Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling

We compare different finite-difference schemes for two-dimensional (2-D) acoustic frequency-domain forward modelling. The schemes are based on staggered-grid stencils of various accuracy and grid rotation strategies to discretize the derivatives of the wave equation. A combination of two
$\mathcal O(\Delta x^2)$
staggered-grid stencils on the classical Cartesian coordinate system and the 45° rotated grid is the basis of the so-called mixed-grid stencil. This method is compared with a parsimonious staggered-grid method based on a fourth-order approximation of the first derivative operator. Averaging of the mass acceleration can be incorporated in the two stencils. Sponge-like perfectly matched layer absorbing boundary conditions are also examined for each stencil and shown to be effective. The deduced numerical stencils are examined for both the wavelength content and azimuthal variation. The accuracy of the fourth-order staggered-grid stencil is slightly superior in terms of phase velocity dispersion to that of the mixed-grid stencil when averaging of the mass acceleration term is applied to the staggered-grid stencil. For fourth-order derivative approximations, the classical staggered-grid geometry leads to a stencil that incorporates 13 grid nodes. The mixed-grid approach combines only nine grid nodes. In both cases, wavefield solutions are computed using a direct matrix solver based on an optimized multifrontal method. For this 2-D geometry, the staggered-grid strategy is significantly less efficient in terms of memory and CPU time requirements because of the enlarged bandwidth of the impedance matrix and increased number of coefficients in the discrete stencil. Therefore, the mixed-grid approach should be suggested as the routine scheme for 2-D acoustic wave propagation modelling in the frequency domain.