The misfit of a local plane-wave approximation to a seismic image region - the topic of this chapter - is my most successful discontinuity attribute (Figures 33 and 34). An image location is considered free of a discontinuity if it is locally well-approximated by a plane wave. Discontinuities violate this local plane-wave assumption.
I compute the discontinuity attribute volume in a series of steps. First I partition the image volume into small sub-volumes, patches. Next, I estimate the best-fitting plane wave for each patch. The misfit between the plane-wave approximation and the original patch yields a local discontinuity attribute. Finally, I merge all local discontinuity patches into a single discontinuity volume comparable to the input image volume. A later chapter discusses this patching of nonstationary images in greater details.
The three-dimensional plane-wave approximation within a patch estimates the plane-wave slope by minimizing a cross-product expression of partial derivatives. The plane wave's one-dimensional waveform is approximated by stacking along the estimated plane-wave slope. Claerbout 1992 describes an equivalent approach for dip picking in two-dimensional seismic sections. The approach can be extended to a more precise iterative estimation ()
This chapter details the estimation of a best-fitting plane-wave function to a given image patch. The subsequent section discusses various measures of plane-wave fit based on either the residual of the plane-wave dip estimation or the residual of the plane-wave waveform estimation.