Broadband constant‐coefficient propagators

Abstract

A B S T R A C TThe phase error between the real phase shift and the Gazdag background phase shift, due to lateral velocity variations about a reference velocity, can be decomposed into axial and paraxial phase errors. The axial phase error depends only on velocity perturbations and hence can be completely removed by the split-step Fourier method. The paraxial phase error is a cross function of velocity perturbations and propagation angles. The cross function can be approximated with various differential operators by allowing the coefficients to vary with velocity perturbations and propagation angles. These variable-coefficient operators require finite-difference numerical implementation. Broadband constant-coefficient operators may provide an efficient alternative that approximates the cross function within the split-step framework and allows implementation using Fourier transforms alone. The resulting migration accuracy depends on the localization of the constant-coefficient operators. A simple broadband constant-coefficient operator has been designed and is tested with the SEG/EAEG salt model. Compared with the split-step Fourier method that applies to either weakcontrast media or at small propagation angles, this operator improves wavefield extrapolation for large to strong lateral heterogeneities, except within the weak-contrast region. Incorporating the split-step Fourier operator into a hybrid implementation can eliminate the poor performance of the broadband constant-coefficient operator in the weak-contrast region. This study may indicate a direction of improving the split-step Fourier method, with little loss of efficiency, while allowing it to remain faster than more precise methods such as the Fourier finite-difference method.

Steep Dipping Structure Depth Imaging Using Wave Equation Based Migration Schemes on 3‐D Sparse Dataset

“…The first formal hybrid domain one-way operator used in seismic imaging is the split-step Fourier [8] . Since then the Fourier finite-difference [9] , generalized-screen [10] , broad band constant-coefficient [11] , and optimum split-step Fourier [12,13] have been developed by introducing higher-order corrections to the split-step Fourier operator. The excellence of the split-step Fourier method is its accuracy of wave amplitude and high computation efficiency in horizontally weakly inhomogenous medium, other methods can accommodate strong transverse velocity variation at the price of adding computational cost.…”

3D Fourier finite‐difference migration by alternating‐direction‐implicit plus interpolation

“…This can be seen from the parameter b = 0.5v/ω, where b is proportional to the real velocity v. As to the Fourier finite-difference method, the background velocity is handled by a phase shift associated with v 0 (z), and the phase errors caused by introducing background velocity are divided into axial and paraxial parts (Fu 2005). The time-delay correction accommodates all axial errors (Fu 2005), while the finite-difference correction term accounts for most of the paraxial errors. This can be seen from parameter b = 0.5(v − v 0 )/ω, where b is proportional to the velocity variation (v − v 0 ) rather than the real velocity v.…”

Modeling 3‐D Scalar Waves Using the Fourier Finite‐Difference Method

“…The first term in the right side of Eq. (1) is the phase-shift operator, which handles the reference velocity and performs in wavenumber domain; the second term is the time-delay operator, which handles the correction for slowness perturbations; and the third term is the finite-difference correction operator, which handles the higher order slowness perturbation for large velocity contrast and steep dip angle [23] . The first two terms consist the split-step Fourier method [5] , or the phase screen method [6] , which is widely used in small angle and weak lateral variation medium.…”