Full waveform inversion using triangular waveform adapted meshes (original) (raw)
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Geophysical Journal International, 2010
To delineate the Earth's structure precisely by time-or frequency-domain waveform inversions, an appropriate initial model that contains long-wavelength structures of the true model is essential. A Laplace-domain waveform inversion, which can provide a smooth velocity model that is equivalent to a long-wavelength velocity model from scratch, was recently proposed. The Laplace-domain waveform inversion can be used for obtaining a good initial model for full-waveform inversions. To improve the applicability of the 2-D Laplace-domain waveform inversion to field data, we adopted the adaptive finite element not only to deal with sources and receivers at shallow depths but also to reduce the computational burden using larger elements below the sources and receivers. For the boundary condition on the bounded domain for the Laplace-domain modelling and inversion, the unique property of the Laplace-domain wavefield, which was damped rapidly from a source, was taken into account. The computational domain was extended further from a source and then the Dirichlet boundary condition was applied to the outer edges of the extended model. With these two adaptations, our Laplacedomain waveform inversion could then be tested on a very large 2004 BP velocity-analysis benchmark data set. Numerical tests of the BP model, which contains three salt domes of high-velocity contrast, were conducted on a self-made synthetic data set by Laplace-domain modelling and a time-domain original data set. For the two data sets, a Laplace-domain waveform inversion using the adaptive mesh successfully provided a smooth velocity model that contained long-wavelength structures of the true velocity model. The smooth velocity model obtained by the Laplace-domain waveform inversion was used as a good initial model for the frequency-domain waveform inversion of the original BP data set. The resulting frequency-domain inversion recovered almost every feature of the salt domes except for some subsalt structures.
Geophysical Journal International, 2011
We present an extension of the 3-D spectral element method (SEM), called the Gaussian quadrature grid (GQG) approach, to simulate in the frequency-domain seismic waves in 3-D heterogeneous anisotropic media involving a complex free-surface topography and/or subsurface geometry. It differs from the conventional SEM in two ways. The first is the replacement of the hexahedral element mesh with 3-D Gaussian quadrature abscissae to directly sample the physical properties or model parameters. This gives a point-gridded model which more exactly and easily matches the free-surface topography and/or any sub-surface interfaces. It does not require that the topography be highly smooth, a condition required in the curved finite difference method and the spectral method. The second is the derivation of a complex-valued elastic tensor expression for the perfectly matched layer (PML) model parameters for a general anisotropic medium, whose imaginary parts are determined by the PML formulation rather than having to choose a specific class of viscoelastic material. Furthermore, the new formulation is much simpler than the time-domain-oriented PML implementation. The specified imaginary parts of the density and elastic moduli are valid for arbitrary anisotropic media. We give two numerical solutions in full-space homogeneous, isotropic and anisotropic media, respectively, and compare them with the analytical solutions, as well as show the excellent effectiveness of the PML model parameters. In addition, we perform numerical simulations for 3-D seismic waves in a heterogeneous, anisotropic model incorporating a free-surface ridge topography and validate the results against the 2.5-D modelling solution, and demonstrate the capability of the approach to handle realistic situations.
Efficient 3D elastic FWI using a spectral-element method on Cartesian-based mesh
SEG Technical Program Expanded Abstracts 2017, 2017
Full Waveform Inversion offers the possibility to extract highresolution quantitative multi-parameters models of the subsurface from seismic data. Heretofore, most of FWI applications at the crustal scale have been performed under the acoustic approximation, generally for marine environments. When considering challenging land problems, efficient strategies are required for moving toward elastic inversion. We present such approach for 3D elastic time-domain inversion, based on spectral element methods designed on cartesian-based meshes. The proposed workflow integrates an easy and accurate cartesianbased mesh building with high-order shape functions to capture rapid topography variations and an efficient workflow for the incident and adjoint fields computation. A nonstationary and anisotropic structure-oriented smoothing filter is implemented directly on the spectral element mesh, for preconditioning FWI by incorporating prior geological information such as coherent lengths, dip and azimuth angles. Numerical illustrations on Marmousi and SEAM II benchmarks illustrate the importance of each ingredient we have developed for making efficient and flexible elastic FWI for land applications.
Journal of Applied Geophysics, 2013
Refraction-traveltime tomography is the most common approach and widely used for estimating velocity models with rugged topography and strongly variant near-surface geology. However, for complex geographical structures, there is often a restriction to the application of the conventional approach because the refracted energy can be trapped by the near-surface structure, which leads to limited depth penetration. To solve this problem, we propose a velocity estimation algorithm for foothill areas using Laplace-domain full waveform inversion (FWI) with irregular finite elements. Because the Laplace-domain FWI uses wavefields damped exponentially in time, the acoustic wave equation can be applied to foothill datasets without suppressing various types of elastic noise. In this study, irregular finite elements are generated to depict complicated surface topography using a Delaunay triangulation and tetrahedralization algorithm. Furthermore, adaptive mesh generation that formulates larger size elements with greater depth is used for minimizing the intensive computational costs in solving the full wave equation in the 2D and 3D domains. The validity of our proposed algorithm is demonstrated for 2D and 3D synthetic datasets and a 2D real exploration dataset acquired in the complex Aquio field foothill area in Bolivia.
Geophysical Journal International, 2009
This paper is concerned with a finite element method that uses hybrid meshes of multiresolution structured and unstructured meshes to simulate earthquake ground motion in large basins. While standard octree-based mesh methods use cubic elements and rely on mesh refinement to model topography, the present hybrid meshes use tetrahedral elements to model complicated layer interfaces and above-ground surfaces and to provide a transition between cubic elements of different sizes. The hybrid meshes avoid the step-like character inherent to the cubic meshes and the resulting concomitant loss of accuracy. Several numerical experiments of increasing complexity are carried out to verify and illustrate the proposed technique.
An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling
Geophysical Journal International, 2010
We present a discontinuous Galerkin finite-element method (DG-FEM) formulation with Convolutional Perfectly Matched Layer (CPML) absorbing boundary condition for 3-D elastic seismic wave modelling. This method makes use of unstructured tetrahedral meshes locally refined according to the medium properties (h-adaptivity), and of approximation orders that can change from one element to another according to an adequate criterion (p-adaptivity). These two features allow us to significantly reduce the computational cost of the simulations. Moreover, we have designed an efficient CPML absorbing boundary condition, both in terms of absorption and computational cost, by combining approximation orders in the numerical domain. A quadratic interpolation is typically used in the medium to obtain the required accuracy, while lower approximation orders are used in the CPMLs to reduce the total computational cost and to obtain a well-balanced workload over the processors. While the efficiency of DG-FEMs have been largely demonstrated for high approximation orders, we favour the use of low approximation orders as they are more appropriate to the applications we are interested in. In particular, we address the issues of seismic modelling and seismic imaging in cases of complex geological structures that require a fine discretization of the medium. We illustrate the efficiency of our approach within the framework of the EUROSEISTEST verification and validation project, which is designed to compare highfrequency (up to 4 Hz) numerical predictions of ground motion in the Volvi basin (Greece). Through the tetrahedral meshing, we have achieved fine discretization of the basin, which appears to be a sine qua non condition for accurate computation of surface waves diffracted at the basin edges. We compare our results with predictions computed with the spectral element method (SEM), and demonstrate that our method yields the same level of accuracy with computation times of the same order of magnitude.
Advances in Modelling and Inversion of Seismic Wave Propagation
High Performance Computing in Science and Engineering, Garching/Munich 2009, 2010
We report on progress in modelling and inversion of seismic waveforms. This involves in particular the simulation of wave propagation through Earth models with complex geometries (i.e., internal interfaces or topography) using numerical solutions based on tetrahedral meshes. In addition, efficient solvers in 3-D based on a regular-grid spectral element method allow for the simulation of many Earth models and for the inversion (i.e., for the fit) of observed seismograms using adjoint techniques. We present an application of this approach to the Australian continent. Furthermore results are presented on exploiting ideas from reverse acoustics to estimate finite source properties of large earthquakes and to constrain crustal scattering through modeling joint observations of rotational and translational ground motions.
Modular and flexible spectral-element waveform modelling in two and three dimensions
Geophysical Journal International, 2018
In this paper, we present a series of mathematical abstractions for seismologically relevant wave equations discretized using finite-element methods, and demonstrate how these abstractions can be implemented efficiently in computer code. Our motivation is to mitigate the combinatorial complexity present when considering geophysical waveform modelling and inversion, where a variety of spatial discretizations, material models, and boundary conditions must be considered simultaneously. We accomplish this goal by first considering three distinct classes of abstract mathematical models: (1) those representing the physics of an underlying wave equation, (2) those describing the discretization of the chosen equation onto a finitedimensional basis and (3) those describing any spatial transforms. A full representation of the discrete wave equation can then be constructed using a hierarchical nesting of models from each class. Additionally, each class is functionally orthogonal to the others, and with certain restrictions models within one class can be interchanged independently from changes in another. We then show how this recasting of the relevant equations can be implemented concisely in computer software using an abstract object-oriented design, and discuss how recent developments in the numerical and computational sciences can be naturally incorporated. This builds to a set of results where we demonstrate how the developments presented can lead to an implementation capable of multiphysics waveform simulations in completely unstructured domains, on both hypercubical and simplical spectral-element meshes, in both two and three dimensions, while remaining concise, efficient and maintainable.
Seismic waveform simulation for models with fluctuating interfaces
Scientific Reports
The contrast of elastic properties across a subsurface interface imposes a dominant influence on the seismic wavefield, which includes transmitted and reflected waves from the interface. Therefore, for an accurate waveform simulation, it is necessary to have an accurate representation of the subsurface interfaces within the numerical model. Accordingly, body-fitted gridding is used to partition subsurface models so that the grids coincide well with both the irregular surface and fluctuating interfaces of the Earth. However, non-rectangular meshes inevitably exist across fluctuating interfaces. This nonorthogonality degrades the accuracy of the waveform simulation when using a conventional finitedifference method. Here, we find that a summation-by-parts (SBP) finite-difference method can be used for models with non-rectangular meshes across fluctuating interfaces, and can achieve desirable simulation accuracy. The acute angle of non-rectangular meshes can be relaxed to as low as 47°. The cell size rate of change between neighbouring grids can be relaxed to as much as 30%. Because the non-orthogonality of grids has a much smaller impact on the waveform simulation accuracy, the model discretisation can be relatively flexible for fitting fluctuating boundaries within any complex problem. Consequently, seismic waveform inversion can explicitly include fluctuating interfaces within a subsurface velocity model.
Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes
Geophysical Journal International, 2011
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 Fréchet 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.