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Tariq Alkhalifah - One of the best experts on this subject based on the ideXlab platform.
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Multi‐parameter reflection waveform inversion for acoustic transversely Isotropic Media with a vertical symmetry axis
Geophysical Prospecting, 2020Co-Authors: Tariq AlkhalifahAbstract:ABSTRACT Full waveform inversion in transversely Isotropic Media with a vertical symmetry axis provides an opportunity to better match the data at the near and far offsets. However, multi‐parameter full waveform inversion, in general, suffers from serious cycle‐skipping and trade‐off problems. Reflection waveform inversion can help us recover a background model by projecting the residuals of the reflected wavefield along the reflection wavepath. Thus, we extend reflection waveform inversion to acoustic transversely Isotropic Media with a vertical symmetry axis utilizing the proper parameterization for reduced parameter trade‐off. From a radiation patterns analysis, an acoustic transversely Isotropic Media with a vertical symmetry axis is better described by a combination of the normal‐moveout velocity vn and the anIsotropic parameters η and δ for reflection waveform inversion applications. We design a three‐stage inversion strategy to construct the optimal resulting model. In the first stage, we only invert for the background vn by matching the simulated reflected wavefield from the perturbations of vn and δ with the observed reflected wavefield. In the second stage, the background vn and η are optimized simultaneously and the far‐offset reflected wavefield mainly contribute to their updates. We perform Born modelling to compute the reflected wavefield for the two stages of reflection waveform inversion. In the third stage, we perform full waveform inversion for the acoustic transversely Isotropic Media with a vertical symmetry axis to delineate the high‐wavenumber structures. For this stage, the medium is described by a combination of the horizontal velocity vh, η and e instead of vn, η and δ. The acoustic multi‐parameter full waveform inversion utilizes the diving waves to improve the background as well as utilizes reflection for high‐resolution information. Finally, we test our inversion algorithm on the modified Sigsbee 2A model (a salt free part) and a two‐dimensional line from a three‐dimensional ocean bottom cable dataset. The results demonstrate that the proposed reflection waveform inversion approach can recover the background model for acoustic transversely Isotropic Media with a vertical symmetry axis starting from an Isotropic model. This recovered background model can mitigate the cycle skipping of full waveform inversion and help the inversion recover higher resolution structures.
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An efficient Helmholtz solver for acoustic transversely Isotropic Media
GEOPHYSICS, 2018Co-Authors: Tariq AlkhalifahAbstract:ABSTRACTThe acoustic approximation, even for anIsotropic Media, is widely used in current industry imaging and inversion algorithms mainly because P-waves constitute most of the energy recorded in seismic exploration. The resulting acoustic formulas tend to be simpler, resulting in more efficient implementations, and they depend on fewer medium parameters. However, conventional solutions of the acoustic-wave equation with higher-order derivatives suffer from S-wave artifacts. Thus, we separate the quasi-P-wave propagation in anIsotropic Media into the elliptic anIsotropic operator (free of the artifacts) and the nonelliptic anIsotropic components, which form a pseudodifferential operator. We then develop a separable approximation of the dispersion relation of nonelliptic-anIsotropic components, specifically for transversely Isotropic Media. Finally, we iteratively solve the simpler lower-order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the ...
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Research Note: The sensitivity of surface seismic P-wave data in transversely Isotropic Media to reflector depth
Geophysical Prospecting, 2016Co-Authors: Tariq AlkhalifahAbstract:The leading component of the high-frequency asymptotic description of the wavefield, given by the travel time, is governed by the eikonal equation. In anIsotropic Media, traveltime measurements from seismic experiments conducted along one surface cannot constrain the long-wavelength attribute of the medium along the orthogonal-to-the-surface direction, as anisotropy introduces an independent parameter controlling wave propagation in the orthogonal direction. Since travel times measured on the Earth's surface in transversely Isotropic Media with a vertical symmetry axis are mainly insensitive to the absolute value of the anIsotropic parameter responsible for relating these observations to depth δ, the travel time was perturbed laterally to investigate the traveltime sensitivity to lateral variations in δ. This formulation can be used to develop inversion strategies for lateral variations in δ in acoustic transversely Isotropic Media, as the surface-recorded data are sensitive to it even if the model is described by the normal moveout velocity and horizontal velocity, or the anellipticity parameter η. Numerical tests demonstrate the enhanced sensitivity of our data when the model is parameterised with a lateral change in δ
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The offset-midpoint traveltime pyramid in 3D transversely Isotropic Media with a horizontal symmetry axis
GEOPHYSICS, 2015Co-Authors: Qi Hao, Alexey Stovas, Tariq AlkhalifahAbstract:Analytic representation of the offset-midpoint traveltime equation for anisotropy is very important for prestack Kirchhoff migration and velocity inversion in anIsotropic Media. For transversely Isotropic Media with a vertical symmetry axis, the offset-midpoint traveltime resembles the shape of a Cheops’ pyramid. This is also valid for homogeneous 3D transversely Isotropic Media with a horizontal symmetry axis (HTI). We extended the offset-midpoint traveltime pyramid to the case of homogeneous 3D HTI. Under the assumption of weak anellipticity of HTI Media, we derived an analytic representation of the P-wave traveltime equation and used Shanks transformation to improve the accuracy of horizontal and vertical slownesses. The traveltime pyramid was derived in the depth and time domains. Numerical examples confirmed the accuracy of the proposed approximation for the traveltime function in 3D HTI Media.
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Mapping of moveout in tilted transversely Isotropic Media
Geophysical Prospecting, 2013Co-Authors: Alexey Stovas, Tariq AlkhalifahAbstract:The computation of traveltimes in a transverse Isotropic medium with a tilted symmetry axis tilted transversely Isotropic is very important both for modelling and inversion. We develop a simple analytical procedure to map the traveltime function from a transverse Isotropic medium with a vertical symmetry axis (vertical transversely Isotropic) to a tilted transversely Isotropic medium by applying point-by-point mapping of the traveltime function. This approach can be used for kinematic modelling and inversion in layered tilted transversely Isotropic Media. © 2013 European Association of Geoscientists & Engineers
Maarten V. De Hoop - One of the best experts on this subject based on the ideXlab platform.
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Recovery of material parameters in transversely Isotropic Media
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Maarten V. De Hoop, Gunther Uhlmann, András VasyAbstract:In this paper we show that in anIsotropic elasticity, in the particular case of transversely Isotropic Media, under appropriate convexity conditions, knowledge of the qSH wave travel times determines the tilt of the axis of isotropy as well as some of the elastic material parameters, and the knowledge of qP and qSV travel times conditionally determines a subset of the remaining parameters, in the sense that if some of the remaining parameters are known, the rest are determined, or if the remaining parameters satisfy a suitable relation, they are all determined, under certain non-degeneracy conditions. Furthermore, we give a precise description of the additional issues, which are a subject of ongoing work, that need to be resolved for a full treatment.
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scalar generalized screen algorithms in transversely Isotropic Media with a vertical symmetry axis
Geophysics, 2001Co-Authors: Jerome Le Rousseau, Maarten V. De HoopAbstract:The scalar generalized‐screen method in Isotropic Media is extended here to transversely Isotropic Media with a vertical symmetry axis (VTI). Although wave propagation in a transversely Isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI Media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the Isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a partic...
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Scalar generalized‐screen algorithms in transversely Isotropic Media with a vertical symmetry axis
GEOPHYSICS, 2001Co-Authors: Jérôme Le Rousseau, Maarten V. De HoopAbstract:The scalar generalized‐screen method in Isotropic Media is extended here to transversely Isotropic Media with a vertical symmetry axis (VTI). Although wave propagation in a transversely Isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI Media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the Isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a partic...
Alexey Stovas - One of the best experts on this subject based on the ideXlab platform.
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S-wave in 2D acoustic transversely Isotropic Media with a tilted symmetry axis
Geophysical Prospecting, 2019Co-Authors: Song Jin, Alexey StovasAbstract:In an acoustic transversely Isotropic medium, there are two waves that propagate. One is the P-wave and another one is the S-wave (also known as S-wave artefact). This paper is devoted to analyse the S-wave in two-dimensional acoustic transversely Isotropic Media with a tilted symmetry axis. We derive the S-wave slowness surface and traveltime function in a homogeneous acoustic transversely Isotropic medium with a tilted symmetry axis. The S-wave traveltime approximations in acoustic transversely Isotropic Media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely Isotropic Media with a vertical symmetry axis. We consider a layered two-dimensional acoustic transversely Isotropic medium with a tilted symmetry axis to analyse the S-wave moveout. We also illustrate the behaviour of the moveout for reflected S-wave and converted waves.
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The offset-midpoint traveltime pyramid in 3D transversely Isotropic Media with a horizontal symmetry axis
GEOPHYSICS, 2015Co-Authors: Qi Hao, Alexey Stovas, Tariq AlkhalifahAbstract:Analytic representation of the offset-midpoint traveltime equation for anisotropy is very important for prestack Kirchhoff migration and velocity inversion in anIsotropic Media. For transversely Isotropic Media with a vertical symmetry axis, the offset-midpoint traveltime resembles the shape of a Cheops’ pyramid. This is also valid for homogeneous 3D transversely Isotropic Media with a horizontal symmetry axis (HTI). We extended the offset-midpoint traveltime pyramid to the case of homogeneous 3D HTI. Under the assumption of weak anellipticity of HTI Media, we derived an analytic representation of the P-wave traveltime equation and used Shanks transformation to improve the accuracy of horizontal and vertical slownesses. The traveltime pyramid was derived in the depth and time domains. Numerical examples confirmed the accuracy of the proposed approximation for the traveltime function in 3D HTI Media.
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The offset-midpoint travel-time pyramid for P-wave in 2D transversely Isotropic Media with a tilted symmetry axis
Geophysical Prospecting, 2014Co-Authors: Qi Hao, Alexey StovasAbstract:For pre-stack phase-shift migration in homogeneous Isotropic Media, the offsetmidpoint travel time is represented by the double-square-root equation. The travel time as a function of offset and midpoint resembles the shape of Cheops’ pyramid. This is also valid for transversely Isotropic Media with a vertical symmetry axis. In this study, we extend the offset-midpoint travel-time pyramid to the case of 2D transversely Isotropic Media with a tilted symmetry axis. The P-wave analytical travel-time pyramid is derived under the assumption of weak anelliptical property of the tilted transverse isotropy Media. The travel-time equation for the dip-constrained transversely Isotropic model is obtained from the depth-domain travel-time pyramid. The potential applications of the derived offset-midpoint travel-time equation include pre-stack Kirchhoff migration, anIsotropic parameter estimation, and travel-time calculation in transversely Isotropic Media with a tilted symmetry axis.
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Mapping of moveout in tilted transversely Isotropic Media
Geophysical Prospecting, 2013Co-Authors: Alexey Stovas, Tariq AlkhalifahAbstract:The computation of traveltimes in a transverse Isotropic medium with a tilted symmetry axis tilted transversely Isotropic is very important both for modelling and inversion. We develop a simple analytical procedure to map the traveltime function from a transverse Isotropic medium with a vertical symmetry axis (vertical transversely Isotropic) to a tilted transversely Isotropic medium by applying point-by-point mapping of the traveltime function. This approach can be used for kinematic modelling and inversion in layered tilted transversely Isotropic Media. © 2013 European Association of Geoscientists & Engineers
Ilya Tsvankin - One of the best experts on this subject based on the ideXlab platform.
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Image‐domain wavefield tomography for tilted transversely Isotropic Media
Geophysical Prospecting, 2019Co-Authors: Ilya Tsvankin, Antoine Guitton, Hui WangAbstract:ABSTRACT Transversely Isotropic models with a tilted symmetry axis have become standard for imaging beneath dipping shale formations and in active tectonic areas. Here, we develop a methodology of wave‐equation‐based image‐domain tomography for acoustic tilted transversely Isotropic Media. We obtain the gradients of the objective function using an integral wave‐equation operator based on a separable dispersion relation that takes the symmetry‐axis tilt into account. In contrast to the more conventional differential solutions, the integral operator produces only the P‐wavefield without shear‐wave artefacts, which facilitates both imaging and velocity analysis. The model is parameterized by the P‐wave zero‐dip normal‐moveout velocity, the Thomsen parameter δ, anellipticity coefficient η and the symmetry‐axis tilt θ. Assuming that the symmetry axis is orthogonal to reflectors, we study the influence of parameter errors on energy focusing in extended (space‐lag) common‐image gathers. Distortions in the anellipticity coefficient η introduce weak linear defocusing regardless of reflector dip, whereas δ influences both the energy focusing and depth scale of the migrated section. These results, which are consistent with the properties of the P‐wave time‐domain reflection moveout in tilted transversely Isotropic Media, provide important insights for implementation of velocity model‐building in the image‐domain. Then the algorithm is tested on a modified anticline section of the BP 2007 benchmark model.
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migration velocity analysis for tilted transversely Isotropic Media
Geophysical Prospecting, 2009Co-Authors: Laxmidhar Behera, Ilya TsvankinAbstract:Tilted transversely Isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold-and-thrust belts) and in subsalt exploration. Here, we intro- duce a methodology for P-wave prestack depth imaging in tilted transversely Isotropic Media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parametersand δ and linearly varying symmetry- direction velocity VP0 controlled by the vertical (kz )a nd lateral (kx) gradients. Since determination of tilt from P-wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity VP0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients kz and kx and the anisotropy parametersand δ in the layer-stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry- axis direction is fixed and VP0 is known, the parameters kz, kx, � and δ can be resolved from reflection data. It should be emphasized that estimation ofin tilted transversely Isotropic Media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely Isotropic Media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30 ◦ ). The ability of our
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Fowler DMO and time migration for transversely Isotropic Media
GEOPHYSICS, 1996Co-Authors: John E. Anderson, Tariq Alkhalifah, Ilya TsvankinAbstract:The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anIsotropic Media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely Isotropic Media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely Isotropic Media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity Vnmo(0) and the anIsotropic parameter η (η reduces to the difference between Thomsen parameters e and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the val...
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Moveout Analysis for Transversely Isotropic Media with a Tilted Symmetry Axis
57th EAEG Meeting, 1995Co-Authors: Ilya TsvankinAbstract:Most existing work on reflection moveouts in anIsotropic Media is restricted to a relatively simple model - laterally homogeneous transversely Isotropic Media with a vertical symmetry axis (VTi). Recently, it has been recognized that anisotropy may distort not only moveout from horizontal reflectors, but also the dip dependence of normal-moveout (NMO) velocity and, therefore, the resuIts of dip-moveout (DMO) processing.
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Fowler DMO and time migration for transversely Isotropic Media with explicit operators
1994Co-Authors: J. Anderson, T. Alkhalifah, Ilya TsvankinAbstract:In this report, the authors devise a Fowler-type DMO algorithm for transversely Isotropic Media using the analytic expression for normal-moveout velocity given by Tsvankin (1995a). Alkhalifah and Tsvankin (1995) have shown that in transversely Isotropic Media with a vertical axis of symmetry (VTI) the P-wave normal-moveout (NMO) velocity as a function of ray parameter can be fully described by just two parameters: the zero-dip NMO velocity V{sub nmo}(O) and the anIsotropic parameter {eta}. In the authors extension of Fowler DMO, resampling in the frequency-wavenumber domain makes it possible to obtain the values of V{sub nmo}(O) and {eta} by inspecting zero-offset (stacked) panels for different pairs of the two parameters. The simplest way to reduce the range of solutions in this two-dimensional search is to obtain V{sub nmo}(O) from conventional NMO velocity analysis. Since most of the computing time is spent on generating constant-velocity stacks, the added computational effort due to the presence of anisotropy is relatively minor. Synthetic and field-data examples demonstrate that the Isotropic Fowler DMO technique fails to generate an accurate zero-offset section and obtain the zero-dip NMO velocity for non-elliptical VTI models. In contrast, their anIsotropic algorithm allows one to find the values of the parameters V{sub more » nmo}(O) and {eta}, and correct for the influence of transverse isotropy in the DMO processing. Combined with poststack F-K Stolt migration, this method represents a complete inversion-processing sequence capable of recovering the effective parameters of transversely Isotropic Media and producing migrated images for the best-fit homogeneous anIsotropic model. Although the current implementation is limited to transversely Isotropic Media with a vertical axis of symmetry (VTI), it can be generalized for more complicated anIsotropic models. « less
Yimin Zhong - One of the best experts on this subject based on the ideXlab platform.
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a fast algorithm for radiative transport in Isotropic Media
Journal of Computational Physics, 2019Co-Authors: Kui Ren, Rongting Zhang, Yimin ZhongAbstract:Abstract Constructing efficient numerical solution methods for the equation of radiative transfer (ERT) remains as a challenging task in scientific computing despite of the tremendous development on the subject in recent years. We present in this work a simple fast computational algorithm for solving the ERT in Isotropic Media. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly-averaged ERT solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the ERT solution. The algorithm does not require the underlying medium be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous Media.
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A fast algorithm for radiative transport in Isotropic Media
arXiv: Numerical Analysis, 2016Co-Authors: Kui Ren, Rongting Zhang, Yimin ZhongAbstract:We propose in this work a fast numerical algorithm for solving the equation of radiative transfer (ERT) in Isotropic Media. The algorithm has two steps. In the first step, we derive an integral equation for the angularly averaged ERT solution by taking advantage of the isotropy of the scattering kernel, and solve the integral equation with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the original ERT solution. Numerical simulations are presented to demonstrate the performance of the algorithm for both homogeneous and inhomogeneous Media.
