Scattering theory approach to the identification of the Helmholtz equation: A nearfield solution

Expressing the Helmholtz equation as a Schrödinger-type equation with a frequency-dependent potential allows one to use the techniques of inverse scattering to determine the potential from farfield scattering information. These techniques were designed to accomodate farfield experimental data which was generated by a plane-wave probe. Here, we consider the problem of finding the potential when (1) the scattering information is restricted to the nearfield and (2) the incident probes are assumed to be arbitrary, unknown, and not necessarily reproducible from experiment to experiment. The basic idea is to derive the on-shell T-matrix elements (or scattering amplitudes), which are required by the farfield inverse scattering techniques, from nearfield scattering information. Once the T matrix is known, the potential follows from the techniques of inverse scattering. [See V. A. Marchenko, Math. Rev. 17, 740 (1956) or M. Razavy, J. Acoust. Soc. Am. 58, 956–963 (1975) for the one-dimensional case, and A. B. Weglein, W. E. Boyse, and J. E. Anderson, to be published in Geophysics (1981) or R. G. Newton, Phys. Rev. Lett. 43, 541 (1979) for the three-dimensional case.] Recently, M. T. Silvia and A. B. Weglein [J. Acoust. Soc. Am. 69, 478–482 (1981)], presented a nearfield inverse scattering solution to the acoustic wave equation for the case when the probes were arbitrary, unknown, and not necessarily reproducible. There, the scattering measurements had to include both the field and its normal derivative. In this paper, a more direct approach to the nearfield inverse scattering problem is presented. Furthermore, for this new technique, we demonstrate that when the incident probe is known, only the field on a closed surface is required for a nearfield solution.