Fast least-squares reverse time migration via a superposition of Kronecker products

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. S115-S134
Author(s):  
Wenlei Gao ◽  
Gian Matharu ◽  
Mauricio D. Sacchi
Keyword(s):  
Least Squares ◽  
Computational Cost ◽  
Low Rank ◽  
Fast Method ◽  
Kronecker Products ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Major Drawback ◽  
Image Domain ◽  
Time Migration

Least-squares reverse time migration (LSRTM) has become increasingly popular for complex wavefield imaging due to its ability to equalize image amplitudes, attenuate migration artifacts, handle incomplete and noisy data, and improve spatial resolution. The major drawback of LSRTM is the considerable computational cost incurred by performing migration/demigration at each iteration of the optimization. To ameliorate the computational cost, we introduced a fast method to solve the LSRTM problem in the image domain. Our method is based on a new factorization that approximates the Hessian using a superposition of Kronecker products. The Kronecker factors are small matrices relative to the size of the Hessian. Crucially, the factorization is able to honor the characteristic block-band structure of the Hessian. We have developed a computationally efficient algorithm to estimate the Kronecker factors via low-rank matrix completion. The completion algorithm uses only a small percentage of preferentially sampled elements of the Hessian matrix. Element sampling requires computation of the source and receiver Green’s functions but avoids explicitly constructing the entire Hessian. Our Kronecker-based factorization leads to an imaging technique that we name Kronecker-LSRTM (KLSRTM). The iterative solution of the image-domain KLSRTM is fast because we replace computationally expensive migration/demigration operations with fast matrix multiplications involving small matrices. We first validate the efficacy of our method by explicitly computing the Hessian for a small problem. Subsequent 2D numerical tests compare LSRTM with KLSRTM for several benchmark models. We observe that KLSRTM achieves near-identical images to LSRTM at a significantly reduced computational cost (approximately 5–15× faster); however, KLSRTM has an increased, yet manageable, memory cost.

From acoustic to elastic inverse extended Born modeling: a first insight in the marine environment

Geophysics ◽  
2021 ◽  
pp. 1-73
Author(s):  
Milad Farshad ◽  
Hervé Chauris
Keyword(s):  
Least Squares ◽  
Marine Environment ◽  
Computational Cost ◽  
Physical Parameters ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Physical Density ◽  
Acoustic Approximation

Elastic least-squares reverse time migration is the state-of-the-art linear imaging technique to retrieve high-resolution quantitative subsurface images. A successful application requires many migration/modeling cycles. To accelerate the convergence rate, various pseudoinverse Born operators have been proposed, providing quantitative results within a single iteration, while having roughly the same computational cost as reverse time migration. However, these are based on the acoustic approximation, leading to possible inaccurate amplitude predictions as well as the ignorance of S-wave effects. To solve this problem, we extend the pseudoinverse Born operator from acoustic to elastic media to account for the elastic amplitudes of PP reflections and provide an estimate of physical density, P- and S-wave impedance models. We restrict the extension to marine environment, with the recording of pressure waves at the receiver positions. Firstly, we replace the acoustic Green's functions by their elastic version, without modifying the structure of the original pseudoinverse Born operator. We then apply a Radon transform to the results of the first step to calculate the angle-dependent response. Finally, we simultaneously invert for the physical parameters using a weighted least-squares method. Through numerical experiments, we first illustrate the consequences of acoustic approximation on elastic data, leading to inaccurate parameter inversion as well as to artificial reflector inclusion. Then we demonstrate that our method can simultaneously invert for elastic parameters in the presence of complex uncorrelated structures, inaccurate background models, and Gaussian noisy data.

Elastic least-squares reverse time migration in the image domain

2017 ◽  
Author(s):  
Bruno Pereira-Dias ◽  
André Bulcão ◽  
Djalma Soares Filho ◽  
Roberto Dias ◽  
Felipe Duarte ◽  
...  
Keyword(s):  
Least Squares ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Image Domain ◽  
Time Migration

Mini-batch Least-Squares Reverse Time Migration In A Deep Learning Framework

Geophysics ◽  
2020 ◽  
pp. 1-61
Author(s):  
Janaki Vamaraju ◽  
Jeremy Vila ◽  
Mauricio Araya-Polo ◽  
Debanjan Datta ◽  
Mohamed Sidahmed ◽  
...  
Keyword(s):  
Deep Learning ◽  
Least Squares ◽  
Spatial Resolution ◽  
Data Science ◽  
Computational Cost ◽  
Computation Time ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Learning Framework ◽  
Time Migration

Migration techniques are an integral part of seismic imaging workflows. Least-squares reverse time migration (LSRTM) overcomes some of the shortcomings of conventional migration algorithms by compensating for illumination and removing sampling artifacts to increase spatial resolution. However, the computational cost associated with iterative LSRTM is high and convergence can be slow in complex media. We implement pre-stack LSRTM in a deep learning framework and adopt strategies from the data science domain to accelerate convergence. The proposed hybrid framework leverages the existing physics-based models and machine learning optimizers to achieve better and cheaper solutions. Using a time-domain formulation, we show that mini-batch gradients can reduce the computation cost by using a subset of total shots for each iteration. Mini-batch approach does not only reduce source cross-talk but also is less memory intensive. Combining mini-batch gradients with deep learning optimizers and loss functions can improve the efficiency of LSRTM. Deep learning optimizers such as the adaptive moment estimation are generally well suited for noisy and sparse data. We compare different optimizers and demonstrate their efficacy in mitigating migration artifacts. To accelerate the inversion, we adopt the regularised Huber loss function in conjunction. We apply these techniques to 2D Marmousi and 3D SEG/EAGE salt models and show improvements over conventional LSRTM baselines. The proposed approach achieves higher spatial resolution in less computation time measured by various qualitative and quantitative evaluation metrics.

scholarly journals Q-compensated least-squares reverse time migration using low-rank one-step wave extrapolation

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. S271-S279 ◽  
Author(s):  
Junzhe Sun ◽  
Sergey Fomel ◽  
Tieyuan Zhu ◽  
Jingwei Hu
Keyword(s):  
Convergence Rate ◽  
Least Squares ◽  
Seismic Attenuation ◽  
Low Rank ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Attenuation Compensation ◽  
Time Migration ◽  
Rank One ◽  
One Step

Attenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging. In viscoacoustic media, reverse time migration (RTM) can be performed with [Formula: see text]-compensation, which is also known as [Formula: see text]-RTM. Least-squares RTM (LSRTM) has also been shown to be able to compensate for attenuation through linearized inversion. However, seismic attenuation may significantly slow down the convergence rate of the least-squares iterative inversion process without proper preconditioning. We have found that incorporating attenuation compensation into LSRTM can improve the speed of convergence in attenuating media, obtaining high-quality images within the first few iterations. Based on the low-rank one-step seismic modeling operator in viscoacoustic media, we have derived its adjoint operator using nonstationary filtering theory. The proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation. Recognizing that, in viscoacoustic media, the wave-equation Hessian may become ill-conditioned, we propose to precondition LSRTM with [Formula: see text]-compensated RTM. Numerical examples showed that the preconditioned [Formula: see text]-LSRTM method has a significantly faster convergence rate than LSRTM and thus is preferable for practical applications.

Improved subsalt images with least-squares reverse time migration

2017 ◽  
Vol 5 (3) ◽  
pp. SN25-SN32 ◽  
Author(s):  
Ping Wang ◽  
Shouting Huang ◽  
Ming Wang
Keyword(s):  
Least Squares ◽  
Field Data ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Lateral Velocity ◽  
Image Domain ◽  
Time Migration ◽  
Iterative Inversion ◽  
Velocity Variations ◽  
Subsalt Imaging

Complex overburdens often distort reservoir images in terms of structural positioning, stratigraphic resolution, and amplitude fidelity. One prime example of a complex overburden is in the deepwater Gulf of Mexico, where thick and irregular layers of remobilized (i.e., allochthonous) salt are situated above prospective reservoir intervals. The highly variant salt layers create large lateral velocity variations that distort wave propagation and the illumination of deeper reservoir targets. In subsalt imaging, tools such as reflection tomography, full-waveform inversion, and detailed salt interpretation are needed to derive a high-resolution velocity model that captures the lateral velocity variations. Once a velocity field is obtained, reverse time migration (RTM) can be applied to restore structural positioning of events below and around the salt. However, RTM by nature is unable to fully recover the reflectivity for desired amplitudes and resolution. This shortcoming is well-recognized by the imaging community, and it has propelled the emergence of least-squares RTM (LSRTM) in recent years. We have investigated how current LSRTM methods perform on subsalt images. First, we compared the formulation of data-domain versus image-domain least-squares migration, as well as methods using single-iteration approximation versus iterative inversion. Then, we examined the resulting subsalt images of several LSRTM methods applied on the synthetic and field data. Among our tests, we found that image-domain single-iteration LSRTM methods, including an extension of an approximate inverse Hessian method in the curvelet domain, not only compensated for amplitude loss due to poor illumination caused by complex salt bodies, but it also produced subsalt images with fewer migration artifacts in the field data. In contrast, an iterative inversion method showed its potential for broadening the bandwidth in the subsalt, but it was less effective in reducing migration artifacts and noise. Based on our understanding, we evaluated the current state of LSRTM for subsalt imaging.

3D acoustic least-squares reverse time migration using the energy norm

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. S261-S270 ◽  
Author(s):  
Daniel Rocha ◽  
Paul Sava ◽  
Antoine Guitton
Keyword(s):  
Least Squares ◽  
Convergence Rates ◽  
Energy Norm ◽  
Ocean Bottom ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Data Set ◽  
Image Domain ◽  
Time Migration ◽  
And Migration

We have developed a least-squares reverse time migration (LSRTM) method that uses an energy-based imaging condition to obtain faster convergence rates when compared with similar methods based on conventional imaging conditions. To achieve our goal, we also define a linearized modeling operator that is the proper adjoint of the energy migration operator. Our modeling and migration operators use spatial and temporal derivatives that attenuate imaging artifacts and deliver a better representation of the reflectivity and scattered wavefields. We applied the method to two Gulf of Mexico field data sets: a 2D towed-streamer benchmark data set and a 3D ocean-bottom node data set. We found LSRTM resolution improvement relative to RTM images, as well as the superior convergence rate obtained by the linearized modeling and migration operators based on the energy norm, coupled with inversion preconditioning using image-domain nonstationary matching filters.

Substituting smoothing with low-rank decomposition — Applications to least-squares reverse time migration of simultaneous source and incomplete seismic data

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. S267-S283 ◽  
Author(s):  
Yangkang Chen ◽  
Min Bai ◽  
Yatong Zhou ◽  
Qingchen Zhang ◽  
Yufeng Wang ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Low Rank ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Decomposition Operator ◽  
Rank Decomposition ◽  
Simultaneous Source ◽  
Low Rank Decomposition

Seismic migration can be formulated as an inverse problem, the model of which can be iteratively inverted via the least-squares migration framework instead of approximated by applying the adjoint operator to the observed data. Least-squares reverse time migration (LSRTM) has attracted more and more attention in modern seismic imaging workflows because of its exceptional performance in obtaining high-resolution true-amplitude seismic images and the fast development of the computational capability of modern computing architecture. However, due to a variety of reasons, e.g., insufficient shot coverage and data sampling, the image from least-squares inversion still contains a large amount of artifacts. This phenomenon results from the ill-posed nature of the inverse problem. In traditional LSRTM, the minimum least-squares energy of the model is used as a constraint to regularize the inverse problem. Considering the residual noise caused by the smoothing operator in traditional LSRTM, we regularize the model using a powerful low-rank decomposition operator, which can better suppress the migration artifacts in the image during iterative inversion. We evaluate in detail the low-rank decomposition operator and the way to apply it along the geologic structure of seismic reflectors. We comprehensively analyze the performance of our algorithm in attenuating crosstalk noise caused by simultaneous source acquisition and migration artifacts caused by insufficient space sampling via two synthetic examples and one field data example. Our results indicate that compared to the conventional smoothing operator, our low-rank decomposition operator can help obtain a cleaner LSRTM image and obtain a slightly better edge-preserving performance.

Comparison between Born and Kirchhoff operators for least-squares reverse time migration and the constraint of the propagation of the background wavefield

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. R725-R739 ◽  
Author(s):  
Kai Yang ◽  
Jianfeng Zhang
Keyword(s):  
Least Squares ◽  
Born Approximation ◽  
Computational Cost ◽  
Stopping Time ◽  
Step Length ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Recording Time ◽  
Time Migration ◽  
Computational Strategy

The Born approximation and the Kirchhoff approximation are two frameworks that are extensively used in solving seismic migration/inversion problems. Both approximations assume a linear relationship between the primary reflected/scattered data to the corresponding physical model. However, different approximations result in different behaviors. For least-squares reverse time migration (LSRTM), most of the algorithms are constructed based on Born approximation. We have constructed a pair of Kirchhoff modeling and migration operators based on the Born modeling operator and the connection between the perturbation model and the reflectivity model, and then we compared the different performances between Born and Kirchhoff operators for LSRTM. Numerical examples on Marmousi model and SEAM 2D salt model indicate that LSRTM with Kirchhoff operators is a better alternative to that with Born operators for imaging complex structures. To reduce the computational cost, we also investigate a strategy by restricting the propagation of the background wavefield to a stopping time rather than the maximum recording time. And this stopping time can be chosen as half of the maximum recording time. This computational strategy can be used in LSRTM procedures of predicting the primary reflected data, calculating the step length, and computing the gradient. Theoretical analyses and numerical experiments are given to justify this computational strategy for LSRTM.

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