Constitutive model and wave equations for linear, viscoelastic, anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (2) ◽  
pp. 537-548 ◽  
Author(s):  
Jose M. Carcione
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Time Domain ◽  
Wave Equations ◽  
Superposition Principle ◽  
Fourier Method ◽  
Rock Properties ◽  
Orthorhombic Symmetry ◽  
Principal Axes ◽  
The Time Domain

Rocks are far from being isotropic and elastic. Such simplifications in modeling the seismic response of real geological structures may lead to misinterpretations, or even worse, to overlooking useful information. It is useless to develop highly accurate modeling algorithms or to naively use amplitude information in inversion processes if the stress‐strain relations are based on simplified rheologies. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and anelastic behavior of rocks. This work presents a new constitutive relation and the corresponding time‐domain wave equation to model wave propagation in inhomogeneous anisotropic and dissipative media. The rheological equation includes the generalized Hooke’s law and Boltzmann’s superposition principle to account for anelasticity. The attenuation properties in different directions, associated with the principal axes of the medium, are controlled by four relaxation functions of viscoelastic type. A dissipation model that is consistent with rock properties is the general standard linear solid. This is based on a spectrum of relaxation mechanisms and is suitable for wavefield calculations in the time domain. One relaxation function describes the anelastic properties of the quasi‐dilatational mode and the other three model the anelastic properties of the shear modes. The convolutional relations are avoided by introducing memory variables, six for each dissipation mechanism in the 3-D case, two for the generalized SH‐wave equation, and three for the qP − qSV wave equation. Two‐dimensional wave equations apply to monoclinic and higher symmetries. A plane analysis derives expressions for the phase velocity, slowness, attenuation factor, quality factor and energy velocity (wavefront) for homogeneous viscoelastic waves. The analysis shows that the directional properties of the attenuation strongly depend on the values of the elasticities. In addition, the displacement formulation of the 3-D wave equation is solved in the time domain by a spectral technique based on the Fourier method. The examples show simulations in a transversely‐isotropic clayshale and phenolic (orthorhombic symmetry).

A new approach to modeling the electromagnetic response of conductive media

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1180-1192 ◽  
Author(s):  
K. H. Lee ◽  
G. Liu ◽  
H. F. Morrison
Keyword(s):  
Wave Equation ◽  
Wave Field ◽  
Time Domain ◽  
Wave Equations ◽  
Absorbing Boundary Conditions ◽  
Free Form ◽  
Scalar Wave ◽  
Coupled Equations ◽  
Em Field ◽  
The Time Domain

We introduce a new and potentially useful method for computing electromagnetic (EM) responses of arbitrary conductivity distributions in the earth. The diffusive EM field is known to have a unique integral representation in terms of a fictitious wave field that satisfies a wave equation. We show that this integral transform can be extended to include vector fields. Our algorithm takes advantage of this relationship between the wave field and the actual EM field. Specifically, numerical computation is carried out for the wave field, and the result is transformed back to the EM field in the time domain. The proposed approach has been successfully demonstrated using two‐dimensional (2‐D) models. The appropriate TE‐mode diffusion equation in the time domain for the electric field is initially transformed into a scalar wave equation in an imaginary q domain, where q is a time‐like variable. The corresponding scalar wave field is computed numerically using an explicit q‐stepping technique. Standard finite‐difference methods are used to approximate the fields, and absorbing boundary conditions are implemented. The computed wave field is then transformed back to the time domain. The result agrees fairly well with the solution computed directly in the time domain. We also present an approach for general three‐dimensional (3‐D) EM problems for future studies. In this approach, Maxwell’s equations in the time domain are first transformed into a system of coupled first‐order wave equations in the q domain. These coupled equations are slightly modified and then cast into a “symmetric” and “divergence‐free” form. We show that it is to this particular form of equations that numerical schemes developed for solving wave equations can be applied efficiently.

Time Domain Simulations on a Single Point Moored Submerged Sphere of Variable Radius

2005 ◽  
Author(s):  
Joa˜o M. B. P. Cruz ◽  
Anto´nio J. N. A. Sarmento
Keyword(s):  
Wave Propagation ◽  
Time Domain ◽  
Physical Phenomenon ◽  
Single Point ◽  
Vertical Direction ◽  
Equation Of Motion ◽  
Hydrodynamic Coefficients ◽  
Energy Converter ◽  
Submerged Sphere ◽  
The Time Domain

This paper presents a different approach to the work developed by Cruz and Sarmento (2005), where the same problem was studied in the frequency domain. It concerns the same sphere, connected to the seabed by a tension line (single point moored), that oscillates with respect to the vertical direction in the plane of wave propagation. The pulsating nature of the sphere is the basic physical phenomenon that allows the use of this model as a simulation of a floating wave energy converter. The hydrodynamic coefficients and diffraction forces presented in Linton (1991) and Lopes and Sarmento (2002) for a submerged sphere are used. The equation of motion in the angular direction is solved in the time domain without any assumption about its output, allowing comparisons with the previously obtained results.

Dip moveout in the Radon domain

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 278-288
Author(s):  
Chengshu Wang
Keyword(s):  
Time Domain ◽  
Three Dimensional ◽  
Fourier Method ◽  
Real Data ◽  
Processing Technique ◽  
The Common ◽  
The Time Domain

I consider a new dip‐moveout (DMO) processing technique in the Radon domain called Radon DMO. The Radon DMO operator directly maps data from the NMO-corrected time domain to the DMO wavefield in the Radon domain. The method is built upon a process that transforms a single NMO-corrected trace into multiple traces spread along hyperbolas in the Radon domain. These hyperbolas are a linear Radon map of the DMO ellipses in the time domain. In this paper, I introduce the amplitude‐preserving Radon DMO and compare some examples of Radon DMO and Fourier DMO for both synthetic and real data. I also show the better frequency preservation properties of the Radon DMO method. Three‐dimensional data are often irregularly sampled with respect to fold, azimuth, and offset. Population deficiencies are exaggerated in the common‐offset domain. Radon DMO does not require that input traces belong to one common‐offset bin as does the Fourier method. Input traces can be organized from multiple offset bins grouping to perform Radon DMO, which is well used in 3-D surveys. Some synthetic and real data examples show these properties.

scholarly journals A new iterative solver for the time-harmonic wave equation

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E57-E63 ◽  
Author(s):  
C. D. Riyanti ◽  
Y. A. Erlangga ◽  
R.-E. Plessix ◽  
W. A. Mulder ◽  
C. Vuik ◽  
...  
Keyword(s):  
Wave Equation ◽  
Linear System ◽  
Time Domain ◽  
Multigrid Method ◽  
Harmonic Wave ◽  
Computational Time ◽  
Full Time ◽  
Iterative Solver ◽  
Time Harmonic ◽  
The Time Domain

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

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