scholarly journals Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

2014 ◽  
Vol 16 (4) ◽  
pp. 913-955 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Finite Difference ◽  
Elastic Wave ◽  
Far Field ◽  
Computational Domain ◽  
Elastic Wave Equation ◽  
Numerical Accuracy ◽  
Coordinate Mapping ◽  
Field Boundaries

AbstractWe develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

Absorbing boundary condition for the elastic wave equation

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 611-624 ◽  
Author(s):  
C. J. Randall
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Grazing Incidence ◽  
Scalar Fields ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Elastic Wave Equation ◽  
Absorbing Boundary

Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Geophysics ◽  
2022 ◽  
pp. 1-71
Author(s):  
Shu-Li Dong ◽  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Elastic Wave ◽  
Rotation Angle ◽  
Surface Expression ◽  
Coordinate Systems ◽  
Elastic Wave Equation ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Grid Points

Effective frequency-domain numerical schemes were central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme was a highly used finite-difference numerical scheme. This scheme made a weighted average of the derivative terms of the elastic wave equations in the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it could simulate S-waves better and had higher accuracy at nearly the same computational cost. Nevertheless, this scheme limited the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions needed to be equal. Otherwise, the grid points would not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems did not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis found that our scheme could effectively reduce the required number of grid points per shear wavelength for equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improved with the increase of Poisson’s ratio. Three numerical examples demonstrated that our scheme could provide more accurate results than the classical nine-point scheme in terms of the internal and the free-surface grid points.

scholarly journals Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ke-Yang Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Computing Time ◽  
Staggered Grid ◽  
Difference Operators ◽  
Elastic Wave Equation ◽  
Absorbing Boundary ◽  
Wave Separation ◽  
Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

Sign in / Sign up

Export Citation Format

Share Document