Hyperbolic Radon Research Articles

Inversion-Based Deblending in Common Midpoint Domain Using Time Domain High-Resolution Radon

We implement an inversion-based deblending method in the common midpoint gathers (CMP) as an alternative to the standard common receiver gather (CRG) domain methods. The primary advantage of deblending in the CMP domain is that reflections from dipping layers are centred around zero offsets. As a result, CMP gathers exhibit a simpler structure compared to common receiver gathers (CRGs), where these reflections are apex-shifted. Consequently, we can employ a zero-offset hyperbolic Radon operator to process CMP gathers. This operator is a computationally more efficient alternative to the apex-shifted hyperbolic Radon required for processing CRG gathers. Sparse transforms, such as the Radon transform, can stack reflections and produce sparse models capable of separating blended sources. We utilize the Radon operator to develop an inversion-based deblending framework that incorporates a sparse model constraint. The inclusion of a sparsity constraint in the inversion process enhances the focusing of the transform and improves data recovery. Inversion-based deblending enables us to account for all observed data by incorporating the blending operator into the cost function. Our synthetic and field data examples demonstrate that inversion-based deblending in the CMP domain can effectively separate blended sources.

Seismic Data Reconstruction Using a Phase-Shift-Plus-Interpolation-Based Apex-Shifted Hyperbolic Radon Transform

The apex-shifted hyperbolic Radon transform (ASHRT) based on the Stolt-stretch operator can be implemented in the frequency domain, which accelerates the computation efficiency of ASHRT. However, the Stolt-stretch operator has limitations when it comes to velocity variations. Therefore, this paper introduces a new ASHRT approach based on post-stack phase shift plus interpolation (PSPI) imaging and modeling operators. This new approach is designed to better adapt to changes in medium velocity and enhance the quality of data reconstruction. When combining this novel transformation with sparsity constraints for model testing and real data applications, the experimental results indicate that it is an effective data reconstruction tool, with superior data reconstruction results compared to traditional ASHRT based on the Stolt-stretch operator.

Fast High-Resolution Hyperbolic Radon Transform

Due to its time-variant nature, the computationally expensive hyperbolic Radon transform (RT) is not easy to be accelerated, e.g., based on the convolution theorem in the frequency domain. However, the hyperbolic RT better matches the trajectories of reflection events in prestack gathers than other time-invariant RTs, e.g., linear or parabolic RTs. Hence, despite its large computational cost, the time-domain hyperbolic RT is still preferred in many seismic processing applications. We propose a fast high-resolution hyperbolic RT (HRHRT) with a fast butterfly algorithm. The forward and adjoint RTs can be greatly accelerated based on a fast butterfly algorithm by reformulating the time-space domain Radon operator as a frequency-domain Fourier integral operator (FIO). The fast butterfly algorithm solves the FIO problem by a blockwise low-rank approximation scheme. The single-step hyperbolic RT can be much faster (e.g., hundreds of times faster for a large problem) than the traditional implementation, resulting in a significant computational boost when the transform is taken in an iterative fashion to estimate the high-resolution Radon coefficients. We demonstrate the similar performance and the much different computational efficiencies between the proposed fast HRHRT and the traditional method over several different problems, i.e., random noise suppression, big-gap seismic reconstruction, and multiples attenuation.

A Fast Sparse Hyperbolic Radon Transform Based on Convolutional Neural Network and Its Demultiple Application

The hyperbolic Radon transform (RT) is a widely used demultiple method in the seismic data processing. But this transformation faces two major defects. The limited aperture of acquisition leads to the scissor-like diffusion in the Radon domain, which introduces separation difficulties between primaries and multiples. In addition, the computation of large matrices inversion involved in hyperbolic RT reduces the processing efficiency. In this letter, a specific Convolutional Neural Network (CNN) is designed to conduct a Fast Sparse Hyperbolic Radon Transform (FSHRT). Two techniques are incorporated into CNN to find the sparse solution. One is the coding-decoding structure, which captures the sparse feature of Radon parameters. The other is the soft threshold activation function followed by the end of neural networks, which suppresses the small parameters and further improves the sparsity. Thus, the network realizes the direct mapping between the adjoint solution and the sparse solution. Furthermore, synthetic and field demultiple experiments are carried out to demonstrate the rapidity and effectiveness of the proposed method.

Radon Transform Based on Waveform for AVO-Preserving Data Construction

The traditional hyperbolic Radon transform suffers from the major problem of how to both obtain a high resolution and preserve the amplitude variation with offset (AVO). In the Radon domain, high resolution (sparseness) is a valid criterion. However, if a sparse model is obtained in the Radon domain due to averaging along the offset direction, then it is not possible to preserve the AVO in the inversion data. In addition, hyperbolic Radon transform has a time-variant kernel based on a traditional iterative algorithm, the conjugate gradient (CG), which requires significant computation time. To solve these problems, we propose a Radon transform based on waveform that contains both cycle and amplitude characteristics of seismic waves. The new transform entails creating an upper envelope for the seismic data and computing a preliminary forward Radon transform in the time domain. The forward Radon transform incorporates a priori information by measuring the energy of each slowness (p) trace to obtain the high-resolution result of the Radon domain. For AVO preserving, the proposed method uses polynomials to describe the AVO characteristics in the inverse Radon transform based on the least-squares inversion. Besides amplitude preserving and high resolution, the proposed method avoids using CG and greatly reduces the cost of computing hyperbolic Radon transform in the time domain. In applications to both synthetic and field data, waveform Radon transform (WRT) has a better performance than the conjugate gradient Radon transform (CGRT).

Multichannel Seismic Reflection Data Processing: Multiple Attenuation of Line 250 at the Coast of Southern California Using Hyperbolic Radon Transform

Multichannel seismic reflection data processing of line 250 at the Coast of Southern California has been done by using Echos®. The purpose of this research is to separate multiple reflections from the primary reflection. Multiple reflections yield dramatic effect especially on marine seismic survey. Multiple reflections in marine seismic data is the noise caused by waves that trapped in a layer of sea water or rock layers. One of geophysics exploration problem is removing multiple reflections from primary reflection. Multiple reflections often destructively interfere with the primary reflection making interpretation difficult, so it’s needed to get seismic section free from multiple reflections. There are some methods to attenuate the multiple reflections, one of them is Hyperbolic Radon transform. Hyperbolic Radon transform method works by transforming the data from time-offset domain into the Radon domain. Primary reflection and multiple reflections can be separated into Radon domain. In the end of this research, seismic section before and after multiple attenuation using hyperbolic Radon transform were compared. Based on the result of processing, it can be concluded that hyperbolic Radon transform is effective to attenuate multiple reflections at 0,24-l,25s TWT. But, it can eliminate some of data and makes seismic section seen unclear.

Microseismic denoising assessment by polarization histograms

We have performed a comparison of microseismic data denoising methods based on their effect on the polarization attributes of 3C microseismic signals. The compared denoising methods include the classic band-pass filtering, and three recently proposed denoising techniques: restricted-domain hyperbolic Radon transform denoising, singular value decomposition (SVD)-based reduced-rank filtering, and empirical mode decomposition denoising. In order to draw the comparison, we denoised 3C synthetic data contaminated with noise extracted from field data records; calculated their rectilinearity, azimuth, and dip polarization attributes; and arranged them into histograms. The comparison has been drawn by measuring the distances between the polarization histograms of the clean and denoised data, assuming that one method outperforms another if the aforementioned distance is smaller. This strategy allows us to quantify the improvement in the calculated polarization attributes due to the different denoising processes. In addition, we also calculated the quality factor (QF) of the denoised signals, which adds value and robustness to the comparison. Our results indicated that the method based on SVD preserves the original polarization attributes better than the other techniques tested in this work. Moreover, it also retrieved the denoised signal with the highest QF. Finally, we tested the methods with field data and assessed their performance qualitatively on the basis of the insight gained from numerical tests with synthetic data.

Steered sample algorithm for acoustic source localization.

High-precision source localization depends on many factors, including a suitable location method. Beamforming-based methods, such as the steered response power (SRP), are a common type of acoustic localization methods. However, these methods have low spatial resolution. The SRP method with phase transform (SRP-PHAT) improves the spatial resolution of SRP and is one of the most effective and robust methods for source localization. However, the introduction of a phase transform to SRP might amplify the power of the noise and result in many local extrema in the SRP space, which has a negative impact on source localization. In this paper, a steered sample algorithm (SSA) based on the reciprocity of wave propagation for acoustic source localization is proposed. The SSA localization process is similar to the hyperbolic Radon transform, which is theoretically analyzed and is the most essential difference form the SRP/SRP-PHAT. Compared with the SRP-PHAT, the experimental results demonstrate that the SSA perform better when it comes to array signal positioning with limited array elements and narrow azimuth signal, where SSA can achieve high precision positioning with lower SNR.

Inversion of Higher Dimensional Radon Transforms of Seismic-Type

We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of the independent variables, and are imposed some symmetry with respect to the axes. These transforms are higher dimensional version of generalization of the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.

Inversion‐based deblending using migration operators

ABSTRACTIn this paper, we compare the denoising‐ and inversion‐based deblending methods using Stolt migration operators. We use Stolt operator as a kernel to efficiently compute apex‐shifted hyperbolic Radon transform. Sparsity promoting transforms, such as Radon transform, can focus seismic data into a sparse model to separate signals, remove noise or interpolate missing traces. Therefore, Radon transforms are a suitable tool for either the denoising‐ or the inversion‐based deblending methods. The denoising‐based deblending treats blending interferences as random noise by sorting the data into new gathers, such as common receiver gather. In these gathers, blending interferences exhibit random structures due to the randomization of the source firing times. Alternatively, the inversion‐based deblending treats blending interferences as a signal, and the transform models this signal by incorporating the blending operator to formulate an inversion problem. We compare both methods using a robust inversion algorithm with sparse regularization. Results of synthetic and field data examples show that the inversion‐based deblending can produce more accurate signal separation for highly blended data.

Inversion of seismic-type Radon transforms on the plane

ABSTRACT We study integral transforms mapping a function on the Euclidean plane to the family of its integration on plane curves, that is, a function of plane curves. The plane curves we consider in the present paper are given by the graphs of functions with a fixed axis of the independent variable and are imposed some symmetry with respect to the axes. These transforms contain the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.

Automatic detection and attenuation of the external source interference noise by using a time-invariant hyperbolic Radon transform

Automatic detection and attenuation of the external source interference noise by using a time-invariant hyperbolic Radon transform

The Shuey-Radon transform

The traditional hyperbolic Radon transform (RT) decomposes seismic data into a sum of constant amplitude basis functions. This limits the performance of the transform when dealing with real data in which the reflection amplitudes include the amplitude variation with offset (AVO) variations. We adopted the Shuey-Radon transform as a combination of the RT and Shuey’s approximation of reflectivity to accurately model reflections including AVO effects. The new transform splits the seismic gather into three Radon panels: The first models the reflections at zero offset, and the other two panels add capability to model the AVO gradient and curvature. There are two main advantages of the Shuey-Radon transform over similar algorithms, which are based on a polynomial expansion of the AVO response. (1) It is able to model reflections more accurately. This leads to more focused coefficients in the transform domain and hence provides more accurate processing results. (2) Unlike polynomial-based approaches, the coefficients of the Shuey-Radon transform are directly connected to the classic AVO parameters (intercept, gradient, and curvature). Therefore, the resulting coefficients can further be used for interpretation purposes. The solution of the new transform is defined via an underdetermined linear system of equations. It is formulated as a sparsity-promoting optimization, and it is solved efficiently using an orthogonal matching pursuit algorithm. Applications to different numerical experiments indicate that the Shuey-Radon transform outperforms the polynomial and conventional RTs.

Fast hyperbolic Radon transform using chirp-z transform

The hyperbolic Radon transform plays an important role in seismic data processing for its ability to focus seismic events in the transform domain. Traditional algorithms based on direct implementations, however, are inefficient with limited applications for processing large data sets. A new algorithm is presented for fast computation of the hyperbolic Radon transform and its sparse calculations. It uses interpolation procedures to stretch the data along time axis and efficiently computes the summation paths in the new coordinates via the chirp-z transform which is carried out by fast Fourier transform (FFT). The proposed fast algorithm is then used within the deconvolutive form of the Radon transform and iterative sparse algorithms for effective decomposition of CMP gathers with an improved temporal resolution, compared to the traditional Radon transforms. The effectiveness of the new algorithm are confirmed on sparse velocity-stack inversion, primary and multiple separation, high-quality stacking, and automatic velocity model building. The tests show that sparse velocity-stack inversion using the new algorithm is even more efficient than the traditional velocity scan, both in resolution and speed. Furthermore, numerical tests show the superiority of the proposed algorithm over the state-of-the-art fast algorithms, based on butterfly scheme and log-polar convolutions, demanding less computational complexity.

Fast hyperbolic deconvolutive Radon transform using generalized Fourier slice theorem

ABSTRACTThe hyperbolic Radon transform has a long history of applications in seismic data processing because of its ability to focus/sparsify the data in the transform domain. Recently, deconvolutive Radon transform has also been proposed with an improved time resolution which provides improved processing results. The basis functions of the (deconvolutive) Radon transform, however, are time‐variant, making the classical Fourier based algorithms ineffective to carry out the required computations. A direct implementation of the associated summations in the time–space domain is also computationally expensive, thus limiting the application of the transform on large data sets. In this paper, we present a new method for fast computation of the hyperbolic (deconvolutive) Radon transform. The method is based on the recently proposed generalized Fourier slice theorem which establishes an analytic expression between the Fourier transforms associated with the data and Radon plane. This allows very fast computations of the forward and inverse transforms simply using fast Fourier transform and interpolation procedures. These canonical transforms are used within an efficient iterative method for sparse solution of (deconvolutive) Radon transform. Numerical examples from synthetic and field seismic data confirm high performance of the proposed fast algorithm for filling in the large gaps in seismic data, separating primaries from multiple reflections, and performing high‐quality stretch‐free stacking.

Surface-Wave Extraction Based on Morphological Diversity of Seismic Events

It is essential to extract high-fidelity surface waves in surface-wave surveys. Because reflections usually interfere with surface waves on X components in multicomponent seismic exploration, it is difficult to extract dispersion curves of surface waves. To make matters worse, the frequencies and velocities of higher-mode surface waves are close to those of PS-waves. A method for surface-wave extraction is proposed based on the morphological differences between surface waves and reflections. Frequency-domain high-resolution linear Radon transform (LRT) and time-domain high-resolution hyperbolic Radon transform (HRT) are used to represent surface waves and reflections, respectively. Then, a sparse representation problem based on morphological component analysis (MCA) is built and optimally solved to obtain high-fidelity surface waves. An advantage of our method is its ability to extract surface waves when their frequencies and velocities are close to those of reflections. Furthermore, the results of synthetic and field examples confirm that the proposed method can attenuate the distortion of surface-wave dispersive energy caused by reflections, which contributes to extraction of accurate dispersion curves.

Simultaneous reconstruction of seismic reflections and diffractions using a global hyperbolic Radon dictionary

We have developed a new transform with basis functions that closely resemble seismic reflections and diffractions. The new transform is an extension of the classic hyperbolic Radon transform and accounts for the apex shifts of the seismic reflection hyperbolas and the asymptote shifts of the seismic diffraction hyperbolas. The adjoint and forward operators of the proposed transform are computed using Stolt operators in the frequency domain to increase the computational efficiency of the transform. This new transform is used, in conjunction with a sparse inversion algorithm, to reconstruct common-shot gathers. Our tests indicate that this new transform is an efficient tool for interpolating coarsely sampled seismic data in cases in which one cannot use small data windows to validate the linear event assumption that is often made by Fourier-based reconstruction methods.

Automatic velocity analysis using high-resolution hyperbolic Radon transform

Velocity analysis is crucial in reflection seismic data processing and imaging. Velocity picking is widely used in the industry for building the initial velocity model. When the size of the seismic data becomes extremely large, we cannot afford the corresponding human endeavor that is required by the velocity picking. In such situations, an automatic velocity-picking algorithm is highly demanded. We have developed a novel automatic velocity-analysis algorithm that is based on the high-resolution hyperbolic Radon transform. We formulate the automatic velocity-analysis problem as a constrained optimization problem. To solve the optimization problem with a hard constraint on the sparsity and distribution of the velocity spectrum, we relax it to a more familiar L1-regularized optimization problem in two steps. We use the iterative preconditioned least-squares method to solve the L1-regularized problem, and then we apply the hard constraint of the target optimization during the iterative inversion. Using synthetic and field-data examples, we determine the successful performance of our algorithm.

Localization of marine seismic vibrator based on hyperbolic Radon transform

In marine seismic surveys to explore seafloor resources, the structure below the seafloor is estimated from the obtained sound waves, which are emitted by a marine seismic sound source and reflected or refracted between the layers below the seafloor. In order to estimate the structure below the seafloor from returned waves, information of the sound source position and the sound speed are needed. Marine seismic vibrators, which are one of the marine seismic sound sources, have some advantages such as high controllability of the frequency and phase of the sound, and oscillation at a high depth. However, when the sound source position is far from the sea surface, it becomes difficult to specify the exact position. In this paper, we propose a method to estimate the position of a marine seismic vibrator and the sound speed from obtained seismic data by formulating an optimization problem via hyperbolic Radon transform. Numerical simulations confirmed that the proposed method almost achieves theoretical lower bounds for the variances of the estimations.

The conjugate gradient method

The conjugate gradient method can be used to solve many large linear geophysical problems — for example, least-squares parabolic and hyperbolic Radon transform, traveltime tomography, least-squares migration, and full-waveform inversion (FWI) (e.g., Witte et al., 2018 ). This tutorial revisits the “Linear inversion tutorial” ( Hall, 2016 ) that estimated reflectivity by deconvolving a known wavelet from a seismic trace using least squares. This tutorial solves the same problem using the conjugate gradient method. This problem is easy to understand, and the concepts apply to other applications. The conjugate gradient method is often used to solve large problems because the least-squares algorithm is much more expensive — that is, even a large computer may not be able to find a useful solution in a reasonable amount of time.