Compressive least-squares migration with on-the-fly Fourier transforms

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. R655-R672 ◽  
Author(s):  
Philipp A. Witte ◽  
Mathias Louboutin ◽  
Fabio Luporini ◽  
Gerard J. Gorman ◽  
Felix J. Herrmann
Keyword(s):  
Objective Function ◽  
Least Squares ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Reverse Time ◽  
Helmholtz Equations ◽  
Data Movement ◽  
Computational Costs ◽  
Time Migration ◽  
Least Squares Migration

Least-squares reverse time migration is a powerful approach for true-amplitude seismic imaging of complex geologic structures, but the successful application of this method is currently hindered by its enormous computational cost, as well as its high memory requirements for computing the gradient of the objective function. We have tackled these problems by introducing an algorithm for low-cost sparsity-promoting least-squares migration using on-the-fly Fourier transforms. We formulate the least-squares migration objective function in the frequency domain (FD) and compute gradients for randomized subsets of shot records and frequencies, thus significantly reducing data movement and the number of overall wave equations solves. By using on-the-fly Fourier transforms, we can compute an arbitrary number of monochromatic FD wavefields with a time-domain (TD) modeling code, instead of having to solve individual Helmholtz equations for each frequency, which becomes computationally infeasible when moving to high frequencies. Our numerical examples demonstrate that compressive imaging with on-the-fly Fourier transforms provides a fast and memory-efficient alternative to TD imaging with optimal checkpointing, whose memory requirements for a fixed background model and source wavelet are independent of the number of time steps. Instead, the memory and additional computational costs grow with the number of frequencies and determine the amount of subsampling artifacts and crosstalk. In contrast to optimal checkpointing, this offers the possibility to trade the memory and computational costs for image quality or a larger number of iterations and is advantageous in new computing environments such as the cloud, where computing is often cheaper than memory and data movement.

Least-squares migration of multisource data with a deblurring filter

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. R135-R146 ◽  
Author(s):  
Wei Dai ◽  
Xin Wang ◽  
Gerard T. Schuster
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Computational Efficiency ◽  
Computational Cost ◽  
Processing Technique ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Migration Method ◽  
Least Squares Migration

Least-squares migration (LSM) has been shown to be able to produce high-quality migration images, but its computational cost is considered to be too high for practical imaging. We have developed a multisource least-squares migration algorithm (MLSM) to increase the computational efficiency by using the blended sources processing technique. To expedite convergence, a multisource deblurring filter is used as a preconditioner to reduce the data residual. This MLSM algorithm is applicable with Kirchhoff migration, wave-equation migration, or reverse time migration, and the gain in computational efficiency depends on the choice of migration method. Numerical results with Kirchhoff LSM on the 2D SEG/EAGE salt model show that an accurate image is obtained by migrating a supergather of 320 phase-encoded shots. When the encoding functions are the same for every iteration, the input/output cost of MLSM is reduced by 320 times. Empirical results show that the crosstalk noise introduced by blended sources is more effectively reduced when the encoding functions are changed at every iteration. The analysis of signal-to-noise ratio (S/N) suggests that not too many iterations are needed to enhance the S/N to an acceptable level. Therefore, when implemented with wave-equation migration or reverse time migration methods, the MLSM algorithm can be more efficient than the conventional migration method.

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.

An exact adjoint operation pair in time extrapolation and its application in least-squares reverse-time migration

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. H27-H33 ◽  
Author(s):  
Jun Ji
Keyword(s):  
Least Squares ◽  
Adjoint Operator ◽  
Synthetic Data ◽  
Forward Modeling ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Product Test ◽  
Least Squares Migration ◽  
Time Extrapolation

To reduce the migration artifacts arising from incomplete data or inaccurate operators instead of migrating data with the adjoint of the forward-modeling operator, a least-squares migration often is considered. Least-squares migration requires a forward-modeling operator and its adjoint. In a derivation of the mathematically correct adjoint operator to a given forward-time-extrapolation modeling operator, the exact adjoint of the derived operator is obtained by formulating an explicit matrix equation for the forward operation and transposing it. The programs that implement the exact adjoint operator pair are verified by the dot-product test. The derived exact adjoint operator turns out to differ from the conventional reverse-time-migration (RTM) operator, an implementation of wavefield extrapolation backward in time. Examples with synthetic data show that migration using the exact adjoint operator gives similar results for a conventional RTM operator and that least-squares RTM is quite successful in reducing most migration artifacts. The least-squares solution using the exact adjoint pair produces a model that fits the data better than one using a conventional RTM operator pair.

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.

Inversion velocity analysis in the subsurface-offset domain

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. R189-R200 ◽  
Author(s):  
Jie Hou ◽  
William W. Symes
Keyword(s):  
Objective Function ◽  
Inverse Operator ◽  
Real Data ◽  
Velocity Analysis ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Image Volume ◽  
Least Squares Migration ◽  
Data Frequency

Optimization-based migration velocity analysis updates long-wavelength velocity information by minimizing an objective function that measures the violation of a focusing criterion, applied to an image volume. Differential semblance optimization forms a smooth objective function in velocity and data, regardless of the data-frequency content. Depending on how the image volume is formed, however, the objective function may not be minimized at a kinematically correct velocity, a phenomenon characterized in the literature (somewhat inaccurately) as “gradient artifacts.” We find that the root of this pathology is imperfect image volume formation resulting from reverse time migration (RTM), and that the use of linearized inversion (least-squares migration) more or less eliminates it. A synthetic Marmousi example and a 2D real data example are used to demonstrate that an approximate inverse operator, a little more expensive than RTM, leads to recovery of a kinematically correct velocity.

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.

Prestack correlative least-squares reverse time migration

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. S159-S172 ◽  
Author(s):  
Xuejian Liu ◽  
Yike Liu ◽  
Huiyi Lu ◽  
Hao Hu ◽  
Majid Khan
Keyword(s):  
Objective Function ◽  
Least Squares ◽  
Step Length ◽  
Inversion Algorithm ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Initial Image ◽  
Data Set ◽  
Time Migration ◽  
Gradient Descent Algorithm

In the correlative least-squares reverse time migration (CLSRTM) scheme, a stacked image is updated using a gradient-based inversion algorithm. However, CLSRTM experiences the incoherent stacking of different shots during each iteration due to the use of an imperfect velocity, which leads to image smearing. To reduce the sensitivity to velocity errors, we have developed prestack correlative least-squares reverse time migration (PCLSRTM), in which a gradient descent algorithm using a newly defined initial image and an efficiently defined analytical step length is developed to separately seek the optimal image for each shot gather before the final stacking. Furthermore, a weighted objective function is also designed for PCLSRTM, so that the data-domain gradient can avoid a strong truncation effect. Numerical experiments on a three-layer model as well as a marine synthetic and a field data set reveal the merits of PCLSRTM. In the presence of velocity errors, PCLSRTM shows better convergence and provides higher quality images as compared with CLSRTM. With the newly defined initial image, PCLSRTM can effectively handle observed data with unbalanced amplitudes. By using a weighted objective function, PCLSRTM can provide an image with enhanced resolution and balanced amplitude while avoiding many imaging artifacts.

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.

Least-squares migration with primary- and multiple-guided weighting matrices

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. S171-S185 ◽  
Author(s):  
Chuang Li ◽  
Jianping Huang ◽  
Zhenchun Li ◽  
Han Yu ◽  
Rongrong Wang
Keyword(s):  
Free Surface ◽  
Least Squares ◽  
Seismic Data ◽  
Synthetic Data ◽  
Velocity Model ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Multiple Reflections ◽  
Time Migration ◽  
Least Squares Migration

Least-squares migration (LSM) of seismic data is supposed to produce images of subsurface structures with better quality than standard migration if we have an accurate migration velocity model. However, LSM suffers from data mismatch problems and migration artifacts when noise pollutes the recorded profiles. This study has developed a reweighted least-squares reverse time migration (RWLSRTM) method to overcome the problems caused by such noise. We first verify that spiky noise and free-surface multiples lead to the mismatch problems and should be eliminated from the data residual. The primary- and multiple-guided weighting matrices are then derived for RWLSRTM to reduce the noise in the data residual. The weighting matrices impose constraints on the data residual such that spiky noise and free-surface multiple reflections are reduced whereas primary reflections are preserved. The weights for spiky noise and multiple reflections are controlled by a dynamic threshold parameter decreasing with iterations for better results. Finally, we use an iteratively reweighted least-squares algorithm to minimize the weighted data residual. We conduct numerical tests using the synthetic data and compared the results of this method with the results of standard LSRTM. The results suggest that RWLSRTM is more robust than standard LSRTM when the seismic data contain spiky noise and multiple reflections. Moreover, our method not only suppresses the migration artifacts, but it also accelerates the convergence.

An efficient step-length formula for correlative least-squares reverse time migration

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. S221-S238 ◽  
Author(s):  
Youshan Liu ◽  
Jiwen Teng ◽  
Tao Xu ◽  
Zhiming Bai ◽  
Haiqiang Lan ◽  
...  
Keyword(s):  
Objective Function ◽  
Least Squares ◽  
Step Length ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Step Size ◽  
Time Migration ◽  
Seismic Records ◽  
Complex Models ◽  
Analytical Step

In correlative least-squares reverse time migration (CLSRTM), the estimation of the optimal step size is usually determined by fitting a parabola and finding its minimum; it involves at least two times extra reading of all seismic records, which significantly lowers the efficiency of the algorithm. To improve the efficiency of the CLSRTM algorithm, we have deduced an analytical step-length (ASL) formula based on the linear property of the demigration operator. Numerical examples performed with the data synthetized by the Marmousi and Sigsbee2A models were used to test its validity. In complex models with imperfect migration velocity, such as the Sigabee2A model, our formula makes the value of the objective function converges to a much smaller minimum. Additional numerical tests performed with the data either acquired irregularly or contaminated by different noise levels verify the robustness of the ASL formula. Compared with the commonly used parabolic search method, the ASL formula is much more efficient because it is free from an extra estimation of the value of the objective function.

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