First-order systems for elastic and acoustic variable-tilt TI media

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T157-T170 ◽  
Author(s):  
Kenneth P. Bube ◽  
Tamas Nemeth ◽  
Joseph P. Stefani ◽  
Wei Liu ◽  
Kurt T. Nihei ◽  
...  
Keyword(s):  
Shear Wave ◽  
Lower Order ◽  
Elastic System ◽  
Transversely Isotropic ◽  
Order System ◽  
Wave Speeds ◽  
First Order ◽  
Vti Media ◽  
Ti Media ◽  
Lower Order Terms

We derive and compare first-order wave propagation systems for variable-tilt elastic and acoustic tilted transversely isotropic (TTI) media. Acoustic TTI systems are commonly used in reverse-time migration. Starting initially with homogeneous vertical transversely isotropic (VTI) media, and then extending to heterogeneous variable-tilt TI media, we derive a pseudoacoustic [Formula: see text] first-order system of differential equations by setting the shear-wave speeds to zero and simplifying the full-elastic system accordingly. This [Formula: see text] system conserves a complete energy, but only when the anelliptic anisotropy parameter [Formula: see text]. For [Formula: see text] (including isotropic media), the system allows linearly time-growing and spatially nonpropagating nonphysical solutions frequently taken for numerical noise. We modified this [Formula: see text] acoustic first-order system by changing the stress variables to obtain a system that stays stable for [Formula: see text]. This system for homogeneous VTI media is generalized to heterogeneous variable-tilt TI media by rotating the stress and strain variables in the full elastic system before setting the shear-wave speeds to zero; the system obtained can be greatly simplified by combining the rotational terms, resulting in only one rotation and extra lower-order terms compared to the [Formula: see text] first-order acoustic system for VTI media. This new system can be simplified further by neglecting the lower-order terms. Both systems (with and without lower-order terms) conserve the same complete energy. Finally, the corresponding [Formula: see text] full elastic system for variable-tilt acoustic TI media can be used for the purposes of benchmarking.

On the instability in second-order systems for acoustic VTI and TTI media

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T171-T186 ◽  
Author(s):  
Kenneth P. Bube ◽  
Tamas Nemeth ◽  
Joseph P. Stefani ◽  
Ray Ergas ◽  
Wei Liu ◽  
...  
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Differential Equations ◽  
Shear Wave ◽  
Transversely Isotropic ◽  
Second Order ◽  
Wave Speeds ◽  
Tti Media ◽  
Second Order Systems ◽  
Vti Media

We studied second-order wave propagation systems for vertical transversely isotropic (VTI) and tilted transversely isotropic (TTI) acoustic media with variable axes of symmetry that have their shear-wave speeds set to zero. Acoustic TTI systems are commonly used in reverse-time migration, but these second-order systems are susceptible to instablities appearing as nonphysical stationary noise growing linearly in time, particularly in variable-tilt TTI media. We found an explanation of the cause of this phenomenon. The instabilities are not caused only by the numerical schemes; they are inherent to the differential equations. These instabilities are present even in homogeneous VTI media. These instabilities are caused by zero wave speeds at a wide variety of wavenumbers — a direct consequence of setting the shear-wave speeds to zero — coupled with the second time derivative in these systems. Although the second-order isotropic wave equation allows smooth time-growing solutions, a larger class of time-growing solutions exists for the second-order acoustic TI systems, including nonsmooth solutions. Boundary conditions appear to be less effective in controlling these time-growing solutions than they are for the isotropic wave equation. These systems conserve an incomplete energy that does not prevent the instabilities. The corresponding steady-state systems are no longer elliptic differential equations and can have nonsmooth solutions that are related to the instabilities. We started initially with homogeneous VTI media, and then extended these results to heterogeneous variable-tilt TTI media. We also developed a second-order acoustic system for heterogeneous variable-tilt TTI media derived directly from the full-elastic system for heterogeneous variable-tilt TTI media. All second-order systems with a dispersion relation obtained by setting the shear-wave speeds to zero in the elastic dispersion relation allowed these nonphysical time-growing solutions; however, knowing the cause of these instabilities, it may be possible to prevent or control the activation of these solutions.

Velocity analysis using nonhyperbolic moveout in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1839-1854 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
D Values ◽  
Two Parameters ◽  
Vti Media ◽  
Ti Media

P‐wave reflections from horizontal interfaces in transversely isotropic (TI) media have nonhyperbolic moveout. It has been shown that such moveout as well as all time‐related processing in TI media with a vertical symmetry axis (VTI media) depends on only two parameters, [Formula: see text] and η. These two parameters can be estimated from the dip‐moveout behavior of P‐wave surface seismic data. Alternatively, one could use the nonhyperbolic moveout for parameter estimation. The quality of resulting estimates depends largely on the departure of the moveout from hyperbolic and its sensitivity to the estimated parameters. The size of the nonhyperbolic moveout in TI media is dependent primarily on the anisotropy parameter η. An “effective” version of this parameter provides a useful measure of the nonhyperbolic moveout even in v(z) isotropic media. Moreover, effective η, [Formula: see text], is used to show that the nonhyperbolic moveout associated with typical TI media (e.g., shales, with η ≃ 0.1) is larger than that associated with typical v(z) isotropic media. The departure of the moveout from hyperbolic is increased when typical anisotropy is combined with vertical heterogeneity. Larger offset‐to‐depth ratios (X/D) provide more nonhyperbolic information and, therefore, increased stability and resolution in the inversion for [Formula: see text]. The X/D values (e.g., X/D > 1.5) needed for obtaining stability and resolution are within conventional acquisition limits, especially for shallow targets. Although estimation of η using nonhyperbolic moveouts is not as stable as using the dip‐moveout method of Alkhalifah and Tsvankin, particularly in the absence of large offsets, it does offer some flexibility. It can be applied in the absence of dipping reflectors and also may be used to estimate lateral η variations. Application of the nonhyperbolic inversion to data from offshore Africa demonstrates its usefulness, especially in estimating lateral and vertical variations in η.

scholarly journals Thermoelastic wave propagation in a rotating elastic medium without energy dissipation

2005 ◽  
Vol 2005 (1) ◽  
pp. 99-107 ◽  
Author(s):  
S. K. Roychoudhuri ◽  
Nupur Bandyopadhyay
Keyword(s):  
Energy Dissipation ◽  
Dispersion Equation ◽  
Shear Wave ◽  
Wave Speed ◽  
Wave Speeds ◽  
First Order ◽  
Thermoelasticity Theory ◽  
General Dispersion ◽  
Without Energy Dissipation ◽  
Thermoelastic Waves

A study is made of the propagation of time-harmonic plane thermoelastic waves of assigned frequency in an infinite rotating medium using Green-Naghdi model (1993) of linear thermoelasticity without energy dissipation. A more general dispersion equation is derived to examine the effect of rotation on the phase velocity of the modified coupled thermal dilatational shear waves. It is observed that in thermoelasticity theory of type II (Green-Naghdi model), the modified coupled dilatational thermal waves propagate unattenuated in contrast to the classical thermoelasticity theory, where the thermoelastic waves undergo attenuation (Parkus, Chadwick, and Sneddon). The solutions of the more general dispersion equation are obtained for small thermoelastic coupling by perturbation technique. Cases of high and low frequencies are also analyzed. The rotation of the medium affects both quasielastic dilatational and shear wave speeds to the first order inωfor low frequency, while the quasithermal wave speed is affected by rotation up to the second power inω. However, for large frequency, rotation influences both the quasidilatational and shear wave speeds to first order inωand the quasithermal wave speed to the second order in1/ω.

On: “Shear‐wave splitting in cross‐hole surveys: Modeling” by Enru Liu, Stuart Crampin, and David C. Booth (GEOPHYSICS, 54, 57–65, January, 1989)

Geophysics ◽  
1989 ◽  
Vol 54 (11) ◽  
pp. 1503-1504
Author(s):  
Don Winterstein
Keyword(s):  
Shear Wave ◽  
A Priori ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
S Wave ◽  
Horizontal Symmetry ◽  
Wave Splitting ◽  
Wave Travel ◽  
Ti Media ◽  
Travel Modeling

Liu et al. extended consideration of shear‐wave (S‐wave) polarization patterns in anisotropic media from the usual vertical to a predominantly horizontal direction of wave travel. Modeling was for transversely isotropic (TI) media with horizontal symmetry axes. The exposition was accurate in major concepts, but the authors could have been more precise in presenting a couple of incidental properties of TI media. These properties have to do with S‐wave polarizations and certain a priori predictions one can make about them from symmetry considerations. I state and prove two predictions here in a tutorial mode, partly to demonstrate the simplicity and power of symmetry concepts as tools for understanding wave behavior in anisotropic media.

Acoustic VTI modeling using high-order finite differences

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. T67-T73 ◽  
Author(s):  
Stig Hestholm
Keyword(s):  
Finite Differences ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
High Order ◽  
Isotropic Layer ◽  
Natural Form ◽  
Near Surface ◽  
First Order ◽  
Seismic Reflectors ◽  
Vti Media

Two second-order wave equations for acoustic vertical transversely isotropic (VTI) media are transformed to six first-order coupled partial differential equations for a more straighforward numerical implementation of the derivatives. The resulting first-order equations have a more natural form for discretization by any finite-difference, pseudospectral, or finite-element method. I discretized the new equations by high-order finite differences and used synthetic seismograms and snapshots for anisotropic and isotropic cases. The relative merits of placing the source deep and close to a free surface are assessed, illustrating advantages of exciting the source inside or outside of a near-surface, thin, isotropic layer. Results show that traveltimes from deep seismic reflectors can remain virtually unaffected when near-surface isotropic layers are included in acoustic VTI media.

scholarly journals 2-D depth migration in transversely isotropic media using explicit operators

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1819-1829 ◽  
Author(s):  
Omar Uzcategui
Keyword(s):  
Least Squares ◽  
Taylor Series ◽  
Least Squares Method ◽  
Transversely Isotropic ◽  
Downward Continuation ◽  
Series Method ◽  
Maximum Angle ◽  
Taylor Series Method ◽  
Vti Media ◽  
Ti Media

Stable, explicit depth‐extrapolation filters can be used to propagate plane waves corresponding to the qP and qSV (quasi‐P and quasi‐SV propagation) modes for transversely isotropic (TI) media. Here, I discuss and compare results of two different methods for obtaining the filters for TI media with a vertical axis of symmetry (VTI). The first, a modified Taylor series method, is used to calculate the N‐coefficients of a finite‐length filter such that the Taylor expansion around vertical propagation matches the spatial Fourier transform of the downward‐continuation operator for VTI media. Second, a least‐squares method is used to calculate the filter coefficients such that the amplitude and phase departures from the ideal response of the downward‐continuation operator for VTI media are minimized over a range of frequencies and propagation angles. In both methods, the amplitude response of the filter is forced to be less than unity in the evanescent region to achieve stability. In general, as exemplified in all the cases studied here, the constrained least‐squares method produced filters with accurate wavefield extrapolation for a wider range of propagation angles than that obtained for the modified Taylor series method. In both methods, the maximum angle that can be accurately propagated depends on the ratio of frequency to vertical phase velocity [Formula: see text], and on the length of the filter. However, for a fixed filter length and for a given ratio [Formula: see text], the maximum angle propagated with accuracy depends on the elastic constants of the medium. The accuracy of the filters degrades as the degree of anisotropy becomes more extreme.

An Adjustment of Reference Tracking PID Controllers for a First-order System with a Dead Time

2016 ◽  
Vol 136 (5) ◽  
pp. 676-682 ◽  
Author(s):  
Akihiro Ishimura ◽  
Masayoshi Nakamoto ◽  
Takuya Kinoshita ◽  
Toru Yamamoto
Keyword(s):  
Dead Time ◽  
Order System ◽  
Pid Controllers ◽  
Reference Tracking ◽  
First Order

scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

scholarly journals Existence results for nonlinear degenerate elliptic equations with lower order terms

2020 ◽  
Vol 10 (1) ◽  
pp. 301-310
Author(s):  
Weilin Zou ◽  
Xinxin Li
Keyword(s):  
Lower Order ◽  
Elliptic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Initial Boundary Value ◽  
Lower Order Terms ◽  
Nonlinear Degenerate

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

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