Nonlinear structure-enhancing filtering using plane-wave prediction

Structure prediction

The method of plane-wave destruction (Claerbout, 1992) uses a local plane-wave model for characterizing the structure of seismic data. It finds numerous applications in seismic imaging and data processing (Fomel, 2002). Letting a seismic section, $\mathbf{s}$, be a collection of traces, $\mathbf{s} = \left[\mathbf{s}_1 \; \mathbf{s}_2 \; \ldots \; \mathbf{s}_N\right]^T$, the plane-wave destruction operation can be defined in a linear operator notation as

$$\mathbf{d} = \mathbf{D(\sigma)\,s}\;,$$ (1)

where $\mathbf{d}$ is the destruction residual and $\mathbf{D}$ is the nonstationary plane-wave destruction operator defined as follows

$$\left[\begin{array}{c} \mathbf{d}_1 \\ \mathbf{d}_2 \\... ...{s}_3 \\ \cdots \\ \mathbf{s}_N \\ \end{array}\right]\;,$$ (2)

where $\mathbf{I}$ stands for the identity operator, $\sigma_i$ is local dip pattern, and $\mathbf{P}_{i,j}(\sigma_i)$ is an operator for prediction of trace $j$ from trace $i$ according to the dip pattern $\sigma_i$. A trace is predicted by shifting it according to the local seismic event slopes. Minimizing the destruction residual $\mathbf{d}$ provides a method of estimating the local slopes $\sigma$ (Fomel, 2002).

The least-squares minimization of $\mathbf{d}$ is achieved by using iterative conjugate-gradient (CG) method and smooth regularization. Local dip at a fault position cannot be accurately estimated, its value will depend on the initial slope estimate and regularization.

Prediction of a trace from a distant neighbor can be accomplished by simple recursion, i.e., predicting trace $k$ from trace $1$ is simply

$$\mathbf{P}_{1,k} = \mathbf{P}_{k-1,k}\, \cdots\,\mathbf{P}_{2,3}\,\mathbf{P}_{1,2}\;.$$ (3)

Fomel (2008) applied plane-wave prediction to predictive painting of seismic images. In this paper, we use a similar construction to recursively predict a trace from its neighbors.

An example is shown in Figure 1a. The input data is borrowed from Claerbout (2008): a synthetic seismic image containing dipping beds, an uncomformity, and a fault. Figure 1b shows the same image with Gaussian noise added. Figure 1c shows local slopes measured from the noisy image by plane-wave destruction. The estimated slope field correctly depicts the constant slope in the top part of the image and the sinusoidal variation of slopes in the bottom. In the next step, we predict every trace from its neighbor traces according to the local slope, as described by Fomel (2008). We chose a total of 14 prediction steps (7 from the left and 7 from the right), which, with the addition of the original section, generated a data volume (Figure 1d). The prediction axis corresponds to index $k$ in equation 3. The volume is flat along the prediction direction, which confirms the ability of plane-wave destruction to follow the local structure. However, it still contains some discontinuous information because of the faults. In the next step, we apply nonlinear structure-enhancing filtering to process the data along the prediction direction.

sigmoid1,gnoise1,ndip1,cube
Figure 1. Noise-free synthetic image (a), noisy image (b), local slopes estimated from Figure 1b (c), and predictive data volume, where every trace is supplemented with predictions from its neighbors (d).

Nonlinear structure-enhancing filtering using plane-wave prediction