A reversible transform for seismic data processing

Introduction

It is well known that the Fourier transform has a variety of properties that make it useful for signal processing in geophysics (Yilmaz, 2001). The Fourier transform essentially rearranges the data an input function into its complex frequency components (Karl, 1989). This can be done because continuous functions and their discretely-sampled equivalents can be expressed in terms of the Fourier basis. The organization of data in the Fourier domain allows many otherwise complex operations to be applied easily. Properties described by the Faltung, Weiner-Khintchine, and shift theorems (Sneddon, 1995) can exploit this organization because of another property, described by the inversion theorem (Sneddon, 1995), which states that the Fourier transform is exactly reversible.

Many individual seismic data processing steps also simply rearrange input data, and can therefore be viewed as transforms. Barring destructive processes such as muting or band-limited frequency filtering, even an entire data processing flow is itself simply transformations the input data into a corrected set. However, conventional methods used to perform processing steps of this type are usually not implemented as transforms. Instead, they are implemented with data-mapping algorithms that require interpolation (Harlan, 1982).

Most conventional interpolation schemes contradict the primary assumption of the Fourier transform, as they do not assume the continuous equivalent of the recorded data is constructed with the Fourier basis. Instead, they assume some other basis in order to design efficient but approximate interpolants (Karl, 1989). These methods not only have the risk of immediately altering the data, but they are also inherently irreversible. Applying or removing processing steps in this way causes a small amount of information loss. With quantitative analysis methods as a goal of modern seismic data processing, it is important to preserve as much of the input information as possible during the flow.

In order to address the problem of data loss due to interpolation-based methods, we propose a reversible one-dimensional transform for the implementation of seismic data processing steps. The proposed transform provides a general framework for unitary and pseudounitary operators common to seismic data processing and imaging (Biondi and Claerbout, 1985; Claerbout, 1992). This framework is developed in the context of nonstationary filtering (Margrave, 1998). Implementing imaging steps as reversible transforms does not require interpolation in the continuous case, and correctly assumes the Fourier basis in agreement with the Fourier transform for interpolation in the discrete case. Although several seismic data processing operations have been individually described by transforms of exactly this type (Claerbout, 1992; Margrave and Ferguson, 1999; Margrave, 1998,2001), the transform here is the general form (Burnett and Ferguson, 2008c; Burnett, 2007).

Since the Fourier basis is used, the forward transform is theoretically exactly invertible. Just as the inversion theorem quantitatively justifies the Fourier domain as a valid processing domain, a reversible transform for data processing steps justifies the use of corrected data sets as if they are in valid processing domains themselves. Any processing step that can be viewed as a nonstationary shift can be easily implemented by this transform. Seismic imaging steps are naturally useful for separating signal from noise, so they offer familiar, exploitable organizations of data (McMechan and Sun, 1991; Yu et al., 2005). Therefore, the proposed reversible transform for seismic data processing offers a useful set of quantitatively valid domains in which to work.

The general transform we derive is based on classical Fourier transform theory (Sneddon, 1995; Papoulis, 1962), and the theory of nonstationary filtering (Margrave, 1998). We first develop the general form for the forward data processing transform, and then derive the inverse of the transform. Following the transform development, we implement the normal moveout correction as an example, and then we discuss how conventional interpolation methods relate to the proposed transform.

A reversible transform for seismic data processing