We propose a new numerical solution to the first-order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time-steps than traditional finite-difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the firstorder linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post-stack migration results are also shown using the proposed method.

We developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method (REM). This solution can be used for seismic modeling and reverse-time migration (RTM). In seismic modeling and RTM, spatial derivatives are usually calculated by finite differences (FDs) or by the Fourier method, and the time evolution is normally obtained by a second-order FD approach. If the spatial derivatives are computed by higher order FD schemes, then the time step needs to be small enough to avoid numerical dispersion, therefore increasing the computational time. However, by using REM with the Fourier method for the spatial derivatives, we can apply the proposed method to propagate the wavefield for larger time steps. Moreover, if the appropriate number of expansion terms is chosen, this method is unconditionally stable and propagates seismic waves free of numerical dispersion. The use of a symplectic numerical scheme provides the solution of the wave equation and its first time derivative at the current time step. Thus, the Poynting vector can also be computed during the time extrapolation process at very low computational cost. Based on the Poynting vector information, we also used a new methodology to separate the wavefield in its upgoing and downgoing components. Additionally, Poynting vector components can be used to compute common gathers in the reflection angle domain, and the stack of some angle gathers can be used to eliminate low-frequency noise produced by the RTM imaging condition. We numerically evaluated the applicability of the proposed method to extrapolate a wavefield with a time step larger than the ones commonly used by symplectic methods as well as the efficiency of this new symplectic method combined with REM to successfully handle the Poynting vector calculation.

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We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

We derive a governing second‐order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite‐difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.