Non-local structured adaptive dictionary learning method for seismic waveform inversion

SUMMARY Full waveform inversion (FWI) is a high-precision subsurface imaging technique that inverts subsurface parameter models by minimizing the discrepancy between observed and synthetic seismic data. However, complicated wave propagation mechanisms, non-convexity of the loss function and limited seismic acquisition system necessitate the incorporation of sufficient prior and physical constraints to alleviate the ill-posedness and cycle-skipping. Although the implicit FWI (IFWI) can encode implicit spectral bias (i.e. inverting model parameters from low frequencies to high frequencies) to reduce the dependence on an accurate initial model, its limited high-frequency inversion capability results in thousands of iterations for the final results. In this paper, we indicate that the frequency hyperparameters of the sine activation function in IFWI modulate the spectral bias, making a trade-off between inversion accuracy and stability, that is, lower frequencies yield robust FWI but lower accuracy, while higher frequencies achieve higher accuracy on the premise of an accurate initial model. To improve both the stability and accuracy of IFWI, we propose a novel implicit FWI method with an adaptive Fourier reparametrization strategy (termed FR-IFWI), which explicitly encodes multifrequency information by reparametrizing the network weights using a learnable coefficient matrix and fixed Fourier frequency bases. The role of learnable matrices in neural networks can evolve from determining frequencies in IFWI to actively selecting frequencies from fixed frequency bases through FR-IFWI, which alleviates the dependence on activation function frequency and obtains more robust and accurate inversion results. Extensive numerical experiments on the modified Marmousi, 2-D SEG/EAGE Salt and Overthrust models confirm that FR-IFWI successfully achieves superior inversion efficiency and accuracy compared with conventional FWI and IFWI methods.

In a probabilistic formulation, the solution to an inverse problem can be expressed an a posteriori probability density function (pdf) that combines the independent states of information provided by data and a priori information. Here, we define an a posteriori probability density function that defines the solution to a tomographic full waveform inverse problem, which provides a means of obtaining an uncertainty estimate. Unfortunately, no explicit formulation of the solution to this problem can be defined. Therefore, the a posteriori probability density function has to be sampled. The full waveform inverse problem is known to be computationally very hard and is, traditionally, considered out of reach for Monte Carlo sampling strategies. We show that by means of informative a priori information this problem become tractable for a sampling strategy anyways. We outline the theoretical framework for a full waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling, which allows for arbitrary complex a priori information to be incorporated. At the same time we show how temporally correlated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground penetrating radar full waveform data using multiplepoint based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full waveform inverse problem.

Full-waveform inversion (FWI) is a highly nonlinear and ill-posed inverse problem, which needs proper regularization to produce reliable results. Recently, sparsity and overcompleteness have been successfully applied to seismic data processing. In this study, we propose a novel adaptive sparsity-promoting regularization for FWI which combines the L-BFGS algorithm with an adaptive overcomplete dictionary learning method. The dictionary is learned from many small imaging patches taken from the optimal velocity model that is obtained by previous L-BFGS iterations. Our dictionary learning method tries to exploit the 2D geometric structure of the training patches in a more direct way and is simple to implement. We test our proposed method on a smoothed Marmousi model, a BG Compass model, and a SEG/EAGE salt model. Since total variation (TV) regularization plays an important role in FWI, the inversion results using the TV regularization method are also presented for comparison purposes. From these experiments, we conclude that the proposed method can achieve better performance than the FWI with the TV regularization method.

SummaryWith the development of Internet of Things (IoT)‐related technologies and artificial intelligence (AI) technologies, various IoT services are becoming more intelligent, and their use range is increasing and diversifying. IoT hardware and IoT software must support AI‐related functions to provide an intelligent IoT service. In general, IoT devices powered by batteries have limited computing performance when compared to general computing environments. Therefore, it is essential to provide AI‐related functions at low power in IoT devices to implement and offer various intelligent services. Neuromorphic computing devices or neuromorphic computing architectures can operate with low power energy consumption. If applied to IoT devices, AI‐related functions can be implemented in a resource‐constrained IoT device environment. The proposed neuromorphic architecture abstraction (NAA) model dynamically selects the proper neuromorphic architecture by comparing the parameter size of a given SNN model. It also considers the specifications and error probability of the available neuromorphic architecture. We also implement the proposed model in a real IoT computing environment and show that the proposed NAA model and dynamic selection scheme can reduce the execution time for training and inferencing. It reduces the training and inferencing time of a given model compared with the method of randomly specifying the neuromorphic architecture.

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.

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.

<p>Seismic full waveform inversion (FWI) produces high resolution images of the subsurface directly from seismic waveforms, and has been applied at global, regional and industrial spatial scales. FWI is traditionally solved using optimization methods, by iteratively updating a model of the subsurface so as to minimize the misfit between observed waveforms and those predicted by linearised (approximate) physics. Due to the nonlinearity of the physical relation between model parameters and seismic waveforms, a good starting model (derived from other methods) and hence strong prior information is required in order that the linearised physical relationships are reasonable in the vicinity of the true solution. Such linearised methods cannot provide accurate estimates of uncertainties, which are required if we are to understand and interpret the results appropriately.</p><p>To estimate uncertainties more accurately, nonlinear Bayesian methods have been deployed to solve the FWI problem. Monte Carlo sampling is one such algorithm but it is computationally expensive, and all Markov chain Monte Carlo-based methods are difficult to parallelise fully. Variational inference provides an efficient, fully parallelisable alternative methodology. This is a class of methods that optimize an approximation to a probability distribution describing post-inversion parameter uncertainties. Both Monte Carlo and variational full waveform inversion (VFWI) have been applied previously to solve FWI problems, but only with strong prior information about the velocity structure to limit the space of possible models. Unfortunately such strong information is almost never available in practice. In addition, VFWI has only been applied to wavefield transmission problems in which seismic data are recorded on a receiver array that lies above the structure to be imaged, given known, double-couple (earthquake-like) sources located underneath the same structure. In practice, knowledge of such sources is never definitive, and usually depends circularly on the unknown structure itself. In this study, we present the first application of VFWI to seismic reflection data acquired from known (deliberately fired) near-surface sources. We also apply variational inference (specifically, Stein variational gradient descent) to solve FWI problems with realistic prior information. We perform multiple inversions using data from different frequency ranges, and show that VFWI produces high resolution mean and uncertainty models using both low and high frequency data, and given only weak prior information. This is usually impossible using traditional linearised methods. We conclude that VFWI may be a useful method to produce high resolution images and reliable uncertainties, at least in smaller FWI problems.</p>

We present a new target-enclosing full waveform inversion (FWI) method based on a interferometric objective function, where the inversion is driven by the misfit between the wavefields reconstructed in a subdomain of interest via the convolution and correlation representation formulas. The method is fully local in the sense that it does not depend on the reconstruction of the physical properties outside the local domain, and only requires a kinematic velocity model estimate for redatuming. We compare the proposed method to another fully local FWI method based on the convolution representation formula and to full-model surface-data FWI. We demonstrate the potential of the proposed interferometric full waveform inversion method to achieve higher resolution images at a comparable or lower cost than the other methods.

The quantitative reconstruction of subsurface earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field, but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and it defines the full reciprocity-gap waveform inversion (FRgWI) method. This misfit functional provides additional features compared to the more traditional least-squares approaches, in particular, in that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be chosen arbitrarily. Based on 3D experiments, FRgWI is shown to behave better than full-waveform inversion in the same context. It allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. However, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance, with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to crosstalk than least-squares shot stacking.

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

Ground penetrating radar (GPR) is one of the most commonly used technologies for non-destructive testing (NDT). With the development of GPR signal processing methodologies, researchers are becoming more concerned not just with the detection of the target itself, but also with their physical properties and main features. In general, full waveform inversion algorithm is required to achieve this aim. But, full waveform inversion problem is a costive approach which need a huge computation. Thus, the ultimate goal of this study is to explore an effective strategy for estimating the relative permittivity of the target using polarimetric GPR data. We have investigated the relation between relative permittivity and polarimetric alpha angle based on the data collected by dualpolarization antennas GPR system. Laboratory experiments that measures different moisture sand targets (simulating for different relative permittivity target) in tree trunk holes have been carried out, taken as analog models for the physiological process representing decays in trees. After signal processing, the rough results that alpha angle versus with relative permittivity were obtained. The results show that for a dry sand the polarimetric alpha angle is small and the polarimetric alpha angle increases with increasing water content.

In this paper we proposed an accurate and efficient numerical algorithm for solving the acoustic wave equation in three-dimensional heterogeneous media. Numerical solution of the wave equation has been used in various science and engineering applications, such as the seismic full waveform inversion (FWI) problem. FWI is a computationally intensive procedure, in which the acoustic wave equation is numerically solved (forward modelling) repeatedly during the iterative process. Therefore, efficiency and accuracy of the numerical method for solving the acoustic wave equation is critical in the success of seismic full waveform inversion. The new method is obtained by combining the Pade approximation and a novel algebraic manipulation with the Alternative Directional Implicit (ADI) method. Numerical experiments have shown that the new method is accurate, efficient and stable.

Full waveform inversion (FWI) is an established precise velocity estimation tool for seismic exploration. Machine learning-based FWI could plausibly circumvent the long-standing cycle-skipping problem of traditional model-driven methods. The physics-guided self-supervised FWI is appealing in that it avoids having to make tedious efforts in terms of label generation for supervised methods. One way is to employ an inversion network to convert the seismic shot gathers into a velocity model. The objective function is to minimize the difference between the recorded seismic data and the synthetic data by solving the wave equation using the inverted velocity model. To further improve the efficiency, we propose a two-stage training strategy for the self-supervised learning FWI. The first stage is to pretrain the inversion network using a simultaneous source for a large-scale velocity model with high efficiency. The second stage is switched to modeling the separate shot gathers for an accurate measurement of the seismic data to invert the velocity model details. The inversion network is a partial convolution attention modified UNet (PCAMUNet), which combines local feature extraction with global information integration to achieve high-resolution velocity model estimation from seismic shot gathers. The time-domain 2D acoustic wave equation serves as the physical constraint in this self-supervised framework. Different loss functions are used for the two stages, that is, the waveform loss with time weighting for the first stage (simultaneous source) and the hybrid waveform with time weighting and logarithmic envelope loss for the second stage (separate source). Comparative experiments demonstrate that the proposed approach improves both inversion accuracy and efficiency on the Marmousi2 model, Overthrust model, and BP model tests. Moreover, the method exhibits excellent noise resistance and stability when low-frequency data component is missing.

In recent years the elastic full waveform inversion (FWI) was successfully applied to synthetic and field data to compute high resolution velocity models. While seismic velocities are derived from recorded phase information, density models can be estimated from the amplitudes. However, due to the complexity of the inverse problem a long wavelength initial model is required for a good reconstruction of the density. The inclusion of gravity data into the FWI concept can solve this problem. In this study a two-step hierarchic joint inversion of seismic waveforms and gravity data is tested using the Marmousi-II model. In step 1 FWI is performed for all elastic parameters. Gaussian filtered velocity models of the true model and a constant halfspace density model (CDH) are used as initial models. While the velocities can be reconstructed well, the density shows large deviations from the true model. In step 2 joint inversion is applied to optimize only the density model, while the velocity inversion results of the first step and the CDH are used as initial models. The results of this combined approach show a significant improvement of the density model compared to the results of a pure FWI.

Summary Ultra-long offset stationary-receiver acquisitions implemented with multi-component Ocean Bottom Nodes (OBN) are emerging as a suitable technology for deep offshore subsalt imaging by full waveform inversion (FWI). Ultra-long offsets allow to undershoot the deepest targeted structures with diving waves. In this framework, first-arrival traveltime + slope tomography (FASTT) provides a simple and efficient tool to build kinematically-accurate initial velocity model for full waveform inversion (FWI). In particular, the incorporation of slopes in the first-arrival tomography in addition to traveltimes removes artifacts in poorly illuminated areas and better reconstruct structural dips. Either the FASTT or the FWI velocity models can be used as reliable background models for prestack depth migration of co-incident towed-streamer data. However, the high resolution of the FWI velocity model facilitates the integrated interpretation of the reflectivity mapped by migration and the intermediate-scale velocity variations reconstructed by waveform inversion. We illustrate the workflow combining first-arrival tomography + FWI of ultra-long offset OBN data and migration of co-incident towed-streamer data with a challenging deep crustal case study in the eastern Nankai subduction zone.