scholarly journals Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C75-C83 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Numerical Solution ◽  
Taylor Series Expansion ◽  
Pure Mode ◽  
Isotropic Layer ◽  
Weak Anisotropy ◽  
Reflected Waves ◽  
Quartic Equation ◽  
S Waves ◽  
Conversion Point ◽  
Quartic Term

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

Modeling and inversion of PS-wave moveout asymmetry for tilted TI media: Part I — Horizontal TTI layer

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D107-D121 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Parameter Estimates ◽  
Pure Mode ◽  
Velocity Analysis ◽  
Nonlinear Inversion ◽  
Isotropic Layer ◽  
Weak Anisotropy ◽  
Time Asymmetry

One of the distinctive features of mode-converted waves is their asymmetric moveout (i.e., the PS-wave traveltime in general is different if the source and receiver are interchanged) caused by lateral heterogeneity or elastic anisotropy. If the medium is anisotropic, the PS-wave moveout asymmetry contains valuable information for parameter estimation that cannot be obtained from pure reflection modes. Here, we generalize the so-called [Formula: see text] method, which is designed to replace reflected PS modes in velocity analysis with pure (unconverted) SS-waves, by supplementing the output SS traces with the moveout-asymmetry attributes of PS-waves. The time-asymmetry attribute [Formula: see text] is computed in the slowness domain as the difference between the paired traveltimes of the PS arrivals corresponding to ray parameters (horizontal slownesses) of equal magnitude but opposite sign. Another useful asymmetry attribute is the offset [Formula: see text] of the PS-wave traveltime minimum on a common-midpoint (CMP) gather. We demonstrate the effectiveness of the developed algorithmand the importance of including the asymmetry attributes of PS-waves in anisotropic velocity analysis for a horizontal transversely isotropic layer with a tilted symmetry axis (or TTI) medium. Simple analytic expressions for the moveout asymmetry of PSV-waves, derived in the weak-anisotropy approximation, are verified by anisotropic ray tracing. The attribute [Formula: see text] is proportional to the anellipticity parameter [Formula: see text] and reaches its maximum when the symmetry axis deviates by 20°–30° from the vertical or horizontal direction. All relevant parameters of a TTI layer can be estimated by a nonlinear inversion of the NMO velocities and zero-offset traveltimes of PP- and SS- (SVSV) waves combined with the moveout-asymmetry attributes of the PSV-wave. The inversion of pure-mode (PP and SS) moveouts alone is nonunique, while the addition of the attributes [Formula: see text] and [Formula: see text] yields stable parameter estimates from 2D data acquired in the vertical symmetry-axis plane. If the TTI model is formed by obliquely dipping fractures, the anisotropic parameters can be inverted further for the fracture orientation and compliances.

Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

Scalar reverse‐time depth migration of prestack elastic seismic data

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1519-1527 ◽  
Author(s):  
Robert Sun ◽  
George A. McMechan
Keyword(s):  
Seismic Waves ◽  
Velocity Model ◽  
P Wave ◽  
Common Source ◽  
Reverse Time ◽  
S Wave ◽  
Wave Source ◽  
Reflected Waves ◽  
S Waves ◽  
Elastic Data

Reflected P‐to‐P and P‐to‐S converted seismic waves in a two‐component elastic common‐source gather generated with a P‐wave source in a two‐dimensional model can be imaged by two independent scalar reverse‐time depth migrations. The inputs to migration are pure P‐ and S‐waves that are extracted by divergence and curl calculations during (shallow) extrapolation of the elastic data recorded at the earth’s surface. For both P‐to‐P and P‐to‐S converted reflected waves, the imaging time at each point is the P‐wave traveltime from the source to that point. The extracted P‐wave is reverse‐time extrapolated and imaged with a P‐velocity model, using a finite difference solution of the scalar wave equation. The extracted S‐wave is reverse‐time extrapolated and imaged similarly, but with an S‐velocity model. Converted S‐wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources. Synthetic examples show that the algorithm gives satisfactory results for laterally inhomogeneous models.

Inversion of reflection traveltimes for transverse isotropy

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
Ilya Tsvankin ◽  
Leon Thomsen
Keyword(s):  
Wave Velocity ◽  
Vertical Velocity ◽  
Joint Inversion ◽  
Transverse Isotropy ◽  
Taylor Series Expansion ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Quartic Term

In anisotropic media, the short‐spread stacking velocity is generally different from the root‐mean‐square vertical velocity. The influence of anisotropy makes it impossible to recover the vertical velocity (or the reflector depth) using hyperbolic moveout analysis on short‐spread, common‐midpoint (CMP) gathers, even if both P‐ and S‐waves are recorded. Hence, we examine the feasibility of inverting long‐spread (nonhyperbolic) reflection moveouts for parameters of transversely isotropic media with a vertical symmetry axis. One possible solution is to recover the quartic term of the Taylor series expansion for [Formula: see text] curves for P‐ and SV‐waves, and to use it to determine the anisotropy. However, this procedure turns out to be unstable because of the ambiguity in the joint inversion of intermediate‐spread (i.e., spreads of about 1.5 times the reflector depth) P and SV moveouts. The nonuniqueness cannot be overcome by using long spreads (twice as large as the reflector depth) if only P‐wave data are included. A general analysis of the P‐wave inverse problem proves the existence of a broad set of models with different vertical velocities, all of which provide a satisfactory fit to the exact traveltimes. This strong ambiguity is explained by a trade‐off between vertical velocity and the parameters of anisotropy on gathers with a limited angle coverage. The accuracy of the inversion procedure may be significantly increased by combining both long‐spread P and SV moveouts. The high sensitivity of the long‐spread SV moveout to the reflector depth permits a less ambiguous inversion. In some cases, the SV moveout alone may be used to recover the vertical S‐wave velocity, and hence the depth. Success of this inversion depends on the spreadlength and degree of SV‐wave velocity anisotropy, as well as on the constraints on the P‐wave vertical velocity.

Multi-Integral Method for Solving the Forward Dynamics of Stiff Multibody Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Numerical Solution ◽  
State Vector ◽  
Taylor Series Expansion ◽  
High Order ◽  
The State ◽  
System State ◽  
Time Step ◽  
Entire Interval ◽  
Free Parameters ◽  
Derivatives Of

The Hermite–Obreshkov–Padé (HOP) procedure is an implicit method for the numerical solution of a system of ordinary differential equations (ODEs) applicable to stiff dynamical systems. This procedure applies an Obreshkov condition to multiple derivatives of the system state vector, both at the start and end of a time step in the numerical solution. That condition is shown to be satisfied by the Hermite interpolating polynomial that matches the state vector and its derivatives, also at the start and end of a time step. The Hermite polynomial, in turn, can be specified in terms of the system state and its derivatives at the start of a step together with a collection of free parameters. Adjusting these free parameters to minimize magnitudes of the ODE residual and its derivatives at the end of a step serves as a proxy for matching the system state and its derivatives. A high-order Taylor expansion at the start of a time step interval models the residual and its derivatives over the entire interval. A variant of this procedure adjusts those parameters to match integrals of the system state over the duration of that interval. This is done by minimizing magnitudes of integrals of the ODE residual calculated from the extrapolating Taylor-series expansion, a process that avoids the need to determine integration constants for multiple integrals of the state. This alternative method eliminates the calculation of high-order derivatives of the system state and hence avoids loss in accuracy from floating-point round off. Numerical performance is evaluated on a dynamically unbalanced constant-velocity (CV) coupling having a high spring rate constraining shaft deflection.

Reflection and refraction of elastic waves at a corrugated interface

1966 ◽  
Vol 56 (1) ◽  
pp. 201-221
Author(s):  
Shuzo Asano
Keyword(s):  
Wave Amplitude ◽  
Normal Incidence ◽  
P Wave ◽  
Irregular Waves ◽  
Angle Of Incidence ◽  
S Wave ◽  
Reflected Waves ◽  
S Waves ◽  
Significant Difference ◽  
Almost All

abstract The effect of a corrugated interface on wave propagation is considered by using the method that was first applied to acoustical gratings by Rayleigh. The problem is what happens when a plane P wave is incident on a corrugated interface that separates two semi-infinite media. As is well known, there are irregular (scattered) waves as well as regular waves. By assuming both the amplitude and the slope of a corrugated interface to be small, quantities of the order of the square of corrugation amplitude are taken into account. In the case of normal incidence for three models considered, the effect of corrugation on reflection is larger than the effect of corrugation on refraction; the amplitude of the regularly reflected waves decreases, and that of the regularly refracted waves and of the irregular waves increases, as the corrugation amplitude becomes larger. Generally, the larger the velocity contrast, the larger the variation of wave amplitude with the wavelength and the amplitude of corrugation. The S wave component generally becomes larger as the wavelength of corrugation becomes smaller. Boundary waves exist, depending upon the ratio of wavelength of corrugation to that of the incident wave. For a specified interface, it is possible that there is a significant difference in wave amplitude as a function of the elastic constants. In the case of oblique incidence, computation was carried out for angles of incidence smaller than 15° for one model. For these small angles of incidence, almost all results for the case of normal incidence still hold. Furthermore, it can be concluded that the effect of the angle of incidence on reflected S waves is larger than for the other waves and that large differences in the amplitudes of waves at different angles of incidence may be expected for the irregular waves.

Nonhyperbolic reflection moveout for horizontal transverse isotropy

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1738-1753 ◽  
Author(s):  
AbdulFattah Al‐Dajani ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Single Layer ◽  
Azimuthal Velocity ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Analytic Description ◽  
Pure Mode ◽  
Quartic Term ◽  
Hti Media ◽  
Vertical Transverse Isotropy

The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear‐wave splitting to describe fractured formations with a single system of parallel vertical penny‐shaped cracks. Here, we present an analytic description of longspread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. The hyperbolic moveout equation parameterized by the exact normal‐moveout (NMO) velocity is sufficiently accurate for P-waves on conventional‐length spreads (close to the reflector depth), although the NMO velocity is not, in general, usable for converting time to depth. However, the influence of anisotropy leads to the deviation of the moveout curve from a hyperbola with increasing spread length, even in a single‐layer model. To account for nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion [t2(x2)] valid for any pure mode in an HTI layer. The quartic moveout coefficient and the NMO velocity are then substituted into the nonhyperbolic moveout equation of Tsvankin and Thomsen, originally designed for vertical transverse isotropy (VTI). Numerical examples for media with both moderate and uncommonly strong nonhyperbolic moveout show that this equation accurately describes azimuthally dependent P-wave reflection traveltimes in an HTI layer, even for spread lengths twice as large as the reflector depth. In multilayered HTI media, the NMO velocity and the quartic moveout coefficient reflect the influence of layering as well as azimuthal anisotropy. We show that the conventional Dix equation for NMO velocity remains entirely valid for any azimuth in HTI media if the group‐velocity vectors (rays) for data in a common‐midpoint (CMP) gather do not deviate from the vertical incidence plane. Although this condition is not exactly satisfied in the presence of azimuthal velocity variations, rms averaging of the interval NMO velocities represents a good approximation for models with moderate azimuthal anisotropy. Furthermore, the quartic moveout coefficient for multilayered HTI media can also be calculated with acceptable accuracy using the known averaging equations for vertical transverse isotropy. This allows us to extend the nonhyperbolic moveout equation to horizontally stratified media composed of any combination of isotropic, VTI, and HTI layers. In addition to providing analytic insight into the behavior of nonhyperbolic moveout, these results can be used in modeling and inversion of reflection traveltimes in azimuthally anisotropic media.

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