Time-varying boundary conditions in simulation of seismic wave propagation

Geophysics ◽  
2011 ◽  
Vol 76 (1) ◽  
pp. A1-A6 ◽  
Author(s):  
Robin P. Fletcher ◽  
Johan O. Robertsson
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Seismic Wave Propagation ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Reverse Time ◽  
Time Migration

We propose two new boundary conditions to regulate coherent reflections from the model boundaries in numerical solutions of wave equations. Both boundary conditions have the common feature that the boundary condition is varied with respect to time. The first boundary condition expands or contracts the computational model during a modeling simulation. The effect is to cause a Doppler shift in the reflected wavefield that can be used to shift energy outside a frequency band of interest. In addition, when the computational domain is expanding, the range of possible incidence angles on the boundary is restricted. This can be used to increase the effectiveness of many existing absorbing boundary conditions that are more effective for incidence angles close to normal. The second boundary condition is an extension of random boundaries. By carefully changing the realization of a random boundary over time, a more diffusive wavefield can be simulated. We show results with 2D numerical simulations of the scalar wave equation for both these boundary conditions. The first boundary condition has application to modeling, but both these boundary conditions have potential application within algorithms that rely upon modeling kernels, such as reverse-time migration and full-waveform inversion.

Reducing computation cost by Lax-Wendroff methods with fourth-order temporal accuracy

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T109-T119 ◽  
Author(s):  
Hongwei Liu ◽  
Houzhu Zhang
Keyword(s):  
Wave Equations ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Second Order ◽  
Numerical Dispersion ◽  
Reverse Time ◽  
Time Migration ◽  
L2 Norm ◽  
Time Marching

Explicit time-marching finite-difference stencils have been extensively used for simulating seismic wave propagation, and they are the most computationally intensive part of seismic forward modeling, reverse time migration, and full-waveform inversion. The time-marching step, determined by both the stability condition and numerical dispersion, is a key factor in the computational cost. In contrast with the widely used second-order temporal stencil, the Lax-Wendroff stencil is more cost effective because the time-marching step can be much larger. It can be proved, using theory and numerical tests, that the Lax-Wendroff stencil does enable larger time steps. In terms of numerical dispersion, the time steps for second-order and Lax-Wendroff stencils are functions of the number of shortest wavelengths away from the source location. These functions are derived by evaluating the relative L2-norm of the differences between the analytical and numerical solutions. The method for determining the time-marching step is model adaptive and easy to implement. We use the pseudo spectral method for the computation of spatial derivatives, and the wave equations that we solved are for isotropic media only, but the described principles can be easily implemented for more complicated types of media.

scholarly journals A Generating - Absorbing Boundary Condition Applied to Wave - Current Interactions Using the Method of Fundamental Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Loukili ◽  
Kamila Kotrasova ◽  
Amine Bouaine
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Fundamental Solutions ◽  
Absorbing Boundary Conditions ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Numerical Wave Tank ◽  
Absorbing Boundary ◽  
Wave Properties

Abstract The purpose of this work is to study the feasibility and efficiency of Generating Absorbing Boundary Conditions (GABCs), applied to wave-current interactions using the Method of Fundamental Solutions (MFS) as radial basis function, the problem is solved by collocation method. The objective is modeling wave-current interactions phenomena applied in a Numerical Wave Tank (NWT) where the flow is described within the potential theory, using a condition without resorting to the sponge layers on the boundaries. To check the feasibility and efficiency of GABCs presented in this paper, we verify accurately the numerical solutions by comparing the numerical solutions with the analytical ones. Further, we check the accuracy of numerical solutions by trying a different number of nodes. Thereafter, we evaluate the influence of different aspects of current (coplanar current, without current, and opposing current) on the wave properties. As an application, we take into account the generating-absorbing boundary conditions GABCs in a computational domain with a wavy downstream wall to confirm the efficiency of the adopted numerical boundary condition.

Tensorial elastodynamics for isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. T359-T373
Author(s):  
Jeffrey Shragge ◽  
Tugrul Konuk
Keyword(s):  
Free Surface ◽  
Surface Topography ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Real Life ◽  
Staggered Grid ◽  
Surface Boundary ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

True-amplitude linearized waveform inversion with the quasi-elastic wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R827-R844 ◽  
Author(s):  
Zongcai Feng ◽  
Gerard Schuster
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Elastic Wave Equation ◽  
P Waves ◽  
Time Migration

We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the same as that for the elastic wave equation, and only requires the solution of two acoustic wave equations for each shot gather. This means that the quasi-elastic wave equation can be used for true-amplitude linearized waveform inversion (also known as least-squares reverse time migration) of elastic PP reflections, in which the corresponding misfit gradients are with respect to the perturbations of density and the P- and S-wave impedances. The perturbations of elastic parameters are iteratively updated by minimizing the [Formula: see text]-norm of the difference between the recorded PP reflections and the predicted pressure data modeled from the quasi-elastic wave equation. Numerical tests on synthetic and field data indicate that true-amplitude linearized waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude linearized waveform inversion provides images with a wider bandwidth and fewer artifacts because the PP amplitudes are accurately explained. We also determine the 2D scalar quasi-elastic wave equation for P-SV reflections and the 3D vector equation for PS reflections.

Implementation of the Kirchhoff integral for elastic waves in staggered‐grid modeling schemes

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1894-1901 ◽  
Author(s):  
Rune Mittet
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elastic Waves ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Reverse Time ◽  
S Waves ◽  
Time Migration ◽  
Delta Functions ◽  
Band Limited

Implementation of boundary conditions in finite‐difference schemes is not straightforward for the elastic wave equation if a staggered grid formulation is used. Reverse time migration of VSP data requires a proper description of the recording surface so as not to excite false P‐ and S‐waves. Such contributions may cause artifacts in the imaging procedure. The boundary conditions for the elastic stress tensor can be implemented numerically in a staggered coarse grid modeling scheme by using band‐limited spatial delta‐functions and band‐limited first‐order derivatives of these spatial delta‐functions. A representation theorem for elastic waves is derived to test the implementation of the spatial part of the boundary condition. The implementation is tested in a 2-D numerical experiment for a closed, but curved, boundary S enclosing a volume V. The test condition is that within the volume V, the difference between the forward modeled field and the retropropagated field should be equal to zero. Both P‐ and S‐waves are properly recovered in a 2-D reverse time modeling example. The numerical artifacts related to the proposed spatial approximation of the boundary condition are found to be negligible.

Viscoacoustic reverse time migration using a time-domain complex-valued wave equation

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. S505-S519 ◽  
Author(s):  
Jidong Yang ◽  
Hejun Zhu
Keyword(s):  
Wave Equation ◽  
Time Domain ◽  
Wave Equations ◽  
Seismic Wave Propagation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Reverse Wave ◽  
Time Migration ◽  
Complex Valued ◽  
Dispersion Term

During seismic wave propagation, intrinsic attenuation inside the earth gives rise to amplitude loss and phase dispersion. Without appropriate correction strategies in migration, these effects degrade the amplitudes and resolution of migrated images. Based on a new time-domain viscoacoustic wave equation, we have developed a viscoacoustic reverse time migration (RTM) approach to correct attenuation-associated dispersion and dissipation effects. A time-reverse wave equation is derived to extrapolate the receiver wavefields, in which the sign of the dissipation term is reversed, whereas the dispersion term remains unchanged. The difference between the forward and time-reverse wave equations is consistent with the physical insights of attenuation compensation during wavefield backpropagation. Due to the introduction of an imaginary unit in the dispersion term, the forward and time-reverse wave equations are complex valued. They are similar to the time-dependent Schrödinger equation, whose real and imaginary parts are coupled during wavefield extrapolation. The analytic property of the extrapolated source and receiver wavefields allows us to explicitly separate up- and downgoing waves. A causal imaging condition is implemented by crosscorrelating downgoing source and upgoing receiver wavefields to remove low-wavenumber artifacts in migrated images. Numerical examples demonstrate that our viscoacoustic RTM approach is capable of producing subsurface reflectivity images with correct spatial locations as well as amplitudes.

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

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