The Transition Between the Scale Domains of Ray and Effective Medium Theory and Anisotropy: Numerical Models

Abstract

The anisotropy of a periodically layered isotropic medium is numerically modeled in order to study the effect of the scale of heterogeneity on seismic observations. An important motivation is to delineate the wavelength ranges over which a pulse propagating obliquely through the structure will be described by either ray (short wavelength) or effective medium (long wavelength) theory. The same band-limited pulse is propagated obliquely at a variety of incidence angles through a compositionally uniform layered structure as a function of the layer thicknesses. The resulting seismograms display similar behavior to that encountered for normal incidence including the effects of stop-and pass-bands. Velocities determined from time picks on these seismograms show a large difference in velocities between the long and short wavelength limits as has been previously demonstrated for normal incidence propagation. The bulk of the transition between these two limits is independent of incidence angle and occurs when the ratio between the wavelength and the layering thickness is near a value of 10. Two more geologically reasonable models show that these effects are diminished with smaller contrasts between the layers.

Seismic properties and anisotropy of the continental crust: Predictions based on mineral texture and rock microstructure

“…Backus [1962] also proposed a long wavelength (λ lim ) limit where his average could be considered as an effective homogenous media with λ lim > average layer thickness (h ave ). More recent work [e.g., Melia and Carlson, 1984;Marion et al, 1994;Carcione et al, 1991;Lui and Schmitt, 2006] introduce a wide range of parameters, including layer thickness, anisotropy of layers, wavelength, and incidence angle to layering. Both numerical modeling and experimental measurements found that ratio R = λ lim /h ave should be at least 10, to be well described by an effective media.…”

Elastic anisotropy of Tambo gneiss from Promontogno, Switzerland: a comparison of crystal orientation and microstructure-based modeling and experimental measurements

“…The 'threshold' ratio of wavelength to the scale of heterogeneities of the material (meaning not only grains of different minerals composing polyphase rock, but also grains with different orientations), over which the application of the effective medium theory is valid, clearly depends on the acoustic impedance contrasts and proportions of materials (e.g. Liu & Schmitt 2006). It is difficult to determine this value for a general 3-D multiphase anisotropic bulk material.…”

7. Seismic, rock physics, spatial models, and their integration in reservoir geophysics

“…The kinematics of the seismic wave is similar over certain spatial volumes re lated to the wavelength, thus properties of fine heterogeneities within this volume only affect the motion through their spatial average. The ef fect of small heterogeneities on the reflectivity and propagation velocities of limited bandwidth seismic waves has been the object of numerous studies from ray theory to effective medium theory (e.g., Marion et al, 1994;Mukerji et al, 1995) and finite frequency wave theory (e.g., Cerveny and Soares, 1992;Shapiro and Hubral, 1996;Spetzler and Snieder, 2004;Liu and Schmitt, 2006); another section will extend this discussion with common averages for upscaling elastic properties in stratified medium. Hence resolution of layers with thickness much small er than that corresponding to the seismic wave is not achievable with reflection seismic inver sion.…”

“…This makes V rt > V emt . The regimes spanning ray theory and effective medium limits, and the scaledependence of wave propagation have been studied theoretically and experimentally by various authors (e.g., Helbig, 1984;Melia and Carlson, 1984;Banik et al, 1985;Carcione et al, 1991;Kerner, 1992;Marion et al, 1994;Mukerji et al, 1995;Shapiro and Hubral, 1996;Imhof, 2003;Liu and Schmitt, 2006). Such scaledepen dent effects on travel times and amplitudes are important when integrating core (~1 Mhz), log (~20 kHz), and seismic (~100 Hz) data.…”