Reconstruction of fluid motion in acoustic diffraction tomography

Abstract

This paper shows how the technique of diffraction tomography can be applied for reconstructing the magnitudes and the directions of the velocity vectors as well as stationary sound-speed perturbation in a fluid flow. The analysis is based on the wave equation describing the propagation of the sound in an inhomogeneous moving fluid medium. For plane-wave irradiation and scattered field detection by a linear array of receivers, the Fourier diffraction theorem for the fluid vorticity has been obtained. Similar to the scalar case, the theorem relates the Fourier transform of the measured forward scattered data to the spatial spectrum of the vorticity and leads to the values of the two-dimensional transform of the velocity vector curl along a semicircular arc in the frequency domain. Utilizing the nonreciprocity of sound scattering from moving fluid (as opposed to the reciprocity of the elastic scattering on the scalar potential), a procedure for the separation of scalar and moving components of the flow has been proposed. It has been shown that nonreciprocity of the sound scattering by flow offers additional information which can be used to separate two independent components of the velocity vector and thus to solve the vector reconstruction problem uniquely for an arbitrary 2-D vector field without any additional constraints concerned with the flow model, what was impossible in previous algorithms based on the time-of-flight measured data. The results offer independent confirmation of those derived by Rouseff and Winters [D. Rouseff and K. B. Winters, ‘‘Two-dimensional vector flow inversion by diffraction tomography,’’ Inverse Problems 10, 687–697 (1994)].