Velocity, attenuation, and quality factor in anisotropic viscoelastic media: A perturbation approach

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. D63-D73 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Elastic Medium ◽  
Order Approximation ◽  
Linear Functions ◽  
Perturbation Approach ◽  
Nonlinear Inversion ◽  
Slowness Surface ◽  
Simple Modification ◽  
First Order ◽  
Slowness Vector ◽  
Velocity Attenuation

Velocity, attenuation, and the quality [Formula: see text] factor of waves propagating in homogeneous media of arbitrary anisotropy and attenuation strength are calculated in high-frequency asymptotics using a stationary slowness vector, the vector evaluated at the stationary point of the slowness surface. This vector is generally complex-valued and inhomogeneous, meaning that the real and imaginary parts of the vector have different directions. The slowness vector can be determined by solving three coupled polynomial equations of the sixth order or by a nonlinear inversion. The procedure is simplified if perturbation theory is applied. The elastic medium is viewed as a background medium, and the attenuation effects are incorporated as perturbations. In the first-order approximation, the phase and ray velocities and their directions remain unchanged, being the same as those in the background elastic medium. The perturbation of the slowness vector is calculated by solving a system of three linear equations. The phase attenuation and phase [Formula: see text]-factor are linear functions of the perturbation of the slowness vector. Calculating the ray attenuation and ray [Formula: see text]-factor is even simpler than calculating the phase quantities because they are expressed in terms of perturbations of the medium without the need to evaluate the perturbation of the slowness vector. Numerical modeling indicates that the perturbations are highly accurate; the errors are less than 0.3% for a medium with a [Formula: see text]-factor of 20 or higher. The accuracy can be enhanced further by a simple modification of the first-order perturbation formulas.

Weak anisotropy-attenuation parameters

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB203-WB213 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Quality Factor ◽  
Propagation Velocity ◽  
Elastic Anisotropy ◽  
Polarization Vector ◽  
Linear Functions ◽  
Weak Anisotropy ◽  
Simple Modification ◽  
Velocity Anisotropy ◽  
Slowness Vector ◽  
Lower Accuracy

Velocity anisotropy and attenuation in weakly anisotropic and weakly attenuating structures can be treated uniformly using weak anisotropy-attenuation (WAA) parameters. The WAA parameters are constructed in a way analogous to weak anisotropy (WA) parameters designed for weak elastic anisotropy. The WAA parameters generalize WA parameters by incorporating attenuation effects. They can be represented alternatively by one set of complex values or by two sets of real values. Assuming high-frequency waves and using the first-order perturbation theory, all basic wave quantities such as the slowness vector, the polarization vector, propagation velocity, attenuation, and the quality factor are linear functions of WAA parameters. Numerical modeling shows that perturbation equations have different accuracy for different wave quantities. The propagation velocity usually is calculated with high accuracy. However, the attenuation and quality factor can be reproduced with appreciably lower accuracy. This happens mostly when the strength of velocity anisotropy is higher than 10% and attenuation is moderate or weak [Formula: see text]. In this case, the errors of the attenuation or [Formula: see text]-factor can attain values comparable to the strength of anisotropy or even higher. A simple modification of the equations by including some higher-order perturbations improves accuracy by three to four times.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

FIRST-ORDER APPROXIMATION OF THE NUMBER-PROJECTED SO(2N) TAMM-DANCOFF EQUATION AND ITS REDUCTION BY THE SCHUR FUNCTION

1999 ◽  
Vol 08 (05) ◽  
pp. 461-483
Author(s):  
SEIYA NISHIYAMA
Keyword(s):  
Wave Function ◽  
Order Approximation ◽  
Approximate Equation ◽  
Variation Principle ◽  
Schur Function ◽  
Soliton Theory ◽  
First Order ◽  
Fermion Systems ◽  
Group Manifold ◽  
First Order Approximation

First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group manifold.

Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

1995 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj
Keyword(s):  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Amplitude Equations ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Quadratic Nonlinearities ◽  
First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

scholarly journals Analytical quality assessment of iteratively reweighted least-squares (IRLS) method

2014 ◽  
Vol 20 (1) ◽  
pp. 132-141 ◽  
Author(s):  
Jianfeng Guo
Keyword(s):  
Least Squares ◽  
Order Approximation ◽  
Diagnostic Tools ◽  
Iteratively Reweighted Least Squares ◽  
First Order ◽  
Iterative Reweighting ◽  
First Order Approximation ◽  
Detectable Bias ◽  
Theoretical Analyses ◽  
Exact Analytical Method

The iteratively reweighted least-squares (IRLS) technique has been widely employed in geodetic and geophysical literature. The reliability measures are important diagnostic tools for inferring the strength of the model validation. An exact analytical method is adopted to obtain insights on how much iterative reweighting can affect the quality indicators. Theoretical analyses and numerical results show that, when the downweighting procedure is performed, (1) the precision, all kinds of dilution of precision (DOP) metrics and the minimal detectable bias (MDB) will become larger; (2) the variations of the bias-to-noise ratio (BNR) are involved, and (3) all these results coincide with those obtained by the first-order approximation method.

Rising and falling film flows: Viewed from a first-order approximation

1992 ◽  
Vol 47 (3) ◽  
pp. 683-694 ◽  
Author(s):  
H.S. Kheshgi ◽  
S.F. Kistler ◽  
L.E. Scriven
Keyword(s):  
Order Approximation ◽  
Falling Film ◽  
First Order ◽  
First Order Approximation ◽  
Film Flows
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