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

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. S539-S553 ◽  
Author(s):  
Jidong Yang ◽  
Hejun Zhu ◽  
George McMechan ◽  
Houzhu Zhang ◽  
Yang Zhao
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Transversely Isotropic ◽  
Image Resolution ◽  
Total Variation Regularization ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Isotropic Media ◽  
Band Limited

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.

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.

Viscoacoustic least-squares reverse time migration using a time-domain complex-valued wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. S479-S499 ◽  
Author(s):  
Jidong Yang ◽  
Hejun Zhu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Time Domain ◽  
Inverse Operator ◽  
Total Variation Regularization ◽  
Imaging Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Complex Valued

With limited recording apertures, finite-frequency source functions, and irregular subsurface illuminations, traditional imaging methods have been insufficient to produce satisfactory reflectivity images with high resolution and amplitude fidelity. This is because most traditional imaging approaches are commonly formulated as the adjoint instead of the inverse operator with respect to the forward-modeling operator. In addition, intrinsic attenuation introduces amplitude loss and phase dispersion during wave propagation. Without considering these effects, migrated images might be kinematically and dynamically incorrect. We have developed a viscoacoustic least-squares reverse time migration (LSRTM) method based on a time-domain complex-valued wave equation. According to the Born approximation, we first linearized the viscoacoustic wave equation and derived a demigration operator. Then, using the complex-valued Lagrange multiplier method, we derived the adjoint viscoacoustic wave equation and corresponding sensitivity kernel. With the forward and adjoint operators, a linear inverse problem is formulated to estimate the subsurface reflectivity model. A total-variation regularization scheme is introduced to enhance the robustness of our viscoacoustic LSRTM, and a diagonal Hessian is used as the preconditioner to accelerate the convergence. Three synthetic examples are used to demonstrate that our approach enables us to compensate attenuation effects, improve imaging resolution, and enhance amplitude fidelity in comparison with the adjoint imaging method.

Reverse time migration in tilted transversely isotropic media

2020 ◽  
Vol 38 (2) ◽  
Author(s):  
Razec Cezar Sampaio Pinto da Silva Torres ◽  
Leandro Di Bartolo
Keyword(s):  
Seismic Anisotropy ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Computer Clusters ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Tti Media

ABSTRACT. Reverse time migration (RTM) is one of the most powerful methods used to generate images of the subsurface. The RTM was proposed in the early 1980s, but only recently it has been routinely used in exploratory projects involving complex geology – Brazilian pre-salt, for example. Because the method uses the two-way wave equation, RTM is able to correctly image any kind of geological environment (simple or complex), including those with anisotropy. On the other hand, RTM is computationally expensive and requires the use of computer clusters. This paper proposes to investigate the influence of anisotropy on seismic imaging through the application of RTM for tilted transversely isotropic (TTI) media in pre-stack synthetic data. This work presents in detail how to implement RTM for TTI media, addressing the main issues and specific details, e.g., the computational resources required. A couple of simple models results are presented, including the application to a BP TTI 2007 benchmark model.Keywords: finite differences, wave numerical modeling, seismic anisotropy. Migração reversa no tempo em meios transversalmente isotrópicos inclinadosRESUMO. A migração reversa no tempo (RTM) é um dos mais poderosos métodos utilizados para gerar imagens da subsuperfície. A RTM foi proposta no início da década de 80, mas apenas recentemente tem sido rotineiramente utilizada em projetos exploratórios envolvendo geologia complexa, em especial no pré-sal brasileiro. Por ser um método que utiliza a equação completa da onda, qualquer configuração do meio geológico pode ser corretamente tratada, em especial na presença de anisotropia. Por outro lado, a RTM é dispendiosa computacionalmente e requer o uso de clusters de computadores por parte da indústria. Este artigo apresenta em detalhes uma implementação da RTM para meios transversalmente isotrópicos inclinados (TTI), abordando as principais dificuldades na sua implementação, além dos recursos computacionais exigidos. O algoritmo desenvolvido é aplicado a casos simples e a um benchmark padrão, conhecido como BP TTI 2007.Palavras-chave: diferenças finitas, modelagem numérica de ondas, anisotropia sísmica.

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

Geophysics ◽  
2021 ◽  
pp. 1-65
Author(s):  
Yingming Qu ◽  
Yixin Wang ◽  
Zhenchun Li ◽  
Chang Liu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Surface Topography ◽  
Density Perturbation ◽  
Second Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Time Migration ◽  
Grid Scheme

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. Besides, 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 a field data sets 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.

Separating quasi-P-wave in transversely isotropic media with a vertical symmetry axis by synthesized pressure applied to ocean-bottom seismic data elastic reverse time migration

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. C295-C307 ◽  
Author(s):  
Pengfei Yu ◽  
Jianhua Geng ◽  
Chenlong Wang
Keyword(s):  
Transversely Isotropic ◽  
P Wave ◽  
Ocean Bottom ◽  
Receiver Side ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Isotropic Media ◽  
Wave Imaging ◽  
Obs Data

Quasi-P (qP)-wavefield separation is a crucial step for elastic P-wave imaging in anisotropic media. It is, however, notoriously challenging to quickly and accurately obtain separated qP-wavefields. Based on the concepts of the trace of the stress tensor and the pressure fields defined in isotropic media, we have developed a new method to rapidly separate the qP-wave in a transversely isotropic medium with a vertical symmetry axis (VTI) by synthesized pressure from ocean-bottom seismic (OBS) data as a preprocessing step for elastic reverse time migration (ERTM). Another key aspect of OBS data elastic wave imaging is receiver-side 4C records back extrapolation. Recent studies have revealed that receiver-side tensorial extrapolation in isotropic media with ocean-bottom 4C records can sufficiently suppress nonphysical waves produced during receiver-side reverse time wavefield extrapolation. Similarly, the receiver-side 4C records tensorial extrapolation was extended to ERTM in VTI media in our studies. Combining a separated qP-wave by synthesizing pressure and receiver-side wavefield reverse time tensorial extrapolation with the crosscorrelation imaging condition, we have developed a robust, fast, flexible, and elastic imaging quality improved method in VTI media for OBS data.

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.

Preconditioned acoustic least-squares two-way wave-equation migration with exact adjoint operator

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. S1-S13 ◽  
Author(s):  
Linan Xu ◽  
Mauricio D. Sacchi
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Adjoint Operator ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Product Test ◽  
Dot Product ◽  
The Time Domain ◽  
Adjoint Operators

We have investigated the problem of designing the forward operator and its exact adjoint for two-way wave-equation least-squares migration. We study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm. We derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot-product test. We also found that the exact adjoint operator is not equal to the classic reverse time migration algorithm. For instance, one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot-product test. Forward and adjoint operators are adopted to solve the so-called least-squares reverse time migration problem via the method of conjugate gradients. We also examine a preconditioning strategy to invert extended images.

Q-compensated reverse time migration in tilted transversely isotropic media

Geophysics ◽  
2020 ◽  
pp. 1-79
Author(s):  
Ali Fathalian ◽  
Daniel O. Trad ◽  
Kristopher A. Innanen
Keyword(s):  
Time Domain ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Modeling And Analysis ◽  
Seismic Amplitude ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
The Time Domain

Anisotropy and absorption are critical to the modeling and analysis of seismic amplitude,phase, and traveltime data. To neglect any of these phenomena, which are often bothoperating simultaneously, degrades the resolution and interpretability of migrated images.However, a full accounting of anisotropy and anelasticity is computationally complex andexpensive. One strategy for accommodating these aspects of wave propagation, while keepingcost and complexity under control, is to do so within an acoustic approximation. Weset up a procedure for solving the time-domain viscoacoustic wave equation for tilted transverselyisotropic (TTI) media, based on a standard linear solid model and, from this, developa viscoacoustic reverse time migration (Q-RTM) algorithm. In this approach, amplitudecompensation occurs within the migration process through a manipulation of attenuationand phase dispersion terms in the time domain differential equations. Specifically, theback-propagation operator is constructed by reversing the sign only of the amplitude lossoperators, but not the dispersion-related operators, a step made possible by reformulatingthe absorptive TTI equations such that the loss and dispersion operators appear separately.The scheme is tested on synthetic examples to examine the capacity of viscoacoustic RTM to correct for attenuation, and the overall stability of the procedure.

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