## 1. Introduction

[2] Analysis and modeling of space-time rainfall [e.g., *Schertzer and Lovejoy*, 1987; *Olsson et al.*, 1993; *Marsan et al.*, 1996; *Veneziano et al.*, 1996; *Cârsteanu and Foufoula-Georgiou*, 1996; *Venugopal and Foufoula-Georgiou*, 1996; *Harris et al.*, 1998; *Deidda et al.*, 1999; *Menabde and Sivapalan*, 2000] (see also *Foufoula-Georgiou* [1997] for a review) have been influenced by statistical theories of turbulence aiming at understanding the partition of energy at different scales. A breakthrough in the statistical theory of turbulence came about with the observation that the support of the transfer of energy is spatially intermittent, i.e., that the energy associated with the small scales in a turbulent flow is not distributed uniformly in space [*Meneveau and Sreenivasan*, 1991; *Frisch*, 1995]. This observation prompted a shift from the global Fourier-based analysis of Kolmogorov to a local analysis aimed at characterizing the nature of the very abrupt variations in velocity fluctuations. This led to the so-called multifractal formalism introduced by *Parisi and Frisch* [1985]. Loosely speaking, this formalism relates the scale dependence of the statistical moments of turbulent velocity fluctuations to the intermittent and multifractal nature of the points at which abrupt local increases of velocities exist.

[3] More specifically, *Parisi and Frisch* [1985] computed from experimental data, as a function of displacement ℓ, the average value of the *q*th power of the change in the turbulent velocity, that is, the average value of ∣*v*(*x* + ℓ) − *v*(*x*)∣^{q} and found that it varied as power law ℓ^{ζ(q)}, where the exponent ζ(*q*) depended nonlinearly on *q*. They interpreted the nonlinear behavior of ζ(*q*) as an indication (or direct consequence) of the existence of spatial heterogeneity in the local regularity of the velocity field. Namely, by calling *D*(*h*) the Hausdorff dimension of the set of points for which the increase in velocity acts as ℓ^{h} (points of singularity *h*), they showed that the contribution of these “singularities of order *h*” to the average value of ∣*v*(*x* + ℓ) − *v*(*x*)∣^{q} is of the order of magnitude of the product ℓ^{qh} · ℓ^{1−D(h)}; the second factor is the probability that a circle of radius ℓ intersects a fractal set of dimension *D*(*h*). This forms the basis of the multifractal formalism, as one notes that when ℓ tends to 0, the dominant term in the above expression is the one with the smallest possible exponent, giving rise to the so-called Legendre transform ζ(*q*) = [*qh* +1 − *D*(*h*)]. The nonlinearity of ζ(*q*) therefore indicates that the velocity fluctuations display multifractal scaling as characterized by a suite of *h* values given by *h*(*q*) = ∂ζ(*q*)/∂*q*.

[4] Having understood in a heuristic manner the intimate connection between ζ(*q*) and *D*(*h*), we point out that the traditional multifractal analysis of turbulence [*Parisi and Frisch*, 1985; *Frisch*, 1995] or other geophysical signals, including rainfall, starts by estimating ζ(*q*) via a moment analysis (power law decay of ∣*v*(*x* + ℓ) − *v*(*x*)∣^{q} with ℓ) and then estimating *D*(*h*) from the Legendre transform. It is noted however, that to conclusively infer the nonlinearity of ζ(*q*) versus *q*, higher-order moments are typically needed which presents a problem for small sample sizes, or when the range of scaling is short. In addition, there are other shortcomings of this traditional methodology [*Muzy et al.*, 1993, 1994], such as the inability to access the whole range of singularity exponents, as will be discussed in detail in the following section.

[5] Motivated by recent advances in multifractal analysis of turbulence velocity signals [e.g., *Arneodo et al.*, 1998c, 1999; *Delour et al.*, 2001], this paper proposes to use an alternative methodology for diagnosing and estimating the multifractal structure of rainfall. The centerpiece of this estimation is access to the whole spectrum of singularities *D*(*h*) which fully characterizes the intermittent structure of rainfall fluctuations and therefore the scaling of their statistical moments, or the scaling of the whole probability density function (PDF) using the so-called “propagator” method [*Castaing et al.*, 1990; *Arneodo et al.*, 1997b, 1998c]. As will be formally presented later, a natural tool for unraveling local singularities of a signal is the wavelet transform [e.g., *Muzy et al.*, 1994; *Mallat*, 1998] which acts as a microscope and by zooming locally at the signal can characterize the nature of its abrupt local fluctuations. A multifractal formalism based on wavelets has been well established in the turbulence literature [e.g., *Muzy et al.*, 1991; *Bacry et al.*, 1993; *Muzy et al.*, 1994; *Arneodo et al.*, 1995a], but has not been adequately explored yet for geophysical signals. One of the goals of this paper is to introduce the wavelet-based multifractal formalism [see also *Davis et al.*, 1994] and the related magnitude cumulant analysis method [*Delour et al.*, 2001] in a pedagogical way such that it can motivate further exploration of these powerful methodologies in geophysics. In particular, we will emphasize the investigation of the correlations of the logarithms of the wavelet coefficients (the so-called “magnitude”) as a powerful test of the existence of a possible underlying multiplicative structure [*Arneodo et al.*, 1998a, 1998b]. The second goal is to revisit the multiscaling analysis of high-resolution temporal rainfall and offer new insights on its multifractal structure.

[6] The paper is structured as follows. In section 2 we present a brief but self-contained review of the wavelet-based multifractal formalism. We start with the continuous wavelet transform and show how it can be used to extract singularities of a signal. We then demonstrate that by concentrating on critically selected points only (the maxima lines pointing to singularities), i.e., working with the Wavelet Transform Modulus Maxima (WTMM) coefficients, results in a more robust estimate of the singularity spectrum *D*(*h*). We also note that by using the WTMM coefficients instead of the CWT coefficients, we have access to the whole range of singularities including the decaying part of the *D*(*h*) curve, which can only be resolved by computing the scaling of negative moments (not possible in the typical structure function or CWT multifractal analysis as the PDFs of those fluctuations are centered around zero). In section 3 we introduce the one- and two-point cumulant analysis method and explain its advantages versus the CWT and WTMM standard multifractal analysis. In section 4 we present the results of applying the proposed methodology to four high-resolution storms sampled every 5 s over the midwestern U.S. For illustrative purposes we present and compare the results of the structure function, CWT, WTMM and cumulant analysis on one data set. We pay special attention to using wavelets of increasing order, i.e., increasing number of vanishing moments, to (1) properly remove nonstationarities in the signal and define (instead of impose) the “fluctuations” whose scaling properties we characterize and (2) to have confidence that the scaling behavior and scaling exponent estimates do not depend on the chosen wavelet. Section 5 presents the results of a timescale magnitude correlation analysis of rainfall fluctuations and poses the hypothesis of a local (within storm pulse) multiplicative cascade. This hypothesis is further tested via a numerical experiment. Section 6 presents the theory of probability density function rescaling via the so-called “propagator” approach and demonstrates its application to the rainfall intensity series. A summary of the inferences about the multiscaling structure of temporal rainfall and concluding remarks are made in section 7.