On the inverse scattering problem for the Helmholtz equation in one-dimension

Miscellaneous · Thu Dec 31 00:00:00 EST 1992

OSTI ID:121351

Chen, Yu

Interest in the numerical solution of acoustic inverse scattering problems arises in 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. The principal advantage of the procedure is that if the scatterer to be reconstructed has m {>=} 1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/a{sup m}, where a is the highest frequency for which scattering data are available. Thus, a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n{sup 2}), where n is the number of features to be recovered and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several numerical examples. Generalizations of this approach in two dimensions are discussed.

• Research Organization: Yale Univ., New Haven, CT (United States)
• OSTI ID: 121351
• Resource Relation: Other Information: TH: Thesis (Ph.D.); PBD: 1992
• Country of Publication: United States
• Language: English

Related Subjects

99 MATHEMATICS
COMPUTERS
INFORMATION SCIENCE
MANAGEMENT
LAW
MISCELLANEOUS
66 PHYSICS
42 ENGINEERING NOT INCLUDED IN OTHER CATEGORIES
ACOUSTICS
NUMERICAL SOLUTION
SOUND WAVES
SCATTERING
ONE-DIMENSIONAL CALCULATIONS
ALGORITHMS
GEOPHYSICAL SURVEYS
DIAGNOSTIC TECHNIQUES
EARTHQUAKES
NONLINEAR PROBLEMS
CONVERGENCE