  The one dimensional wave equation serves as a basic model for imaging
modalities such as seismic which utilize acoustic data reflected back from a
layered medium. In 1955 Peterson et al. described a single scattering
approximation for the one dimensional wave equation that relates the reflection
Green's function to acoustic impedance. The approximation is simple, fast to
compute and has become a standard part of seismic theory. The present paper
re-examines this classical approximation in light of new results concerning the
(exact) measurement operator for reflection imaging of layered media, and shows
that the classical approximation can be substantially improved. We derive an
alternate formula, called the refined impedance transform, that retains the
simplicity and speed of computation of the classical estimate, but which is
qualitatively more accurate and applicable to a wider range of recorded data.
The refined impedance transform can be applied to recorded data directly
(without the need to deconvolve the source wavelet), and solves exactly the
inverse problem of determining the value of acoustic impedance on the far side
of an arbitrary slab of unknown structure. The results are illustrated with
numerical examples.
