Asymptotic Ultrasonic Inversion Based on Using more than One Frequency

Abstract

Let the spacial sound pressure u in a body B be governed by the equation ${{\nabla }^{2}}u + {{\left\{ \omega /c\left( {\bar{r}} \right) \right\}}^{2}}u = 0$ where $c\left( {\bar{r}} \right)$ denotes the speed of sound at the point $\bar{r}$ in IR³. Given either plane wave excitation exp$\left\{ i\bar{k}\centerdot \bar{r} - i\omega t \right\}$ or spherical wave excitation exp $\left\{ i\bar{k}\left| \bar{r}-{{{\bar{r}}}_{s}} \right| \right\}/\left\{ 4\pi \left| \bar{r}-{{{\bar{r}}}_{s}} \right| \right\}$, if k = w/c₀ is large, where c₀ denotes the speed of sound in the medium surrounding B, Rytov’s approximation yields a simple expression for u in terms of f where $f\left( {\bar{r}} \right)=c_{0}^{2}/{{\mathrm{c}}^{\mathrm{2}}}\left( {\bar{r}} \right)-1$ . While the reconstruction of f based on varying the direction of $\bar{k}$ (or varying the source point ${{\bar{r}}_{s}}$ for the case of a spherical wave source) and reading u or surfaces that enclose B is a three dimensional problem, this three dimensional problem is reduced to a one-dimensional problem (i.e., “x-ray inversion”) using more than one frequency. That is, the diffraction effects may be eliminated using more than one frequency.

Research supported by U. S. Army Research Contract No. DAAG 2980-K-0089.