Stochastic inversion for scaling geology

SUMMARY We have examined acoustic, density resistivity, gamma-ray and neutron logs from a number of boreholes in both sedimentary and igneous sequences. We show that the power spectra of these geophysical variables obey a scaling law, that is, the power spectra are proportional to some power of the frequency. In general, the power spectra are approximately inversely proportional to the frequency. This suggests that frequency-dependent noise models are more appropriate for modelling the spatial variation of geophysical parameters than the widely assumed white noise (frequency-independent) model and should be incorporated into the inversion for these variables, through a priori parameter covariances. the covariance of a scaling variable is simply obtained from the power spectrum. It is independent of the absolute value of the lag, that is, there is no preferred length scale, but is dependent upon the sample length. We demonstrate the advantage of scaling noise covariances with the inversion of DC resistivity sounding data both with the exact covariance and with the approximate case of inverse proportionality. Adoption of a frequency-dependent noise model leads to a reduction in the a posteriori parameter variances and to solutions exhibiting a degree of smoothness commensurate with measured spatial variations of these parameters.