In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.@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