PURE MODE P AND S WAVE PHASE VELOCITY EQUATIONS IN ELASTIC ORTHORHOMBIC MEDIA

Geophysics ◽  
2021 ◽  
pp. 1-82
Author(s):  
Alexey Stovas ◽  
Yuriy Roganov ◽  
Vyacheslav Roganov
Keyword(s):  
Wave Equations ◽  
Approximate Equation ◽  
Approximate Solutions ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Elastic Model ◽  
Pure Mode ◽  
S Wave ◽  
Practical Applications ◽  
Independent Parameters

In an elastic model with orthorhombic symmetry, there are nine independent stiffness coefficients that control the propagation of all intrinsically coupled wave modes. For practical applications in P-wave modeling and inversion, it is important to derive the approximate solutions that support propagation of P waves only and depends on fewer independent parameters. Due to the increasing interest in shear-wave propagation in anisotropic media, we also derive an approximate equation that supports propagation of S waves only. However, the reduction in number of independent parameters for the S wave equation is not possible. We derive pure P and S wave equations in an elastic orthorhombic model and show that the accuracy is sufficient for practical applications.

Wave characteristics in elliptical orthorhombic media

Geophysics ◽  
2021 ◽  
pp. 1-52
Author(s):  
Alexey Stovas ◽  
Yuriy Roganov ◽  
Vyacheslav Roganov
Keyword(s):  
Anisotropic Medium ◽  
Wave Equations ◽  
Reference Model ◽  
Relative Magnitude ◽  
P Wave ◽  
S Wave ◽  
Wave Characteristics ◽  
Singularity Point ◽  
Wave Properties ◽  
Wave Modes

An elliptical anisotropic medium is defined as a simplified representation of anisotropy in which the anelliptic parameters are set to zero in all symmetry planes. Despite of the fact that this model is rather seldom observed for real rocks, it is often used as a reference model. The P-wave equations for an elliptical anisotropic medium is well known. However, the S-wave equations have not been derived. Thus, we define all wave modes in elliptical orthorhombic models focusing mostly on the S-wave properties. We show that all wave modes in elliptical orthorhombic model are generally coupled and analyze the effect of additive coupling term. As the result, there is an S wave fundamental singularity point located in one of the symmetry planes depending on the relative magnitude of S wave stiffness coefficients.

Decoupling approximation of P- and S-wave phase velocities in orthorhombic media

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Bowen Li ◽  
Alexey Stovas
Keyword(s):  
Phase Velocity ◽  
P Wave ◽  
Wave System ◽  
S Wave ◽  
Phase Velocities ◽  
Wave Phase ◽  
Algebraic Expressions ◽  
Wave Phase Velocity ◽  
Independent Parameters ◽  
Velocity Approximation

Characterizing the kinematics of seismic waves in elastic orthorhombic media involves nine independent parameters. All wave modes, P-, S1-, and S2-waves, are intrinsically coupled. Since the P-wave propagation in orthorhombic media is weakly dependent on the three S-wave velocity parameters, they are set to zero under the acoustic assumption. The number of parameters required for the corresponding acoustic wave equation is thus reduced from nine to six, which is very practical for the inversion algorithm. However, the acoustic wavefields generated by the finite-difference scheme suffer from two types of S-wave artifacts, which may result in noticeable numerical dispersion and even instability issues. Avoiding such artifacts requires a class of spectral methods based on the low-rank decomposition. To implement a six-parameter pure P-wave approximation in orthorhombic media, we develop a novel phase velocity approximation approach from the perspective of decoupling P- and S-waves. In the exact P-wave phase velocity expression, we find that the two algebraic expressions related to the S1- and S2-wave phase velocities play a negligible role. After replacing these two algebraic expressions with the designed constant and variable respectively, the exact P-wave phase velocity expression is greatly simplified and naturally decoupled from the characteristic equation. Similarly, the number of required parameters is reduced from nine to six. We also derive an approximate S-wave phase velocity equation, which supports the coupled S1- and S2-waves and involves nine independent parameters. Error analyses based on several orthorhombic models confirm the reasonable and stable accuracy performance of the proposed phase velocity approximation. We further derive the approximate dispersion relations for the P-wave and the S-wave system in orthorhombic media. Numerical experiments demonstrate that the corresponding P- and S-wavefields are free of artifacts and exhibit good accuracy and stability.

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.

Synthesis of multicomponent quasi-P and converted quasi-P-S seismograms for intersecting fracture systems

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1261-1271 ◽  
Author(s):  
Andrey A. Ortega ◽  
George A. McMechan
Keyword(s):  
Shear Wave ◽  
Transverse Isotropy ◽  
Normal Incidence ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
S Wave ◽  
Stiffness Constant ◽  
Wave Splitting ◽  
Vertical Fractures

Dynamic ray shooting with interpolation is an economical way of computing approximate Green’s functions in 3-D heterogeneous anisotropic media. The amplitudes, traveltimes, and polarizations of the reflected rays arriving at the surface are interpolated to synthesize three‐component seismograms at the desired recording points. The algorithm is applied to investigate kinematic quasi-P-wave propagation and converted quasi-P-S-wave splitting variations produced in reflections from the bottom of a layer containing two sets of intersecting dry vertical fractures as a function of the angle between the fracture sets and of the intensity of fracturing. An analytical expression is derived for the stiffness constant C16 that extends Hudson’s second‐order scattering theory to include tetragonal-2 symmetry systems. At any offset, the amount of splitting in nonorthogonal (orthorhombic symmetry) intersecting fracture sets is larger than in orthogonal (tetragonal-1 symmetry) systems, and it increases nonlinearly as a function of the intensity of fracturing as offset increases. Such effects should be visible in field data, provided that the dominant frequency is sufficiently high and the offset is sufficiently large. The amount of shear‐wave splitting at vertical incidence increases nonlinearly as a function of the intensity of fracturing and increases nonlinearly from zero in the transition from tetragonal-1 anisotropy through orthorhombic to horizontal transverse isotropy; the latter corresponds to the two crack systems degenerating to one. The zero shear‐wave splitting corresponds to a singularity, at which the vertical velocities of the two quasi‐shear waves converge to a single value that is both predicted theoretically and illustrated numerically. For the particular case of vertical fractures, there is no P-to-S conversion of vertically propagating (zero‐offset) waves. If the fractures are not vertical, the normal incidence P-to-S reflection coefficient is not zero and thus is a potential diagnostic of fracture orientation.

Analysis of wave-induced pore pressure changes recorded during the 1980 Mammoth Lakes, California, earthquake sequence

1984 ◽  
Vol 74 (4) ◽  
pp. 1395-1407
Author(s):  
Gerald M. Mavko ◽  
Ed Harp
Keyword(s):  
Pore Pressure ◽  
Vertical Surface ◽  
P Wave ◽  
Earthquake Sequence ◽  
Surface Velocity ◽  
Elastic Model ◽  
S Wave ◽  
Porous Saturated Medium ◽  
Wave Induced ◽  
Mammoth Lakes

Abstract Acceleration and wave-induced pore pressure were recorded in a saturated sand during the 1980 Mammoth Lakes, California, earthquake sequence. For the largest event recorded, the pore pressure was observed to be proportional to vertical surface acceleration during the P-wave arrivals and proportional to horizontal surface velocity during the S-wave arrivals. The results can be quantitatively explained with a linear elastic model of a porous saturated medium, such that pore pressure depends on dilatation and is independent of shear strain. A slight frequency dependence in the ratio of pore pressure to dilatation indicates local fluid flow on the scale of individual pores. The good agreement between observations and theory indicates that the deformation was primarily linear, even though maximum shear strains were close to the typical thresholds for liquefaction.

Acoustic approximations for processing in transversely isotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 623-631 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Elastic Model ◽  
S Wave ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Two Parameters ◽  
Vti Media

When transversely isotropic (VTI) media with vertical symmetry axes are characterized using the zero‐dip normal moveout (NMO) velocity [[Formula: see text]] and the anisotropy parameter ηinstead of Thomsen’s parameters, time‐related processing [moveout correction, dip moveout (DMO), and time migration] become nearly independent of the vertical P- and S-wave velocities ([Formula: see text] and [Formula: see text], respectively). The independence on [Formula: see text] and [Formula: see text] is well within the limits of seismic accuracy, even for relatively strong anisotropy. The dependency on [Formula: see text] and [Formula: see text] reduces even further as the ratio [Formula: see text] decreases. In fact, for [Formula: see text], all time‐related processing depends exactly on only [Formula: see text] and η. This fortunate dependence on two parameters is demonstrated here through analytical derivations of time‐related processing equations in terms of [Formula: see text] and η. The time‐migration dispersion relation, the NMO velocity for dipping events, and the ray‐tracing equations extracted by setting [Formula: see text] (i.e., by considering VTI as acoustic) not only depend solely on [Formula: see text] and η but are much simpler than the counterpart expressions for elastic media. Errors attributed to this use of the acoustic assumption are small and may be neglected. Therefore, as in isotropic media, the acoustic model arising from setting [Formula: see text], although not exactly true for VTI media, can serve as a useful approximation to the elastic model for the kinematics of P-wave data. This approximation can boost the efficiency of imaging and DMO programs for VTI media as well as simplify their description.

scholarly journals In-Reservoir Waveform Retrieval for Monitoring at Groningen—Seismic Interferometry with Active and Passive Deep Borehole Data

2021 ◽  
Vol 13 (15) ◽  
pp. 2928
Author(s):  
Muhammad F. Akbar ◽  
Ivan Vasconcelos ◽  
Hanneke Paulssen ◽  
Wen Zhou
Keyword(s):  
Synthetic Data ◽  
P Wave ◽  
Elastic Model ◽  
Detailed Knowledge ◽  
Virtual Source ◽  
Test Bed ◽  
S Wave ◽  
Borehole Data ◽  
Gas Field ◽  
Reservoir Monitoring

The Groningen gas field in the Netherlands is an ideal test bed for in-situ reservoir monitoring techniques because of the availability of both active and passive in-reservoir seismic data. In this study, we use deconvolution interferometry to estimate the reflection and transmission response using active and passive borehole data within the reservoir at ∼3-km depth and separate up- and downgoing P- and S-wave fields by f-k filtering. We validate the results using synthetic data of a 1D elastic model built from sonic logs recorded in the well. The estimated full-waveform reflection response for a virtual source at the top geophone is consistent with the synthetic response. For the virtual source at the bottom geophone, the reflection response appears to be phase delayed, though its arrivals are consistent with the local subsurface geology. Similarly, the first-order estimated local transmission response successfully approximates that of the P-wave velocity in the reservoir. The study shows that reliable subsurface information can be obtained from borehole interferometry without detailed knowledge of the medium parameters. In addition, the method could be used for passive reservoir monitoring to detect velocity, attenuation, and/or interface time-lapse variations.

Some comments on common-asymptotic-conversion-point (CACP) sorting of converted-wave data in isotropic, laterally inhomogeneous media

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. U29-U36 ◽  
Author(s):  
Mirko van der Baan
Keyword(s):  
Wave Velocity ◽  
Velocity Ratio ◽  
P Wave ◽  
Pure Mode ◽  
S Wave ◽  
Converted Wave ◽  
Wave Data ◽  
Conversion Point ◽  
Isotropic Media ◽  
Zero Offset

Common-midpoint (CMP) sorting of pure-mode data in arbitrarily complex isotropic or anisotropic media leads to moveout curves that are symmetric around zero offset. This greatly simplifies velocity determination of pure-mode data. Common-asymptotic-conversion-point (CACP) sorting of converted-wave data, on the other hand, only centers the apexes of all traveltimes around zero offset in arbitrarily complex but isotropic media with a constant P-wave/S-wave velocity ratio everywhere. A depth-varying CACP sorting may therefore be required to position all traveltimes properly around zero offset in structurally complex areas. Moreover, converted-wave moveout is nearly always asymmetric and nonhyperbolic. Thus, positive and negative offsets need to be processed independently in a 2D line, and 3D data volumes are to be divided in common azimuth gathers. All of these factors tend to complicate converted-wave velocity analysis significantly.

A finite-difference approach for solving pure quasi-P-wave equations in transversely isotropic and orthorhombic media

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. C161-C172 ◽  
Author(s):  
Xueyan Li ◽  
Hejun Zhu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Differential Operators ◽  
Transversely Isotropic ◽  
Coupled Systems ◽  
P Wave ◽  
Finite Difference Schemes ◽  
S Wave ◽  
P Waves ◽  
Pseudo Differential Operators

Starting from the dispersion relation and setting S-wave velocity along symmetry axes to zero, pseudoacoustic-wave equations have been proposed to describe the kinematics of acoustic wavefields in transversely isotropic (TI) and orthorhombic media. To date, the numerical stability of the pseudoacoustic-wave equations has been improved by developing coupled systems of wave equations; however, most simulations still suffer from S-wave artifacts that are the fundamental solutions of the fourth- and sixth-order partial differential equations. Pure quasi-P-wave equations accurately describe the traveltimes of P-waves in TI and orthorhombic media and are free of S-wave artifacts. However, it is difficult to directly solve the pure quasi-P-wave equations using conventional finite-difference schemes due to the presence of pseudo-differential operators. We approximated these pseudo-differential operators by algebraic expressions, whose coefficients can be determined by minimizing differences between the true and approximated values of the pseudo-differential operators in the wavenumber domain. The derived new coupled systems involve modified acoustic-wave equations and a Poisson’s equation that can be solved by conventional finite-difference stencils and fast Poisson’s solver. Several 2D and 3D numerical examples demonstrate that the simulations based on the new systems are free of S-wave artifacts and have correct kinematics of quasi-P-waves in TI and orthorhombic media.

scholarly journals Reflection Full Waveform Inversion with Decoupled Elastic-wave Equations in Inhomogeneous Medium

2021 ◽  
Vol 64 (1) ◽  
Author(s):  
Zhanyuan Liang ◽  
Guochen Wu ◽  
Xiaoyu Zhang ◽  
Qingyang Li
Keyword(s):  
Elastic Wave ◽  
Inhomogeneous Medium ◽  
Wave Velocity ◽  
Wave Equations ◽  
Waveform Inversion ◽  
P Wave ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Full Waveform ◽  
Elastic Wave Equations

Reflection full-waveform inversion (RFWI) can reduce the nonlinearity of inversion providing an accurate initial velocity model for full-waveform inversion (FWI) through the tomographic components (low-wavenumber). However, elastic-wave reflection full-waveform inversion (ERFWI) is more vulnerable to the problem of local minimum due to the complicated multi-component wavefield. Our algorithm first divides kernels of ERFWI into four subkernels based on the theory of decoupled elastic-wave equations. Then we try to construct the tomographic components of ERFWI with only single-component wavefields, similarly to acoustic inversions. However, the S-wave velocity is still vulnerable to the coupling effects because of P-wave components contained in the S-wavefield in an inhomogeneous medium. Therefore we develop a method for decoupling elastic- wave equations in an inhomogeneous medium, which is applied to the decomposition of kernels in ERFWI. The new decoupled system can improve the accuracy of S-wavefield and hence further reduces the high-wavenumber crosstalk in the subkernel of S-wave velocity after kernels are decomposed. The numerical examples of Sigsbee2A model demonstrate that our ERFWI method with decoupled elastic-wave equations can efficiently recover the low-wavenumber components of the model and improve the precision of the S-wave velocity.

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