Brian DeFacio

The connection between time‐ and frequency‐domain three‐dimensional inverse scattering methods

We relate three‐dimensional scattering theory for the time‐independent Schrodinger equation without spherical symmetry to scattering theory for the plasma‐wave equation (PWE). We review a recent inverse scattering method for the PWE and find the corresponding method for the Schrodinger equation. We then review Newton’s three‐dimensional Marchenko method for the Schrodinger equation and transform it to the corresponding PWE method. The resulting time‐domain hyperbolic method clarifies the role of causality in Newton’s important recent work.

Three-dimensional inverse scattering: plasma and variable velocity wave equations

Exact equations governing three‐dimensional time‐domain inverse scattering are derived for the plasma wave equation and the variable velocity classical wave equation. This derivation was announced for the plasma wave equation in a short note by the authors. That work was motivated by Newton’s three‐dimensional generalization of Marchenko’s equation. This paper gives the details of the new derivation and extends it to the classical wave equation. For the time domain derivation in this paper, the scattering region is assumed to have compact support and smoothly joins the surrounding three‐dimensional infinite medium. The derivation contains the following ingredients: (1) a representation of the solution at a point in terms of its values on a large sphere, (2) the far‐field form of the Green’s function, (3) causality, and (4) information carried in the wave front of the solution. The derivation of the classical wave inverse scattering equation requires that the velocity in the scattering region be less than ...

A new equation of scattering theory and its use in inverse scattering

Abstract This paper considers scattering for the Schrodinger equation and for the wave equation with variable speed. A new integral equation, which we call the unifying equation, is derived. This unifying equation is used to remove a difficulty in earlier work on the inverse scattering problem for the wave equation. In addition it allows a new representation for the speed in terms of scattering quantities.

A new mathematical formulation of accelerated observers in general relativity. II

The observation of a general vector field based on exp* is employed to obtain formulas for the coordinate velocity and coordinate acceleration of a geodesic particle. Our results are shown to reduce to those based on a parallel transport definition of observation in special relativity. In general relativity the difference between the expressions for the coordinate velocity and coordinate acceleration derived from the two definitions of observation is given in terms of the Riemann curvature tensor.

WAVELETS AND THEIR APPLICATION TO DIGITAL SIGNAL PROCESSING IN ULTRASONIC NDE

As the use of digital based ultrasonic testing systems becomes more prevalent, there will be an increased emphasis on the development of digital signal processing techniques. In the past, various Fourier based digital signal processing approaches have been formulated and applied in the ultrasonic nondestructive evaluation (NDE) research community. In many cases, the inherent inability of Fourier methods to handle non-stationary signals has been exposed as the Fourier methods are applied to non-stationary ultrasonic signals. Our intent is to investigate the application of wavelet based signals processing techniques to a variety of problems in ultrasonic NDE. Wavelet methods have a number of potential advantage over Fourier methods including the inherent ability of wavelets to deal with non-stationary signals.

A Perturbation Method for Inverse Scattering in Three-Dimensions Based on the Exact Inverse Scattering Equations

The detection and characterization of macroscopic flaws, such as cracks in solids are fundamental goals of nondestructive evaluation. Many inspection methods use scattered electromagnetic or ultrasonic waves. These methods rely explicitly on the development of inverse scattering theory. This theory seeks to determine the geometrical and material properties of flaws from scattering data.

Physical basis of three-dimensional inverse scattering for the plasma wave equation

Exact inverse-scattering equations are derived for the time-domain plasma-wave equation. Care is taken to motivate each step of the derivation by elementary physical arguments. The inverse method in this formulation is shown to depend on (1) causality, (2) the far-field properties of the Green function, and (3) the representation theorem.

An Application of Wavelet Signal Processing to Ultrasonic Nondestructive Evaluation

In this paper we present a flaw signature estimation approach which utilizes the Wiener filter [1–5] along with a wavelet based procedure [6–15] to achieve both deconvolution and reduction of acoustic noise. In related ealier work by Patterson et al. [6], the wavelet transform was applied to certain components of the Wiener filter, and coefficient chopping was used to reduce acoustic noise. In the approach that we present here, the wavelet transform is applied individually to the real part and to the imaginary part of the scattering amplitude estimate determined by application of a sub-optimal form of the Wiener filter. This wavelet transform takes the real and imaginary parts, respectively, from the typical Fourier frequency domain to a wavelet phase space. In this new space, the acoustic noise shows significant separation from the flaw signature making selective pruning of wavelet coefficients an effective means of reducing the acoustic noise. The final estimates of the real and imaginary parts of the scattering amplitude are determing via an inverse wavelet transform.

Feynman’s Operational Calculus As A Generalized Path Integral

Feynman’s heuristic prescription for forming functions of noncommuting operators is discussed along with methods for making his ideas rigorous. The emphasis is on one method and on the extent to which Feynman’s operational calculus can be viewed as a generalized path integral.

Classical, linear, electromagnetic impedance theory with infinite integrable discontinuities

The impedance theory is formulated for classical, linear electromagnetic scattering from a compact obstacle with a finite number of nonintersecting boundaries. The boundaries are allowed to support infinite, integrable discontinuities in electromagnetic response and the compact regions can depend on space and time. The direct scattering problem is discussed, generalizing recent results by Sabatier and collaborators for the scalar impedance acoustic problem to classical electromagnetism. A chain of Maxwell scattering equations are derived for the direct scattering problem. Two kinds of ambiguities of electromagnetism at a fixed angle of incidence are found to arise, one from discontinuities in electromagnetic material properties, and the other is from time dispersion. Cases are mentioned when parts of the scattering medium are allowed to have time‐dependent motions. This is in contrast to the case of scalar acoustics where ambiguities are intrinsic to certain infinite families of values of Young’s modulii.