Seismic acoustic impedance (AI) inversion is essential for reservoir prediction and characterization. In recent years, deep learning has shown immense potential as a data-driven approach in seismic data processing, inversion, and interpretation. As a data-driven method, deep learning-based seismic inversion results better when sufficient labeled data are provided. Overfitting and poor generalization often occur when labels are insufficient. Due to the lack of labeled data in seismic inversion problems, the difficulty of inversion increases, leading to unstable and poor generalization of prediction results. To partially address this issue, we propose a constrained seismic inversion strategy. Since seismic records are time series, we exploit the convolutional neural network (CNN) and bidirectional LSTM (Bi-LSTM) network structures that are more applicable to time series. We combine the physical model and the initial model as constraints to improve the network stability and generalization ability, and impose sparse constraints on the reflection coefficient to further improve the prediction accuracy. The network structure transformation improves the efficiency and stability of the training process. Through numerical experiments and real data tests, it is proved that the proposed method improves the vertical resolution and geological reliability, providing a more stable and efficient method for seismic inversion under conditions of limited labeled data. The overall performance improved by 2% through comparative analysis.

Petrophysical seismic inversion is a challenging problem due to its intrinsic nonlinearity and ill-posedness. Deep learning emerges as a promising solution to tackle this intricate problem, with semi-supervised learning proving particularly valuable in scenarios with limited labeled data. However, many existing semi-supervised learning approaches applied to reservoir parameters inversion are unidimensional or focus on single model parameters, potentially hindering the attainment of highly accurate predictions for multiple petrophysical parameters. To this end, we introduce a novel knowledge-driven semi-supervised deep learning approach for multidimensional petrophysical seismic inversion. This framework features a lightweight 2-D UNet, incorporating prior knowledge about the range of model parameters, to parameterize the set of pseudo-inverse operators, enabling effective multitask learning. By leveraging the low-frequency porosity as the sole initial model input, our approach enhances the information-sharing capabilities of the neural network. We also introduce Hermite cubic splines to parameterize source wavelets varying with angles, ensuring smooth and compactly supported waveforms. In addition, we develop a semi-supervised training loss function that integrates deterministic forward operators and sampling operators, allowing simultaneous updating of weights in both forward and pseudo-inverse operators. The proposed method facilitates the simultaneous inversion of wavelets, porosity, water saturation, and clay volume. Synthetic and field data tests are conducted to validate our approach, demonstrating that it significantly enhances inversion accuracy compared to 1-D semi-supervised deep learning methods.

Ground-penetrating radar (GPR) is a pivotal noninvasive tool that yields subsurface images critical to archeology, near-surface characterization, geotechnical studies, and disaster response. The antenna central frequency of the GPR system has a significant impact on penetration depth and resolution. Lower antenna frequencies penetrate deeper but at lower resolutions, while higher frequencies offer detailed images at reduced depths. Therefore, improving the resolution of low-frequency radar with increased detection depth is an essential research focus. Inspired by image super-resolution advancements, supervised deep learning methods that rely on strictly paired training data have achieved remarkable success. However, acquiring such paired samples in practical scenarios is often a formidable challenge. To tackle this, we propose a novel resolution enhancement technique through weakly supervised learning, effectively addressing the scarcity of strictly paired samples in real-world situations. We utilize two sets of antennas with different central frequencies to construct our training data, with a low-frequency antenna as input and a high-frequency antenna as the learning target. A cycle-consistent generative adversarial network (Cycle-GAN) is trained to discern the mapping between low-resolution inputs and unpaired high-resolution data. The refined network is then employed to improve low-frequency GPR data resolution. Our work is validated on synthetic and real-world datasets. The proposed method effectively strengthens critical high-frequency details for finer imaging and broadens the frequency bandwidth. Significantly, it enhances resolution without compromising the detection depth of low-resolution GPR data, marking a substantial advancement in subsurface imaging technology.

In marine exploration, conventional migration algorithms based on primary reflections often have poor angular illumination, especially in shallow areas and complex salt boundaries. Source density is the primary controller of angular illumination. It is well-known that source density is relatively sparse in a typical marine seismic acquisition, such as in a towed streamer survey. In contrast, imaging of multiples treats each receiver as a virtual source, mimicking a high-density source survey. Imaging of multiples can provide additional angular illumination and help to improve subsurface imaging. The extra illumination provided by multiple reflections enhances angle-domain common-image gathers (ADCIGs), a component of velocity model building and amplitude versus angle (AVA) analysis. Our main objective is to illustrate the advantages of angular illumination from imaging of multiples in the angular domain. To achieve this purpose, we propose a workflow for calculating high-quality ADCIGs for imaging of multiples. The workflow mainly contains up-going and down-going wavefield decomposition and the stabilized Poynting vectors for calculating more accurate imaging angles. To illustrate the accuracy of the proposed workflow, a simple model is used to calculate angle gathers for imaging of multiples. Then, the Sigsbee2b model and a field dataset from the Gulf of Mexico are used to compute angle gathers by imaging primaries and multiples to demonstrate the improved angular illumination achieved when multiples are also used for imaging. The comparison results show that the angle gathers obtained by imaging of multiple has better angular illumination, especially in shallow areas and complex salt boundaries.

Convolutional dictionary learning (CDL) can represent signals and images via the superposition of components given by the convolution of sparse coefficients (features) and the elements of a dictionary (filters). The filters represent universal signals that can model different images, whereas the coefficients are intrinsic to one particular image. Estimating the coefficients and the filters from a set of observed signals is similar to a blind deconvolution problem where we aim to simultaneously represent a signal via the convolution of two unknown signals. Classical CDL provides data-dependent filters that, in the seismic data processing case, might not have a solid resemblance to typical waveforms that one observes in seismic records. This limits the dictionary’s representation and discriminability, thus suffering from suboptimal denoising or reconstruction results. To address this challenge, we propose a new CDL algorithm. The proposed approach introduces a parametric constraint to enforce simplicity on the filters, guiding the learning process toward a more efficient and structured representation of the data. Specifically, we restrict each filter to include one single waveform parametrizable via a second-order traveltime curve and a seismic wavelet. The learned dictionary comprises linear and parabolic events that adapt adequately to observed seismic waveforms and resemble local Radon transform basis functions. The alternating direction method of multipliers (ADMMs) is adopted to solve the proposed parametric convolutional learning problem. The experimental results demonstrate that the proposed method achieves superior reconstruction results compared to the existing convolutional and patch-based dictionary learning methods.

Seismic acquisition costs are directly associated with the number of sensors used in the survey. Limiting the number of sensors in a seismic survey can be beneficial, especially when sensors are expensive to purchase, deploy, and maintain. This work explores an optimal design method for ocean bottom node (OBN) detector deployment. The proposed method is based on a reinforcement learning (RL) approach. We assume access to an initial dataset over the area of study. These data are used to extract an overcomplete prelearned basis library via the proper orthogonal decomposition (POD) method, which leads to a fast least-squares seismic data reconstruction algorithm. Then, the sensor selection procedure entails using$Q$-learning to find the sensor configuration that maximizes the reconstruction quality.

Multicomponent seismic exploration as a powerful geophysical technology has attracted more and more attention in seeking and identifying subtle oil and gas reservoirs. However, like traditional single-component acquisition, multicomponent seismic data acquisition also encounters sparse and irregular sampling problems. The conventional reconstruction methods treat multicomponent data as independent components and recover the missing traces in a component-by-component manner. These componentwise approaches ignore the internal mutual relationships among different components and damage the vector characteristics of the seismic wavefield. Quaternion algebra provides an effective vector representation tool for multicomponent data. Upon that, we present a new vector Projection Onto Convex Sets (POCS) method with biquaternion Fourier transform to reconstruct the irregularly missing traces of 3-D and three-component (3-D-3C) seismic data. Compared to the current real-valued quaternion reconstruction methods implemented in the time domain, the proposed method performed in the frequency domain can fully maintain the conjugate symmetry property of the biquaternion data and only reconstruct the positive or negative frequency slices of the observed 3-D-3C data. Besides, the new method can simultaneously interpolate the 3C or 4C data with different missing patterns. We compare the proposed vector POCS method with the scalar POCS method through experiments with a synthetic 3-D-3C dataset and a field 3-D-3C volume. Both experiments demonstrate the effectiveness and superiority of the proposed method.

In the above article[1], in Fig. 5(b), the Amplitude Error subfigures of Y and Z components are the same as the Amplitude Error subfigure of X component. They are incorrect. They have been replaced by the correct Amplitude Error subfigures of Y and Z components inFig. 1.

The attenuation of coherent noise in land seismic data, specifically ground roll and near-surface scattered energy, remains a longstanding challenge. Although recent advances in deep learning have improved signal separation from coherent noise, supervised methods are limited by the necessity for realistic training samples. To circumvent this issue, we propose an unsupervised deep learning approach to attenuate ground roll and scattered energy, eliminating the requirement for training labels. Our method leverages the inherent low-frequency bias of a generator network, which is naturally prone to learn self-similar features during training. This empowers the network to extract the desired component exhibiting self-similarity in the time-space domain, while disregarding unwanted components. Notably, horizontal components in seismic data exhibit pronounced self-similarity. To enhance the self-similarity of ground roll, we apply a linear moveout (LMO) correction to horizontally align it and utilize the generator network for separation. Additionally, for scattered energy attenuation, we employ the generator network to extract flattened reflections after normal moveout (NMO) correction. Our strategy distinctively merges model-driven procedures, specifically NMO and LMO, anchored in the geological velocity model. The synergy between data-driven deep learning and model-driven processes underscores the success of our approach. We demonstrate the validity of our proposed method using both synthetic and field shot data. The field data examples highlight the superior attenuation capabilities of our method, surpassing conventional denoising techniques by effectively reducing both random and coherent noise.

A robust vector MSSA algorithm for denoising seismic data is introduced, and initial results of applying this algorithm to denoise synthetic and real seismic data records are presented. In particular, the MSSA algorithm, originally applied to denoise scalar seismic wavefields, is generalized to denoise vector seismic wave fields. We also introduce a robust rank reduction algorithm within the MSSA denoising algorithm which attenuates erratic signals in the input wavefield without distorting the output.

We solve the simultaneous source separation problem by adopting the projected gradient descent (PGD) method to iteratively estimate the data one would acquire via a conventional seismic acquisition. The projection operator is a windowed robust singular spectrum analysis (SSA) filter that suppresses source interferences in the$f-x$(frequency-space) domain. We reformulate the SSA filter as a robust optimization problem solved via a bifactored gradient descent (BFGD) algorithm. Robustness becomes achievable by adopting Tukey’s biweight loss function for the design of the robust SSA filter. The SSA filter requires breaking down common-receiver gathers or common offset gathers into small overlapping windows. The traditional SSA method needs the filter rank as an input parameter, which can vary from window to window. The latter has been a shortcoming for the application of classical SSA filtering to complex seismic data processing. The proposed robust SSA filter is less sensitive to rank-selection, making it appealing for deblending applications that require windowing. Additionally, the robust SSA projection provides an effective attenuation of random source interferences during the initial iterations of the PGD method. Comparing classical and robust SSA filters, we also report an acceleration of the PGD method convergence when we adopt the robust SSA filter. Finally, we provide synthetic and real data examples, and discuss heuristic strategies for parameter selection.

Five-dimensional seismic reconstruction is receiving increasing attention and can be viewed as a tensor completion problem, which involves reconstructing a low-rank tensor from a partially observed tensor. Tensor train (TT) decomposition and tensor ring (TR) decomposition are two powerful tensor networks for solving this problem. However, updating core tensors leads to high computational costs in practical applications. We propose two efficient methods to exploit low TT-rank and low TR-rank structures by theoretically establishing the relationship between tensor ranks and matrix unfoldings, respectively. Specifically, the former uses a well-balanced matricization scheme, and the latter employs a tensor circular unfolding. Furthermore, we utilize the randomized parallel matrix factorization to accelerate the solution of these problems. Both synthetic and real data experiment demonstrates that the proposed algorithm can also achieve remarkable reconstruction performance; in the meantime, the computational cost is significantly reduced.

Seismic data acquired by geophones are processed to estimate images of the earth's interior that are used to explore, develop, and monitor resources and to study the shallow structure of the crust for geological, environmental, and geotechnical purposes. These multidimensional data sets are often irregularly sampled and incomplete in the so-called midpoint and offset acquisition coordinates. Multidimensional seismic data reconstruction can be viewed as a low-rank matrix or tensor completion problem. In this paper, we introduce a fast and efficient low-rank tensor completion algorithm named parallel square matrix factorization (PSMF) and adopt it to reconstruct seismic data in the typical seismic data processing coordinates: frequency, midpoint, and offset. For each frequency slice, we establish a tensor minimization model composed of a low-rank constrained term and a data misfit term. Then we adopt the PSMF algorithm for the recovery of the missing samples. In the PSMF method, we avoid using unbalanced “long strip” matrices that result from conventional tensor unfolding. Instead, the tensor is unfolded into almost square or square matrices that are low rank. We also compare the proposed PSMF method with other completion methods. Experiments via synthetic data and field data sets validate the effectiveness of the proposed algorithm.

We describe a method that allows for blind surface consistent estimation of the source and receiver wavelets of seismic signals. This is very relevant for surface-consistent deconvolution where current processing standards focus on the removal of the source and receiver effects under the minimum phase assumption. The proposed method, which is an extension of the Euclid deconvolution method, employs an iterative algorithm that simultaneously estimates the source and receiver wavelets that are consistent with the data. Unlike most deconvolution methods, the algorithm requires no prior phase assumptions. Another important feature of the algorithm is that we questioned the Gaussian density assumption of the reflectivity series and instead implemented a sparse regularizer to constrain the solution space of our desired reflectivity series. In other words, we assume that the reflectivity series can be cast as a sparse vector with few nonzero coefficients.

In recent years, efforts have been made in designing simultaneous-source strategies that permit to save seismic acquisition costs. Seismic sources are fired with time overlap producing seismic records that contain a mixture of sources. These records need to be unmixed before seismic imaging. The unmixing process can be written as an inverse problem where one attempts to solve a linear system of equations to estimate the unmixed seismic data. This article describes a source separation process where we assume that source interferences can be modelled via an erratic noise process. In addition, the ideal unmixed data are assumed to be sparse in the Hyperbolic Radon transform domain. Therefore, the source separation problem is posed as an inverse problem where one seeks to retrieve a sparse model from observations contaminated with erratic (sparse) noise. We present a modification of the fast iterative shrinkage-thresholding algorithm that permits to cope with the simultaneous estimation of sparse Radon coefficients that are required to synthesize the unmixed data. The algorithm is also utilized to estimate the erratic noise caused by source interferences.

We consider the problem of multidimensional seismic data signal recovery and noise attenuation. These data are multi-dimensional signals that can be described via a low-rank fourth-order tensor in the frequency-space domain. Tensor completion strategies can be used to recover unrecorded observations and to improve the signal-to-noise ratio of seismic data volumes. Tensor completion is posed as an inverse problem and solved via a convex optimization algorithm where a misfit function is minimized in conjunction with the nuclear norm of the tensor. This formulation offers automatic rank determination. We illustrate the performance of the algorithm with a synthetic example and with a real data set obtained by an onshore seismic survey.

When improving the Earth's description of seismic data via deconvolution, the spatial coherency of the information that can be extracted may be damaged through suboptimal trace-by-trace processing. Furthermore, the source function is usually not known or is known only approximately. This article presents an efficient multichannel blind deconvolution for addressing these problems and restoring three-dimensional (3D) seismic data based on a variational approach. To overcome the ill-posedness of the deconvolution problem, appropriate regularizers are used for the reflectivity and source. A new sparsity and continuity-promoting regularizer is introduced which is able to promote temporal sparsity and at the same time continuity along 3D singularities in reflectivity space. Sparsity in a wavelet domain is used as source prior. To find a solution for the overall problem, we alternate between two subproblems: 3D reflectivity estimation and updating the source. Each subproblem consists of solving a convex constrained optimization which is carried out very fast and efficiently via the alternating split Bregman iteration. We use the generalized cross validation criterion for determining the optimum number of Bregman iterations. We illustrate the performance and optimality of our blind deconvolution with simulated and field seismic data.

We present an efficient deconvolution method to retrieve sparse reflectivity series from seismic data in the presence of additive Gaussian and non-Gaussian noise. The problem is first formulated as an unconstrained optimization including a mixedlp-l1measure for the data misfit and for the model regularization term, respectively. An efficient algorithm based on the alternating split Bregman technique is developed, and a numerical procedure based on the generalized cross-validation (GCV) technique is presented for the selection of the corresponding regularization parameter. To circumvent excessive computations of multiple optimizations to determine the minimizer of GCV curve, we formulate the deconvolution problem in the frequency domain as a basis pursuit denoising and solve it using the split Bregman algorithm with computational complexityO(Nlog(N)). Apart from significant stability against outliers in the data, the main advantage of such formulation is that the GCV curve can be generated during the iterations of the optimization procedure. The minimizer of the GCV curve is then used to properly determine the error bound in the data and hence the optimum number of iterations. Numerical experiments show that the proposed method automatically generates high-resolution solutions by only a few iterations needless of any prior knowledge about the noise in the data.

Although data parallelism is a well-known computational model, there are few programming systems that are both easy to program (for simple applications) and able to work across administrative domains. For data sets (e.g., collections of image data) that are often inherently distributed, there is a need for a simple data-parallel programming system. We describe the design, implementation, and an evaluation of Trellis-SDP, a simple data-parallel programming system that facilitates the rapid development of data- intensive applications. Trellis-SDP is layered on top of the Trellis infrastructure, a software system for creating overlay metacomputers: user-level aggregations of computer systems. Trellis-SDP is based on file-level data parallelism and provides a Master-Worker programming framework in which the worker components can run self-contained, new or existing binary applications. We evaluate our programming system with a non-trivial seismic data processing application.