Abstract
<strong class="journal-contentHeaderColor">Abstract.</strong> In this article, we introduce spyro, a software stack to solve wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite-element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite-element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time domain FWI approach that uses meshes composed of variably sized triangular elements to discretize the domain is detailed. To resolve both the forward and adjoint-state equations and to calculate a mesh-independent gradient associated with the FWI process, a fully explicit, variable higher-order (up to degree <span class="inline-formula"><i>k</i>=5</span> in 2D and <span class="inline-formula"><i>k</i>=3</span> in 3D) mass-lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wave field (e.g., local <span class="inline-formula"><i>P</i></span>-wave speed) and by leveraging higher-order basis functions, the number of degrees of freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.
Highlights
The construction of models consistent with observations of Earth’s physical properties can be posed mathematically as solving 15 an inverse problem referred to as full waveform inversion (FWI) (Lines and Newrick, 2004; Virieux and Operto, 2009; Fichtner, 2011; Brittan et al, 2013)
We introduce spyro, a software stack to solve acoustic wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite element method
The aim of this paper is to address the issues associated with the application of triangular, unstructured finite element methods (FEM) to perform FWI with the higher-order mass lumped elements of Chin-Joe-Kong et al (1999) and Geevers et al (2018b)
Summary
The construction of models consistent with observations of Earth’s physical properties can be posed mathematically as solving 15 an inverse problem referred to as full waveform inversion (FWI) (Lines and Newrick, 2004; Virieux and Operto, 2009; Fichtner, 2011; Brittan et al, 2013). Recognizing this issue, many advanced software packages have been put forward, separating the concerns between low-level programming/implementation and the high-level mathematical formulation to more confidently write FEM codes for various application domains (Krischer et al, 2015; Modrak et al, 2018; Alnæs et al, 2015; Witte et al, 2019; Cockett et al, 2015; Rücker et al, 2017; Louboutin et al, 2019; Rathgeber et al, 2017) These approaches 85 often present a programming environment in which data objects correspond to higher-level mathematical objects inherent to inverse problems and/or numerical discretizations such as the finite difference, finite element or finite volume methods. We demonstrate computational results in both 2D and 3D, discuss and conclude the work
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