the effect of geometrical attenuation present in real NAH problems. Again, using the geometrical representation, it is shown how, from an improved propagation model together with a given measurement noise level in the hologram, one can guarantee a certain quality level of the inverse procedure. Finally, numerical results show, in a preliminary way, how the identified source strength converges towards the exact velocity when the estimated propagation model tends to the exact propagation model.