Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime

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 this paper we proposed an accurate and efficient numerical algorithm for solving the acoustic wave equation in three-dimensional heterogeneous media. Numerical solution of the wave equation has been used in various science and engineering applications, such as the seismic full waveform inversion (FWI) problem. FWI is a computationally intensive procedure, in which the acoustic wave equation is numerically solved (forward modelling) repeatedly during the iterative process. Therefore, efficiency and accuracy of the numerical method for solving the acoustic wave equation is critical in the success of seismic full waveform inversion. The new method is obtained by combining the Pade approximation and a novel algebraic manipulation with the Alternative Directional Implicit (ADI) method. Numerical experiments have shown that the new method is accurate, efficient and stable.

Full waveform inversion (FWI) is an optimization procedure which minimizes the waveform difference between the observed and calculated seismic data. The method, through an iterative forward modeling and inversion based on the acoustic or elastic wave equation, has recently gained wide interest with specific application in exploration and producing. FWI is wellknown for its ability to obtain high resolution and accurate subsurface models in areas with complex geological structures related to the presence of salt bodies encountered in deep water exploration. However, conventional FWI algorithms require seismic data that contains low frequencies (<5Hz). Lack of low frequencies results in unreliable velocity models for seismic imaging. In this paper, we aim to relax the low-frequency requirement of FWI and propose a technique which employs principles from sparse deconvolution in order to recover the missing low frequencies. Through a series of synthetic examples, it is shown that our proposed methodology is able to preserve and utilize the traveltime information of the original data; hence, it allows convergence of the FWI. The major advantage of this method is that it is fully automatic, involving no residual moveout or traveltime difference analysis.

Full-waveform inversion (FWI) is firmly established within our industry as a powerful velocity model building tool. FWI carries significant theoretical advantages over conventional velocity model building methods such as refraction and reflection tomography. Specifically, by solving a nonlinear inverse problem through the wave equation, FWI is able to recover a broadband velocity model containing both high and low spatial wavenumbers, thus extending the approximation of residual moveout correction inherent in traditional velocity model building approaches. Moreover, FWI is capable of inverting information from the entire wavefield (i.e., early arrivals, reflections, refractions, and multiple energy) rather than from a subset as in conventional approaches (i.e., first break and primary reflections), thereby availing itself of more information to better constrain its model estimate. However, these theoretical benefits cannot be realized easily in practice because various complexities of real seismic data often conspire to violate algorithmic assumptions, leading to unsatisfactory results. Dynamic matching FWI (DMFWI) is a newly developed algorithm that solves an inversion problem that maximizes the cross correlation of two dynamically matched data sets — one recorded and the other synthetic. Dynamic matching of the two data sets de-emphasizes the amplitude impact, which allows the algorithm to focus on minimizing their kinematic differences rather than amplitude in the data-fitting process. The multichannel correlation makes the algorithm robust for data with low signal-to-noise ratio. Applications of DMFWI across different types of acquisition and geologic settings demonstrate that this novel FWI approach can resolve complex velocity errors and provide high-quality migrated images that exhibit a high degree of geologic plausibility. Additionally, reflectivity images can be obtained in a straightforward manner as natural byproducts through computation of the directional derivative of the inverted FWI velocity models.

Full Waveform Inversion (FWI) is a successful and well-established inverse method for reconstructing material models from measured wave signals. In the field of seismic exploration, FWI has proven particularly successful in the reconstruction of smoothly varying material deviations. By contrast, non-destructive testing (NDT) often requires the detection and specification of sharp defects in a specimen. If the contrast between materials is low, FWI can be successfully applied to these problems as well. However, so far, the method is not fully suitable for reconstructing homogeneous Neumann boundaries such as hidden backsides of walls, crack-like defects, or internal voids, which are characterized by an infinite contrast in the material parameters. Inspired by fictitious domain methods, we introduce a dimensionless scaling function γ to model void regions in the forward and inverse scalar wave equation problem. Applying the scaling function γ to the material parameters in different ways results in three distinct formulations for mono-parameter FWI and one for two-parameter FWI. The resulting problems are solved by first-order optimization, where the gradient is computed using the adjoint state method. The corresponding Fréchet kernels are derived for each approach and the associated minimization is performed using an L-BFGS algorithm. A comparison between the different approaches shows that scaling the density with γ is most promising for parametrizing void material in forward and inverse problems. Finally, in order to consider arbitrary complex geometries known a priori, this approach is combined with an immersed boundary method, the finite cell method (FCM).

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

The quantitative reconstruction of subsurface earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field, but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and it defines the full reciprocity-gap waveform inversion (FRgWI) method. This misfit functional provides additional features compared to the more traditional least-squares approaches, in particular, in that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be chosen arbitrarily. Based on 3D experiments, FRgWI is shown to behave better than full-waveform inversion in the same context. It allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. However, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance, with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to crosstalk than least-squares shot stacking.

An accurate earth model is key to any successful depth imaging effort. Full waveform inversion is an advanced velocity model building process which uses the full two way wave equation. Existing methods use a ray tracing approach to distribute velocity errors within the model. In this presentation, we will show examples of seismic data processed with the latest technology, including earth model building with full waveform inversion. Introduction The industry has moved to using two-way wave equation migrations(commonly known as reverse time migration), especially in areas of complex geology such as the salt bodies in the Gulf of Mexico and offshore Brazil. The velocity model, including velocity anisotropy, is key to any depth migration effort. The natural next step is to use the two-way wave equation for velocity model building. One of the most advanced tools for velocity model building using the two-way wave equation is Full Waveform Inversion(FWI). Full-waveform inversion uses computer intensive forward modeling of the seismic measurement combined with residual wavefield back propagation to iterate to a final velocity model, which can provide greater detail than tomographic ray tracing approaches. Velocity inversion using full waveform inversion has been used on a number of projects around the world including the GOM, Brazil and the North Sea.

This study aims at designing optimal crosshole seismic approach for CO2EOR (enhance oil recovery by CO2 injection) monitoring in Abu Dhabi. It is not easy to perform successful reservoir monitoring for carbonate reservoirs in the Middle East because rock properties may not change much associated with fluid replacement. We investigate advantage of newly available elastic full waveform inversion (FWI) for detection of CO2 flooded zones. To investigate advantage, applicability and future problem of time-lapse FWI for CO2EOR monitoring, we conducted rock physics study, elastic seismic simulation, and test application of FWI to synthetic data. We first made realistic velocity models based on Gassmann's equation and actual well logs obtained before and after CO2 injection at carbonate reservoirs in Abu Dhabi. The in-situ light oil at the target reservoirs was replaced by CO2 (Secondary mode CO2EOR). We prepared models with different width and thickness of CO2 flooded zones and analyzed detectability of seismic waveform change. We also conducted test application of elastic full waveform inversion to synthetic data for delineation of CO2 flooded zones. Results of the rock physics study show 1.6% decrease in P-wave velocity, little change in S-wave velocity and density associated with secondary mode CO2EOR. The small property changes are caused by the rigid rock frame and small density difference in light oil and supercritical CO2 at the reservoir condition. Next we performed synthetic seismic analysis based on the rock property models, available crosshole seismic source and realistic well distances. Because of the small property change, seismic traveltime difference and waveform change are quite small when well distance is 116ft. When well distance is 700 ft or greater, guided waves, or trapped energy inside low velocity layers can be observed; relatively large waveform change can be observed in the guided waves due to CO2 injection. Although guided waves have not been used for reservoir monitoring, they are sensitive to small Vp and thus can be an indicator of the CO2 flooding. The elastic FWI result for a model with well distance of 1160 ft demonstrated good representation of the differential Vp with vertical resolution of 20-30 ft and some horizontal resolution. Although we need further investigation on repeatability and signal transmittability using actual field data, our synthetic study indicates possibility of identifying layers with CO2 using guided waves and elastic FWI.

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.

In a probabilistic formulation, the solution to an inverse problem can be expressed an a posteriori probability density function (pdf) that combines the independent states of information provided by data and a priori information. Here, we define an a posteriori probability density function that defines the solution to a tomographic full waveform inverse problem, which provides a means of obtaining an uncertainty estimate. Unfortunately, no explicit formulation of the solution to this problem can be defined. Therefore, the a posteriori probability density function has to be sampled. The full waveform inverse problem is known to be computationally very hard and is, traditionally, considered out of reach for Monte Carlo sampling strategies. We show that by means of informative a priori information this problem become tractable for a sampling strategy anyways. We outline the theoretical framework for a full waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling, which allows for arbitrary complex a priori information to be incorporated. At the same time we show how temporally correlated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground penetrating radar full waveform data using multiplepoint based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full waveform inverse problem.

The second order scattering information provided by the Hessian matrix and its inverse plays an important role in both, parametric inversion and uncertainty quantification. On the one hand, for parameter inversion, the Hessian guides the descent direction such that the cost function minimum is reached with less iterations. On the other hand, it provides a posteriori information of the probability distribution of the parameters obtained after full waveform inversion, as a function of the a priori probability distribution information. Nevertheless, the computational cost of the Hessian matrix represents the main obstacle in the state-of-the-art for practical use of this matrix from synthetic or real data. The second order adjoint state theory provides a strategy to compute the exact Hessian matrix, reducing its computational cost, because every column of the matrix can be obtained by performing two forward and two backward propagations. In this paper, we first describe an approach to compute the exact Hessian matrix for the acoustic wave equation with constant density. We then provide an analysis of the use of the Hessian matrix for uncertainty quantification of the full waveform inversion of the velocity model for a synthetic example, using the 2D acoustic and isotropic wave equation operator in time.

Ultrasound computed tomography based on full waveform inversion with source directivity calibration

Full-waveform inversion (FWI) has evolved in recent years from a technique for recovering long- to intermediate-wavelength updates of acoustic overburden velocities as part of a broader model building workflow to a standalone tool for high-fidelity seismic imaging using raw seismic data – with ever greater resolution and ever more sophisticated physics being sought. In doing so, it has moved notably closer to fulfilling the vision of its original inventors in the late 70s and early 80s. In this work, two novel approaches to FWI along with enabling technologies for their effective deployment are considered. The first of these allows for robust long-wavelength updates to be generated from raw seismic data in the absence of diving wave energy, providing reliable estimates of acoustic velocity at depths equivalent to or greater than the longest offsets available in the data – thereby enabling more effective imaging with data of limited quality. The second allows for the estimation of elastic amplitude-versus-angle behaviour while still using an acoustic wave equation, thereby mitigating the significant computational burden typically associated with elastic FWI. Finally, it is demonstrated that these approaches can be deployed in a reliable and cost-effective manner via the considered use of public cloud resources, with learnings relevant for analogous massively parallel scientific applications. Case studies from two marine environments are provided for discussion, one being from deep-water US Gulf of Mexico and the other from intermediate water depths offshore Western Australia.

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.