Temporal dispersion correction for wave-propagation modelling with a series approach

Temporal dispersion correction of second-order finite-difference time stepping for numerical wave propagation modelling exploits the fact that the discrete operator is exact but for the wrong frequencies. Mapping recorded traces to the correct frequencies removes the numerical error. Most of the implementations employ forward and inverse Fourier transforms. Here, it is noted that these can be replaced by a series expansion involving higher time derivatives of the data. Its implementation by higher-order finite differencing can be sensitive to numerical noise, but this can be suppressed by enlarging the stencil. Tests with the finite-element method on a homogeneous acoustic problem with an exact solution show that the method can achieve the same accuracy as higher-order time stepping, similar to that obtained with Fourier transforms. The same holds for an inhomogeneous problem with topography where the solution on a very fine mesh is used as reference. The series approach costs less than dispersion correction with the Fourier method and can be used on the fly during the time stepping. It does, however, require a wavelet that is sufficiently many times differentiable in time.