Seislet transform and seislet frame

Introduction

Wavelet transforms have found many applications in science and engineering (Mallat, 2009), including geophysics (Kazemeini et al., 2009; Dessing, 1997; Foster et al., 1994; Wapenaar et al., 2005). The power of wavelet transforms, in comparison with the classic Fourier transform, lies in their ability to represent non-stationary signals. As a result, wavelets can provide a compact basis for non-stationary data decomposition. Having a compact basis is useful both for data compression and for designing efficient numerical algorithms.

A number of wavelet-like transforms that explore directional characteristics of images have been proposed in the image analysis literature (Welland, 2003). Among those transforms are bandelets (Pennec and Mallat, 2005), contourlets (Do and Vetterli, 2005), curvelets (Starck et al., 2002), directionlets (Velisavljevic, 2005), shearlets (Guo and Labate, 2007), etc. Unlike isotropic wavelets, directional transforms attempt to design basis functions so that they appear elongated anisotropically along 2-D curves or 3-D surfaces, which might be characteristic for an image. Therefore, these transforms achieve better accuracy and better data compression in representing non-stationary images with curved edges. Curvelets are particularly appropriate for seismic data because they provide a provably optimal decomposition of wave-propagation operators (Candés and Demanet, 2005). Application of the curvelet transform to seismic imaging and seismic data analysis has been an area of active research (Chauris and Nguyen, 2008; Herrmann and Hennenfent, 2008; Douma and de Hoop, 2007; Herrmann et al., 2007).

Although the wavelet theory originated in seismic data analysis (Morlet, 1981), none of the known wavelet-like transforms were designed specifically for seismic data. Even though some of the transforms are applicable for representing seismic data, their original design was motivated by different kinds of data, such as piecewise-smooth images. In this paper, we investigate the possibility of designing a transform tailored specifically for seismic data. In analogy with the previous naming games, we call such a transform the seislet transform (Fomel, 2006).

In seismic data analysis, it is common to represent signals as sums of sinusoids (in 1-D) or plane waves (in 2-D) with the help of the digital Fourier transform (DFT). Certain methods for seismic data regularization, such as the anti-leakage Fourier transform (Xu et al., 2005), the Fourier reconstruction method (Zwartjes and Gisolf, 2007; Zwartjes and Sacchi, 2007; Zwartjes and Gisolf, 2006), or POCS (Abma and Kabir, 2006) rely on the ability to represent signals sparsely in the transform domain. The digital wavelet transform (DWT) is often preferred to the Fourier transform for characterizing digital images, because of its ability to localize events in both time and frequency domains (Mallat, 2009; Jensen and la Cour-Harbo, 2001). However, DWT may not be optimal for describing data that consist of individual sinusoids or plane waves. It is for those kinds of data that the seislet transform attempts to achieve an optimally compact representation.

The approach taken in this paper follows the general recipe for digital wavelet transform construction known as the lifting scheme (Sweldens, 1995). The lifting scheme provides a convenient and efficient construction for digital wavelet transforms of different kinds. The key ingredients of this scheme are a prediction operator and an update operator defined at different digital scales. The goal of the prediction operator is to predict regular parts of the input data so that they could be subtracted from the analysis. The goal of the update operator is to carry essential parts of the input data to the next analysis scale. Conventional wavelet transforms use prediction and update operators designed for characterizing locally smooth images. In this paper, we show how designing prediction and update tailored for seismic data can improve the effectiveness of the transform in seismic applications. In 1-D, our prediction and update operators focus on predicting sinusoidal signals with chosen frequencies. In 2-D and 3-D, we use predictions along locally dominant event slopes found by the method of plane-wave destruction (Claerbout, 1992; Fomel, 2002). One can extend the idea of the seislet transform further by changing the definition of prediction and update operators in the lifting scheme (Liu and Fomel, 2009).

The seislet transform decomposes a seismic image into an orthogonal basis which is analogous to the wavelet basis but aligned along dominant seismic events. In 1-D, the classic wavelet transform is equivalent to the seislet transform with a zero frequency. In 2-D, the wavelet transform in the horizontal direction is equivalent to the seislet transform with a zero slope. When more than one frequency or more than one slope field are employed for analysis, the seislet transform turns into an overcomplete representation or a tight frame.

The paper is organized as follows. We start by reviewing the digital wavelet transform and the lifting scheme. Next, we modify the lifting scheme to define 1-D and 2-D seislet transforms. Finally, we generalize the transform construction to a frame. We illustrate applications of both the seislet transform and the seislet frame with synthetic and field data examples.

Seislet transform and seislet frame