Visco-acoustic Full Waveform Inversion Research Articles

Advancing attenuation estimation through integration of the Hessian in multiparameter viscoacoustic full-waveform inversion

Accurate seismic attenuation models of subsurface structures not only enhance subsequent migration processes by improving fidelity, resolution, and facilitating amplitude-compliant angle gather generation but also provide valuable constraints on subsurface physical properties. Leveraging full-wavefield information, multiparameter viscoacoustic full-waveform inversion ( Q-FWI) simultaneously estimates seismic velocity and attenuation ( Q) models. However, a major challenge in Q-FWI is the contamination of crosstalk artifacts, where inaccuracies in the velocity model are mistakenly mapped to the inverted attenuation model. While incorporating the Hessian is expected to mitigate these artifacts, the explicit implementation is prohibitively expensive due to its formidable computational cost. In this study, we formulate and develop a Q-FWI algorithm via the Newton-conjugate gradient (CG) framework, where the search direction at each iteration is determined through an internal CG loop. In particular, the Hessian is integrated into each CG step in a matrix-free fashion using the second-order adjoint-state method. We find through synthetic experiments that our Newton-CG Q-FWI significantly mitigates crosstalk artifacts compared with the limited-memory Broyden-Fletcher-Goldfarb-Shanno method and the CG method, albeit with a notable computational cost. In the discussion of several key implementation details, we also determine the significance of the approximate Gauss-Newton Hessian, the second-order adjoint-state method, and the two-stage inversion strategy.

Inversion strategies for Q estimation in viscoacoustic full-waveform inversion

Estimation of an attenuation parameter, represented by the quality factor Q, holds paramount importance in seismic exploration. One of the main challenges in Q estimation through viscoacoustic full-waveform inversion (FWI) is effectively decoupling Q from velocity. In this study, our objective is to enhance Q inversion by addressing critical aspects, such as gradient preconditioning, workflow, and misfit design. By developing a new preconditioner that approximates the diagonal of the Hessian, we facilitate automatic parameter tuning across different classes, ensuring comparable magnitudes of preconditioned gradients for velocity and Q. Moreover, our investigations confirm the efficacy of the two-stage hierarchical strategy in mitigating velocity- Q trade-offs, enabling more accurate Q estimation by first focusing on velocity reconstruction before jointly estimating velocity and Q. The analysis and numerical examples also highlight the importance of broadband data and long-offset acquisition for a reliable Q estimation. In addition, leveraging amplitude information can improve Q estimation to some extent, but careful consideration of frequency band and noise effects is necessary. We explore two misfit functions that capture amplitude variation with frequency in the time-frequency domain, noting their sensitivity to noise. To address this, we develop a differential strategy that can effectively mitigate the effects of low-frequency noise. This comprehensive study on enhancing Q estimation in viscoacoustic FWI offers valuable insights for multiparameter inversion in realistic scenarios.

Unbalanced optimal transport for full waveform inversion in visco-acoustic media

Abstract As a high-precision parameter inversion method, visco-acoustic full waveform inversion (QFWI) is widely used in the inversion of parameters such as velocity and quality factor Q in visco-acoustic media. Conventional QFWI, using the L2 norm as the objective function, is susceptible to face the cycle-skipping problem, especially with inaccurate initial models. Lately, adopting the optimal transportation (OT) distance as the objective function in QFWI (OT-QFWI) has become one of the most promising solutions. In OT-QFWI, converting oscillatory seismic data into a probability distribution that satisfies equal-mass and non-negativity conditions is essential. However, seismic data in visco-acoustic media face challenges in meeting the equal-mass assumption, primarily due to the attenuation effect (amplitude attenuation and phase distortion) associated with the quality factor Q. Unbalanced optimal transportation (UOT) has shown potential in solving equal-mass assumption. It offers the advantage of relaxing equal-mass requirements through entropy regularization. Owing to this advantage, UOT can mitigate the attenuation effect caused by inaccurate quality factor Q during the inversion. Simultaneously, the Sinkhorn algorithm can quickly solve the UOT distance through CUDA programming. Accordingly, we propose a UOT-based QFWI (UOT-QFWI) method to improve the accuracy of two-parameter inversion. The proposed method mitigates the impact of inaccurate quality factor Q by introducing the UOT distance to calculate the objective function, thereby helping to obtain more accurate inverted parameters. Experimental tests on the 1D Ricker wavelet and 2D synthetic model are used to validate the effectiveness and robustness of our proposed method.

Data-assimilated time-lapse visco-acoustic full-waveform inversion: Theory and application for injected CO2 plume monitoring

Continuous seismic monitoring for quantifying CO2 plume migration and detection of any potential leakages in the subsurface is essential for the security of long-term anthropogenic carbon dioxide geologic storage. Traditional time-lapse full-waveform inversion (TLFWI) methods aim to map the CO2 distribution by estimating seismic velocity changes, but recent studies find that CO2-induced attenuation is an important complement to seismic velocity for tracking the CO2 plumes and even quantifying the CO2 saturation. We have developed a novel data-assimilated TLFWI method to construct high-resolution time-lapse velocity and attenuation changes from dense time-lapse monitoring data. This method consists of two theoretical developments: visco-acoustic full-waveform inversion (QFWI) and multiparameter hierarchical matrix-powered extended Kalman filter (mHiEKF). The method is capable of (1) posing temporal constraints to retrieve time-lapse information from dense monitoring data by using mHiEKF, (2) accurately recovering high-spatial-resolution velocity and attenuation perturbations using first-order equation system-based QFWI, and (3) providing the model uncertainty by estimating their model standard deviation. With numerical examples, we first find the effectiveness of the new QFWI on estimating accurate velocity and attenuation models simultaneously. Then, a CO2 leakage case and a realistic Frio-II CO2 monitoring case are presented to find the advantages and applicability of our data-assimilated QFWI method for estimating time-lapse changes using dense time-lapse monitoring surveys. By assimilating time-lapse seismic monitoring data over time, our data-assimilated QFWI method can improve the resolution of velocity and attenuation changes and decrease their model uncertainties.

Visco-acoustic full waveform inversion: From a DG forward solver to a Newton-CG inverse solver

Full waveform inversion (FWI) entails the ill-posed reconstruction of material parameters (such as sound speed and attenuation) from measurements of complete wave fields (full seismograms). In this paper we present a novel framework for FWI in the visco-acoustic regime. The new framework is based on a new elegant derivation of the system of state and adjoint PDEs which are approximated by the discontinuous Galerkin (DG) method. The inverse problem is then solved by the well established regularization scheme CG-REGINN which has not yet been applied in the context of FWI. For the DG discretization we provide a preconditioner for the efficient computation of the time steps by GMRES which yields optimal convergence estimates in space and time and which is confirmed by numerical tests. The inverse solver expresses the required Fréchet derivative and its adjoint in the DG setting. Successful reconstructions in a simplified cross-well setting serve as a proof of concept for our framework and demonstrate the applicability of our new combination of DG method and inverse solver. Some of the inversion experiments use seismograms generated by an independent finite difference time domain forward solver to avoid inverse crime.

Visco-acoustic full waveform inversion using decoupled fractional Laplacian constant-Q wave equation and optimal transport-based misfit function

Full waveform inversion (FWI) has the potential to recover high-resolution physical parameters of the Earth from seismic data. Considering the attenuation effects of the subsurface, we develop a new visco-acoustic FWI method in the time domain with a known Q model. The seismic wavefield is modelled by solving a decoupled fractional Laplacian visco-acoustic wave equation, which can better describe amplitude attenuation and phase distortion during wave propagation. To weaken the initial model dependence of visco-acoustic FWI, we introduce the optimal transport-based misfit functional which can measure the data residual globally. Numerical examples on the 2D Marmousi model demonstrate the superiority of the proposed method compared with the L2 norm-based method which measures the misfit of observed and synthetic seismic data locally, generating cycle-skipping issue. The inversion results of seismic data without low-frequency components further demonstrate the validity and effectiveness of the proposed method, which is insensitive towards low-frequency components, and may achieve reasonable inversion results for field data application.

A sequential inversion with outer iterations for the velocity and the intrinsic attenuation using an efficient wavefield inversion

Full-waveform inversion (FWI) has become a popular method to retrieve high-resolution subsurface model parameters. It is a highly nonlinear optimization problem based on minimizing the misfit between the observed and predicted data. For intrinsically attenuating media, wave propagation experiences significant loss of energy. Thus, for better data fitting, it is sometimes crucial to consider attenuation in FWI. Viscoacoustic FWI aims at achieving a joint inversion of the velocity and attenuation models. However, multiparameter FWI imposes additional challenges including expanding the null space and facing parameter trade-off issues. Theoretically, an ideal way to mitigate the trade-off issue in multiparameter FWI is to apply the inverse Hessian operator to the parameter gradients. However, it is often not practical to calculate the full Hessian and its matrix inverse because this will be extremely expensive. To improve the computational efficiency and mitigate the trade-off issue, we have used an efficient wavefield inversion (EWI) method to invert for the velocity and the intrinsic attenuation. This approach is implemented in the frequency domain, and the velocity, in this case, is complex-valued in the viscoacoustic EWI. We evaluate a sequential update strategy for the velocity and the intrinsic attenuation, and we repeat the separate optimizations, which we refer to as outer iterations, until the convergence is achieved. Because viscoacoustic EWI is able to recover an accurate velocity model, the velocity update leakage to the [Formula: see text] model is largely reduced. We determine the effectiveness of this approach using synthetic data generated for the viscoacoustic Marmousi and Overthrust models. To further demonstrate the validity of our approach, we generate data in the time domain using a viscoelastic wave equation solver and obtain reasonable inversion results in the frequency domain using the viscoacoustic approximation.

Estimating P Wave Velocity and Attenuation Structures Using Full Waveform Inversion Based on a Time Domain Complex‐Valued Viscoacoustic Wave Equation: The Method

AbstractTo complement velocity distributions, seismic attenuation provides additional important information on fluid properties of hydrocarbon reservoirs in exploration seismology, as well as temperature distributions, partial melting, and water content within the crust and mantle in earthquake seismology. Full waveform inversion (FWI), as one of the state‐of‐the‐art seismic imaging techniques, can produce high‐resolution constraints for subsurface (an)elastic parameters by minimizing the difference between observed and predicted seismograms. Traditional waveform inversion for attenuation is commonly based on the standard‐linear‐solid (SLS) wave equation, in which case the quality factor (Q) has to be converted to stress and strain relaxation times. When using multiple attenuation mechanisms in the SLS method, it is difficult to directly estimate these relaxation time parameters. Based on a time domain complex‐valued viscoacoustic wave equation, we present an FWI framework for simultaneously estimating subsurface P wave velocity and attenuation distributions. Because Q is explicitly incorporated into the viscoacoustic wave equation, we directly derive P wave velocity and Q sensitivity kernels using the adjoint‐state method and simultaneously estimate their subsurface distributions. By analyzing the Gauss‐Newton Hessian, we observe strong interparameter crosstalk, especially the leakage from velocity to Q. We approximate the Hessian inverse using a preconditioned L‐BFGS method in viscoacoustic FWI, which enables us to successfully reduce interparameter crosstalk and produce accurate velocity and attenuation models. Numerical examples demonstrate the feasibility and robustness of the proposed method for simultaneously mapping complex velocity and Q distributions in the subsurface.

Parameter crosstalk and modeling errors in viscoacoustic seismic full-waveform inversion

Simultaneous use of data within relatively broad frequency bands is essential to discriminating between velocity and [Formula: see text] errors in the construction of viscoacoustic full-waveform inversion (QFWI) updates. Individual frequencies or narrow bands in isolation cannot provide sufficient information to resolve parameter crosstalk issues in a surface seismic acquisition geometry. At the same time, too broad a frequency band introduces significant problems in the presence of modeling errors. The risk of modeling errors arising in QFWI is high because of the range of very different geologic contributors to attenuation and dispersion and the variety of available mathematical descriptions. We perform numerical tests that suggest that by relaxing the typical requirement that the frequency dependence of the assumed intrinsic attenuation model be self-consistent across the full spectrum, significant improvement in the fidelity of the inversion results can be obtained in cases where the attenuation model assumed in the inversion differs substantially from the true subsurface behavior. We find that the size of the frequency bands used in this inversion approach is a useful hyperparameter controlling trade-off between crosstalk reduction and flexibility in coping with uncertain physics.

Semiglobal viscoacoustic full-waveform inversion

Full-waveform inversion deals with estimating physical properties of the earth’s subsurface by matching simulated to recorded seismic data. Intrinsic attenuation in the medium leads to the dispersion of propagating waves and the absorption of energy — media with this type of rheology are not perfectly elastic. Accounting for that effect is necessary to simulate wave propagation in realistic geologic media, leading to the need to estimate intrinsic attenuation from the seismic data. That increases the complexity of the constitutive laws leading to additional issues related to the ill-posed nature of the inverse problem. In particular, the joint estimation of several physical properties increases the null space of the parameter space, leading to a larger domain of ambiguity and increasing the number of different models that can equally well explain the data. We have evaluated a method for the joint inversion of velocity and intrinsic attenuation using semiglobal inversion; this combines quantum particle-swarm optimization for the estimation of the intrinsic attenuation with nested gradient-descent iterations for the estimation of the P-wave velocity. This approach takes advantage of the fact that some physical properties, and in particular the intrinsic attenuation, can be represented using a reduced basis, substantially decreasing the dimension of the search space. We determine the feasibility of the method and its robustness to ambiguity with 2D synthetic examples. The 3D inversion of a field data set for a geologic medium with transversely isotropic anisotropy in velocity indicates the feasibility of the method for inverting large-scale real seismic data and improving the data fitting. The principal benefits of the semiglobal multiparameter inversion are the recovery of the intrinsic attenuation from the data and the recovery of the true undispersed infinite-frequency P-wave velocity, while mitigating ambiguity between the estimated parameters.

Accelerating full-waveform inversion with attenuation compensation

The calculation of the gradient in full-waveform inversion (FWI) usually involves crosscorrelating the forward-propagated source wavefield and the back-propagated data residual wavefield at each time step. In the real earth, propagating waves are typically attenuated due to the viscoelasticity, which results in an attenuated gradient for FWI. Replacing the attenuated true gradient with a [Formula: see text]-compensated gradient can accelerate the convergence rate of the inversion process. We have used a phase-dispersion and an amplitude-loss decoupled constant-[Formula: see text] wave equation to formulate a viscoacoustic FWI. We used this wave equation to generate a [Formula: see text]-compensated gradient, which recovers amplitudes while preserving the correct kinematics. We construct an exact adjoint operator in a discretized form using the low-rank wave extrapolation technique, and we implement the gradient compensation by reversing the sign of the amplitude-loss term in the forward and adjoint operators. This leads to a [Formula: see text]-dependent gradient preconditioning method. Using numerical tests with synthetic data, we demonstrate that the proposed viscoacoustic FWI using a constant-[Formula: see text] wave equation is capable of producing high-quality velocity models, and our [Formula: see text]-compensated gradient accelerates its convergence rate.

New geophysical models for subsurface velocity structure in the Nechako–Chilcotin plateau from 2.5-D waveform tomography

Seismic inversion is applied to generate physical property models (P-wave velocity and numerical attenuation) for four profiles in the Nechako–Chilcotin plateau region of south-central British Columbia, Canada. A newly developed method that combines three-dimensional (3-D) travel-time inversion and 2.5-dimensional (2.5-D) viscoacoustic full-waveform inversion was applied to generate the geophysical models from vibroseis data acquired along the preexisting crooked roads. These models are useful for the characterization of rock types in terms of their positions and thicknesses, which may be used in conjunction with geological ground truth to infer the extent of lithostratigraphic units in the subsurface. The velocity structures also may be used for future reprocessing of the seismic reflection data to derive improved images based on the better near-surface velocity models. The subsurface geology of the Nechako–Chilcotin plateau region is complex, resulting from multiple stages of tectonic compression and extension, contemporaneous with the deposition of sediments and volcanic material. Several basin structures are identified from the joint interpretation of the waveform tomography velocity models and post-stack time migration images. The combination of these results enables the extrapolation and characterization of geological structures to ∼3 km depth, particularly within the Cenozoic volcanic units that dominate near-surface stratigraphy. Based on the seismic profiles, a fence-diagram geological interpretation that extends to ∼3 km depth illustrates the complex structure of the Jurassic to Neogene stratigraphic sequence.

Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 2: imaging compressive-wave and shear-wave velocities

Multiparameter elastic full waveform inversion (FWI) is a promising technology that allows inferences to be made on rock and fluid properties, which thus narrows the gap between seismic imaging and reservoir characterization. Here, we assess the feasibility of 2-D vertical transverse isotropic visco-elastic FWI of wide-aperture multicomponent ocean-bottom-cable data from the Valhall oil field. A key issue is to design a suitable hierarchical data-driven and model-driven FWI workflow, the aim of which is to reduce the nonlinearity of the FWI. This nonlinearity partly arises because the shear (S) wavespeed can have a limited influence on seismic data in marine environments. In a preliminary stage, visco-acoustic FWI of the hydrophone component is performed to build a compressional (P)-wave velocity model, a density model and a quality-factor model, which provide the necessary background models for the subsequent elastic inversion. During the elastic FWI, the P and S wavespeeds are jointly updated in two steps. First, the hydrophone data are inverted to mainly update the long-to-intermediate wavelengths of the S wavespeeds from the amplitude-versus-offset variations of the P-P reflections. This S-wave velocity model is used as an improved starting model for the subsequent inversion of the better-resolving data recorded by the geophones. During these two steps, the P-wave velocity model is marginally updated, which supports the relevance of the visco-acoustic FWI results. Through seismic modelling, we show that the resulting visco-elastic model allows several P-to-S converted phases recorded on the horizontal-geophone component to be matched. Several elastic quantities, such as the Poisson ratio, and the ratio and product between the P and S wavespeeds, are inferred from the P-wave and S-wave velocity models. These attributes provide hints for the interpretation of an accumulation of gas below lithological barriers.

Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation

Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds.

High-resolution seismic attenuation imaging from wide-aperture onshore data by visco-acoustic frequency-domain full-waveform inversion

SUMMARY Here we assess the potential of the visco-acoustic frequency domain full-waveform inversion (FWI) to reconstruct P-wave velocity (VP) and P-wave attenuation factor (Q) from surface onshore seismic data. First, we perform a sensitivity analysis of the FWI based upon a grid search analysis of the misfit function and several synthetic FWI examples using velocity and Q models of increasing complexity. Subsequently, we applied both the acoustic and viscoacousticFWItorealsurfacewide-apertureonshoreseismicdatafromthePolishBasin,wherea strongattenuationoftheseismicdataisobserved.Thesensitivityanalysisofthevisco-acoustic FWI suggests that the FWI can jointly reconstruct the velocity and the attenuation factor if the signatureoftheattenuationissufficientlystronginthedata.Asyntheticexamplecorresponding to a homogeneous background model with an inclusion shows a reliable reconstruction of VP and Q in the inclusion, when Q is as small as 90 and 50 in the background model and in the inclusion, respectively. A first application of acoustic FWI to real data shows that a heuristic normalization of the data with offset allows us to compensate for the effect of the attenuation in the data and reconstruct a reliable velocity model. Alternatively, we show that visco-acoustic FWI allows us to reconstruct jointly both a reliable velocity model and a Q model from the true-amplitude data. We propose a pragmatical approach based upon seismic modelling and source wavelet estimation to infer the best starting homogeneous Q model for visco-acoustic FWI. We find the source wavelet estimation quite sensitive to the quality of the velocity and attenuation models used for the estimation. For example, source-to-source wavelets are significantly more consistent when computed in the final FWI model than in the initial one. A good kinematic and amplitude match between the early-arriving phases of the real and time-domain synthetic seismograms computed in the final FWI model provides an additional evidence of the reliability of the final FWI model. We find the recovered velocity and attenuation models consistent with the expected lithology and stratigraphy in the study area. We link high-attenuation zones with the increased clay content and the presence of the mineralized fluids.