Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-[Formula: see text] model to describe wave propagation in heterogeneous media. We have developed two efficient modeling schemes to solve the decoupled fractional Laplacian viscoacoustic wave equation. Both schemes can cope with spatial variable-order fractional Laplacians conveniently, and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media. Both schemes are based on fast Fourier transform, and have a spectral accuracy in space. The first scheme solves a modified wave equation with constant-order fractional Laplacians instead of spatial variable-order fractional Laplacians. Due to separate discretization of space and time, the first scheme has only first-order accuracy in time. Differently, the second scheme is based on an analytical wave propagator, and has a higher accuracy in time. To increase computational efficiency of the second modeling scheme, we have adopted the low-rank decomposition in heterogeneous media. We also evaluated the feasibility of applying an empirical approximation to approximate the fractional Laplacian that controls amplitude loss during wave propagation. When the empirical approximation is applied, our two modeling schemes become more efficient. With the help of numerical examples, we have verified the accuracy of our two modeling schemes with and without applying the empirical approximation, for a wide range of seismic quality factor ([Formula: see text]). We also compared computational efficiency of our two modeling schemes using numerical tests.

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SUMMARY In seismic data processing, the median filter is usually applied along the structural direction of seismic data in order to attenuate erratic or spike-like noise. The performance of a structure-oriented median filter highly depends on the accuracy of the estimated local slope from the noisy data. When local slope contains significant error, which is usually the case for noisy data, the structure-oriented median filter will still cause severe damages to useful energy. We propose a type of structure-oriented median filter that can effectively attenuate spike-like noise even when the local slope is not accurately estimated, which we call structure-oriented space-varying median filter. A structure-oriented space-varying median filter can adaptively squeeze and stretch the window length of the median filter when applied in the locally flattened dimension of an input seismic data in order to deal with the dipping events caused by inaccurate slope estimation. We show the key difference among different types of median filters in detail and demonstrate the principle of the structure-oriented space-varying median filter method. We apply the structure-oriented space-varying median filter method to remove the spike-like blending noise arising from the simultaneous source acquisition. Synthetic and real data examples show that structure-oriented space-varying median filter can significantly improve the signal preserving performance for curving events in the seismic data. The structure-oriented space-varying median filter can also be easily embedded into an iterative deblending procedure based on the shaping regularization framework and can help obtain much improved deblending performance.