Application of minimum variance filters in power-line noise attenuation

Noise is a factor that influences the reliability of the seismic data to provide a better understanding of the hydrocarbon reservoir characteristics. Coherent and random (incoherent) noises are two important types of noise that contaminate the seismic reflection data. Power-line noise falls in the coherent type of noise (Yilmaz, 2001; Sheriff and Geldart, 1995). It is a class of noise often existing in land acquisition in populated areas. It produces a characteristic 50 or 60 Hz sinusoidal noise on seismic reflection traces and covers the seismic data from reflectors. Its amplitude is time invariant, whereas the seismic data amplitude decays with time (Xia and Miller, 2000; Dingus, 2010).
Coherent noise attenuation, particularly power-line attenuation, is one of the problems in seismic reflection data processing. Several methods have been introduced for elimination of power-line noise from seismic reflection data. Xia and Miller (1998 and 2000) predicted the amplitude and phase of a harmonic using the Levenberg-Marquart method and subtracted the harmonic constructed by the predicted amplitude and phase from the seismic trace to attenuate the power-line noise.  
A notch filter is a commonly used method for power-line noise attenuation. A notch filter is a filter that contains one or more deep notches or, ideally, perfect nulls in its frequency response characteristic (Proakis and Manolakis, 2007). This filter is usually designed in a Z domain by putting the zeros of the filter’s transfer function on the unit circle with the angular frequency of the power-line noise. For resonance of the notch filter’s frequency response, poles are put on the same angular frequency of zeros near the unit circle. The transfer function of the notch filter in a Z domain to eliminate the angular frequency can be written as:
                                                        
where the value of  is chosen in a way that the maximum absolute value of  is one.
In this study, we use a minimum variance filter to attenuate the power-line noise from reflection seismic data. The basic idea of the minimum variance filter is to pass the frequency - for which the filter is designed- through the filter without distortion; and to minimize the variance of the output signal for all other frequencies. The frequency response of such a filter adapts to the spectral components of the input signal on each frequency of interest. Therefore, we could compute the filter coefficient as:
                                                                    
where  is the filter length. and  are computed as:
                                         
Because of the adaptive characteristics of the minimum variance filter, it is expected that the proposed method to produce better results than a notch filter. Therefore, to evaluate the efficiency of the minimum variance filter method to eliminate the power-line noise, we test the proposed method on both synthetic and real reflection seismic data. The output of our method is compared with the results of a notch filter. The results showed that the minimum variance filter method can attenuate the mono-frequency power-line noise from seismic traces better than the notch filter method. The results are compared in both time and frequency domains, and an improvement in power-line noise attenuation is observed using our method in both time and frequency domains.