## 1. Introduction

[2] In view of the space-time scales at which modeling and understanding are sought, many problems of surface water hydrology are not amenable to the classic deterministic methods of physics and mathematics. A prominent example being the PUB problem, an acronym coined for Prediction from Ungauged Basins. Broadly speaking one seeks to predict the discharge from river basins using information gleaned from geographic (topographic) maps and local climate patterns, but in the absence of streamflow gauges and other such network measuring devices [see *Gupta and Waymire*, 1997, 1998, and references therein].

[3] The salient fact is that many of the problems for surface water hydrology fall within John von Neumann's “medium scale range” for fluid motions. More specifically, in contemplating the future impact of computational technology at a historical Conference on the Application of Numerical Integration Techniques to the Problem of the General Circulation held in the mid-1950s, *von Neumann* [1960] called attention to a “medium-range” scale which is too long from the point of view of an initial value problem, and too short for a steady state theory to be applicable. In particular, this is a scale range over which statistical fluctuations cannot be ignored.

[4] It is not surprising that stochastic models would find such widespread use as found in hydrology during the past fifty years. In the case of rainfall, many of the early attempts toward classes of models with a hierarchical structure fall within the broad framework introduced by *LeCam* [1961]. This broad class of random fields continues to provide a useful framework for modeling and data analysis in hydrology. However, there are problems such as the PUB problem for which such models cannot be adequately calibrated. These are problems for which one may attempt to identify and exploit certain statistical multiscaling structure, self-similarity, and fractal geometry; see *Gupta* [2004], *Rodriquez-Iturbe et al.* [1998b], *Reggiani et al.* [2001], and *Burd et al.* [2000] for recent general formulations and illustrative results in the context of river basin hydrology.

[5] Continued progress in connection with the PUB problem, for example, will depend on further research on both landform and atmospheric processes, and their physical and biological interactions. While outside the scope of this article, the development of the fractal geometry of river networks and flows illustrates a related facet of the overarching direction of such research; for example, see *Rodriguez-Iturbe and Rinaldo* [1997] and *Veitzer et al.* [2003] for general observations and specific results. For rainfall, one may cite the work by *Lovejoy* [1982] and *Lovejoy and Mandelbrot* [1985] as an early identification of scaling structure in rain fields that has continued to be explored and refined [see *Lovejoy and Schertzer*, 1985, 1990; *Schertzer et al.*, 1991; *Gupta and Waymire*, 1993; *Over and Gupta*, 1994; *Rodriguez-Iturbe et al.*, 1998a]. Also see *Catrakis and Domotakis* [1998] for new ideas in turbulence related to the main topic of this paper. In particular, it is here that rainfall and turbulence theory have found a common home as highly singular and intermittent data structures.

[6] This paper has an intentionally narrow technical focus and is far from comprehensive. Certainly exciting new ideas not mentioned here continue to be explored in the literature. The main objective of this paper is simply to highlight a mathematical notion which has proven to be powerful for analysis of singular structures and, with a broader aim, toward engaging the physical scientists in further developing its applications to data structures of the type found in rain and turbulence. To this end, in section 2 we very briefly recall the multiplicative cascade model. This is followed by section 3 in which we focus on the role of “size biasing” from a mathematical perspective. In section 4 its significance is discussed within statistical turbulence theory.

[7] As remarked by *LeCam* [1961, p. 165] in connection with rainfall models: “A true model should take into account the laws of fluid mechanics and thermodynamics.” The same can of course be said for turbulence models, where the basic physics is thought to be captured by the Navier-Stokes equations. However in neither case are the problems solved. In the final section we describe some recent mathematical efforts to analyze Navier-Stokes equations by means of certain natural multiplicative cascades recently uncovered in Fourier frequency space. For simplicity of the mathematical exposition, the main ideas behind this representation are explained in terms of Burgers equation. Interested readers are referred to *Waymire* [2005, and references therein] for the corresponding exposition in the context of three-dimensional incompressible Navier-Stokes equations. However it must be emphasized that neither the mathematical theory nor the physics of these ‘stochastic cascade representations’ to Navier-Stokes in Fourier frequency space is sufficiently well developed to claim any connection with the multiplicative cascade model in physical space; also see *Ossiander* [2005] for recent efforts to develop a corresponding multiplicative representation in physical space, but subject to the same caveat.