Geophysical data rarely shows any smoothness at any scale and this often makes an honest comparison with theoretical model output difficult since models often build in a variety of smoothness assumptions. 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; non-differentiable and even discontinuous models are therefore in order. We need methods of statistically analyzing and co-analyzing geophysical data, whether measured in situ or remotely sensed, as well as model output that are adapted to these characteristics. In this respect, multifractals (i.e., scale-invariant structures) currently provide a productive framework.

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 non-negative stationary field derived from the data, leading to a characterization of the intermittency in this (generally gradient-related) field. In contrast, the approach based on Structure Functions uses the nonstationary signal itself, seeking power laws for the statistical moments of absolute increments over arbitrarily large scales. This leads 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, respectively "roughness" and "sparseness," are illustrated and the associated analytical (differentiability and continuity) properties are discussed. As an example, the two techniques are applied to some 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 for describing geophysical data exhibiting scale-invariance with many applications including data intercomparison, (dynamical or stochastic) model and field retrieval validations.