Two common properties of empirical moments shared by spatial rainfall, river flows, and turbulent velocities are identified: namely, the log‐log linearity of moments with spatial scale and the concavity of corresponding slopes with respect to the order of the moments. A general class of continuous multiplicative processes is constructed to provide a theoretical framework for these observations. Specifically, the class of log‐Levy‐stable processes, which includes the lognormal as a special case, is analyzed. This analysis builds on some mathematical results for simple scaling processes. The general class of multiplicative processes is shown to be characterized by an invariance property of their probability distributions with respect to rescaling by a positive random function of the scale parameter. It is referred to as (strict sense) multiscaling. This theory provides a foundation for studying spatial variability in a variety of hydrologic processes across a broad range of scales.

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The channel network and the overland flow regions in a river basin satisfy Horton's empirict,• morphologic laws when ordered according to the Strahler ordering scheme. This setting is presently employed in a kinetic theoretic framework for obtaining an explicit mathematical representation for the instantaneous unit hydrograph (iuh) at the basin outlet. Two examples are developed which lead to explicit formulae for the iuh. These examples are formally analogous to the solutions that would result if a basin is represented in terms of linear reservoirs and channels, respectively, in series and in parallel. However, this analogy is only formal, and it does not carry through physically. All but one of the parameters appearing in the iuh formulae are obtained in terms of Horton's bifurcation ratio, stream length ratio, and stream area ratio. The one unknown parameter is obtained through specifying the basin mean lag time independently. Three basins from Illinois are selected to check the theoretical results with the observed direct surface runoff hydrographs. The theory provided excellent agreement for two basins with areas of the order of 1100 mi 2 (1770 km 2) but underestimates the peak flow for the smaller basin with 300-mi 2 (483-km 2) area. This relative lack of agreement for the smaller basin may be used to question the validity of the linearity assumption in the rainfall runoff transformation which is embedded in the above development.

It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a loginfinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems. PACS numbers: 47.27.Gs, 47.27.TeThe statistics of fully developed turbulence exhibit certain universal features as a result of strong nonlinear interactions. One set of intriguing quantities characterizing universal behavior of turbulent flows is a set of scaling exponents for two-point correlation functions, e.g. , g"of the velocity structure functions defined by an expressionset of quantities is an exponent~p for pth-order moment of locally averaged energy dissipation e over a ball of size 4: (et") -8". The range of the length scale 8 for the above power-law behavior to be valid is called an inertial range. Kolmogorov's refined similarity hypothesis[1] provides a relation between these two sets of quantities: g"= p/3 + r"t3. This Letter addresses a predictive model [2] of g"and r". Note that g2 characterizes the scaling for the kinetic energy fluctuations and is directly related to the exponent for the kinetic energy spectrum E(k) -k:n =1+ j2.During the past half century since Kolmogorov's 1941 seminal work [3] that predicts r"= 0 and g"= p/3, there has been a continual effort to experimentally determine the values of g"or r". There is now strong evidence [4 -7] that g"W p/3, which is usually referred to as the intermittency effects or anomalous scaling exponents. Many theoretical models have been proposed to address this phenomenon. Some approaches start with an ansatz of probability density function (PDF) of et such as the log-normal model [2] or a log-stable model [8], for example. Others [9 -12] propose discrete random multiplicative processes (RMP) modeling the energy cascade. The stochastic process that generates the energy cascade is furnished by random multiplicative factors Wt, t, relating the fluctuations of e~at two different length scales 8]and Zz. et, = Wq, t, et, . By construction, the multiplicative factor is independent of ez, . The probability distribution of Wt, t, determines r", since it is required that log(Wt", t, )/ log(82/Zi) = r". Existing cascade models of random multiplicative processes [10 -12] are obtained by making an ad hoc ansatz for the PDF of lVt, t, being composed of a certain number of discrete atoms described by one or more adjustable parameters. The parameters in the models are difficult to determine by plausible phys...

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