A unified approach for de-noising and interpolation of seismic data in the frequency-wavenumber (f-k) domain is introduced. First an angular search in the f-k domain is carried out to identify a sparse number of dominant dips. Then, an angular mask function is designed based on the identified dominant dips. The mask function is utilized with the least-squares fitting principle for optimal de-noising or interpolation of data. Synthetic and real data examples are provided to examine the performance of the proposed method.
Noise elimination and interpolation of seismic data are important topics of research in seismic data processing community. Several important signal processing techniques have been utilized for de-noising and interpolation purposes. For instance, the prediction filters are used by Canales (1984) and Spitz (1991) in the frequency-space (f-x) domain for de-noising and interpolation of data, respectively. Other methods such as projection filters (Soubaras, 1994), Singular Value Decomposition (Trickett, 2003), Cadzow de-noising (Cadzow and Ogino, 1981; Trickett and Burroughs, 2009), and Singular Spectrum Analysis (Oropeza and Sacchi, 2009) have also been used for random noise attenuation in the f-x domain. All of the f-x de-noising methods are based on the assumption that the spatial signals at each single frequency are composed of a sum of a limited number of complex harmonics. The Fourier transform plays a substantial role in most de-noising and interpolation methods. A frequently used domain for seismic data denoising is the frequency-wavenumber (f-k) domain. One of the prominent attributes of the f-k domain is the separation of signals according to their dip. This property makes the f-k domain a suitable option for ground-roll elimination due to the distinctive dip information of ground-roll and reflection seismic data. The most commonly used f-k domain de-noising method is the dip filtering method. The spatial interpolation of seismic records can also be interpreted as a noise elimination problem in the f-k domain. Depending on the sampling function, noise introduced by decimation of data in the f-k domain can be considered as incoherent (for random spatial sampling) or coherent (for regular spatial sampling). The f-x domain seismic-trace interpolation methods have been proposed by Spitz (1991) and Porsani (1999). These methods utilize the low frequency portion of data for a robust and alias-free interpolation of high frequencies. Gulunay (2003) have introduced the f-k equivalent of f-x interpolation methods by creating a mask function from low frequencies. In both f-x and f-k domain interpolation methods, the given frequencies are interpolated independently. Recently, Curry (2009) has introduced an f-k interpolation method using a Fourierradial adaptive thresholding strategy in an attempt to utilize the continuity of events along the frequency axis. This article introduces another f-k domain method which utilizes information from all desired frequencies for robust interpolation or de-noising of seismic data. The proposed f-k method in this paper consists of three steps. The first step is an angular search for a range of dips, over all of the frequencies, to identify the dominant energy dips.