STOCHASTIC-ANALYSIS OF SONIC LOGS FROM THE UPPER CRYSTALLINE CRUST - METHODOLOGY

To relate local fluctuations observed in sonic logs to small-scale vel ocity fabric dong boreholes, both filtering effects and noise introduc ed by the logging procedure must be taken into account. Sonic log velo cities are represented as a time series consisting of a large-scale de terministic and a small-scale stochastic component. The deterministic trend, approximated by a low-order polynomial best-fit, contains infor mation on the average velocity structure, whereas the small-scale stoc hastic variations consist of noise plus in situ velocity variations co nvolved with the logging system response. The velocity fluctuations of the sonic data considered here are zero-mean and have quasi-Gaussian probability density functions. Therefore, they are well characterised by their second statistical moment, i.e. their autocovariance function . Tests on synthetic data indicate that the autocovariance function co rresponding to this data model may be used to extract information on t he second-order statistics of the in situ velocity variations along th e borehole and to constrain the level of white noise in sonic logs. Ig noring the presence of filtering effects and noise in sonic logs may r esult in seriously flawed estimates of the second-order statistics of the actual velocity structure. Assuming a von Karman autocovariance fu nction for the in situ velocity variations, this model provides a good match to the autocovariance functions of sonic log data from the Silj an Ring (Sweden) and Sudbury areas (Canada). Although differing signif icantly in their noise content these two data sets yield similar resul ts for the small-scale velocity structure, which is modelled as a band limited self-affine time series. For the Siljan Ring borehole we found a close relation between small-scale variations of the borehole diame ter as determined from caliper logs and the level of uncorrelated nois e present in the sonic log data.