Simulating 2D Waves Propagation in Elastic Solid Media Using Wavelet Based Adaptive Method

In this study, an improved wavelet-based adaptive-grid method is presented for solving the second order hyperbolic Partial Differential Equations (PDEs) for describing the waves propagation in elastic solid media. In this method, the multiresolution adaptive threshold-based approach is incorporated with smoothing splines as denoiser of spurious oscillations. This smoothing method is fast, stable, less sensitive to noise, and directly applicable to unequally sampled data. However, the conventional methods can not be directly applied to estimate the smoothing parameters; therefore the optimum ranges are captured through trial-and-error efforts. Here, the spatial derivatives are directly calculated in a non-uniform grid by Fornberg fast method. The derivatives are calculated in 2D simulations, applying antisymmetric end padding method to minimize Gibb’s phenomenon, caused by the edge effects. Therefore, stable moving front is achieved. In the realistic source modeling, time dependent thresholding method, introduced here, is an efficient and cost effective adaptive scheme as well. Furthermore, level-dependent thresholding scheme is used to diminish the effects of non-physical long period waves reflected by absorbing boundaries. Finally, several 2D finite, infinite and semi-infinite numerical examples are simulated. These examples have fixed, free and absorbing boundary conditions. Here, the robustness of proposed method is demonstrated.