The time-domain behavior of power-law noises. [of many geophysical phenomena]

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).

[1]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[2]  T. Barnett.  Recent changes in sea level and their possible causes , 1990 .

[3]  F. Wyatt.  Measurements of coseismic deformation in southern California: 1972–1982 , 1988 .

[4]  Stephen R. Brown,et al.  Roughness of natural fault surfaces , 1987 .

[5]  W. Munk,et al.  Tidal spectroscopy and prediction , 1966, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[6]  L. E. Gilbert.  Are topographic data sets fractal? , 1989 .

[7]  F. Snodgrass.  Precision Digital Tide Gauge: Vibrating wire pressure transducers provide sea level measurements to the nearest one-tenth millimeter. , 1964, Science.

[8]  J. Rutman.  Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress , 1978, Proceedings of the IEEE.

[9]  W. Press.  Flicker noises in astronomy and elsewhere. , 1978 .

[10]  W. Munk,et al.  Propagation of ocean swell across the Pacific , 1966, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  M. S. Keshner.  1/f noise , 1982, Proceedings of the IEEE.

[12]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .

[13]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[14]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[15]  A. Malinverno.  Testing linear models of sea-floor topography , 1989 .

[16]  Chak Ming Chie,et al.  Theory of oscillator instability based upon structure functions , 1976, Proceedings of the IEEE.

[17]  Frank K. Wyatt,et al.  Shallow borehole tilt: A reprise , 1988 .

[18]  W. Sturges.  Large-Scale Coherence of Sea Level at Very Low Frequencies , 1987 .