My investigations

In this chapter I explore three discontinuity attributes: standard edge enhancement, plane-wave misfit, and prediction error.

I illustrate each approach on one synthetic (Figure 4) and two seismic images (Figures  7 and 11). The first seismic image depicts a Gulf salt dome. The faults of the image are sharp and well defined. Bahorich et al illustrates the original coherency attribute at the same image. The second seismic image shows a North Sea horst and is the more challenging test case. The faults are smooth and difficult to find in the seismic time slice. Bednar used the same image in the publication that describes his discontinuity attribute.

 

Label Name Computation Output format 4l 4lGradient methods

$$\vert \nabla g(x) \vert$$ Grad Gradient magnitude scalar at each pixel

$$\nabla^2 g(x)$$ Lap Laplace scalar at each pixel 4l 4lDip misfit methods

$${\bf r} = {\bf p} \times \nabla g$$ DXG Cross-product 3 component vector at each pixel

$$\vert {\bf r} \vert$$ DXGLN Magnitude of r scalar at each pixel

$$({\bf p} \times \nabla) \cdot ({\bf p} \times \nabla) g$$ DXGAA Backprojection of r scalar at each pixel 4l 4lWaveform misfit methods

$$Cor_z[g({\bf x}), f({\bf p} \cdot {\bf x})]$$ PPC Correlation of estimate scalar for each trace

and original of patch 4l 4lPrediction-error methods

$$A \, g$$ Pef3d 3-D PEF scalar at each pixel

$${\bf \epsilon} = [A_x,A_y,A_z] g$$ XPef 3 2-D PEF 3 component vector at each pixel

$$\epsilon \vert \epsilon \vert$$ XPefLN scalar at each pixel

$$\epsilon [A_x,A_y,A_z] \cdot \epsilon$$ XPefAA scalar at each pixel

The Table 1 lists the various methods that I include in this publication. The standard edge enhancement techniques fail to differentiate sedimentary beds and boundaries of sedimentary packages (Figure 16). However, two-dimensional edge enhancement within a time slice takes advantage of the dominant horizontal dip of sedimentary strata and yields reasonable fault maps (Figure 23).

Of all my discontinuity attributes, the trace correlation of an image region and its locally best-fitting plane wave consistently yields the clearest fault maps (Figures 33 and 34). Other measures of plane-wave misfit that I tested generated only inferior fault images (e.g., Figure 27).

The algorithm of my plane-wave correlation is, similar to the computation of the commercial coherency attribute by Bahorich and Farmer 1995. Both processes certainly generate a similar discontinuity map of the Gulf salt dome. My plane-wave estimation is related to Symes'  differential semblance. I derived the plane-wave estimation by generalizing a dip estimation scheme by Claerbout 1994. Bednar derives his discontinuity attribute from Claerbout's dip estimation, too.

Prediction-error is an unreliable and noisy discontinuity attribute. The discontinuity map of the salt dome (Figure 44) shows the radiating faults, and even resolves features within the salt. Unfortunately, the image is obscured by noise. Even worse, the prediction error filter predicts and removes almost all features of the smoother North Sea horst image (Figure 45).

Prediction-error images tend to be white Jain (1989), since the filter removes the predictable correlated components of its input. Claerbout 1992 introduced a combination of two two-dimensional filters that detect and suppress a three-dimensional plane-wave. I added a third prediction-error filter, which generalizes Claerbout's intuitive, original formulation. The three prediction-errors are linked to the cross-product expression of my plane-wave formulation.

The prediction-error filters' removal of all image features in the North Sea case suggests limiting the predictive power of the filters. In earlier studies Schwab et al. (1996); Schwab (1997), I limited the number of iterations, and I enlarged and pre-whitened the training patches. However, the prediction-error maps were still white.

The data dependent attribute measures - misfit of local plane wave and the prediction error - depend on local estimation within stationary local image patches. Since patches that contain a discontinuity are nonstationary, the attributes usually yield a large output anywhere within such a patch. (Figures 26 or 36. Note, that the image is composed of about 300 patches. A side of the cubic patch is 12 pixels long; about one seventh of the total data cube edge of 80 pixels.) The schemes detect if a patch contains a discontinuity, but do not pinpoint the discontinuity within the patch. Consequently, the data dependent schemes are limited to a resolution the size of their patches.