Full Waveform Inversion Process Research Articles

Combined non-convex second-order total variation with overlapping group sparsity for full waveform inversion

Full waveform inversion (FWI) can provide an accurate velocity model by matching observed and simulated seismograms. Mathematically, FWI is a highly ill-posed inverse problem that the observed data are independent of the model and cannot be inverted accurately. For a stable and reasonable inversion, proper regularization methods have to be taken into account. We propose a novel composite regularization for frequency-domain FWI problem, which uses a non-convex second-order total variation (TV) term and an overlapping group sparse TV (OGS-TV) regularization term. Compared with the conventional TV regularization, our method has better accuracy and robustness, and could effectively make use of the sparsity in the velocity model. Furthermore, the alternating direction multiplier algorithm with the adaptive selection of the penalty parameters is developed to solve this composite constraint problem, which can improve the stability of the FWI process. To illustrate the superior method both visually and quantitatively, we experimentally compare the proposed method with the conventional TV regularized FWI and the second-order total generalized variation regularized FWI and overlapping group sparsity regularized FWI on the well-known geological models.

Seismic Full Waveform Inversion Accelerated by Overlapping Data Input and Computation

Seismic full waveform inversion (FWI) is a powerful technology to obtain high-precision and high-resolution images of subsurface structures. However, FWI is a data-intensive algorithm that needs to read extensive seismic data from disks, which significantly affects its performance. We proposed a portable parallel framework to improve FWI by overlapping data input and computation (ODIC). The framework is based on POSIX threads (Pthreads), which is a standard thread API library and can create a parent thread and a child thread in the FWI process. The former is used to perform computation and the latter to read data from disks, both running simultaneously. This framework has two attractive features. First, it is broadly applicable; it can run on almost any computer from a laptop to a supercomputer. Second, it is easy to implement; it can be readily applied to existing FWI programs. A 3D FWI example shows that the framework speeds up FWI considerably.

Localized adaptive waveform inversion: theory and numerical verification

SUMMARYCorrectly interpreting phase events thanks to data processing techniques based on correlation or deconvolution has been the focus of numerous studies in the field of high-resolution seismic imaging using full-waveform inversion. To mitigate the non-convexity of the misfit function and the risk to converge towards non-informative local minima, correlation and deconvolution techniques make it possible to focus on phase information instead of amplitude information and to design more convex misfit function, alleviating the dependency of the full-waveform inversion process on the accuracy of initial models. Such techniques however rely on the assumption that phase events can be compared one by one, or that all the phase events are shifted in time in a similar way. This assumption is not satisfied in practice, which limits the effectiveness of these correlation/deconvolution-based methods. To overcome this issue, we propose to account for the non-stationary relation between observed and predicted data through a local in-time deconvolution technique, based on time–frequency analysis of the signal using a Gabor transform. This makes it possible to estimate instantaneous time-shift between locally coherent phase events. This strategy generalizes the conventional normalized deconvolution technique, which has been popularized under the name of adaptive waveform inversion. To support the introduction of our novel method, we compare it with four misfit functions based respectively on classical cross-correlation, penalized cross-correlation, penalized deconvolution, and adaptive waveform inversion. We analyse the behaviour of these methods on specific scenarios, and then propose a comparison on 2-D synthetic benchmarks. We show how our ‘localized’ adaptive waveform inversion applies in these realistic tests and overcomes some of the limitations of the aforementioned techniques.

Spyro: a Firedrake-based wave propagation and full-waveform-inversion finite-element solver

Abstract. In this article, we introduce spyro, a software stack to solve wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite-element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite-element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time domain FWI approach that uses meshes composed of variably sized triangular elements to discretize the domain is detailed. To resolve both the forward and adjoint-state equations and to calculate a mesh-independent gradient associated with the FWI process, a fully explicit, variable higher-order (up to degree k=5 in 2D and k=3 in 3D) mass-lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wave field (e.g., local P-wave speed) and by leveraging higher-order basis functions, the number of degrees of freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.

Frequency-domain acoustic full waveform inversion with an embedded boundary method for irregular topography

In the implementation of full waveform inversion (FWI) to identify subsurface velocity distributions with land seismic data, which are often acquired in regions with irregular topography, wave equation-based modelling requires caution. In particular, when using the finite difference method (FDM), unwanted scattered waves are generated because irregular surfaces crossing a rectangular grid are discretized via a staircase approximation; hence, if the problems caused by this staircase approximation are disregarded, FDM-based FWI may fail due to the presence of undesirable wavefields. To resolve this problem, this study develops a 2D frequency-domain acoustic FWI technique using a 9-point FDM-based modelling scheme that includes an embedded boundary method (EBM). This study suggests a workflow for the whole EBM-based FWI process from the calculation of coefficients for the EBM-based 9-point FDM modelling to applying it to FWI for proper velocity updates. In numerical examples, using velocity models with a tilted surface and an arbitrarily fluctuating surface, we synthesize seismic data and verify the accuracy of EBM-based 9-point FDM modelling and its superiority over the conventional FDM by comparing it with wavefields derived from the spectral element method. Then, we show that our EBM-based FWI is able to estimate subsurface velocity distributions even though the model has irregular topography, which spoils the result of the conventional FWI.

Frequency-domain reflection waveform inversion with generalized internal multiple imaging

Full-waveform inversion (FWI) has the potential to provide a high-resolution detailed model of the earth’s subsurface, but it often fails to do so if the starting model differs significantly from the true one. Reflection waveform inversion (RWI) is a popular method for building a sufficiently accurate initial model for FWI. In traditional RWI, the low-wavenumber updates are always computed and captured by smearing the data misfit along the reflection path with the help of migration/demigration. However, the success of RWI relies heavily on accurately reproducing the data in demigration. Thus, we have introduced a new generalized internal multiple imaging-based RWI (GIMI-RWI) implementation, in which we avoid the Born modeling and update the primary reflection kernel directly. In GIMI-RWI, we store one reflection kernel for each source-receiver pair, preserving the unique wavepath for every single source-receiver trace. Subsequently, the convolution between the data residuals and the corresponding reflection kernel can build the tomographic velocity updates. In this situation, the long-wavelength tomographic updates are free of migration footprints and will contribute a smoother background velocity to reduce the cycle-skipping risk and stabilize the followed FWI process. In addition, the GIMI-RWI method is source independent because it entirely relies on the data. Using a synthetic example extracted from the Sigsbee2A model, we find the reliable performance of the GIMI-RWI technique.

Exact line search method for using the L1-norm misfit function in full waveform inversion

Full waveform inversion (FWI) is a non-linear inverse problem that can be sensitive to noise. The tolerance of the noise-interference characteristics depends on the types of misfit functions. To date, different misfit functions, such as the least-squares norm (L2), the least-absolute-value norm (L1), and combinations of the two (e.g., the Huber and hybrid criteria), have been applied to FWI. The L2 norm is highly sensitive to non-Gaussian errors in the data and gives rise to high-amplitude artifacts in reconstructed models. For non-Gaussian noise data, the L1 norm and the Huber and hybrid criteria always reliably reconstruct models. However, the Huber and hybrid criteria require tedious error investigations to estimate their threshold criterion. Thus, the L1 norm is adopted here to improve the anti-noise ability of the FWI. The step length is closely related to the misfit function, and an optimal step-length estimation method can rapidly make the FWI algorithm reach the global minimum, with a reduced number of iterations and fewer extra forward modeling simulations during each iteration. The step length can usually be obtained using the exact or inexact line search method. Generally, the exact line search method is faster than the inexact one. Therefore, we derived an exact line search method for the L1 norm in the FWI process. Its effectiveness was tested using noise-free data from Overthrust and the SEG/EAGE salt models. The results demonstrate that this method can recover high-resolution velocity models with low computational costs. Numerical tests using the synthetic Overthrust model contaminated by strong noise were used to further validate the robustness of this exact line search method.

Multistage elastic full-waveform inversion for tilted transverse isotropic media

SUMMARYSeismic anisotropy is an important physical phenomenon that significantly affects wave propagation in complex sedimentary basins. When geological structures exhibit steep dips or severe folding, the symmetry axis of the transversely isotropic (TI) representation of the region can be rotated, leading to tilted transversely isotropic (TTI) media. We seek to find the optimal full-waveform inversion (FWI) strategy to estimate both the seismic velocities and the anisotropic parameters, including the tilt angle, in the presence of elastic TTI media. We first formulate the forward and inverse problems for elastic TTI media and analyse the radiation patterns of the model parameters. Based on the analyses of the radiation patterns, we propose two similar multistage FWI strategies that add inversion parameters over three stages, beginning with the isotropic parameters (horizontal P- and vertical S-wave velocity) and moving to the anisotropic parameters; the tilt angle is directly inverted in the last stage. Since diving waves, which are useful for providing long-wavelength updates, are mainly controlled by horizontal motion in anisotropic media, it is reasonable to choose the horizontal P-wave velocity rather than the vertical P-wave velocity. Then, the anisotropic parameters are inverted mainly using the reflected waves based on the isotropic background model built in the first stage. The main difference between the two multistage FWI strategies is whether the anisotropic parameter η is inverted. Comparing the two multistage FWI strategies with the simultaneous inversion strategy for a downsized version of the synthetic BP TTI model, we confirm that the multistage FWI strategies yield better inversion results than the simultaneous inversion strategy. When we compare the two multistage FWI strategies with each other for surface seismic data, ignoring η during the FWI process (focused multistage FWI) yields better inversion results for the tilt angle than those obtained with the inversion of η because η has less influence on the FWI than the other parameters and is not recovered well, which plays a role in degrading the tilt angle. Numerical examples support our conclusions that the focused multistage FWI strategy (neglecting η) is the optimal FWI strategy for TTI media and achieves computational efficiency for surface seismic data.

Estimation of velocity and borehole receiver location via full waveform inversion of vertical seismic profile data

ABSTRACTBecause a vertical seismic profile (VSP) survey includes the installation of geophones into a borehole, the results of data processing are influenced by the deviation error of the borehole trajectory. In particular, when full waveform inversion (FWI) is applied to VSP data, receiver location errors distort the results of velocity inversion. Under this circumstance, we attempt to estimate the exact locations of receivers and resolve the distortion of the inverted velocity model by adding receiver locations to the inversion parameters. This makes the FWI process a multi-parametric inverse problem. In this case, it is desirable to invert each parameter group alternatively. Because receiver coordinates depend on borehole trajectory, the inverse problem is regularised by adding a term measuring the curvature of the borehole trajectory gradient to obtain a physically meaningful solution. In addition, we propose a grid search method to improve the estimation of receiver coordinates. In the case of VSP data, the inversion of receiver coordinates using the grid search method is more efficient and robust than using a local descent method. Through numerical examples, we verify that the proposed inversion algorithm can improve FWI results and mitigate velocity distortion. We also demonstrate that the convergence of FWI can be improved by applying the grid search method to the estimation of receiver coordinates.

Mono-component multiparameter acoustic full waveform inversion in vertically transverse isotropic media using converted vector wavefields

In multiparameter full-waveform inversion (FWI) with acoustic-approximation for vertically transverse isotropic (VTI) media, it is important to choose an appropriate modeling technique for computational efficiency and numerical stability. In addition, because the gradient is determined by the modeling algorithm used in the FWI process, we need to examine if the modeling algorithm yields a gradient similar to that obtained from the elastic scattering theory. The decomposed equation with mono-component has been proposed because it is computationally efficient and does not generate shear-wave artifacts. However, the decomposed equation has the limitation that its numerical scattering potentials for the perturbations of anisotropic parameters do not properly simulate the theoretical elastic scattering potentials. To overcome the limitation, we propose using the vector virtual sources to compute the gradient in the FWI algorithm by converting the mono-component (pressure) source and receiver wavefields into the two-component particle displacements using the equation of motion on the staggered grids. Unlike the conventional method, our approach properly simulates PP radiation patterns for anisotropic parameters obtained under the elastic assumption, while retaining computational efficiency achieved by using the mono-component. Numerical examples show that anisotropic parameters are updated in the correct gradient direction by our method in the FWI process.

Improving estimates of buried pipe diameter and infilling material from ground-penetrating radar profiles with full-waveform inversion

Ground-penetrating radar (GPR) is a widely used tool for the detection and location of buried utilities. Buried pipes generate characteristic diffraction hyperbolas in raw GPR data. Current methods for analyzing the shapes and timing of the diffraction hyperbolas are very effective for locating pipes, but they are less effective for determining the diameter of the pipes, particularly when the pipes are smaller than the radar wavelengths, typically a few tens of centimeters. A full-waveform inversion (FWI) method is described for improving estimates of the diameter of a pipe and confirming the infilling material (air/water/etc.) for the simple case of an isolated diffraction hyperbola on a profile run perpendicular to a pipe with antennas in broadside mode (parallel to the pipe). The technique described here can improve a good initial guess of the pipe diameter (within 30%–50% of the true value) to a better estimate (less than approximately 8% misfit). This method is developed by combining two freely available software packages with a deconvolution method for GPR effective source wavelet estimation. The FWI process is run with the PEST algorithm (model-independent parameter estimation and uncertainty analysis). PEST iteratively calls the gprMax software package for forward modeling of the GPR signal as the model for the pipe and surrounding soil is refined.

Comparison of weighting techniques for acoustic full waveform inversion

To reconstruct long-wavelength structures in full waveform inversion (FWI), the wavefield-damping and weighting techniques have been used to synthesize and emphasize low-frequency data components in frequency-domain FWI. However, these methods have some weak points. The application of wavefield-damping method on filtered data fails to synthesize reliable low-frequency data; the optimization formula obtained introducing the weighting technique is not theoretically complete, because it is not directly derived from the objective function. In this study, we address these weak points and present how to overcome them. We demonstrate that the source estimation in FWI using damped wavefields fails when the data used in the FWI process does not satisfy the causality condition. This phenomenon occurs when a non-causal filter is applied to data. We overcome this limitation by designing a causal filter. Also we modify the conventional weighting technique so that its optimization formula is directly derived from the objective function, retaining its original characteristic of emphasizing the low-frequency data components. Numerical results show that the newly designed causal filter enables to recover long-wavelength structures using low-frequency data components synthesized by damping wavefields in frequency-domain FWI, and the proposed weighting technique enhances the inversion results.

Full‐wavefield migration: using surface and internal multiples in imaging

ABSTRACTMultiple scattering is usually ignored in migration algorithms, although it is a genuine part of the physical reflection response. When properly included, multiples can add to the illumination of the subsurface, although their crosstalk effects are removed. Therefore, we introduce full‐wavefield migration. It includes all multiples and transmission effects in deriving an image via an inversion approach. Since it tries to minimize the misfit between modeled and observed data, it may be considered a full waveform inversion process. However, full‐wavefield migration involves a forward modelling process that uses the estimated seismic image (i.e., the reflectivities) to generate the modelled full wavefield response, whereas a smooth migration velocity model can be used to describe the propagation effects. This separation of modelling in terms of scattering and propagation is not easily achievable when finite‐difference or finite‐element modelling is used. By this separation, a more linear inversion problem is obtained. Moreover, during the forward modelling, the wavefields are computed separately in the incident and scattered directions, which allows the implementation of various imaging conditions, such as imaging reflectors from below, and avoids low‐frequency image artefacts, such as typically observed during reverse‐time migration. The full wavefield modelling process also has the flexibility to image directly the total data (i.e., primaries and multiples together) or the primaries and the multiples separately. Based on various numerical data examples for the 2D and 3D cases, the advantages of this methodology are demonstrated.

Combining full wavefield migration and full waveform inversion, a glance into the future of seismic imaging

The next generation seismic migration and inversion technology considers multiple scattering as vital information, allowing the industry to derive significantly better reservoir models — with more detail and less uncertainty—while requiring a minimum of user intervention. Three new insights have been uncovered with respect to this fundamental transition. Unblended or blended multiple scattering can be included in the seismic migration process, and it has been proposed to formulate the imaging principle as a minimization problem. The resulting process yields angle-dependent reflectivity and is referred to as recursive full wavefield migration (WFM). The full waveform inversion process for velocity estimation can be extended to a recursive, optionally blended, anisotropic multiple-scattering algorithm. The resulting process yields angle-dependent velocity and is referred to as recursive full waveform inversion (WFI). The mathematical equations of WFM and WFI have an identical structure, but the physical meaning behind the expressions is fundamentally different. In WFM the reflection process is central, and the aim is to estimate reflection operators of the subsurface, using the up- and downgoing incident wavefields (including the codas) in each gridpoint. In WFI, however, the propagation process is central and the aim is to estimate velocity operators of the subsurface, using the total incident wavefield (sum of up- and downgoing) in each gridpoint. Angle-dependent reflectivity in WFM corresponds with angle-dependent velocity (anisotropy) in WFI. The algorithms of WFM and WFI could be joined into one automated joint migration-inversion process. In the resulting hybrid algorithm, being referred to as recursive joint migration inversion (JMI), the elaborate volume integral solution was replaced by an efficient alternative: WFM and WFI are alternately applied at each depth level, where WFM extrapolates the incident wavefields and WFI updates the velocities without any user interaction. The output of the JMI process offers an integrated picture of the subsurface in terms of angle-dependent reflectivity as well as anisotropic velocity. This two-fold output, reflectivity image and velocity model, offers new opportunities to extract accurate rock and pore properties at a fine reservoir scale.