The elimination of linear and nonlinear noises such as the multiple refractions and the surface waves in the pre-stack data is generally done separately, which decreases the fidelity of the signal and the data processing efficiency. Because the wavelet function is localized in the time and frequency domain, we used a least-square adaptive subtraction method (LMS) to attenuate linear and nonlinear noises simultaneously in the 2D wavelet domain. In the LMS method, we first used part of the 2D wavelet coefficients to predict the noise model, and then subtracted it from the wavelet coefficients of the data to get the coefficients corresponding to the signal. Finally the signal is estimated by reconstructing the subtracted wavelet coefficients. We tested this method to both synthetic and real data and successfully removed those linear and nonlinear noises.
There are different kinds of strong linear and nonlinear noises such as the multiple refractions and surface waves in the pre-stack data in northwestern China. In the routine data processing, different methods have often to be used to eliminate surface waves and different linear noises. Since the linear noises often have different or even opposite slopes, the noise attenuation process has to be conducted in several passes, each for a certain slope. This is very time-consuming and potentially becomes a bottle beck for the ever-increasing seismic data volumes. And furthermore, this "cascaded" processing could severely harm the signals, and thus further affect the subsequent processing.
For the above mentioned problem, we proposed a leastsquare adaptive subtraction method (LMS) to the wavelet coefficients processing of the seismic data. The purpose was to simultaneously attenuate noises such as the multiple refractions and surface waves in the 2D wavelet domain to keep signal and improve the processing efficiency. The traditional discrete wavelet transform (DWT) exhibits different time shifts caused by the data decimation at different scales (Yu et al., 2004), which could potentially affect the adaptive subtraction. Therefore we used the timeinvariant stationary wavelet transform (SWT) (Nason et al., 1995) instead of the DWT to solve the time shift problem
The implementation of the 2D DWT is similar to the F-K transform while the 2D DWT has better characteristics in the temporal and spatial analysis. We can split the seismic data into wavelet coefficients corresponding to different frequency bands and wave numbers. Figure 1 shows the scheme of the 2D DWT. The data can be split into 4 sub bands at each scale. In this figure, the first letter represents the temporal frequency, the second letter represents spatial wave number, and the number is the decomposing scale. As the 2D DWT coefficients of the sub-bands are created by the dyadic decimation in the timespace domain, the size of each sub-band is 1/4 of the upper band, and this causes time shift variance in different subbands for different scales. 2D SWT has the same characteristic of temporal and spatial analysis as the 2D DWT, but each sub band after the 2D SWT has the same size as the upper ones, and thus it avoids the disadvantage of the time shift variance.
