The separation of upgoing and downgoing waves in vertical seismic profiling (VSP) data is crucial for subsequent imaging, interpretation, and inversion. The interweaving of upgoing and downgoing waves and the presence of noise complicate the entire wave field, making it difficult to separate upgoing and downgoing waves. Various model-driven separation methods including low-rank approximation (LRA) and data-driven methods have achieved promising results. However, the performance of pure model-driven methods, such as F-K filtering and Radon transform, rely on domain transformation sparse representation for both upgoing and downgoing wavefield. Additionally, pure data-driven methods require a large number of precisely separated signals as training samples, which is a nontrivial task. To overcome these difficulties, this article proposes a model-data-driven VSP wavefield separation framework that iteratively completes the task of wavefield separation in an unsupervised manner. The key to this model is to use the powerful feature representation capability of the deep convolutional autoencoder (DCAE) to model the downgoing waves and use LRA to model the upgoing waves. By accurately modeling the upgoing and downgoing waves, we integrate the model-driven and data-driven methods together, while protecting both the upgoing and downgoing waves, and inheriting the advantages of the DCAE and LRA methods. Subsequently, we also proposed an alternating minimization optimization strategy to optimize the parameters of the model and iteratively obtain high-quality solutions. Comparative experiments on synthetic data and real data show that our method can achieve effective separation results while suppressing Gaussian noise.

The popular iterative Vertical seismic profiling (VSP) wavefield separation method has higher precision wavefield separation results than the conventional VSP scalar wavefield separation method, but it still faces challenges in achieving desirable high-precision wavefield separation results due to the time-variant wavelet, the complex wavefield, and the precision of wavefield flattening. In this paper, in order to minimize the impact of the factors mentioned above, we propose a trace-by-trace iterative VSP wavefield separation method (TISM) based on the gradually changing characteristics of VSP data in adjacent traces, the cross-correlation, and the iterative VSP wavefield separation method. TISM includes the flowchart of TISM, the target trace guided sub dataset automatic generation (TDG), the cross-correlation guided sub dataset optimization (CDO), and the cross-correlation guided wavefield separation result optimization (CWO). TDG aims to automatically generate the sub VSP dataset corresponding to the target trace and minimize the impact of the time-variant wavelet and complex wavefield. CDO aims to minimize the effect of wavefield flattening and form the high-precision wavefield flattened sub dataset for scalar wavefield separation. CWO aims to obtain the high-precision wavefield separation result. Synthetic and actual VSP data applications demonstrate that TISM can minimize the impact of the time-variant wavelet, the complex wavefield and wavefield flattening, thereby obtaining high-precision VSP wavefield separation results.

The 3-D seismic multihorizon extraction is crucial for 3-D sequence stratigraphy analysis and reservoir modeling. Deep neural networks (DNNs) often cause dislocated horizons in regions with complex geological structures, such as faults and unconformities. This issue arises from two main factors. First, obtaining horizon labels from real seismic data is subjective and expensive, resulting in existing DNNs lacking field seismic data labels, which limits their ability to extract multiple horizons across complex geological structures. Second, directly extracting multiple horizons using only seismic data reduces precision in discontinuous areas with faults and unconformities. To address these issues, this article proposes a domain adaptation layer based on multikernel maximum mean discrepancy (MK-MMD) and designs a domain adaptive DNN (DA-DNN) for seismic multihorizon extraction. We map synthetic and field seismic data to the reproducing kernel Hilbert space (RKHS) and use MK-MMD to minimize feature differences between them. Unlike the traditional multiscale Gaussian kernel function used in MK-MMD, this article constructs a hybrid kernel function that integrates multiscale Gaussian and multiscale Laplacian kernels. The multiscale Gaussian kernel evaluates local-to-global feature differences in continuous areas, whereas the multiscale Laplacian kernel captures rapid feature variations in complex geological structures. Finally, a few seismic horizons and fault attributes guide the training process of DA-DNN, further improving the prediction accuracy of multihorizon extraction. Synthetic and field seismic examples show our model can extract seismic multiple horizons more accurately in field seismic data and performs better in discontinuous areas.

The popular deep-learning-based horizon tracking methods heavily relies on large volumes of well-labeled horizon data, which face significant challenges in achieving high-precision horizon tracking with limited label (or few sample), especially when encountering complex seismic data and geological structures with 1-D limited label. In this article, we propose a limited-label multiscale multihorizon tracking method (LMMT) based on the multimodal deep learning and (1-D limited label. In this method, the horizon is characterized in the seismic trace (1-D), the seismic profile (2-D), and the horizon slice (3-D). LMMT is comprised of the flowchart of LMMT, the 1-D convolution kernel single-modal multihorizon tracking method (OMT), the high-precision high-continuity horizon and strata optimization (HHO), and the 2-D (or 3-D) convolution kernel multimodal multihorizon tracking method (TMT). OMT takes the input 1-D limited horizon labels as its labels and utilizes a 1-D convolution kernel for strata division and multihorizon tracking. HHO aims to generate the 2-D (or 3-D) high-precision and high-continuity horizon and strata based on the 3-D horizon tracking results obtained by OMT or TMT, thereby providing high-precision high-continuity horizon labels and strata for TMT. TMT incorporates the 2-D (or 3-D) high-precision high-continuity horizon obtained by HHO as its labels, integrates the random masking result of the high-precision high-continuity strata obtained by HHO as the reference information, and utilizes a 2-D (or 3-D) convolution kernel for high-precision multihorizon tracking. Two 3-D seismic dataset applications demonstrate that LMMT achieves high-precision multihorizon tracking results with limited labels.

Due to the non-Gaussian distribution of erratic noise, conventional Gaussian denoising methods often encounter substantial challenges and pressures when suppressing this kind of noise. To overcome this challenge, several state-of-the-art (SOTA) schemes, for instance, robust low-rank approximation (LRA) and deep learning (DL) methods, have been designed and achieved promising results in the treatment of erratic noise. However, these SOTA denoising methods focus mainly on matrix-based modeling representations and fail to fully reflect the correlations associated with erratic noise and valid signals in the spatial dimension and thus may display suboptimal performance. As an alternative, a robust tensor DL (RTDL) denoising method for unsupervised 3-D seismic erratic noise suppression that involves the use of a reasonable combination of tensor sparse representation (SR) and a tensor neural network (tNN) is proposed in this study. The key to RTDL is to introduce a robust tensor sparse norm for erratic noise to exhaustively exploit the spatial tubular sparse distribution properties in 3-D space; notably, adding a tensor sparse norm to the tNN model yields a new data-driven model with 3-D erratic noise reduction capabilities. To find the optimized parameters of the new model, an efficiency tensor optimization method is established on the basis of alternating minimization (Alt), the aim of which is to alternately solve two subproblems involving tensor SR and a tNN. This paper presents well-designed experiments and satisfactory results compared with those of SOTA methods based on both synthetic and real field datasets.

Due to the poorly coupled geophones present in boreholes, vertical seismic profiling (VSP) data are known to suffer from intense coupling noise, which causes severe VSP image deterioration and significantly hinders subsequent processing. Thus, diverse denoising approaches are indispensable preprocessing steps for suppressing this kind of noise to achieve good results. Among them, robust principal component analysis (RPCA) is a common signal and noise separation model that is generally considered a highly promising method for removing intense coupling noise; however, handcrafted priors have limited denoising ability, especially for the low-rank assumption of useful signals. As an alternative, following the RPCA framework, this article proposes an unsupervised iterative robust deep convolutional autoencoder (IRDCAE) model to suppress intense VSP coupling noise without any assumptions regarding valuable signals. The key to the IRDCAE approach is the use of weighted column sparsity (WCS) to characterize the behavior of the intense coupling noise, where the weight prior is derived from the pure noise component before the first break. By adding a WCS regularization term to the conventional deep convolutional autoencoder (DCAE), our IRDCAE method transforms the model from an entirely data-driven model to a model+data driven approach. Thus, the IRDCAE approach has the advantages of both RPCA and DCAE, resulting in the ability to separate intense coupling noise from useful signals in an unsupervised manner by optimizing the IRDCAE model via an alternating minimization algorithm. The exceptional performance of the IRDCAE model is exhibited with synthetic and field VSP data.

A key challenge of 3-D seismic horizon tracking lies in effectively utilizing the appropriate seismic attributes to enhance tracking precision. Numerous existing horizon tracking methods based on models or deep learning (DL), hinge on mathematical or data-driven mapping relationships between seismic data and the target horizons.This mapping utilizes only seismic data or a single attribute, resulting in inaccurate horizon tracking across discontinuities. As an alternative approach, we propose a multiple attribute regression network (MARN) for 3-D seismic horizon tracking, which leverages multiple seismic attributes to achieve precise and robust tracking results. In this study, we initiate by formulating the problem as a multiple attribute regression model, subsequently the system state equation is introduced to establish temporal relationship. To address this nonlinear regression model, the deep convolutional autoencoder (DCAE), which possesses the capability to automatically learning spatial correlation from attributes, thereby extracting deep features. In this case, these features from DCAE serve as input sequences, enabling long short-term memory (LSTM) to effectively model the spatial-temporal relationships between multiple seismic attributes and target horizons. The proposed MARN is thoroughly compared with the single attribute regression network (SARN) on two real field 3-D datasets.

The volumetric local slope, which indicates the orientation of seismic events, plays a prominent role in the subsequent geological interpretation typically including horizon tracking, seismic facies analysis, and fault interpretation. Although numerous existing estimation methods are available, they still suffer from the challenge of reaching a balance between resolution preservation and resisting the heavy random noise. As an alternative, this letter proposes a seismic volumetric local slope estimation method named the multiscale gradient structure tensor (MGST), combining GST with the 3-D multiscale Gaussian pyramid (GP). In this regard, to preserve the details of the original resolution and fully exploit the unique information at different scales, the GP is reconstructed in 3-D space by decomposing the data into multiple scales. After that, we attempt to employ the GST to derive the local slopes in two directions at each scale, along with a corresponding quality metric. Finally, within the Kalman filter framework, the local slope of each scale is sequentially integrated using the quality metric as the weighting mechanism, resulting in an accurate and robust estimation. Experiments on both synthetic and real field datasets indicate that the proposed MGST method outperforms the traditional GST and plane-wave destruction (PWD) methods.

Seismic facies analysis (SFA) plays a pivotal role in the interpretation of subsurface structures, with a pressing need to develop automated techniques for analyzing 4-D prestack seismic data. Various automated SFA methods, encompassing both supervised and unsupervised paradigms, have shown encouraging potential in fulfilling this demand. Nonetheless, supervised methods heavily hinge upon precious labeled seismic datasets of high caliber, and unsupervised methods handle all seismic samples indiscriminately during training, resulting in pronounced biases toward the majority classes of seismic data. As an alternative, this letter proposes class-imbalanced deep embedding clustering (CDEC), an unsupervised deep clustering methodology devised to analyze seismic data with class-imbalanced facies distributions. Within CDEC, a focal loss meticulously tailored to address class imbalance challenges is seamlessly integrated into the classic deep convolutional embedding clustering (DCEC). By balancing weights between the minority and majority seismic samples during network training, this approach adeptly attenuates biases toward the majority classes while concurrently bolstering the efficacy of SFA. Experimental evaluations conducted on synthetic and real field datasets compellingly underscore the effectiveness and utility of the proposed CDEC method.

Random plus footprint noise provokes severe seismic image deterioration and makes it challenging for interpreters to recognize and analyze accurate subsurface responses. Thus, as an elementary and indispensable preprocessing step, diverse footprint removal approaches, including filtering in the frequency or time–frequency domain and dictionary learning (DL), have been documented to achieve promising results in tackling this challenge. However, the prevailing denoising methods tend to treat 3-D seismic data as images for processing, but such flattening or matricization operations inevitably obliterate the 3-D image structures concealed in noisy observational seismic data, which hinders the removal performance of these approaches. To resolve this issue, this article proposes a new tensor model for 3-D seismic random plus footprint noise suppression, and this model is based on unidirectional total variation regularized low-rank tensor approximation (UTV-LRTA). In this model, UTV regularization is imposed to obtain the innately structural and directional behavior of the acquisition footprint. In this way, the footprint is removed by effectively decomposing the footprint-contaminated seismic image into a footprint-free image and a footprint component by UTV. In contrast, random noise is mitigated by regularizing the low rankness of the third-order seismic tensors using the tensor nuclear norm. Moreover, a simple and powerful optimization algorithm based on the split Bregman iteration is introduced to resolve the proposed UTV-LRTA model. The suggested model is thoroughly assessed on synthetic and field datasets and significantly surpasses the state-of-the-art approaches quantitatively and qualitatively evaluated in the analyzed field examples.

Seismic acquisition footprints appear as stably faint and dim structures and emerge fully spatially coherent, causing inevitable damage to useful signals during the suppression process. Various footprint removal methods, including filtering and sparse representation (SR), have been reported to attain promising results for surmounting this challenge. However, these methods, e.g., SR, rely solely on the handcrafted image priors of useful signals, which is sometimes an unreasonable demand if complex geological structures are contained in the given seismic data. As an alternative, this article proposes a footprint removal network (dubbed FR-Net) for the unsupervised suppression of acquired footprints without any assumptions regarding valuable signals. The key to the FR-Net is to design a unidirectional total variation (UTV) model for footprint acquisition according to the intrinsically directional property of noise. By strongly regularizing a deep convolutional autoencoder (DCAE) using the UTV model, our FR-Net transforms the DCAE from an entirely data-driven model to a prior-augmented approach, inheriting the superiority of the DCAE and our footprint model. Subsequently, the complete separation of the footprint noise and useful signals is projected in an unsupervised manner, specifically by optimizing the FR-Net via the backpropagation (BP) algorithm. We provide qualitative and quantitative evaluations conducted on three synthetic and field datasets, demonstrating that our FR-Net surpasses the previous state-of-the-art (SOTA) methods.

Side window filtering (SWF) can effectively capture detailed image edges and is widely applied in image processing. However, its fixed-scale (or fixed-size) filter kernel cannot adapt to complex images, and the final output at the target pixel is only determined by the side window output with the minimum error functional, limiting its filtering capability. To further enhance the filtering capability of SWF, we first extend the traditional side windows with fixed-scale filter kernel to multiscale side windows by introducing the multiscale filter kernels, which leads to better complex image matching. Then, we further introduce an adaptively weighted parameter, which is inversely proportional to the error functional, to fully consider the contributions of all multiscale side windows to the final output. We finally propose the multiscale adaptive SWF (MASWF). Synthetic and field seismic data examples demonstrate that MASWF is a good potential technique for seismic data random noise attenuation and can be widely used in digital signal processing fields.

Erratic seismic noise, following a (known or unknown) non-Gaussian distribution, poses a formidable challenge to conventional methods of random noise attenuation. Many erratic noise cancellation methods, for instance, robust reduced-rank and sparsity-promoting filtering, have been proven to achieve promising results in overcoming this challenge. Among them, deep learning (DL) methods require no assumptions about the underlying clear seismic image and are also more robust against erratic and random noise. However, the success of existing DL-based denoising methods strongly depends on supervised learning from a large number of ground-truth seismic images affected by erratic noise and their clean counterparts, which are typically unavailable in a real-world setting. As an alternative, this article presents an unsupervised DL method for erratic-plus-Gaussian noise removal based on a robust deep convolutional autoencoder (RDCAE). In the RDCAE, the mean squared error (mse) loss in a classic DCAE is replaced by the smooth Welsch function to exploit the concept of robust image denoising. In this way, the erratic noise is downweighted by means of a curbed weight defined in terms of the Welsch function. In contrast, the random noise is diluted by combining the mean square in the Welsch function and the total variation (TV). Subsequently, the training procedures required for solving the RDCAE are derived on the basis of the backpropagation (BP) algorithm for a neural network. Experiments conducted on both synthetic and real field datasets are reported to illustrate the efficacy of the proposed method.

The core challenge of seismic data interpolation is how to capture latent spatial-temporal relationships between unknown and known traces in 3-D space. The prevailing tensor-based interpolation schemes seek a globally low-rank approximation to mine the high-dimensional relationships hidden in 3-D seismic data. However, when the low-rank assumption is violated for data involving complex geological structures, the existing interpolation schemes fail to precisely capture the trace relationships, which may influence the interpolation results. As an alternative, this article presents a basic deep tensor autoencoder (DTAE) and two variants to implicitly learn a data-driven, nonlinear, and high-dimensional mapping to explore the complicated relationship among traces without the need for any underlying assumption. Then, tensor backpropagation (TBP), which can be essentially viewed as a tensor version of traditional backpropagation (BP), is introduced to solve for the new model parameters. For ease of implementation, a mathematical relationship between tensor and matrix autoencoders is constructed by taking advantage of the properties of a tensor–tensor product. Based on the derived relationship, the DTAE weight parameters are inferred by applying a matrix autoencoder to each frontal slice in the discrete cosine transform (DCT) domain, and this process is further summarized into a general theoretical and practical framework. Finally, the performance benefits of the proposed DTAE-based method are demonstrated in experiments with both synthetic and real field seismic data.

The inherent challenge of 3-D seismic noise attenuation is determining how to uncover high-dimensional concise structures that only exist in true signals to eliminate random noise. The prevailing deep learning (DL) denoising methods have achieved promising performance in revealing the compact structures underlying contaminated seismic data. However, as clean ground-truth seismic data are generally unavailable in real-world settings, most existing matrix-based DL denoising schemes fail to automatically describe this type of high-dimensional structure in an unsupervised manner, potentially rendering them unable to effectively perform 3-D seismic data denoising tasks. To tackle this challenge, this article presents a tensor convolutional neural network (TCNN)-based data denoising scheme using Stein’s unbiased risk estimate (SURE) (called SURE-TCNN) to learn intrinsic high-dimensional structures without ground-truth seismic data. Considering that SURE provides an almost unbiased estimate of the mean squared error (MSE), SURE-TCNN has the potential to provide similar results to those of the supervised MSE-based TCNN with ground-truth data. For ease of implementation, the properties of a transform-based tensor-tensor product (t-product) are followed to establish a solid theoretical connection between the SURE-TCNN tensor and matrix. Derived from this connection, the SURE-TCNN weight parameters are determined by implementing matrix-based SURE-convolutional neural networks (CNNs) on each frontal slice in the time-frequency domain (e.g., the wavelet domain). Synthetic and field data examples demonstrate the superior performances of the proposed model against three state-of-the-art (SOTA) methods.

Multidimensional (M-D) seismic data denoising is cast as an underdetermined inverse problem whose solution hinges on effective image priors extracted from machine learning knowledge. However, modeling seismic image priors is challenging due to the M-D nature of seismic images. Among the most promising prevailing image prior techniques is learning prior knowledge of the underlying structure by various 2-D or 3-D deep learning (DL)-based methods. However, for higher-dimensional seismic data such as 4-D prestack data, these DL denoising schemes undoubtedly fail to capture the complete image structure in the absence of the flattening operation. To address this challenge, we present a framelet-based order-ptensor neural network (dubbed the FPTNN) model to implicitly learn the priors reflecting the typical behavior of clear M-D seismic images in a data-driven manner. First, motivated by the supremacy of the framelet transform over the Fourier transform, replacing the Fourier transform with the framelet gives a new definition with respect to the order-ptensor-tensor product (t-product). Then, through the redefined order-pt-product, the order-ptNN framework is a straightforward extension of the tNN with a standard t-product for M-D seismic denoising. By exploiting the fact that the order-pt-product can be computed through matrix multiplication in the framelet domain, we can readily reach the optimal weighted parameters in the FPTNN via DL on a set of transformed matrix frontal slices. The experiments on both synthetic and real field seismic datasets comprehensively demonstrate the advantages of our method against other state-of-the-art (SOTA) methods.

The transfer learning between the source and target domain has already achieved significant success in machine learning areas. However, the existing methods can not achieve satisfactory result when solving the two distant domains transfer learning problem. In the worst case, it could lead to the negative transfer. In this paper, we propose a novel framework called transitive transfer sparse coding (TTSC) to solve the two distant domains transfer learning problem. On the one hand, as an extension of the sparse coding, the TTSC framework constructs a robust and high-level dictionary across three different domains and simultaneously obtains three good feature sparse representations. On the other hand, TTSC utilizes the intermediate domain as a strong bridge to transfer valuable knowledge between the source domain and target domain. Empirical studies validated that the TTSC framework significantly could outperform state-of-the-art methods.

The 3-D seismic data reconstruction can be understood as an underdetermined inverse problem, and thus, some additional constraints need to be provided to achieve reasonable results. A prevalent scheme in 3-D seismic data reconstruction is to compute the best low-rank approximation of a formulated Hankel matrix by rank-reduction methods with a rank constraint. However, the predefined Hankel structure is easily damaged by the low-rank approximation, which leads to harming its recovery performance. In this article, we present a structured tensor completion (STC) framework to simultaneously exploit both the Hankel structure and the low-tubal-rank constraint to further enhance the performance. Unfortunately, under the assumption of elementwise sampling used by existing methods, STC is intractable to be solved since Hankel constraints cannot be expressed as linear tensor equations. Instead, tubal sampling is proposed to describe the missing trace behavior more accurately and further build a bridge between tensor and matrix completion (MC) to overcome the solving issue in two aspects: through the bridge from tensor to MC, STC can be solved efficiently using MC from random samplings of each frontal slice in the Fourier domain. Through the bridge from matrix to tensor completion, various tensor models within the framework can be developed from noise-specific MC to meet the need for data reconstruction in changeable noise environments. Moreover, alternating-minimization and alternating-direction methods of multipliers are developed to solve the proposed STC. The superior performance of STC is demonstrated in both synthetic and field seismic data.

In this paper, we propose a novel method for generating high-granularity three-dimensional (3D) seismic data from low-granularity data based on tensor sparse coding, which jointly trains a high-granularity dictionary and a low-granularity dictionary. First, considering the high-dimensional properties of seismic data, we introduce tensor sparse coding to seismic data interpolation. Second, we propose that the dictionary pairs trained by low-granularity seismic data and high-granularity seismic data have the same sparse representation, which are used to recover high-granularity data with the high-granularity dictionary. Finally, experiments on the seismic data of an actual field show that the proposed method effectively perform seismic trace interpolation and can improve the resolution of seismic data imaging.

Localization technology is important for the development of indoor location-based services (LBS). Global Positioning System (GPS) becomes invalid in indoor environments due to the non-line-of-sight issue, so it is urgent to develop a real-time high-accuracy localization approach for smartphones. However, accurate localization is challenging due to issues such as real-time response requirements, limited fingerprint samples and mobile device storage. To address these problems, we propose a novel deep learning architecture: Tensor-Generative Adversarial Network (TGAN). We first introduce a transform-based 3D tensor to model fingerprint samples. Instead of those passive methods that construct a fingerprint database as a prior, our model applies artificial neural network with deep learning to train network classifiers and then gives out estimations. Then we propose a novel tensorbased super-resolution scheme using the generative adversarial network (GAN) that adopts sparse coding as the generator network and a residual learning network as the discriminator. Further, we analyze the performance of TGAN and implement a trace-based localization experiment, which achieves better performance. Compared to existing methods for smartphones indoor positioning, that are energy- consuming and high demands on devices, TGAN can give out an improved solution in localization accuracy, response time and implementation complexity.

Radio-frequency (RF) tomographic imaging is a promising technique for inferring multi-dimensional physical space by processing RF signals traversed across a region of interest. However, conventional RF tomography schemes are generally based on vector compressed sensing, which ignores the geometric structures of the target spaces and leads to low recovery precision. The recently proposed transform-based tensor model is more appropriate for sensory data processing, as it helps exploit the geometric structures of the three-dimensional target and improve the recovery precision. In this paper, we propose a novel tensor sensing approach that achieves highly accurate estimation for real-world three-dimensional spaces. First, we use the transform-based tensor model to formulate a tensor sensing problem, and propose a fast alternating minimization algorithm called Alt-Min. Secondly, we drive an algorithm which is optimized to reduce memory and computation requirements. Finally, we present evaluation of our Alt-Min approach using IKEA 3D data and demonstrate significant improvement in recovery error and convergence speed compared to prior tensor-based compressed sensing.