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ABSTRACT We demonstrate that implicit time integration methods for second-order-in-time wave equations can be derived from rational expansions to the cosine function of pseudo-differential operators. Using the scalar wave equation as an example, we give complete stability condition and grid dispersion analysis results for general implicit time integration methods. Furthermore, we propose an optimization method to develop unconditionally stable implicit time stepping schemes.
ABSTRACT We propose a novel approach for frequency response modeling with the rapid expansion method (REM). This new approach is derived from a running summation algorithm of time domain wavefields based on discrete Fourier transform (DFT). Unlike other time domain frequency response modeling methods, which use wavefield snapshots from time marching simulations, our approach directly operates on the Chebyshev expansion polynomials. When combined with the pseudospectral method in space, this new approach can produce spectrally accurate frequency responses with high efficiency.
ABSTRACT The pseudo-analytical method relies on pseudo-Laplacians to compensate for time stepping errors caused by the second-order time stepping scheme. Pseudo-Laplacian slowly varies with the compensation velocity which makes it well suited for models with mild velocity variations. For models with high velocity variations, the pseudo-analytical method becomes difficult because high compensation velocities cause over-compensations to wavefields in low velocity areas which can bring significant artifacts into the simulation results. To tackle this problem, I propose to use spatially varying normalized pseudo-Laplacians, which are determined by actual velocity variations in space, to locally compensate for time stepping errors. This new implementation of the pseudo-analytical method involves two steps. The first step applies local compensations using adaptive normalized pseudo-Laplacians, computed either in wavenumber domain or in space domain. The second step carries out the second-order time marching computations, which can be realized by any numerical schemes and not limited to the wavenumber domain method. I use numerical experiments to demonstrate that the proposed method can produce highly accurate results with relaxed stability conditions compared to the conventional pseu-dospectral method.
ABSTRACT One fundamental shortcoming of the conventional pseudo-acoustic approximation is that it only prevents shear wave propagation along the symmetry axis of anisotropy and not in other directions. This problem leads to the presence of unwanted shear waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudo-acoustic wave equations become unstable for anisotropy parameters e