Pseudospectral scheme

The pseudospectral method is proposed by Kosloff and Baysal (1982), which uses Fourier transformation, multiplication by $ik$ in the wavenumber domain, and inverse Fourier transformation back to the spatial domain to compute the spatial derivatives. Differential operators $V$ , $H$ and $T$ in equation 4.9 are written in the wavenumber domain and are easily evaluated there with a pseudospectral method. Meanwhile, as in the pseudospectral method, performing the operations in the wavenumber domain guarantees that it will not suffer from numerical dispersion.

Substituting equations 4.5 and 4.9 into equation 4.2, I write the TTI pure P-wave equation as

$$u(\vecx ,t+\dt ) = 2u(\vecx ,t) - u(\vecx ,t-\dt ) - \dt ^2 \bigg[ v_v^2V + v_h^2H + (v_n^2-v_h^2)T \bigg] u(\vecx ,t).$$ (410)

From the above equations, we can see that at each time step of a 3D simulation, the evaluation of the differential operator $V$ demands at least a 3D forward FFT of the wavefield plus six 3D inverse FFTs. A similar analysis applies to the differential operator $T$ as well. Therefore, a total of fourteen 3D FFTs are required to simulate the pure P-wave wavefield at each time step in a TTI medium. When it comes to 2D, all $k_y$ terms are eliminated, and thus only eight 2D FFTs are needed.