SUMMARY

The separation of signal and noise is a key issue in seismic data processing. By noise we refer to the incoherent noise that is present in the data. We use the recently introduced multi-scale and multidirectional curvelet transform for suppression of random noise. The curvelet transform decomposes data into directional plane waves that are local in nature. The coherent features of the data occupy the large coefficients in the curvelet domain, whereas the incoherent noise lives in the small coefficients. In other words, signal and noise have minimal overlap in the curvelet domain. This gives us a chance to use curvelets to suppress the noise.

INTRODUCTION

In seismic data, recorded wave fronts (i.e. reflections), arise from the interaction of the incident wave field with inhomogeneities in the Earth’s subsurface. The wave fronts can get contaminated with noise during aquisition or even due to processing problems. The forward problem of seismic denoising can be written as:

where y is the known noisy data, m is the unknown model (noise-free data) and n is the noise. Our objective is to recover m. As seismic data are contaminated with random noise, many methods have been developed to suppress such incoherent noise. Some of these techniques discriminate between signal and noise based on their frequency content (bandpass filter) or they use some sort of prediction filter (F-X Deconvolution). Such methods do remove the random noise but at the same time they may remove some of the signal energy (Neelamani et al., 2008). More recently, wavelet transform based processing has been applied for noise suppression. The wavelet transform is good for point-like events (one-dimensional singularity). However, for higher dimensions (e.g seismic data), wavelets fail to give a parsimonious representation. In this work, we use the recently introduced curvelet transform to suppress random noise. The signal and noise have minimal overlap in the curvelet domain (Neelamani et al., 2008). This makes the curvelet transform the ideal choice for detecting wave fronts and suppressing noise. For this work, we cast the denoising problem as an inverse problem. We compare this method with hard thresholding and soft thresholding of curvelet coefficients for a fixed threshold. We start with a brief introduction to curvelets, followed by a presentation of our algorithm and application to synthetic data.

CURVELETS

Curvelets are amongst one of the latest members of the family of multiscale and multidirectional transforms (Candés et al., 2006). They are tight frames (energy preserving transform) with moderate redundancy. A curvelet is strictly localized in frequency and pseudo-localized in space; i.e., it has a rapid spatial decay. In the physical domain, curvelets look like little plane waves that are oscillatory in one direction and smooth in perpendicular directions. In the F-K domain each curvelet lives in an angular wedge. Each curvelet is associated with a position, frequency bandwidth and an angle. Different curvelets at different frequencies, angles and positions are shown in Fig. 1. The construction of curvelets is such that any object with wavefront-like structure (e.g seismic images) can be represented by relatively few significant transform coefficients (Candés et al., 2006).

This content is only available via PDF.
You can access this article if you purchase or spend a download.