Least-squares reverse time migration via linearized waveform inversion using a Wasserstein metric

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. S411-S423
Author(s):  
Peng Yong ◽  
Jianping Huang ◽  
Zhenchun Li ◽  
Wenyuan Liao ◽  
Luping Qu
Keyword(s):  
Least Squares ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Gradient Algorithm ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Wasserstein Metric ◽  
Time Migration ◽  
Imaging Results ◽  
Primal Dual

Least-squares reverse time migration (LSRTM), an effective tool for imaging the structures of the earth from seismograms, can be characterized as a linearized waveform inversion problem. We have investigated the performance of three minimization functionals as the [Formula: see text] norm, the hybrid [Formula: see text] norm, and the Wasserstein metric ([Formula: see text] metric) for LSRTM. The [Formula: see text] metric used in this study is based on the dynamic formulation of transport problems, and a primal-dual hybrid gradient algorithm is introduced to efficiently compute the [Formula: see text] metric between two seismograms. One-dimensional signal analysis has demonstrated that the [Formula: see text] metric behaves like the [Formula: see text] norm for two amplitude-varied signals. Unlike the [Formula: see text] norm, the [Formula: see text] metric does not suffer from the differentiability issue for null residuals. Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that, compared to the [Formula: see text] norm, the hybrid [Formula: see text] norm and [Formula: see text] metric can accelerate LSRTM and are less sensitive to non-Gaussian noise. For the field data application, the [Formula: see text] metric produces the most reliable imaging results. The hybrid [Formula: see text] norm requires tedious trial-and-error tests for the judicious threshold parameter selection. Hence, the more automatic [Formula: see text] metric is recommended as a robust alternative to the customary [Formula: see text] norm for time-domain LSRTM.

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.

Least-squares reverse time migration in TTI media using a pure qP-wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. S199-S216
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Jidong Yang ◽  
Xu Guo ◽  
Yundong Guo
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Seismic Imaging ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
The Matrix ◽  
The Stability

Anisotropy is a common phenomenon in subsurface strata and should be considered in seismic imaging and inversion. Seismic imaging in a vertical transversely isotropic (VTI) medium does not take into account the effects of the tilt angles, which can lead to degraded migrated images in areas with strong anisotropy. To correct such waveform distortion, reduce related image artifacts, and improve migration resolution, a tilted transversely isotropic (TTI) least-squares reverse time migration (LSRTM) method is presented. In the LSRTM, a pure qP-wave equation is used and solved with the finite-difference method. We have analyzed the stability condition for the pure qP-wave equation using the matrix method, which is used to ensure the stability of wave propagation in the TTI medium. Based on this wave equation, we derive a corresponding demigration (Born modeling) and adjoint migration operators to implement TTI LSRTM. Numerical tests on the synthetic data show the advantages of TTI LSRTM over VTI RTM and VTI LSRTM when the recorded data contain strong effects caused by large tilt angles. Our numerical experiments illustrate that the sensitivity of the adopted TTI LSRTM to the migration velocity errors is much higher than that to the anisotropic parameters (including epsilon, delta, and tilted angle parameters), and its sensitivity to the epsilon model and tilt angle is higher than that to the delta model.

Diffraction imaging for fault-karst structure by least-squares reverse time migration

2020 ◽  
pp. 1-40
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Liyun Fu ◽  
Shikai Jian ◽  
Bing Hu ◽  
...  
Keyword(s):  
Least Squares ◽  
Western China ◽  
Small Scale ◽  
Imaging Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Imaging Results ◽  
Heterogeneous Structures ◽  
Karst Reservoir

The fault-karst reservoir, which evolved from the deformation and karstification of carbonate rock, is one of the most important reservoir types in western China. Along the deep-seated fault zones, there are a lot widely spread and densely distributed fractures and vugs. The energy of the diffractions generated by heterogeneous structures, such as faults, fractures and vugs, are much weaker than that of the reflections produced by continuous formation interface. When using conventional full wavefield imaging method, the imaging results of continuous layers usually cover small-scale heterogeneities. Given that, we use plane-wave destruction (PWD) filter to separate the diffractions from the full data and image the separated diffractions using least-squares reverse time migration (LSRTM) method. We use several numerical examples to demonstrate that the newly developed diffractions LSRTM (D-LSRTM) can improve the definition of the heterogeneous structures, characterize the configuration and internal structure of the fault-karst structure well and enhance the interpretation accuracy for fault-karst reservoir.

Full Wavefield Least-Squares Reverse Time Migration

Geophysics ◽  
2021 ◽  
pp. 1-48
Author(s):  
Mikhail Davydenko ◽  
Eric Verschuur
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Input Data ◽  
Waveform Inversion ◽  
Secondary Source ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Internal Multiples ◽  
The One

Waveform inversion based on Least-Squares Reverse Time Migration (LSRTM) usually involves Born modelling, which models the primary-only data. As a result the inversion process handles only primaries and corresponding multiple elimination pre-processing of the input data is required prior to imaging and inversion. Otherwise, multiples left in the input data are mapped as false reflectors, also known as crosstalk, in the final image. At the same time the developed Full Wavefield Migration (FWM) methodology can handle internal multiples in an inversion-based imaging process. However, because it is based on the framework of the one-way wave equation, it cannot image dips close to and beyond 90 degrees. Therefore, we aim at upgrading LSRTM framework by bringing functionality of FWM to handle internal multiples. We have discovered that the secondary source term, used in the original formulation of FWM to define a wavefield relationship that allows to model multiple scattering via reflectivity, can be injected into a pressure component when simulating the two-way wave equation using finite-difference modelling. We use this modified forward model for estimating the reflectivity model with automatic crosstalk supression and validate the method on both synthetic and field data containing visible internal multiples.

Least-squares reverse time migration of multiples

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. S11-S21 ◽  
Author(s):  
Dongliang Zhang ◽  
Gerard T. Schuster
Keyword(s):  
Free Surface ◽  
Least Squares ◽  
Synthetic Data ◽  
Time History ◽  
Ocean Bottom ◽  
Virtual Source ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Sea Bed

The theory of least-squares reverse time migration of multiples (RTMM) is presented. In this method, least squares migration (LSM) is used to image free-surface multiples where the recorded traces are used as the time histories of the virtual sources at the hydrophones and the surface-related multiples are the observed data. For a single source, the entire free-surface becomes an extended virtual source where the downgoing free-surface multiples more fully illuminate the subsurface compared to the primaries. Since each recorded trace is treated as the time history of a virtual source, knowledge of the source wavelet is not required and the ringy time series for each source is automatically deconvolved. If the multiples can be perfectly separated from the primaries, numerical tests on synthetic data for the Sigsbee2B and Marmousi2 models show that least-squares reverse time migration of multiples (LSRTMM) can significantly improve the image quality compared to RTMM or standard reverse time migration (RTM) of primaries. However, if there is imperfect separation and the multiples are strongly interfering with the primaries then LSRTMM images show no significant advantage over the primary migration images. In some cases, they can be of worse quality. Applying LSRTMM to Gulf of Mexico data shows higher signal-to-noise imaging of the salt bottom and top compared to standard RTM images. This is likely attributed to the fact that the target body is just below the sea bed so that the deep water multiples do not have strong interference with the primaries. Migrating a sparsely sampled version of the Marmousi2 ocean bottom seismic data shows that LSM of primaries and LSRTMM provides significantly better imaging than standard RTM. A potential liability of LSRTMM is that multiples require several round trips between the reflector and the free surface, so that high frequencies in the multiples suffer greater attenuation compared to the primary reflections. This can lead to lower resolution in the migration image compared to that computed from primaries. Another liability is that the multiple migration image is more down-dip limited than the standard primaries migration image. Finally, if the surface-related multiple elimination method is imperfect and there are strong multiples interfering with the primaries, then the resulting LSRTMM image can be significantly degraded. We conclude that LSRTMM can be a useful complement, not a replacement, for RTM of primary reflections.

Joint least-squares reverse time migration of primary and prismatic waves

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. S29-S40 ◽  
Author(s):  
Jizhong Yang ◽  
Yuzhu Liu ◽  
Yunyue Elita Li ◽  
Arthur Cheng ◽  
Liangguo Dong ◽  
...  
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Velocity Model ◽  
Migration Velocity ◽  
Gradient Algorithm ◽  
Normal Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Recorded Data

Direct imaging of the steeply dipping structures is challenging for conventional reverse time migration (RTM), especially when there are no strong reflectors in the migration velocity model. To address this issue, we have enhanced the imaging of the steeply dipping structures by incorporating the prismatic waves. We formulate the imaging problem in a nonlinear least-squares optimization framework because the prismatic waves cannot be linearly mapped from the model perturbation. Primary and prismatic waves are jointly imaged to provide a single consistent image that includes structures illuminated by both types of waves, avoiding the complexities in scaling and/or interpreting primary and prismatic images separately. A conjugate gradient algorithm is used to iteratively solve the least-squares normal equation. This inversion procedure can become unstable if directly using the recorded data for migration because it is hindered by the crosstalk caused by imaging primary waves with the prismatic imaging operator. Therefore, we isolate the prismatic waves from the recorded data and image them with the prismatic imaging operator. Our scheme only requires a kinematically accurate and smooth migration velocity model, without the need to explicitly embed the strong reflectors in the migration velocity model. Realistic 2D numerical examples demonstrate that our method can resolve the steeply dipping structures much better than conventional least-squares RTM of primary waves.

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.

Elastic least-squares reverse time migration with hybrid l1/l2 misfit function

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. S271-S291 ◽  
Author(s):  
Bingluo Gu ◽  
Zhenchun Li ◽  
Peng Yang ◽  
Wencai Xu ◽  
Jianguang Han
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Synthetic Data ◽  
Adjoint Equations ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Wave Image

We have developed the theory and synthetic tests of elastic least-squares reverse time migration (ELSRTM). In this method, a least-squares reverse time migration algorithm is used to image multicomponent seismic data based on the first-order elastic velocity-stress wave equation, in which the linearized elastic modeling equations are used for forward modeling and its adjoint equations are derived based on the adjoint-state method for back propagating the data residuals. Also, we have developed another ELSRTM scheme based on the wavefield separation technique, in which the P-wave image is obtained using P-wave forward and adjoint wavefields and the S-wave image is obtained using P-wave forward and S-wave adjoint wavefields. In this way, the crosstalk artifacts can be minimized to a significant extent. In general, seismic data inevitably contain noise. We apply the hybrid [Formula: see text] misfit function to the ELSRTM algorithm to improve the robustness of our ELSRTM to noise. Numerical tests on synthetic data reveal that our ELSRTM, when compared with elastic reverse time migration, can produce images with higher spatial resolution, more-balanced amplitudes, and fewer artifacts. Moreover, the hybrid [Formula: see text] misfit function makes the ELSRTM more robust than the [Formula: see text] misfit function in the presence of noise.

scholarly journals Least-squares reverse-time migration with sparsity constraints

2021 ◽  
Vol 18 (2) ◽  
pp. 304-316
Author(s):  
Di Wu ◽  
Yanghua Wang ◽  
Jingjie Cao ◽  
Nuno V da Silva ◽  
Gang Yao
Keyword(s):  
Side Effects ◽  
Least Squares ◽  
Synthetic Data ◽  
Image Resolution ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
L1 Norm ◽  
Constrained Least Squares ◽  
Time Migration ◽  
Sparsity Constraints

Abstract Least-squares reverse-time migration (RTM) works with an inverse operation, rather than an adjoint operation in a conventional RTM, and thus produces an image with a higher resolution and more balanced amplitude than the conventional RTM image. However, least-squares RTM introduces two side effects: sidelobes around reflectors and high-wavenumber migration artifacts. These side effects are caused mainly by the limited bandwidth of seismic data, the limited coverage of receiver arrays and the inaccuracy of the modeling kernel. To mitigate these side effects and to further boost resolution, we employed two sparsity constraints in the least-squares inversion operation, namely the Cauchy and L1-norm constraints. For solving the Cauchy-constrained least-squares RTM, we used a preconditioned nonlinear conjugate-gradient method. For solving the L1-norm constrained least-squares RTM, we modified the iterative soft thresholding method. While adopting these two solution methods, the Cauchy-constrained least-squares RTM converged faster than the L1-norm constrained least-squares RTM. Application examples with synthetic data and laboratory modeling data demonstrated that the constrained least-squares RTM methods can mitigate the side effects and promote image resolution.

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