Time-frequency analysis of seismic data using local attributes

Introduction

Time-frequency decomposition maps a 1D signal into a 2D signal of time and frequency, and describes how the spectral content of the signal changes with time. Time-frequency analysis has been used extensively in seismic data processing and interpretation, including attenuation measurement (Reine et al., 2009), direct hydrocarbon detection (Castagna et al., 2003), and stratigraphic mapping (Partyka et al., 1998). The widely used short-time Fourier transform (STFT) method produces a time-frequency spectrum by taking the Fourier transform of data windows (Cohen, 1995), which leads to a tradeoff between temporal and spectral resolution.

Over the past two decades, wavelet-based methods have been applied to time-frequency analysis of seismic data. Chakraborty and Okaya (1995) compare wavelet-based with Fourier-based methods for performing time-frequency analysis on seismic data and showed that the wavelet-based method improves spectral resolution. The continuous-wavelet transform (CWT) provides a time-scale map, known as a scalogram (Rioul and Vetterli, 1991), rather than a time-frequency spectrum. Because a scale represents a frequency band, Hlawatsch and Boudreaux-Bartels (1992) choose their scale to be inversely proportional to the center frequency of the wavelet, which allowed them to transform their scalogram into a time-frequency map. Sinha et al. (2005) provide a novel method of obtaining such a time-frequency map by taking a Fourier transform of the inverse CWT.

Wigner-Ville distribution (WVD) (Wigner, 1932) represents time-frequency components by using the reverse of the signal as an analysis window function. WVD varies resolution in the time-frequency plane by providing good temporal resolution at high frequencies and good frequency resolution at a low-frequency (Cohen, 1989). Applications of WVD are hindered by cross-term interference, which can be suppressed by using appropriate kernel functions. A smoothed pseudo-WVD with independent time and frequency functions as kernels achieves a relatively good resolution in both time and frequency (Li and Zheng, 2008). Wu and Liu (2009) employed smoothed pseudo-WVD with a Gaussian kernel function to reduce cross-term interference.

The S-transform is an invertible time-frequency analysis technique that combines elements of wavelet transforms and short-time Fourier transforms. The S-transform with an arbitrary and varying shape was applied to seismic data analysis by Pinnegar and Mansinha (2003). Matching pursuit (Mallat and Zhang, 1993), which was also applied in seismic signal analysis (Chakraborty and Okaya, 1995; Wang, 2007; Liu et al., 2004; Castagna et al., 2003; Liu and Marfurt, 2005), decomposes a seismic trace into a series of wavelets that belong to a comprehensive dictionary of functions. These wavelets are selected so as to best match signal structures. The spectrum of the signal is then the time-shifted sum of each of the wavelets.

Local attributes (Fomel, 2007a) can adaptively measure timevarying seismic signal characteristics in the neighborhood of each data point. Local attributes have been successfully applied to seismic image registration (Fomel and Jin, 2009), phase detection (van der Baan and Fomel, 2009; Fomel and van der Baan, 2009), and stacking (Liu et al., 2011b,2009). A natural extension of local attributes, regularized nonstationary regression, decomposes input data into a number of nonstationary components (Fomel, 2009; Liu et al., 2011a; Liu and Fomel, 2010).

In this paper, we propose a new method of time-frequency analysis in which we use time-varying Fourier coefficients to define a time-frequency map. As in regularized nonstationary regression, shaping regularization (Fomel, 2007b) constrains continuity and smoothness of the coefficients. Given the close connection between Fourier transforms and the least-squares norm, the least-squares approach to time-frequency analysis is not new, having been used previously, for example, by Youn and Kim (1985). What is novel about our approach is the use of regularization for explicitly controlling the time resolution of time-frequency representations.

The paper is organized as follows.We first describe the proposed method for time-frequency analysis. Then we show how to compute the time-varying average frequency from the time-frequency map. We use benchmark synthetic examples to test the performance of the proposed method. Finally, we apply the proposed method to channel detection and low-frequency anomaly detection in field seismic data.

Time-frequency analysis of seismic data using local attributes