Waveform Inversion Research Articles

WUS324: Multiscale Full Waveform Inversion Approaching Convergence Improves Waveform Fits While Imaging Seismic Structure of the Western United States

AbstractWe report a new model of radially anisotropic crustal and upper mantle structure of the western United States (WUS324) obtained from full waveform inversion of earthquake data. We ran three multiscale inversion stages beyond model WUS256 (Rodgers et al., 2022, https://doi.org/10.1029/2022jb024549) allowing them to approach convergence to fit a larger data set to a shorter minimum period of 16 s. WUS324 is based on 324 total iterations from its starting model, significantly more (16 times) than previous studies. Waveform misfit reductions are 66%–70% for both the inversion data and an independent validation data set providing confidence in the predictive power of the model. WUS324 provides much better fits and reveals shear wavespeed, vS, structure of this large region with more detail than previous waveform tomography models. We show representative images demonstrating the resolution of diverse seismic structure across this highly heterogeneous region including oceanic lithosphere, subducting slabs and continental magmatism.

Physics-guided full waveform inversion using Encoder-Solver convolutional neural networks

Abstract Full Waveform Inversion (FWI) is an inverse problem for estimating the wave velocity distribution in a given domain, based on observed data on the boundaries. The inversion is computationally demanding because we are required to solve multiple forward problems, either in time or frequency domains, to simulate data that are then iteratively fitted to the observed data. We consider FWI in the frequency domain, where the Helmholtz equation is used as a forward model, and its repeated solution is the main computational bottleneck of the inversion process. To ease this cost, we integrate a learning process of an Encoder-Solver preconditioner that is based on convolutional neural networks (CNNs). The Encoder-Solver is trained to effectively precondition the discretized Helmholtz operator given velocity medium parameters. Then, by re-training the CNN between the iterations of the optimization process, the Encoder-Solver is adapted to the iteratively evolving velocity medium as part of the inversion. Without retraining, the performance of the solver deteriorates as the medium changes. Using our light retraining procedures, we obtain the forward simulations effectively throughout the process. We demonstrate our approach to solving FWI problems using 2D geophysical models with high-frequency data.

FWI-WF: Robust full-waveform inversion using hybrid Wasserstein-1 and Fourier metrics

The performance of full-waveform inversion (FWI) in constructing high-resolution subsurface models is closely related to the design of mismatch functions. The least-squares norm ([Formula: see text]) is commonly used; however, it is prone to local minima when high-quality initial guess and low-frequency data are unavailable. The Wasserstein-1 ([Formula: see text]) metric captures time shifts more effectively, but it may be plagued by imprecise deep structures. The Fourier metric leverages power spectra from simulated and observed data, offering higher-resolution updates near solutions. We develop a progressive waveform inversion method called FWI-WF using [Formula: see text] and Fourier metrics. Specifically, in the early stage of inversion, we apply greater weight to the [Formula: see text] metric for constructing a good background model and avoiding falling into local minima. Then, the Fourier metric gradually dominates to refine edges and deep structures, providing high-resolution inversion results. During the optimization process, we use automatic differentiation to improve inversion efficiency. Experimental results on three baseline geologic models indicate that FWI-WF outperforms three state-of-the-art methods.

Multi‐Method Structural Investigation of the Schneiderberg–Baalberge Burial Mound (Saxony‐Anhalt, Germany) Including Seismic Full‐Waveform Inversion (FWI)

ABSTRACTThe construction history and subsequent usage of burial mounds are an important testimony for socio‐economic transformation in prehistoric societies. The Baalberge–Schneiderberg burial mound, subject of the presented study, falls in this category as it is considered as an important monument that indicates the emergence of early social stratification during the Chalcolithic period in central Europe. This hypothesis relies on the chronological development of the burial mound, which is not fully understood until now. Therefore, a reconstruction of the complex stratigraphy of the burial mound including construction phases and later alterations is highly relevant for archaeological research, but the required excavations would be onerous and inconsistent with preservation efforts. In this paper, we demonstrate that non‐invasive geophysical prospection, especially seismic sounding with shear and Love waves, is suitable to obtain the required stratigraphic information, if seismic full waveform inversion (FWI) and reflection imaging are applied. Complementary information on the preservation state of the mound is obtained through Electrical Resistivity Tomography (ERT) and Electromagnetic Induction (EMI) measurements. To support the seismic and geoelectric results, we utilize Dynamic Testing (DynP), geoarchaeological corings, 14C‐Dating and archaeological records. Our investigations reveal two construction phases of the Baalberge–Schneiderberg mound. The 14C‐Dating yields dates for the older burial mound that are contemporary to the Chalcolithic Baalberge group (4000–3400 bc). During the Early Bronze Age (EBA), the mound was enlarged to its final size by people of the Aunjetitz/Únětice society (2300–1600 bc). However, both seismic and geoelectric depth sections show an extensive disturbance of the original stratigraphy due to former excavations. For this reason, the exact shape of the older burial mound cannot be determined exactly. Based on our data, we estimate that its height was below 2 m. In consequence, the original Baalberge burial mound was less monumental as until now assumed, which potentially prompting a revision of its significance as indicator for social differentiation.

Optimising seismic imaging design parameters via bilevel learning

Abstract Full waveform inversion (FWI) is a standard algorithm in seismic imaging. It solves the inverse problem of computing a model of the physical properties of the earth’s subsurface by minimising the misfit between actual measurements of scattered seismic waves and numerical predictions of these, with the latter obtained by solving the (forward) wave equation. The implementation of FWI requires the a priori choice of a number of ‘design parameters’, such as the positions of sensors for the actual measurements and one (or more) regularisation weights. In this paper we describe a novel algorithm for determining these design parameters automatically from a set of training images, using a (supervised) bilevel learning approach. In our algorithm, the upper level objective function measures the quality of the reconstructions of the training images, where the reconstructions are obtained by solving the lower level optimisation problem—in this case FWI. Our algorithm employs (variants of) the BFGS quasi-Newton method to perform the optimisation at each level, and thus requires the repeated solution of the forward problem—here taken to be the Helmholtz equation. This paper focuses on the implementation of the algorithm. The novel contributions are: (i) an adjoint-state method for the efficient computation of the upper-level gradient; (ii) a complexity analysis for the bilevel algorithm, which counts the number of Helmholtz solves needed and shows this number is independent of the number of design parameters optimised; (iii) an effective preconditioning strategy for iteratively solving the linear systems required at each step of the bilevel algorithm; (iv) a smoothed extraction process for point values of the discretised wavefield, necessary for ensuring a smooth upper level objective function. The algorithm also uses an extension to the bilevel setting of classical frequency-continuation strategies, helping avoid convergence to spurious stationary points. The advantage of our algorithm is demonstrated on a problem derived from the standard Marmousi test problem.

Assessing the value of combined use of distributed acoustic sensing (DAS) and multicomponent vertical seismic profiling (VSP) data with network-based full waveform inversion and uncertainty quantification: A case study in Alberta, Canada

Distributed acoustic sensing (DAS) technology, deployed in a vertical seismic profiling (VSP) experimental configuration, has emerged as a candidate for non-disruptive and low-cost seismic monitoring of CO2 geostorage and plume evolution. Full waveform inversion (FWI) has likewise received significant attention because it uses relatively complete physical models of wave propagation and because of its sample-by-sample incorporation of data information. Recent artificial neural network-based FWI algorithms (built with, for instance, recursive neural networks, or RNNs) have added to FWI a range of flexible and efficient tools for gradient computation and options for uncertainty assessment and initial model proxies. An important current research area for the use of DAS data is to better understand how they change our confidence levels in models selected through FWI. In particular, we seek to understand whether either DAS data or conventional geophone data alone are optimal for FWI model selection in the CO2 problem, and if not, to what degree they complement each other. The Snowflake 4D VSP dataset, which includes multi-offset and multi-azimuth broadband sources illuminating both fiberoptic cable and densely sampled accelerometer phones in the borehole, was acquired by our group to directly address these questions. In this study, we quantify uncertainty (UQ) by sampling the posterior model covariance matrix from the inverse Hessian matrix at the end of RNN-FWI runs on the Snowflake baseline data, invoking a velocity-density parameterization, and involving mixtures of accelerometer and DAS data. In this UQ context, the complementary effect of combining accelerometer and DAS data is evident in the Vp and Vs models. Our confidence in the approach is bolstered by its independent prediction of a region of known high uncertainty. In the overall pursuit of reliable and low-cost monitoring tools, this supports continued consideration of a multicomponent sensors supported by DAS approach.

Seismic Evidence for Velocity Heterogeneity Along ∼40 Ma Old Oceanic Crustal Segment Formed at the Slow‐Spreading Equatorial Mid‐Atlantic Ridge From Full Waveform Inversion of Ocean Bottom Seismometer Data

AbstractIn slow spreading environments, oceanic crust is formed by a combination of magmatic and tectonic processes. Using full waveform inversion applied to active‐source ocean bottom seismometer data, we reveal the presence of a strong lateral variability in the 40–48 Ma old oceanic crust formed at the slow‐spreading Mid‐Atlantic Ridge in the equatorial Atlantic Ocean. Over a 120 km‐long section between the St Paul fracture zone (FZ) and the Romanche transform fault (TF), we observe four distinct 20–30 km long crustal segments. The segment affected by the St Paul FZ consists of three layers, an ∼2 km thick layer with a P‐wave velocity <6 km/s, a 1.5 km thick middle crust with a velocity of 6–6.5 km/s, and an underlying layer where velocity is ∼7 km/s, representing the lower crust. The segment associated with an abyssal hill morphology contains a high velocity of ∼7 km/s at 2–2.5 km below the basement, indicating the presence of primitive gabbro or serpenized peridotite. The segment associated with a low basement morphology seems to have 5.5–6.5 km/s velocity starting near the basement extending down to ∼4 km depth, indicating chemically distinct crust. The segment close to the Romanche TF, a velocity 4.5–5 km/s near the seafloor increasing to 7 km/s at 4 km depth indicates a magmatic origin. The four distinct crustal segments have a good correlation with the overlying seafloor morphology. These observed strong crustal heterogeneities could result from alternate tectonic and magmatic processes along the ridge axis, possibly modulated by thermal and/or chemical variations in the mantle during their formation along the ridge segment.

Modeling Rayleigh wave in viscoelastic media with constant Q model using fractional time derivatives

The propagation of seismic waves within the near-surface weathering layers, characterized by their low-quality factors (Q), is often accompanied by strong attenuation and dispersion phenomena. Among these, the Rayleigh wave, with its sensitivity to dispersion, has proven to be a powerful tool for near-surface exploration. We propose a novel approach for simulating Rayleigh wave propagation in such low-Q media. Our method uses the time-domain fractional wave equation with memory effect, based on Kjartansson's constant-Q (CQ) model, for accurate characterization of the propagation process. To solve numerically the wave equation with the fractional derivatives, we employ a finite-difference method combined with the auxiliary differential equation-perfectly matched layer (ADE-PML) and the acoustic-elastic boundary approach (AEA). The algorithm's high computational accuracy is verified through comparison with the conventional integer-order wave equation based on the nearly constant-Q (NCQ) models in strong attenuation media. The research in this paper deepens our understanding of the propagation characteristics of Rayleigh waves in strongly weathering layers. This new method strongly supports those seismic imaging and inversion methods depending on seismic modeling, including the reverse time migration and the full waveform inversion of the internal structure of low-Q media.

Enhancing Full Waveform Inversion of Field GPR Data: A Source-Independent Approach with Dynamic Reference Selection via SE-Wave-U-Net

Ground Penetrating Radar (GPR) is a widely-utilized non-destructive detection tool. Full Waveform Inversion (FWI) offers a reliable algorithm for accurately imaging GPR data. Despite its potential, FWI's effectiveness is often compromised by unknown excitation sources, which can lead to computational failures and/or diminished imaging accuracy. Addressing this challenge, this paper introduces a convolution-type objective function designed to mitigate the detrimental effects of these unknown excitation sources on FWI imaging. Critical to the success of this function is the precise selection of reference traces. Further refining the capability for signal separation, we have integrated Squeeze-and-Excitation (SE) modules into the traditional Wave-U-Net architecture, culminating in the development of the SE-Wave-U-Net framework. This framework is adept at dynamically extracting accurate reference traces from field GPR data, which are essential for the accurate FWI imaging. The efficacy, accuracy, and practical applicability of proposed methods are validated through extensive analysis of numerical, laboratory experiments, and field GPR data. The proposed algorithm not only establishes a high-precision computational framework for FWI but also enhances the reliability of GPR imaging in complex environments.

Greedy selection of optimal location of sensors for uncertainty reduction in seismic moment tensor inversion

We address an optimal sensor placement problem through Bayesian experimental design for seismic full waveform inversion for the recovery of the associated moment tensor. The objective is that of optimally choosing the location of the sensors (stations) from which to collect the observed data. The Shannon expected information gain is used as the objective function to search for the optimal network of sensors. A closed form for such objective is available due to the linear structure of the forward problem, as well as the Gaussian modeling of the observational errors and prior distribution. The resulting problem being inherently combinatorial, a greedy algorithm is deployed to sequentially select the sensor locations that form the best network for learning the moment tensor. Numerical results are presented to display the optimal network of sensors and how the uncertainty in the inferred seismic moment tensor contracts. The scenario of full three-dimensional velocity models or unknown earthquake source locations is treated as nuisance uncertainty, contributing to the overall uncertainty without being the focus of the inversion. This is addressed using a consensus approach over a set of realizations of the nuisance parameter. We analyzed the resulting network of stations for the moment tensor inversion under model misspecification, which reflects realistic data-generating processes.

Full waveform inversion with smoothing of dilated convolutions

Full waveform inversion with smoothing of dilated convolutions

Deep neural helmholtz operators for 3D elastic wave propagation and inversion

Summary Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full waveform inversion, which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent works in seismology have shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We employ a Helmholtz neural operator for 2D and 3D elastic wave modeling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly on the surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3D full-waveform inversion, reducing the computation time by a factor of 350.

Soil water content estimation by using ground penetrating radar data full waveform inversion with grey wolf optimizer algorithm

AbstractSoil water content (SWC) estimation is important for many areas including hydrology, agriculture, soil science, and environmental science. Ground penetrating radar (GPR) is a promising geophysical method for SWC estimation. However, at present, most of the studies are based on partial information of GPR, like travel time or amplitude information, to invert the SWC. Full waveform inversion (FWI) can use the information of the entire waveform, which can improve the accuracy of parameter estimation. This study proposes a novel SWC estimation scheme by using the FWI of GPR, optimized by the grey wolf optimizer (GWO) algorithm. The proposed scheme includes a petrophysical relationship to link the SWC with the relative dielectric permittivity, 1D GPR forward modeling, and a GWO optimization algorithm. First, numerical modeling was carried out, and the proposed scheme was applied to both noise‐free and noisy data to verify its applicability. Then, the proposed method was applied to data collected from a field experimental site. These results, derived from both synthetic and real datasets, show that the proposed inversion scheme resulted in a good match between the observed and calculated GPR data. In the numerical modeling, it was observed that the SWC could be inverted accurately, even when noise was present in the data. These demonstrate that the GWO method can be applied for the quantitative interpretation of GPR data. The proposed scheme shows potential for SWC estimation by using GPR full waveform data.

Bayesian time‐lapse full waveform inversion using Hamiltonian Monte Carlo

AbstractTime‐lapse images carry out important information about dynamic changes in Earth's interior, which can be inferred using different full waveform inversion schemes. The estimation process is performed by manipulating more than one seismic dataset, associated with the baseline and monitors surveys. The time‐lapse variations can be so minute and localized that quantifying the uncertainties becomes fundamental to assessing the reliability of the results. The Bayesian formulation of the full waveform inversion problem naturally provides confidence levels in the solution, but evaluating the uncertainty of time‐lapse seismic inversion remains a challenge due to the ill‐posedness and high dimensionality of the problem. The Hamiltonian Monte Carlo can effectively sample over high‐dimensional distributions with affordable computational efforts. In this context, we explore the sequential approach in a Bayesian fashion for time‐lapse full waveform inversion using the Hamiltonian Monte Carlo method. The idea relies on integrating the baseline survey information as prior knowledge to the monitor estimation. We compare this methodology with a parallel scheme in perfect and a simple perturbed acquisition geometry scenario considering the Marmousi and a typical Brazilian pre‐salt velocity model. We also investigate the correlation effect between baseline and monitor samples on the propagated uncertainties. The results show that samples between different surveys are weakly correlated in the sequential case, while the parallel strategy provides time‐lapse images with lower dispersion. Our findings demonstrate that both methodologies are robust in providing uncertainties even in non‐repeatable scenarios.

Variational prior replacement in Bayesian inference and inversion

SUMMARY Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically by calculating the posterior probability distribution function. Prior information is often described by a prior probability distribution. Situations arise in which we wish to change prior information during the course of a scientific project. However, estimating the solution to any single Bayesian inference problem is often computationally costly, as it typically requires many model samples to be drawn, and the data set that would have been recorded if each sample was true must be simulated. Recalculating the Bayesian inference solution every time prior information changes can therefore be extremely expensive. We develop a mathematical formulation that allows the prior information that is embedded within a solution, to be changed using variational methods, without recalculating the original Bayesian inference. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2-D seismic full waveform inversion example, in which VPR provides similar posterior solutions to those obtained by solving independent inference problems using different prior distributions. The former can be completed within minutes on a laptop computer, whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information: uniform, smoothing and geological prior distributions.

Envelope normalized reflection waveform inversion

AbstractThe reflection waveform inversion has the capability to reconstruct the background velocity model using only the reflection data by employing a migration/demigration process. Utilizing the waveform discrepancy to update the background velocity model, the conventional reflection waveform inversion method heavily relies on the true‐amplitude migration/demigration technique to reproduce the primary amplitude information from the observed reflections. We can reproduce the amplitude of observed reflections by performing least‐squares reverse time migration to estimate the reflectivity in each iteration. However, this strategy is quite time‐consuming. To avoid the need for the true‐amplitude migration/demigration or least‐squares reverse time migration, we develop an amplitude‐independent reflection waveform inversion method that uses an envelope‐normalized objective function. The envelope‐normalized waveform difference can extract the phase residuals accurately as a function of time. Compared with the global energy–normalized misfit, our proposed envelope‐normalized objective function is essentially a phase‐matched measurement. At the same time, due to the amplitude independence of our proposed objective function, the subsequent weak reflections contribute with a similar weight to the total value of the misfit as the strong early reflections do. This makes it possible to recover the deep subsurface velocity. Synthetic data of the Sigsbee model and marine streamer field data applications validate that our amplitude‐independent reflection waveform inversion method can further improve the resolution and accuracy by aligning the reflection events of synthetic and observed data phase to phase without the need to perform true‐amplitude migration/demigration or least‐squares reverse time migration as in conventional reflection waveform inversion.

Truncated Newton full waveform inversion method for the human brain imaging

Abstract It has been shown that full-waveform inversion (FWI) method can be a competitive alternative for medical imaging problems. It offers high-resolution results while delivering the advantages of being fast, safe, portable, and affordable, compared to magnetic resonance imaging (MRI) and x-ray computed tomography. However, to the best of our knowledge, FWI applications in medical imaging only use the first-order derivative information. In this case, the high parameter contrasts between different tissues of human body and multi-scatterings problems may lead FWI to local minima. Thus, we present a competitive truncated Newton method for high-resolution imaging of the human brain. This truncated Newton method, based on the efficient linear solver MINRES-QLP, can make full use of multiple scattering wavefield information. Compared with the truncated Newton method based on conjugate gradient, the MINRES-QLP iterative method presents various advantages when solving linear systems, such as the capacity to handle both non-singular and singular systems, less computational cost, and efficiency even for ill-conditioned systems. Numerical experiments for imaging the brain with and without the skull are conducted. Numerical results indicate that, compared with the truncated Newton method based on conjugate gradient, the truncated Newton method based on MINRES-QLP exhibits computational efficiency while maintaining the same level of accuracy.

Airframe Icing Properties and Geometry Reconstruction using Ultrasonic Full Waveform Inversion Method

Abstract Recently, Non-destructive Ultrasonic Techniques have been utilized to detect aircraft icing. Traditional methods like ultrasound pulse-echo, guided waves, and optics are unsuitable due to ice’s complex properties and shape. Pulse-echo ultrasound, reliant on surface layer travel time, becomes ineffective with thin ice layers and variable ice properties. This research aimed to reconstruct ice properties and geometry on aircraft surfaces using ultrasonic full waveform inversion (FWI). For simulation COMSOL software is used to verify wave propagation through a nonlinear, thinly layered model. Wave propagation is then modelled using the 2D acoustic equation and finite differential methods, generating synthetic data. The FWI algorithm implemented, utilizes this synthetic data to reconstruct material properties and ice shape. Crucially, ultrasound transducers are discreetly positioned beneath the aircraft skin to non-destructively gather ice information without affecting aerodynamics.

Envelope Inversion with Source encoding for Ultrasound Computed Tomography

Abstract Ultrasound computed tomography (USCT) is a noninvasive and non-ionizing imaging technique for soft tissue and limb bones. Full waveform inversion (FWI) has received increased interest due to its high resolution. The difficulties in FWI are twofold: high computational burden and poor convergence. To address these issues, an envelope inversion with source coding (EI-SE) algorithm is proposed in this paper. The envelope-based objective function measures the matching degree between the predicted and measured envelopes. It can provide the low-frequency components of the signal that are not available to FWI. Therefore, the EI can provide an accurate solution even when the initial model is far from the ground truth and low-frequency data are missing. The simulations are conducted to demonstrate the validity of the proposed algorithm. The root mean squared error (RMSE) of the proposed algorithm is much smaller than that of conventional FWI. The results show that EI-SE is effective in avoiding cycle skipping.

Full Waveform Inversion Based on Dynamic Time Warping and Application to Reveal the Crustal Structure of Western Yunnan, Southwest China

Full Waveform Inversion Based on Dynamic Time Warping and Application to Reveal the Crustal Structure of Western Yunnan, Southwest China