A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

Yang, Pengliang; Brossier, Romain; Métivier, Ludovic; Virieux, Jean

doi:10.1093/gji/ggw262

Cited by 91 publications

(39 citation statements)

References 71 publications

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“…(1) is not selfadjoint. Therefore, applying the adjoint state method to it would lead to different set of equations for the forward and adjoint wavefields (see Yang et al 2016). We instead transform eq.…”

Section: Forward Modellingmentioning

confidence: 99%

“…Note that l = R −1 R. Hence, our transformation, which was first presented by Fabien-Ouellet et al (2016), is equivalent the approach of Yang et al (2016), that is, we multiply the wave equation by the inverse of the stiffness tensor, or the compliance.…”

Section: Transformation To Self-adjoint Operatorsmentioning

confidence: 99%

“…Furthermore, the finite difference solution is not affected by the sign changes and both forward and adjoint modelling share the same stability criteria. Our adjoint equations are different from (Tarantola 1988;Tromp et al 2004) that are given in the displacement formulation, or from (Yang et al 2016) that results from the direct application of the adjoint state method to the original forward model, giving asymmetric forward and adjoint equations. However, all those formulations are equivalent, as they all share the same forward and adjoint second order wave equation for particle displacements:…”

Section: Adjoint State Equationsmentioning

confidence: 99%

“…However, the only modification required to the forward modelling code to calculate the adjoint wavefield is minor. In comparison, the direct application of the adjoint state method to the viscoelastic wave equation leads to two distinct sets of partial differential equations that must be solved by two different modelling codes (Yang et al 2016). This fact is not negligible, considering that a highly optimized modelling code can be very time consuming to produce.…”

Section: O N C L U S I O N Smentioning

confidence: 99%

“…Hence, the results obtained for the displacement formulation cannot be used directly with the viscoelastic equations of Robertsson et al (1994) and the like. A recent study by Yang et al (2016) derived the forward and adjoint equations for this formulation, which are asymmetric because the first order viscoelastic wave equation is not self-adjoint. Hence, two different systems of partial differential equations must be solved for the forward and adjoint variables.…”

Section: Introductionmentioning

confidence: 99%

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Time domain viscoelastic full waveform inversion

Fabien‐Ouellet¹,

Gloaguen²,

Giroux³

2017

Geophysical Journal International

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S U M M A R YViscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity-stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity-stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO 2 geological sequestration. We show that FWI based on our formulation can indeed recover P-and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO 2 plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

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Section: Forward Modellingmentioning

confidence: 99%

Section: Transformation To Self-adjoint Operatorsmentioning

confidence: 99%

Section: Adjoint State Equationsmentioning

confidence: 99%

Section: O N C L U S I O N Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time domain viscoelastic full waveform inversion

Fabien‐Ouellet¹,

Gloaguen²,

Giroux³

2017

Geophysical Journal International

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Characterization of silty to fine‐sandy sediments with SH waves: full waveform inversion in comparison with other geophysical methods

Schwardt

Köhn

Wunderlich

et al. 2020

Near Surface Geophysics

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We apply seismic full waveform inversion to SH‐ and Love‐wave data for investigating the near‐surface lithology at an archaeological site. We evaluate the resolution of the applied full waveform inversion algorithm through ground truthing in the form of an excavation and sediment core studies. Thereby, we investigate the benefits of full waveform inversion in comparison with other established methods of near‐surface prospecting in terms of resolution capabilities and interpretation security. The study is performed in a presumed harbour area of the ancient Thracian city of Ainos. The exemplary target is the source of a linear magnetic anomaly oriented perpendicular to the coast, which was found in a previous magnetic gradiometry survey, suggesting a mole. The SH‐wave full waveform inversion recovered a subsurface SH‐wave velocity model with submeter resolution showing lateral and vertical velocity variation between 40 and 150 m/s. To tame the non‐linearity of the full waveform inversion, a sequential inversion of frequency bands has to be combined with time‐windowing in order to separate the Love wave from the reflected SH wavefield. We compare the full waveform inversion results with multichannel analysis of surface waves, standard seismic reflection imaging, electrical resistivity tomography and electromagnetic induction. It turns out that the respective depth sections are correlated to a certain degree with the full waveform inversion results. However, the structural resolution of the other geophysical methods is significantly lower than for the full waveform inversion. An exception is the reflection seismic imaging, which shows the same resolution as full waveform inversion but can only be interpreted together with the full waveform inversion–based velocity model. An archaeological excavation as well as coring data allows ground truthing and a direct understanding of the geophysical structures. The results show that the target was a sort of near‐surface trench of about 3–4 m width and 0.8 m to 1.0 m depth, filled with silty sediment, which differs from the layered surrounding in colour and composition. The ground truthing revealed that only SH‐wave full waveform inversion and seismic reflection imaging could image the trench and sediment structure with satisfying lateral and depth resolution. We emphasize that the velocity distribution from SH‐wave full waveform inversion agrees closely with the excavated subsurface structures, and that the discovered changes in seismic velocity correlate with changes in the sand content in the respective sediment facies sequences. The study demonstrated that SH‐wave full waveform inversion is capable to image structural and lithological changes in the near subsurface at scales as low as 0.5 m, thus providing the high resolution needed for archaeological and geoarchaeological prospection.

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Numerical Fréchet derivatives of the displacement tensor for 2.5‐D frequency‐domain seismic full‐waveform inversion in viscoelastic TTI media

Yang,

Zhou,

Engsig

et al. 2023

Near Surface Geophysics

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Derivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Fréchet derivatives (or sensitivity kernels), are a key ingredient for seismic full‐waveform inversion with a local‐search optimization algorithm. They provide a quantitative measure of the expected changes in the seismograms due to perturbations of the subsurface model parameters for a given survey geometry. Since 2.5‐D wavefield modeling involves a real point source in a 2‐D geological model with 3D (spherical) wave properties, it yields synthetic data much closer to the actual practical field data than the commonly used 2‐D wave simulation does, which uses an unrealistic line source in which the waves spread cylindrically. Based on our recently developed general 2.5‐D wavefield modeling scheme, we apply the perturbation method to obtain explicit analytic expressions for the derivatives of the displacement tensor for 2.5‐D/2‐D frequency‐domain seismic full‐waveform inversion in general viscoelastic anisotropic media. We then demonstrate the numerical calculations of all these derivatives in two common cases: (i) viscoelastic isotropic and (ii) viscoelastic tilted transversely isotropic (TTI) solids. Examples of the differing sensitivity patterns for the various derivatives are investigated and compared for four different homogeneous models involving 2‐D and 2.5‐D modeling. Also, the numerical results are verified against the analytic solutions for homogeneous models. We further validate the numerical derivatives in a 2‐D heterogeneous viscoelastic TTI case by conducting a synthetic data experiment of frequency‐domain full‐waveform inversion to individually recover the twelve independent model parameters (density, dip angle, five elastic moduli, and five corresponding Q‐factors) in a simple model comprising an anomalous square box target embedded in a uniform background. Another 2.5‐D multi‐target model experiment presenting impacts from four common seismic surveying geometries validates the Fréchet derivatives again.This article is protected by copyright. All rights reserved

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A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

Cited by 91 publications

References 71 publications

Time domain viscoelastic full waveform inversion

Time domain viscoelastic full waveform inversion

Characterization of silty to fine‐sandy sediments with SH waves: full waveform inversion in comparison with other geophysical methods

Numerical Fréchet derivatives of the displacement tensor for 2.5‐D frequency‐domain seismic full‐waveform inversion in viscoelastic TTI media

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