Traditional Full Waveform Inversion Research Articles

Gabor-wavelet-activation implicit neural learning for full waveform inversion

Seismic full waveform inversion (FWI) is an efficient imaging method for estimating subsurface physical parameters. However, when the initial models are inaccurate or seismic data lack low frequencies, most traditional discrete grid-based FWI algorithms often encounter local minimum problems. Additionally, the inversion performance and computational cost are closely related to spatial resolution. Reparameterizing velocity models using neural networks (NNs) effectively mitigates local minimum issues. However, most NN-FWI approaches perform well when overparameterized and are limited to fixed grid-like representations. In contrast, implicit neural learning (INL) enables the representation of models at any resolution using a multilayer perception (MLP) network that learns a continuous function from discrete coordinates. To enhance the capability of INL, we developed a general Gabor-wavelet-activation INL approach and applied it to FWI, referred to as wavelet implicit neural learning FWI (WinFWI), using only the vertical particle-velocity. The constructed MLP was lightweight, which reduced the computational overhead. Numerical experiments on a synthetic block model and Marmousi2 model demonstrate the robustness of the proposed method to challenges such as parameter crosstalk. This was further validated using the Chevron 2014 blind test data. All comparisons show that our method is more general and robust than traditional and INL-based FWI algorithms. Moreover, additional feature visualization and numerical analysis illustrate the potential advantages of WinFWI in balancing computational cost and inversion accuracy.

Ground-penetrating radar dual-parameter full waveform inversion based on finite domain method

Abstract Ground penetrating radar (GPR) is a crucial method for shallow geophysical exploration method. GPR full waveform inversion (FWI) utilizes all available information including dynamics and kinematics, and theoretically achieving the highest imaging accuracy. However, a significant challenge for FWI is its low computational efficiency, which fails to meet the real-time requirements in actual exploration. To address this issue, we propose a finite domain method (FDM) based on the attenuation characteristic of high frequency GPR electromagnetic waves. This approach limits the calculation to a finite domain, thereby mitigating the excessive computational burden. We determine the size of the finite domain from absorption attenuation, wavefront diffusion, and scattering loss. Numerical experiments demonstrate that compared with traditional FWI, FWI based on FDM reduces the calculation time by more than 5 times.

Real-time dual-parameter full-waveform inversion of GPR data based on robust deep learning

SUMMARY Ground penetrating radar (GPR) is becoming an increasingly important tool for understanding the shallow electrical structures of the Earth and planets due to its adaptability to harsh detection environments, efficient data acquisition and accurate detection results. GPR full-waveform can simultaneously constrain the permittivity and resistivity of the medium, providing more comprehensive geophysical information and reducing the non-uniqueness of inversion. However, given the highly non-linear inverse problem and the massive data resulted from high temporal and spatial samplings, traditional full-waveform inversion algorithms are prohibitively costly. Inspired by Google's vision semantic segmentation system, we develop a robust deep learning-guided network that integrates geology and geophysics knowledge to support the real-time translation of zero-offset GPR data into dual-parameter electrical structures. We test our proposed network using synthetic data, which demonstrates that the algorithm can provide an accurate dual-parameter electrical model from a GPR sounding in milliseconds on a common laptop PC, exhibiting high robustness and adaptability to noise interference and extreme values of model parameters. We also apply our network to field data gathered for pollutant investigation in the United States. The resulting dual-parameter structure provides a more comprehensive and realistic depiction of subsurface electrical properties and reveals the migration and ageing of pollutants. Our algorithm's real-time and accurate advantages are expected to further unleash the potential of GPR technology and enable it to play a more significant role in earth and planetary exploration.

Elastic full-waveform inversion using tools of neural networks

In this paper, we investigate the full-waveform inversion (FWI) for elastic wave equation as training a neural network. The forward modeling of the elastic wave equation in the time domain by the staggered-grid difference schemes can be reformulated as a process of a recurrent neural network (RNN). As a result, the FWI problem is equivalent to neural network training, and the parameter of RNN coincides with the model parameter of inversion. Furthermore, a variety of stochastic optimizers including Adgrad, RMSprop, Adam, Nadam and Admax in neural networks can be applied in the training process. The gradient of the objective function with the model parameters is computed by the technique of automatic differentiation instead of the adjoint-state method in the traditional FWI. A new objective function of FWI is also proposed. Compared to the traditional FWI methods, the developed FWI using tools of neural networks has a relatively good robustness. Numerical computations and comparisons with Marmousi model for two and three parameters simultaneous inversion are completed. The results show that the algorithms except Adgrad can yield good inversion results. The FWI framework developed in this paper has potential applications for other complex partial equations.

Inexact augmented Lagrangian method-based full-waveform inversion with randomized singular value decomposition

Abstract The main advantage of full-waveform inversion (FWI) is the ability to obtain useful subsurface structure information, such as velocity and density, from complex seismic data. We have developed a novel inversion algorithm to improve the capability of FWI to achieve high-resolution imaging, even under complex conditions caused by random noise contamination, initial model dependence, or the selection of parameters to be estimated. Our algorithm considers an effective image processing and dimension reduction tool, randomized singular value decomposition-weighted truncated nuclear norm regularization, for embedding FWI to achieve high-resolution imaging results. This algorithm obtains a truncated matrix approximating the original matrix by reducing the rank of the velocity increment matrix, thus achieving the truncation of noisy data, with the truncation range controlled by weighted truncated nuclear norm regularization. Subsequently, we employ an inexact augmented Lagrangian method algorithm in the optimization to compress the solution space range, thus relaxing the dependence of FWI and randomized singular value decomposition-weighted truncated nuclear norm regularization on the initial model and accelerating the convergence rate of the objective function. We tested on one set of synthetic data, and the results show that, compared with traditional FWI, our method can more effectively suppress the impact of random noise, thus obtaining higher resolution and more accurate subsurface model information. This work indicates that the combination of randomized singular value decomposition-weighted truncated nuclear norm regularization and FWI is an effective imaging strategy that can help to solve the challenges faced by traditional FWI.

A fast wavefield reconstruction inversion solution in the frequency domain

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

Progressive multitask learning for high-resolution prediction of reservoir elastic parameters

Reservoir elastic parameters play an important role in resource exploration; however, the band-limited characteristics of seismic data and the ill-posed nature of seismic inversion significantly affect inversion accuracy. To alleviate this problem, a high-resolution prediction method for reservoir elastic parameters based on the progressive multitask learning network (PMLN) is proposed. Our network consists of three parts: network 1 for low-frequency extension (LFE), network 2 for reservoir parameter inversion, and network 3 for image superresolution (SR). Taking the seismic frequency band as the link, network 1 is first used to predict the low-frequency information of seismic data. Then, the nonlinear mapping relationship between the high-pass-filtered seismic data (and its envelope) and full-frequency seismic data is established. Second, network 2 directly predicts the reservoir elastic parameters using the seismic data after LFE. Finally, the SR of the inversion results is achieved from the image perspective based on network 3. The three networks have a progressive relationship and can share network features, which is beneficial for improving computing efficiency. As the features extracted by the network represent different contributions to the prediction target, a channel attention mechanism is introduced. Furthermore, the loss function of network 2 is improved using dip constraints to obtain high-precision reservoir parameters. Synthetic and field data analyses find that all three networks are competent for their respective tasks, and the PMLN can obtain high-resolution prediction results of reservoir elastic parameters. Compared with traditional full-waveform inversion, the PMLN effectively improves prediction accuracy. Therefore, the PMLN is expected to become a powerful tool for predicting the elastic parameters of reservoirs.

Joint FWI of Active Source Data and Passive Virtual Source Data Reconstructed Using an Improved Multidimensional Deconvolution

Traditional full waveform inversion (FWI) highly depends on sufficient low-frequency data or a good initial model. Passive seismic data contain rich low-frequency components, and passive seismic FWI using virtual source data by seismic interferometry (SI) is a promising method. However, the distribution of passive sources in the subsurface is always inhomogeneous, which will lead to artifacts in the reconstruction results by SI using cross-correlation (CC). SI by multidimensional deconvolution (MDD) can counteract the source inhomogeneity but requires the separation of the reference wavefields, which is difficult to achieve for noise source data. To mitigate this problem, we propose an improved SI method by linear Radon transform based multidimensional deconvolution (LRTMDD). The interferometric point-spread function can be extracted more accurately and efficiently in the linear Radon domain, thus improving the reconstruction results. The passive virtual source full waveform inversion (PVSFWI) based on LRTMDD is further proposed, which can effectively use the low-frequency information in the virtual source data to invert the macroscopic velocity structures even in the case of inhomogeneous source distributions, and without the need to estimate the virtual source wavelets. A joint multi-source FWI strategy is proposed to solve the problem of missing low-frequency data suffered by active source FWI. Numerical experiments on the Marmousi model and the SEG/EAGE overthrust model show that the proposed methods can fully combine the respective advantages of multisource seismic data to stably achieve high-accuracy velocity models in the case of inhomogeneous passive source distributions and the lack of low-frequency data in active seismic data.

Wavefield Reconstruction Inversion Based on the Multi-Scale Cumulative Frequency Strategy for Ground-Penetrating Radar Data: Application to Urban Underground Pipeline

High-precision detection of the underground pipelines is an indispensable part of the development and construction of cities. At present, the inversion technology for ground-penetrating radar (GPR) data is an effective means of realizing shallow-underground-space visualization in the field of geophysical exploration. However, the traditional full-waveform inversion (FWI) method usually faces the problems of strong nonlinearity of the objective function, high dependence on the initial model, and huge calculation cost. For improving the accuracy and efficiency of GPR data inversion, a wavefield reconstruction inversion (WRI) strategy is used for GPR data imaging to reduce the nonlinearity of the inversion problem and the dependence on the initial model. Then, the frequency weighting strategy and the multi-scale method are adopted to avoid the high-frequency component data dominating the optimization process and enhance the stability of inversion. In this paper, two numerical experiments of pipeline models with different materials and spacing or buried depths verified that the proposed method can effectively reconstruct the subsurface pipelines, and further performance of our algorithm on the field data verified the reliability of high-precision imaging of urban underground pipelines, which shows great potential of application in the field of high-precision detection of the urban underground pipelines.

Time-Domain Wavefield Reconstruction Inversion Solutions in the Weighted Full Waveform Inversion Form

Full Waveform Inversion (FWI) methods evaluate subsurface media properties by minimizing the misfit between synthetic and observed data. However, its derivation process omits measurement errors and physical assumptions in modeling, resulting in many problems in practical utilizations. Wavefield Reconstruction Inversion (WRI) can handle non-physical data by combining theoretical and measurement errors but is computationally expensive in large-scale or 3D cases. An alternative formula for WRI in the form of traditional FWI (WRI in the FWI form, FWRI) was developed. The new form includes a medium-dependent weight function or a data domain Hessian matrix. Based on the data domain Hessian matrix, a feasible accurate time-domain solution is provided for 2D and 3D FWRI. Specifically, the point spread function (PSF) method and an-isotropic TV regularization were used to calculate the weighted residuals. Furthermore, an approximate solution based on the forward operator is provided, considering the efficient application in 3D. Numerical tests using the homogeneous model show that accurately computed residuals mitigate the effects of "cycle-skipping" and non-physical data. Sensitivity kernel analysis demonstrates the low wavenumber update mechanism of FWRI and verifies the robustness of the data domain Hessian to "cycle-skipping". The robustness of the proposed algorithm is demonstrated using a 2D Marmousi model with partially elastic perturbations and a visco-acoustic Red Sea model. Furthermore, the 3D Overthrust model is solved with an approximate solution to demonstrate the feasibility of the proposed method in 3D applications.

Deep learning for velocity model building with common-image gather volumes

SUMMARY Subsurface velocity model building is a crucial step for seismic imaging. It is a challenging problem for conventional methods such as full-waveform inversion (FWI) and wave equation migration velocity analysis (WEMVA), due to the highly nonlinear relationship between subsurface velocity values and seismic responses. In addition, traditional FWI and WEMVA methods are often computationally expensive. In this paper, we propose to apply a deep learning technique to construct subsurface velocity models automatically from common-image gather (CIG) volumes. In our method, pairs of synthetic velocity models and CIG volumes are generated to train a convolutional neural network. Our proposed network achieves promising results on different synthetic data sets. The training performance of several commonly used loss functions is also studied.

A gradient-based Markov chain Monte Carlo method for full-waveform inversion and uncertainty analysis

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the diagonal approximate Hessian of the local log posterior. Our algorithm is called the gradient-based Markov chain Monte Carlo (GMCMC) method. The GMCMC FWI method can quantify inversion uncertainties with estimated posterior distribution given sufficiently long Markov chains. By directly sampling the posterior distribution, we obtain a global view of the model space. Theoretically speaking, statistical assessments do not depend on starting models. Our method is applied to the 2D Marmousi model with the frequency-domain FWI setting. Numerical results suggest that our method can be readily applied to 2D cases with affordable computational efforts.

3D aquifer characterization of the Hermalle-sous-Argenteau test site using crosshole ground-penetrating radar amplitude analysis and full-waveform inversion

To improve the understanding of flow and transport processes in the critical zone, high-resolution and accurate estimation of the small-scale heterogeneity is essential. Preferential flow paths related to high-porosity layers and clay lenses in gravel aquifers greatly affect flow and transport processes in the subsurface, and their high electrical contrast to their surrounding matrix and limited extent can act as low-velocity electromagnetic waveguides. In the past decade, time-domain full-waveform inversion (FWI) of crosshole ground-penetrating radar (GPR) data has shown to provide 2D decimeter-scale resolution images of relative permittivity and electrical conductivity of the subsurface, which can be related to porosity and soil texture. Most studies using crosshole GPR FWI resolved high-porosity zones that were identified by an amplitude analysis approach. But clay lenses or zones with higher electrical conductivity that act as low-velocity waveguides are hard to distinguish in the measured data and amplitude analysis because of the absence of characteristic wave-propagation features. We have investigated a set of nine crosshole GPR data sets from a test site in Hermalle-sous-Argenteau near the Meuse River in Belgium to characterize the aquifer within a decimeter-scale resolution and to improve the understanding of a previously performed heat tracer experiment. Thereby, we extend the amplitude analysis to identify two different types of low-velocity waveguides either caused by an increased porosity or a higher electrical conductivity (and higher porosity). Combining the GPR amplitude analysis for low-velocity waveguide zones with the standard FWI results provided information on waveguide zones, which modified the starting models and further improved the FWI results. Moreover, an updated effective source wavelet is estimated based on the updated permittivity starting models. In comparison with the traditional FWI results, the updated FWI results present smaller gradient of the medium properties and smaller root-mean-squared error values in the final inversion results. The nine crosshole sections are used to generate a 3D image of the aquifer and allowed a detailed analysis of the porosity distribution along the different sections. Consistent structures of the permittivity and electrical conductivity show the robustness of the updated FWI results. The aquifer structures obtained by the FWI results agree with those results of the heat tracer experiment.

Ultrasound Computed Tomography of Knee Joint

Ultrasound computed tomography (UCT) is considered to have great potential for the early diagnosis of knee joint injuries. Combining prior knowledge and sound speeds of tissues, UCT can diagnose a variety of symptoms and the extent of tissue lesion. However, no existing studies can reconstruct the sound speed distributions of knee joints in UCT. Therefore, in this study, sound speed distributions of knee joints are reconstructed to provide an imaging basis for the formulation of a treatment plan. In addition, Full waveform inversion (FWI) methods are utilized to estimate highresolution images. However, for media with large variations in sound speeds, traditional FWI methods may lead to a cycle-skipping phenomenon and incorrect convergence direction. Hence, this study proposes a multi-centerfrequency source based FWI method to overcome the above limitations. A penalized least-squares optimization problem is constructed to obtain a numerical solution of the sound speed distribution. Computer simulations are conducted to prove the effectiveness and robustness of the proposed method. Furthermore, the reasons for selecting the center frequencies of sources are analyzed. Two numerical knee joint phantoms are used to evaluate the performance. The results suggest that the biases of the reconstructed images are less than 1% under a 5% noise condition.

Reflection intensity waveform inversion

Traditional iteration-based full-waveform inversion (FWI) methods encounter serious challenges if the initial velocity model is far from the true model or if the observed data are lacking low-frequency content. As such, the optimization algorithm may be trapped in local minima and fail to go to a global optimal model. In addition, the traditional FWI method requires long-offset data to update the deep structure of a velocity model with diving waves. To overcome the disadvantages of traditional FWI under these circumstances, we have developed a reflection intensity waveform inversion method. This method aims to minimize the seismic intensity differences between the modeled reflection data and field data. Our method is less dependent on the starting model, and long-offset data are no longer required. The wave intensity, proportional to the square of the original data amplitude, can have a low-frequency band and a higher frequency band, even for waveforms without initial low-frequency content. Our multiscale intensity inversion starts from the low-frequency information in the intensity data, and it can largely avoid the cycle-skipping problem. Synthetic and field data examples demonstrate that our method is able to overcome cycle skipping in handling data with no low-frequency information.

Controlled-order multiple waveform inversion

Traditional full-waveform inversion (FWI) seeks to find the best model by minimizing an objective function defined as the difference between the model-predicted and observed data in amplitude and phase. In principle, FWI should fit all wave types including direct waves, diving waves, primaries, and multiples. However, when an initial model is far from the true model, FWI will encounter difficulties in matching multiples. Physically, multiples may contain more subsurface information compared to primary and diving waves. Multiples cover a wide range of reflection angles during wave propagation and offer the advantage of imaging the shadow zones that cannot be reached or are poorly illuminated by primary reflections. We have developed a new method of waveform inversion using multiples. We first separate the multiples into different orders. The objective function we seek to minimize consists of the data difference between the modeled data using a lower order multiple as the source and the higher order multiple as data. This method is called controlled-order multiple waveform inversion (CMWI). Our numerical examples determined that the CMWI is a promising method to improve velocity updates.

Progressive matching optimisation method for FWI

AbstractTo eliminate the dependency on a good initial model of the traditional full waveform inversion (FWI) method, we propose an optimisation method combining a derivative free optimisation method of modified particle swarm with gradient descent search. We worked with the acoustic wave approximation, in two dimensions, with the synthetic Marmousi velocity model as the test case. We were able to obtain a high-precision inversion of this model, comparable to traditional FWI methods, with the distinct advantage of not using an initial model close to the global optimal, as would usually be required. For this result, we used a progressive inversion scheme by consecutive layers, and a modified particle swarm optimisation algorithm where we introduced the gradient of the misfit function as a local search guide, and other regularization terms.

Subsurface velocity inversion from deep learning-based data assimilation

Data assimilation has widespread applications in meteorology and oceanography. This paper applies data assimilation for seismic exploration. Comparing with full waveform inversion, the distrust of the prior velocity information is preserved, to avoid the local minima caused by its inaccuracy. Prior velocity information is scarce in seismic exploration, but the introduction of deep learning methods makes it possible, to correct the major defects in the application of data assimilation in seismic velocity inversion. Finally, we design a simple salt body model, and we introduce deep learning to obtain prior velocity information and drive data assimilation for high-precision inversion of subsurface velocity. It is compared with the traditional full waveform inversion, to prove the effectiveness and advantages of this new process.

Improving reconstruction of tunnel lining defects from ground-penetrating radar profiles by multi-scale inversion and bi-parametric full-waveform inversion

Complex irregular defects of tunnel linings under complicated geological conditions cannot be accurately reconstructed with the traditional full-waveform inversion (FWI) method due to their irregular geometrical characteristics and complex dielectric properties. Because of extensive inversion calculations and high memory requirements, the traditional FWI method is very sensitive to the initial model and plunge into a local minimum or cycle skipping. To solve this problem, a novel ground-penetrating radar (GPR) FWI method involving two parameters (i.e. permittivity and conductivity) is proposed for improving reconstruction accuracy of lining defects using the total variation (TV) regularization. First, the Delaunay unstructured triangular mesh in finite element time-domain (FETD) method is employed to perform GPR forward modeling, and then the total-variation model constraint and multi-scale inversion strategy are implemented during execution of the conjugate gradient (CG) algorithm, which facilitates the quick search for the global optimal minimum value, thus guaranteeing the avoidance of the ill-posed problem during the inversion process. Accordingly, the detailed features of lining defects can be characterized and reconstructed even for those complicated geological conditions, more specifically, the results show that, with fewer iterations (up to 59 times less), the proposed method present a lower reconstruction error both on permittivity (up to 12.05% lower) and conductivity (up to 7.35% lower). From a comparison of the inversion results and the model, it can be concluded that the proposed FWI algorithm can effectively eliminate the non-physical oscillation and artifacts in the image reconstruction, which may significantly improving the accuracy of defects interpretation and assessing the severity of complex defects.

Application of ground penetrating radar to detect tunnel lining defects based on improved full waveform inversion and reverse time migration

ABSTRACTGround penetrating radar is a popular approach to detect defects in tunnel lining. However, the interpretation is usually based on the original image, which is very different from the real shape of the lining defects. Full waveform inversion and reverse time migration are helpful to solve this problem. Full waveform inversion can invert the relative permittivity distribution and reverse time migration can migrate reflection events to their proper locations. Traditional full waveform inversion method is only applicable to cross‐hole ground penetrating radar data or surface multi‐offset ground penetrating radar data. We propose an improved full waveform inversion method which offers satisfactory inversion result for surface common‐offset radar. The forward modelled waveform and the objective function curve show that our new full waveform inversion method is much more accurate than traditional full waveform inversion for common‐offset radar. Traditional reverse time migration has weaker amplitude with increasing depth; we use an energy matrix to improve the imaging effect. Moreover, our reverse time migration is based on the relative permittivity distribution obtained from full waveform inversion, which provides more accurate imaging result. Through several numerical and engineering examples, we discuss the application of both methods in tunnel lining inspection. The results show that for tunnel lining without rebars, the combined methods can give satisfactory imaging results. But the image quality deteriorates rapidly when dealing with rebars.