Physics-Informed Neural Networks (PINNs) have gained attention for solving partial differential equations, including the scattered Helmholtz equation, due to their flexibility and mesh-free formulation. However, their performance suffers from low-frequency bias, particularly in high-frequency wavefield simulations, limiting convergence speed and accuracy. To address this, we propose a novel and simplified PINN framework that incorporates explicit, trainable Gabor basis functions to efficiently capture the localized and oscillatory nature of wavefields. Unlike previous Gabor-based PINNs that rely on multiplicative filters or auxiliary networks to learn Gabor parameters, our approach redefines the network's task as learning a nonlinear mapping from input coordinates to a custom Gabor coordinate system, where a Gabor function captures the dominant oscillatory behavior of the wavefield. This formulation absorbs the effect of two Gabor parameters into the learned mapping, reducing computational complexity and eliminating the need for manual tuning of hyperparameters. We also present an efficient formulation for incorporating a Perfectly Matched Layer (PML) into the training by deriving real-valued loss components and introducing an analytical expression for the background wavefield. Numerical experiments on various velocity models show that our Gabor-PINN achieves faster convergence, higher accuracy, and greater robustness to architectural design and initialization compared to both traditional PINNs and prior Gabor-based methods. The improvement lies not in adding architectural complexity-as is common in enhanced PINNs-but in absorbing this complexity into the learned coordinate transformation, making the method both simpler and more effective. Our implementation is publicly available to support reproducibility and future research.

The concept of neural network (NN)-based seismic resolution enhancement has gained a lot of traction recently. Yet, the majority of works on the topic rely on training NNs on synthetic data via a supervised learning strategy, often encountering generalization issues on real data. To address this problem, we develop a self-supervised learning (SSL) method for seismic resolution enhancement. Specifically, we reinterpret seismic resolution enhancement as a frequency extension task, particularly focusing on the reconstruction of high-frequency components. Initially, we warm up the NN using the original/available band-limited data as pseudolabels, with input data derived from filtering out high-frequency elements from the data. Subsequently, the network undergoes iterative data refinement (IDR), where pseudolabels are predicted from the NN trained in the previous epoch, and input data are obtained by filtering out high-frequency components from these predictions. Based on this strategy, we also present a hybrid framework for simultaneous seismic denoising and resolution enhancement. During the whole training, we used multiloss constraints to enhance the network performance. The efficacy of our method is demonstrated through tests on both synthetic and field data.

Accurate separation of P- and S-waves is essential for multi-component seismic data processing, as it helps eliminate interference between wave modes during imaging or inversion, which leads to high-accuracy results. Traditional methods for separating P- and S-waves rely on the Christoffel equation to compute the polarization direction of the waves in the wavenumber domain, which is computationally expensive. Although machine learning has been employed to improve the computational efficiency of the separation process, most methods still require supervised learning with labeled data, which is often unavailable for field data. To address this limitation, we propose a wavefield separation technique based on Physics-Informed Neural Networks (PINNs), which leverage automatic differentiation to compute partial derivatives. We formulate the P- and S-wave separation equations as loss functions to train a neural network that learns functional solutions to these equations. The network takes spatial coordinates as input and outputs the corresponding separated P- and S-wavefields. Once trained, it enables near-instantaneous evaluation of the separated wavefields at any spatial location. This unsupervised machine learning approach is applicable to unlabeled data. Numerical tests demonstrate that the proposed PINN-based separation method can accurately separate P- and S-waves in both homogeneous and heterogeneous media.

Image enhancement is crucial for improving the resolution of seismic images obtained from band-limited data. While machine learning techniques, particularly the U-Net model, have shown significant progress in this area, they often require substantial computational resources and time. To address these challenges, we introduce a transformer-based approach for enhancing seismic image resolution, which incorporates convolutional layers, an average pooling layer, and an efficient transformer (ET). The ET leverages efficient multihead attention (EMHA) to capture long-term dependencies among image blocks, focusing on the pixels within their contextual surroundings. In our proposed model, we use a combined loss function consisting of the mean square error (mse) and the structural similarity (SSIM) to enhance the network’s learning capability. By training the model on synthetic seismic data, we observe improved structural features, enhanced resolution, and effective denoising. Notably, our approach outperforms the U-Net model in terms of SSIM and the peak signal-to-noise ratio (SNR). Furthermore, we evaluate the pretrained model on several field datasets, yielding promising results compared to the benchmark method. This demonstrates the potential applicability and effectiveness of our proposed approach in real-world scenarios.

Solutions of wave equations (e.g., wavefields) are invaluable to imaging and inverting the subsurface. The main challenge in attaining such solutions is the exponential increase in computational cost with finer discretization. Being discretization invariant, the recently developed physics-informed neural networks (PINNs) framework offers accurate and more flexible PDE solutions than conventional solvers. However, they are challenged by the relatively slow convergence and the need to perform additional training for other PDE parameters (velocity models). To address this limitation, we introduce a PINN framework that utilizes latent representations of the PDE parameters (velocity models) as additional inputs into the PINN model and performs training over a distribution of viable velocity models. We use a two-stage training scheme in which, we first learn a latent representation for a distribution of velocity models. Then, we train a physics-informed neural network over inputs given by randomly drawn samples from the coordinate space within the solution domain and samples from the learned latent representation of the velocity models. Through numerical tests and benchmarking against several existing algorithms, we demonstrate that the proposed framework provides up to three times the speed up and an order of magnitude accuracy improvement. The proposed framework retains the flexibility and accuracy features of the functional representation of PINN solutions while gaining a generalization feature to adapt to various velocity models efficiently.

Seismic data often contain gaps due to various obstacles in the investigated area and recording instrument failures. Deep-learning techniques offer promising solutions for reconstructing missing data parts by utilizing existing data. Nonetheless, self-supervised methods frequently struggle with capturing under-represented features such as weaker events, crossing dips, and higher frequencies. To address these challenges, we propose a novel ensemble deep model (EDM) along with a tailored self-supervised training approach for reconstructing seismic data with consecutive missing traces. Our model comprises two branches of U-nets, each fed from distinct data transformation modules aimed at amplifying under-represented features and promoting diversity among learners. Our loss function minimizes relative errors at the outputs of individual branches and the entire model, ensuring accurate reconstruction of various features while maintaining overall data integrity. Additionally, we employ masking while training to enhance sample diversity and memory efficiency. Applications on two benchmark synthetic datasets and two real datasets demonstrate improved accuracy compared to a conventional U-net, successfully reconstructing weak events, diffractions, higher frequencies, and reflections covered by groundroll. Despite these advancements, our method does incur three times the training cost compared to a simple U-net.

Microseismic source imaging plays a significant role in passive seismic monitoring. However, such a process is prone to failure due to aliasing when dealing with sparsely measured data. Thus, we propose a direct microseismic imaging framework based on physics-informed neural networks (PINNs), which can generate focused source images, even with very sparse recordings. We use the PINNs to represent a multi-frequency wavefield and then apply inverse Fourier transform to extract the source image. To be more specific, we modify the representation of the frequency-domain wavefield to inherently satisfy the boundary conditions (the measured data on the surface) by means of a hard constraint, which helps to avoid the difficulty in balancing the data and PDE losses in PINNs. Furthermore, we propose the causality loss implementation with respect to depth to enhance the convergence of PINNs. The numerical experiments on the Overthrust model show that the method can admit reliable and accurate source imaging for single- or multiple- sources and even in passive monitoring settings. Compared with the time-reversal method, the results of the proposed method are consistent with numerical methods but less noisy. Then, we further apply our method to hydraulic fracturing monitoring field data, and demonstrate that our method can correctly image the source with fewer artifacts.

Traditional supervised denoising networks learn network weights through “black box” (pixel-oriented) training, which requires clean training labels. The inability of such denoising networks to interpret their behavior and the requirement for clean data as labels limit their applicability in real-case scenarios. Deep unfolding methods unroll an optimization process into Deep Neural Networks (DNNs), improving the interpretability of networks. Also, modifiable filters in DNNs allow us to embed the prior information of the desired signals to be extracted, in order to remove noise in a self-supervised manner. Thus, we propose a Gabor-based learnable sparse representation network to suppress different noise types in a self-supervised fashion through constraints/bounds applied to the parameters of the Gabor filters of the network during the training stage. The effectiveness of the proposed method is demonstrated on two noise type examples, pseudo-random noise and ground roll, on synthetic and real data.

The use of convolutional neural networks (CNNs) in seismic interpretation tasks, like facies classification, has garnered a lot of attention for its high accuracy. However, its drawback is usually poor generalization when trained with limited training data pairs, especially for noisy data. Seismic images are dominated by diverse wavelet textures corresponding to seismic facies with various petrophysical parameters, which can be suitably represented by Gabor functions. Inspired by this fact, we propose using learnable Gabor convolutional kernels in the first layer of a CNN network to improve its generalization. The modified network combines the interpretability features of Gabor filters and the reliable learning ability of the original CNN. It replaces the pixel nature of conventional CNN filters with a constrained function form that depends on five parameters that are more in line with seismic signatures. This allows us, in training, to constrain the angle and wavelength of the Gabor kernels to specific ranges to help enhance the seismic features and reduce noise. We, also, test this modified CNN using various kernels on salt&pepper and speckle noise. The experiments on the Netherland F3 dataset show that we obtain the best generalization and robustness of the CNN to noise when Gabor kernels are used in the first layer.

The computation of the seismic wavefield by solving the Helmholtz equation is crucial to many practical applications, e.g., full waveform inversion. Physics-informed neural networks (PINNs) provide functional wavefield solutions represented by neural networks (NNs), but their convergence is slow. To address this problem, we propose a modified PINN using multiplicative filtered networks, which embeds some of the known characteristics of the wavefield in training, e.g., frequency, to achieve much faster convergence. Specifically, we use the Gabor basis function due to its proven ability to represent wavefields accurately and refer to the implementation as GaborPINN. Meanwhile, we incorporate prior information on the frequency of the wavefield into the design of the method to mitigate the influence of the discontinuity of the represented wavefield by GaborPINN. The proposed method achieves up to a two-magnitude increase in the speed of convergence as compared with conventional PINNs.

The high-resolution waveform inversion for seismic velocities is gaining increasing interest as we start to deal with complex structures. Although full waveform inversion (FWI) has been used for several years, obtaining high-resolution velocity models still presents many obstacles, such as the high computational cost and the limited bandwidth of the data. Thus, we propose a deep learning (DL)-based algorithm to build high-resolution velocity models using low-resolution velocity models, migration images, and well-log velocities as inputs. The well information, specifically, helps enhance the resolution with ground-truth information, especially around the well. These three inputs are fed to an improved neural network, a variant of U-Net, as three channels to predict the corresponding true velocity models, which serve as labels in the training. The incorporation of well velocities from several locations is crucial for improving the resolution of the output model. Numerical experiments on complex models demonstrate the robust performance of this network and the crucial role that well information plays, especially in generalizing the approach to models that differ from the trained ones and achieving superior performance compared with FWI.

Seismic resolution enhancement is a key step for subsurface structure characterization. Although many have proposed the use of deep learning (DL) for resolution enhancement, these are typically hindered by the limitations in the application of synthetically trained networks onto real datasets. Domain adaptation (DA) offers an approach to reduce this disparity between training and inference data, aiming through the application of data transformations to bring the distributions of both data closer to each other. We propose a simple DA procedure, termed MLReal-Lite (the light version of the earlier introduced MLReal), that mainly relies on linear operations, namely convolution and correlation; these transformations introduce aspects of the field data into the synthetic data prior to training, and vice-versa with regard to the inference stage. Taking 1-D and 2-D resolution enhancement tasks as examples, we show how the inclusion of MLReal-Lite improves the performance of neural networks. Not only do the results demonstrate notable improvements in seismic resolution, they also exhibit a higher signal-to-noise ratio (SNR) and better continuity of events, in comparison to the tests without MLReal-Lite. Finally, while illustrated on a resolution enhancement task, our proposed methodology is applicable for any seismic data of dimensions N-D, offering a DA applicable from well ties through to 3-D seismic volumes, and beyond.

In salt provinces, full-waveform inversion (FWI) is most likely to fail when starting with a poor initial model that lacks the salt information. Conventionally, salt bodies are included in the FWI starting model by interpreting the salt boundaries from seismic images, which is time-consuming and prone to error. Studies show that FWI can improve the interpreted salt provided that the data have long offsets, and low frequencies, which is not always the case. Thus, we develop an approach to invert for the salt body starting from a poor initial model, limited data offsets, and the absence of low frequencies. We leverage deep learning to apply multi-stage flooding and unflooding of the velocity model. Specifically, we apply a multi-scale FWI using three frequency bandwidths.We apply a network after each frequency scale. After the first two bandwidths, the networks are trained to flood the salt, while the network after the last frequency bandwidth is trained to unflood it. We follow the unflooding step, with a final FWI. We verify the method on the synthetic BP 2004 salt model benchmark. We only use the synthetic data of short offsets up to 6 km and remove frequencies below 3 Hz. We also apply the method to real vintage data acquired in the Gulf of Mexico region. The real data lack frequencies below 6 Hz and the streamer length is only 4.8 km. With these limitations, we manage to recover the salt body and verify the result by using them to image the data and analyze the resulting angle gathers.

The ocean bottom node (OBN) seismic acquisition system is designed to gather high-fidelity, wide-azimuth, and long-offset four-component (4C) data, which includes shear waves and enables the use of the elastic assumption in imaging and inversion. However, deploying geophysical instruments on the seafloor is difficult and costly, leading to the usual adoption of sparse node spacing. This can, however, lead to poor illumination and imaging challenges, especially in the shallow subsurface near the seafloor. To address these issues in the context of 4C elastic imaging, we propose a deep learning-based method using a multi-scale convolutional neural network (Ms-CNN) to improve the imaging quality of OBN surveys with sparse data acquisition. As an alternative to interpolating the sparse seismic data in the data domain, which can be a challenging task due to the limitations attributed to sampling theorem and the often larger amounts of data compared to the image, we train an Ms-CNN in a supervised fashion to map from sparse data images of PP and PS sections produced by 4C Gaussian beam migration to the equivalent dense data images, allowing for the direct processing of sparse data to improve imaging quality. Here, we combine the mean absolute error and multiscale structure similarity index measure in the loss function to optimize the network’s training process, and to help improve the performance. The effectiveness of the method is demonstrated through experiments on synthetic and field data, resulting in improved event continuity and reduced noise in migration results from sparse OBN acquisitions.

Building subsurface models is a very important but challenging task in hydrocarbon exploration and development. The subsurface elastic properties are usually sourced from seismic data and well logs. Thus, we design a deep learning (DL) framework using Vision Transformer (ViT) as the backbone architecture to build the subsurface model using well log information as we apply full waveform inversion (FWI) on the seismic data. However, training a ViT network from scratch with limited well log data can be difficult to achieve good generalization. To overcome this, we implement an efficient self-supervised pre-training process using a masked autoencoder (MAE) architecture to learn important feature representations in seismic volumes. The seismic volumes required by the pre-training are randomly extracted from a seismic inversion, such as an FWI result. We can also incorporate reverse time migration (RTM) image into the seismic volumes to provide additional structure information. The pre-training task of MAE is to reconstruct the original image from the masked image with a masking ratio of 75%. This pre-training task enables the network to learn the high-level latent representations. After the pre-training process, we then fine-tune the ViT network to build the optimal mapping relationship between 2D seismic volumes and 1D well segments. Once the fine-tuning process is finished, we apply the trained ViT network to the whole seismic inversion domain to predict the subsurface model. At last, we use one synthetic data set and two field data sets to test the performance of the proposed method. The test results demonstrate that the proposed method effectively integrates seismic and well information to improve the resolution and accuracy of the velocity model.

Considering the conventional seismic wavelength, and the nature of the Earth layering, seismic waves experience, in many parts of the subsurface, considerable anisotropy, and with the effect of gravity on sedimentation, the anisotropy tends to be of a transversely isotropic with a vertical axis of symmetry (VTI) nature. Inverting for such a model of the Earth using waveforms, we face considerable nonlinearity and parameter trade-off. A recently introduced optimal transport of the matching filter (OTMF) provided us with a robust misfit function for reducing cycle-skipping in Full-Waveform Inversion (FWI). We apply a VTI FWI using the OTMF misfit on a field dataset from offshore Australia, comparing its performance to that of conventional FWI using the L2-norm misfit in a variety of circumstances. Due to strong anisotropy in this region, an isotropic inversion though can fit the record, leading to common image gathers with sizable linear moveouts. Thus, in an anisotropic VTI setup, starting the inversion from 3 Hz, both the L2 norm and the OTMF misfit functions can generate a geologically meaningful model, and recover similar anisotropy anomalies. We demonstrate that the OTMF misfit, in some sense, can address the nonlinearity of FWI due to its intrinsic global updating features. Compared to the results from isotropic full-waveform inversion, the improvements in the RTM image and the common image gathers further demonstrate the benefits of including anisotropy in the FWI inversion engine, as well as, the good performance of the OTMF in mitigating cycle-skipping.

Full waveform inversion (FWI) has the potential to provide high-resolution subsurface model estimations. However, due to limitations in observation, e.g., regional noise, limited aperture, and band-limited data, it is hard to obtain the desired high-resolution model with FWI. To address this challenge, we propose a new paradigm for FWI regularized by generative diffusion model. Specifically, we pre-train a diffusion model in a fully unsupervised manner on a prior velocity model distribution that represents our expectations of the subsurface and then adapt it to the seismic observations by incorporating the FWI into the sampling process of the generative diffusion models. What makes diffusion models uniquely appropriate for such an implementation is that the generative process retains the form and dimensions of the velocity model. Numerical examples demonstrate that our method can outperform the conventional FWI with only negligible additional computational cost. Even in cases of very sparse observations or observations with strong noise, the proposed method could still reconstruct a high-quality subsurface model. Thus, we can incorporate our prior expectations of the solutions in an efficient manner. We further test this approach on field data, which demonstrates the effectiveness of the proposed method.

Imaging is a crucial inversion-based task in fields ranging from medical to structure investigations, and Earth discovery. The exploding reflector assumption provides a direct imaging approach for zero-offset (coincident source-receiver) data, like ground penetrating radar (GPR) data. In imaging, however, we face aliasing problems when the data are coarsely sampled. Formulating the corresponding frequency-domain wavefield as a neural network (NN) function of the lateral and depth coordinates, as well as frequency, allows us to use the physics-informed neural network (PINN) framework to obtain images of the subsurface. In this case, we use a modified Helmholtz equation that incorporates the data on the Earth surface (hard constraint) as the loss function to optimize the NN function. This modified Helmholtz formulation allows us to avoid the inherent weaknesses that PINN has in handling boundary conditions, like the data on the surface as an additional loss term. The frequency dimension allows for image reconstruction by directly summing the wavefield over frequencies (the zero-time imaging condition). This zero-offset implementation serves as a proof of concept for later extensions to prestack data using the double square-root equation.

Physics-informed neural networks (PINNs) can offer approximate multidimensional functional solutions to the Helmholtz equation that are flexible, require low memory, and have no limitations on the shape of the solution space. However, the neural network (NN) training can be costly and the cost dramatically increases as we train for multi-frequency wavefields by adding frequency as an additional input to the NN multi-dimensional function. In this case, the often large variation of the wavefield features (specifically wavelength) with frequency adds more complexity to the NN training. Thus, we propose a new loss function for the NN multidimensional input training that allows us to seamlessly include frequency as a dimension. We specifically utilize the linear relation between frequency and wavenumber (the wavefield space representation) to incorporate a reference frequency scaling to the loss function. As a result, the effective wavenumber of the wavefield solution as a function of frequency remains almost stationary, which reduces the learning burden on the NN function. We demonstrate the effectiveness of this modified loss function on a layered model.

Full waveform inversion (FWI) plays a central role in the field of exploration geophysics due to its potential in recovering the properties of the subsurface at a high resolution. A starting model with ample long wavelength components is essential for the success of most FWI algorithms. Reflection waveform inversion (RWI) is one popular way to invert for the long wavelength velocity components from the short offset seismic data by decomposing the gradient of FWI into migration and tomographic terms. However, the transmitted part of Born scattering in conventional RWI still produces high-wavenumber artifacts, which would hinder its convergence. Thus, in this letter, an efficient nontransmission energy norm Born scattering is used in RWI to overcome the drawbacks of conventional RWI. Finally, we use numerical examples to show that the energy norm Born scattering can provide clean reflection energy from the reflector and enhance the low-wavenumber information in the RWI gradient.

Machine learned tasks on seismic data are often trained sequentially and separately, even though they utilize the same features (i.e. geometrical) of the data. We present StorSeismic, as a dataset centric framework for seismic data processing, which consists of neural network pre-training and fine-tuning procedures. We, specifically, utilize a neural network as a preprocessing tool to extract and store seismic data features of a particular dataset for any downstream tasks. After pre-training, the resulting model can be utilized later, through a fine-tuning procedure, to perform different tasks using limited additional training. Used often in Natural Language Processing (NLP) and lately in vision tasks, BERT (Bidirectional Encoder Representations from Transformer), a form of a Transformer model, provides an optimal platform for this framework. The attention mechanism of BERT, applied here on a sequence of traces within the shot gather, is able to capture and store key geometrical features of the seismic data. We pre-train StorSeismic on field data, along with synthetically generated ones, in the self-supervised step. Then, we use the labeled synthetic data to fine-tune the pre-trained network in a supervised fashion to perform various seismic processing tasks, like denoising, velocity estimation, first arrival picking, and NMO (normal moveout). Finally, the fine-tuned model is used to obtain satisfactory inference results on the field data.

Time-lapse (TL) seismic monitoring plays a vital role in reservoir characterization and management. Elastic full-waveform inversion (EFWI) has been applied to time-lapse seismic data to allow for a quantitative estimation of time-varying elastic properties. However, the high-resolution inversion can be computationally intense and ill-posed. To estimate the high-resolution time-lapse changes at a reasonable cost, we utilize two key techniques for the inversion: 1) we develop an elastic redatuming approach to retrieve the virtual elastic data for both base and monitor data at the target level using mainly a kinematically accurate velocity, thus, reducing the computational cost by focusing the high-resolution inversion on the target zone; 2) We integrate high-resolution well information and seismic data in the target-oriented inversion, where a high-resolution prior model is predicted by deep learning to regularize the inversion. A deep neural network (DNN) is capable of learning the mappings between the time-lapse seismic estimation and the facies interpreted from well information after the training process. Thus, we can derive a prior model for time-lapse changes by mapping the facies characterized by the property changes to the target inversion domain. We then implement the target-oriented TLEFWI regularized by the prior model, where the redatumed time-lapse elastic data and the prior model jointly contributes to the inversion result. The numerical examples validate that the proposed approach enables us to retrieve the time-lapse changes of elastic property in the target zone with improved resolution and well consistency.

Low-frequency signal content in seismic data as well as a realistic initial model are key ingredients for robust and efficient full-waveform inversions. However, acquiring low-frequency data is challenging in practice for active seismic surveys. Data-driven solutions show promise to extrapolate low-frequency data given a high-frequency counterpart. While being established for synthetic acoustic examples, the application of bandwidth extrapolation to field datasets remains non-trivial. Rather than aiming to reach superior accuracy in bandwidth extrapolation, we propose to jointly reconstruct low-frequency data and a smooth background subsurface model within a multi-task deep learning framework. We automatically balance data, model and trace-wise correlation loss terms in the objective functional and show that this approach improves the extrapolation capability of the network. We also design a pipeline for generating synthetic data suitable for field data applications. Finally, we apply the same trained network to synthetic and real marine streamer datasets and run an elastic full-waveform inversion from the extrapolated dataset.

Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI is often implemented by solving for the frequency-domain representation of the wavefield using the finite-difference method. The approach requires matrix inversions and affords limited flexibility to accommodate irregular model geometries. On the other hand, the physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN) to provide flexible continuous functional approximations of the solutions without matrix inversions. By including a data-constrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network, and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

The eikonal equation plays an important role across multidisciplinary branches of science and engineering. In geophysics, the eikonal equation, and its characteristics, are used in addressing two fundamental questions pertaining to seismic waves: what paths do the seismic waves take (its spreading)? and how long do they take? There have been numerous attempts to solve the eikonal equation, which can be broadly categorized as finite-difference and physics informed neural network (PINN) based approaches. While the former has been developed and optimized over the years, it still inherits some numerical inaccuracies and also the cost scales exponentially with the velocity model size. More importantly, it requires upwind calculations to satisfy the viscosity solution. PINNs, on the other hand, have shown great promise due to several features allowing for higher accuracy and scalability than conventional approaches. In this paper, we demonstrate another unique feature of PINN solutions, specifically its flexibility resulting from the global nature of its NN functional optimization, allowing for functional gradients referred to as automatic differentiation. This feature allows us to overcome the inability of conventional methods to handle large areas of missing information (gap) in the velocity model. We find empirically that the PINNs interpolation-extrapolation inherent capability enables us to circumvent a scenario when traveltime modelling is performed on velocity models containing gaps. Such a capability is crucial when performing traveltime modelling using the global tomographic Earth velocity model.

The microseismic monitoring technique is widely applied to petroleum reservoirs to understand the process of hydraulic fracturing. Geophones continuously record the microseismic events triggered by fluid injection on the Earth’s surface or in monitoring wells. The microseismic event localization precision has a large impact on the performance of the technique. Deep learning has achieved significant progress in computer vision and natural language processing in recent years. We propose to use a deep convolutional neural network (CNN) to directly map the field records to their event locations. The biggest advantage of deep learning methods over conventional methods is that they can efficiently predict the characteristics of a huge amount of recorded data without human intervention. Thus, we use a CNN to predict the event location of field microseismic data that were recorded during a hydraulic fracturing process of a shale gas play in Oklahoma, the United States. We use synthetic data with extracted field noise from the records to train CNN. The synthetic training data allow us to produce the corresponding labels, and the extracted noise from the field data reduces the difference between the field and synthetic data. We use a correlation preprocessing step to avoid the need for event detection and picking of arrivals. We demonstrate that the proposed approach provides accurate microseismic event locations at a much faster speed than traditional imaging methods, such as time-reversal imaging. Comparison with an existing study on the same data is presented to evaluate the performance of the trained neural network.

Full-waveform inversion (FWI) is popularly used to retrieve a high-resolution velocity model that maximizes the data fitting directly. It is a highly nonlinear optimization problem, and thus, FWI can easily fall into a local minimum. Wavefield reconstruction inversion (WRI) allows us to relax the wave equation constraint to provide a larger search space. However, it requires a high computational cost to update the velocity in each selected frequency through many expensive iterations. By recasting a linear optimization problem in terms of a modified source function (which includes the original source and secondary sources) and relying on the background velocity model, we end up with cheap inner iterations for inverting the wavefield. We refer to this setup as an efficient wavefield inversion (EWI). However, like WRI, EWI cannot mitigate the cycle-skipping problem completely when the background velocity model is far from the true one and low-frequency components in the data are missing. In this case, we propose to use additional outer iterations to better recover the velocity model. In the salt body inversion, we utilize a total variation (TV) regularization to constrain the inverted velocity model at each outer iteration. We demonstrate these features on a modified Marmousi model and a central part of the BP salt model. The application on a 2-D real data set also demonstrates the effectiveness of the proposed method.

A full-waveform inversion (FWI) is a highly nonlinear inversion methodology. FWI tends to converge to a local minimum rather than a global one. We refer to this phenomenon as “cycle skipping” in FWI. A cost-effective solution for resolving this issue is to design a more convex misfit function for the optimization problem. A global comparison based on using a matching filter (MF) admits more robust misfit functions. In this case, we would compute an MF first by deconvolving the predicted data from the measured ones. When the velocity model is accurate, the predicted data resemble the measured ones, and the resulting MF would be an approximated Dirac delta function. If the velocity produces data that are different from the observed ones, a misfit function can be formulated by penalizing the energy away from the zero-lag time (the center). Here, we develop a general mechanism for an evolution of the MF to our objective in FWI. Specifically from the statistics point of view, rather than using a penalty, we propose a novel misfit by minimization of the mean and information entropy of the MF distribution. We show that the resulting misfit function can mitigate the “cycle skipping” as well as reduce the mean and variance of the resulting MF distribution. We use a modified Marmousi example to demonstrate the features of the proposed misfit. We also evaluate the robustness of the proposed method using inaccurate (rotation in phase) source wavelets and measured data with different levels of Gaussian random noise.