In this paper, we review the finite-difference implementation of a narrow-angle, one-way vector wave equation for elastic, 3D media. Extrapolation is performed in the frequency domain, where the second-order-accurate lateral spatial-difference operators are sufficiently accurate for narrow-angle propagation. We perform a numerical analysis of the finite-difference scheme to highlight the stability and dispersion characteristics. The von Neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme, and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths, source pulse spectral content, and angular range of forward propagation of interest. Although the algorithm is reasonable, its computational efficiency is limited by the second-order-accurate extrapolation step; therefore, the extrapolation scheme can be improved. We extend the Cartesian narrow-angle formulation to curvilinear coordinates, where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the Cartesian propagator. An example of curvilinear extrapolation for a simple model consisting of a high-velocity sphere within a homogeneous background velocity structure shows that the narrow-angle propagator is capable of modeling frequency-dependent geometric spreading and diffraction effects in curvilinear coordinates.
Numerical Approximation of the Space Fractional Cahn-Hilliard Equation
