On the Inverse Scattering Problem for the Helmholtz Equation in One Dimension
Abstract
Interest in the numerical solution of acoustic inverse scattering problems arises in a number of areas. Examples include medical diagnostics, non- destructive industrial, testing, geophysical prospecting for petroleum and minerals, and detection of earthquakes. The highly nonlinear and oscillatory nature of the problem is one of the major difficulties one encounters in the construction of effective inversion algorithms. Schemes based on global or local linearization methods, or nonlinear optimization techniques, tend to work only when the index of refraction is almost constant. They develop serious convergence problems whenever the perturbation of the index of refraction increases. Limited successes in the solution of the inverse problems have been achieved only in one dimensional cases (Gelfand-Levitan and layer striping methods are among the most notable). These methods are generally unstable numerically since the procedures used to calculate the index of refraction are ill-conditioned. We present a method for the solution of inverse problems for the one dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the index of refraction, and can be viewed as a frequency domain version of the layer-stripping approach.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 22, 1992
- Accession Number
- ADA253130
Entities
People
- Yu Chen
Organizations
- Yale University