MULTIFRACTAL CHARACTERIZATIONS OF NONSTATIONARITY AND INTERMITTENCY IN GEOPHYSICAL FIELDS - OBSERVED, RETRIEVED, OR SIMULATED

Geophysical data rarely show any smoothness at any scale, and this oft en makes comparison with theoretical model output difficult. However, highly fluctuating signals and fractal structures are typical of open dissipative systems with nonlinear dynamics, the focus of most geophys ical research. High levels of variability are excited over a large ran ge of scales by the combined actions of external forcing and internal instability. At very small scales we expect geophysical fields to be s mooth, but these are rarely resolved with available instrumentation or simulation tools; nondifferentiable and even discontinuous models are therefore in order. We need methods of statistically analyzing geophy sical data, whether measured in situ, remotely sensed or even generate d by a computer model, that are adapted to these characteristics. An i mportant preliminary task is to define statistically stationary featur es in generally nonstationary signals. We first discuss a simple crite rion for stationarity in finite data streams that exhibit power law en ergy spectra and then, guided by developments in turbulence studies, w e advocate the use of two ways of analyzing the scale dependence of st atistical information: singular measures and qth order structure funct ions. In nonstationary situations, the approach based on singular meas ures seeks power law behavior in integrals over all possible scales of a nonnegative stationary field derived from the data, leading to a ch aracterization of the intermittency in this (gradient-related) field. In contrast, the approach based on structure functions uses the signal itself, seeking power laws for the statistical moments of absolute in crements over arbitrarily large scales, leading to a characterization of the prevailing nonstationarity in both quantitative and qualitative terms. We explain graphically, step by step, both multifractal statis tics which are largely complementary to each other. The geometrical ma nifestations of nonstationarity and intermittency, ''roughness'' and ' 'sparseness'', respectively, are illustrated and the associated analyt ical (differentiability and continuity) properties are discussed. As a n example, the two techniques are applied to a series of recent measur ements of liquid water distributions inside marine stratocumulus decks ; these are found to be multifractal over scales ranging from almost-e qual-to 60 m to almost-equal-to 60 km. Finally, we define the ''mean m ultifractal plane'' and show it to be a simple yet comprehensive tool with many applications including data intercomparison, (dynamical or s tochastic) model and retrieval validations.