We consider an inverse obstacle scattering problem for acoustic waves in a stratified medium. Consider a bounded obstacle described by a domain imbedded in a stratified medium.

By stratified medium here we mean that the refraction index of the medium is a function of one variable.

Suppose we send in a number of incident waves, and measure the scattered waves in one or more location. Can we determined (reconstruct) the shape of the obstacle (scatterer)? Many similar inverse problems in wave propagation assume that the medium is homogeneous. However, in a stratified medium sound waves may be trapped by acoustic ducts and caused to propagate horizontally. Therefore, the scattered energy flux is not spread out spherically. Instead there are free-wave far-field and guided-wave far-field. (See Waves in a homogeneous medium for a demonstration of continuous waves from a point source in a homogeneous medium, and Waves in a stratified mediumfor a demonstration of continuous waves from a point source in a stratified medium, respectively.)

Due to the nature of stratified medium, some results which are valid for inverse scattering in a homogeneous medium may not be true for a stratified medium. For example, a scatterer is determined uniquely by far-field patterns in an open set corresponding to infinitely many linearly independent incident waves. This may not be true for stratified medium, unless the far-field is detected in a window large enough to contain both free-waves and guided-waves. Or two open sets are needed; one for free-wave far-field and one for guided-wave far-field.

Our research concerns the problems of determining the shape of scatterer from incomplete far-field data in a stratified medium. We have developed a theory for scattering of acoustical waves in a stratified medium. Theoretical analysis shows that incident waves and scattered waves have a reciprocal relation which provides a way to compensate the incomplete far-field data.

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