Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

Abstract

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.

A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration

“…where Δ𝑡 denotes the time step to extrapolate wavefields and 𝐿 = 𝑐 √ −∇ 2 . If we take for cos(Δ𝑡𝐿), its second-order (1 − (Δ𝑡𝐿) 2 ∕2) Taylor-series expansion and substitute in Equation ( 2), we will obtain the second-order standard finite-difference (FD) scheme, that is (Araujo & Pestana, 2019;Etgen, 1986)…”

“…The waveform obtained by the Lie scheme using Equation ( 19) to Δ𝑡 = 1 ms and k = 1 term is used as reference. employ, for example, the Taylor-series, Chebyshev, Hermite and Legendre polynomial expansions (or any other expansion) for the differential operator cos(Δ𝑡 fd 𝐿) (Araujo & Pestana, 2019). For instance, using the second-order Taylor approximation for the cos(Δ𝑡 fd 𝐿), the operator H in Equation ( 6) becomes…”

“…with Δ𝑥 and Δ𝑧 are the spatial sampling and 𝑐 max is the maximum velocity of a given model (Araujo & Pestana, 2019).…”

“…Soubaras and Zhang (2008), Zhang and Zhang (2009) and Fomel et al (2013) have proposed numeri-cal solutions based on polynomial expansion to improve time integration in the wave equation. Recently, Araujo and Pestana (2019) have shown a numerical scheme that employs a multiple-angle formula and Taylor-series expansion to propagate the wavefields.…”

Wavefield separation in VTI media with multiple-angle expansion

“…On this basis, Kosloff et al proposed the rapid-expansion method [2] , which was later developed by Pestana et al applied to reverse time migration method [3] . Revelo and Pestana extended the RE method to the separation of up and down going waves of seismic wavefield [4] , reverse time migration imaging of anisotropic media, and noise suppression [5] , and realized the application of the method in the field of anisotropic media imaging [6] . Araujo and Pestana introduced perfectly matched layer (PML) absorption boundary conditions into the rapidexpansion method, which significantly reduced boundary reflection interference [7] .…”