The optimized expansion based low-rank method for wavefield extrapolation

Abstract

Spectral methods are fast becoming an indispensable tool for wavefield extrapolation, especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation. However, for inhomogeneous media, we face difficulties in dealing with the mixed space-wavenumber domain extrapolation operator efficiently. To solve this problem, we evaluated an optimized expansion method that can approximate this operator with a low-rank variable separation representation. The rank defines the number of inverse Fourier transforms for each time extrapolation step, and thus, the lower the rank, the faster the extrapolation. The method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low-rank representation. As a result, we obtain lower rank representations compared with the standard low-rank method within reasonable accuracy and thus cheaper extrapolations. Additional bounds set on the range of propagated wavenumbers to adhere to the physical wave limits yield unconditionally stable extrapolations regardless of the time step. An application on the BP model provided superior results compared to those obtained using the decomposition approach. For transversely isotopic media, because we used the pure P-wave dispersion relation, we obtained solutions that were free of the shear wave artifacts, and the algorithm does not require that η > 0. In addition, the required rank for the optimization approach to obtain high accuracy in anisotropic media was lower than that obtained by the decomposition approach, and thus, it was more efficient. A reverse time migration result for the BP tilted transverse isotropy model using this method as a wave propagator demonstrated the ability of the algorithm.

Efficient anisotropic quasi-Pwavefield extrapolation using an isotropic low-rank approximation

“…This decomposition can provide more accurate extrapolation results with an increase of computational cost (especially for tilted media) since the background wavefield is no longer isotropic. The number of FFTs required by the isotropic low-rank approximation proposed here can be reduced further using the optimized expansion method introduced by Wu & Alkhalifah (2014). The proposed method has an approximation (equation 6) when calculating the phase directions.…”

“…In this case, the approach becomes relatively expensive. Wu & Alkhalifah (2014) proposed an optimized expansion method to reduce the required FFTs per time step. Still, the dependency on the complexity of the model limits their practical applications in complex anisotropic media.…”

The optimized gradient method for full waveform inversion and its spectral implementation

“…This optimized expansion method provides us with unconditionally stable and dispersion free wavefields (Wu & Alkhalifah 2014a). Its coefficients depend on the velocity range, mesh sampling and maximum frequency only, which will allow us to pre-compute the coefficients before the FWI procedure only once.…”

“…The optimized expansion based low-rank method (Wu & Alkhalifah 2014a) tries to get the best a ij , v i , k j by solving the following minimization problem…”

“…Optimized expansion based on the low-rank method was recently introduced to solve the isotropic and anisotropic wave equation (Wu & Alkhalifah 2014a). It admits dispersion free and unconditionally stable wavefield solutions.…”