Summary

In this abstract we propose a new method to solve the twoway wave equation which we call the “Explicit Marching” (EM) algorithm. By introducing a square-root operator, the two-way wave equation can be formulated as a first-order PDE in time which is similar to the one-way wave equation. To solve the new wave equation, we use a stable explicit extrapolation method in the time direction and handle the lateral velocity variations in both the space and wavenumber domains. Unlike the conventional explicit finite-difference schemes, this new method does not suffer from numerical stability or numerical dispersion problems. Therefore, it can be used to design a cost-effective and high quality reverse-time migration.

Introduction

Reverse-time migration based on directly solving the twoway wave equation (Whitmore, 1983; Baysal et al., 1983) provides a natural way to deal with large lateral velocity variation and imposes no dip limitations on the images. Recently it has attracted considerable attention and is considered to be a method of choice for imaging complex subsurface structures. Reverse-time migration has historically been more expensive than one-way wave equation migration (Claerbout, 1971) because of large memory requirements and a larger number of computations. In the literature, reverse-time migration is implemented by solving the twoway wave equation with different finite-difference schemes, mainly divided into two categories: implicit finite-difference method and explicit finite-difference method. A conventional implicit finite-difference scheme requires solving a matrix having a size equal to the product of the dimensions which greatly increases the memory usage and computational cost. Therefore, explicit finitedifference schemes are almost exclusively used in 2D and 3D reverse-time migrations. Although explicit finitedifference schemes are easy to solve, theoretical analysis shows that they are only conditionally stable which imposes a limit on the marching time step size. On the other hand, both finite-difference methods suffer from numerical dispersion problems. To overcome these problems, either high-order schemes are used or grid size and time step size are reduced. In either case there is an increase in the computational cost. In this abstract, we propose a new way of solving the twoway wave equation, which we call “Explicit Marching” (EM) method. It differs from conventional finite-difference methods in that it does not suffer from stability and numerical dispersion problems. The new method is based on reformulating the two-way wave equation by introducing a complex (analytic) pressure wavefield. The solution of the new equation can be symbolically expressed and then computed by a stable explicit extrapolator derived from an optimized separable approximation (OSA). Numerical examples show that EM based reverse-time migration has the capability to image steeply dipping reflectors and complex structures.

Theory

To perform a prestack reverse time migration (RTM), we first temporally extrapolate the forward wavefield by solving the two-way wave equation in time and store this four-dimensional wavefield. Then we propagate the recorded seismic data backwards in time and generate the backward wavefield. During this process, we apply a proper imaging condition to the two wavefields to produce the image.

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