Propagating the prior from far to near offset: A self-supervised diffusion framework for progressively recovering near-offsets of towed-streamer data

In marine towed-streamer seismic acquisition, the nearest hydrophone is often two hundred meter away from the source resulting in missing near-offset traces, which degrades critical processing workflows such as surface-related multiple elimination, velocity analysis, and full-waveform inversion. Existing reconstruction methods, like transform-domain interpolation, often produce kinematic inconsistencies and amplitude distortions, while supervised deep learning approaches require complete ground-truth near-offset data that are unavailable in realistic acquisition scenarios. To address these limitations, we propose a self-supervised diffusion-based framework that reconstructs missing near-offset traces without requiring near-offset reference data. Our method leverages overlapping patch extraction with single-trace shifts from the available far-offset section to train a conditional diffusion model, which learns offset-dependent statistical patterns governing event curvature, amplitude variation, and wavelet characteristics. At inference, we perform trace-by-trace recursive extrapolation from the nearest recorded offset toward zero offset, progressively propagating learned prior information from far to near offsets. The generative formulation further provides uncertainty estimates via ensemble sampling, quantifying prediction confidence where validation data are absent. Controlled validation experiments on synthetic and field datasets show substantial performance gains over conventional parabolic Radon transform baselines. Operational deployment on actual near-offset gaps demonstrates practical viability where ground-truth validation is impossible. Notably, the reconstructed waveforms preserve realistic amplitude-versus-offset trends despite training exclusively on far-offset observations, and uncertainty maps accurately identify challenging extrapolation regions.

LAViG-FLOW: Latent Autoregressive Video Generation for Fluid Flow Simulations

Modeling and forecasting subsurface multiphase fluid flow fields underpin applications ranging from geological CO2 sequestration (GCS) operations to geothermal production. This is essential for ensuring both operational performance and long-term safety. While high fidelity multiphase simulators are widely used for this purpose, they become prohibitively expensive once many forward runs are required for inversion purposes and quantify uncertainty. To tackle this challenge we propose LAViG-FLOW, a latent autoregressive video generation diffusion framework that explicitly learns the coupled evolution of saturation and pressure fields. Each state variable is compressed by a dedicated 2D autoencoder, and a Video Diffusion Transformer (VDiT) models their coupled distribution across time. We first train the model on a given time horizon to learn their coupled relationship and then fine-tune it autoregressively so it can extrapolate beyond the observed time window. Evaluated on an open-source CO2 sequestration dataset, LAViG-FLOW generates saturation and pressure fields that stay consistent across time while running orders of magnitude faster than traditional numerical solvers.

Diffusion priors enhanced velocity model building from time-lag images using a neural operator

Velocity model building serves as a crucial component for achieving high precision subsurface imaging. However, conventional velocity model building methods are often computationally expensive and time consuming. In recent years, with the rapid advancement of deep learning, particularly the success of generative models and neural operators, deep learning based approaches that integrate data and their statistics have attracted increasing attention in addressing the limitations of traditional methods. In this study, we propose a novel framework that combines generative models with neural operators to obtain high resolution velocity models efficiently. Within this workflow, the neural operator functions as a forward mapping operator to rapidly generate time lag reverse time migration (RTM) extended images from the true and migration velocity models. In this framework, the neural operator is acting as a surrogate for modeling followed by migration, which uses the true and migration velocities, respectively. The trained neural operator is then employed, through automatic differentiation, to gradually update the migration velocity placed in the true velocity input channel with high resolution components so that the output of the network matches the time lag images of observed data obtained using the migration velocity. By embedding a generative model, trained on a high-resolution velocity model distribution, which corresponds to the true velocity model distribution used to train the neural operator, as a regularizer, the resulting predictions are cleaner with higher resolution information. Both synthetic and field data experiments demonstrate the effectiveness of the proposed generative neural operator based velocity model building approach.

Diffusion Model-Based Posterior Sampling in Full Waveform Inversion

Bayesian full waveform inversion (FWI) offers uncertainty-aware subsurface models; however, posterior sampling directly on observed seismic shot records is rarely practical at the field scale because each sample requires numerous wave-equation solves. We aim to make such sampling feasible for large surveys while preserving calibration, that is, high uncertainty in less illuminated areas. Our approach couples diffusion-based posterior sampling with simultaneous-source FWI data. At each diffusion noise level, a network predicts a clean velocity model. We then apply a stochastic refinement step in model space using Langevin dynamics under the wave-equation likelihood and reintroduce noise to decouple successive levels before proceeding. Simultaneous-source batches reduce forward and adjoint solves approximately in proportion to the supergather size, while an unconditional diffusion prior trained on velocity patches and volumes helps suppress source-related numerical artefacts. We evaluate the method on three 2D synthetic datasets (SEG/EAGE Overthrust, SEG/EAGE Salt, SEAM Arid), a 2D field line, and a 3D upscaling study. Relative to a particle-based variational baseline, namely Stein variational gradient descent without a learned prior and with single-source (non-simultaneous-source) FWI, our sampler achieves lower model error and better data fit at a substantially reduced computational cost. By aligning encoded-shot likelihoods with diffusion-based sampling and exploiting straightforward parallelization over samples and source batches, the method provides a practical path to calibrated posterior inference on observed shot records that scales to large 2D and 3D problems.

Inspired by machine learning optimization: can gradient-based optimizers solve cycle skipping in full waveform inversion given sufficient iterations?

Full waveform inversion (FWI) iteratively updates the velocity model by minimizing the difference between observed and simulated data. Due to the high computational cost and memory requirements associated with global optimization algorithms, FWI is typically implemented using local optimization methods. However, when the initial velocity model is inaccurate and low-frequency seismic data (e.g., below 3 Hz) are absent, the mismatch between simulated and observed data may exceed half a cycle, a phenomenon known as cycle skipping. In such cases, local optimization algorithms (e.g., gradient-based local optimizers) tend to converge to local minima, leading to inaccurate inversion results. In machine learning, neural network training is also an optimization problem prone to local minima. It often employs gradient-based optimizers with a relatively large learning rate (beyond the theoretical limits of local optimization that are usually determined numerically by a line search), which allows the optimization to behave like a quasi-global optimizer. Consequently, after training for several thousand iterations, we can obtain a neural network model with strong generative capability. In this study, we also employ gradient-based optimizers with a relatively large learning rate for FWI. Results from both synthetic and field data experiments show that FWI may initially converge to a local minimum; however, with sufficient additional iterations, the inversion can gradually approach the global minimum, slowly from shallow subsurface to deep, ultimately yielding an accurate velocity model. Furthermore, numerical examples indicate that, given sufficient iterations, reasonable velocity inversion results can still be achieved even when low-frequency data below 5 Hz are missing.

Physics-informed waveform inversion using pretrained wavefield neural operators

Full waveform inversion (FWI) is crucial for reconstructing high-resolution subsurface models, but it is often hindered, considering the limited data, by its null space resulting in low-resolution models, and more importantly, by its computational cost, especially if needed for real-time applications. Recent attempts to accelerate FWI using learned wavefield neural operators have shown promise in efficiency and differentiability, but typically suffer from noisy and unstable inversion performance. To address these limitations, we introduce a novel physics-informed FWI framework to enhance the inversion in accuracy while maintaining the efficiency of neural operator-based FWI. Instead of relying only on the L2 norm objective function via automatic differentiation, resulting in noisy model reconstruction, we integrate a physics constraint term in the loss function of FWI, improving the quality of the inverted velocity models. Specifically, starting with an initial model to simulate wavefields and then evaluating the loss over how much the resulting wavefield obeys the physical laws (wave equation) and matches the recorded data, we achieve a reduction in noise and artifacts. Numerical experiments using the OpenFWI and Overthrust models demonstrate our method's superior performance, offering cleaner and more accurate subsurface velocity than vanilla approaches. Considering the efficiency of the approach compared to FWI, this advancement represents a significant step forward in the practical application of FWI for real-time subsurface monitoring.

An effective physics-informed neural operator framework for predicting wavefields

Solving the wave equation is fundamental for geophysical applications. However, numerical solutions of the Helmholtz equation face significant computational and memory challenges. Therefore, we introduce a physics-informed convolutional neural operator (PICNO) to solve the Helmholtz equation efficiently. The PICNO takes both the background wavefield corresponding to a homogeneous medium and the velocity model as input function space, generating the scattered wavefield as the output function space. Our workflow integrates PDE constraints directly into the training process, enabling the neural operator to not only fit the available data but also capture the underlying physics governing wave phenomena. PICNO allows for high-resolution reasonably accurate predictions even with limited training samples, and it demonstrates significant improvements over a purely data-driven convolutional neural operator (CNO), particularly in predicting high-frequency wavefields. These features and improvements are important for waveform inversion down the road.

Least-Squares-Embedded Optimization for Accelerated Convergence of PINNs in Acoustic Wavefield Simulations

Physics-Informed Neural Networks (PINNs) have shown promise in solving partial differential equations (PDEs), including the frequency-domain Helmholtz equation. However, standard training of PINNs using gradient descent (GD) suffers from slow convergence and instability, particularly for high-frequency wavefields. For scattered acoustic wavefield simulation based on Helmholtz equation, we derive a hybrid optimization framework that accelerates training convergence by embedding a least-squares (LS) solver directly into the GD loss function. This formulation enables optimal updates for the linear output layer. Our method is applicable with or without perfectly matched layers (PML), and we provide practical tensor-based implementations for both scenarios. Numerical experiments on benchmark velocity models demonstrate that our approach achieves faster convergence, higher accuracy, and improved stability compared to conventional PINN training. In particular, our results show that the LS-enhanced method converges rapidly even in cases where standard GD-based training fails. The LS solver operates on a small normal matrix, ensuring minimal computational overhead and making the method scalable for large-scale wavefield simulations.

Full waveform inversion with CNN-based velocity representation extension

Full waveform inversion (FWI) updates the velocity model by minimizing the discrepancy between observed and simulated data. However, discretization errors in numerical modeling and incomplete seismic data acquisition can introduce noise, which propagates through the adjoint operator and affects the accuracy of the velocity gradient, thereby impacting the FWI inversion accuracy. To mitigate the influence of noise on the gradient, we employ a convolutional neural network (CNN) to refine the velocity model before performing the forward simulation, aiming to reduce noise and provide a more accurate velocity update direction. We use the same data misfit loss to update both the velocity and network parameters, thereby forming a self-supervised learning procedure. We propose two implementation schemes, which differ in whether the velocity update passes through the CNN. In both methodologies, the velocity representation is extended (VRE) by using a neural network in addition to the grid-based velocities. Thus, we refer to this general approach as VRE-FWI. Synthetic and real data tests demonstrate that the proposed VRE-FWI achieves higher velocity inversion accuracy compared to traditional FWI, at a marginal additional computational cost of approximately 1%.

Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields

Physics-Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh-free formulation. However, their low-frequency bias limits their accuracy and convergence speed for high-frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor-based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN-based seismic applications.