Simultaneous Inversion of Velocity and Density Profiles
MASSACHUSETTS INST OF TECH CAMBRIDGE LAB FOR INFORMATION AND DECISION SYSTEMS
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The multidimensional inverse scattering problem for an acoustic medium is considered within the homogeneous background Born approximation. The objective is to reconstruct simultaneously the velocity and density profiles of the medium. The medium is probed by wide-band plane-wave sources, and the time traces observed at the receivers are appropriately filtered to obtain generalized projections of the velocity and density scattering potentials, which are related to the velocity and density variations in the medium. The generalized projections are weighted integrals of the scattering potentials in the two-dimensional geometry the weighting functions are concentrated along parabolas. The reconstruction problem for the generalized projections is formulated in a way similar to the problem of x-ray, or straight- line tomography. The solution is expressed as a back-projection operation followed by a two dimensional space-invariant filtering operation. In the Fourier domain, the resulting image is a inlinear combination of the velocity and density scattering potentials, where the coefficients depend on the angle of incidence of the probing wave. Therefore, two or more different angles of incidence are necessary to solve for the velocity and density scattering potentials separately. The technique of defining a back-projection operator and relating it to the unknown medium for the case of zero-offset problems,, where projections over circles arise, was introduced by Fawcett 1985. With a similar technique, Ozbek Levy 1987 solved the velocity inversion problem in constant-density acoustic media under plane-wave illumination, where parabolic projections are the data. This work extends this work to the joint reconstruction of velocity and density. Only the 2D case is presented here, for the 3D case and more detailed development, see Ozbek Levy 1988.
- Numerical Mathematics