f–k filtering and Radon transform (RT) are classical methods for processing 2-D seismic signals. They assume that target signal events exhibit specific trajectories in the time–offset domain (linear, parabolic, and hyperbolic) and process them accordingly. In fact, seismic signals are nonstationary, with geometric and dynamic characteristics changing pointwise. This makes that these methods suboptimal for processing such signals. The 2-D nonstationary convolution filtering conducted in the time–space domain allows filter variations point-by-point, adapting to the nonstationarity of seismic signals. However, obtaining the filter factors involves complex calculations, making it not widely applicable. This article proposes an efficient method based on a two-dimensional nonstationary convolution model (TNCM) and termed two-dimensional nonstationary filtering by operator scaling (TNFOS). Leveraging the resemblance among filters, specific filter factors can be easily acquired by extracting and scaling the reference filter factors. Through the cascading of filters, TNFOS effectively fulfills nearly all design requirements for 2-D velocity or dip filtering, enabling the broad implementation of 2-D nonstationary filtering. Using synthetic and field experiments, we tested the efficiency of TNFOS and provided four successful applications in suppressing seismic interference.
Improving the resolution of nonstationary reflection seismic data without well data is challenging, as it requires prior estimation of the Q -value. The coupling of reflectivity sequence and wavelet introduces spectral fluctuations, rendering the extraction of Q highly unstable. While some statistical wavelet amplitude estimation methods, such as spectral shaping (SS) and contraction operator mapping (COM), can yield smooth wavelet amplitude spectra, they do not contribute to Q analysis as they lack a physical mechanism for absorption attenuation. Therefore, we propose an absorption-constrained wavelet power spectrum inversion (AWPSI) method based on the segmented stationary convolution model (SSCM). The absorption constraint (AC) term incorporates prior knowledge of the medium’s Q -effect and enables simultaneous inversion of wavelet amplitude spectra for all windows. Incorporating AWPSI into COM and SS methods leads to AWPSI-COM and AWPSI-SS methods, which yield higher precision wavelet power spectra and remove the spectral fluctuations caused by reflectivity sequences. We demonstrate that AWPSI enables COM to extract wavelet amplitude spectra more accurately while preserving the attenuation characteristics caused by Q -effect. Using the equivalent Q field obtained by the AWPSI-COM method, an inverse Q -filter can be performed to generate accurate high-resolution seismic profiles. Synthetic and actual stacked seismic data verify the effectiveness of the proposed method.
Extracting a reliable model of quality factor Q from the spectral information of seismic signals is a significantly important step for seismic imaging and reservoir characterization. However, the conventional Q estimation approach based on the frequency domain is susceptible to the high oscillative spectrum created by the unavoidable random and coherent noise in the observed signal. To address this issue, we introduce an absorption-constrained wavelet power spectrum inversion (AWPSI) method. The inversion involves absorption constraint (AC) and spectrum constraint (SC), where the AC term is a novel constraint for estimating the wavelet’s amplitude spectrum or power spectrum. By treating medium absorption as a physical prior and utilizing information from all waveforms rather than individual ones, AWPSI can yield high-quality wavelet power spectra and ensure stable Q estimation results. In addition, the proposed method has no limitations on the type of wavelet. Since its inversion target is wavelet power spectra, AWPSI is broadly applicable to various frequency-based Q estimation approaches. We validate the effectiveness of the proposed method using both synthetic and actual vertical seismic profile (VSP) data.
In the zero-offset vertical seismic profile (VSP), if the data noise is weak, the spectral ratio method (SRM) is considered to be the most reliable and commonly used method for estimating the medium Q factor. However, due to the possible picking error in the first-arrival time of the downgoing waveform, the estimated Q value fluctuates, which affects the subsequent attenuation compensation and lithology analysis. In order to overcome this fluctuation, we present an iterative maximum posterior probability (IMAP) method that uses zero-offset VSP data to invert the medium Q factor. Based on SRM, this method gives the maximum posterior probability (MAP) method for inverting the quality factor Q , establishes the update formulas of quality factor Q and travel time difference Δt , and performs alternate iterations on them. Since the prior information of the formation is introduced and Δt is corrected in each iteration, the inversion result can eliminate the fluctuations in the Q value and velocity caused by the Δt error. Two synthetic experimental examples and one field experiment verify the effectiveness and superiority of the proposed method.
The attenuation of seismic field noise using self-supervised deep learning has gained attention due to its label-free training process. However, common self-supervised methods are limited by the pixelwise independence assumption, which does not align with field seismic noise characteristics, and suffer from signal leakage due to receptive fields containing inherent blind spots or traces. In this paper, we propose a self-supervised random noise attenuation method based on the non-pixelwise independence assumption. By considering the spatial correlation map of field noise, we extend the blind spot to a generalized blind neighborhood, ensuring that the prediction pixel is not influenced by neighboring pixels with noise correlation greater than zero. The blind neighborhood size controls how much spatial correlation is disrupted, allowing our method to handle random noise with varying spatial correlation. Since larger blind neighborhoods may lead to signal loss, we introduce an automatic trade-off between noise correlation disruption and signal preservation during training. Experiments on real seismic noise attenuation (including random and tracewise coherent noise) demonstrate the superiority of our method in destroying the spatial coherence of noise and preventing useful signal leakage.
Improving the resolution of nonstationary reflection seismic data without well data is challenging, as it requires prior estimation of the Q -value. The coupling of reflectivity sequence and wavelet introduces spectral fluctuations, rendering the extraction of Q highly unstable. While some statistical wavelet amplitude estimation methods, such as spectral shaping (SS) and contraction operator mapping (COM), can yield smooth wavelet amplitude spectra, they do not contribute to Q analysis as they lack a physical mechanism for absorption attenuation. Therefore, we propose an absorption-constrained wavelet power spectrum inversion (AWPSI) method based on the segmented stationary convolution model (SSCM). The absorption constraint (AC) term incorporates prior knowledge of the medium’s Q -effect and enables simultaneous inversion of wavelet amplitude spectra for all windows. Incorporating AWPSI into COM and SS methods leads to AWPSI-COM and AWPSI-SS methods, which yield higher precision wavelet power spectra and remove the spectral fluctuations caused by reflectivity sequences. We demonstrate that AWPSI enables COM to extract wavelet amplitude spectra more accurately while preserving the attenuation characteristics caused by Q -effect. Using the equivalent Q field obtained by the AWPSI-COM method, an inverse Q -filter can be performed to generate accurate high-resolution seismic profiles. Synthetic and actual stacked seismic data verify the effectiveness of the proposed method.
Self-Supervised Learning (SSL) has become a powerful solution to extract rich representations from unlabeled data. Yet, SSL research is mostly focused on clean, curated and high-quality datasets. As a result, applying SSL on noisy data remains a challenge, despite being crucial to applications such as astrophysics, medical imaging, geophysics or finance. In this work, we present a fully self-supervised framework that enables noise-robust representation learning without requiring a denoiser at inference or downstream fine-tuning. Our method first trains an SSL denoiser on noisy data, then uses it to construct a denoised-to-noisy data curriculum (i.e., training first on denoised, then noisy samples) for pretraining a SSL backbone (e.g., DINOv2), combined with a teacher-guided regularization that anchors noisy embeddings to their denoised counterparts. This process encourages the model to internalize noise robustness. Notably, the denoiser can be discarded after pretraining, simplifying deployment. On ImageNet-1k with ViT-B under extreme Gaussian noise ($\sigma=255$, SNR = 0.72 dB), our method improves linear probing accuracy by 4.8% over DINOv2, demonstrating that denoiser-free robustness can emerge from noise-aware pretraining. The code is available at https://github.com/wenquanlu/noisy_dinov2.
Foundation models, as a mainstream technology in artificial intelligence, have demonstrated immense potential across various domains in recent years, particularly in handling complex tasks and multimodal data. In the field of geophysics, although the application of foundation models is gradually expanding, there is currently a lack of comprehensive reviews discussing the full workflow of integrating foundation models with geophysical data. To address this gap, this paper presents a complete framework that systematically explores the entire process of developing foundation models in conjunction with geophysical data. From data collection and preprocessing to model architecture selection, pre-training strategies, and model deployment, we provide a detailed analysis of the key techniques and methodologies at each stage. In particular, considering the diversity, complexity, and physical consistency constraints of geophysical data, we discuss targeted solutions to address these challenges. Furthermore, we discuss how to leverage the transfer learning capabilities of foundation models to reduce reliance on labeled data, enhance computational efficiency, and incorporate physical constraints into model training, thereby improving physical consistency and interpretability. Through a comprehensive summary and analysis of the current technological landscape, this paper not only fills the gap in the geophysics domain regarding a full-process review of foundation models but also offers valuable practical guidance for their application in geophysical data analysis, driving innovation and advancement in the field.
Dense ground displacement measurements are crucial for geological studies but are impractical to collect directly. Traditionally, displacement fields are estimated using patch matching on optical satellite images from different acquisition times. While deep learning-based optical flow models are promising, their adoption in ground deformation analysis is hindered by challenges such as the absence of real ground truth, the need for sub-pixel precision, and temporal variations due to geological or anthropogenic changes. In particular, we identify that deep learning models relying on explicit correlation layers struggle at estimating small displacements in real-world conditions. Instead, we propose a model that employs iterative refinements with explicit warping layers and a correlation-independent backbone, enabling sub-pixel precision. Additionally, a non-convex variant of Total Variation regularization preserves fault-line sharpness while maintaining smoothness elsewhere. Our model significantly outperforms widely used geophysics methods on semi-synthetic benchmarks and generalizes well to challenging real-world scenarios captured by both medium- and high-resolution sensors. Project page: https://jbertrand89.github.io/microflow/.
Full Waveform Inversion (FWI) is an important geophysical technique considered in subsurface property prediction. It solves the inverse problem of predicting high-resolution Earth interior models from seismic data. Traditional FWI methods are computationally demanding. Inverse problems in geophysics often face challenges of non-uniqueness due to limited data, as data are often collected only on the surface. In this study, we introduce a novel methodology that leverages Deep Operator Networks (DeepONet) to attempt to improve both the efficiency and accuracy of FWI. The proposed DeepONet methodology inverts seismic waveforms for the subsurface velocity field. This approach is able to capture some key features of the subsurface velocity field. We have shown that the architecture can be applied to noisy seismic data with an accuracy that is better than some other machine learning methods. We also test our proposed method with out-of-distribution prediction for different velocity models. The proposed DeepONet shows comparable and better accuracy in some velocity models than some other machine learning methods. To improve the FWI workflow, we propose using the DeepONet output as a starting model for conventional FWI and that it may improve FWI performance. While we have only shown that DeepONet facilitates faster convergence than starting with a homogeneous velocity field, it may have some benefits compared to other approaches to constructing starting models. This integration of DeepONet into FWI may accelerate the inversion process and may also enhance its robustness and reliability.
This review explores the integration of deep learning (DL) with full-waveform inversion (FWI) for enhanced seismic imaging and subsurface characterization. It covers FWI and DL fundamentals, geophysical applications (velocity estimation, deconvolution, tomography), and challenges (model complexity, data quality). The review also outlines future research directions, including hybrid, generative, and physics-informed models for improved accuracy, efficiency, and reliability in subsurface property estimation. The synergy between DL and FWI has the potential to transform geophysics, providing new insights into Earth's subsurface.
Magnetic data inversion is an important tool in geophysics, used to infer subsurface magnetic susceptibility distributions from surface magnetic field measurements. This inverse problem is inherently ill-posed, characterized by non-unique solutions, depth ambiguity, and sensitivity to noise. Traditional inversion approaches rely on predefined regularization techniques to stabilize solutions, limiting their adaptability to complex or diverse geological scenarios. In this study, we propose an approach that integrates variable dictionary learning and scale-space methods to address these challenges. Our method employs learned dictionaries, allowing for adaptive representation of complex subsurface features that are difficult to capture with predefined bases. Additionally, we extend classical variational inversion by incorporating multi-scale representations through a scale-space framework, enabling the progressive introduction of structural detail while mitigating overfitting. We implement both fixed and dynamic dictionary learning techniques, with the latter introducing iteration-dependent dictionaries for enhanced flexibility. Using a synthetic dataset to simulate geological scenarios, we demonstrate significant improvements in reconstruction accuracy and robustness compared to conventional variational and dictionary-based methods. Our results highlight the potential of learned dictionaries, especially when coupled with scale-space dynamics, to improve model recovery and noise handling. These findings underscore the promise of our data-driven approach for advance magnetic data inversion and its applications in geophysical exploration, environmental assessment, and mineral prospecting. The code is publicly available at: https://github.com/ahxmeds/magnetic-inversion-dictionary.git.
In recent years, deep learning (DL) has emerged as a promising alternative approach for various seismic processing tasks, including primary estimation (or multiple elimination), a crucial step for accurate subsurface imaging. In geophysics, DL methods are commonly based on supervised learning from large amounts of high-quality labelled data. Instead of relying on traditional supervised learning, in the context of free-surface multiple elimination, we propose a method in which the DL model learns to effectively parameterize the free-surface multiple-free wavefield from the full wavefield by incorporating the underlying physics into the loss computation. This, in turn, yields high-quality estimates without ever being shown any ground truth data. Currently, the network reparameterization is performed independently for each dataset. We demonstrate its effectiveness through tests on both synthetic and field data. We employ industry-standard Surface-Related Multiple Elimination (SRME) using, respectively, global least-squares adaptive subtraction and local least-squares adaptive subtraction as benchmarks. The comparison shows that the proposed method outperforms the benchmarks in estimation accuracy, achieving the most complete primary estimation and the least multiple energy leakage, but at the cost of a higher computational burden.
First-break picking is a pivotal procedure in processing microseismic data for geophysics and resource exploration. Recent advancements in deep learning have catalyzed the evolution of automated methods for identifying first-break. Nevertheless, the complexity of seismic data acquisition and the requirement for detailed, expert-driven labeling often result in outliers and potential mislabeling within manually labeled datasets. These issues can negatively affect the training of neural networks, necessitating algorithms that handle outliers or mislabeled data effectively. We introduce the Simultaneous Picking and Refinement (SPR) algorithm, designed to handle datasets plagued by outlier samples or even noisy labels. Unlike conventional approaches that regard manual picks as ground truth, our method treats the true first-break as a latent variable within a probabilistic model that includes a first-break labeling prior. SPR aims to uncover this variable, enabling dynamic adjustments and improved accuracy across the dataset. This strategy mitigates the impact of outliers or inaccuracies in manual labels. Intra-site picking experiments and cross-site generalization experiments on publicly available data confirm our method's performance in identifying first-break and its generalization across different sites. Additionally, our investigations into noisy signals and labels underscore SPR's resilience to both types of noise and its capability to refine misaligned manual annotations. Moreover, the flexibility of SPR, not being limited to any single network architecture, enhances its adaptability across various deep learning-based picking methods. Focusing on learning from data that may contain outliers or partial inaccuracies, SPR provides a robust solution to some of the principal obstacles in automatic first-break picking.
Machine learning methods have shown great success in various scientific areas, including fluid mechanics. However, reconstruction problems, where full velocity fields must be recovered from partial observations, remain challenging. In this paper, we propose a novel operator learning framework for solving reconstruction problems by using the Energy Transformer (ET), an architecture inspired by associative memory models. We formulate reconstruction as a mapping from incomplete observed data to full reconstructed fields. The method is validated on three fluid mechanics examples using diverse types of data: (1) unsteady 2D vortex street in flow past a cylinder using simulation data; (2) high-speed under-expanded impinging supersonic jets impingement using Schlieren imaging; and (3) 3D turbulent jet flow using particle tracking. The results demonstrate the ability of ET to accurately reconstruct complex flow fields from highly incomplete data (90\% missing), even for noisy experimental measurements, with fast training and inference on a single GPU. This work provides a promising new direction for tackling reconstruction problems in fluid mechanics and other areas in mechanics, geophysics, weather prediction, and beyond.
Recently, the advent of generative AI technologies has made transformational impacts on our daily lives, yet its application in scientific applications remains in its early stages. Data scarcity is a major, well-known barrier in data-driven scientific computing, so physics-guided generative AI holds significant promise. In scientific computing, most tasks study the conversion of multiple data modalities to describe physical phenomena, for example, spatial and waveform in seismic imaging, time and frequency in signal processing, and temporal and spectral in climate modeling; as such, multi-modal pairwise data generation is highly required instead of single-modal data generation, which is usually used in natural images (e.g., faces, scenery). Moreover, in real-world applications, the unbalance of available data in terms of modalities commonly exists; for example, the spatial data (i.e., velocity maps) in seismic imaging can be easily simulated, but real-world seismic waveform is largely lacking. While the most recent efforts enable the powerful diffusion model to generate multi-modal data, how to leverage the unbalanced available data is still unclear. In this work, we use seismic imaging in subsurface geophysics as a vehicle to present ``UB-Diff'', a novel diffusion model for multi-modal paired scientific data generation. One major innovation is a one-in-two-out encoder-decoder network structure, which can ensure pairwise data is obtained from a co-latent representation. Then, the co-latent representation will be used by the diffusion process for pairwise data generation. Experimental results on the OpenFWI dataset show that UB-Diff significantly outperforms existing techniques in terms of Fr\'{e}chet Inception Distance (FID) score and pairwise evaluation, indicating the generation of reliable and useful multi-modal pairwise data.
Full waveform inversion (FWI) often faces challenges due to inadequate seismic observations, resulting in band-limited and geologically inaccurate inversion results. Incorporating prior information from potential velocity distributions, well-log information, and our geological knowledge and expectations can significantly improve FWI convergence to a realistic model. While diffusion-regularized FWI has shown improved performance compared to conventional FWI by incorporating the velocity distribution prior, it can benefit even more by incorporating well-log information and other geological knowledge priors. To leverage this fact, we propose a geological class and well-information prior-assisted FWI using conditional diffusion models. This method seamlessly integrates multi-modal information into FWI, simultaneously achieving data fitting and universal geologic and geophysics prior matching, which is often not achieved with traditional regularization methods. Specifically, we propose to combine conditional diffusion models with FWI, where we integrate well-log data and geological class conditions into these conditional diffusion models using classifier-free guidance for multi-modal prior matching beyond the original velocity distribution prior. Numerical experiments on the OpenFWI datasets and field marine data demonstrate the effectiveness of our method compared to conventional FWI and the unconditional diffusion-regularized FWI.
Inverse problems, which involve estimating parameters from incomplete or noisy observations, arise in various fields such as medical imaging, geophysics, and signal processing. These problems are often ill-posed, requiring regularization techniques to stabilize the solution. In this work, we employ Flow Matching (FM), a generative framework that integrates a deterministic processes to map a simple reference distribution, such as a Gaussian, to the target distribution. Our method DAWN-FM: Data-AWare and Noise-informed Flow Matching incorporates data and noise embedding, allowing the model to access representations about the measured data explicitly and also account for noise in the observations, making it particularly robust in scenarios where data is noisy or incomplete. By learning a time-dependent velocity field, FM not only provides accurate solutions but also enables uncertainty quantification by generating multiple plausible outcomes. Unlike pre-trained diffusion models, which may struggle in highly ill-posed settings, our approach is trained specifically for each inverse problem and adapts to varying noise levels. We validate the effectiveness and robustness of our method through extensive numerical experiments on tasks such as image deblurring and tomography.
The decomposition of a signal is a fundamental tool in many fields of research, including signal processing, geophysics, astrophysics, engineering, medicine, and many more. By breaking down complex signals into simpler oscillatory components we can enhance the understanding and processing of the data, unveiling hidden information contained in them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in handling mono-dimensional stationary signals struggle with non-stationary data sets and they require, this is the case of the wavelet, the selection of predefined basis functions. In contrast, the Empirical Mode Decomposition (EMD) method and its variants, such as Iterative Filtering (IF), have emerged as effective nonlinear approaches, adapting to signals without any need for a priori assumptions. To accelerate these methods, the Fast Iterative Filtering (FIF) algorithm was developed, and further extensions, such as Multivariate FIF (MvFIF) and Multidimensional FIF (FIF2), have been proposed to handle higher-dimensional data. In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique, an innovative method that extends FIF to handle data that vary simultaneously in space and time. This new algorithm is capable of extracting Intrinsic Mode Functions (IMFs) from complex signals that vary in both space and time, overcoming limitations found in prior methods. The potentiality of the proposed method is demonstrated through applications to artificial and real-life signals, highlighting its versatility and effectiveness in decomposing multidimensional and multivariate nonstationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis across many scientific and engineering disciplines.
Recovering the useful signal from seismic field data is critical in seismic data processing. Seismic field data are usually coupled by a useful signal and field noise (random noise with unknown distribution), making them difficult to decouple. Unfortunately, subject to the assumption biases of data prior and noise prior, different random noise attenuation methods may have different performance biases. Suppose data-driven supervised deep learning methods have plenty of labeled [real noisy (field) and clean (useful)] data pairs. In that case, they will learn useful information from the labeled dataset and relax the biases of the data-prior and noise-prior assumptions. To this end, we first use the invertible neural network (INN) to disentangle the field data in observational space into the latent variable in latent space. Then, by manipulating the latent variable’s partitions encoding high- and low-frequency information, INN can generate quality-controlled fake field data and decouple useful signal and field noise parts from field data in its backward pass. To gain decoupling and generative capabilities, the training of our INN only requires a relatively small labeled dataset containing field–useful data pairs. Sampling in latent space, the trained INN can generate an infinite number of paired (fake field and useful) samples. Experiments show that our method can effectively decouple useful signal and field noise, and the noise of the fake field data is close to field noise. The generated paired dataset can benefit downstream tasks, such as field noise attenuation.