We have developed a new method for solving the acoustic full-wave equation, which we call the one-step wave extrapolation (OSE) matrix method. In our method, the wave equation is redefined by introducing a complex (analytic) wavefield and reformulating the traditional acoustic full-wave equation as a first-order partial differential equation in time. Afterward, the analytical wavefield is separated to its real and imaginary parts, and the resulting first-order coupled set of equations is solved by the Tal-Ezer’s technique, which consists of using the Chebyshev polynomial expansion to approximate the matrix exponential operator. The matrix is antisymmetrical with a square-root pseudodifferential operator, which is computed using the Fourier method. In this way, the implementation of the proposed method is straightforward and if the appropriate number of Chebyshev polynomial expansion terms is chosen, the proposed numerical algorithm is unconditionally stable and propagates seismic waves free of numerical dispersion for any seismic velocity variation in a recursive manner. Moreover, in our method, the number of Fourier transforms is explicitly determined and it is a function of the maximum eigenvalue of the matrix operator and time-step size. A numerical modeling example is shown to demonstrate that the proposed method has the capability to extrapolate waves using a time stepping up to Nyquist limit. We have also developed a reverse time migration example with illumination compensation. The migration results based on the OSE method demonstrate the capability of this new method to image complex structures in the presence of strong velocity contrasts.
Low-Rank Updates of Matrix Functions II: Rational Krylov Methods
