This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1 Surface-Consistent Sparse Multichannel Blind Deconvolution of Seismic Signals Nasser Kazemi, Emmanuel Bongajum, and Mauricio D. Sacchi Abstract—We describe a method that allows for blind surface consistent estimation of the source and receiver wavelets of seismic signals. This is very relevant for surface-consistent deconvolution where current processing standards focus on the removal of the source and receiver effects under the minimum phase assumption. The proposed method, which is an extension of the Euclid deconvo- lution method, employs an iterative algorithm that simultaneously estimates the source and receiver wavelets that are consistent with the data. Unlike most deconvolution methods, the algorithm requires no prior phase assumptions. Another important feature of the algorithm is that we questioned the Gaussian density assumption of the reflectivity series and instead implemented a sparse regularizer to constrain the solution space of our desired reflectivity series. In other words, we assume that the reflectivity series can be cast as a sparse vector with few nonzero coefficients. Index Terms—Blind deconvolution, nonminimum phase, optimization, sparsity, surface consistent. I. I NTRODUCTION L AND seismic data suffer from near-surface effects. Lateral variations in near-surface heterogeneity contribute toward the variations in the reflection amplitudes and time delays. These distortions in the seismic record can be primarily viewed as a compound effect of the heterogeneous environment in the vicinity of the source and receiver locations. Consequently, surface-consistent deconvolution (SCD) has been commonly used for processing land seismic data in order to estimate and remove source and receiver effects. Contrary to single-trace deconvolution methods, SCD is a multichannel deconvolution approach. Multichannel deconvolution methods take advantage of the statistical properties of the signal of interest, which has been recorded under different channels, and results in a more reliable and stable solution [1]–[3]. Various types of SCD methods exist, with each using different assumptions and de- compositions of the seismic traces in order to take advantage of data redundancy [4]. While some decompositions use midpoint binning [4], others use reciprocity of the medium response [5]. Manuscript received March 18, 2015; revised August 4, 2015 and November 5, 2015; accepted December 22, 2015. N. Kazemi and M. D. Sacchi are with the Department of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada (e-mail: kazemino@ualberta.ca; msacchi@ualberta.ca). E. Bongajum was with the Department of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada. He is now with Schlumberger, Calgary, AB T2G 0P6, Canada (e-mail: ebongajum@slb.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2513417 SCD can be performed in the log/Fourier domain [4]–[8] or in the time domain [9]. Reference [9] also argues that the validity of the surface-consistent decomposition model that is adopted is very important in order to obtain more stable and correct estimates of the individual deconvolution operators. When SCD is performed in the log/Fourier domain, the problem becomes linear with respect to the amplitude spectrum of source and receiver components. The log/Fourier domain allows for the separation of the amplitude spectrum of each component. How- ever, SCD in the log/Fourier domain does not have control on phase estimation, and the minimum phase assumption must be invoked. Most authors have focused on the surface-consistent decom- position of the spectral amplitudes [4]–[8], where the minimum phase assumption is usually used. A comprehensive SCD ap- proach warrants the correct estimation of the amplitude and phase information of the individual deconvolution operators. Unfortunately, very little focus has been directed at the surface- consistent decomposition of the phase spectra [7]. Our work expands the Euclid deconvolution [3], [10]–[12] to the case of SCD. In other words, the homogeneous system of equations arising in Euclid deconvolution is reformulated in terms of SCD, and an alternating optimization algorithm is proposed to estimate source and receiver wavelets. The estimated operators, in turn, permit the computation of inverse filters to equalize the prestack volume. II. THEORY In order to explain the surface-consistent extension of the Euclid deconvolution, let us consider a simple acquisition geometry containing two shots (i.e., i and j ) and two receivers (i.e., m and n). The z -transform of the noise-free seismic trace d im created by source i and recorded by receiver m can be written as D im (z )= S i (z )G m (z )R im (z ) (1) where S i (z ), G m (z ), and R im (z ) represent the z -transforms of the source function, receiver response, and medium response, respectively. Following the decomposition model in (1), one can also write the z -transform of another trace within same shot gather as D in (z )= S i (z )G n (z )R in (z ). (2) Dividing (1) by (2) leads to D in (z )G m (z )R im (z ) − D im (z )G n (z )R in (z )=0. (3) 0196-2892 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.