Vector AR filters: Extending f‐ x random noise attenuation to the multicomponent case

Abstract

SUMMARYWe utilize vector autoregressive (VAR) modeling for multicomponentseismic record noise elimination. VAR modeling of multicomponentsignals can be used to identify potential coherencies between each pairof components. This can be effectively utilized to further improve sig-nal processing concepts such as de-noising. This article proposes anextension of traditional f kxrandom noise attenuation to 3-componentseismic records. For this purpose, we develop a 3-component VARmodel in the f x domain. Examples are used to show the effective-ness of the proposed method.INTRODUCTIONIn exploration seismology we often deal with multicomponent seismicrecords. For instance, 3-component geophones simultaneously recordtwo horizontal components and one vertical component of the incidentwave-field. The common approach in dealing with multicomponentdata is to process each component separately. However, Vector autore-gressive (VAR) modeling presents a promising application for process-ing of multicomponent data. It can provide not only a robust analysisof each individual component but also valuable information about thecoherency between each component (Pagano, 1978; Hrafnkelsson andNewton, 2000). In this article we will investigate the application ofVAR modeling for random noise elimination of seismic records.A large class of de-noising methods utilize Fourier transform. The un-derlying assumption behind Fourier-based de-noising methods is thesearch for a few dominant energy harmonics. This is achieved bypreserving a finite number of frequency or wavenumber componentsin the Fourier domain. Frequency-space (f-x) domain methods com-prise a very large group of seismic data interpolation and de-noisingmethods. For instance, prediction filters are used by Canales (1984)and Spitz (1991) in the frequency-space (f-x) domain for de-noisingand interpolation of data, respectively. Other methods such as projec-tion filters (Soubaras, 1994), Singular Value Decomposition (Trickett,2003), Cadzow de-noising (Cadzow, 1988; 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 thespatial signals at each single frequency are composed of a superposi-tion of a limited number of complex harmonics.The sparse Fourier assumptions for the single component data arealso valid for each individual component of the multicomponent data.Therefore, one can process each of those components separately basedon their own merits. However, independent processing of each com-ponent ignores the inherent coherency between them. Therefore, uti-lizing methods such as VAR modeling, which can exploit this depen-dency and coherency between components, can improve the data pro-cessing output to a great extent.In this article we adapt VAR modeling for noise elimination of multi-component signals. The details of computing VAR models and theirspectral interpretations are presented. Next, we introduce an approachfor noise elimination using the VAR operators. Finally, the proposedde-noising method is applied to seismic records in the f-xdomain. Syn-thetic 1D and seismic examples are provided to examine the perfor-mance of the proposed VAR de-noising method.THEORYVector Autoregressive (VAR) operatorsFor a multicomponent signal with length N, we define the M-orderforward VAR model as (Leonard and Kennett, 1999)g =