Tensorial formulation of the wave equation for modelling curved interfaces

Abstract

Many situations of practical interest involving seismic wave modelling require curved interfaces and free-surface topography to be taken into account. Collocation methods, for instance pseudospectral or finite-difference algorithms, are attractive approaches for modelling wave propagation through these complex realistic models, particularly in view of their ease of implementation. Nonetheless, these methods formulated in Cartesian coordinates are not well suited to such models because the sharp interfaces and free surface do not coincide with grid lines. This leads to a slow convergence rate, resulting in visible artefacts such as diffractions from staircase discretizations of interfaces and the free surface. Such problems can be overcome through the use of curved grids whose lines follow sharp interfaces and whose density increases in the vicinity of these interfaces. One approach is to solve the wave equation in Cartesian coordinates by using the chain rule to express the Cartesian partial derivatives in terms of derivatives computed in the new coordinate system. However, it is more natural to solve the tensorial form of the wave equation directly in the desired curvilinear coordinate system, making use of a transformation of a square grid onto the curved grid. The tensorial approach, which is independent of the coordinate system, requires the same number of derivatives to be computed as in the Cartesian case, whereas the chain rule approach requires 25 per cent more in 2-D and 50 per cent more in 3-D. While the tensorial approach is less computationally expensive than the chain rule method, it requires more memory. Numerical tests validate the tensorial approach by comparing the results with the analytical solution of the tilted Lamb problem. Other numerical experiments demonstrate the ability of the tensorial formulation to model wave propagation in the presence of free-surface topography. Mode conversions between Rayleigh and body waves are observed when bumps on the free surface are encountered.

Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

“…A less straightforward issue using pseudospectral differential operators is to model the freesurface boundary condition. While in finite-element methods the implementation of traction-free boundary conditions is natural -simply do not impose any constraint at the surface nodes -finitedifference and pseudospectral methods require a particular boundary treatment [14,23,25,26].…”

“…Each of these terms, integrated over an element Ω e , is easily computed by substituting the expansion of the fields (14), computing gradients using (17) and the chain rule (18), and using the GLL integration rule (16). After this spatial discretization with spectral elements, imposing that (12) holds for any test vector w N , as in a classical FEM, we have to solve an ordinary differential equation in time.…”

“…After introducing the piecewise-polynomial approximation (14), the integrals in (12) can be approximated at the element level using the GLL integration rule:…”

“…The computations are based on two different numerical techniques, namely, the Fourier-Chebyshev pseudospectral method (PSM) [12][13][14] and the spectral finite-element method (SEM) [15][16][17][18][19][20]. The first is based on global differential operators in which the field is expanded in terms of Fourier and Chebyshev polynomials, while the second is an extension of the finite-element method that uses Legendre polynomials as interpolating functions.…”

Accurate seabed modeling using finite difference methods

“…To achieve this, the continuous model is first sampled along any problematic interfaces to form a curved grid; the curved grid is then mapped to an orthogonal grid via coordinate transform. There are two widely used methods: the curvilinear coordinate method [16,17] and the body-fitted grid method [22,38,55]. Both these methods are commonly used to model irregular topography of the free surface.…”