Level Set--Based Shape Optimization Approach for Sharp-Interface Reconstructions in Time-Domain Full Waveform Inversion

Viscous 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 CO₂ 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 CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Time-domain multi-scale full waveform inversion (FWI) is an important method for seismic imaging. Using different frequency contents of seismic data, this method mitigates the effect of the local-minima problem by sequential inversion. One of the important factors determining the convergence rate of the nonlinear inversion is the precondition of the gradient. Virtual-source precondition has been successfully applied to Laplace-domain and Laplace-Fourier-domain FWI. In this paper, we will examine the application of the virtual-source precondition to time-domain multi-scale FWI. The research results reveal that, compared to the gradient method without precondition, the virtual-source-based gradient method achieves much greater convergence rate while its additional computational cost is very low. Therefore, virtual-source precondition is very effective for time-domain multi-scale FWI. keywords—virtual-source; precondition; full-waveform inversion

Velocity macromodel building is an essential step of the seismic imaging workflow. Indeed, obtaining acceptable results through migration or full waveform inversion is highly dependent on the kinematic accuracy of the background/initial velocity model. Two decades ago, stereotomography was proposed as an alternative to reflection traveltime tomography, the first relying on semi-automatic picking of locally coherent events associated with small reflection or diffraction segments tied to scatterers in depth by a pair of rays, while the latter on interpretive picking of laterally continuous reflections. The flexibility of stereotomography paved the way for many developments that have shown the efficiency of the method whilst emphasizing on the complementary information carried out by traveltimes and slopes of locally coherent events. A recent formulation recast stereotomography under a matrix-free formulation based on eikonal solvers and the adjoint-state method. In the latter, like in the previous works, the scatterer positions and the velocity field are updated jointly to tackle the ill-famed velocity–position coupling in reflection tomography. Following on from this adjoint-state formulation, we propose a new parsimonious formulation of slope tomography that offers the chance to restrain the problem to minimizing the residuals of a single data class being a slope, in search of a sole parameter class being the subsurface velocity field. This parsimonious formulation results from a variable projection, which is implemented by enforcing a consistency between the scatterer coordinates and the velocity macromodel through migration of kinematic attributes. We explain why the resulting reduced-parametrization inversion is more suitable for tomographic problems than the most common joint inversion strategy. We benchmark our method against the complex Marmousi model along with a validation through time domain full waveform inversion and then present the results of a field data case study.

Time-domain full waveform inversion (FWI) in the acoustic regime comprises a parameter identification problem for the acoustic wave equation: pressure waves are initiated by sources, get scattered by the earth’s inner structure, and their reflected parts are picked up by receivers located on the surface. From these reflected wave fields the two parameters, density and sound speed, have to be reconstructed. Mathematically, FWI reduces to the solution of a nonlinear and ill-posed operator equation involving the parameter-to-solution map of the wave equation. Newton-like iterative regularization schemes are well suited and well analyzed to tackle this inverse problem. Their convergence results are often based on an assumption about the nonlinear map known as tangential cone condition. In this paper we verify this assumption for a semi-discrete version of FWI where the searched-for parameters are restricted to a finite dimensional space. As a byproduct we establish that the semi-discrete seismic inverse problem is locally Lipschitz stable, in particular, it is conditionally well-posed.

In seismic exploration, 3D acoustic full waveform inversion (FWI) is nowadays commonly used to estimate large-to-intermediate scale velocity features with low-frequency and wide-aperture data. Velocity maps estimated with this technique can have higher resolution than the ones obtained by travel-time inversion. However, since full waveform inversion fits both the phases and the amplitudes of seismic data, the acoustic assumption may not be adequate to reproduce the data's amplitude versus offset variations in presence of hard velocity contrasts. This could limit the applications of acoustic FWI and put some constrains on the choice of the initial model reducing model's resolution. To better understand the behaviour of acoustic FWI in presence of hard layers and study the potential elastic effects, we built a synthetic model that mimics the geology of North Oman. Then we inverted the data generated with this model starting from an almost exact initial model and also from a smooth one. We show that acoustic FWI diverges when starting from an initial model with too high resolution. The inversion over-fits the elastic data leading to erroneous velocity updates. This synthetic experiment illustrates the limitations of acoustic FWI when used to obtain a high resolution velocity image in complex geology.

<p>In the context of seismic imaging, the full waveform inversion (FWI) is more and more 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 medium for which FWI can be useful. In this work, we show that the general acoustic inverse problem based on limited frequency band data is intrinsically non-unique, making any general acoustic FWI impossible. Our work is based on two tools: particle relabelling and homogenization. On the one hand, the particle relabelling method shows it is possible to deform a true medium based on a smooth mapping into a new one without changing the signal recorded at seismic stations. This is a potentially strong source of non-uniqueness for an inverse problem based a seismic data. Nevertheless, in the elastic case, the deformed medium loses the elastic tensor minor symmetries and, in the acoustic case, it implies density anisotropy. It is therefore not a source of non-uniqueness for elastic or isotropic acoustic inverse problems, but it is for the anisotropic acoustic case. On the other hand, the homogenization method shows that any fine-scale medium can be up-scaled into an effective medium without changing the waveforms in a limited frequency band. The effective media are in general anisotropic, both in the elastic and acoustic cases, even if the true media are isotropic at a fine scale. It implies that anisotropy is in general present and needs to be inverted. Therefore, acoustic anisotropy can not be avoided in general. We conclude, based on a particle relabelling and homogenization arguments, that the acoustic FWI solution is in general non-unique. We show, in 2-D numerical FWI examples based on the Gauss-Newton iterative scheme, the effects of this non-uniqueness in the local optimization context. We numerically confirm that the acoustic FWI is in general non-unique and that finding a physical solution is not possible.</p>

<p>The existing conventional inversion algorithm does not provide satisfactory results due to the complexity of propagated wavefield though the gas cloud. Acoustic full waveform inversion has been developed and applied to a realistic synthetic offshore shallow gas cloud feature with Student-t approach, with and without simultaneous sources encoding. As a modeling operator, we implemented the grid based finite-difference method in frequency domain using second order elastic wave equation. Jacobin operator and its adjoint provide a necessary platform for solving full waveform inversion problem in a reduced Hessian matrix. We invert gas cloud model in 5 frequency band selected from 1 to 12 Hz, each band contains 3 frequencies. The inversion results are highly sensitive to the misfit. The model allows better convergence and recovery of amplitude losses. This approach gives better resolution then the existing least-squares approach. In this paper, we implement the full waveform inversion for low frequency model with minimum number of iteration providing a better resolution of inversion results.</p>

Full Waveform Inversion (FWI) is a seismic imaging technique that utilizes the entire content of seismic trace. This method aims to minimizes differences between real and synthetic seismograms. FWI consists of three main steps: forward modelling to simulate synthetic seismograms, adjoint simulation to calculate the gradient misfit, and optimization to obtain an appropriate model perturbation. FWI is an extremely non-linear inversion technique and computationally intensive since it involves many model parameters. In the context of geophysics, finding parameter models that describe the structure of the earth’s subsurface is the main goal. Geophysical measurements in the field only obtain data that is influenced by the parameter model, in this case seismic recording data, therefore mathematical techniques are studied and applied to determine the correct parameter model from the data obtained, also known as the inversion process. In this study, main research area is on optimization step which include mathematical approach to find perturbation model effectively. The Newton method can be used in the optimization step of FWI. However, its implementation is not promising due to high computational cost in calculating Hessian matrix. To reduce computation cost, an identity matrix is chosen to replace the Hessian matrix, as it usually done in the steepest descent method. Since FWI has a high level of non-linearity, steepest descent method requires many iterations to minimize the objective function. Therefore, this study uses a conjugate gradient method to make the data misfit minimized more quickly. Apart from that, the Line search technique is used to find an appropriate step-length at each iteration on both methods. The FWI workflow is applied to the synthetic acoustic velocity model. We conduct FWI using steepest descent and conjugate gradient methods. The results show that the conjugate gradient method requires fewer iterations than the steepest descent method to minimize the objective function. FWI model result using conjugate gradient method is closer with the true model compared to steepest descent method, showed on misfit result and model image. By comparing these methods we can see that conjugate gradient method can be chosen for application on FWI optimization step for better result on seismic imaging. On the industry scale, better results provide less uncertainty on the model parameters that useful for energy exploration or engineering.

We present an extension of the adjoint method that allows us to compute the second derivatives of seismic data functionals with respect to Earth model parameters. This work is intended to serve as a technical prelude to the implementation of Newton-like optimization schemes and the development of quantitative resolution analyses in time-domain full seismic waveform inversion.
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\nThe Hessian operator H applied to a model perturbation δm can be expressed in terms of four different wavefields. The forward field u excited by the regular source, the adjoint field u† excited by the adjoint source located at the receiver position, the scattered forward field δu and the scattered adjoint field δu†. The formalism naturally leads to the notion of Hessian kernels, which are the volumetric densities of Hδm. The Hessian kernels appear as the superposition of (1) a first-order influence zone that represents the approximate Hessian, and (2) second-order influence zones that represent second-order scattering.
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\nTo aid in the development of physical intuition we provide several examples of Hessian kernels for finite-frequency traveltime measurements on both surface and body waves. As expected, second-order scattering is efficient only when at least one of the model perturbations is located within the first Fresnel zone of the Fréchet kernel. Second-order effects from density heterogeneities are generally negligible in transmission tomography, provided that the Earth model is parameterized in terms of density and seismic wave speeds.
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\nWith a realistic full waveform inversion for European upper-mantle structure, we demonstrate that significant differences can exist between the approximate Hessian and the full Hessian—despite the near-optimality of the tomographic model. These differences are largest for the off-diagonal elements, meaning that the approximate Hessian can lead to erroneous inferences concerning parameter trade-offs. The full Hessian, in contrast, allows us to correctly account for the effect of non-linearity on model resolution.

To improve the understanding of flow and transport processes in the critical zone, high-resolution and accurate estimation of the small-scale heterogeneity is essential. Preferential flow paths related to high-porosity layers and clay lenses in gravel aquifers greatly affect flow and transport processes in the subsurface, and their high electrical contrast to their surrounding matrix and limited extent can act as low-velocity electromagnetic waveguides. In the past decade, time-domain full-waveform inversion (FWI) of crosshole ground-penetrating radar (GPR) data has shown to provide 2D decimeter-scale resolution images of relative permittivity and electrical conductivity of the subsurface, which can be related to porosity and soil texture. Most studies using crosshole GPR FWI resolved high-porosity zones that were identified by an amplitude analysis approach. But clay lenses or zones with higher electrical conductivity that act as low-velocity waveguides are hard to distinguish in the measured data and amplitude analysis because of the absence of characteristic wave-propagation features. We have investigated a set of nine crosshole GPR data sets from a test site in Hermalle-sous-Argenteau near the Meuse River in Belgium to characterize the aquifer within a decimeter-scale resolution and to improve the understanding of a previously performed heat tracer experiment. Thereby, we extend the amplitude analysis to identify two different types of low-velocity waveguides either caused by an increased porosity or a higher electrical conductivity (and higher porosity). Combining the GPR amplitude analysis for low-velocity waveguide zones with the standard FWI results provided information on waveguide zones, which modified the starting models and further improved the FWI results. Moreover, an updated effective source wavelet is estimated based on the updated permittivity starting models. In comparison with the traditional FWI results, the updated FWI results present smaller gradient of the medium properties and smaller root-mean-squared error values in the final inversion results. The nine crosshole sections are used to generate a 3D image of the aquifer and allowed a detailed analysis of the porosity distribution along the different sections. Consistent structures of the permittivity and electrical conductivity show the robustness of the updated FWI results. The aquifer structures obtained by the FWI results agree with those results of the heat tracer experiment.

Full-waveform inversion (FWI) includes migration and tomography modes. The tomographic component of the gradient from reflection data is usually much weaker than the migration component. To use the tomography mode to fix background velocity errors, it is necessary to extract the tomographic component from the gradient. Otherwise, the inversion will be dominated by the migration mode. We have developed a method based on nonstationary smoothing to extract the tomographic component from the raw gradient. By analyzing the characteristics of the scattering angle filtering, the wavenumber of the tomographic component at a given frequency is seen to be smaller than that of the migration component. Therefore, low-wavenumber-pass filtering can be applied to extract the tomographic component. The low-wavenumber-pass smoothing filters are designed with Gaussian filters that are determined by the frequency of inversion, the model velocity, and the minimum scattering angle. Thus, this filtering is nonstationary smoothing in the space domain. Because this filtering is carried out frequency by frequency, it works naturally and efficiently for FWI based on frequency-domain modeling. Furthermore, because the maximum opening angle of the reflections in a typical acquisition geometry is much smaller than the minimum scattering angle for the tomographic component, which is generally set at 160°, there is a relatively large gap between the wavenumbers of the tomographic and migration components. In other words, the nonstationary smoothing can be applied once to a group of frequencies for time-domain FWI without leaking the migration component into the tomographic component. Analyses and numerical tests indicate that two frequency groups are generally sufficient to extract the tomographic component for the typical frequency range of time-domain FWI. The numerical tests also demonstrate that the nonstationary smoothing method is effective and efficient at extracting the tomographic component for reflection waveform inversion.

Full-waveform inversion (FWI) has become the centerpiece of velocity model building (VMB) in seismic data processing in recent years. It has proven capable of significantly improving the velocity model and, thus, the migration image for different acquisition types and geologic settings, including complex environments such as salt. With the advent of FWI imaging, the scope of FWI applications has extended further from VMB into the imaging landscape. However, current FWI applications in the industry prevalently employ the acoustic approximation. One common problem of acoustic FWI (A-FWI) is the apparent salt halos at the salt-sediment interface in the resulting FWI velocity and FWI image, the presence of which hinders direct interpretation and imaging focusing around salt bodies. With synthetic and field data examples, we demonstrate that this salt halo is caused mainly by the large mismatch between the elastic recorded data and the acoustic modeled data, particularly at middle to long offsets. To overcome limitations imposed by acoustic assumptions, we developed an elastic FWI (E-FWI) algorithm that combines an elastic modeling engine with the time-lag cost function, which we call elastic time-lag FWI (E-TLFWI). With a more accurate modeling engine, E-TLFWI significantly reduces the salt halo observed in its acoustic counterpart. However, we also observe that the images migrated using the A-FWI and E-FWI velocity models remain similar overall, with some slight improvements around and beneath salt boundaries, particularly near steep salt flanks, as a result of the reduced salt halo. By contrast, FWI images derived from E-TLFWI show considerable benefits over those from acoustic time-lag FWI, such as improved event focusing, better structural continuity, and higher signal-to-noise ratio. The sharpened salt boundaries and enhanced quality of the FWI images reveal the significant value of E-FWI and provide the justification for its greatly increased cost.

As one of the ultrasound computed tomography (UCT) algorithms, full-waveform inversion (FWI) can recover clear and detailed internal structures of bones by iteratively minimizing the misfit between observed and numerical data. However, time domain full-waveform inversion (TDFWI) often lacks sufficient low-frequency information, and the computation of full-time data imposes a significant computational burden. Besides, there are few studies on whole-medium reconstruction that jointly invert the bone and surrounding soft tissues. In this study, an FWI-based frequency-domain UCT (FD-UCT) is utilized for simultaneously imaging the sound speed of bones and soft tissues, which facilitates the inversion efficiency while ensuring the existence of low-frequency information. Simulated studies show that the FD-UCT can both reconstruct bones with large acoustic impedance and distinguish subtle impedance differences within soft tissues to reveal lesions.

Full waveform inversion (FWI) is a seismic imaging method suitable for wide aperture / azimuth acquisitions thanks to its ability to account for waves propagating over a broad range of incidence angles. Wide aperture / azimuth acquisitions makes, however, the imaging quite sensitive to anisotropy because of the intrinsic difference between vertical and horizontal velocities in vertical transversely isotropic (VTI) media. In this study, we present a numerical procedure to define the best parametrization for acoustic VTI FWI. High-resolution velocity model for either the vertical, the horizontal or the normal move out (NMO) velocity can be developed by monoparameter FWI, when the considered wave speed is combined with the Thomsen’s parameters δ and e in the parametrization, and when available smooth background models of δ and e are kept fixed during FWI. An alternative is to jointly reconstruct two wavespeeds (for example, the vertical and the horizontal velocities) using a model parametrization that combines these two wavespeeds and δ . However, the spatial resolution of the two resulting velocity models is more limited that of the velocity model obtained with the first mentioned parametrization.

It is necessary to account for anisotropy in full waveform inversion (FWI) of wide-azimuth and wide-aperture seismic data in most geologic environments, for correct depth positioning of reflectors, and for reliable estimations of wave speeds as a function of the direction of propagation. In this framework, choosing a suitable anisotropic subsurface parameterization is a central issue in monoparameter and multiparameter FWI. This is because this parameterization defines the influence of each physical parameter class on the data as a function of the scattering angle, and hence the resolution of the parameter reconstruction, and on the potential trade-off between different parameter classes. We apply monoparameter and multiparameter frequency-domain acoustic vertical transverse isotropic FWI to synthetic and real wide-aperture data, representative of the Valhall oil field. We first show that reliable monoparameter FWI can be performed to build a high-resolution velocity model (for the vertical, the horizontal, or normal move-out velocity), provided that the background models of two Thomsen parameters describe the long wavelengths of the subsurface sufficiently accurately. Alternatively, we show the feasibility of the joint reconstruction of two wave speeds (e.g., the vertical and horizontal wave speeds) with limited trade-off effects, while Thomsen parameter [Formula: see text] is kept fixed during the inversion. The influence of the wave speeds on the data for a limited range of scattering angles when combined each other can, however, significantly hamper the resolution with which the two wave speeds are imaged. These conclusions inferred from the application to the real data are fully consistent with those inferred from the theoretical parameterization analysis of acoustic vertical transverse isotropic FWI performed in the companion report.