Efficient wave-mode separation in vertical transversely isotropic media

Abstract

Wave-mode separation can be achieved by projecting elastic wavefields onto mutually orthogonal polarization directions. In isotropic media, because the P-wave’s polarization vectors are consistent with wave vectors, the isotropic separation operators are represented by divergence and curl operators, which are easy to realize. In anisotropic media, polarization vectors deviate from wave vectors based on local anisotropic strength and separation operators lose their simplicity. Conventionally, anisotropic wave-mode separation is implemented either by direct filtering in the wavenumber domain or nonstationary filtering in the space domain, which are computationally expensive. Moreover, in conventional anisotropic separation, correcting for amplitude and phase changes of waveforms by applying separation operators is also more difficult than in an isotropic case. We have developed new operators for efficient wave-mode separation in vertical transversely isotropic (VTI) media. Our separation operators are constructed by local rotation of wave vectors to directions where the quasi-P (qP) wave is polarized. The deviation angles between the wave vectors and the qP-wave’s polarization vectors are explicitly estimated using the Poynting vectors. Obtaining polarization directions by rotating wave vectors yields separation operators in VTI media with the same forms as divergence and curl operators, except that the spatial derivatives are now rotated to implement wavefield projections in accurate polarization directions. The main increase in computational cost relative to isotropic separation operators is the estimation of the Poynting vectors, which is relatively small within elastic-wave extrapolation. As a result, applying the proposed operators is efficient. In the meantime, the waveforms corrections for divergence and curl operators can be directly extended for our new operators due to the similarities between these operators. By numerical exercises, we have determined that wave modes can be well-separated with small numerical cost using the present separation operators. The conservation of energy in wave-mode separation by applying waveform corrections was also verified.

Poynting and polarization vectors based wavefield decomposition and their application on elastic reverse time migration in 2D transversely isotropic media

“…However, this method cannot handle overlapping events (Tang et al . 2016; Zhou and Wang ; Lu and Liu ), which we will demonstrate in the following numerical experiment. The elastic Poynting vectors can be expressed as (Červený ) where i and j represent the x ‐ or z ‐component of the elastic Poynting vectors s , respectively, and are the stress tensor and particle velocity, respectively.…”

“…() developed an efficient method to propagate and decouple the elastic waves using the low‐rank approximations for anisotropic media. Zhou and Wang () introduced efficient anisotropic separation operators that are performed by locally rotating wave vectors to the qP‐wave polarization directions. They first calculate the deviation angles between these two vectors, which are estimated using the Poynting vectors.…”

“…This procedure can be used in both the structured and unstructured mesh with high computing efficiency. Moreover, the proposed method preserves the amplitude and phase information compared with conventional pseudo‐derivative methods with the correction of the waveform changes (Zhou and Wang ).…”

Elastic wave‐mode separation in 2D transversely isotropic media using optical flow

“…In the Fourier domain, the expressions are given as follows (Yan and Sava, 2009): where and are the qP‐ and qSV‐wavefields in the wavenumber domain, respectively; and are the elastic vector wavefields in the wavenumber domain; is the polarization vector; is the polarization angle; is the circular frequency; and is the phase velocity. All algorithms based on the spatial derivatives are under the control of this equation, such as the divergence and curl operators in isotropic media (Aki and Richards, 1980) and the separation operators of Zhou and Wang (2017). Taking the partial derivative with respect to or in the space domain is equivalent to multiplying the wavefield by the spatial wavenumber in the wavenumber domain ...…”

“…In isotropic media, the polarization vector is consistent with the wave vector for the P‐wave. In the anisotropic case, the polarization direction can be obtained by rotating the wave vector (Zhou and Wang, 2017), and the deviation angle is a function of the ratio of P‐wave velocity to S‐wave velocity , Thomsen parameter , and phase angle .…”