scholarly journals Free Wave Propagation in Plates of General Anisotropic Media

1989 ◽  
Vol 56 (4) ◽  
pp. 881-886 ◽  
Author(s):  
Adnan H. Nayfeh ◽  
D. E. Chimenti
Keyword(s):  
Anisotropic Plate ◽  
Secular Equation ◽  
Formal Analysis ◽  
Transversely Isotropic ◽  
Wave Dispersion ◽  
Wave Mode ◽  
Higher Symmetry ◽  
Free Wave ◽  
Axis Of Symmetry ◽  
Free Wave Propagation

We develop the analysis for the propagation of free waves in a general anisotropic plate. We begin with a formal analysis of waves in a plate belonging to the triclinic symmetry group. The calculation is then carried forward for the slightly more specialized case of a monoclinic plate. We derive the secular equation for this case in closed form and isolate the mathematical conditions for symmetric and antisymmetric wave mode propagation in completely separate terms. Material systems of higher symmetry, such as orthotropic, transversely isotropic, cubic, and isotropic are contained implicitly in our analysis. We also demonstrate that the particle motions for Lamb and SH modes uncouple if propagation occurs along an in-plane axis of symmetry. We present numerical free-wave dispersion results drawn from concrete examples of materials belonging to several of these symmetry groups.

Free-Wave Dispersion Curves of a Multi-Supported String

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Benjamin A. Cray ◽  
Andrew J. Hull ◽  
Albert H. Nuttall
Keyword(s):  
Material Properties ◽  
Wave Dispersion ◽  
Dispersion Curves ◽  
Viscous Damping ◽  
Free Wave ◽  
Support Set ◽  
Dispersion Formula ◽  
General Dispersion ◽  
Support Sets ◽  
Free Wave Propagation

Free-wave propagation of an infinite, tensioned string, supported along its length by repeating segments of multiple spring-mass connections, is examined. The segments can consist of an arbitrary number of different support sets and be of any overall length. Periodicity is intrinsic, since the segments repeat; the goal, though, is to examine what effect variations within the segments have on dispersion. The formulation reveals an unexpected amount of complexity for such a simply posed system. Each support set has independent mass, stiffness, and viscous damping, and the sets are allowed to be offset from one another. A free-wave dispersion formula is derived for two sets of supports (Q = 2) and compared to the well-known ideally periodic expression (Q = 1). A means to obtain general dispersion formulas, for any Q, is discussed. It is shown that the systems’ dispersion curves are primarily governed by the material properties of the string and by the location of the supports.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Experimental study of aerosol oscillations in an open tube in the shock-free wave mode

2013 ◽  
Vol 51 (6) ◽  
pp. 873-875 ◽  
Author(s):  
D. A. Gubaidullin ◽  
R. G. Zaripov ◽  
L. A. Tkachenko
Keyword(s):  
Experimental Study ◽  
Wave Mode ◽  
Free Wave ◽  
Open Tube

scholarly journals An efficient wavefield inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R195-R206 ◽  
Author(s):  
Chao Song ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Optimization Problem ◽  
Source Function ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Highly Nonlinear ◽  
Isotropic Media ◽  
Axis Of Symmetry

Conventional full-waveform inversion (FWI) aims at retrieving a high-resolution velocity model directly from the wavefields measured at the sensor locations resulting in a highly nonlinear optimization problem. Due to the high nonlinearity of FWI (manifested in one form in the cycle-skipping problem), it is easy to fall into local minima. Considering that the earth is truly anisotropic, a multiparameter inversion imposes additional challenges in exacerbating the null-space problem and the parameter trade-off issue. We have formulated an optimization problem to reconstruct the wavefield in an efficient matter with background models by using an enhanced source function (which includes secondary sources) in combination with fitting the data. In this two-term optimization problem to fit the wavefield to the data and to the background wave equation, the inversion for the wavefield is linear. Because we keep the modeling operator stationary within each frequency, we only need one matrix inversion per frequency. The inversion for the anisotropic parameters is handled in a separate optimization using the wavefield and the enhanced source function. Because the velocity is the dominant parameter controlling the wave propagation, it is updated first. Thus, this reduces undesired updates for anisotropic parameters due to the velocity update leakage. We find the effectiveness of this approach in reducing parameter trade-off with a distinct Gaussian anomaly model. We find that in using the parameterization [Formula: see text], and [Formula: see text] to describe the transversely isotropic media with a vertical axis of symmetry model in the inversion, we end up with high resolution and minimal trade-off compared to conventional parameterizations for the anisotropic Marmousi model. Application on 2D real data also indicates the validity of our method.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

The effect of transverse isotropy on isotropic traveltime inversion of vertical seismic profile data—A modeling study

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1235-1241 ◽  
Author(s):  
Jan Douma
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Seismic Profile ◽  
Isotropic Layer ◽  
Low Velocity ◽  
Traveltime Inversion ◽  
Axis Of Symmetry ◽  
Transversely Isotropic Layer

Traveltime inversion of multioffset VSP data can be used to determine the depths of the interfaces in layered media. Many inversion schemes, however, assume isotropy and consequently may introduce erroneous structures for anisotropic media. Synthetic traveltime data are computed for layered anisotropic media and inverted assuming isotropic layers. Only the interfaces between these layers are inverted. For a medium consisting of a horizontal isotropic low‐velocity layer on top of a transversely isotropic layer with a horizontal axis of symmetry (e.g., anisotropy due to aligned vertical cracks), 2-D isotropic inversion results in an anticline. For a given axis of symmetry the form of this anticline depends on the azimuth of the source‐borehole direction. The inversion result is a syncline (in 3-D a “bowl” structure), regardless of the azimuth of the source‐borehole direction for a vertical axis of symmetry of the transversely isotropic layer (e.g., anisotropy due to horizontal bedding).

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