Green's function for three‐dimensional elastic wave equation with a moving point source on the free surface with applications

2020 ◽  
Vol 68 (4) ◽  
pp. 1281-1290
Author(s):  
Jing‐Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Elastic Wave ◽  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Elastic Wave Equation

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 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.

scholarly journals A Green's function for diffraction by a rational wedge

1989 ◽  
Vol 105 (1) ◽  
pp. 185-192 ◽  
Author(s):  
A. D. Rawlins
Keyword(s):  
Wave Equation ◽  
Point Source ◽  
Green’S Function ◽  
Half Plane ◽  
Green's Function ◽  
General Result ◽  
Spherical Wave ◽  
Source Terms ◽  
Angular Sector ◽  
Rational Multiple

AbstractIn this paper we derive an expression for the point source Green's function for the reduced wave equation, valid in an angular sector whose angle is equal to a rational multiple of π. This Green's function can be used to find new expressions for the field produced by the diffraction of a spherical wave by a wedge whose angle can be expressed as a rational multiple of π. The expressions obtained will be in the form of source terms and real integrals representing the diffracted field. The general result obtained is used to derive a new representation for the solution of the problem of diffraction by a mixed hard–soft half plane.

Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

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