scholarly journals Explicit coupling of acoustic and elastic wave propagation in finite-difference simulations

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. T293-T308
Author(s):  
Longfei Gao ◽  
David Keyes
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Elastic Wave ◽  
Seismic Exploration ◽  
Difference Operators ◽  
Continuous Systems ◽  
Elastic Model ◽  
Wave System ◽  
Energy Conserving ◽  
Acoustic Interface

We present a mechanism to explicitly couple the finite-difference discretizations of 2D acoustic and isotropic elastic-wave systems that are separated by straight interfaces. Such coupled simulations allow for the application of the elastic model to geological regions that are of special interest for seismic exploration studies (e.g., the areas surrounding salt bodies), with the computationally more tractable acoustic model still being applied in the background regions. Specifically, the acoustic wave system is expressed in terms of velocity and pressure while the elastic wave system is expressed in terms of velocity and stress. Both systems are posed in first-order forms and are discretized on staggered grids. Special variants of the standard finite-difference operators, namely, operators that possess the summation-by-parts property, are used for the approximation of spatial derivatives. Penalty terms, which are also referred to as the simultaneous approximation terms, are designed to weakly impose the elastic-acoustic interface conditions in the finite-difference discretizations and couple the elastic and acoustic wave simulations together. With the presented mechanism, we are able to perform the coupled elastic-acoustic wave simulations stably and accurately. Moreover, it is shown that the energy-conserving property in the continuous systems can be preserved in the discretized systems with carefully designed penalty terms.

scholarly journals ENERGY-CONSERVING AND RECIPROCAL SOLUTIONS FOR HIGHER-ORDER PARABOLIC EQUATIONS

2001 ◽  
Vol 09 (01) ◽  
pp. 183-203 ◽  
Author(s):  
DMITRY MIKHIN
Keyword(s):  
Finite Difference ◽  
Energy Conservation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Finite Difference Scheme ◽  
Flow Reversal ◽  
Difference Operators ◽  
Energy Conserving ◽  
The One ◽  
Moving Media

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Padé approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

Acoustic Wave Equation Modeling with New Time-Space Domain Finite Difference Operators

2013 ◽  
Vol 56 (6) ◽  
pp. 840-850 ◽  
Author(s):  
LIANG Wen-Quan ◽  
YANG Chang-Chun ◽  
WANG Yan-Fei ◽  
LIU Hong-Wei
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Equation Modeling ◽  
Difference Operators ◽  
Acoustic Wave Equation ◽  
Space Domain ◽  
Time Space ◽  
New Time

2-D elastic wave modeling with frequency-space 25-point finite-difference operators

2009 ◽  
Vol 6 (3) ◽  
pp. 259-266 ◽  
Author(s):  
Jianping Liao ◽  
Huazhong Wang ◽  
Zaitian Ma
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Difference Operators ◽  
Wave Modeling ◽  
Frequency Space

Elastic wave propagation using fully vectorized high order finite‐difference algorithms

Geophysics ◽  
1992 ◽  
Vol 57 (2) ◽  
pp. 218-232 ◽  
Author(s):  
A. Vafidis ◽  
F. Abramovici ◽  
E. R. Kanasewich
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Second Order ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Two Dimensional ◽  
Elastic Wave Equation ◽  
The Matrix ◽  
Comparison Of The Results ◽  
High Order Finite Difference

Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

2-D elastic wave forward modeling with frequency-space 25-point finite-difference operators

2018 ◽  
Author(s):  
Hongliang Li ◽  
Shoudong Wang ◽  
Doudou Wang ◽  
Mingxiao Cui ◽  
Kailong Su
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Forward Modeling ◽  
Difference Operators ◽  
Frequency Space

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.

Anisotropic wave propagation through finite‐difference grids

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1203-1216 ◽  
Author(s):  
Heiner Igel ◽  
Peter Mora ◽  
Bruno Riollet
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Three Dimensions ◽  
Difference Operators ◽  
Elastic Model ◽  
General Description ◽  
Staggered Grids ◽  
Convolution Sums ◽  
Group And Phase Velocities ◽  
Wave Properties

An algorithm is presented to solve the elastic‐wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in Cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency‐dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.

A new family of finite-difference schemes to solve the heterogeneous acoustic wave equation

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T187-T199 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Cleberson Dors ◽  
Webe J. Mansur
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Staggered Grid ◽  
Order System ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Memory Usage ◽  
Acoustic Wave Equation ◽  
New Family

Equivalent staggered grid scheme (ESG) is a new family of schemes based on the finite-difference method (FDM). The method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic FDM approaches. The main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations (SSG), but with the same memory requirements of simple grid schemes. Hence, it results in a reduction of memory usage by 33% in 2D and 50% in 3D problems, compared to the memory usage of SSG. The first-order system of equations in terms of pressure and velocity is not used here. Instead, the formulation is based on applying new central difference operators to the second-order acoustic wave equation in terms of pressure, obtaining the same level of accuracy and stability as the SSG schemes. The equivalence between the ESG and SSG is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed.

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