Accelerated plane-wave destruction

Introduction

Local slope fields have been widely used in geophysical applications, such as wave-field separation and denoising (Fomel et al., 2007; Harlan et al., 1984), antialiased seismic interpolation (Bardan, 1987), seislet transform (Fomel and Liu, 2010), velocity-independent NMO correction and imaging (Cooke et al., 2009; Fomel, 2007b), predictive painting (Fomel, 2010), seismic attribute analysis (Marfurt et al., 1999), etc.

Several tools exist for local slope estimation: local slant stack (Harlan et al., 1984; Ottolini, 1983), complex trace analysis (Barnes, 1996), multiwindow dip search (Marfurt, 2006), local structure tensor (Fehmers and Höcker, 2003; Hale, 2007), and plane-wave destruction (Claerbout, 1992; Fomel, 2002). Plane-wave destruction (PWD) approximates the local wave-field by a local plane wave, and models it using a linear differential equation (Claerbout, 1992).

When plane-wave destruction is applied on discrete sampled seismic signals, the corresponding differential equation needs to be discretized by finite differences. Claerbout (1992) used explicit finite differences. In this method, plane-wave approximation of the wavefield can be seen as applying a linear finite impulse reponse (FIR) filter to the wavefield. Slope estimation is equivalent to estimating a parameter of the FIR filter. A least-squares estimator of the local slope can be obtained by minimizing the prediction error of the filter. To improve estimation performance of the explicit finite difference scheme, Schleicher et al. (2009) proposed total least-squares estimation.

The implicit finite difference scheme was applied to the differential equation by Fomel (2002). Using an infinite impulse response (IIR) filter, known as the Thiran allpass filter (Thiran, 1971), to approximate the phase-shift operator, the plane-wave destruction equation becomes a nonlinear equation of the slope. An iterative algorithm was designed to estimate the slope. In order to improve stability in the iterative algorithm, a smoothing regularization (Fomel, 2007a) of the increment can be applied at each iteration. Iterations of regularization can be time consuming, however, particularly in the 3D case.

In this paper, we prove the fact that the plane-wave destruction equation is a polynomial equation of an unkown slope. In the case of a three-point approximation of Thiran's filter, the convergence results of the iterative algorithm can be analytically analyzed. In this case, we obtain an analytical estimator of the local slope and show that the smoothing regularization can be applied on the final estimator only once. This approach reduces the computational time significantly. We present both 2D and 3D examples, which demonstrate that the proposed algorithm can obtain a slope-estimation result faster than the iterative algorithm, with a similar or even better accuracy.

Accelerated plane-wave destruction