Discussion

Two main research tracks of deblending are the filtering method, which is fast and easy to control, and the inversion method, which is more computationally demanding but can obtain better result. Two main existing challenges for deblending are 1) damaging signals for the filtering method and 2) lacking extra constraints during inversion in order to deal with high blending fold (Mahdad, 2012). The paper mainly discusses the filtering based deblending methods, so let us first focus on the drawbacks of the filtering method. To date, the filtering based deblending methods are mainly based on some types of median filtering, e.g., Huo et al. (2012), Chen (2015), Gan et al. (2016a), and etc. Median filtering is a classic method. However, when we dig into the median filtering method further and further, we find that the implementation of a traditional median filter affects the result significantly. For example, Huo et al. (2012) propose to apply the median filter to the dip direction of the seismic events, which can preserve the useful energy well. Chen (2015) proposes to apply the median filter with different filter lengths, which can also preserve the useful energy well. Gan et al. (2016a) propose a similar method as Huo et al. (2012) but with a different structural filtering strategy and a different slope/dip estimation strategy. However, both Huo et al. (2012) and Gan et al. (2016a) depend on accurate slope/dip estimation. What if slope/dip is not estimated accurately, e.g., in extremely noisy blended data or not in CMP gathers (as required by Gan et al. (2016a))? Then, both methods fail. In addition, for Chen (2015)'s method, what if there exists steeply dipping events? No matter what thresholds of the method we use, it fails. The structure-oriented space-varying median filter is not a seemingly simple concatenation. It relieves the influence of slope/dip estimation on the following median filtering (traditional). So the dependency of an accurate slope/dip estimation no longer exists. Besides, because of the flattening (or approximate flattening when slope/dip is not accurate), the events are guaranteed to be not steeply dipping. The combination of the two methods is elegant, which solves two long-existing and fundamental problems for the “median filtering” based deblending methods. The straightforward benefit is that we can deal with the massive blended data in a very fast and amplitude-preserving way.

The structure-oriented space-varying median filter is not computationally expensive. To understand the computational cost of the structure-oriented space-varying median filter better, we can first split structure-oriented space-varying median filter into two components: space-varying median filter and structure-oriented median filter. Compared with the traditional median filter, the space-varying median filter uses a median filter twice, each with different filter length criterion, thus the computational cost of space-varying median filter is about the twice of the cost of the median filter plus the calculation of the local similarity between the initially filtered data and a removed noise section. Calculation of the local similarity is also an efficient algorithm thanks to the fast implementation of the shaping regularization framework (Fomel, 2007b). The structure-oriented median filter adds an extra flattening (or trace prediction) step into the traditional median filter and the extra computational cost is composed of local slope calculation and spatial neighbor trace prediction. According to our experience, the 2D calculation is still acceptable. More details of slope calculation are referred to Fomel (2002) and more explanations regarding the trace prediction can be found in Liu et al. (2010). The main computational cost of structure-oriented space-varying median filter comes from calculation of the local slope, trace prediction, and local similarity calculation required by space-varying median filter in each flattened local gather. All these extra computational steps can be implemented via efficient algorithms based on shaping regularization (Fomel, 2007a), thus the application of the structure-oriented space-varying median filter is not expensive. Here, we compare the computational cost in time for four different filtering methods in Table 1. The computation is done on a PC station equipped with an Intel Core i7 CPU clocked at 3.1 GHz and 16 GB of RAM. With a small data size, the computational cost for four methods are comparable. For a large data size (e.g., $800 \times 400$), the computational cost for the structure-oriented space-varying median filter is roughly 3 times that of the structure-oriented median filter. It is also worth noting that the actual calculation time is usually data dependent. The exact calculation time for the local slope estimation and similarity computation may vary according to the input data but are not expensive.

Table 1: Comparison of computational time. MF denotes median filter. SVMF denotes space-varying median filter. SOMF denotes structure-oriented median filter. SOSVMF denotes structure-oriented space-varying median filter.

Model size	50$\times$50	100$\times$100	200$\times$200	800$\times$ 400
MF (s)	0.442	0.460	0.471	0.591
SVMF (s)	0.473	0.491	0.527	0.962
SOMF (s)	0.491	0.508	0.600	1.525
SOSVMF (s)	0.660	0.848	1.769	4.237

The structure-oriented space-varying median filter is close to an adaptive method. The only parameter that needs to be chosen is the filter length. Once the filter length $L$ is fixed, the filter can adaptively adjust according to the complicated seismic structure via the two-step cascaded structure-oriented filtering and variable window strategy. Traditionally, the performance of the structure-oriented filtering highly depends on the local slope calculated from the noisy data. While adjusting the the parameters for the plane-wave destruction algorithm, e.g., the smoothing radius, regularization parameters, and the number of iterations, can somewhat improve the performance of slope estimation, it is still almost impossible to output an optimal slope map. As shown in the beginning part of the paper, a roughly calculated local slope map will result in unflattened local gathers and are not suitable for a simple median filtering. The structure-oriented space-varying median filter eases the dependency of the performance on the input slope field in that the following space-varying median filter can deal with the unflattened energy via the variable window strategy. From a different aspect, the traditional space-varying median filter does not account for the structure patterned existing in seismic data and is only effective to events with smaller dip angle. The local flattening process in structural filtering helps the space-varying median filter prepare events with small dip angle. Combining the strategies of structural filtering and variable window length, the structure-oriented space-varying median filter is adaptive to almost any input data with an arbitrary level of structural complexity. For example, in the supplementary data, we show the results of a crossing-event dataset, where we can find that the proposed method also adapts to dataset containing crossing events.