Seismic modelingSeismic modeling

Seismic modeling, processing, and inversion often require the calculation of the seismic response resulting from a suite of closely related seismic models. Even though changes to the model may be restricted to a small subvolume, we need to perform simulations for the full model. We present a new finite‐difference method that circumvents the need to resimulate the complete model for local changes. By requiring only calculations in the subvolume and its neighborhood, our method makes possible significant reductions in computational cost and memory requirements. In general, each source/receiver location requires one full simulation on the complete model. Following these pre‐computations, recalculation of the altered wavefield can be limited to the region around the subvolume and its neighborhood. We apply our method to a 2-D time‐lapse seismic problem, thereby achieving a factor of 15 reduction in computational cost. Potential savings for 3-D are far greater.

An efficient method for calculating finite-difference seismograms after model alterations

We theoretically study the scattering ofP, SV andSH waves by a zonal distribution of cracks, which simulates a fault fracture zone. An investigation is conducted how the geometrical properties of the crack distribution and the frictional characteristics of the crack surface are reflected in the attenuation and dispersion of incident waves, as well as in the amplitudes of the transmitted and reflected waves from the zone. If the crack distribution within the fault zone changes temporally during the preparation process of the expected earthquake, it will be important for earthquake prediction to monitor it, utilizing the scattering-induced wave phenomena. We consider the two-dimensional problem. Aligned cracks with the same length are assumed to be randomly distributed in a zone with a finite width, on which elastic waves are assumed to be incident. The distribution of cracks is assumed to be homogeneous and sparse. The crack surface is assumed to be stress-free, or to undergo viscous friction; the latter case simulates fluid-filled cracks. The opening displacement of the crack is assumed to be negligibly small. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed. When the crack surface is stress-free, it is commonly observed for every wave mode (P, SV andSH) that the attenuation coefficientQ
−1 peaks aroundka∼1, the phase velocity is almost independent ofk in the rangeka 1, wherek is the intrinsicS wavenumber anda is the half length of the crack. The effect of the friction is to shift the peak ofQ
−1 and the corner of the phase velocity curve to the low wavenumber range. The high wavenumber asymptote ofQ
−1 is proportional tok
−1 independently of model parameters and the wave modes. If the seismological observation thatQ
−1 ofS waves has a peak at around 0.5 Hz in the earth's crust is combined with our results, the upper limit of crack size within the crust is estimated about 4 km. The information regarding the transmitted and reflected waves, such as the high wavenumber limit of the amplitude of the transmitted wave etc., allows estimation of the strength of the friction.

Scattering of elastic waves by a fracture zone containing randomly distributed cracks

This paper is concerned with the use of leaky Lamb waves for the nondestructive evaluation (NDE) of damage in anisotropic materials such as fiber-reinforced composites. Two fundamental acoustic properties of the material, namely, the wave speed and attenuation have been measured. Stiffness is deduced from the wave speed. The damage mode selected for this study is matrix cracking. As expected, the in-plane stiffness decreases and the attenuation increases with an increase in the linear crack density.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1087&context=aere_pubs

Leaky Lamb waves in an anisotropic plate. II: Nondestructive evaluation of matrix cracks in fiber-reinforced composites

https://repository.tudelft.nl/islandora/object/uuid%3A0057895e-9bf9-499e-b89e-3ab4d42da6dd/datastream/OBJ/download

Reflection and Transmission of Elastic-Waves by the Spatially Periodic Interface between 2 Solids (Theory of the Integral-Equation Method)

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a crack of finite width is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered field together with the constitutive equation, it is shown that the resulting integral equations uncouple for this kind of obstacle. In them, the amount by which the components of particle displacement jump across the crack occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.

Scattering of plane, elastic waves by a plane crack of finite width

A formulation of elastodynamic diffraction problems for sinusoidally in time varying disturbances in a linearly elastic medium is presented. Starting with the elastodynamic reciprocity relation, an integral representation for the particle displacement is derived. In it, the particle displacement and the traction at the boundary of the obstacle occur. From the integral representation, an associated integral equation is obtained by letting the point of observation approach the boundary of the obstacle. The “obstacle” may be either a rigid body, a void, or a body with elastic properties differing from those of its environment, or a combination of these. The integral equation thus obtained is well-suited for numerical treatment, when obstacles up to a few wavelengths in maximum diameter are considered.

Scattering of elastic waves by elastically transparent obstacles (integral-equation method)

The diffraction of two-dimensional elastic waves by a cylindrical obstacle is investigated. In two-dimensional wave motions, the geometrical configuration as well as the field quantities involved are assumed to be independent of one of the Cartesian coordinates. As such, the two-dimensional theory can be considered as a special case of the three-dimensional one. Following this line of reasoning, the integral-equation formulation of two-dimensional elastodynamic diffraction problems is derived. For a number of configurations, the resulting integral equations are solved numerically. Also, numerical results pertaining to normalized power scattering characteristics and extinction cross-sections are presented.

Diffraction of time-harmonic elastic waves by a cylindrical obstacle

In the present paper, a radiation condition for time-harmonic elastic waves in a homogeneous, isotropic, perfectly elastic medium is presented. It is based upon certain physical considerations pertaining to the elastodynamic radiation generated by a source of finite extent. In it, the particle displacement on and the traction across a spherical surface around the source occur. Separate conditions are imposed on the radial and the angular parts of these quantities. It is shown that the resulting far-field radiation characteristics for the particle displacement are in accordance with the properties that could be expected from the physical considerations we have started with.

T. H. Tan

Papers

Scattering of plane, elastic waves by a plane crack of finite width

Scattering of elastic waves by elastically transparent obstacles (integral-equation method)

Diffraction of time-harmonic elastic waves by a cylindrical obstacle

Far-field radiation characteristics of elastic waves and the elastodynamic radiation condition