Imaging of the SEG/EAGE Salt Model Seismic Data Using Sparse $f$– $x$ Finite-Impulse-Response Wavefield Extrapolation Filters

Abstract

In this paper, a stable explicit depth wavefield extrapolation is obtained using sparse frequency-space (f-x) finite impulse response (FIR) digital filters. The ideal impulse response of the wavefield extrapolation filter is nonsparse in nature, as will be shown. The problem of designing such filters to obtain stable images of the famous challenging Society of Exploration Geophysicists/European Association of Geoscientists and Engineers (SEG/EAGE) salt model seismic data was formulated as an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization that is convex with a quadratic constraint. Then, sparse filters were obtained by employing hard thresholding to the filter coefficients' magnitude. An ad hoc threshold value was set to the minimum of the bin location values that were used to calculate the histogram of the f-x FIR filter coefficients' magnitude. Thus, the f-x FIR filter coefficients (both real and imaginary parts), at which the filter coefficients' magnitude is below the threshold, were set equal to zero. Poststack depth imaging of the SEG/EAGE salt model data set was then performed using the explicit depth wavefield extrapolation with the proposed nonsparse and sparse L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization-based algorithms. Taking into account the amount of improved computational efficiency and compared with the results of other methods such as the phase shift plus interpolation migration algorithm, the results of the proposed sparse L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization method showed practically stable images of the salt data set. The proposed method can therefore be used to design high accuracy sparse extrapolation filters.