Multifractal characterizations of nonstationarity and intermittency in geophysical fields: Observed, retrieved, or simulated

Abstract

Geophysical data rarely show any smoothness at any scale, and this often 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 geophysical research. High levels of variability are excited over a large range of scales by the combined actions of external forcing and internal instability. At very small scales we expect geophysical fields to be smooth, 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 geophysical data, whether measured in situ, remotely sensed or even generated by a computer model, that are adapted to these characteristics. An important preliminary task is to define statistically stationary features in generally nonstationary signals. We first discuss a simple criterion for stationarity in finite data streams that exhibit power law energy spectra and then, guided by developments in turbulence studies, we advocate the use of two ways of analyzing the scale dependence of statistical information: singular measures and qth order structure functions. In nonstationary situations, the approach based on singular measures seeks power law behavior in integrals over all possible scales of a nonnegative stationary field derived from the data, leading to a characterization 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 increments 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 statistics which are largely complementary to each other. The geometrical manifestations of nonstationarity and intermittency, “roughness” and “sparseness”, respectively, are illustrated and the associated analytical (differentiability and continuity) properties are discussed. As an example, the two techniques are applied to a series of recent measurements of liquid water distributions inside marine stratocumulus decks; these are found to be multifractal over scales ranging from ≈60 m to ≈60 km. Finally, we define the “mean multifractal plane” and show it to be a simple yet comprehensive tool with many applications including data intercomparison, (dynamical or stochastic) model and retrieval validations.

Multi-fractal thermal characteristics of the southwestern GIN sea upper layer

“…1, we use I'HosptalÕs rule to define a straightforward measure of inhomogeneity in the sense of singular measure [4]:…”

“…Both parameters have geometrical interpretations as co-dimensions: information dimension and graph dimension [4]. The information dimension, represented by (1 À C 1 ), is a first-order estimate of sparseness of strong gradient distributed in the system.…”

“…10) in terms of C 1 and H 1 We find larger variation in H 1 (0.11-0.56) than in C 1 (0.08-0.10). Identification of ðH 1 ; C 1 Þ ranges helps to select realistic model to describe the field [4]. …”

Intermittent‐type temporal fluctuations in seismicity of the Irpinia (southern Italy) Region

“…The idea of intermittency qualifies the notion of inhomogeneity, in the sense that a geophysical signal is characterized by a spiky dynamics, with sudden and intense bursts of high frequency activity [Davis et al, 1994]. Intermittent-type behaviour has been observed in a wide range of experimental and numerical studies of dynamical systems [Seuront et al, 1996;Schertzer et al, 1998;Bruno et al, 19991. A seismic process can be characterized by a fluctuating behavior, that is temporal phases of low activity are interspersed between those with relatively large density of the events.…”

“…Intermittent-type behaviour has been observed in a wide range of experimental and numerical studies of dynamical systems [Seuront et al, 1996;Schertzer et al, 1998;Bruno et al, 19991. A seismic process can be characterized by a fluctuating behavior, that is temporal phases of low activity are interspersed between those with relatively large density of the events. This <<sparseness>> can be viewed as the geometrical manifestation of intermittency [Davis et al, 1994].…”