Least-squares Reverse Time Migration Method 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.

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.

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.

Mesh-free least-squares reverse-time migration

AbstractThe reflectivity of the subsurface can be precisely determined using least-squares reverse-time migration (LSRTM). As LSRTM necessitates solving the wave equation, the numerical solution method of the wavefield directly determines the quality of the migration image. The conventional LSRTM method usually uses the finite difference method based on a regular grid to calculate the wavefield. Due to the stepwise approximation of an irregular surface with a regular grid, scattering noise may occur in the propagation of the wavefield, which affects the quality of the image. In addition, the conventional LSRTM cannot effectively handle the models with rugged topography. The finite difference method generated by radial basis functions (FD-RBF) is a mesh-free method and can construct interpolation functions to solve the wave equations numerically according to arbitrarily distributed spatial coordinate points. Therefore, we use the FD-RBF method to develop a mesh-free LSRTM approach to eliminate the influence of the inherent limitation of a regular grid on the imaging. Numerical examples show that the mesh-free LSRTM method can better represent the curved or steep interface within the model and is also suitable for models with rugged topography. The LSRTM method can provide higher-quality images and effectively reduce the memory required for calculations.

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.

Quasielastic least-squares reverse time migration of PS reflections

Conventional analysis of amplitude variation with offset for elastic PS reflections is based on analytical reflection coefficients in a layered medium, and wave-equation-based PS migration is mainly used to produce a structural image. To overcome this problem, we have developed a least-squares reverse time migration (LSRTM) method for elastic PS reflections based on a quasielastic wave equation. The quasielastic wave equation can accurately model PS reflections with elastic amplitudes under the first-order Born approximation. Our LSRTM method inverts for perturbations of the S-wave velocity and density by minimizing the [Formula: see text] norm of the difference between recorded and predicted PS reflections modeled using a quasielastic wave equation. We refer to our new method as quasielastic LSRTM of PS reflections. Numerical tests on synthetic and field data indicate that our method can properly handle the amplitudes of elastic PS reflections and provides an accurate estimate of the perturbations of S-wave velocity and density. Extending the method to the 3D case is not straightforward and might require incorporating certain PS data processing techniques into the inversion itself.

Least-Squares Reverse Time Migration for Reflection-Angle-Dependent Reflectivity

Least-squares reverse time migration (LSRTM) can estimate high-quality reflectivity of subsurface medium from seismic data. However, the subsurface reflectivity depends on reflection angles, and its variations over reflection angles are extremely important because they can be used to estimate sub-surface physical properties for seismic interpretation. We present a new formulation of the LSRTM method that can estimate reflection-angle-dependent reflectivity from seismic data. We derive a forward modeling operator which predicts the reflection data without calculating the reflection angles, and verify that it approximately equals to the reflection-angle-dependent wave-equation-based Kirchhoff modeling operator under the assumption that the velocity perturbation is small and the reflection angle is smaller than the critical angle. Based on the proposed modeling operator associated with the adjoint of the angle-dependent wave-equation-based Kirchhoff modeling operator, we reformulate LSRTM as an inverse problem to invert for reflection-angle-dependent reflectivity using a preconditioned conjugate gradient algorithm. The algorithm uses a low-rank filter as the preconditioner to attenuate migration artifacts. Imaging tests on synthetic and field seismic data are used to verify validity and superiority of the proposed method. The tests illustrate that the proposed method can produce the reflection-angle-dependent reflectivity with much higher signal-to-noise ratio, resolution and amplitude fidelity than reverse time migration. Compared with conventional LSRTM, it can produce more focused stacked image when the migration velocity contains errors. Moreover, conventional LSRTM only produces the angle-independent reflectivity, whereas the proposed method has the feasibility to produce the reflection-angle-dependent reflectivity.

3-D Least-Squares Reverse Time Migration in Curvilinear-τ Domain

Curvilinear-grid-based least-squares reverse time migration (LSRTM) can produce an accurate image of complex subsurface structures. However, a huge amount of computational cost of LSRTM makes it difficult in real data applications, especially in 3-D cases. We propose a wavefield continuation operator in a new curvilinear- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> domain to make the sampling space in the vertical direction to be uniform by stretching and compressing the low- and high-velocity zones, respectively. An objective function based on a conical wave encoding and student’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> distribution is constructed to improve the computational efficiency and robustness of LSRTM. The gradient formula is derived based on the objective function, and to correct the unbiased random estimation error, the idea of random optimization is introduced to obtain a weighted gradient. Demigration and adjoint wave equations in the curvilinear- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> domain are derived to calculate the synthetic records and the backward-propagated wavefields. Numerical examples on synthetic and field datasets suggest that the proposed 3-D LSRTM method produces better images with a higher signal-to-noise ratio (SNR), more improved resolution, and more balanced amplitude than the conventional 3-D LSRTM and greatly improves the computational efficiency of 3-D LSRTM.

Reflection angle/azimuth-dependent least-squares reverse time migration

Seismic images under complex overburdens such as salt are strongly affected by illumination variations due to overburden velocity variations and imperfect acquisition geometries, making it difficult to obtain reliable image amplitudes. Least-squares reverse time migration (LSRTM) addresses these issues by formulating full wave-equation imaging as a linear inverse problem and solving for a reflectivity model that explains the recorded seismic data. Because subsurface reflection coefficients depend on the incident angle, and possibly on the azimuth, quantitative interpretation under complex overburdens requires LSRTM with output in terms of image gathers, e.g., as a function of the reflection angle or angle and azimuth. We have developed a reflection angle- or angle/azimuth-dependent LSRTM method aimed at obtaining physically meaningful image amplitudes interpretable in terms of angle- or angle/azimuth-dependent reflection coefficients. The method is formulated as a linear inverse problem solved iteratively with the conjugate gradient method. It requires an adjoint pair of linear operators for reflection angle/azimuth-dependent migration and demigration based on full wave-equation propagation. We implement these operators in an efficient way by using a mapping approach between migrated shot gathers and subsurface reflection angle/azimuth gathers. To accelerate convergence of the iterative inversion, we apply image-domain preconditioning operators computed from a single de-remigration step. An angle continuity constraint and a structural dip constraint, implemented via shaping regularization, are used to stabilize the solution in the presence of limited illumination and to control the effects of coherent noise. We examine the method on a synthetic data example and on a wide-azimuth streamer data set from the Gulf of Mexico, where we find that angle/azimuth-dependent LSRTM can achieve significant uplift in subsalt image quality, with overburden- and acquisition-related illumination variation effects on angle/azimuth-dependent image amplitudes largely removed.

Q least-squares reverse time migration based on the first-order viscoacoustic quasidifferential equations

Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. In addition, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and one field data set indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.

Shear reflectivity compensation in full-waveform inversion using least-squares reverse-time migration

SUMMARY The computational cost of elastic-waveform inversion is too high for inverting PP reflections, while using acoustic full-waveform inversion (FWI) is inaccurate because it does not depend on the shear modulus/velocity/impedance that affects elastic PP wavefield amplitudes. To solve this problem, we develop a waveform inversion method that uses acoustic least-squares reverse-time migration (LSRTM) to compensate the shear reflectivity for acoustic FWI. Our method is based on the quasi-elastic-wave equation developed by Chapman et al. (2014). The quasi-elastic-wave equation uses a linearized acoustic-wave equation with shear modulus μ as a virtual source to correct the acoustic PP wavefield amplitudes toward elastic ones. Our waveform inversion method inverts for elastic parameters by minimizing the L2 norm of the difference between recorded and predicted PP reflections modelled using the quasi-elastic-wave equation. Numerical tests on synthetic and field data show that our method can properly handle the amplitudes of elastic PP reflections and provides an accurate estimate of the P- and S-wave velocities/impedances and, in some cases, the density. The method does not need the computationally expensive numerical solution to the elastic-wave equation. It also gives a better estimate of elastic parameters than a pure LSRTM method for elastic PP reflections.

Least-squares reverse time migration method using the factorization of the Hessian matrix

Least-squares reverse time migration method using the factorization of the Hessian matrix

Least-squares reverse time migration with sparse regularization in the 2D wavelet domain

The inadequate sampling of seismic data in the spatial dimension results in migration artifacts. Conventional least-squares reverse time migration (LSRTM) could improve the image quality. However, even LSRTM will not work in some inadequately sampling situations. To mitigate the impact of migration artifacts, we have developed a new LSRTM method with a sparse regularization, which takes advantage of the effective sparse representation of the subsurface reflectivity model in the 2D undecimated wavelet transform (UWT) domain. Different from other sparse regularizations, a sparseness constraint in the 2D UWT domain is applied on the angle slices of the image. To efficiently solve the least-squares inversion problem, we employ an inversion scheme using the conjugate gradient method that uses a soft threshold method to achieve sparse constraint in updating the conjugate gradient direction. Compared with the sparse constraint based on the discrete wavelet transform (DWT), the threshold in this method is angle-dependent and is determined according to the energy distribution of each angle slice. To avoid overregularization that can lead to instability and increase the number of iterations, we also apply an exponential threshold strategy. Numerical tests on synthetic datasets demonstrate that our method is capable of improving the image quality by enhancing the resolution and suppressing migration artifacts caused by inadequately sampled seismic data. The method can converge more rapidly than conventional LSRTM. Because this method performs sparse regularization on several slopes, it achieves better performance on enhancing complex structures with discontinuities such as the steeply dipping faults compared to DWT-based regularization.

Periodic plane-wave least-squares reverse time migration for diffractions

Diffraction imaging is important for high-resolution characterization of small subsurface heterogeneities. However, due to geometry limitations and noise distortion, conventional diffraction imaging methods may produce low-quality images. We have adopted a periodic plane-wave least-squares reverse time migration method for diffractions to improve the image quality of heterogeneities. The method reformulates diffraction imaging as an inverse problem using the Born modeling operator and its adjoint operator derived in the periodic plane-wave domain. The inverse problem is implemented for diffractions separated by a plane-wave destruction filter from the periodic plane-wave sections. Because the plane-wave destruction filter may fail to eliminate hyperbolic reflections and noise, we adopt a hyperbolic misfit function to minimize a weighted residual using an iteratively reweighted least-squares algorithm and thereby reduce residual reflections and noise. Synthetic and field data tests show that the adopted method can significantly improve the image quality of subsalt and deep heterogeneities. Compared with reverse time migration, it produces better images with fewer artifacts, higher resolution, and more balanced amplitude. Therefore, the adopted method can accurately characterize small heterogeneities and provide a reliable input for seismic interpretation in the prediction of hydrocarbon reservoirs.

Elastic least-squares reverse time migration in vertical transverse isotropic media

Using adjoint-based elastic reverse time migration, it is difficult to produce high-quality reflectivity images due to the limited acquisition apertures, band-limited source time function, and irregular subsurface illumination. Through iteratively computing the Hessian inverse, least-squares migration enables us to reduce the point-spread-function effects and improve the image resolution and amplitude fidelity. By incorporating anisotropy in the 2D elastic wave equation, we have developed an elastic least-squares reverse time migration (LSRTM) method for multicomponent data from the vertically transversely isotropic (VTI) media. Using the perturbed stiffness parameters [Formula: see text] and [Formula: see text] as PP and PS reflectivities, we linearize the elastic VTI wave equation and obtain a Born modeling (demigration) operator. Then, we use the Lagrange multiplier method to derive the corresponding adjoint wave equation and reflectivity kernels. With linearized forward modeling and adjoint migration operators, we solve a linear inverse problem to estimate the subsurface reflectivity models for [Formula: see text] and [Formula: see text]. To reduce the artifacts caused by data over-fitting, we introduce total-variation regularization into the reflectivity inversion, which promotes a sparse solution in terms of the model derivatives. To accelerate the convergence of LSRTM, we use source illumination to approximate the diagonal Hessian and use it as a preconditioner for the misfit gradient. Numerical examples help us determine that our elastic VTI LSRTM method can improve the spatial resolution and amplitude fidelity in comparison to adjoint migration.

Multiparameter least-squares reverse time migration for acoustic—elastic coupling media based on ocean bottom cable data

In marine seismic exploration, ocean bottom cable technology can record multicomponent seismic data for multiparameter inversion and imaging. This study proposes an elastic multiparameter lease-squares reverse time migration based on the ocean bottom cable technology. Herein, the wavefield continuation operators are mixed equations: the acoustic wave equations are used to calculate seismic wave propagation in the seawater medium, whereas in the solid media below the seabed, the wavefields are obtained by P- and S-wave separated vector elastic wave equations. At the seabed interface, acoustic-elastic coupling control equations are used to combine the two types of equations. P- and S-wave separated elastic migration operators, demigration operators, and gradient equations are derived to realize the elastic least-squares reverse time migration based on the P- and S-wave mode separation. The model tests verify that the proposed method can obtain high-quality images in both the P- and S-velocity components. In comparison with the traditional elastic least-squares reverse time migration method, the proposed method can readily suppress imaging crosstalk noise from multiparameter coupling.

Fast image-domain target-oriented least-squares reverse time migration

We have developed a fast image-domain target-oriented least-squares reverse time migration (LSRTM) method based on applying the inverse or pseudoinverse of a target-oriented Hessian matrix to a migrated image. The image and the target-oriented Hessian matrix are constructed using plane-wave Green’s functions that are computed by solving the two-way wave equation. Because the number of required plane-wave Green’s functions is small, the proposed method is highly efficient. We exploit the sparsity of the Hessian matrix by computing only a couple of off-diagonal terms for the target-oriented Hessian, which further improves the computational efficiency. We examined the proposed LSRTM method using the 2D Marmousi model. We demonstrated that our method correctly recovers the reflectivity model, and the retrieved results have more balanced illumination and higher spatial resolution than traditional images. Because of the low cost of computing the target-oriented Hessian matrix, the proposed method has the potential to be applied to large-scale problems.

One-step data-domain least-squares reverse time migration

Least-squares reverse time migration (LSRTM) is an iterative inversion algorithm for estimating the broadband-wavenumber reflectivity model. Although it produces superior results compared with conventional reverse time migration (RTM), LSRTM is computationally expensive. We have developed a one-step LSRTM method by considering the demigrated and observed data to design a deblurring preconditioner in the data domain using the Wiener filter. For the Wiener filtering, we further use a stabilized division algorithm via the Taylor expansion. The preconditioned observed data are then remigrated to obtain a deblurred image. The total cost of this method is about two RTMs. Through synthetic and real data experiments, we see that one-step LSRTM is able to enhance image resolution and balance source illumination at low computational costs.

Fast least-squares reverse time migration of VSP free-surface multiples with dynamic phase-encoding schemes

For vertical seismic profile (VSP), free-surface multiples can provide much wider subsurface illumination when compared with primaries. However, migration of multiple reflections generates not only the desired image of reflection interfaces but also many crosstalk artifacts. Therefore, the least-squares reverse time migration method is used to image the VSP downgoing free-surface multiples (receiver-side ghosts) and iteratively suppress crosstalks, in which full downgoing data (including direct waves) and downgoing multiples are used as sources and observed data, respectively. To reduce the computational cost, we have developed the simultaneous imaging of different common-receiver gathers that are dynamically blended together with iterations through the altered realizations of the phase-encoding function. Relative to the popular encoding function with a combination of random time delays and polarities, only the random polarities can be applied for further increasing the computational efficiency. Synthetic experiments on Sigsbee2B and Pluto1.5 models indicate that the proposed method can effectively eliminate crosstalk artifacts and improve imaging resolution while calculated even more efficiently than reverse time migration of VSP ghosts.

Elastic least-squares reverse time migration using the energy norm

Incorporating anisotropy and elasticity into least-squares migration is an important step toward recovering accurate amplitudes in seismic imaging. An efficient way to extract reflectivity information from anisotropic elastic wavefields exploits properties of the energy norm. We derive linearized modeling and migration operators based on the energy norm to perform anisotropic least-squares reverse time migration (LSRTM) describing subsurface reflectivity and correctly predicting observed data without costly decomposition of wave modes. Imaging operators based on the energy norm have no polarity reversal at normal incidence and remove backscattering artifacts caused by sharp interfaces in the earth model, thus accelerating convergence and generating images of higher quality when compared with images produced by conventional methods. With synthetic and field data experiments, we find that our elastic LSRTM method generates high-quality images that predict the data for arbitrary anisotropy, without the complexity of wave-mode decomposition and with a high convergence rate.