The time-domain behavior of power-law noises



The power spectra of many geophysical phenomena are well approximated by a power-law dependence on frequency or wavenumber. I derive a simple expression for the root-mean square variability of a process with such a spectrum over an interval of time or space. The resulting expression yields the power-law time dependence characteristic of fractal processes, but can be generalized to give the temporal variability for more general spectral behaviors. The method is applied to spectra of crustal strain (to show what size of strain events can be detected over periods of months to seconds) and of sea level (to show the difficulty of extracting long-term rates from short records).