Should We Use the First- or Second-order Formulation with Spectral Elements for Seismic Modelling?

The second-order formulation of the wave equation is often used for spectral-element discretizations. For some applications, however, a first-order formulation may be desirable. It can, in theory, provide much better accuracy in terms of numerical dispersion if the consistent mass matrix is used and the degree of the polynomial basis functions is odd. However, we find in the 1-D case that the eigenvector errors for elements of degree higher than one are larger for the first-order than for the second-order formulation. These errors measure the unwanted cross talk between the different eigenmodes. Since they are absent for the lowest degree, that linear element may perform better in the first-order formulation if the consistent mass matrix is inverted. The latter may be avoided by using one or two defect-correction iterations. Numerical experiments on triangles confirm the superior accuracy of the first-order formulation. However, with a delta-function point source, a large amount of numerical noise is generated. Although this can be avoided by a smoother source representation, its higher cost and the increased susceptibility to numerical noise make the second-order formulation more attractive.Applied Geophysics and Petrophysic