Excitation of Geophysical Systems with Fractal Flicker Noise

Abstract

Geophysicists often model their measurements, derived from natural processes, as the linear superposition of a simple rational system function and a purely random excitation process. For many geophysical processes, the assumption of linearity for its deterministic component is sufficient but the assumption of a purely random excitation often and easily leads to a misidentification of the system function. Many geophysical systems are excited by stochastic processes which appear to be stationary even on geological time scales but which possess a preponderance of long period components. Selfsimilar, fractal stochastic processes form a class of possible geophysical excitations having “power spectrum” of the form 1/ |f |^k. Of this class, flicker-noise processes, for which k = 1 exist, on the boundary between the stationary and evolutionary subsets. No fractal stationary random excitation can provide for greater weighting of long period components.

The Chandler wobble of the earth’s rotation axis can be essentially described as a single-pole linear system. The multitude of natural forces which contribute to its excitation combine as a stochastic process which is heavily weighted in long periods. Because of its basic importance in astronomy, navigation, time-keeping, etc., the wobble has been carefully measured since the turn of this century. Recent advances in geodetic and astronometric technology has provided a reliable, homogeneous data set which can be directly decomposed into a linear, deterministic wobble function and stochastic excitation. The use of the flicker-noise excitation model allows for the direct identification of the theoretically-simple, single-pole resonance. A subset of the pole position record, obtained from the Bureau International de l’Heure in Paris is analysed in terms of a modified autoregressive data model comprising an all-pole system function excited by a minimum-power, flicker-noise process.