A multi-level direct-iterative solver for seismic wave propagation modelling: space and wavelet approaches

We present a new numerical modelling approach for frequency-domain finite-difference (FDFD) wave simulations. The new approach is developed as an extension to standard FDFD modelling schemes, when wave propagation simulations are performed in large-scale 2-D or 3-D models with complex heterogeneous rheology. Partial differential equations are presented in matrix-type form. Wavefield solutions are computed on different coarse- and fine-discretized numerical grids by a combination of a direct solver with an iterative solver. Two different connection strategies are designed. Both compute a coarse- grid wavefield solution using a direct matrix solver. The obtained solution is projected on a fine-discretized grid, which is used as an initial solution for an iterative solver to compute the desired fine-grid solution. The wavefield projection that combines coarse and fine grids, is either based on a space interpolation scheme, called the direct iterative space solver (DISS) or on a multi-scale wavelet expansion, called the direct iterative wavelet solver (DIWS). The DISS scheme mimics a nested iteration scheme of a full multi-grid method, since numerical grids are prolonged by a simple bilinear interpolation scheme. The simple grid combination leads to wavefield solutions that are affected by spatial phase-shift artefacts (aliasing), which may be suppressed by a large number of iteration steps or a standard V- and W-cycles sequence between grids. The actual DIWS matrix construction implementation is computationally more expensive, though the wavelet iteration scheme guarantees fast and stable iterative convergence. Coarse-grid wavefield solutions are combined with fine-grid solutions through the multi-resolution scaling property of a standard orthogonal wavelet expansion. Since the wavelet transformation accounts for grid interactions, phase-shift artefacts are greatly reduced and significantly fewer iteration steps are required for convergence. We demonstrate the performance and accuracy of the DISS and DIWS strategies for two complex 2-D heterogeneous wave simulation examples.