Applications of plane-wave destruction filters

Introduction

Plane-wave destruction filters, introduced by Claerbout (1992), serve the purpose of characterizing seismic images by a superposition of local plane waves. They are constructed as finite-difference stencils for the plane-wave differential equation. In many cases, a local plane-wave model is a very convenient representation of seismic data. Unfortunately, early experiences with applying plane-wave destructors for interpolating spatially aliased data (Claerbout, 1992; Nichols, 1990) demonstrated their poor performance in comparison with that of industry-standard $F$-$X$ prediction-error filters (Spitz, 1991).

For each given frequency, an $F$-$X$ prediction-error filter (PEF) can be thought of as a $Z$-transform polynomial. The roots of the polynomial correspond precisely to predicted plane waves (Canales, 1984). Therefore, $F$-$X$ PEFs simply represent a spectral (frequency-domain) approach to plane-wave destruction This powerful and efficient approach is, however, not theoretically adequate when the plane-wave slopes or the boundary conditions vary both spatially and temporally. In practice, this limitation is addressed by breaking the data into windows and assuming that the slopes are stationary within each window.

Multidimensional $T$-$X$ prediction-error filters (Claerbout, 1999,1992) share the same purpose of predicting local plane waves. They work well with spatially aliased data and allow for both temporal and spatial variability of the slopes. In practice, however, $T$-$X$ filters appear as very mysterious objects, because their construction involves many non-intuitive parameters. The user needs to choose a raft of parameters, such as the number of filter coefficients, the gap and the exact shape of the filter, the size, number, and shape of local patches for filter estimation, the number of iterations, and the amount of regularization. Recently developed techniques for handling non-stationary PEFs (Crawley et al., 1999) performed well in a variety of applications (Guitton et al., 2001; Crawley, 2000), but the large number of adjustable parameters still requires a significant level of human interaction and remains the drawback of the method.

Clapp et al. (1998) have recently revived the original plane-wave destructors for preconditioning tomographic problems with a predefined dip field (Clapp, 2001). The filters were named steering filters because of their ability to steer the solution in the direction of the local dips. The name is also reminiscent of steerable filters used in medical image processing (Freeman and Adelson, 1991; Simoncelli and Farid, 1996).

In this paper, I revisit Claerbout's original technique of finite-difference plane-wave destruction. First, I develop an approach for increasing the accuracy and dip bandwidth of the method. Applying the improved filter design to several data regularization problems, I discover that the finite-difference filters often perform as well as, or even better than, $T$-$X$ PEFs. At the same time, they keep the number of adjustable parameters to a minimum, and the only estimated quantity has a clear physical meaning of the local plane-wave slope. No local windows are required, because the slope is estimated as a smoothly variable continuous function of the data coordinates.

Conventional methods for estimating plane-wave slopes are based on picking maximum values of stacking semblance and other cumulative coherency measures (Neidell and Taner, 1971). The differential approach to slope estimation, employed by plane-wave destruction filters, is related to the differential semblance method (Symes and Carazzone, 1991). Its theoretical superiority to conventional semblance measures for the problem of local plane wave detection has been established by Symes (1994) and Kim and Symes (1998).

Applications of plane-wave destruction filters