Seismic data interpolation using plane-wave shaping regularization

2-D synthetic test

The quarterdome (qdome) 3-D seismic image was created by Claerbout (1993). This model has been used for previous interpolation tests including those using both local plane-wave prediction filters (Fomel, 1999) and steering filters (Clapp et al., 1998; Clapp, 2000). The model and corresponding dip field are displayed in Figure 1 along with the decimated well data and local slope calculation. In order to demonstrate the local properties of PWS, its the impulse response is tested along with PWD and PWC. By adding spikes to 50 random locations in the image and applying the operators, we can see the anisotropic response aligned with the local slope. A plot of the convergence rates is given in Figure 3. PWS converges in far fewer iterations - 6 as compared to the 55 required by PWD and 28 of PWC. By applying 2-D interpolation with PWD, PWC, and PWD; we can see a side-by-side comparison of the three methods after 60 iterations (Figure 4). Each interpolation scheme produces very similar results. The error sections - Figures 4d, 4e, and 4f - indicate comparable accuracy as well.

reflectors2
Figure 1. Synthetic example reproduced from Clapp et al. (2004). A 2-D slice of the quarterdome - qdome - synthetic seismic image is given in (a) from which (b) the decimated data - wells - and (c) the dip field are derived. The nine traces were preserved in the application of the mask to produce the wells, and (c) is found by taking the local slope.

impulses
Figure 2. 2-D impulse response. 50 spikes are placed randomly through qdome and hit with the operators of (a) PWD, (b) PWC, and (c) PWS.

Matrix1Comparison
Figure 3. Convergence rate for 2-D synthetic example. This plot shows a comparison of the model error (2-norm) versus iteration number. The points denoted with "s" are derived from interpolation using PWS. The symbols "d" and "c" correspond to the same calculation using PWD and PWC respectively.

q-compa,q-compb,q-compc,q-compd,q-compe,q-compf
Figure 4. Reconstruction of 2-D synthetic data with interpolation from (a) plane-wave destruction, (b) plane-wave construction, and (c) plane-wave shaping. Parts (d), (e), and (f) correspond to the respective data misfit, computed with $\left \vert\left \vert d-d_i\right \vert\right \vert _2^2$ for the true solution $d$ and reconstruction $d_i$ at iteration $i$. Convergence was reached after 55, 28 and 6 iterations respectively.

Seismic data interpolation using plane-wave shaping regularization