River flows have been known to be scaling for over 40 years and scaling notions have developed rapidly since the 1980s. Using the framework of universal multifractals and time series of rainfall and river runoff for 30 French catchments (basin sizes of 40 km2 to 200 km2) from 1 day to 30 years, we quantify types and extent of the scaling regimes. For both flow and rain series, we observed a scale break at roughly 16 days, which we associate with the “synoptic maximum”; the time scale of structures of planetary spatial extent. For the two scaling regimes in both series, we estimate the universal multifractal parameters as well as the critical exponents associated with multifractal phase transitions. Using these exponents, we perform (causal) multifractal time series simulations and show how a simple (linear) scaling transfer function can be used to relate the low‐frequency rainfall series to the corresponding river flow series. The high‐frequency regime requires nonlinear transforms.

SUMMAR YIn Paper I, we showed how anisotropic scaling spectral (second-order) models of the magnetization (M) were realistic at both high-and intermediate-wavenumber regimes of the surface magnetic field (B). However, in order to produce full stochastic M and surface B models, we need assumptions about statistical moments other than second order. The usual approach is to assume quasi-Gaussian statistics so that all the statistical moments are scaling according to a single exponent. The corresponding fields are monofractal. All structures-both weak and strong-have the same unique fractal dimension, there are no strong anomalies and there are no intermittent transitions from one strata or region to another; such assumptions are quite unrealistic. Using seven surface B surveys, we show that the data are, on the contrary, multifractal, and we characterize their multifractal parameters in both the high-and intermediate-wavenumber regimes with the help of universal multifractal exponents. Using anisotropic (stratified) multifractal models, we deduce the M statistics and produce M and surface B simulations with all statistical exponents quite near to those of the observed surface B field; they are also visually realistic, showing anomalies at all scales. Finally, we analyse the horizontal anisotropy of the surface B fields and use this to infer the M statistics. This enables us to produce anisotropic 3-D M, B models with more realistic texture and morphology of structures. We conclude that both multifractality and scaling anisotropy are indispensable for realistic geophysical models.

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SUMMAR YScaling models of geofields attempt to capture the strong and wide-range variability ubiquitous in geosystems. Unfortunately, they are generally both isotropic (self-similar) and monofractal (non-intermittent, quasi-Gaussian). In this first paper of a two-paper series, we lift the first of these restrictions, arguing that anisotropic scaling is essential for taking into account the stratification of the Earth and its consequences. In particular, at horizontal scales below several thousand kilometres we model the thin or Curiedepth-limited crustal magnetization and the corresponding surface magnetic field (B) by using anisotropic scaling. We show that it generically gives rise to a new intermediate scaling ('red noise') surface B field regime quantitatively very close to that observed on two sets of regional surface B field surveys. This scaling is impossible to explain using standard self-similar models. Using these data as well as horizontal and vertical susceptibility data, we estimate the basic model parameters and show that that model is compatible with the available data. In Paper II we lift the monofractal restriction and perform multifractal analyses; we then extend the anisotropic scaling model to include multifractal B and magnetization fields.

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