Data-driven inversion/depth imaging derived from approximations to one-dimensional inverse acoustic scattering

This paper presents a new mathematical framework based on inverse scattering for the estimation of the scattering potential and its nature of a one-dimensional acoustic layered medium from single scattering data. Given the Born potential associated with constant-velocity imaging of the single scattering data, a closed-form implicit expression for the scattering potential is derived in the WKBJ and eikonal approximations. Adding physical insight, the WKBJ and eikonal solutions can be adjusted so that they conform to the geometrically derived precise solutions of the one-dimensional scattering problem recently suggested by the authors. In a layered medium, the WKBJ and eikonal approximations, in addition to providing an implicit solution for the scattering potential, provide an explicit estimate of the potential, not within the actual potential discontinuities (layer interfaces), but within the Born potential discontinuities derived by the constant-velocity imaging. This estimate of the potential is called the 'squeezed' potential since it mimics the actual potential when the depth axis is squeezed so that the discontinuities of the actual potential match those of the Born potential. It is shown that the squeezed potential can be estimated by amplitude-scaling the Born potential by an amplitude function of the Born potential. The accessibility of the squeezed potential makes the inverse acoustic scattering problem explicit and non-iterative since the estimated squeezed potential can non-linearly be stretched with respect to the depth axis so that the potential discontinuities are moved towards their correct depth location. The non-linear stretch function is a function of the Born potential. The solution is fully data-driven in the respect that no information about the medium other than the Born potential is required.

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