Intrinsic non-uniqueness of the acoustic full waveform inverse problem

Abstract

Summary In the context of seismic imaging, full waveform inversion (FWI) is increasingly popular. Because of its lower numerical cost, the acoustic approximation is often used, especially at the exploration geophysics scale, both for tests and for real data. Moreover, some research domains such as helioseismology face true acoustic media for which FWI can be useful. In this work, an argument that combines particle relabelling and homogenization is used to show that the general acoustic inverse problem based on band-limited data is intrinsically non-unique. It follows that the results of such inversions should be interpreted with caution. To illustrate these ideas, we consider 2-D numerical FWI examples based on a Gauss-Newton iterative inversion scheme, and demonstrate effects of this non-uniqueness in the local optimization context.

Novel Hybrid Numerical Simulation of the Wave Equation by Combining Physical and Numerical Representation Theorems and a Review of Hybrid Methodologies

“…The meshing of the local target model M l 1 is completely independent of the meshing of the global reference model M g 0 . The associated local hybrid simulation is flexible, resulting in the possibility of using a more efficient SEM with very high polynomial degrees (e.g., degrees 12–24; Lyu et al., 2020) to reduce memory requirements, speed up the forward simulation, and use the upscale non‐periodic homogenization (Capdeville & Métivier, 2018; Lyu, Capdeville, Al‐Attar, et al., 2021) to improve the ability of FWI. Although classic SEM applications mostly rely on degrees 4–8 in each direction, higher degrees are often not adopted, primarily owing to the explicit meshing of mechanical discontinuities and exceedingly small available time steps.…”

“…Although classic SEM applications mostly rely on degrees 4–8 in each direction, higher degrees are often not adopted, primarily owing to the explicit meshing of mechanical discontinuities and exceedingly small available time steps. Note that in the recent homogenization method in seismology to smoothen the internal mechanical discontinuity (Capdeville et al., 2010), the smooth models used in forward/backward simulations using FWI (Lyu, Capdeville, Al‐Attar, et al., 2021), the computational complexity analysis of code‐independent features for SEM, and the actual computation time benchmarks all make very high polynomial degrees SEM attractive and competitive (Lyu et al., 2020).…”

“…Although classic SEM applications mostly rely on degrees 4-8 in each direction, higher degrees are often not adopted, primarily owing to the explicit meshing of mechanical discontinuities and exceedingly small available time steps. Note that in the recent homogenization method in seismology to smoothen the internal mechanical discontinuity (Capdeville et al, 2010), the smooth models used in forward/backward simulations using FWI (Lyu, Capdeville, Al-Attar, et al, 2021), the computational complexity analysis of code-independent features for SEM, and the actual computation time benchmarks all make very high polynomial degrees SEM attractive and competitive (Lyu et al, 2020). For the VM hybrid method, the adopted ABC is effective in the local target model M l1 with small structural perturbations (Monteiller et al, 2013;, but it is not sufficiently accurate for the scattered tangential incidence waves in complicated local target models in the hybrid calculation (Clayton & Engquist, 1977;Xie et al, 2014) 2) and waveforms calculated using the four methods.…”

“…The commonly admitted G to obtain sufficient accuracy in a constant‐velocity medium is approximately G ≈ 5 for one element with degree 8 (NGLL = 9, number of GLL points per element per direction; Basabe & Sen, 2007; Lyu et al., 2020; Seriani & Oliveira, 2008). A low G can be important in the FWI context owing to the extensive operations on forward and adjoint wavefields (Komatitsch et al., 2016; Lyu, Capdeville, Al‐Attar, et al., 2021). In our study, the meshing based on NGLL = 8 was adopted for the first global simulations in the global reference model M g 0 .…”

“…Over the past three decades, with the development of effective computer clusters, efficient numerical methods, such as the finite difference method (FDM) and spectral element method (SEM), improved full waveform inversion (FWI) techniques, and extensive global deployment of stations, the imaging of Earth has achieved unprecedented accuracy (Bozdag et al., 2016; Fichtner et al., 2018; French & Romanowicz, 2015; Lei et al., 2020). Faster and more accurate calculations of the elastic wavefield within heterogeneous media are key to developing and improving accurate imaging techniques relying on full waveform analysis (Capdeville & Métivier, 2018; Lyu, Capdeville, Al‐Attar, & Zhao, 2021; Pratt et al., 1998; Tarantola, 1984; Tromp, 2019; Virieux & Operto, 2009). Earth's structure is multiscale, yet capturing such a broad range of complex heterogeneities with seismic wave propagation across the observable frequency band (e.g., ≥1 s) requires thousands of global numerical simulations of seismic/acoustic wave equations, which are still computationally prohibitive.…”