Stable wide‐angle Fourier finite‐difference downward extrapolation of 3‐D wavefields

Abstract

I present an unconditionally stable implicit finite-difference operator that corrects the constant-velocity phase shift operator for lateral velocity variations. My method is based on the Fourier FiniteDifference (FFD) method first proposed by Ristow and Rühl (1994). Contrary to previous results, my correction operator is stable even when the medium velocity has sharp discontinuities, and the reference velocity is higher than the medium velocity. The stability of the new correction enables the definition of a new downward-continuation method that is based on the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction starting from a reference velocity lower than the medium velocity, the second wavefield is obtained by applying the FFD correction starting from a reference velocity higher than the medium velocity. I will refer to the proposed method as Fourier Finite-Difference Plus Interpolation (FFDPI) method, because it combines the advantages of the FFD technique with the advantages of interpolation, I present a simple and economical procedure for defining frequency-dependent interpolation weights. I demonstrate that when the interpolation step is performed using these frequency-dependent 1 interpolation weights, it significantly reduces the residual phase error after interpolation, the frequency dispersion caused by the discretization of the Laplacian operator, and the azimuthal anisotropy caused by splitting.Tests on zero-offset data from the SEG-EAGE salt data set show that the proposed method improves the imaging of a fault reflection with respect to a similar interpolation scheme that employs a split-step correction for adapting to lateral velocity variations.

High Resolution/High Fidelity Seismic Imaging and Parameter Estimation for Geological Structure and Material Characterization

“…It can properly handle wave motion related phenomena. Although the original phase screen method cannot handle large scattering angles under large velocity perturbations, several modifications have been developed to improve its accuracy for wide-angles and large velocity contrasts (e.g., Ristow and Ruhl, 1994;Jin et al, 1998a;Xie and Wu, 1998;Huang et al, 1999;Huang and Fehler, 2000;Biondi, 2002). By processing the wavefield in both space and wavenumber domains, the screen method is also highly efficient.…”

“…However, real situations often contain large velocity contrasts. For scalar waves, several approaches have been employed to improve the accuracy of the screen propagator for large velocity contrasts and wide scattering angles (e.g., Ristow and Ruhl, 1994;Jin et al, 1998a;Xie and Wu, 1998;Huang and Fehler, 2000;Biondi, 2002). A similar method can be adopted for elastic waves (Xie and Wu, 1999).…”

“…The recently developed dual-domain one-way wave equation based methods (e.g., Stoffa et al, 1990;Ristow and Ruhl, 1994;Jin, et al, 1998Xie and Wu, 1998;Huang and Fehler, 2000;Biondi, 2002) provided accurate propagators for seismic wave extrapolation. To apply these propagators to illumination analysis, the central part is extracting angle related information from the propagated wavefield.…”

“…Finally, the corrected results replace up and us in equation (1) to give the modified solution. As for scalar waves, the correction is accomplished by solving an implicit finite-difference problem in the spatial domain (e.g., Ristow and Ruhl, 1994;Jin et al, 1998;Wu, 1998, 1999;Huang and Fehler, 2000;Biondi, 2002), which usually gives better stability than the wavenumber domain method. These processes can fit easily into a dual domain operation for wavefield ex trap o la t io n.…”

Controlled‐aperture wave‐equation migration

“…Many wave-equation migration methods make use of reference slownesses during wavefield downward continuation (Stoffa et al, 1990;Huang et al, 1999a, b;Fehler, 2000b, 2002;Biondi, 2002;Jin et al, 2002). The accuracy of such reference-slowness-based wave-equation migration methods decreases when the migration volume has increasing slowness perturbations.…”