Full Waveform Inversion Method Research Articles

Scale-Aware Edge-Preserving Full Waveform Inversion with Diffusion Filter for Crosshole Sensor Arrays.

Full waveform inversion (FWI) is recognized as a leading data-fitting methodology, leveraging the detailed information contained in physical waveform data to construct accurate, high-resolution velocity models essential for crosshole surveys. Despite its effectiveness, FWI is often challenged by its sensitivity to data quality and inherent nonlinearity, which can lead to instability and the inadvertent incorporation of noise and extraneous data into inversion models. To address these challenges, we introduce the scale-aware edge-preserving FWI (SAEP-FWI) technique, which integrates a cutting-edge nonlinear anisotropic hybrid diffusion (NAHD) filter within the gradient computation process. This innovative filter effectively reduces noise while simultaneously enhancing critical small-scale structures and edges, significantly improving the fidelity and convergence of the FWI inversion results. The application of SAEP-FWI across a variety of experimental and authentic crosshole datasets clearly demonstrates its effectiveness in suppressing noise and preserving key scale-aware and edge-delineating features, ultimately leading to clear inversion outcomes. Comparative analyses with other FWI methods highlight the performance of our technique, showcasing its ability to produce images of notably higher quality. This improvement offers a robust solution that enhances the accuracy of subsurface imaging.

Adaptive structure-based full-waveform inversion

In simultaneous-shot full-waveform inversion (FWI), the re-started limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm can be used to suppress crosstalk, but crosstalk cannot be completely eliminated from the inversion results. To further solve the crosstalk problem caused by the interference among individual shots in simultaneous-shot FWI, adaptive structure-based smoothing is applied to the FWI with the restarted L-BFGS algorithm. Structure-based smoothing can mitigate the crosstalk by highlighting structure boundaries. To perform structure-based smoothing, an implicit anisotropic diffusion equation is solved. We carry out a multiscale inversion strategy. Because the FWI results in the low-frequency band contain less structural information and more crosstalk, structure-based smoothing is applied to the frequency band at frequencies higher than the peak frequency to prevent negative effects from the model structure in the low-frequency band. The estimated structural information is iteratively updated during the inversion process. Furthermore, structure-based smoothing is only added to the iterations with invariant encoding to reduce oversmoothing. The invariant encoding means that the shot encoding remains unchanged between iterations. Numerical experiments with an overthrust model indicate that the proposed adaptive structure-based FWI method can effectively suppress artifacts and provide a high-resolution inversion result, even when the encoded data are contaminated with strong noise. Another advantage of the adaptive structure-based FWI method is that no seismic migration needs to be performed, which makes it more efficient for real data.

The method for precise seismic detection of geological structures in underground coal mines and application

The occurrence of major accidents such as water inrush in coal mines and coal-gas outbursts is closely correlated with the unclear exploration of underground geological structures. Seismic exploration in underground coal mines is not limited by ground conditions and close to the detected target, so it has advantages with respect to the detection of underground geological structures. However, the geological structures leading to accidents are usually characterized by small size and diverse combinations, making it difficult to detect. In coal mine working faces, the use of transmission observation systems, combined with full waveform inversion methods, represents a new attempt at finely detecting geological structures. Therefore, a three-dimensional numerical model of an abnormal combination of faults and collapse columns was established, a physical simulation model of a multi-collapse column combination was built, and transmission and detection experiments were carried out. The results show that the morphology and locations of 1 fault and 4 collapse columns obtained by the FWI algorithm are consistent with the data from the actual numerical model. According to physical simulation results, the background velocity of the model after inversions and the velocity of 5 anomalies are consistent with the actual values. Field experiments were conducted in the actual underground coal mines. After verification, the average error ratios of the long axis, short axis and intersection angle of the 4 collapse columns were 0.19, 0.15 and 0.02. The detection findings and comparison results confirm the feasibility of precise detection of geological structures by transmissive seismic waves based on the FWI method.

An extended Gauss-Newton method for full-waveform inversion

Full-waveform inversion (FWI) is a large-scale nonlinear ill-posed problem for which computationally expensive Newton-type methods can become trapped in undesirable local minima, particularly when the initial model lacks a low-wavenumber component and the recorded data lack low-frequency content. A modification to the Gauss-Newton (GN) method is developed to address these issues. The standard GN system for multisource multireceiver FWI is reformulated into an equivalent matrix equation form, with the solution becoming a diagonal matrix rather than a vector as in the standard system. The search direction is transformed from a vector to a matrix by relaxing the diagonality constraint, effectively adding a degree of freedom to the subsurface offset axis. The relaxed system can be explicitly solved with only the inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions, which simplifies the inversion of the Hessian matrix. When used to solve the extended-source FWI objective function, the extended GN (EGN) method integrates the benefits of model and source extension. The EGN method effectively combines the computational effectiveness of the reduced FWI method with the robustness characteristics of extended formulations and offers a promising solution for addressing the challenges of FWI. It bridges the gap between these extended formulations and the reduced FWI method, enhancing inversion robustness while maintaining computational efficiency. The robustness and stability of the EGN algorithm for waveform inversion are demonstrated numerically.

Wasserstein distance-based full waveform inversion method for density reconstruction

Reliable density information is an essential factor in lithologic interpretation and reservoir evaluation. However, density reconstruction is hampered by a lack of long-wavelength information and the crosstalk between different parameters. The powerful nonlinearity in density inversion constrains the multi-parameter full waveform inversion application. This paper introduces the Wasserstein distance into the full waveform inversion for velocity and density based on the optimal transport theory. We construct an objective function with better convexity to avoid the cycle-skipping caused by the poor initial model. Furthermore, due to the sensitivity of the quadratic Wasserstein distance to low-frequency, we add the low-wavenumber component absent in the conventional density inversion. We acquire a long-wavelength density background using optimal transport inversion and then apply it as an initial model for least-squares inversion. Numerical examples show that our optimal transport inversion effectively builds a reasonable density background model for FWI. Density inversion can be improved by incorporating optimal transport and least-squares theory, leading to a reliable quantitative description for petroleum exploration and development.

Robust joint adaptive multiparameter waveform inversion with attenuation compensation in viscoacoustic media

Full-waveform inversion (FWI) has been proven as an effective method to estimate subsurface parameters by iteratively reducing the data residual between the predictions and the observations. Nevertheless, FWI is greatly dependent on the initial model, and a poor initial model will lead to an incorrect solution. Furthermore, owing to the anelasticity of the earth, seismic waves will attenuate during propagation, which results in an attenuated gradient and makes the convergence rate of FWI even worse in viscoacoustic medium. Commonly, multiparameter (e.g. velocity and Q) waveform inversion can be used to mitigate these problems. Benefiting from the theory of Q-compensated wavefield propagation, we develop a Q-compensated joint multiparameter waveform inversion method to weaken the nonlinearity of the FWI objective function, which enables it to cope with challenges related with attenuation-induced gradient energy loss and cycle skipping simultaneously. We refer to our Q-compensated joint multiparameter FWI scheme as QJMFWI. The main contributions of QJMFWI are as follows: (1) given the difficulty associated with the estimation of velocity and Q simultaneously in viscoacoustic media, QJMFWI provides a straightforward waveform inversion method for velocity and Q model construction, by which we can obtain velocity and Q information with improved accuracy and resolution; and (2) compared with conventional FWI methods, QJMFWI relaxes the requirement for a good initial velocity and Q model, which can avoid trapping into local minima. Numerical and field data examples demonstrate that QJMFWI is an effective method to invert for accurate subsurface parameters in viscoacoustic media.

Full-waveform inversion of pure quasi-P waves in titled transversely isotropic media based on different parameterizations

Full-waveform inversion (FWI) builds subsurface parameter models by minimizing the residuals between the modeled and observed data. Accounting for the effects of anisotropy is critical for high-resolution imaging of complex structures. We develop an acoustic anisotropic FWI method based on a pure quasi-P-wave (qP-wave) equation. The equation coefficients and their derivatives with respect to Thomsen’s anisotropy parameters ([Formula: see text] and [Formula: see text]) are estimated by least-squares optimization. We derive and analyze the radiation patterns for six parameter classes: the velocity along the symmetry axis [Formula: see text], [Formula: see text], and [Formula: see text]; the normal moveout velocity [Formula: see text], the anisotropy parameter [Formula: see text], and [Formula: see text]; the horizontal velocity [Formula: see text], [Formula: see text], and [Formula: see text]; [Formula: see text], [Formula: see text], and [Formula: see text]; [Formula: see text], [Formula: see text], and [Formula: see text]; and [Formula: see text], [Formula: see text], and [Formula: see text]. The parameterization [Formula: see text] has significant trade-off between [Formula: see text] and [Formula: see text] at the intermediate and wide scattering angles. The anisotropy parameters [Formula: see text] and [Formula: see text] are resolvable at the short scattering angles for the parameterizations [Formula: see text] and [Formula: see text], respectively. The parameter crosstalk for the parameterizations [Formula: see text] and [Formula: see text] is more serious than that for other types of parameterizations. We perform FWI of pure qP-waves in vertical transversely isotropic (VTI) and titled transversely isotropic (TTI) media. Inversion results on the overthrust VTI model and the modified BP TTI model show that the velocity, anisotropy parameters, and tilt angle can be individually reconstructed when other parameters are sufficiently accurate. The multiparameter FWI cannot obtain reliable tilt angles for each type of parameterization. The inversion with the parameterization [Formula: see text] produces [Formula: see text], [Formula: see text], and [Formula: see text] models with modest accuracy, whereas the parameterization [Formula: see text] helps to improve the accuracy of [Formula: see text] and [Formula: see text] models.

A Source Independent TV-regularized full waveform inversion method for peripheral imaging around a borehole

A Source Independent TV-regularized full waveform inversion method for peripheral imaging around a borehole

High-Precision Corrosion Detection via SH1 Guided Wave Based on Full Waveform Inversion.

Corrosion detection for industrial settings is crucial for safe and efficient operations. Due to its high imaging resolution, the guided-wave full-waveform inversion tomography technique has significant potential for corrosion detection of plate metals. Limited by the long wavelengths of A0 and S0 mode waves, this method exhibits inadequate detection resolution for the earlier shallow and small corrosion defects. Based on the relatively short wavelength characteristics of the SH1 mode wave, we propose a high-precision corrosion detection method via SH1 guided wave using the full waveform inversion algorithms. By conducting finite element simulations of ultrasonic-guided waves on aluminum plates with varying corrosion defects, a comparison was made to assess the detection precision across A0, S0, and SH1 modes. The comparison results showed that, whether for regular or irregular defects, the SH1 mode wave always exhibited higher imaging accuracy than the A0 and S0 mode waves for shallow and small-sized defects. The corresponding experiments were conducted on an aluminum plate with simple or complex defects. The results of the experiments reconfirmed that the full waveform inversion method using SH1 guided wave can effectively reconstruct the shape and size of small and shallow corrosion defects within aluminum plates.

Robust Elastic Full-Waveform Inversion Based on Normalized Cross-Correlation Source Wavelet Inversion

The elastic full-waveform inversion (EFWI) method efficiently utilizes the amplitude, phase, and travel time information present in multi-component seismic recordings to create detailed parameter models of subsurface structures. Within full-waveform inversion (FWI), accurate source wavelet estimation significantly impacts both the convergence and final result quality. The source wavelet, serving as the initial condition for the wave equation’s forward modeling algorithm, directly influences the matching degree between observed and synthetic data. This study introduces a novel method for estimating the source wavelet utilizing cross-correlation norm elastic waveform inversion (CNEWI) and outlines the EFWI algorithm flow based on this CNEWI source wavelet inversion. The CNEWI method estimates the source wavelet by employing normalized cross-correlation processing on near-offset direct waves, thereby reducing the susceptibility to strong amplitude interference such as bad traces and surface wave residuals. The proposed CNEWI method exhibits a superior computational efficiency compared to conventional L2-norm waveform inversion for source wavelet estimation. Numerical experiments, including in ideal scenarios, with seismic data with bad traces, and with multi-component data, validate the advantages of the proposed method in both source wavelet estimation and EFWI compared to the traditional inversion method.

Generative Adversarial Network Based Inversion of Cross-hole Radar Data

Cross-hole radar is a popular method to characterize subsurface structures, yet traditional data interpretation methods have limitations: tomography method has poor inversion accuracy, while the full waveform inversion method costs huge computing time. In order to solve the above problems, an automatic inversion algorithm based on generative adversarial networks (GAN) is developed in this paper to interpret the permittivity from cross-hole radar B-scan images. This algorithm uses a low-resolution GAN to extract the global features in the cross-hole radar data to reconstruct the dielectric constant distribution map with low resolution, and then adopts a high-resolution GAN to enhance the resolution of the inversion results. The algorithm is trained on 1000 pairs of cross-hole radar data obtained from the finite-difference time-domain (FDTD) method. Finally, 100 pairs of similar data which has never been shown in the network are used to verify the inversion performance of the algorithm. The results show that the inversion accuracy of is greater than 90%, and the structural similarity index measure (SSIM) of the reconstructed image reaches 0.9. In addition, the proposed method also has rapid computing speed.

Full wavefield inversion with multiples: Nonlinear Bayesian inverse multiple scattering theory beyond the Born approximation

Imaging earth’s structures is fundamental to the earth’s interior, yet how to reconstruct earth’s heterogeneities using full-waveform inversion of full wavefield data that includes primary reflections and multiples remains enigmatic. I develop a nonlinear Bayesian approach for full-waveform inversion method with multiple scattering. Instead of using single scattering Born approximation to formulate the sensitivity kernels, I develop multiple scattering sensitivity kernels using multiple scattering-based Green’s functions. It is based on the Lippmann-Schwinger integral and Marchenko methods, for which the Green’s functions are retrieved from reflection data by solving a Marchenko equation. To estimate the uncertainty of velocities, I apply a Bayesian framework to the inverse problem. Our results indicate that the method does not depend on the earth’s scattering potential and the full-waveform inversion method with multiple scattering is a good alternative approach to image the earth’s structures.

An efficient plug-and-play regularization method for full waveform inversion

Abstract Nonlinear inverse problems arise in various fields ranging from scientific computation to engineering technology. Inverse problems are intrinsically ill-posed, and effective regularization techniques are necessary. The core of a suitable regularization method is to introduce the prior information of the model via an explicit or implicit regularization function. Plug-and-play regularization is a flexible framework that integrates the most effective denoising priors into an iterative algorithm, and it has recently shown great potential in the solution of linear ill-posed problems. Unlike traditional regularization methods, plug-and-play regularization does not require an explicit regularization function to represent the prior information of the model. In this work, by using total variation, block-matching and three-dimensional filtering, and fast and flexible denoising convolutional neural network denoisers, we propose a novel iterative regularization algorithm based on the alternating direction method of multipliers method. The combination of total variation and block-matching three-dimensional filtering regularizers can take advantage of the sparsity and nonlocal similarity in the solution of inverse problems. When combined with traditional and novel regularizers, deep neural networks have been shown to be an effective regularization approach, which can achieve state-of-the-art performance. Finally, we apply the proposed algorithm to the full waveform inversion problem to show the effectiveness of our method. Numerical results demonstrate that the proposed algorithm outperforms existing inversion methods in terms of quantitative measures and visual perceptual quality.

Wide‐spectrum reconstruction of a velocity model based on the wave equation reflection inversion method and its application

AbstractMost seismic data used for conventional exploration lack far‐offset signals and effective low‐frequency components. This makes it difficult for conventional full waveform inversion methods to recover the low‐wavenumber components of mid‐ and deep‐layer models. To resolve the bottleneck of ray‐theoretical reflection traveltime tomography, wave equations, such as reflection inversion methods, are receiving increasing attention. We reconstructed a wide‐spectrum velocity model based on the multiscale wave‐equation reflection inversion method for model decomposition. First, using the dynamic image warping method to obtain reflection traveltime residuals, the wave‐equation reflection traveltime inversion was used to recover the low‐wavenumber components of the background model, which also solved the cycle‐skipping problem. Second, the medium‐wavenumber component of the model is then supplemented by the reflection waveform inversion, while the reflectivity model is obtained relying on the least‐squares reverse time migration method. Based on this method, the background and perturbation models are updated alternately and iteratively. At the same time, the stratum structural tensor information was extracted using the perturbed model image to construct a preconditioned operator for the stratum structural constraint, suppress unreasonably high‐wavenumber components in the generalized gradient and improve the geological consistency of the inversion results. Testing of the model using the Sigsbee2b model and seismic field data from the East China Sea showed that, compared with the conventional reflection traveltime tomography method based on prestack depth migration, the wave‐equation reflection inversion strategy with well‐matched information from traveltimes to waveforms significantly improved the accuracy of the middle and deep velocity modelling and enhanced the imaging quality of the whole body.

Inner product preconditioned trust-region methods for frequency-domain full waveform inversion

Full waveform inversion is a seismic imaging method which requires solving a large-scale minimization problem, typically through local optimization techniques. Most local optimization methods can basically be built up from two choices: the update direction and the strategy to control its length. In the context of full waveform inversion, this strategy is very often a line search. We here propose to use instead a trust-region method, in combination with non-standard inner products which act as preconditioners. More specifically, a line search and several trust-region variants of the steepest descent, the limited memory BFGS algorithm and the inexact Newton method are presented and compared. A strong emphasis is given to the inner product choice. For example, its link with preconditioning the update direction and its implication in the trust-region constraint are highlighted. A first numerical test is performed on a 2D synthetic model then a second configuration, containing two close reflectors, is studied. The latter configuration is known to be challenging because of multiple reflections. Based on these two case studies, the importance of an appropriate inner product choice is highlighted and the best trust-region method is selected and compared to the line search method. In particular we were able to demonstrate that using an appropriate inner product greatly improves the convergence of all the presented methods and that inexact Newton methods should be combined with trust-region methods to increase their convergence speed.

Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness

Full waveform inversion (FWI) infers the subsurface structure information from seismic waveform data by solving a non-convex optimization problem. Data-driven FWI has been increasingly studied with various neural network architectures to improve accuracy and computational efficiency. Nevertheless, the applicability of pre-trained neural networks is severely restricted by potential discrepancies between the source function used in the field survey and the one utilized during training. Here, we develop a Fourier-enhanced deep operator network (Fourier-DeepONet) for FWI with the generalization of seismic sources, including the frequencies and locations of sources. Specifically, we employ the Fourier neural operator as the decoder of DeepONet, and we utilize source parameters as one input of Fourier-DeepONet, facilitating the resolution of FWI with variable sources. To test Fourier-DeepONet, we develop three new and realistic FWI benchmark datasets (FWI-F, FWI-L, and FWI-FL) with varying source frequencies, locations, or both. Our experiments demonstrate that compared with existing data-driven FWI methods, Fourier-DeepONet obtains more accurate predictions of subsurface structures in a wide range of source parameters. Moreover, the proposed Fourier-DeepONet exhibits superior robustness when handling data with Gaussian noise or missing traces and sources with Gaussian noise, paving the way for more reliable and accurate subsurface imaging across diverse real conditions.

Full Waveform Inversion for NDT using ultrasonic linear arrays

Ultrasonic (UT) imaging is a widespread technique for nondestructive testing (NDT). The state-of-the-art UT image reconstruction algorithms are based on delay-and-sum (DAS) operations, which assume constant acoustic velocity across the tested objects and also neglects nonlinear effects such as diffractions and multiple reflections. These assumptions limit the reconstruction capabilities of DAS-based algorithms, especially for complex objects composed by several materials. In seismology, Full Waveform Inversion (FWI) methods have been used to obtain subsurface properties based on scattered and transmitted sound waves - which is a very similar problem to ultrasonic imaging in NDT - using the full wave information, showing promising results. In this paper, we present the application of FWI in NDT using simulated data representing acquisitions with a common linear array transducer. We review the theoretical formulation of FWI and discuss some difficulties that arise when it is applied to NDT. Image reconstruction is performed using both a state-of-the-art DAS algorithm, namely the Total Focusing Method, and FWI. The implementation targets a GPU platform, using the CUDA API. This leverages the highly parallelizable nature of the simulations and the FWI algorithm. The obtained results show that FWI can show more internal structures than TFM, with less information about the specimen. To foster further development, all source codes are provided.

Seismic velocity inversion transformer

Velocity model inversion is one of the most challenging tasks in seismic exploration, and an accurate velocity model is essential for high-resolution seismic imaging. Recently, velocity inversion methods based on deep learning (DL), particularly convolutional neural networks (CNNs), have attracted considerable attention from the seismic exploration community. These researchers aim to directly estimate the velocity model from raw seismograms using a well-trained model. Although CNN-based velocity inversion methods have demonstrated remarkable performance in terms of intelligence and automation, their inversion performance is often constrained by a limited long-range dependence. Specifically, when conducting a convolutional operation on raw seismic data using small kernels (i.e., 1 × 1, 3 × 3, 5 × 5, and 7 × 7), CNN-based methods extract only the local features and neglect the weak spatial correlation between different local features that reflect the information of the same interface. This correlation could assist CNN in providing an overview of the seismic data and promote inversion performance when using DL. Furthermore, the time-varying properties of seismic data pose a challenge to the weight sharing of CNNs. Here, we have developed a new DL framework based on a transformer, called the seismic velocity inversion transformer (SVIT), to address the problem of velocity inversion. SVIT uses a self-attention mechanism to capture the long-range dependence of seismic data, rather than stacking multiple convolutional layers as in CNNs. Thus, SVIT can provide more informative remote features for building velocity models. The validity and reliability of the proposed method are demonstrated through numerical experiments using synthetic models. Compared with the conventional full-waveform inversion method and an existing CNN-based velocity inversion method, our SVIT indicates greater consistency with the target in terms of the velocity value, subsurface structures, and geologic interfaces and is expected to provide a new DL-based solution to resolve inversion problems.

The back-and-forth method for the quadratic Wasserstein distance-based full-waveform inversion

The conventional least-squares misfit function compares synthetic data to observed data in a point-by-point style. The Wasserstein distance function, also called the optimal transport function, matches patterns. The kinematic information of seismograms is therefore efficiently extracted. This property makes it more convex than the conventional least-squares function. Computing the 1D Wasserstein function is fast. Processing the 2D or 3D seismic data volume trace by trace, however, loses the interreceiver coherency. The main difficulty of extending to the high-dimensional Wasserstein function is the heavy computation cost. This computational challenge can be alleviated by a back-and-forth method. After explaining the computation strategy and incorporating it into full-waveform inversion, we illustrate the superior performances of the high-dimensional Wasserstein function with a Camembert model and the Marmousi model. The superiority is also demonstrated with the Chevron 2014 blind test.

Improved seismic envelope full-waveform inversion

The envelope full-waveform-inversion method has the potential to greatly improve the estimation of long-wavelength components of subsurface seismic velocity, and simultaneously avoid or reduce inversion cycle skipping, by fitting the envelopes of observed and modeled seismic waveform data, rather than fitting the waveforms directly. However, some issues such as instability of the adjoint source and gradient term make it challenging to use in practice for many seismology applications. We examine the envelope inversion (EI) methods with different envelope function exponent p values from an adjoint source analysis perspective and reveal the adverse effects of the adjoint source weighting term on the data residuals, which degrade or mute important inversion gradient contributions, especially from reflected or scattered phases in the seismic data. We develop the theory and implementation for an improved EI (IEI) method by adding an upshift constant (zero-frequency direct current [DC] bias term) to the seismic waveform data before calculating the envelope functions, which can help solve the instability issue of the adjoint source of the EI with power value p = 1. Our approach is different than the common but simple addition of a “water-level” damping constant in the denominator of the data residual-weighting term, which can ignore important information contained in the data residuals. Our IEI method becomes a mixed envelope-waveform objective function, which introduces a new adjoint source weighting term that can preserve all of the information contained in the data residuals, by boosting the low-amplitude phases of the seismic waveform, while simultaneously preserving the correct envelope (energy shape) of the high-amplitude phases. We test the performance of all three methods in detail using the highly realistic SEAM4D model and data and find the advantages of our new IEI method compared with conventional EI and full-waveform inversion methods.