Least-squares Reverse Time Migration Research Articles

Vertical transversely isotropic elastic least-squares reverse time migration based on elastic wavefield vector decomposition

Anisotropic elastic reverse time migration (RTM) is a promising technique for imaging complex oil and gas reservoirs. However, the migrated images often suffer from low spatial resolution, migration artifacts, wave-mode crosstalk, and unbalanced amplitude response. Conventional vertical transversely isotropic elastic least-squares reverse time migration (VTI-elastic LSRTM) defines stiffness parameter perturbations as elastic images, which have different physical meanings from VTI-elastic RTM images. We have developed a VTI-elastic LSRTM method based on elastic wavefield vector decomposition that is a natural extension of VTI-elastic RTM. More specifically, our method applies least-squares inversion to VTI-elastic RTM and defines the compressional- and shear-wave reflectivity as elastic images (PP, PS, SP, and SS images). When computing the elastic images, we decompose the elastic wavefields into compressional and shear wavefields and cross-correlate the corresponding wave modes. We derive the reverse time demigration operator by taking the adjoint of the RTM operator. Using the migration and demigration operators, we formulate the VTI-elastic LSRTM as a linear inverse problem with the least-squares criterion. The conjugate gradient method is used to solve the optimization problem. Three numerical examples are presented to test the feasibility of our method. The VTI-elastic LSRTM images have higher resolution, fewer migration artifacts and wave-mode crosstalk, and improved amplitude response when compared with VTI-elastic RTM images.

Steeply dipping structural target-oriented viscoacoustic least-squares reverse time migration and its application

Steeply dipping structural target-oriented viscoacoustic least-squares reverse time migration and its application

Least-Squares Reverse-Time Migration of Water-Bottom-Related Multiples

Reverse-time migration of multiples can generate imaging results with wider illumination, higher folds, and broader wavenumber spectra than the conventional migration of primaries. However, the results of the migration of multiples retain heavy crosstalks generated by interactions between unrelated multiples, thereby seriously degrading imaging qualities. To eliminate such crosstalks, we propose a least-squares optimized algorithm of multiples. In this method, different-order water-column multiples and water-bottom-related multiples are extracted using multiple decomposition strategies before migration procedures. The proposed method treats the nth-order water-column multiples as virtual sources for Born modeling to produce the predicted (n+1)th-order water-bottom-related multiples. In each iteration, the gradients are calculated by crosscorrelating the forward-propagated nth-order water-column multiples with the backward-propagated seismic residuals between the observed and predicted (n+1)th-order water-bottom-related multiples. The developed approach is referred to as the least-squares reverse-time migration of water-bottom-related multiples (LSRTM-WM). Numerical experiments on a layered model and the Pluto 1.5 model demonstrate that LSRTM-WM can significantly remove crosstalks and considerably improve spatial resolution.

Target-oriented least-squares reverse-time migration using Marchenko double-focusing: reducing the artefacts caused by overburden multiples

SUMMARY Geophysicists have widely used least-squares reverse-time migration (LSRTM) to obtain high-resolution images of the subsurface. However, LSRTM is computationally expensive and it can suffer from multiple reflections. Recently, a target-oriented approach to LSRTM has been proposed, which focuses the wavefield above the target of interest. Remarkably, this approach can be helpful for imaging below complex overburdens and subsalt domains. Moreover, this approach can significantly reduce the computational burden of the problem by limiting the computational domain to a smaller area. Nevertheless, target-oriented LSRTM still needs an accurate velocity model of the overburden to focus the wavefield accurately and predict internal multiple reflections correctly. A viable alternative to an accurate velocity model for internal multiple prediction is Marchenko redatuming. This method is a novel data-driven method that can predict Green’s functions at any arbitrary depth, including all orders of multiples. The only requirement for this method is a smooth background velocity model of the overburden. Moreover, with Marchenko double-focusing, one can make virtual sources and receivers at a boundary above the target and bypass the overburden. This paper proposes a new algorithm for target-oriented LSRTM, which fits the Marchenko double-focused data with predicted data. The predicted data of the proposed method is modelled by a virtual source term created by Marchenko double-focusing on a boundary above the target of interest. This virtual source term includes all the interactions between the target and the overburden. Moreover, the Marchenko double-focused data and the virtual source term are free of multiples generated in the overburden. Consequently, our target-oriented LSRTM algorithm suppresses the multiples purely generated inside the overburden. Our algorithm correctly accounts for all orders of multiples caused by the interactions between the target and the overburden, resulting in a significant reduction of the artefacts caused by the overburden internal multiple reflections and improves amplitude recovery in the target image compared to conventional LSRTM.

Q-compensated least-squares reverse time migration with velocity-anisotropy correction based on the first-order velocity-pressure equations

Anisotropy and Q attenuation bring great challenges to seismic wave migration. On migrated images, anisotropy creates structural and positioning errors, and Q attenuation leads to weak amplitudes and misplacement of reflectors. A 2D Q-compensated least-squares reverse time migration with velocity-anisotropy correction ( QLSRTM-VA) is proposed through the construction of velocity-anisotropic Q-compensated forward modeling, Q-compensated adjoint, and Q-attenuated demigration operators to simultaneously correct velocity-anisotropy and Q-attenuation in the migration process. The preceding operators are derived using first-order velocity-anisotropic viscoacoustic quasi-differential wave equations with variable densities, which are stable, capable of conveniently dealing with variable density media and are easy to transform between velocity-anisotropic Q-compensation and Q-attenuation versions. As exemplified by two synthetic and field data sets, our QLSRTM-VA method increases the imaging resolution, signal-to-noise ratio, and amplitude preservation in deep regions. Our method is capable of producing better images than viscoacoustic isotropic least-squares reverse time migration (LSRTM) and acoustic anisotropic LSRTM.

Least-Squares Reverse Time Migration of Primary and Internal Multiple Predicted by the High-Order Born Modeling Method

Multiples can cause artifacts in imaging; however, they contain information about underground structures. If the internal multiples are removed as a noise, the information contained by the internal multiple will also be removed. This will cause loss of some useful structures in the image. If the multiples and the primary can be separated from the recorded seismic data for imaging, the information contained by the multiples can be used and the artifacts can be attenuated. Here we developed a method to separate the primary and internal multiples and use them in least squares reverse time migration (LSRTM). This method first separates the primary and the internal multiples in the data residual and predicts the wavefield of the primary and internal multiples in a forward- propagated wavefield. We use the high-order Born modeling method to predict the internal multiples. In this method, the internal multiples can be achieved by forward modeling of three times in the time domain. In the internal multiple prediction process, we get the wavefield of the primary and internal multiples in the forward-propagated wavefield. Then, by introducing the weighting matrix, we established the objective function for imaging of the primary and the internal multiples separately. In the calculation of gradient, we use the correlation of primary in the forward-propagated wavefield with the backward-propagated wavefield of the primary in the data residual, and internal multiples in the forward-propagated with the backward-propagated wavefield of internal multiples in the data residual. In this method, the multiple prediction process provided the internal multiples to suppress the artifacts, and LSRTM constructed the model for the multiple prediction process. Finally, we performed numerical tests using synthetic data, and the results indicated that the LSRTM without the internal multiple can suppress not only the artifacts of internal multiples but also some useful structures below the salt dome. LSRTM with primary and internal multiples can suppress the artifact of internal multiples, and the useful structures below the salt dome are compensated in the image.

Wavefield-reconstructed least-squares reverse time migration with dynamic constraint factor

Wavefield-reconstructed least-squares reverse time migration with dynamic constraint factor

Capability of elastic-wave imaging for monitoring conformance and containment in geologic carbon storage

• We have studied the capability of elastic-wave imaging for monitoring conformance and containment in geologic carbon storage, particularly for monitoring secondary CO 2 plumes, using the waveform information of time-lapse multi-component seismic reflection data for the Kimberlina-2 models. We have described the elastic models of different secondary CO 2 plume scenarios and performed elastic-wave modeling and imaging for these models. Our results show that seismic imaging with elastic least-squares reverse-time migration of multi-component data is able to locate the secondary CO 2 plumes at different depths in the Kimberlina-2 model. With noise-free data, the time-lapse seismic images of CO 2 migration scenarios also reveal the growth of the secondary CO 2 plumes with time. We have also demonstrated that elastic-wave imaging using multi-component data can provide the relative magnitude difference in elastic properties caused by the secondary CO 2 plumes. Even with noisy data, seismic imaging using elastic LSRTM is capable of locating early stages of the secondary CO 2 plumes. Seismic noise has more impact on locating the very early stage of the secondary CO 2 plumes than those in the later stages. Our image noise analysis further suggests that the elastic-wave imaging capability decreases for the acquisition geometry with sparser shots/receivers or shorter recording offsets. The image sign-to-noise ratio is approximately proportional to the square root of the total number of seismic traces used for elastic-wave imaging. This relationship can be use to guide seismic monitoring design. Note that it is difficult to use multi-component data to accurately invert for the time-lapse change of an elastic property if its change is much smaller than those of the other elastic properties. We will study how seismic noises can affect the CO 2 detection threshold in the future. Commercial-scale geologic carbon storage requires monitoring for conformance and containment of the injected CO 2 , including secondary CO 2 plumes. Detecting and locating secondary CO 2 plumes are even more challenging than monitoring the primary CO 2 plumes. We study the capability of elastic-wave imaging for monitoring secondary CO 2 plumes with time-lapse multi-component data from 3D elastic models built using the geologic settings of the Kimberlina site located in the San Joaquin Basin, California. In the numerical simulation of CO 2 migration in these time-lapse Kimberlina-2 models, CO 2 migrates from the storage reservoir along a steeply dipping fault and forms secondary CO 2 plumes in formations at different depths. In our study, we extract a 2D slice through the location of the secondary CO 2 plumes to perform elastic-wave modeling and imaging based on numerical simulation to the elastic-wave equation. We use a finite-difference algorithm to generate time-lapse multi-component synthetic elastic-wave reflection data and employ elastic least-squares reverse-time migration to produce images of P- and S-wave velocities and density for various sizes of the secondary CO 2 plumes after CO 2 migration through the fault. Our imaging results show that, for noise-free data, elastic-wave imaging accurately locates the secondary CO 2 plumes at different depths and reveals the growth of the size of the secondary CO 2 plumes with years of CO 2 migration. The relative amplitude differences of seismic images reveal how secondary CO 2 plumes change elastic parameters differently. We further study the capability of elastic-wave imaging for monitoring the secondary CO 2 plumes using noisy synthetic data containing seismic noise extracted from a field surface seismic dataset. We evaluate elastic-wave imaging capability for monitoring secondary CO 2 plumes using different noise levels and different acquisition geometries. Our results demonstrate that elastic-wave imaging is capable of locating secondary CO 2 plumes even with strong noisy data, and a relatively higher signal-to-noise ratio (SNR) is needed to locate (or image) the secondary CO 2 plumes at earlier stages. The imaging capability decreases for sparser or shorter-offset acquisition geometry. Our results indicate that the image SNR is approximately proportional to the square root of the total trace number used for elastic-wave imaging. This relationship can be used to guide whether a denser or longer-offset acquisition geometry is worth the increase of seismic image quality for improving the capability to locate secondary CO 2 plumes. Note that multi-component data can inform us whether the time-lapse change of one elastic property is much smaller than others, but it is difficult to accurately invert for the time-lapse change of an elastic property if its change is relatively much smaller than the changes of the other elastic properties.

Least-squares reverse time migration based on the viscoacoustic VTI pure qP-wave equation

With the deepening of oil and gas exploration and the increasing complexity of exploration targets, the influence of anisotropy and anelasticity of subsurface media on seismic imaging cannot be ignored. The least-squares reverse time migration is developed on the idea of linear inversion, which can effectively solve the amplitude imbalance, low resolution, and serious imaging noise problems of RTM. In this paper, based on the viscoacoustic pure qP-wave equation, the corresponding demigration operator, adjoint operator, and gradient-sensitive kernel are derived, and the least-square reverse time migration imaging algorithm of viscoacoustic pure qP-wave in VTI medium is proposed. During iterative inversion, the inverse of Hessian is approximately solved to achieve stable attenuation compensation. Finally, we verify the effectiveness and applicability of the proposed viscoacoustic VTI least-squares reverse time migration imaging algorithm through the model tests and field data. The numerical results show that the method can compensate for the amplitude loss and phase distortion caused by attenuation, and correct the anisotropy-induced misalignment of the reflection interfaces, which improves the accuracy and resolution of the imaging profile.

Least-squares reverse time migration via deep learning-based updating operators

Two common issues of least-squares reverse time migration (LSRTM) consist of the many iterations required to produce substantial subsurface imaging improvements and the difficulty of choosing adequate regularization strategies with optimal hyperparameters. We investigate how supervised learning can mitigate these shortcomings by solving the LSRTM problem through an iterative deep learning framework inspired by the projected gradient descent algorithm. In particular, we develop an image-to-image approach interlacing the gradient steps at each iteration with blocks of residual convolutional neural networks (CNNs) that capture the prior information in the training phase. By including the least-squares data-misfit gradient into the learning process, we force the solution to comply with the observed seismic data, while the CNN projections implicitly account for the regularization effects that lead to high-resolution reflectivity updates. After training with 900 randomly generated instances, our network ensemble can estimate accurate reflectivity distributions in only a few iterations. To demonstrate the effectiveness and generalization properties of the method, we consider three synthetic cases: a folded and faulted model, the Marmousi model, and the Sigsbee2a model. We empirically find that it is possible to obtain an improved reflectivity model for out-of-distribution instances by using the learned reconstructions as warm starts for the conjugate gradient algorithm and bridging the gap between the learned and conventional LSRTM schemes. Finally, we apply the proposed network with transfer learning on a 2D towed-streamer Gulf of Mexico field data set, producing high-resolution images comparable with traditional LSRTM but drastically reducing the required number of iterations.

A new scheme of wavefield decomposed elastic least-squares reverse time migration

Elastic least-squares reverse time migration (ELSRTM) describes the reflectivity of the underground media more accurately than acoustic LSRTM in theory while suffering from the P- and S-waves crosstalk artifacts. We propose a new wavefield decomposed ELSRTM scheme to alleviate these crosstalk artifacts, which is different from conventional methods. In our new scheme, we implement the wavenumber domain elastic wavefield vector decomposition equivalently in the time-space domain to decompose source wavefield without Fourier transform, but with high precision. Then we decompose adjoint wavefield by constructing the shear component in a decoupled adjoint wave equation. Finally, based on elastic impedance parameterization, we derive the gradients with respect to elastic reflectivity in the wavefield-decomposed ELSRTM. Numerical examples show that our method is feasible even when applied to models with complex and uncorrelated P- and S-wave velocity structures.

Gradient normalized least-squares reverse-time migration imaging technology

Least-squares reverse-time migration (LSRTM) can overcome the problems of low resolution and unbalanced amplitude energy of deep formation imaging in reverse-time migration (RTM); hence, it can obtain a more accurate imaging profile. In the conventional conjugate gradient LSRTM, the gradient is obtained based on cross correlation without a precondition operator, and the source has a great influence on the gradient, causing the convergence rate to be slow. In the framework of conventional conjugate gradient LSRTM, a normalized cross-correlation of the source wavefield was used in this study to effectively weaken the influence of the source effect and reduce the low-frequency noise. The idea of normalized cross-correlation of the source wavefield was adopted to improve the steepest descent gradient to further accelerate the iterative convergence speed and complete the final migration imaging. Model and field data examples verify the advantages of the proposed methods over conventional methods in reducing source effects, improving convergence speed, and enhancing underground deep illumination.

A fast least-squares reverse time migration method using cycle-consistent generative adversarial network

With high imaging accuracy, high signal-to-noise ratio, and good amplitude balance, least-squares reverse time migration (LSRTM) is an imaging algorithm suitable for deep high-precision oil and gas exploration. However, the computational costs limit its large-scale industrial application. The difference between traditional reverse time migration (RTM) and LSRTM is whether to eliminate the effect of the Hessian operator or not while solving Hessian matrix explicitly or eliminating the effect of the Hessian matrix implicitly has a very high requirement on computation or storage capacity. We simulate the inverse Hessian by training a cycle-consistent generative adversarial network (cycleGAN) to construct a mapping relationship between the RTM results and the true reflectivity models. The trained network is directly applied to the RTM imaging results, which improves the imaging quality while significantly reducing the calculation time. We select three velocity models and two velocity models respectively to generate the training and validation data sets, where the validation data is not involved in the training process. The prediction results on the validation data sets show that the trained network significantly improves the imaging quality with almost no additional in computational effort. Finally, we apply the network trained with only synthetics to the field data. The predicted results confirm the effectiveness and good generalization of the proposed method.

A new elastic least-squares reverse-time migration method based on the new gradient equations

A new elastic least-squares reverse-time migration method based on the new gradient equations

Huber inversion-based reverse-time migration with de-primary imaging condition and curvelet-domain sparse constraint

Least-squares reverse-time migration (LSRTM) formulates reverse-time migration (RTM) in the least-squares inversion framework to obtain the optimal reflectivity image. It can generate images with more accurate amplitudes, higher resolution, and fewer artifacts than RTM. However, three problems still exist: (1) inversion can be dominated by strong events in the residual; (2) low-wavenumber artifacts in the gradient affect convergence speed and imaging results; (3) high-wavenumber noise is also amplified as iteration increases. To solve these three problems, we have improved LSRTM: firstly, we use Huber-norm as the objective function to emphasize the weak reflectors during the inversion; secondly, we adapt the de-primary imaging condition to remove the low-wavenumber artifacts above strong reflectors as well as the false high-wavenumber reflectors in the gradient; thirdly, we apply the L 1 -norm sparse constraint in the curvelet-domain as the regularization term to suppress the high-wavenumber migration noise. As the new inversion objective function contains the non-smooth L 1 -norm, we use a modified iterative soft thresholding (IST) method to update along the Polak-Ribière conjugate-gradient direction by using a preconditioned non-linear conjugate-gradient (PNCG) method. The numerical examples, especially the Sigsbee2A model, demonstrate that the Huber inversion-based RTM can generate high-quality images by mitigating migration artifacts and improving the contribution of weak reflection events.

Deep learning-based point-spread function deconvolution for migration image deblurring

High-resolution seismic imaging is crucial for deriving an accurate geologic section and ensuring successful petroleum exploration. However, traditional high-resolution imaging methods, such as least-squares reverse time migration (LSRTM) and migration deconvolution, usually involve high computational costs for approximating the inverse Hessian matrix with limited improvements in resolution. To obtain high-resolution migration images and reduce computing costs, we present a deep learning-based point-spread function (PSF) deconvolution method. We decompose the large Hessian matrix into small dispersed PSFs and design a convolutional neural network (CNN) to automatically predict the deconvolution operator of every single PSF. The deconvolution operator eliminates the PSF effect and balances the insufficient illumination of the acquisition geometry. To obtain an efficient CNN model for predicting deconvolution operators, we trained 2500 pairs of PSFs and their corresponding deconvolution operators collected from part of a model using the regular PSF deconvolution method. After training the network, we could predict all deconvolution operators in a few seconds for the following processing of migration images. The results on synthetic and field-data applications indicate that our method could provide a deblurred migration image comparable to that of the LSRTM approaches with significantly reduced computational and memory costs.

Robust boundary-preserving multi-source least-squares reverse-time migration with combined sparse constraints

Least-squares reverse time migration (LSRTM) can generate a higher-resolution image than other algorithms but with a higher computation cost. Although the conventional source-encoding algorithm can improve the efficiency, it faces the problems of multi-source crosstalk noise and observed-synthetic data mismatching caused by the unpractical assumption of the fixed acquisition setup. In this paper, we propose a hybrid sparsity-constrained multi-source encoding algorithm to overcome the above bottleneck problems. Harmonic wavelets are used as an encoding operator so that the blended multi-source wavefields can be decomposed as each original single-source wavefield based on the orthogonality of trigonometric functions. Due to that the harmonic frequency selection might lead to an insufficient stack of single-frequency components, there would generate background noise on the image, reducing the image quality and convergence rate. Therefore, we apply a combined-sparsity constraint to improve the robustness and convergence rate of this algorithm. Besides, we use the convolution-norm misfit function to eliminate the influence of incorrect source wavelet. Synthetic data examples show the rationality and feasibility of our algorithm.

Efficient least-squares reverse time migration using local cross-correlation imaging condition

Abstract The data-domain least-squares reverse time migration (LSRTM), a promising imaging method for obtaining a high-resolution reflectivity image, can be implemented by matching the de-migrated data to the observed one. However, LSRTM requires expensive computational costs in its implementation due to tremendous memory usage (direct storage on disk/memory) for saving source wavefields and repeatable forward/backward wavefield simulations with iterations compared to conventional RTM. Although the source wavefield reconstruction technique can be used to reduce the memory usage, additional computational cost is inevitable because it is implemented during backward wavefield simulation. To alleviate the computational burden, we have developed an efficient LSRTM scheme with local cross-correlation imaging condition (LSRTM-LC), which can use pre-saved source wavefields according to the window size. Because the window size is much shorter than the total recording time for wavefield extrapolation, storing source wavefields into computer memory can be feasible. In addition, the procedure for storing source wavefields is implemented only at the first iteration because the background velocity is fixed during iterations of LSRTM. To validate the feasibility of the LSRTM-LC scheme, we have carried out several numerical tests with synthetic datasets. Numerical tests demonstrated that LSRTM-LC can provide us with equivalent de-migrated data from saved source wavefields and the gradient vector obtained from LSRTM-LC compared to those obtained from the conventional LSRTM. As a result, LSRTM-LC can generate the same quality of reflectivity models as conventional LSRTM, however, it requires less computational costs compared to conventional LSRTM.

Q-compensated least-squares reverse time migration in TTI media using the visco-acoustic TTI wave equation based on the SLS model

The anelasticity and anisotropy widely exist in real subsurface media. Strong anelasticity will lead to phase dispersion and amplitude attenuation during seismic wave propagation. Anisotropy cause seismic waves to have obviously different kinematic characteristics from that of isotropy. For seismic migration, ignoring the anelasticity and anisotropy of subsurface media will significantly reduce the quality of migration images, even cause imaging failure. We propose a Q-compensated least-squares reverse time migration (Q-LSRTM) in tilted transversely isotropic (TTI) media to correct these effects. According to the Born approximation of seismic wave equation, a linearized visco-acoustic TTI pure qP-wave modeling operator is derived using a new visco-acoustic TTI wave equation for one standard linear solid (SLS) model, which can deal with the anelasticity and can simulate pure qP-wave steadily in attenuating anisotropic media without qSV-wave artifacts. Then, the corresponding adjoint equation is formulated using the adjoint-state method to calculate the gradient sensitivity kernel for the visco-acoustic TTI media. Because of the least-squares inversion, the Q compensation can be achieved during the iterations, so that the over-amplification of noises can be avoided naturally. In addition, compared with conventional LSRTM, the proposed method compensates for the anelasticity and corrects the anisotropy, so as to produce images with better spatial resolution and amplitude fidelity. Numerical examples demonstrate the feasibility and advantages of the proposed method for the data including strong attenuation effects over conventional LSRTM.

Optimal minibatch shot selection for target-oriented least-squares reverse time migration

We have introduced a strategy to select optimal minibatches for target-oriented least-squares reverse time migration that adopts stochastic optimizers to solve the inverse problem. The proposed method with optimal minibatches can improve image quality at reduced computational costs. We formulate the optimal shot selection criterion using a shot effectiveness metric and clustering techniques. We first calculate the illumination and effectiveness using forward modeling operations for each shot. We then apply k-means clustering on the shot effectiveness to obtain optimal minibatches suitable for stochastic optimization methods. Clustering shots into optimal minibatches can help retain only those necessary shots to image a target zone. These batches are a-priori calculated only once and used for all iterations. We adopt the stochastic Adam optimizer to speed up the algorithm’s convergence. A novel beam migration approach based on frame theory is used to speed up the analysis. This beam method extracts only subsets of beams needed for the migration calculation from the surface data. Efficiency is further enhanced due to the spectral localization of the beams. The presented workflow leads to a robust and efficient inversion algorithm that provides high-resolution images in hard-to-image target zones.