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., ), 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.

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 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.

2020-02-10