To relate local fluctuations observed in sonic logs to small-scale
velocity fabric dong boreholes, both filtering effects and noise
introduced by the logging procedure must be taken into account. Sonic
log velocities are represented as a time series consisting of a large-scale
deterministic and a small-scale stochastic component. The deterministic
trend, approximated by a low-order polynomial best-fit, contains
information on the average velocity structure, whereas the small-scale
stochastic variations consist of noise plus in situ velocity variations
convolved 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
corresponding to this data model may be used to extract information
on the second-order statistics of the in situ velocity variations
along the borehole and to constrain the level of white noise in sonic
logs. Ignoring the presence of filtering effects and noise in sonic
logs may result in seriously flawed estimates of the second-order
statistics of the actual velocity structure. Assuming a von Karman
autocovariance function for the in situ velocity variations, this
model provides a good match to the autocovariance functions of sonic
log data from the Siljan Ring (Sweden) and Sudbury areas (Canada).
Although differing significantly in their noise content these two
data sets yield similar results for the small-scale velocity structure,
which is modelled as a bandlimited self-affine time series. For the
Siljan Ring borehole we found a close relation between small-scale
variations of the borehole diameter as determined from caliper logs
and the level of uncorrelated noise present in the sonic log data.