Seismic data decomposition into spectral components using regularized nonstationary autoregression

Introduction

Decomposing data into components has an immediate application in noise-attenuation problems in cases where signal and noise correspond to different components. The classic Fourier transform, Radon transform (Gardner and Lu, 1991), wavelet transform (Mallat, 2009), curvelet frame (Herrmann and Hennenfent, 2008), and seislet transform and frame (Fomel and Liu, 2010) are some examples of possible decompositions applicable to seismic data. A fundamental characteristic of seismic data is non-stationarity. In 1D (time dimension), seismic data are nonstationary because of wave-attenuation effects. In 2D and 3D (time and space dimensions), non-stationarity is manifested by variable slopes of seismic events. The nonstationary character of seismic data can be captured by EMD (empirical mode decomposition) proposed by Huang et al. (1998). EMD has found a number of important applications in seismic data analysis (Han and van der Baan, 2013; Battista et al., 2007; Magrin-Chagnolleau and Baraniuk, 1999; Bekara and van der Baan, 2009). However, it remains ``empirical'' because its properties are not fully understood. Daubechies et al. (2011) recently proposed an EMD-like decomposition using the continuous wavelet transform and synchrosqueezing (Daubechies and Maes, 1996). Synchrosqueezing improves the analysis but remains an indirect method when it comes to extracting spectral attributes (Herrera et al., 2013).

In this paper, I develop an efficient decomposition algorithm, which explicitly fits seismic data to a sum of oscillatory signals with smoothly varying frequencies and smoothly varying amplitudes. Such a decomposition is close in properties to the one generated by EMD but with explicit controls on the frequencies and amplitudes of different components and on their smoothness. Recently, Hou and Shi (2013,2011) developed an explicit data-adaptive decomposition based on matching-pursuit sparse optimization, an accurate but computationally expensive method. To implement a faster approach, I adopt regularized nonstationary regression, or RNR (Fomel, 2009), a general method for fitting data to a set of basis functions with nonstationary coefficients. RNR was previously applied to time-frequency decomposition over a set of regularly sampled frequencies (Liu and Fomel, 2013). When the input signal is fitted to shifted versions of itself, RNR turns into regularized nonstationary autoregression, or RNAR, and is related to adaptive prediction-error filtering. RNAR was previously applied to data regularization (Liu and Fomel, 2011) and noise removal (Liu and Chen, 2013; Liu et al., 2012). In this paper, I use it for spectral analysis and estimating different frequencies present in the data using a nonstationary extension of Prony's method of autoregressive spectral analysis (Marple, 1987; Bath, 1995). After the frequencies have been identified, I use RNR to determine local, smoothly-varying amplitudes of different components.

The paper opens with a brief review of RNR and RNAR and explains an extension of Prony's method to the nonstationary case. Next, I use simple synthetic and field-data examples to illustrate performance of the proposed technique.

Seismic data decomposition into spectral components using regularized nonstationary autoregression