Complex rays applied to wave propagation in a viscoelastic medium

1990 ◽  
Vol 132 (1-2) ◽  
pp. 401-415 ◽  
Author(s):  
D. J. Hearn ◽  
E. S. Krebes
Keyword(s):  
Wave Propagation ◽  
Viscoelastic Medium ◽  
Complex Rays

On computing ray‐synthetic seismograms for anelastic media using complex rays

Geophysics ◽  
1990 ◽  
Vol 55 (4) ◽  
pp. 422-432 ◽  
Author(s):  
D. J. Hearn ◽  
E. S. Krebes
Keyword(s):  
Surface Layer ◽  
Plane Wave ◽  
Saddle Point ◽  
Viscoelastic Medium ◽  
Observation Point ◽  
Acute Angle ◽  
Synthetic Seismograms ◽  
Propagation Angle ◽  
Source Point ◽  
Complex Rays

A plane wave propagating in a viscoelastic medium is generally inhomogeneous, meaning that the direction in which the spatial rate of amplitude attenuation is maximum is generally different from the direction of travel. The angle between these two directions, which we call the “attenuation angle,” is an acute angle. In order to trace the ray corresponding to a plane wave propagating between a source point and a receiver point in a layered viscoelastic medium, one must know both the initial propagation angle (the angle that the raypath makes with the vertical) and the initial attenuation angle at the source point. In some recent literature on the computation of ray‐synthetic seismograms in anelastic media, values for the initial attenuation angle are chosen arbitrarily; but this approach is fundamentally unsatisfactory, since different choices lead to different results for the computed waveforms. Another approach, which is more deterministic and physically acceptable, is to deduce the value of the initial attenuation angle from the value of the complex ray parameter at the saddle point of the complex traveltime function. This value can be obtained by applying the method of steepest descent to evaluate approximately the integrals giving the exact wave field at the observation point. This well‐known technique results in the ray‐theory limit. The initial propagation angle can also be determined from the saddle point. Among all possible primary rays between source and receiver, each having different initial propagation and attenuation angles, the ray determined by the saddle point, which we call a “stationary ray,” has the smallest traveltime, a result which is consistent with Fermat’s principle of least time. Such stationary rays are complex rays, i.e., the spatial (e.g., Cartesian) coordinates of points on stationary raypaths are complex numbers, whereas the arbitrarily determined rays mentioned above are usually traced as real rays. We compare examples of synthetic seismograms computed with stationary rays with those from some arbitrarily determined rays. If the initial value of the attenuation angle is arbitrarily chosen to be a constant for all initial propagation angles, the differences between the two types of seismograms are generally small or negligible in the subcritical zone, except when the constant is relatively large in value, say, within 10 degrees or so of its upper bound of 90 degrees. In that case, the differences are significant but still not large. However, if the surface layer is highly absorptive, the differences can be quite large and pronounced. For larger offsets, i.e., in the supercritical zone, large phase discrepancies can exist between the waveforms for the stationary rays and those for the arbitrarily determined rays, even if the constant initial attenuation angle is not large and even for moderate absorptivity in the surface layer.

scholarly journals Stress Wave Propagation through Rock Joints Filled with Viscoelastic Medium Considering Different Water Contents

2020 ◽  
Vol 10 (14) ◽  
pp. 4797 ◽  
Author(s):  
Xiaolin Huang ◽  
Shengwen Qi ◽  
Bowen Zheng ◽  
Youshan Liu ◽  
Lei Xue ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Water Content ◽  
Seismic Response ◽  
Transmission Coefficient ◽  
Stress Wave ◽  
Viscoelastic Medium ◽  
Rock Joints ◽  
Stress Wave Propagation ◽  
Multiple Reflections ◽  
Joint Spacing

A rock mass often contains joints filled with a viscoelastic medium of which seismic response is significant to geophysical exploration and seismic engineering design. Using the propagator matrix method, an analytical model was established to characterize the seismic response of viscoelastic filled joints. Stress wave propagation through a single joint highly depended on the water content and thickness of the filling as well as the frequency and incident angle of the incident wave. The increase in the water content enhanced the viscosity (depicted by quality factor) of the filled joint, which could promote equivalent joint stiffness and energy dissipation with double effects on stress wave propagation. There existed multiple reflections when the stress wave propagated through a set of filled joints. The dimensionless joint spacing was the main controlling factor in the seismic response of the multiple filled joints. As it increased, the transmission coefficient first increased, then it decreased instead, and at last it basically kept invariant. The effect of multiple reflections was weakened by increasing the water content, which further influenced the variation of the transmission coefficient. The water content of the joint filling should be paid more attention in practical applications.

Seismic modeling in viscoelastic media

Geophysics ◽  
1993 ◽  
Vol 58 (1) ◽  
pp. 110-120 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Wave Propagation ◽  
Viscoelastic Medium ◽  
Polynomial Interpolation ◽  
Superposition Principle ◽  
Forward Modeling ◽  
Memory Storage ◽  
Three Dimensions ◽  
Bright Spot ◽  
Viscoelastic Media ◽  
Standard Linear

Anelasticity is usually caused by a large number of physical mechanisms which can be modeled by different microstructural theories. A general way to take all these mechanisms into account is to use a phenomenologic model. Such a model which is consistent with the properties of anelastic media can be represented mechanically by a combination of springs and dash‐pots. A suitable system can be constructed by the parallel connection of several standard linear elements and is referred to as the general standard linear solid rheology. Two relaxation functions that describe the dilatational and shear dissipation mechanisms of the medium are needed. This model properly describes the short and long term behaviors of materials with memory and is the basis for describing viscoelastic wave propagation. This work presents two‐dimensional (2-D) and three‐dimensional (3-D) forward modeling in linear viscoelastic media. The theory implements Boltzmann’s superposition principle based on a spectrum of relaxation mechanisms in the time‐domain equation of motion by the introduction of the memory variables. The algorithm uses a polynomial interpolation of the evolution operator for time integration and the Fourier pseudospectral method for computation of the spatial derivatives. This scheme has spectral accuracy for band‐limited functions with no temporal or spatial dispersion, a very important fact in anelastic wave propagation. Examples are given of how to pose typical problems of viscoelastic forward modeling for geophysical problems in two and three dimensions. A model separating an elastic medium of a viscoelastic medium with similar elastic moduli but different attenuations shows that the interface generates appreciable reflected energy. A second example computes the response of a single interface in the presence of highly dissipative sandstone lenses, properly simulating the anelasticity of direct and converted P‐ and S‐waves. A common‐shot reflection survey over a gas cap reservoir indicates that attenuation significantly affects the bright spot response. 3-D viscoelastic modeling requires twice the memory storage of 3-D viscoacoustic modeling when one dissipation mechanism is used for each wave mode. Simulation in a 3-D homogeneous viscoelastic medium shows how the anelastic characteristics of the different modes can be controlled independently. The result of this simulation is compared to the analytical solution indicating sufficient accuracy for many applications.

scholarly journals Wave propagation in a viscoelastic medium with a high concentration of pores

1991 ◽  
Vol 89 (4B) ◽  
pp. 1859-1859
Author(s):  
Y. Ma ◽  
V. K. Varadan ◽  
V. V. Varadan ◽  
C. Audoly
Keyword(s):  
Wave Propagation ◽  
Viscoelastic Medium ◽  
High Concentration

scholarly journals Effect of viscoelastic medium on wave propagation along protein microtubules

AIP Advances ◽  
2019 ◽  
Vol 9 (4) ◽  
pp. 045108 ◽  
Author(s):  
Muhammad Safeer ◽  
M. Taj ◽  
Syed Solat Abbas
Keyword(s):  
Wave Propagation ◽  
Viscoelastic Medium
Sign in / Sign up

Export Citation Format

Share Document