Scale dependence of reflection and transmission coefficients

Abstract

Well logs show that heterogeneities occur at many different depth scales. This study examines the effects of these heterogeneities on the propagation of seismic waves, and specifically the dependence of reflection and transmission on the spatial scale content of the medium. Wavelet transformations are used to filter certain spatial scales from an acoustic sonic log. The scale‐filtered logs are used to construct layerstack models for which reflection and transmission seismograms are computed. The modified logs are also used to calculate frequency dependent reflection and transmission coefficients as functions of scale content. It is observed that features shorter than one‐fourth of the dominant wavelength have little effect on the reflection and transmission of seismic waves. Features larger than the dominant wavelength affect arrival times of individual packets within the wavetrain, but often these features hardly alter the overall appearance of individual wave packets. Reflection and transmission coda are primarily governed by heterogeneity at spatial scales similar to half the propagating wavelength. These scales appear to control the presence and shape of the events within the coda. The study also shows that the arrival times of packets at 1 kHz approach the theoretically expected value obtained from the harmonic velocity average, and the arrival times of packets below 1 Hz approach the theoretical value expected for the Backus average of the velocities.

Damage and seismic velocity structure of pulverized rocks near the San Andreas Fault

“…Thus, the grouping of profiles with respect to frequency content of first break wavelets suggests that the alluvial sediments are responsible for the damping and that the pulverized rocks of the outcrop have similar properties as those below the alluvial sediments. If the expected dispersion for heterogeneous rocks applies to the investigated site, the increase in velocity with depth derived from the survey may be underestimated due to the increase in wavelength with increasing travel distance [see also Imhof , ].…”

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

“…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.…”

Theory and seismic applications of the eigenimage discrete wavelet transform

“…implies that the resolvable models extracted from either well‐log or seismic data will be scale‐dependent (Li, Sacchi and Ulrych 1996; Wapenaar et al . 1997; Verhelst 2000; Song, Zhang and Ulrych 2000; Leary and Al‐Kindy 2002; Imhof 2003). Consequently, the question of scale dependence of seismic wavefields resulting in frequency‐dependent scattering cannot be ignored in seismic processing.…”

“…There are several ways of representing observed traces in terms of their temporal and spatial spectra, amongst which the discrete wavelet transform (DWT) is a very convenient tool for data processing and compression. The pioneering work of Daubechies (1992) has given rise to numerous successful geophysical applications of DWT, including noise attenuation (Deighan and Watts 1997), trace characterization (Grubb and Walden 1997), data compression (Bosman and Reiter 1993), phase identification (Anant and Dowla 1997; Bear and Pavlis 1997; Bear, Pavlis and Bokelmann 1999), instantaneous spectral analysis (Castagna, Sun and Siegfried 2003), upscaling or segmentation of well logs (Verhelst 2000; Imhof 2003), migration (Wapenaar et al . 1997; Song et al .…”