Introduction.- Numerical Solution of the Elastic Wave Equation.- Computing Sensitivity Kernels.- Seismological Data Functionals and their Associated Adjoint Sources.- Iterative Optimisation.- Full Waveform Tomography for Upper-mantle Structure in Australasian Region.- A Comparative Study of Local-scale full Waveform Tomographies.- Source Staking and Data Reduction in Global full Waveform Tomography.

Full Seismic Waveform Modelling and Inversion

We present forward and adjoint spectral-element simulations of coupled acoustic and (an)elastic seismic wave propagation on fully unstructured hexahedral meshes. Simulations benefit from recent advances in hexahedral meshing, load balancing and software optimization. Meshing may be accomplished using a mesh generation tool kit such as CUBIT, and load balancing is facilitated by graph partitioning based on the SCOTCH library. Coupling between fluid and solid regions is incorporated in a straightforward fashion using domain decomposition. Topography, bathymetry and Moho undulations may be readily included in the mesh, and physical dispersion and attenuation associated with anelasticity are accounted for using a series of standard linear solids. Finite-frequency Fr'echet derivatives are calculated using adjoint methods in both fluid and solid domains. The software is benchmarked for a layercake model. We present various examples of fully unstructured meshes, snapshots of wavefields and finite-frequency kernels generated by Version 2.0 'Sesame' of our widely used open source spectral-element package SPECFEM3D.

/pdf/forward-and-adjoint-simulations-of-seismic-wave-propagation-133ae34l3b.pdf

Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes

https://www.mathematik.tu-dortmund.de/lsiii/cms/papers/GoeddekeKomatitschErlebacherMichea2011.pdf

High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster

A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

Computational physics has become an essential research and interpretation tool in many fields. Particularly in reservoir geophysics, ultrasonic and seismic modeling in porous media is used to study the properties of rocks and to characterize the seismic response of geologic formations. We provide a review of the most common numerical methods used to solve the partial differential equations describing wave propagation in fluid-saturated rocks, i.e., finite-difference, pseudospectral, and finite-element methods, including the spectral-element technique. The modeling is based on Biot-type theories of dynamic poroelasticity, which constitute a general framework to describe the physics of wave propagation. We explain the various techniques and discuss numerical implementation aspects for application to seismic modeling and rock physics, as, for instance, the role of the Biot diffusion wave as a loss mechanism and interface waves in porous media.

/pdf/computational-poroelasticity-a-review-wzk6m20vl8.pdf

Computational poroelasticity — A review

Purely numerical methods based on finite-element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms. We present formulas for the grid dispersion and stability criteria for some popular finite-element methods (FEM) for wave propagation, namely, classical and spectral FEM. We develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these FEMs for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high-order polynomials, as is the case for spectral FEM. Analysis reveals that for spectral FEM of order four or greater, dispersion is less than 0.2% at four to five nodes per wavelength, and dispersion is not angle dependent. New results can be compared with grid-dispersion results of some classical finite-difference methods (FDM) used for acoustic or elastic wave propagation. Analysis reveals that FDM and classical FEM require a larger sampling ratio than a spectral FEM to obtain results with the same degree of accuracy. The staggered-grid FDM is an efficient scheme, but the dispersion is angle dependent with larger values along the grid axes. On the other hand, spectral FEM of order four or greater is isotropic with small dispersion, making it attractive for simulations with long propagation times.

https://www.math.purdue.edu/~santos/research/disc_galerkin_ref/DeBasabe-Sen-2007-Geophysics.pdf

Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations

SUMMARY Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to wave propagation. In this work, we investigate the applicability of the interior penalty DGM to elastic wave propagation by analysing it’s grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.

https://www.math.purdue.edu/~santos/research/disc_galerkin_ref/DeBasabe-etal-2008-GJI.pdf

The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion

SUMMARY

We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or elastic wave propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and elastic cases, respectively.

Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

We have formulated and implemented a discontinuous Galerkin method (DGM) for elastic wave propagation that allows for discontinuities in the displacement field to simulate fractures or faults. The approach is based on the interior-penalty formulation of DGM, and the fractures are simulated using the linear-slip model, which is incorporated into the weak formulation by including an additional term that is similar to the penalty term but uses the fracture compliance instead of an arbitrary penalty parameter. We have calibrated our results against an analytic solution of fracture-induced anisotropy for a set of elongated horizontal fractures, and we have evaluated numerical examples that simulate the reflection and transmission of waves at a fracture and at fracture interface waves. This method can further be used with models containing intersecting fractures and multiple fracture sets in 2D or 3D domains.

Elastic wave propagation in fractured media using the discontinuous Galerkin method

The numerical simulation of wave propagation in media with solid and fluid layers is essential for marine seismic exploration data analysis. The numerical methods for wave propagation that are applicable to this physical settings can be broadly classified as partitioned or monolithic: The partitioned methods use separate simulations in the fluid and solid regions and explicitly satisfy the interface conditions, whereas the monolithic methods use the same method in all the domain without any special treatment of the fluid-solid interface. Despite the accuracy of the partitioned methods, the monolithic methods are more common in practice because of their convenience. In this paper, we analyse the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface. The analysis is based on grid-dispersion criteria and numerical examples. The methods studied here include: the classical finite-difference method (FDM) based on the second-order displacement formulation of the elastic wave equation (DFDM), the staggered-grid finite difference method (SGFDM), the velocity-stress FDM with a standard grid (VSFDM) and the spectral-element method (SEM). We observe that among these, DFDM and the first-order SEM have a large amount of grid dispersion in the fluid region which renders them impractical for this application. On the other hand, SGFDM, VSFDM and SEM of order greater or equal to 2 yield accurate results for the body waves in the fluid and solid regions if a sufficient number of nodes per wavelength is used. All of the considered methods yield limited accuracy for the surface waves because the proper boundary conditions are not incorporated into the numerical scheme. Overall, we demonstrate both by analytic treatment and numerical experiments, that a first-order velocity-stress formulation can, in general, be used in dealing with fluid-solid interfaces without using staggered grids necessarily.

/pdf/a-comparison-of-finite-difference-and-spectral-element-l9us7jon0v.pdf

Jonás D. De Basabe

Papers

Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations

The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion

Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

Elastic wave propagation in fractured media using the discontinuous Galerkin method

A comparison of finite-difference and spectral-element methods for elastic wave propagation in media with a fluid-solid interface