Distributional finite-difference modelling of seismic waves

Abstract

Summary This study introduces a Distributional finite-difference Method (DFDM) for modeling the propagation of elastic waves in heterogeneous media in the time domain. DFDM decomposes the modeling domain into multiple elements that can have arbitrary sizes. When large elements are used, the proposed method closely resembles the finite-difference method because the wavefield is updated using operations involving band diagonal matrices only. Thus DFDM is computationally efficient. When smaller elements are employed, DFDM looks closer to the finite-element or the spectral element methods and permits to mesh complicated structures. A complete multi-domain algorithm for modeling elastic wave propagation in arbitrarily heterogeneous media is presented. The algorithm’s stability is discussed, and the usual Courant condition governs the stability of the proposed scheme. Numerical examples show that the proposed algorithm accurately accounts for free surfaces, solid-fluid interfaces and accommodates non-conformal meshes in their basic form. Seismograms obtained using the proposed method are compared to those computed using analytical solutions and the spectral element method. To achieve comparable accuracy, DFDM requires fewer points per wavelength than the spectral element method, for example.

P-SV-wave propagation in heterogeneous media: Velocity-stress distributional finite-difference method

“…A review of these popular modeling methods is carried out in Virieux et al (2011). Recently, Masson (2022) proposed a distributional finite-di↵erence method, denoted by the DFD keyword thereafter. The DFD approach lies somewhere between the pseudo-spectral/finite-di↵erence methods and the finite/spectralelement methods and combines some desirable features of the former methods in a relatively simple manner.…”

“…Thus, special care must be taken to eliminate spurious(non-physical) parasitic modes (see, e.g., New et al, 1998), for example, the odd-even decoupling observed when using centered finite-di↵erence operators. In Masson (2022) a MacCormack type of scheme is proposed to address this issue (see e.g., Zhang and Chen, 2006). In this study, we consider an alternative approach where the bases representing the wavefield are analogous to the staggered grid in which the variables are not defined at the same position (as opposed to the collocated grid).…”

“…The new method we propose preserves the intrinsic staggered property of the block-anti-diagonal first-order system using staggered basis functions for discretization. The specific choice of the shape functions makes the DFD approach very similar to finite-di↵erence techniques: we shall use B-splines in this work and refer the reader DFD modeling to the presentation of Masson (2022) for other alternative functions. The boundary conditions are treated with the usual integration-by-parts procedure.…”

“…Before detailing the content of our manuscript, let us underline the principal aspects in which our approach di↵ers from the general method introduced in Masson (2022). The first aspect is the choice of the basis functions employed for representing the field variables.…”

“…The first aspect is the choice of the basis functions employed for representing the field variables. Masson (2022) represents the displacements in a first basis (with the basis function skewed to the left) and the stresses in a second basis (with basis functions skewed to the right). In this study, we represent the field variables in different bases and organize the basis functions according to the staggered grid.…”