There are two main types of deblending approaches. The first is based on filtering, which simply treats the blending interference as noise and uses denoising algorithms to remove it (Mousavi and Langston, 2016a; Mousavi et al., 2016; Mousavi and Langston, 2017,2016b). These methods include the median filtering based approaches (Huang et al., 2018; Gan et al., 2016a; Huo et al., 2012; Chen, 2015), adaptive subtraction methods (Kim et al., 2009; Spitz et al., 2008), and dip filtering based methods (Hampson et al., 2008). The second is based on inversion. The inversion based algorithms take advantage of iterative solvers to solve the ill-posed inverse problem in deblending. Regularization strategies should be used to constrain the model to be the desired solution. A number of methods based on different types of constraints have been developed in the literature, e.g., the seislet constraint (Chen et al., 2014), the curvelet constraint (Zu et al., 2016; Kumar et al., 2015; Qu et al., 2016), the Fourier constraint (Abma et al., 2015), the rank constraint (Cheng and Sacchi, 2014; Zhou et al., 2017), the hybrid rank-sparsity constraint (Zu et al., 2017c), and the hybrid mask-sparsity constraint (Zhou, 2017). Here “seislet constrint” denote “seislet-domain sparsity constraint”. While the filtering based methods are more computational efficient and easier to implement, the inversion-based approaches usually lead to better deblending performance (van Borselen et al., 2012; Zu et al., 2016).

To date, there are also many reported trials for directly migrating the simultaneous source data. Because of the strong interference noise, the migrated images might contain distinct artifacts if no extra implementation step is introduced into the migration process. Berkhout et al. (2012) discussed the illumination aspect of direct imaging of simultaneous source data. Xue et al. (2016) proposed a new direct imaging method that is based on least-squares reverse time migration and structural enhancing operator (Liu et al., 2010). Different from the preconditioning constraint that was used in Dai and Schuster (2011), Xue et al. (2016) developed a new iterative solver to solve the least-squares reverse time migration related inverse problem, which is based on the shaping regularization framework. To tackle the smearing problem of structural smoothing operator in the least-squares reverse time migration method proposed by Xue et al. (2016), Chen et al. (2017) developed a local singular spectrum analysis (SSA) constraint to preserve the edges and discontinuities on the image. While those imaging methods are promising to make the future imaging framework more flexible, most reported successes are stuck on the synthetic test. A lot of researchers are still working hard to push the direct imaging technique from simulation to practical applications (Dutta, 2017; Li et al., 2018).

In this paper, we are aiming at introducing a novel median filtering method for either separating the simultaneous sources in a fast one-step way, or iteratively improving the traditional shaping regularized deblending approaches by constructing a more powerful shaping operator. Since the new method is by nature a type of median filter, it is worth mentioning some existing methods that use median filtering for deblending. Huo et al. (2012) proposed a multi-dimensional vector median filter for separating simultaneous-source data. A dip scanning step is first performed so that a vector median filter can be applied along the structural direction. Chen (2015) developed a space-varying median filter to deal with curved events in seismic data. The filter length varies spatially so that noise is removed using a long filter length and the signal is preserved using a relatively smaller filter length. Gan et al. (2016a) proposed a structure-oriented median filter to maximize the effectiveness of a median filter when processing a complicated dataset. Both methods in either Huo et al. (2012) or Gan et al. (2016a) can be considered as structure-oriented filtering, which requires an sufficiently-accurate slope estimation. However, when the slope is not properly estimated, e.g., in the case of strong ambient noise and blending noise, the structural filtering strategy would fail because those filters will tend to damage signal energy.

We investigate the reason why the structural filtering fails when the slope is not accurate enough. The principle of the structural filtering is to create a local window along the structural direction, or in other words, to create a locally flattened gather to apply the 1D median or mean filter. When the slope is accurate, e.g., using the plane-wave destruction algorithm (Fomel, 2002), the local gather is exactly flattened. However, when the slope is not precise enough, the local gather still contains curved events. In this case, we take advantage of the spatially varying median filter to deal with the slightly curved events in the flattened dimension. The proposed method is thus capable of preserving more useful energy than the traditional structural filtering method.

The paper is organized as follows, we first introduce the basics of median filtering and structure-oriented filtering. Then, we focus on introducing the proposed structure-oriented space-varying median filtering method. We also introduce the way we imbed the proposed filtering method in the shaping regularized iterative deblending framework. Next, we use several synthetic and real data examples to show the potential of the proposed method, either in obtaining a fast simultaneous source separation, or in improving the deblending performance in an iterative fashion. Finally, we draw some key conclusions at the end of the paper.

2020-02-10