Full-waveform Inversion Workflow Research Articles

Borehole acoustic full-waveform inversion

Full-waveform inversion (FWI) is a technique that has the potential for building high-resolution elastic velocity models. We apply this technique to wireline monopole acoustic logging data to image the near-wellbore formation velocity structures, which can be used in a fluid intrusion evaluation. Our FWI workflow is established in cylindrical coordinates instead of Cartesian coordinates to adapt to the borehole geometry. Assuming that the acoustic logging tool is centralized and formation properties are azimuthally invariant, we can simplify the problem to two dimensions in the inversion. Synthetic tests find that the high-resolution P- and S-wave velocity model around the borehole could be successfully inverted using FWI once a reasonable starting velocity model is given. However, borehole FWI differs from seismic FWI in that strong borehole-guided waves exist near the borehole wall due to the elastic effects. The borehole-guided waves have little sensitivity to velocity away from the borehole wall. In addition, it is difficult to simultaneously match the waveform of the body waves and guided waves in synthetic data and field data. Therefore, for field data applications, we propose to obtain the formation velocity structures around the borehole by FWI using only the first arrived P waves. The field data tests find that the proposed method is applicable for inverting the P-wave velocity structure around the borehole with a similar resolution compared with ray-tracing tomography. FWI may achieve higher resolution in field applications in the future with better assumptions made in the forward modeling and using more information in the field data.

Bayesian inverse scattering theory for multiparameter full-waveform inversion in transversely isotropic media

Full-waveform inversion (FWI) is an effective method that uses kinematic and dynamic information to provide an accurate image of the subsurface target. By applying the Bayesian approaches to FWI workflow, the traditional deterministic solution is replaced with a maximum a posteriori solution. This provides not only a reliable subsurface model for the nonlinear inverse scattering problem but also an uncertainty quantification. The high computational cost of the Bayesian FWI and the complexity of a large-scale anisotropy inversion, however, restrict its application in industrial settings. We have developed an FWI approach for tilted transversely isotropic (TTI) media. The strain field and the particle displacement of the anisotropic media are expressed by an integral method based on the scattering theory and an explicit multiscattering modified Green’s function. To reduce the computational cost, the wavefield is calculated by using the preconditioned generalized minimal residual iterative solver, which approximates the solution in the Krylov subspace. The Fréchet derivatives are expressed by using the distorted Born iterative method. An iterated extended Kalman filter is applied to solve the inverse problem and the posterior covariance matrix frequency-by-frequency. Numerical experiments indicate that the presented methodology can reasonably reconstruct the elastic moduli of the TTI medium. Furthermore, the numerical experiments demonstrate the accuracy and feasibility of our methodology.

A fast wavefield reconstruction inversion solution in the frequency domain

Full-waveform inversion (FWI) aims at estimating subsurface physical parameters by minimizing the misfit between simulated data and observations. FWI relies heavily on an accurate initial model and is less robust to measurement noise and physical assumptions in modeling. Compared with FWI, wavefield reconstruction inversion (WRI) is more robust to these uncertainties but faces high computational costs. To overcome these challenges, we have developed a new form of WRI. This reformulation takes the form of a traditional FWI formula, which includes a medium-dependent weight function, and can be easily incorporated into the current FWI workflow. This weight function contains the covariance matrices to characterize the distribution of uncertainties in measurements and physical assumptions. We discuss various options of the theoretical covariance matrix of the new inversion method and find how they relate to various well-known approaches, including FWI, WRI, and extended FWI. On the basis of the preceding comparison, we develop a theoretical covariance matrix definition based on the source. Numerical experiments demonstrate that our method with a source-dependent theoretical covariance matrix is more computationally efficient than conventional WRI while preserving a certain degree of robustness.

Full waveform inversion of a Brazilian pre-salt OBN dataset using a concentric circle source geometry

We have designed a workflow to apply full-waveform inversion (FWI) to a new type of acquisition recorded on ocean-bottom nodes (OBNs) in the Brazilian presalt area. The data set consists of three large-radius concentric circles of seismic source points recorded on OBN placed within the area of the circles. This geometry provides full azimuth distribution and benefits from long offsets in which the effect of diving waves is strong. These diving waves carry information about the presalt targets of interest. In the FWI workflow, we retrieve the P-wave velocity using the density as a propagation parameter. We test three objective functions: ℓ2-norm, ℓ1-norm, and nonparametric residuals objective function (NPR-FWI), which have different abilities to handle low signal-to-noise ratio data. In our inversion tests, the ℓ1-norm and NPR-FWI retrieve more information about the presalt reservoir than the ℓ1-norm. We find through several inversion experiments that we can recover structures related to the presalt reservoir, demonstrating the potential for this new type of OBN circular source geometry. We also discuss implications for low-cost data acquisition that take advantage of the sparse nature of sources and receivers needed in this survey geometry, including its extension to time-lapse target-oriented reservoir monitoring.

Full-waveform inversion with adversarial losses via deep learning

Full-waveform inversion (FWI) is a powerful technology for determining geophysical properties, nevertheless, with high computational cost and difficulty in convergence. Motivated by the powerful tools available in modern deep learning system to deal with complex non-linear inverse problems, we implement the FWI workflow with deep neural networks and propose the use of two different neural networks with adversarial losses to provide fast and accurate solutions. Specifically, a deep neural network is used to generate simulated seismic data based on the finite-difference method, and another fully connected neural network is used to fit the Wasserstein distance between simulated and observed seismic data. The Wasserstein distance and the L2 metric together form an adversarial loss function. The parallel-layer model and the SEAM model are used to verify the benefits of the proposed method and its performance in the face of seismic data lack low-frequency components and noise interference. Numerical experiments demonstrate that, compared with other algorithms, the adaptive gradient (Adagrad) algorithm performs best. In addition, the Adagrad algorithm with adversarial losses has faster convergence speed and more inverting details than with L2 losses. The proposed method provides reliable inversion results, even when seismic data without low-frequency components. Moreover, the adversarial losses are less sensitive to noise, although still difficult to converge.

Love wave full-waveform inversion for archaeogeophysics: From synthesis tests to a field case

High precision detection of the shallow-surface structure is of great significance to geotechnical engineering and archaeological geophysical research. Surface wave Full waveform inversion (FWI) is attracting more and more attention in shallow seismic exploration. The FWI of Love waves is more straightforward and has a higher precision and resolution compared with Rayleigh waves. In the classic gradient-based FWI algorithm, the gradient amplitude does not scale with increasing depth due to the coupling effect and double scattering; thus, the inversion image cannot represent the significant small-scale characteristic structures in archaeogeophysics. To overcome the above limitations, a pre-conditioned FWI approach was presented, and a complete elastic Love waves FWI workflow was developed for archaeogeophysics. The superior performance of the developed method was verified through synthesis tests and a field experiment. The synthetic tests on a tomb model show that the pre-conditioned approach has a better parameter reconstruction capability, can characterize small-scale anomalies, improves the convergence rate, and has a strong anti-noise ability. In the field experiment, the pre-conditioned approach-based Love waves FWI workflow was adopted to invert the recorded Love waves datasets gathered in Xianyang City, China. The target of the survey was to determine the location and approximate size of a buried ancient tomb to provide reliable guidance for further excavation. The obtained FWI image revealed an underground profile with a depth of 9.5 m, and a low-velocity anomaly with a height of about 1.6 m and a length of about 4.7 m was delineated, which we identified as the tomb. The subsequent excavation confirmed this conclusion. The above results show that the complete Love wave FWI workflow developed has the ability to perform accurate imaging under the condition of few shots.

Sensitivity kernels for receiver function misfits in a full waveform inversion workflow

SUMMARY Receiver functions have been used for decades to study the Earth’s major discontinuities by focusing on converted waves. Deconvolution, which is the mathematical backbone of the method, is assumed to remove the source time function and the far-field dependence on structure, making it a useful method to map the nearby Earth structure and its discontinuities. Ray theory, a plane incoming wavefield, and a sufficiently well-known near-receiver background velocity model are conventionally assumed to map the observations to locations in the subsurface. Many researchers are aware of the shortcoming of these assumptions and several remedies have been proposed for mitigating their consequences. Adjoint tomography with a quasi-exact forward operator is now within reach for most researchers, and we believe is the way forward in receiver function studies. A first step is to calculate adjoint sensitivity kernels for a given misfit function. Here, we derive the adjoint source for a receiver function waveform misfit. Using a spectral element forward code, we have calculated sensitivity kernels for P-to-S converted waves using several 2-D models representing an average crust with an underlying mantle. The kernels show profound differences between P- and S-wave speed sensitivity. The sensitivity to P-wave speed is wide-ranging and related to the scattered P-wavefield which interferes with that of the P-to-S converted wave. The S-wave speed sensitivity is more local and mostly associated to potential locations of P-to-S conversion, although more distant sensitivity is also observed. Notably, there is virtually no sensitivity to impedance. We further observe the well-known trade-off between depth of the discontinuity and wave speed, but find that considering a longer waveform that includes more surface reverberations reduces this trade-off significantly.

Nordkapp Topseis/Node Acquisition — Lessons from a Modelling Study

Based on an extensive 3D modelling study utilising full-wavefield Finite-Difference modelling and Full-Waveform Inversion (FWI) we demonstrate that the TopSeis/OBN hybrid acquisition acquired from May to August 2021 in the Nordkapp basin in the Barents Sea has the potential to image salt flanks and sedimentary details, given an accurate initial model in the shallow and a carefully designed deblending and FWI workflow. As a part of this we demonstrate that the large offsets and multi-azimuth recorded by the ocean bottom nodes are crucial to map the complex salt diapirism in the area including the steeply dipping flanks.

Challenges in shallow target reconstruction by 3D elastic full-waveform inversion — Which initial model?

Elastic full-waveform inversion (FWI) is a powerful tool for high-resolution subsurface multiparameter characterization. However, 3D FWI applied to land data for near-surface applications is particularly challenging because the seismograms are dominated by highly energetic, dispersive, and complex-scattered surface waves (SWs). In these conditions, a successful deterministic FWI scheme requires an accurate initial model. Our study, primarily focused on field data analysis for 3D applications, aims at enhancing the resolution in the imaging of complex shallow targets, by integrating devoted SW analysis techniques with a 3D spectral-element-based elastic FWI. From dispersion curves, extracted from seismic data recorded over a sharp-interface shallow target, we build different initial S-wave ([Formula: see text]) and P-wave ([Formula: see text]) velocity models (laterally homogeneous and laterally variable), using a specific data transform. Starting from these models, we carry out 3D FWI tests on synthetic and field data, using a relatively straightforward inversion scheme. The field data processing before FWI consists of band-pass filtering and muting of noisy traces. During FWI, a weighting function is applied to the far-offset traces. We test 2D and 3D acquisition layouts, with different positions of the sources and variable offsets. The 3D FWI workflow enriches the overall content of the initial models, allowing a reliable reconstruction of the shallow target, especially when using laterally variable initial models. Moreover, a 3D acquisition layout guarantees a better reconstruction of the target’s shape and lateral extension. In addition, the integration of model-oriented (preliminary monoparametric FWI) and data-oriented (time windowing) strategies into the main optimization scheme has produced further improvement of the FWI results.

Full-waveform inversion for full-wavefield imaging: Decades in the making

Seismic imaging using full-wavefield data that includes primary reflections, transmitted waves, and their multiples has been the holy grail for generations of geophysicists. To be able to use the full-wavefield data effectively requires a forward-modeling process to generate full-wavefield data, an inversion scheme to minimize the difference between modeled and recorded data, and, more importantly, an accurate velocity model to correctly propagate and collapse energy of different wave modes. All of these elements have been embedded in the framework of full-waveform inversion (FWI) since it was proposed three decades ago. However, for a long time, the application of FWI did not find its way into the domain of full-wavefield imaging, mostly owing to the lack of data sets with good constraints to ensure the convergence of inversion, the required compute power to handle large data sets and extend the inversion frequency to the bandwidth needed for imaging, and, most significantly, stable FWI algorithms that could work with different data types in different geologic settings. Recently, with the advancement of high-performance computing and progress in FWI algorithms at tackling issues such as cycle skipping and amplitude mismatch, FWI has found success using different data types in a variety of geologic settings, providing some of the most accurate velocity models for generating significantly improved migration images. Here, we take a step further to modify the FWI workflow to output the subsurface image or reflectivity directly, potentially eliminating the need to go through the time-consuming conventional seismic imaging process that involves preprocessing, velocity model building, and migration. Compared with a conventional migration image, the reflectivity image directly output from FWI often provides additional structural information with better illumination and higher signal-to-noise ratio naturally as a result of many iterations of least-squares fitting of the full-wavefield data.

Refraction waves full waveform inversion of deep reflection seismic profiles in the central part of Lhasa Terrane

The deep reflection seismic method is an effective technique for detecting the fine structures of the lithosphere and solving deep geological problems. At present, the methods for obtaining the underground velocity from land deep reflection seismic profile mainly rely on the travel-time information based velocity analysis and tomography, which theoretically limit the resolution of the imaging results. To achieve the goal of using deep reflection seismic profile for crustal-scale high-precision velocity imaging, we must finally overcome the problem of full waveform inversion (FWI) using full wavefield data. In this paper, we successfully conduct FWI using the early-arrival wavefields (mainly primary and multiple refraction wavefields) in the deep reflection seismic profile, which can be seen as the first important step towards the goal of crustal-scale high-precision velocity imaging. We propose a robust refraction waves FWI workflow with global normalized cross-correlation, multiple refraction matching, reverse time source estimation and wavenumber domain gradient filtering as its core algorithm. The application of the proposed FWI method to the refraction waves of the deep reflection seismic profiles in the central part of Lhasa Terrane shows that the refraction FWI images have a higher resolution to underground structures both in horizontal and vertical directions than the first-arrival tomography images. The lateral velocity change can be well-matched with the typical geological characteristics of the work area. The FWI result can better describe the lateral boundaries of the strata in different eras. At a depth of less than 3 km from the surface, the refraction FWI can correct the interface morphology of the tomography results and recover some small-scale velocity structures and high-velocity anomalies.

Elastic vertically transversely isotropic full-waveform inversion using cross-gradient constraints — An application toward high-level radioactive waste repository monitoring

Anisotropic seismic full-waveform inversion (FWI) is a challenging task. In the case of 2D vertically transversely isotropic (VTI) media, there are five independent model parameters. This relatively large number of different parameter types imposes significant trade-off issues and makes the inversion parameterization a challenging task. The problem is less severe in a crosshole configuration, in which a wider angular coverage of the region of interest is available. There exist many suggestions for suitable inversion parameterizations. We have determined that, for a crosshole configuration, a relatively simple velocity-based parameterization provides a good FWI reconstruction of the subsurface. Furthermore, considerable improvements of the tomographic images can be achieved by supplying structural similarity constraints using cross gradients to the inversion problem. With two synthetic data sets, we determine that a structurally constrained VTI FWI workflow produces sharper subsurface images without adversely affecting the parameter trade-off issue. With a second synthetic experiment, we find that structurally constrained VTI FWI is robust to major differences in anomaly locations for different parameter types. We successfully applied the methodology to a crosshole data set acquired to image a downscaled version of a high-level radioactive waste repository. The resulting tomograms allowed a narrow highly fractured zone to be imaged.

Accelerating numerical wave propagation by wavefield adapted meshes. Part II: full-waveform inversion

SUMMARYWe present a novel full-waveform inversion (FWI) approach which can reduce the computational cost by up to an order of magnitude compared to conventional approaches, provided that variations in medium properties are sufficiently smooth. Our method is based on the usage of wavefield adapted meshes which accelerate the forward and adjoint wavefield simulations. By adapting the mesh to the expected complexity and smoothness of the wavefield, the number of elements needed to discretize the wave equation can be greatly reduced. This leads to spectral-element meshes which are optimally tailored to source locations and medium complexity. We demonstrate a workflow which opens up the possibility to use these meshes in FWI and show the computational advantages of the approach. We provide examples in 2-D and 3-D to illustrate the concept, describe how the new workflow deviates from the standard FWI workflow, and explain the additional steps in detail.

Seismic wavefield imaging of Earth’s interior across scales

Seismic full-waveform inversion (FWI) for imaging Earth’s interior was introduced in the late 1970s. Its ultimate goal is to use all of the information in a seismogram to understand the structure and dynamics of Earth, such as hydrocarbon reservoirs, the nature of hotspots and the forces behind plate motions and earthquakes. Thanks to developments in high-performance computing and advances in modern numerical methods in the past 10 years, 3D FWI has become feasible for a wide range of applications and is currently used across nine orders of magnitude in frequency and wavelength. A typical FWI workflow includes selecting seismic sources and a starting model, conducting forward simulations, calculating and evaluating the misfit, and optimizing the simulated model until the observed and modelled seismograms converge on a single model. This method has revealed Pleistocene ice scrapes beneath a gas cloud in the Valhall oil field, overthrusted Iberian crust in the western Pyrenees mountains, deep slabs in subduction zones throughout the world and the shape of the African superplume. The increased use of multi-parameter inversions, improved computational and algorithmic efficiency, and the inclusion of Bayesian statistics in the optimization process all stand to substantially improve FWI, overcoming current computational or data-quality constraints. In this Technical Review, FWI methods and applications in controlled-source and earthquake seismology are discussed, followed by a perspective on the future of FWI, which will ultimately result in increased insight into the physics and chemistry of Earth’s interior. Advances in numerical methods and high-performance computing have enabled imaging of Earth’s interior across scales using the physics of seismic-wave propagation. This Technical Review summarizes the use of full-waveform inversion in controlled-source and earthquake seismology.

Fast 2D full-waveform modeling and inversion using the Schur complement approach

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.

Land FWI in the Delaware Basin, west Texas: A case study

As part of a wide-reaching full-waveform inversion (FWI) research program, FWI is applied to an onshore seismic data set collected in the Delaware Basin, west Texas. FWI is routinely applied on typical marine data sets with high signal-to-noise ratio (S/N), relatively good low-frequency content, and reasonably long offsets. Land seismic data sets, in comparison, present significant challenges for FWI due to low S/N, a dearth of low frequencies, and limited offsets. Recent advancements in FWI overcome limitations due to poor S/N and low frequencies making land FWI feasible to use to update the shallow velocities. The chosen area has contrasting and variable near-surface conditions providing an excellent test data set on which to demonstrate the workflow and its challenges. An acoustic FWI workflow is used to update the near-surface velocity model in order to improve the deeper image and simultaneously help highlight potential shallow drilling hazards.

Preconditioning full-waveform inversion with efficient local correlation operators

Full-waveform inversion (FWI) is an iterative locally linearized data-fitting technique. The FWI method attempts to move from an initial low-wavenumber representation of earth parameters to a broader representation of the medium. An issue with the method is that FWI is an ill-posed problem, oversampled for the numerical forward discretization. The success of the model parameter reconstruction can often be greatly affected by external factors such as the presence of noise in the input field data or other artifacts arising from the imaging condition present in the FWI gradient computation. We have developed a strategy for mitigating against the influence of such external factors by preconditioning the discrete data gradient using an efficient differential approach instead of the often used integral formulation. Such an application of a smoothing correlation operator allows one to use prior information to locally filter along expected geological dips while being consistent with faults. The application of this preconditioning strategy to real and synthetic 2D data sets illustrates how this incremental additional step makes the FWI workflow less sensitive to noise and spatial aliasing artifacts. Nothing prevents a possible 3D acoustic extension, thanks to the small added computer cost for this local filter application. Three-dimensional elastic FWI may require this inexpensive filtering strategy due to the prohibitive forward-modeling costs that could be partially mitigated by using a coarse shot increment in conjunction with gradient preconditioning.

Suprasalt model building using full-waveform inversion

The application of full-waveform inversion (FWI) to bring high resolution to the velocity model is becoming a standard approach in the velocity model-building workflow. Diving wave FWI in conjunction with reflection FWI (RFWI) has been widely used in the Gulf of Mexico (GOM) to optimize the suprasalt model. Accuracy of a velocity model from tomography is dependent on residual moveout (RMO) picking accuracy. In a good signal-to-noise ratio area, the confidence of RMO picking is high. But gathers in areas affected by gas exhibit poor event continuity, which makes it difficult to get accurate RMO picks. In such a geologic regime, FWI can improve the velocity model and therefore the final image quality. There are two main components of a velocity model from the GOM area: the first is the sediment, and the second is salt geometry. In the beginning of the model-building cycle, it is most likely that salt geometry is not accurately defined. This inaccuracy leads to a big mismatch between synthetic and observed data for both diving wave FWI and RFWI. One way to handle this situation is to start with the salt model and iteratively adjust the salt interpretation as FWI model building progresses from lower to higher frequencies. Another approach could be eliminating the salt-related energy from the input and then using the sediment-only model for FWI. We are proposing a desalt approach in which we try to eliminate or reduce the salt-related energy from the input data and then use a sediment-only velocity model as a starting model for the entire suprasalt FWI workflow. We will present a case study in which, by adapting the desalt workflow, we could manage to do more FWI iterations by eliminating salt interpretation.

Tunnel detection at Yuma Proving Ground, Arizona, USA — Part 1: 2D full-waveform inversion experiment

We have applied time domain 2D full-waveform inversion (FWI) to detect a known 10 m deep wood-framed tunnel at Yuma Proving Ground, Arizona. The acquired seismic data consist of a series of 2D survey lines that are perpendicular to the long axis of the tunnel. With the use of an initial model estimated from surface wave methods, a void-detection-oriented FWI workflow was applied. A straightforward [Formula: see text] quotient masking method was used to reduce the inversion artifacts and improve confidence in identifying anomalies that possess a high [Formula: see text] ratio. Using near-surface FWI, [Formula: see text] and [Formula: see text] velocity profiles were obtained with void anomalies that are easily interpreted. The inverted velocity profiles depict the tunnel as a low-velocity anomaly at the correct location and depth. A comparison of the observed and simulated waveforms demonstrates the reliability of inverted models. Because the known tunnel has a uniform shape and for our purposes an infinite length, we apply 1D interpolation to the inverted [Formula: see text] profiles to generate a pseudo 3D (2.5D) volume. Based on this research, we conclude the following: (1) FWI is effective in near-surface tunnel detection when high resolution is necessary. (2) Surface-wave methods can provide accurate initial S-wave velocity [Formula: see text] models for near-surface 2D FWI.

A numerical study of 3D frequency-domain elastic full-waveform inversion

We have developed a 3D elastic full-waveform inversion (FWI) algorithm with forward modeling and inversion performed in the frequency domain. The Helmholtz equation is solved with a second-order finite-difference method using an iterative solver equipped with an efficient complex-shifted incomplete LU-based preconditioner. The inversion is based on the minimization of the data misfit functional and a total variation regularization for the unknown model parameters. We implement the Gauss-Newton method as the optimization engine for the inversions. The codes are parallelized with a message passing interface based on the number of shots and receivers. We examine the performance of this elastic FWI algorithm and workflow on synthetic examples including surface seismic and vertical seismic profile configurations. With various initial models, we manage to obtain high-quality velocity images for 3D earth models.