Parsimonious finite-volume frequency-domain method for 2D P-SV-wave modeling

A new numerical technique for solving 2D elastodynamic equations based on a ﬁnite-volume frequency-domain approach is proposed. This method has been developed as a tool to perform two-dimensional (2D) elastic frequency-domain full-waveform inversion. In this context, the system of linear equations that results from the discretisation of the elastodynamic equations is solved with a direct solver, allowing efﬁcient multiple-source simulations at the partial expense of the memory requirement. The discretisation of the ﬁnite-volume approach is through triangles. Only ﬂuxes with the required quantities are shared between the cells, re-laxing the meshing conditions, as compared to ﬁnite-element methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 ﬁnite-volume stencil, the parsimonious approach allows the minimising of core memory requirements for the simulation. Efﬁcient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O (∆ x 2 ) staggered-grid ﬁnite-difference formulations when considering structured triangular meshes. The validation has been performed with analytical solutions of several canonical problems and with numerical solutions computed with a well-established ﬁnite-difference time-domain method in heterogeneous media. In the presence of a free surface, the ﬁnite-volume method requires ten triangles per wavelength for a ﬂat topography, and ﬁfteen triangles per wavelength for more complex shapes, well below the criteria required by the staircase approximation of O (∆ x 2 ) ﬁnite-difference methods. Comparisons between the frequency-domain ﬁnite-volume and the O (∆ x 2 ) rotated ﬁnite-difference methods also show that the former is faster and less memory demanding for a given accuracy level, an attractive feature for frequency-domain seismic inversion. We have thus developed an efﬁcient method for 2D P-SV-wave modelling on structured triangular meshes as a tool for frequency-domain full-waveform inversion. Further work is required to improve the accuracy of the method on unstructured meshes. hyperbolic elastodynamic system in the frequency domain, as is usual for FV formulations (Remaki, 2000). We will deduce the discretised system of linear equations to be solved only for velocity components, using the parsimonious strategy (Luo and Schuster, 1990). The numerical dispersion behaviour of this scheme will be analysed before discussing the perfectly matched layer (PML) conditions (Berenger, 1994). The implementation of the source will require speciﬁc attention before going on to numerical validations against both the analytical solutions and the numerical solutions obtained by other numerical methods. For these examples, we will show the respective numerical costs of frequency-domain FV and O (∆ x 2 ) FD methods. We will conclude with the potential of this FV approach in the frequency domain for the modelling of seismic 2D P-SV waves and the perspectives for inversion.