One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results@en

In land applications, topography variation may impact the imaging if not taken into account. With low-frequency and wide-aperture data, the long-to-intermediate wavelength components of the velocity model can be recovered by full-waveform inversion. Standard static corrections to handle the topography do not work satisfactorily on long-offset data. We present a method to handle 3-D free-surface topography for acoustic FWI by directly modelling the effect of the topography with a finite-difference scheme for the first-order wave equation introduced earlier (Huiskes et al., 2016; Mulder and Huiskes, 2017). The method is based on an immersed interface approach (Lombard et al., 2008) in which the free surface does not have to coincide with the discretization grid, avoiding a staircase-like representation of the topography (Bohlen and Saenger, 2006). We refer to Huiskes et al. (2016) for a more detailed overview of methods for simulating seismic waves interacting with free-surface topography. First, we review the main elements of the numerical scheme that is designed specifically for discretization on standard staggered grids (SSG) using high-order derivative operators, which are modified based on their relative position to the free surface. Next, we extend the approach to vertical transverse isotropic (VTI) media to be able to model velocity anisotropy required in long-offset inversions. In relatively simple synthetic experiments, we then reproduce topography artefacts we have seen on real land fullwaveform inversions, allowing us to quantify the effect of elevation variation on the inversion accuracy. @en

We propose a finite-difference scheme for the simulation of seismic waves interacting with 3-D freesurface topography. The intended application is velocity model building by acoustic full-waveform inversion (FWI). The scheme follows an immersed boundary approach for wave equations in the firstorder stress-velocity formulation, discretized on a standard staggered grid. Our scheme employs modified 1-D stencils rather than a full 3-D field wavefield extension at the free surface. Although this decreases the accuracy, it improves the scheme's simplicity and robustness. To avoid stability problems, points close to the zero-pressure boundary must be excluded. The scheme, and its adjoint, have been tested by tilted geometry tests and by comparison to a finite-element method. We present a first test result of full-waveform inversion with the new scheme. @en