Full Waveform Inversion Research Articles

GAN-Based Inversion of Crosshole GPR Data to Characterize Subsurface Structures

The crosshole ground-penetrating radar (GPR) technique is widely used to characterize subsurface structures, yet the interpretation of crosshole GPR data involves solving non-linear and ill-posed inverse problems. In this work, we developed a generative adversarial network (GAN)-based inversion framework to translate crosshole GPR images to their corresponding 2D defect reconstruction images automatically. This approach uses fully connected layers to extract global features from crosshole GPR images and employs a series of cascaded U-Net structures to produce high-resolution defect reconstruction results. The feasibility of the proposed framework was demonstrated on a synthetic crosshole GPR dataset created with the finite-difference time-domain (FDTD) method and real-world data from a field experiment. Our inversion network obtained recognition accuracy of 91.36%, structural similarity index measure (SSIM) of 0.93, and RAscore of 91.77 on the test dataset. Furthermore, comparisons with ray-based tomography and full-waveform inversion (FWI) suggest that the proposed method provides a good balance between inversion accuracy and efficiency and has the best generalization when inverting actual measured crosshole GPR data.

Full-waveform inversion by model extension: Practical applications

Producing reliable acoustic subsurface velocity models still remains the main bottleneck of the oil and gas industry’s traditional imaging sequence. In complex geologic settings, the output of conventional ray-based or wave-equation-based tomographic methods may not be sufficiently accurate for full-waveform inversion (FWI) to converge to a geologically satisfactory earth model. We create a new method referred to as full-waveform inversion by model extension (FWIME) in which a wave-equation migration velocity analysis (WEMVA) technique is efficiently paired with a modified version of FWI. We find that our method is more powerful than applying WEMVA and FWI sequentially, and that it can converge to accurate solutions without the use of a good initial guess or low-frequency energy. We determine FWIME’s potential on five realistic and challenging numerical examples that simulate complex geologic scenarios often encountered in hydrocarbon exploration. We guide the reader step by step throughout the optimization process. We find that our method can simultaneously invert all wave types with the same simple mechanism and without the need for a user-intensive hyperparameter tuning process. In an open-source online repository, we provide our C++/compute unified device architecture (CUDA) numerical implementation accelerated with graphics processing units, encapsulated in a Python interface. All the numerical examples developed are accessible through Python notebooks and fully reproducible.

Full-waveform inversion by model extension: Theory, design, and optimization

We describe a new method, full-waveform inversion by model extension (FWIME), that recovers accurate acoustic subsurface velocity models from seismic data when conventional methods fail. We leverage the advantageous convergence properties of wave equation migration velocity analysis (WEMVA) with the accuracy and high-resolution nature of acoustic full-waveform inversion (FWI) by combining them into a robust mathematically consistent workflow with minimal need for user inputs. The novelty of FWIME resides in the design of a new cost function and a novel optimization strategy to combine the two techniques, making our approach more efficient and powerful than applying them sequentially. We observe that FWIME mitigates the need for accurate initial models and low-frequency long-offset data, which can be challenging to acquire. Our new objective function contains two components. First, we modify the forward mapping of the FWI problem by adding a data-correcting term computed with an extended demigration operator, whose goal is to ensure phase matching between predicted and observed data, even when the initial model is inaccurate. The second component, which is a modified WEMVA cost function, allows us to progressively remove the contributions of the data-correcting term throughout the inversion process. The coupling between the two components is handled by the variable projection method, which reduces the number of adjustable hyperparameters, thereby making our solution simple to use. We devise a model-space multiscale optimization scheme by reparameterizing the velocity model on spline grids to control the resolution of the model updates. We generate three cycle-skipped 2D synthetic data sets, each containing only one type of wave (transmitted, reflected, or refracted), and we analyze how FWIME successfully recovers accurate solutions with the same procedure for all three cases. In a second paper, we apply FWIME to challenging realistic examples where we simultaneously invert all wave modes.

Frequency-domain full-waveform inversion-based musculoskeletal ultrasound computed tomography.

Recently, full-waveform inversion (FWI) has become a promising tool for ultrasound computed tomography (USCT). However, as a computationally intensive technique, FWI suffers from computational burden, especially in conventional time-domain full-waveform inversion (TDFWI). On the contrary, frequency-domain full-waveform inversion (FDFWI) provides a relatively high computational efficiency as the propagation of discrete frequencies is much cheaper than full time-domain modeling. FDFWI has already been applied in soft tissue imaging, such as breast, but for the musculoskeletal model with high impedance contrast between hard and soft tissues, there is still a lack of an effective source estimation method. In this paper, a water-referenced data calibration method is proposed to address the source estimation challenge in the presence of bones, which achieves consistency between the measured and simulated data before the FDFWI procedure. To avoid the cycle-skipping local minimum effect and facilitate the algorithm convergence, a starting frequency criterion for musculoskeletal FDFWI is further proposed. The feasibility of the proposed method is demonstrated by numerical studies on retrieving the anatomies of the leg models and different musculoskeletal lesions. The study extends the advanced FDFWI method to the musculoskeletal system and provides an alternative solution for musculoskeletal USCT imaging with high computational efficiency.

Self-Attention Fully Convolutional DenseNets for Automatic Salt Segmentation.

3-D salt segmentation is important for many research topics spanning from exploration geophysics to structural geology. In seismic exploration, 3-D salt segmentation is directly related to the velocity modeling building that affects many processing steps, such as seismic migration and full waveform inversion. Manually picking the salt boundary becomes prohibitively time-consuming when the data size is too large. Here, we develop a highly generalized fully convolutional DenseNet for automatic salt segmentation. A squeeze-and-excitation network is used as a self-attention mechanism for guiding the proposed network to extract the most significant information related to the salt signals and discard the others. The proposed framework is a supervised technique and shows robust performance when applied to a new dataset using transfer learning and a small amount of training data. We test the robustness of the proposed framework on the Kaggle TGS salt segmentation dataset. To demonstrate the generalization ability of the framework, we further apply the trained model to an independent dataset synthesized from the 3-D SEAM model. We apply transfer learning to finely tune the trained model from the TGS dataset using only a small percentage of data from the 3-D SEAM dataset and obtain satisfactory results.

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

Spectral recomposition for optimizing starting points in Full-Waveform Inversion

By optimizing the wavelet fitting of the arrivals in a reflection event, it is possible to perform significant enhancements for Full-Waveform Inversion (FWI) predictions and its processing time. For this reason, approaches capable of reducing the number of iterations for FWI — without decreasing the quality of the prediction — are of interest. Once the initial guess for FWI is better estimated, it is possible to predict the velocity model within a determined accuracy with fewer iterations. We thus propose an approach which can perform this estimation, based on spectral recomposition of seismic data. We design an inversion scheme to reconstruct the seismic spectrum of wavelets of a reflection event, which subsequently allows estimating the position in time of each wavelet in a seismogram. After finding the position in time of each wavelet, we can guide the calculated wavelet to fit the corresponding observed signal, starting from a closer initial point. Our approach leads to quite accurate predictions of velocity models with fewer iterations.

Reweighted variational full-waveform inversions

Starting from an initial model and predefined priors, a variational full-waveform inversion (VFWI) seeks posterior distributions of model parameters via optimization using a Bayesian theorem. Thus, VFWI is useful in estimating uncertainties in full-waveform inversions (FWIs). However, the resolution of the inverted models from VFWIs is usually not as good as those from conventional FWIs. We decompose the loss function in a VFWI into two terms: the complexity cost and the data misfit. The data variance, which balances these two terms, is expected to decrease during an iterative optimization to gradually put higher weight on the data misfit. We develop and compare two reweighting schemes in VFWI that can produce high-resolution velocity models and the associated uncertainties. Inversions are performed efficiently in a deep-learning framework to take advantage of automatic differentiation. Tests using synthetic data and blind data indicate that reweighted VFWIs are less sensitive to initial models and can generate inversion results that have higher resolution than those from conventional FWIs.

Expectation-maximization framework-based image-domain least-squares migration

High-resolution and high-fidelity seismic inversion imaging is one of the core issues of lithologic reservoir exploration. As a linearization subproblem of full-waveform inversion, least-squares migration (LSM) can obtain inversion results with more reliable amplitude, better resolution, and fewer artifacts that can be solved in either the data domain or image domain. For data-domain LSM, the calculation of the data-domain iteration can be very computationally intensive. In practical situations, the convexity of the cost function of LSM could be poor, resulting in slow or even nonconvergence iteration, so the ideal inversion imaging results cannot be reached. The image-domain LSM (ID-LSM) follows the logic of image deblurring, which realizes the effects of LSM in another way. The key problem of ID-LSM is to calculate a reliable Hessian matrix that can be effectively replaced by a set of point-spread functions (PSFs). Affected by the unknown source wavelet, the accuracy of the background velocity, and the accuracy of the forward/migration operator, the calculation results of the Hessian/PSF cannot be accurate. We first illustrate the impact of incorrect Hessian/PSF on ID-LSM through numerical experiments. To eliminate the influence of the inaccuracy of PSF, we develop the ID-LSM problem as a high-dimensional image deconvolution problem. An alternate update strategy of PSF and imaging results is given under the framework of the expectation-maximization algorithm by additionally introducing prior information about PSF and the inversion result. Numerical examples of synthetic data and field data verify that our method significantly improves image quality.

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.

Estimación de propiedades eléctricas del subsuelo usando GPR e inversión de onda completa: un caso de estudio en Colombia

A method of Full Waveform Inversion on GPR data for the estimation of subsurface electrical properties such as relative permittivity and conductivity is proposed in this paper. The GPR radar antenna used for subsurface data acquisition is a B-scan acquisition and it operates at a center frequency of 400 MHz. B-scan acquisitions are a challenge in the subsurface parameter estimation process due to lack of illumination. In addition, B-scan acquisitions are more sensitive to the starting point in estimating subsurface parameters in comparison to multiple offset acquisitions. However, despite the challenges, this type of acquisition is used because it allows portability in areas of difficult access and quick data collection, reducing processing times and costs. In this work, Full Waveform Inversion with cost function constraints was evaluated to estimate subsurface relative permittivity and conductivity using B-scan acquisitions. The proposed methods were evaluated using data collected in a study area located in Mogotes, Santander, Colombia. From the results obtained, it can be concluded that the use of regularization in the inversion process gives smoother subsurface models, also preserving discontinuities. In addition, the incoherent noise in the data is reduced by Gaussian regularization, allowing a better interpretation of the study area.

Time- Versus Frequency-Domain Inverse Elastic Scattering: Theory and Experiment

This study formally adapts the time-domain linear sampling method (TLSM) for ultrasonic imaging of stationary and evolving fractures in safety-critical components. The TLSM indicator is then applied to the laboratory test data of [F. Pourahmadian and H. Yue, Inverse Problems, 37 (2021), 055012; F. Pourahmadian, Mech. Syst. Signal Process., 161 (2021), 108029] and the obtained reconstructions are compared to their frequency-domain counterparts. The results highlight the unique capability of the time-domain imaging functional for high-fidelity tracking of evolving damage, and its relative robustness to sparse and reduced-aperture data at moderate noise levels. A comparative analysis of the TLSM images against the multifrequency LSM maps of [F. Pourahmadian and H. Yue, Inverse Problems, 37 (2021), 055012] further reveals that thanks to the full-waveform inversion in time and space, the TLSM generates images of remarkably higher quality with the same dataset.

Efficient 1.5D full waveform inversion in the Laplace-Fourier domain

Accurate near-surface characterization and velocity model building are important for a number of geotechnical and geophysical applications. 3D full waveform inversion (FWI) can be used to generate a detailed velocity model, but must be provided with a good initial velocity model. A method for producing such an initial velocity model is explored in this paper. Assuming that the near-subsurface can be locally approximated by an effective flat-layered medium, a specialized form of acoustic FWI is proposed. The inversion is based on regularized least-squares in the Laplace-Fourier domain as a measure to mitigate the cycle-skipping problem. Forward modeling is carried out by solving a number of independent finite-difference problems in parallel, for a set of horizontal wavenumbers. The set is determined adaptively, using a pair of algorithms discussed in detail. A numerical implementation of the inverse Hankel transform, used to synthesize data in the frequency-offset domain, is also described. The forward modeling scheme is validated against analytic solutions of the Helmholtz equation, showing good agreement between the two. FWI is tested by inverting a synthetic dataset consisting of flat layers with embedded velocity anomalies. Important features are recovered after inversion at a small number of complex frequencies, providing a detailed initial model for successive 3D FWI.

Helmholtz-equation solution in nonsmooth media by a physics-informed neural network incorporating quadratic terms and a perfectly matching layer condition

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep-learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are nonsmooth and describe structurally complex media. Thus, we solve the acoustic and visco-acoustic scattered-field (Lippmann-Schwinger) wave equation in the frequency domain with PINN instead of the wave equation to remove the source singularity. We first illustrate that nonsmooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function to better couple the real and imaginary parts of the wavefield. Moreover, we design new neurons by replacing the classical affine function with a quadratic function in the argument of the activation function to better capture the nonsmooth features of the wavefields. We find that the PML condition and the quadratic functions improve the results including handling attenuation and discuss the reason for this improvement. We also illustrate that a network trained to predict a wavefield for a specific medium can be used as an initial model of the neural network for predicting other wavefields corresponding to PDE-coefficient alterations and improving the convergence speed accordingly. This pretraining strategy should find applications in iterative full-waveform inversion and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments is small.

An improved elastic wavefield separation method based on the Helmholtz decomposition

Elastic wave-mode separation plays an important role in elastic reverse time migration and elastic full-waveform inversion. It helps to remove crosstalk artifacts and improve imaging quality. A class of efficient methods for elastic wave-mode separation in isotropic elastic media is the Helmholtz decomposition technique. Although this kind of approach produces pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high especially for 3D large-scale problems. By making use of the relationships among the divergence, curl, gradient, and exterior derivative operations, we develop an improved elastic wave-mode separation based on the Helmholtz decomposition. We also need to solve a Poisson equation, but the Laplace operator operates on a scalar function rather than a vector function. Thus, for multidimensional (2D or 3D) problems, the Poisson equation only needs to be solved once for the vector P and S wavefields. This allows us to reduce the computational cost of the conventional Helmholtz decomposition method by a factor of two for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of three. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via the Fourier transform and use a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P and S waves with the same phases and amplitudes as the input-coupled wavefields but also significantly reduces the computational cost. Numerical tests indicate the efficiency of the proposed method and confirm the theoretical results.

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.

Finite-frequency sensitivity kernels and hierarchical traveltime and waveform inversion of direct and reflected waves from vertical seismic profile data

Wave-equation-based traveltime or waveform inversion updates the velocity model along finite-frequency sensitivity kernels and exhibits higher accuracy than ray-based tomography. Owing to special geometry, the kernels for vertical seismic profile (VSP) data are different from those for surface seismic data. We have used the Born approximation to compute the traveltime and waveform kernels for direct and reflected waves from VSP data. Based on the property of sensitivity kernels for different information and arrivals, we develop a hierarchical inversion scheme: joint wave-equation traveltime inversion (JWETI) of direct and reflected waves, joint waveform inversion (JWI) of direct and reflected waves, and full-waveform inversion (FWI) of the reflected wave. The traveltime kernels, the waveform kernels, and the migration isochrones are used to retrieve the long-, intermediate-, and short-wavelength components of the velocity model, respectively. Numerical tests on synthetic and real walkway VSP data reveal that the JWETI stage obtains a plausible velocity macromodel, which can be treated as a starting model for subsequent JWI and FWI stages. The JWI stage is a transition between JWETI and FWI and helps to improve the structures of the deep layers of the model. The FWI stage offers some high-wavenumber contents to the inverted model. For the VSP survey, the transmitted direct waves can be used to reduce the dependence of FWI on the initial model. However, the conventional reflection-based wave-equation traveltime inversion and FWI still suffer from cycle skipping when the initial model is far away from the real model. In contrast, the hierarchical inversion scheme can yield reasonable inversion results around the borehole, even when large velocity errors are present in the starting model.

Salt and subsalt structure recovery — An envelope-based waveform inversion with local angle domain illumination compensation and L-BFGS

Correct subsalt structure recovery is important for subsalt reservoir exploration. However, high-precision subsalt structure retrieval using full-waveform inversion (FWI) is difficult, which is caused by the shielding from the salt body and the depth of the target. To obtain correct subsalt structures, techniques that can effectively work with deep targets are desired. In addition, the correct recovery of the salt body is critical for the recovery of the subsalt structure. However, the salt body recovery itself is also challenging in FWI, especially when a reasonable starting model is not available. We consider the recovery of the salt structure and subsalt structure. A local angle domain illumination-compensated direct envelope inversion with the local resolution function is developed to provide a large-scale background salt structure with an enhanced and more balanced recovery of the salt body in the deep part. Then, FWI with angle domain illumination compensation is used to improve the recovery of the salt and subsalt structures. To further enhance the quality of the inversion, model updating with the L-BFGS method is embedded into the whole process. Numerical examples with the SEG/EAGE salt model and the Sigsbee model demonstrate that the proposed method could retrieve the salt structure as well as the subsalt structure effectively with high precision starting from a linear initial model.

Multi-objective particle swarm optimization for Rayleigh wave full waveform inversion

Conventional full waveform inversion (FWI) of Rayleigh wave updates the earth model iteratively by minimizing the difference between measured and synthetic data using single-objective function. However, FWI of Rayleigh wave is challenging due to its high nonlinearity and multimodality. Thus, single-objective inversion may lead to incorrect results, and multi-objective inversion is attractive to obtain global optimal solution. In this paper, we propose a multi-objective particle swarm optimization (MOPSO) scheme for FWI of Rayleigh wave. During the MOPSO inversion, the model is optimized iteratively by using multiple criteria simultaneously. We adopt four objective functions in our proposed MOPSO strategy to characterize the observed data: waveform, peak time, waveform envelope and phase velocity spectrum. The concept of Pareto optimal solution set is introduced in the MOPSO strategy to avoid the contradiction of multi-objective functions. Each non-dominated solution is recorded on the Pareto optimal set for model iteration. Both synthetic data and real field data demonstrate the effectiveness and practicability of the MOPSO strategy.