Localization and backscattering spectrum of seismic waves in stratified lithology

Abstract

Using four sonic well logs from diverse geological environments, we analyze the statistics of lithological layers relevant to seismic wave propagation. The autocorrelation functions are found to be well approximated by exponentials with correlation lengths generally in the 1.5 to 3 m range. We use localization theory to calculate the apparent attenuation caused by random scattering as a function of frequency. This attenuation has a peak at 50 to 150 Hz with a corresponding localization length of 1.6 to 4.8 km (1 to 3 mi) and an apparent Q of 120 to 450. At seismic frequencies (20 Hz) the attenuation is smaller, with localization lengths in the 16 to 32 km (10 to 20 mi) range and an apparent Q of 300 to 700. These values of apparent Q are consistent with observations of previous authors who used well‐log calculations. Using finite differences we compute synthetic traces for a 20-Hz pulse with multiple backscattering from the lithology given by each of the well logs. The power spectral density of the trace is computed for different time gates and compared with theoretical predictions. The nonwhite, [Formula: see text] dependence of the spectrum as predicted by theory is confirmed. Furthermore, when plotted as a function of the scaled variable inversely proportional to the localization length, the average of the four power spectra exhibits excellent agreement with the dimensionless scaling function derived using localization theory.