Variable Background Born Inversion by Wavefield Backpropagation

Abstract

The inverse scattering problem for an acoustic medium is formulated by using the variable background Born approximation. A constant density acoustic medium is probed by a wide-band point source, and the scattered field is observed along a curved receiver array located outside the region where the medium velocity is different from the assumed background velocity function. The solution proposed relies on the introduction of a backpropagated field. This field is obtained by using a finite-difference scheme backward in time to backpropagate into the medium the scattered field observed along the receiver array. The backpropagated field is imaged at the source travel times, giving an image of the same type as obtained by reverse-time finite-difference migration techniques. The gradient of this image is then taken along rays linking the source to points in the medium and, after scaling, this gives the reconstructed potential. To relate the reconstructed potential to the true scattering potential, high frequency asymptotics and an additional approximation introduced by Beylkin [1] are used. These approximations reduce the validity of the reconstruction procedure to the high wavenumber region. With these approximations it is shown that, at a given point, the reconstructed potential corresponds to the convolution of the true potential with a weighting function obtained by partially reconstructing an impulse from its projections inside a cone. The angular range of this cone is totally determined by the geometry of the receiver array and by the relative location of the source with respect to the point considered. In the special case when the receiver array surrounds the domain where the scattering potential is located, it is found that, within the Born approximation, the reconstructed potential recovers exactly the high wavenumber part of the Fourier transform of the true potential. It is expected that, for a wide class of problems, the reconstruction technique described in this paper will be computationally more efficient than the generalized Radon transform (GRT) inversion method proposed by Beylkin [1] and Miller, Oristaglio, and Beylkin [2], [3].