Noise Attenuation: A Hybrid Approach Based on Wavelet Transform

Field seismic data can contain many sorts of noise and interferences. If the noise is stronger in order of magnitude than the signal, most techniques often applied in seismic processing will be severely affected. Denoising methods, even very robust schemes such as physical wavelet frame denoising (PWFD), are not exceptions. In this paper, we present a robust, data adaptive and fast 1D wavelet transform (WT) method to attenuate this kind of noise and combine it with the 2D PWFD method to achieve a hybrid strategy for noise attenuation. Introduction Strong noise in field seismic data sometimes can be several orders of magnitude larger than the signal. Attenuating this kind of noise is very important for the afterwards processing. The wavelet transform (WT) has been largely used to attenuate noise in image and signal processing, including geophysics. In most cases it is used to depress random noise, following a thresholding theory developed by Donoho (1993). The idea is that, given a signal with random noise, this noise will map mainly to small WT coefficients because its energy is distributed along all scales. The signal, on the contrary, maps to large WT coefficients because it maps to some scales more than others, due to its coherence. Therefore a simple scheme to attenuate noise is to compute adequate thresholds for every scale and zero out or downweight the coefficients below these thresholds. This procedure has been successfully applied in many fields and several statistical estimators for the thresholds have been formulated for the case of Gaussian noise. However, if the noise is not Gaussian, coherent and especially of strong amplitude, the above criteria will break. In this situation, we must be careful to explore denoising methods to avoid signal distortion. In this paper, we introduce a hybrid two-step approach to attenuate high amplitude noise in seismic gathers. First, we calculate the 1D WT of the data and, instead of removing low coefficients, we remove or attenuate very large WT coefficients. This filtering process is performed scale by scale, modifying the threshold values at every scale. This kind of filtering has been originally applied to filter impulsive noise in magnetotelluric data [Trad and Travassos2000]. When applying this procedure to seismic data, strong coherent noise is, in general, not totally removed, but partly attenuated leaving signal untouched. By applying in a second step the 2D wavelet frame denoising filtering [Zhang2000] both coherent and random noise are depressed. Discrete WT filtering Wavelets are used to represent a time series in the same way as trigonometric functions in Fourier Analysis. One important difference is that in wavelet analysis the scale in which we look at the data plays a crucial role: wavelets process data at different scales, or resolutions. In wavelet analysis one adopts a wavelet prototype function called the analyzing or mother wavelet. This work uses the Daubechies wavelet [Daubechies1992] as the analyzing wavelet. This is an orthogonal, fractal wavelet with a compact representation. The class of orthogonal wavelets is widely used in multiresolution analysis. As any particular set of wavelets, the Daubechies is specified by a set of coefficients. In particular we have generated this set of wavelets with 4 to 20 coefficients. The best results in this work were obtained with the Daubechies wavelet with 20 coefficients, which is a very smooth wavelet. This is so because the strong interferences in seismic gathers are often surface waves which have smooth waveform and low frequency. Let ψ(t) be the mother wavelet defined in the space of square-integrable functions over the real numbers L(R). Through dilations and translations of ψ(t) one constructs an orthogonal basis