Abstract Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, thedispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-differencecalculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and fourin three dimensions, which decouple in isotropicmedia. Therefore the Lebedev scheme will havetwice or four times the computational requirements, memory, and CPU as the Virieux gridbut can be used with general anisotropy. In twodimensions, the rotated staggered grid is exactlyequivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numericaldispersion in finite-difference grids depends on the direction of propagation and the grid typeand parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computationalrequirements for the two grid types, the dispersion, averaged over propagation direction and mediumvelocity are calculated. Setting the parameters sothe average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field dataand at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

Abstract Numerical modelling of seismic motion in sedimentary basins often has to account for P -wave to S -wave speed ratios as large as five and even larger, mainly in sediments below groundwater level. Therefore,we analyse seven schemes for their behaviour with a varying P -wave to S -wave speed ratio. Four finite-difference (FD) schemes include (1) displacement conventional-grid, (2) displacement-stress partly-staggeredgrid, (3) displacement-stress staggered-grid and (4) velocity–stress staggered-grid schemes. Three displacement finite-element schemes differ in integration: (1) Lobatto four-point, (2) Gauss four-point and (3) Gauss one-point. To compare schemes at the most fundamental level, and identify basic aspects responsible for their behaviours with the varying speed ratio, we analyse 2-D second-order schemes assuming an elastic homogeneous isotropic medium and a uniform grid. We compare structures of the schemes and applied FD approximations. We define (full) local errors in amplitude and polarization in one time step, and normalize them for a unit time. We present results of extensive numerical calculations for wide ranges of values of the speed ratio and a spatial sampling ratio, and the entire range of directions of propagation with respect to the spatial grid. The application of some schemes to real sedimentary basins in general requires considerably finer spatial sampling than usually applied. Consistency in approximating first spatial derivatives appears to be the key factor for the behaviour of a scheme with respect to the P -wave to S -wave speed ratio.

Abstract We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new 3D fourth-order staggered-grid finite-difference (FD) scheme for modeling seismic motion and seismic-wave propagation. We first consider a 1D problem for a welded planar interface of two half-spaces. A simple physical model of the contact of two media and mathematical considerations are shown to give an averaged medium representing the contact of two media. An exact heterogeneous formulation of the equation of motion is a basis for constructing the corresponding heterogeneous FD scheme. In a much more complicated 3D problem we analyze a planar-interface contact of two isotropic media (both with interface parallel to a coordinate plane and interface in general position in the Cartesian coordinate system) and a nonplanar-interface contact of two isotropic media. Because in the latter case 21 elastic coefficients at each point are necessary to describe the averaged medium, we consider simplified boundary conditions for which the averaged medium can be described by only two elastic coefficients. Based on the simplified approach we construct the explicit heterogeneous 3D fourth-order displacement-stress FD scheme on a staggered grid with the volume harmonic averaging of the shear modulus in grid positions of the stress-tensor components, volume harmonic averaging of the bulk modulus in grid positions of the normal stress-tensor components, and volume arithmetic averaging of density in grid positions of the displacement components. Our displacement-stress FD scheme can be easily modified into the velocity-stress or displacement-velocity-stress FD schemes. The scheme allows for an arbitrary position of the material discontinuity in the spatial grid. Numerical tests for 12 configurations in four types of models show that our scheme is more accurate than the staggered-grid schemes used so far. Numerical examples also show that differences in thickness of a soft surface or interior layer smaller than one grid spacing can cause considerable changes in seismic motion. The results thus underline the importance of having a FD scheme with sufficient sensitivity to heterogeneity of the medium.

Summary Differential forms provide an elegant formulation of electromagnetic field theory. Their geometric structure also leads to a self-consistent scheme for discretizing Maxwell’s equations in conducting media. This discretization, which is essentially a staggered-grid scheme, preserves differential operator identities, conservation laws, and physical boundary conditions. The self-consistent scheme has been implemented in a code to model transient electromagnetic fields in a half-space with a step-function current excitation. For small conductivity contrasts results from the self-consistent scheme agree very well with results from a 3-D integral-equation code and from a staggered-grid finite-difference code. For high contrasts, though, the results from the integral-equation code differ. The difference may be a result of improper discretization of the integral equation.