# The refined impedance transform for 1D acoustic reflection data

## TL;DR

This paper introduces a refined impedance transform for 1D acoustic reflection data that improves accuracy over classical methods while maintaining computational simplicity, enabling better imaging of layered media.

## Contribution

The paper derives a new impedance transform that enhances classical seismic approximation, applicable directly to data and capable of accurately reconstructing impedance profiles.

## Key findings

- The refined impedance transform is more accurate than classical methods.
- It can be applied directly to recorded data without deconvolution.
- It exactly solves the inverse problem for impedance on the far side of a layer.

## Abstract

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.