2021/07/13 by Edward Caunt, Caunt, Edward
Earth and Planetary Sciences · Engineering · #FOS: Mathematics #Geophysical Methods and Applications #Numerical Analysis (math.NA) #Seismic Imaging and Inversion Techniques #Seismic Waves and Analysis
paper · pdf · doi:10.48550/arxiv.2107.13525
openalex publication_date 2021/07/13 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Propagation characteristics of a wave are defined by the dispersion\nrelationship, from which the governing partial differential equation (PDE) can\nbe recovered. PDEs are commonly solved numerically using the finite-difference\n(FD) method, with stencils constructed from truncated Taylor series expansions\nwhich, whilst typically providing good approximation of the PDE in the\nspace-time domain, often differ considerably from the original partial\ndifferential in the wavenumber-frequency domain where the dispersion\nrelationship is defined. Consequentially, stable, high-order FD schemes may not\nnecessarily result in realistic wave behavior, commonly exhibiting numerical\ndispersion: lagging high-frequency components as a product of discretization. A\nmethod for optimizing FD stencil weightings via constrained minimization to\nbetter approximate the partial derivative in the wavenumber domain is proposed,\nallowing for accurate propagation with coarser grids than would be otherwise\npossible. This was applied to second derivatives on a standard grid and first\nderivatives on a staggered grid. To evaluate the efficacy of the method, a pair\nof numerical simulations were devised to compare spatially-optimized stencils\nwith conventional formulations of equivalent extent. A spatially-optimized\nformulation of the 1D acoustic wave equation with Dirichlet boundary conditions\nis presented, evaluating performance at a range of grid spacings, examining the\ninterval between the theoretical maximum grid spacings for the conventional and\noptimized schemes in finer detail. The optimized scheme was found to offer\nsuperior performance for undersampled wavefields and heavily oversampled\nwavefields. Staggered-grid first derivative stencils were then applied to the\nP-SV elastic wave formulation, simulating seismic wave propagation for a\ntwo-layer, water-over-rock model.\n