Sonic-log measurements provide detailed 1-D information on the distribution of elastic properties within the upper crystalline crust at scales from about one metre to several kilometres. 10 P-wave sonic logs from six upper-crustal drill sites in Europe and North America have been analysed for their second-order statistics. The penetrated lithological sequences comprise Archean volcanic sequences, Proterozoic mafic layered intrusions, and Precambrian to Phanerozoic gneisses and granites. Despite this variability in geological setting, tectonic history, and petrological composition, there are notable similarities between the various data sets: after removing a large-scale, deterministic component from the observed velocity-depth function, the residual velocity fluctuations of all data sets can be described by autocovariance functions corresponding to band-limited self-affine stochastic processes with quasi-Gaussian probability density functions. Depending on the maximum spatial wavelength present in the stochastic part of the data, the deterministic trend can be approximated either by a low-order polynomial best fit or by a moving-average of the original sonic-log data. The choice of the trend has a significant impact on the correlation length and on the standard deviation of the residual stochastic component, but does not affect the Hurst number. For trends defined by low-order polynomial best fits, correlation lengths were found to range from 60 to 160 m, whereas for trends defined by a moving average the correlation lengths are dominated by the upper cut-off wavenumber of the corresponding filter. Regardless of the trend removed, the autocovariance functions of all data sets are characterised by low Hurst numbers of around 0.1-0.2, or equivalently by power spectra decaying as ∽ 1/k. A possible explanation of this statistical uniformity is that sonic-log fluctuations are more sensitive to the physical state, in particular to the distribution of cracks, than to the petrological composition of the probed rock
