Can Higher-Order Finite-Difference Operators Be across a Material Interface?

It is well known that higher-order and thus longer-stencil finite-difference operators (FDOs) can be advantageously used for evaluating spatial derivatives in the finite-difference schemes applied to smoothly heterogeneous media. This is because they reduce spatial grid dispersion. However, realistic models often include sharp material interfaces. Can high-order long-stencil FDOs be applied across such material interface? We address this question by comparing exact spatial derivatives against derivatives approximated by FDOs with respect to the interface representation, velocity contrast, and order of the FDO. The interface is considered in an arbitrary position with respect to the spatial grid. The material interface exactly represented by the Heaviside step function causes a large error of the FDO spatial derivative near the interface. The maximum error near the interface practically does not depend on the order of the FDO. There are only small differences in errors among FDOs of different orders elsewhere. The larger the velocity contrast, the larger the error. If the material interface is represented using a wavenumber band-limited Heaviside function, the error is smoothed and several times smaller. The error in the wavenumber band-limited model decreases with an increasing order of the FDO. Our findings combined with those by Moczo et al. (2022) lead to the important conclusion: The wavenumber band-limited representation of the material interface is not only a necessary consequence of discretization of the original physical model but also significantly reduces the error in evaluating a spatial derivative using the FDO.