Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering

Introduction

Further development of exploration and production of reservoirs increases the demand for random noise suppression. Current random noise attenuation methods are realized in either the $t-x$ or a transformed domain (Liu et al., 2009c). In the $t-x$ domain, denoising methods include stacking (Yilmaz, 2001; Mayne, 1962; Liu et al., 2009a), polynomial fitting (Liu et al., 2011; Zhong et al., 2006), and median filtering (Liu et al., 2007). All these methods fully utilize the differences of both travel time and apparent velocity between signal and noise in the $t-x$ domain. In transformed domains, denoising methods include $f-x$ predictive filtering (Canales, 1984), the wavelet transform (Zhang and Ulrych, 2003; Gao et al., 2006), the curvelet transform (Neelamani et al., 2008), and the seislet transform (Liu et al., 2009b; Fomel and Liu, 2010). These methods transform the seismic data from the $t-x$ domain to some other domain, where the signal and random noise can be separated. The noise is removed in the transformed domain prior to transformation back to the $t-x$ domain.

Canales (1984) first used $f-x$ predictive filtering to attenuate random noise. Since then, continuous efforts have been made to improve the predictive precision or to modify the conventional version to meet better the requirements set by various applications. Guo et al. (1995) proposed $f-xy$ predictive filtering in order to improve the adaptation to both 2D and 3D post-stack seismic data processing. Su et al. (1998) suggested $f-xyz$ predictive filtering, which is mainly used for random noise attenuation in a pre-stack data set. Kang et al. (2003) proposed a $f-x$ quasi-linear transform method adapted to the non-linear events of seismic data in complex regions. Unfortunately, when the subsurface is extremely complex, $f-x$ predictive filtering does not yield good results because of the large number of dip components that need to be predicted.

Huang et al. (1998) proposed a new signal processing method, which uses empirical mode decomposition (EMD) to prepare stable input for the Hilbert Transform. The essence of EMD is to stabilize a non-stationary signal. That is, to decompose a signal into a series of intrinsic mode functions (IMF). Each IMF has a relatively local-constant frequency. The frequency of each IMF decreases according to the separation sequence of each IMF. EMD is a breakthrough in the analysis of linear and stable spectra. It adaptively separates non-linear and non-stationary signals, which are features of seismic data, into different frequency ranges. Bekara and van der Baan (2009) applied $f-x$ EMD to attenuation of random and coherent noise, with good results. Cai et al. (2011) suggested using $t-f-x$ EMD to denoise seismic data on the basis of a mixed time-frequency analysis. Nevertheless, for the purpose of random noise attenuation, the $f-x$ and $t-f-x$ domain EMD methods can only be applied on NMO corrected or post-stack seismic data. With profiles containing dipping events, these methods will suppress some of the useful energy.

In this paper, we propose a new approach , termed $f-x$ empirical mode decomposition predictive filtering (EMDPF), which combines both $f-x$ EMD and $f-x$ predictive filtering. This new noise attenuation methodology can adapt to more complex seismic profiles than $f-x$ EMD and preserve more useful energy than $f-x$ predictive filtering. $f-x$ EMDPF uses an EMD based dip filter to reduce the dip components for the subsequent predictive filtering in order to improve the predictive precision.

We start this paper by reviewing the conventional $f-x$ predictive filtering theory and point out its high-prediction-error problem when the number of dip components becomes large. Then we review basic EMD theory and its application, both in data processing and the exploration geophysical fields. Finally, we suggest a way to combine the properties of both $f-x$ predictive filtering and $f-x$ EMD in order to form the new denoising algorithm, $f-x$ EMDPF. Three synthetic data sets and one real data set demonstrate that $f-x$ EMDPF can preserve much more useful energy while removing slightly less random noise than $f-x$ EMD and $f-x$ predictive filtering.

Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering