Diffraction tomography for inhomogeneities in layered background medium

Abstract

Diffraction tomography was originally formulated for a constant velocity background medium. A variable background medium, e.g., layered, with embedded finer scale heterogeneities is a more practical model for subsurface reservoirs than the uniform background. The variable background of large scale variations may be determined from well logs or transmission tomography. To image the finer scale heterogeneities, we have developed a Fourier diffraction back‐propagation method for point sources in a layered background. The method is based on the normal mode solution to the acoustic wave equation in cylindrical coordinates. The Fourier spectrum of the scattered fields is first decomposed into contributions from different layers. Then, a selection rule is applied to sort out the heterogeneity spectrum of the individual layers. The selection rule relates the scattered field in diffraction space to the spectrum of the heterogeneities, i.e., a Fourier diffraction theorem for layered media. The theorem differs from its counterpart for a uniform background medium by a matrix filter that reduces to unity as the stratification degenerates to a uniform background. A reconstruction algorithm based on this theorem is implemented and tested for an arbitrary layered background. The theory deals directly with point sources; therefore, the resulting algorithm does not require application of the “2.5-D correction” to field data as required in previously published diffraction tomography algorithms. Results obtained for both synthetic and field data demonstrate that an inversion with spatial resolution on the order of a wavelength can be achieved for crosswell data. The computations involved are much more efficient than those of traveltime tomography or crosswell migration. Unlike migration or CDP mapping, the diffraction tomography algorithm provides quantitative estimates for fine scale velocity.