Median filter along the structural direction

Median filtering has a remarkable capability for attenuating spike-like noise. However, the median filter might still cause some damage to useful components when the data is highly non-stationary in the space direction. Considering that the recorded waveforms may have structural patterns, if the median filter is applied along the structural direction, the damage can be minimized. We utilize a way that is used in reflection seismic data processing to flatten the recorded waveform data in a local manner. The flattening process is based on the predictability between neighbor traces. In the following context, we will introduce in details the flattening process. Then, the median filter can be applied in the flattened dimension to maximize its effectiveness.

Flattening the waveforms is basically a data mapping process. We perform the data mapping by a recursive prediction strategy. We recall that the target of the data mapping is to create a flattened gather in each local window centered at each trace. Let the width of the window to be $2N+1$, then the central trace $\mathbf{d}_i$ can be predicted from its $j$th ($j\le N$) right trace $\mathbf{d}_{i+j}$ in such a recursive manner,

$$\mathbf{d}_i=\mathbf{P}_{i+1,i}\cdots\mathbf{P}_{i+j-1,i+j-2}\mathbf{P}_{i+j,i+j-1}\mathbf{d}_{i+j},$$ (2)

where $\mathbf{P}_{i,j}$ denotes a prediction operator to predict trace $\mathbf{d}_i$ from trace $\mathbf{d}_j$. For predicting trace $\mathbf{d}_i$ from its left trace, we use a similar recursive formula,

$$\mathbf{d}_i=\mathbf{P}_{i-1,i}\cdots\mathbf{P}_{i-j+1,i-j+2}\mathbf{P}_{i-j,i-j+1}\mathbf{d}_{i-j}.$$ (3)

Figure 1 shows a sketch of the trace prediction process for $N=1$. Each trace is associated with a local flattened window. The prediction process transforms the curving event in the local window to a flattened event. More details regarding the trace prediction and the flattening operator can be found in Liu et al. (2010). Figure 2 shows a demonstration of the structure-oriented processing. Figure 2a is the data with clear structural patterns. Figure 2b shows the locally flattened data in the third dimension by predicting each trace from its left and right neighboring traces. The median filter is then applied along the third dimension (prediction axis) to remove erratic or spike-like noise. The 2D section defined by the axes “Time (ms)” and “Position (m)” indicates the original profile so it contains curving events. It is the same as the profile shown in Figure 2a. The 2D section defined by the axes “Time (ms)” and “Prediction (trace)” indicates the flattened data. The 2D section defined by the axes “Prediction (trace)” and “Position (m)” indicates the constant Time slice. The data shown in Figure 2 can be considered as a common offset data or a migrated image. The curving events denote the curving subsurface reflectors. It is worth noting that the flattening strategy also works for crossing events, but the median filters may damage the useful energy due to the anomalous amplitude when crossing events exist.

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Figure 1. Sketch of the trace prediction process for $N=1$.

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Figure 2. A simple demonstration of structure-oriented processing. (a) Input data. (b) Locally flattened domain. The 2D section defined by the axes “Time (ms)” and “Position (m)” indicates the original profile so it contains curving events. The 2D section defined by the axes “Time (ms)” and “Prediction (trace)” indicates the flattened data. The 2D section defined by the axes “Prediction (trace)” and “Position (m)” indicates the constant Time slice.