A robust seismic wavefield modeling method based on minimizing spatial simulation error using L2-norm cost function

To reduce the spatial simulation error generated by the finite difference method, previous researchers compute the optimal finite-difference weights always by minimizing the error of spatial dispersion relation. However, we prove that the spatial simulation error of the finite difference method is associated with the dot product of the spatial dispersion relation of the finite-difference weights and the spectrum of the seismic wavefield. Based on the dot product relation, we construct a $L_2$ norm cost function to minimize spatial simulation error. For solving this optimization problem, the seismic wavefield information in wavenumber region is necessary. Nevertheless, the seismic wavefield is generally obtained by costly forward modeling techniques. To reduce the computational cost, we substitute the spectrum of the seismic wavelet for the spectrum of the seismic wavefield, as the seismic wavelet plays a key role in determining the seismic wavefield. In solving the optimization problem, we design an exhaustive search method to obtain the solution of the $L_2$ norm optimization problem. After solving the optimization problem, we are able to achieve the finite-difference weights that minimize spatial simulation error. In theoretical error analyses, the finite-difference weights from the proposed method can output more accurate simulation results compared to those from previous optimization algorithms. Furthermore, we validate our method through numerical tests with synthetic models, which encompass homogenous/inhomogeneous media as well as isotropic and anisotropic media.