Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C85-C94 ◽  
Author(s):  
Houzhu (James) Zhang ◽  
Hongwei Liu ◽  
Yang Zhao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Velocity ◽  
Wave Equations ◽  
Transverse Isotropy ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Computation Efficiency ◽  
The Stability ◽  
Vertical Transverse Isotropy

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

True-amplitude linearized waveform inversion with the quasi-elastic wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R827-R844 ◽  
Author(s):  
Zongcai Feng ◽  
Gerard Schuster
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Elastic Wave Equation ◽  
P Waves ◽  
Time Migration

We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the same as that for the elastic wave equation, and only requires the solution of two acoustic wave equations for each shot gather. This means that the quasi-elastic wave equation can be used for true-amplitude linearized waveform inversion (also known as least-squares reverse time migration) of elastic PP reflections, in which the corresponding misfit gradients are with respect to the perturbations of density and the P- and S-wave impedances. The perturbations of elastic parameters are iteratively updated by minimizing the [Formula: see text]-norm of the difference between the recorded PP reflections and the predicted pressure data modeled from the quasi-elastic wave equation. Numerical tests on synthetic and field data indicate that true-amplitude linearized waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude linearized waveform inversion provides images with a wider bandwidth and fewer artifacts because the PP amplitudes are accurately explained. We also determine the 2D scalar quasi-elastic wave equation for P-SV reflections and the 3D vector equation for PS reflections.

Elastic wave-equation migration velocity analysis preconditioned through mode decoupling

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R341-R353 ◽  
Author(s):  
Chenlong Wang ◽  
Jiubing Cheng ◽  
Wiktor Waldemar Weibull ◽  
Børge Arntsen
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Velocity ◽  
Seismic Imaging ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Velocity Models ◽  
Migration Velocity Analysis

Multicomponent seismic data acquisition can reveal more information about geologic structures and rock properties than single component acquisition. Full elastic wave seismic imaging, which uses multicomponent seismic to its full potential, is promising because it provides more opportunities to understand the material properties of the earth by the joint use of P- and S-waves. A prerequisite of seismic imaging is the availability of a reliable macrovelocity model. Migration velocity analysis for P-waves, which can fill that requirement for the P-wave velocity, has been well-studied, especially under the acoustic approximation. However, a reliable estimation of the S-wave velocities remains troublesome. Elastic wave-equation migration velocity analysis has the potential to build P- and S-wave velocity models together, but it inevitably suffers from the effects of mode coupling and conversion in the forward and adjoint wavefield reconstructions. We have developed a differential semblance optimization approach to sequentially invert the background P- and S-wave velocity models from extended PP- and PS-images in the subsurface offset domain. Preconditioning of the gradients with respect to the S-wave velocity through mode decoupling can improve the reliability of the optimization. Numerical investigations with synthetic examples demonstrate the effectiveness of gradient preconditioning and the feasibility of our migration velocity analysis approach for elastic wave imaging.

Nonlinear two‐dimensional elastic inversion of multioffset seismic data

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1211-1228 ◽  
Author(s):  
Peter Mora
Keyword(s):  
Wave Equation ◽  
Wave Velocity ◽  
A Priori ◽  
Gaussian Model ◽  
Wave Impedance ◽  
P Wave ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Gradient Direction ◽  
P Wave Velocity

The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude‐offset variations and shearwave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S‐wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P‐wave velocity, S‐wave velocity, and density as well as the P‐wave impedance, S‐wave impedance, and density. These are better resolved than the Lamé parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and the modeled data corresponding to these earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least‐squares algorithm is the gradient direction which tells how to update the model parameters. The gradient direction can be derived directly from the wave equation and it may be computed by several wave propagations. Although in principle any scheme could be chosen to perform the wave propagations, the elastic finite‐ difference method is used because it directly simulates the elastic wave equation and can handle complex, and thus realistic, distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot‐profile migration. However, it has greater power than any migration since it solves for the P‐wave velocity, S‐wave velocity, and density and can handle very general situations including transmission problems. Three main weaknesses of this technique are that it requires fairly accurate a priori knowledge of the low‐ wavenumber velocity model, it assumes Gaussian model statistics, and it is very computer‐intensive. All these problems seem surmountable. The low‐wavenumber information can be obtained either by a prior tomographic step, by the conventional normal‐moveout method, by a priori knowledge and empirical relationships, or by adding an additional inversion step for low wavenumbers to each iteration. The Gaussian statistics can be altered by preconditioning the gradient direction, perhaps to make the solution blocky in appearance like well logs, or by using large model variances in the inversion to reduce the effect of the Gaussian model constraints. Moreover, with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.

An average-derivative optimal scheme for modeling of the frequency-domain 3D elastic wave equation

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T209-T234 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Phase Velocity ◽  
Point Scheme ◽  
Grid Points

Because of its high computational cost, we needed to develop an efficient numerical scheme for the frequency-domain 3D elastic wave equation. In addition, the numerical scheme should be applicable to media with a liquid-solid interface. To address these two issues, we have developed a new average-derivative optimal 27-point scheme with arbitrary directional grid intervals and a corresponding numerical dispersion analysis for the frequency-domain 3D elastic wave equation. The novelty of this scheme is that its optimal coefficients depend on the ratio of the directional grid intervals and Poisson’s ratio. In this way, this scheme can be applied to media with a liquid-solid interface and a computational grid with arbitrary directional grid intervals. For media with a variable Poisson’s ratio, we have developed an effective and stable interpolation method for optimization coefficients. Compared with the classic 19-point scheme, this new scheme reduces the required number of grid points per wavelength for equal and unequal directional grid intervals. The reduction of the number of grid points increases as the Poisson’s ratio becomes larger. In particular, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classic 19-point scheme produces a spurious numerical S-wave phase velocity, as Poisson’s ratio reaches 0.5. We have performed numerical examples to develop the theoretical analysis.

Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. T339-T356 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Phase Velocity ◽  
Elastic Wave ◽  
Frequency Domain ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Classical Scheme ◽  
Wave Phase Velocity ◽  
Grid Points ◽  
Average Derivative

Based on an average-derivative method, we developed a new nine-point numerical scheme for a frequency-domain elastic-wave equation. Compared with the classic nine-point scheme, this scheme reduces the required number of grid points per wavelength for equal and unequal directional spacings. The reduction in the number of grid points increases as the Poisson’s ratio becomes larger. In particular, as the Poisson’s ratio reaches 0.5, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classical scheme produces spurious numerical S-wave phase velocity. Numerical examples demonstrate that this new scheme produces more accurate results than the classical scheme at approximately the same computational cost.

scholarly journals Prior Analysis in Determination of TI Anisotropy Type through Well-Based Modelling: A Case Study on the Deep Water Environment

2021 ◽  
Vol 873 (1) ◽  
pp. 012102
Author(s):  
Madaniya Oktariena ◽  
Wahyu Triyoso ◽  
Fatkhan Fatkhan ◽  
Sigit Sukmono ◽  
Erlangga Septama ◽  
...  
Keyword(s):  
Wave Velocity ◽  
Deep Water ◽  
Initial Conditions ◽  
Transverse Isotropy ◽  
Water Environment ◽  
Horizontal Stress ◽  
P Wave ◽  
S Wave ◽  
Geomechanical Analysis ◽  
Vertical Transverse Isotropy

Abstract The existence of anisotropy phenomena in the subsurface will affect the image quality of seismic data. Hence a prior knowledge of the type of anisotropy is quite essential, especially when dealing with deep water targets. The preliminary result of the anisotropy of the well-based modelling in deep water exploration and development is discussed in this study. Anisotropy types are modelled for Vertical Transverse Isotropy (VTI) and Horizontal Transverse Isotropy (HTI) based on Thomsen Parameters of ε and γ. The parameters are obtained from DSI Logging paired with reference δ value for modelling. Three initial conditions are then analysed. The first assumption is isotropic, in which the P-Wave Velocity, S-Wave Velocity, and Density Log modelled at their in-situ condition. The second and third assumptions are anisotropy models that are VTI and HTI. In terms of HTI, the result shows that the model of CDP Gather in the offset domain has a weak distortion in Amplitude Variation with Azimuth (AVAz). However, another finding shows a relatively strong hockey effect in far offset, which indicates that the target level is a VTI dominated type. It is supported by the geomechanical analysis result in which vertical stress acts as the maximum principal axis while horizontal stress is close to isotropic one. To sum up, this prior anisotropy knowledge obtained based on this study could guide the efficiency guidance in exploring the deep water environment.

Approximation of pure acoustic seismic wave propagation in TTI media

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB97-WB107 ◽  
Author(s):  
Chunlei Chu ◽  
Brian K. Macy ◽  
Phil D. Anno
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Computational Cost ◽  
P Wave ◽  
Anisotropy Axis ◽  
S Wave ◽  
Time Migration ◽  
Tti Media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

Elastic wave-equation datuming

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. U51-U61
Author(s):  
Xufei Gong ◽  
Qizhen Du ◽  
Qiang Zhao ◽  
Pengyuan Sun ◽  
Jianlei Zhang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Time Shift ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Data ◽  
Static Corrections ◽  
The One ◽  
Processing Techniques ◽  
Multicomponent Data

Wave-equation datuming (WED) techniques have demonstrated superiority when waves occur on the acquisition surface nonvertically, and traditional static corrections based on the time shift become inaccurate. Meanwhile, as for multicomponent data, those scalar techniques can hardly maintain the vector characteristics for the following multicomponent data processing flows. Considering this, we have developed an elastic-wave datuming approach to handle the static corrections for multicomponent data. Different from those existing scalar WED techniques, the multicomponent data are first decomposed into multicomponent P- and S-wave data. Then, the decomposed data are transformed into the [Formula: see text]-[Formula: see text] domain, and they are extrapolated from the acquisition surface to the datum using the one-way elastic-wave continuation. Finally, the datumed multicomponent data are reconstructed at the output datum by adding up the datumed P- and S-wave data. This elastic WED can guarantee that the same wave modes on different components are equally datumed, and the data remain multicomponent so that they are still applicable to multicomponent-joint processing techniques. Finally, several test examples involved in this paper have proved our method’s effectiveness in multicomponent data datuming application.

scholarly journals Pseudoelastic pure P-mode wave equation

Geophysics ◽  
2021 ◽  
Vol 86 (6) ◽  
pp. T469-T485
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Waveform Inversion ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Amplitude Variation With Offset ◽  
S Waves ◽  
Mode Wave ◽  
Wave Velocities ◽  
Simulation Engine

We have developed a pseudoelastic wave equation describing pure pressure waves propagating in elastic media. The pure pressure-mode (P-mode) wave equation uses all of the elastic parameters (such as density and the P- and S-wave velocities). It produces the same amplitude variation with offset (AVO) effects as PP-reflections computed by the conventional elastic wave equation. Because the new wave equation is free of S-waves, it does not require finer grids for simulation. This leads to a significant computational speedup when the ratio of pressure to S-wave velocities is large. We test the performance of our method on a simple synthetic model with high-velocity contrasts. The amplitude admitted by the pseudoelastic pure P-mode wave equation is highly consistent with that associated with the conventional elastic wave equation over a large range of incidence angles. We further verify our method’s robustness and accuracy using a more complex and realistic 2D salt model from the SEG Advanced Modeling Program. The ideal AVO behavior and computational advantage make our wave equation a good candidate as a forward simulation engine for performing elastic full-waveform inversion, especially for marine streamer data sets.

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