Least-squares Reverse Time Migration Research Articles

Least-squares reverse time migration of simultaneous sources with deep-learning-based denoising

Least-squares reverse time migration (LSRTM) is currently one of the most advanced migration imaging techniques in the field of geophysics. It uses least-squares inversion to fit the observed data, resulting in high-resolution imaging results with more accurate amplitudes and better illumination compensation than conventional reverse time migration (RTM). However, noise in the observed data and the Born approximation forward operator can result in high-wavenumber artifacts in the final imaging results. Moreover, iteratively solving LSRTM leads to one or two orders of computational cost higher than conventional RTM, making it challenging to apply extensively in industrial applications. Simultaneous source acquisition technology can reduce the computational cost of LSRTM by reducing the number of wavefield simulations. However, this technique also can cause high-wavenumber crosstalk artifacts in the migration results. To effectively remove the high-wavenumber artifacts caused by these issues, simultaneous source and deep learning are combined to speed up LSRTM as well as suppress high-wavenumber artifacts. A deep residual neural network (DR-Unet) is trained with synthetic samples, which are generated by adding field noise to synthesized noise-free migration images. Then, the trained DR-Unet is applied on the gradient of LSRTM to remove high-wavenumber artifacts in each iteration. Compared to directly applying DR-Unet denoising to LSRTM results, embedding DR-Unet denoising into the inversion process can better preserve weak reflectors and improve denoising effects. Finally, the proposed LSRTM method is tested on two synthetic data sets and a land data set. The tests demonstrate that the proposed method can effectively remove high-wavenumber artifacts, improve imaging results, and accelerate convergence speed.

Target-oriented least-squares reverse time migration with Marchenko redatuming and double focusing: Field data application

Recently, the focus of reflection seismologists has shifted to applications for which a high-resolution image of the subsurface is required. Least-squares reverse time migration (LSRTM) is a common tool used to compute such images. Even so, its high computational costs have led seismologists to use target-oriented LSRTM for imaging only a small target of interest within a larger subsurface block. Redatuming the data to the upper boundary of the target of interest is one approach to target-oriented LSRTM, but many redatuming methods cannot account for multiple scattering within the overburden. We apply a target-oriented LSRTM algorithm that integrates Marchenko redatuming and double focusing to a field data set. This redatuming method accounts for all orders of multiple scattering in the overburden, thus improving the accuracy of target-oriented LSRTM. Moreover, we determine the effectiveness of a double-focusing algorithm in reducing the data size by decreasing spatial and temporal dimensions of the model and the data. The algorithm’s performance is evaluated using field data acquired in the Norwegian Sea. The numerical results indicate that the target-oriented LSRTM algorithm with Marchenko double-focusing can reduce the internal multiple effects and increase the resolution of the resulting image.

Q-compensated image-domain least-squares reverse time migration through preconditioned point-spread functions

Image-domain least-squares reverse time migration (IDLSRTM) through point-spread functions (PSFs) has been proven to be a feasible approach to improve the spatial resolution and amplitude fidelity of reflection images recovered by reverse time migration (RTM). However, it usually ignores the earth’s Q-effects, which may lead to an unfocused reflection image with an undesired spatial resolution. In this paper, we develop a Q-compensated IDLSRTM approach (denoted as Q-IDLSRTM) through PSFs, in which we use the viscoacoustic wave equation based on the generalized standard linear solid model to simulate inherent subsurface attenuation and the linear inversion to compensate for the amplitude attenuation. The PSFs are estimated by a round of modeling-migration computation and spatial interpolation on the fly. There are two key points in the developed Q-IDLSRTM approach. The first is that we must apply the deblurring filter as a preconditioner to compensate for the attenuation of image amplitude of PSFs and RTM in a viscoacoustic medium, before the iterative solution. The preconditioned PSFs and RTM images can help us to construct a less ill-posed image-domain inverse problem that can produce an improved image quality and a faster convergence rate, compared with the conventional Q-IDLSRTM approach without the deblurring filter. The second key point is that we can impose the L1-norm constraint and total variation regularization on the reflection image to stabilize the solution of the ill-posed inverse problem. Several 2D and 3D experiments verify that the developed approach can achieve better imaging quality in terms of amplitude fidelity and spatial resolution relative to the conventional Q-IDLSRTM and acoustic IDLSRTM approaches.

Ocean bottom dual-sensor Q-compensated elastic least-squares reverse time migration based on acoustic and separated-viscoelastic coupling equations

The ocean bottom dual-sensor (OBD) observation system places water detectors (pressure sensors) and land detectors on the irregular seabed interface to receive acoustic pressure fields and elastic wavefields, respectively. These data are usually applied to effectively remove ghost waves and ringing interference in the seawater layer, rather than being jointly used for migration. We propose an ocean bottom dual-sensor viscoacoustic and separated-viscoelastic coupled Q-compensated least-squares reverse time migration (OBD-VSV- QLSRTM) in curvilinear coordinates. In this method, to solve the Q-compensated elastic inverse problem, we jointly use the recorded acoustic pressure and elastic multicomponent seismic data from the OBD observation system. A new misfit function in the sense of least-squares inversion is constructed based on viscoacoustic and separated viscoelastic wavefields. Acoustic and separated viscoelastic equations are used to describe the acoustic wavefield in the acoustic medium and the Q-compensated P and S wavefields in the underlying viscoelastic medium, respectively. On the acoustic-viscoelastic interface, we implement a stable governing equation to ensure continuous and steady transmission of Q-compensated stresses in the subseabed and pressure in the seawater. To address the irregular seabed problem, we mesh the acoustic-viscoelastic models onto curvilinear grids and apply coordinate transformations to map the models and equations to a new curvilinear coordinate system. We derive viscoacoustic and separated-viscoelastic coupled adjoint and demigration operators in the curvilinear coordinates, enabling linear acoustic-viscoelastic modeling and stable receiver-side back-propagated wavefield continuation. Numerical examples conducted on two typical acoustic-viscoelastic models demonstrate that our proposed OBD-VSV- QLSRTM in curvilinear coordinates can produce highly accurate P- and S-wave images efficiently.

Efficient least-squares reverse time migration in TTI media using a finite-difference solvable pure qP-wave equation

Abstract The anisotropic characteristics of underground media play a crucial role in seismic wave propagation and should be accounted for during seismic imaging. The least-squares reverse time migration (LSRTM) method achieves ideal imaging results by suppressing migration artifacts and balancing amplitude. Therefore, pure quasi-P (P-qP) wave equations in tilted transversely isotropic (TTI) media have been widely used to implement LSRTM, owing to their ability to produce stable and noise-free wavefields. However, solving the anisotropic P-qP-wave equations typically necessitates the use of spectral-based methods, making them computationally inefficient, especially in 3D applications. In this study, we first develop a P-qP-wave equation in TTI media that can be efficiently computed through the finite-difference (FD) approach. Numerical tests show that, in comparison to the previous TTI P-qP-wave equation, the newly derived FD solvable TTI P-qP-wave equation yields reasonably accurate and highly efficient wavefield simulations. Then, building on our newly derived wave equation, we derive its adjoint migration and demigration operator to implement TTI LSRTM. Two synthetic examples suggest that the newly presented P-qP-wave TTI LSRTM approach effectively correct for the anisotropy effects, providing high-quality imaging results. Additionally, our approach has superior computational efficiency over the conventional P-qP-wave TTI LSRTM technique based on a hybrid FD pseudo-spectral (HFDPS) solver.

GPR Least-Squares Reverse Time Migration Based on the Improved Cross Correlation Window

GPR Least-Squares Reverse Time Migration Based on the Improved Cross Correlation Window

Nonlinear least-squares reverse time migration of prismatic waves for delineating steeply dipping structures

Prismatic reflections in seismic data carry abundant information about subsurface steeply dipping structures, such as salt flanks or near-vertical faults, playing an important role in delineating these structures when effectively used. Conventional linear least-squares reverse time migration (L-LSRTM) fails to use prismatic waves due to the first-order Born approximation, resulting in a blurry image of steep interfaces. We develop a nonlinear LSRTM (NL-LSRTM) method to take advantage of prismatic waves for the detailed characterization of subsurface steeply dipping structures. Compared with current least-squares migration methods of prismatic waves, our NL-LSRTM is nonlinear and thus avoids the challenging extraction of prismatic waves or the prior knowledge of L-LSRTM results. The gradient of NL-LSRTM consists of the primary and prismatic imaging terms, which can accurately project observed primary and prismatic waves into the image domain for the simultaneous depiction of near-horizontal and near-vertical structures. However, we find that the full Hessian-based Newton normal equation has two similar terms, which prompts us to make further comparison between the Newton normal equation and our NL-LSRTM. We determine that the Newton normal equation is problematic when applied to the migration problem because the primary reflections in the seismic records will be incorrectly projected into the image along the prismatic wavepath, resulting in an artifact-contaminated image. In contrast, the nonlinear data-fitting process included in the NL-LSRTM contributes to balancing the amplitudes of primary and prismatic imaging results, thus making NL-LSRTM produce superior images compared with the Newton normal equation. Several numerical tests validate the applicability and robustness of NL-LSRTM for the delineation of steeply dipping structures and illustrate that the imaging results are much better than the conventional L-LSRTM.

Viscoacoustic least squares reverse-time migration using L1-2 norm sparsity constraint

Abstract Least-squares reverse-time migration (LSRTM) has become an advanced technique for complex structures imaging of the subsurface, as it can provide a higher resolution and more balanced amplitude migrated image than conventional reverse-time migration (RTM). However, the intrinsic attenuation of subsurface introduces amplitude attenuation and phase dispersion of seismic wavefield, which leads to the inverted image kinematically and dynamically inexactitude. Moreover, the imperfect geometry, limited bandwidth of seismic data, and inappropriate modeling kernel etc., would inevitably introduce two side-effects in migrated image, resulting in degradation of LSRTM imaging potential. To alleviate above issues, we present a data-domain sparsity constraint viscoacoustic least-squares reverse-time migration algorithm in this paper. In particular, we utilize the decoupled constant Q fractional Laplacians (DFLs) viscoacoustic wave equation as the modeling kernel to describe the attenuation effects of the subsurface, while a model constraint constructed in the misfit function via L1-2 norm is carried out to clear the migrated artefacts and boost the imaging resolution. Thanks to the excellent performance in sparsity, the drawbacks of unconstraint LSRTM can be effectively mitigated by the L1-2 norm-based regularization. In this paper, we adopt the alternating direction of multipliers method (ADMM) to iteratively address the constrained L1-2 minimization problem by implementing a proximal operator, and three synthetic examples are hired to evaluate the effectiveness and practicability of the proposed strategy. Migration results prove that the proposed scheme can effectively compensate the attenuation effects, improve the resolution, and suppress the migration artifacts of inverted images even in the complex imaging situations.

Efficient extended least-squares reverse time migration based on an excitation amplitude imaging condition

In least-squares reverse time migration (LSRTM), which is typically an overdetermined linear inverse problem, the kinematic traveltime error from an inaccurate migration velocity leads to migration image degradation and slow convergence speed. Extended LSRTM (ELSRTM), with an extended imaging condition along the subsurface offset in LSRTM, can effectively mitigate sensitivity to the migration velocity accuracy by adding an extra dimension to the model space. However, for assembling migration images, large imaging condition operations that are proportional to the number of subsurface-offset bins hinder the practical application of ELSRTM. To address this computational problem in ELSRTM, we develop an efficient ELSRTM method based on a modified excitation amplitude (ExA) imaging condition. Furthermore, our ELSRTM method can correctly represent the forward source wavefield by convolving the source wavelet with the ExA. With the forward source wavefield represented this way, Born-modeled data can be simulated efficiently. Thus, wavefield simulation for generating the forward source wavefield is implemented only once at the first iteration. Based on these computational advantages, our ELSRTM method becomes highly efficient. Synthetic data examples on a modified Marmousi2 velocity model demonstrate that our ELSRTM method produces an accurate migration image efficiently even if an inaccurate migration velocity is used. Field data acquired from the Chukchi Sea of the Arctic Ocean also are used to verify the practicality of our ELSRTM method.

Staining algorithm for least-squares reverse time migration

The staining algorithm-based reverse time migration (RTM) technique has shown promise in improving imaging results for certain complex structures, such as subsalt and steeply dipping layers, by incorporating the complex domain. However, RTM suffers from low-resolution images as it utilizes the adjoint of the forward modeling operator rather than its inverse operator. To address this limitation, we propose a new approach called stained least-squares RTM (ST-LSRTM) that enhances the gradient through the integration of the staining algorithm. In ST-LSRTM, we first develop the wave equation and Born approximation, transitioning from the real to the complex domains. To simplify the linear perturbation operator, we constrain the magnitude of the imaginary velocity. Next, we compute the gradient in the complex domain. The enhanced gradient is obtained by replacing the traditional gradient with the normalized stained gradient within the stained region. Finally, we construct a misfit function based on the L2 norm and calculate the step size and gradient direction using the conjugate gradient method. Model tests demonstrate that ST-LSRTM achieves high-resolution imaging results even when the stained area is imprecise. Comparisons of waveform curves and normalized residual curves confirm that ST-LSRTM approaches true reflection coefficient model more closely compared to traditional LSRTM.

Numerical simulations of pure quasi-P-waves in orthorhombic anisotropic media

The accurate simulation of anisotropic media is critical in seismic imaging and inversion. In recent years, some scholars have dedicated efforts to the study of precise elastic waves in anisotropic media; however, it is easy to separate P-wave and S-wave from elastic wave fields in isotropic media but difficult to separate them in anisotropic media. To address this issue, others have proposed pseudo-pure-wave equations based on the theory of wave-mode separation, but shear wave interference still exists. Therefore, we derived the first-order pure quasi-P-wave equation with no shear wave component in orthorhombic anisotropic media (ORT) which is common in the Earth’s crust and has very important research value. The presence of a pseudodifferential operator in the equation poses a challenge for solving. In order to solve the pure wave equation, we decomposed the original pseudodifferential operator into an elliptic differential operator and a scalar operator, both of which are easily solvable. In addition, we extended the equation from ORT media to tilted ORT (TORT) media. The example results indicate that our pure quasi-P-wave equation can yield a more stable and accurate P-wave field. The pure wave equation we propose can be applied in reverse time migration (RTM), the least squares RTM (LSRTM), and even the full waveform inversion (FWI).

Least Squares Reverse Time Migration of Ground Penetrating Radar Data Based on Modified Total Variation

As a fundamental part of ground penetrating radar (GPR) data processing, reverse time migration (RTM) can correctly position reflection waves and focusing diffraction waves on the proper spatial position. Least-squares reverse-time migration (LSRTM) is widely used in the seismic field for its ability to suppress artifacts and generate high-resolution images in comparison to conventional RTM. However, in the particular case of GPR detection methods, LSRTM is extremely susceptible to aliasing artifacts caused by under-sampling. In pursuit of enhanced precision in underground structure characterization, this paper presents the development of a new LSRTM based on modified total variation (MTV) regularization to improve imaging resolution. Initially, the objective function of LSRTM is derived by combining the Born approximation in 2-D transversal magnetic mode. Next, the adjoint equations and their gradients are solved using the Lagrange multiplier method. The objective function is then constrained by MTV regularization to ensure the precision and convergence of the LSRTM, which delivers a refined edge with reconstruction details. In the numerical experiments, in comparison to the conventional LSRTM method, the LSRTM-MTV algorithm demonstrated a 30.4% increase in computational speed and a 21.1% reduction in mean squared error (MSE). The outperformance of the proposed method is verified in detail through the image resolution and amplitude preservation in the test of synthetic data and laboratory data. Future research efforts will center on applying the proposed method to models featuring dispersive or anisotropic media that closely mimic real-world conditions and extending the application to various imaging techniques involving objective function minimization.

Least-squares RTM in nonorthogonal coordinates and applications to VTI media

Least-squares RTM in nonorthogonal coordinates and applications to VTI media

Least squares reverse time migration imaging with illumination preconditioned based on improved PRP conjugate gradients

Least squares reverse time migration (LSRTM) imaging is the one of the most accurate methods for migration imaging at present, and Polak–Ribiere–Polyak conjugate gradient (PRPCG) for LSRTM has the good numerical performance but weak convergence, so we construct an optimization factor to improve the iteration direction of the gradient, which can automatically generate a sufficient descent direction. The improved PRPCG (IPRPCG) can reduce the data residual values and speed up the iteration. And the illumination preconditioned (IP) operator is employed to IPRPCG-LSRTM which solves the problem of low resolution due to the insufficient iterative gradient information. In this paper, the experiments show that the imaging results of the proposed method (IPRPCG-IP-LSRTM) is improved greatly in detail characterization and events continuity, the iterative curve converged faster significantly, and the normalized data residual was reduced by 6.55% on average, which improved the accuracy of migration imaging effectively.

Source-independent least-squares reverse time migration in vertical transversely isotropic media based on the Student’s t-distribution

Abstract Seismic anisotropy exists in various types of strata and should be considered in seismic imaging schemes. Seismic imaging algorithms based on isotropic assumption neglect the effects of anisotropy on seismic data, which cause migration artifacts and waveform distortion. To correct the effects of anisotropy on seismic wave propagation, we propose an imaging algorithm that performs least-squares reverse time migration in vertical transversely isotropic acoustic media. We derive the following operators to implement this algorithm, the de-migration operator, its adjoint migration operator, and the corresponding gradient. However, an inaccurate estimated source wavelet will introduce error in the seismic simulation, and thus increase the mismatch between observed and synthetic data for least-squares reverse time migration. In addition, the noises, especially the noises with abnormal amplitudes in the seismic data, damage the inversion convergence and reduce the imaging resolution. To improve the image quality, we propose to use convolved wavefields between observed and synthetic data so that such a mismatch can be independent of the source wavelets. Also, we use the Student’s t-distribution instead of L2 norm in our inversion scheme to better handle the seismic noise. Its implementation only modifies the gradient of the conventional least-square reverse time migration scheme. Our numerical tests show a clear improvement using our proposed imaging algorithm when compared with the conventional isotropic migration scheme for the anisotropic data. Also, the synthetic examples demonstrate the feasibility and effectiveness of our proposed source-independent algorithm using the Student’s t-distribution.

A tutorial of image-domain least-squares reverse time migration through point-spread functions

Least-squares reverse time migration (LSRTM) has shown great potential to improve the amplitude fidelity and spatial resolution of a reverse time migration (RTM) image. However, the main disadvantage is that it requires significant computational resources for the iterative solution. To ameliorate this problem, the image-domain LSRTM (IDLSRTM) approach through point-spread functions (PSFs) has been proven to be a viable and alternative technique to deconvolve the standard RTM image. In this paper, we develop a tutorial for the numerical implementation of IDLSRTM through PSFs. The Hessian matrix in the standard IDLSRTM approach is estimated through the spatial interpolation of precomputed PSFs on the fly, where the PSFs are computed by one-round forward modeling and migration. However, the resulting IDLSRTM scheme is highly ill-conditioned because of the incomplete acquisition condition, irregular subsurface illumination, and band-limited data. To stabilize inversion and improve the inverted image, we suggest that the PSFs and RTM images should be deblurred by applying a deblurring filter. The deblurring PSFs can reduce the condition number of the standard Hessian matrix and make the inverse problem more well conditioned, thus improving imaging quality and accelerating convergence. Furthermore, the regularization operators should be imposed to produce a reasonable inverted image and avoid the overfitting of the image-matching term. Based on several examples with synthetic and field data, we can conclude two significant points. The first point is that the standard IDLSRTM approach through the conventional PSFs can only recover a reflectivity image similar to the deblurring RTM and nonstationary matching filter (NMF) approaches. The second point is that the deblurring IDLSRTM approach through the deblurring PSFs can retrieve the reflectivity image with higher resolution and better amplitude fidelity than the standard IDLSRTM approach and the standard RTM approach with the deblurring filter or NMF.

Plane-wave least-squares diffraction imaging using short-time singular spectrum analysis

Abstract Diffractions are seismic waves generated by small-scale heterogeneities in the subsurface. These are often superimposed by strong reflections so that they are not visible on the image, leading to misinterpretation and incorrect localization of the scatterers. Therefore, the separation of diffracted and reflected waves is a crucial step in identifying these small-scale diffractors. To realize the separation of diffraction and imaging, a least-squares reverse time migration method of plane waves (PLSRTM) optimized with short-time singular spectrum analysis (STSSA) was developed in this work. The proposed STSSA algorithm exploits the properties of singular spectral analysis (SSA) to separate linear signals. By establishing the Hanning window and the energy compensation function, it also compensates for the shortcomings of SSA in local dip processing and convergence of linear signals. As there is no clear boundary between reflected and diffracted waves, the energy loss during separation leads to a slow convergence rate of the diffraction wave imaging technique. We use STSSA as a constraint for PLSRTM, which greatly improves the imaging quality for diffraction waves. The tests with the Sigsbee2A model and noisy seismic data have shown that our method can effectively improve the resolution of diffraction wave imaging and that the constraint of STSSA increases the robustness to noisy data.

Anisotropic elastic least-squares reverse time migration with density variations in vertical transverse isotropic media

Anisotropic elastic least-squares reverse time migration with density variations in vertical transverse isotropic media

Convolutional Neural-Network-Based Reverse-Time Migration with Multiple Reflections.

Reverse-time migration (RTM) has the advantage that it can handle steep dipping structures and offer high-resolution images of the complex subsurface. Nevertheless, there are some limitations to the chosen initial model, aperture illumination and computation efficiency. RTM has a strong dependency on the initial velocity model. The RTM result image will perform poorly if the input background velocity model is inaccurate. One solution is to apply least-squares reverse-time migration (LSRTM), which updates the reflectivity and suppresses artifacts through iterations. However, the output resolution still depends heavily on the input and accuracy of the velocity model, even more than for standard RTM. For the aperture limitation, RTM with multiple reflections (RTMM) is instrumental in improving the illumination but will generate crosstalks because of the interference between different orders of multiples. We proposed a method based on a convolutional neural network (CNN) that behaves like a filter applying the inverse of the Hessian. This approach can learn patterns representing the relation between the reflectivity obtained through RTMM and the true reflectivity obtained from velocity models through a residual U-Net with an identity mapping. Once trained, this neural network can be used to enhance the quality of RTMM images. Numerical experiments show that RTMM-CNN can recover major structures and thin layers with higher resolution and improved accuracy compared with the RTM-CNN method. Additionally, the proposed method demonstrates a significant degree of generalizability across diverse geology models, encompassing complex thin layers, salt bodies, folds, and faults. Moreover, The computational efficiency of the method is demonstrated by its lower computational cost compared with LSRTM.

Phase-encoding-based least-squares reverse time migration of controlled-order multiples

Reverse time migration of controlled-order multiples (RTM-CM) can significantly remove strong crosstalk artifacts in multiples imaging. However, the computational cost is demanding if all pairs of consecutive-order multiples are considered, and an RTM-CM image retains residual crosstalk. To address these issues, we have developed a phase-encoding-based migration approach. This approach first phase-encodes different-order multiples by randomly modifying time shifts and polarity reversals, and then blends the encoded data into supergathers. As a result of using supergathers, migrations of different-order multiples are simultaneously accomplished, which considerably improves computational efficiency. As the simultaneous migration inherits the advantages of RTM-CM and provides good gradients for least-squares migration (LSM), we introduced an iterative LSM to gradually attenuate residual crosstalks in our approach. Numerical experiments on synthetic and field data sets revealed that our approach can significantly enhance imaging quality by suppressing crosstalks, increasing spatial resolutions, and reducing computational costs.