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.

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.

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Underlying the diversity of life and the complexity of ecology is order that re ects the operation of fundamental physical and biological processes. Power laws describe empirical scaling relationships that are emergent quantitative features of biodiversity. These features are patterns of structure or dynamics that are self-similar or fractal-like over many orders of magnitude. Power laws allow extrapolation and prediction over a wide range of scales. Some appear to be universal, occurring in virtually all taxa of organisms and types of environments. They offer clues to underlying mechanisms that powerfully constrain biodiversity. We describe recent progress and future prospects for understanding the mechanisms that generate these power laws, and for explaining the diversity of species and complexity of ecosystems in terms of fundamental principles of physical and biological science.Keywords: biodiversity; ecology; fractal; power law; scaling; self-similarity BACKGROUNDThe Earth's surface and the living things that inhabit it are incredibly diverse. The Earth presents an abiotic template of geology, physical oceanography and limnology, and climate that varies on a scale from the largest oceans, continents, lakes and rivers to the tiniest microsites. Billions of individual organisms belonging to millions of species are distributed over the Earth. They interact with each other and the abiotic environment on time-scales from microseconds to millennia and on spatial scales from a few micrometres to the entire globe. Underlying this enormous physical and biological diversity, however, are emergent patterns of ecological organization that are precise, quantitative, and universal or nearly so. Examples include the latitudinal, elevational and other gradients of species diversity, the way that species are aggregated into genera and higher taxonomic categories, the body sizes and relative abundances of coexisting species in ecological communities, the way that species diversity changes with sample area, and the successional changes in productivity, biomass and species composition and diversity following disturbance (Williams 1964;MacArthur 1972;Brown 1995).These emergent general features of ecological systems provide powerful clues about the underlying mechanisms that constrain ecological complexity and regulate biodiversity. On the one hand, the emergent patterns represent the * Author for correspondence (jhbrown@unm.edu).One contribution of 11 to a special Theme Issue 'The biosphere as a complex adaptive system'. outcome of the fundamental law-like processes of physics, chemistry and biology. Many of these mechanisms are well understood. They include thermodynamics, conservation of mass and energy, atomic particles and chemical elements, chemical stoicheiometry, geological tectonics and erosion, laws of biological inheritance, evolution by natural selection, and many others. It is obvious that they must play a role in regulating biodiversity. On the other hand, it is far from clear how these fundamental processes act an...

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Following a brief review of relevant theoretical and empirical spatial results, a theory of space‐time rainfall, applicable to fields advecting without deformation of the coordinates, is presented and tested. In this theory, spatial rainfall fields are constructed from discrete multiplicative cascades of independent and identically distributed (iid) random variables called generators. An extension to space‐time assumes that these generators are iid stochastic processes indexed by time. This construction preserves the spatial structure of the cascades, while enabling it to evolve in response to a nonstationary large‐scale forcing, which is specified externally. The construction causes the time and space dimensions to have fundamentally different stochastic structures. The time dimension of the process has an evolutionary behavior that distinguishes between past and future, while the spatial dimensions have an isotropic stochastic structure. This anisotropy between time and space leads to the prediction of the breakdown of G. I. Taylor's hypothesis of fluid turbulence after a short time, as is observed empirically. General, nonparametric, predictions of the theory regarding the spatial scaling properties of two‐point temporal cross moments are developed and applied to a tracked rainfall field in a case study. These include the prediction of the empirically observed increase of correlation times as resolution decreases and the scaling of temporal cross moments, a new finding suggested by this theory.

We study the spatial random field of peak flows indexed by a channel network. Invafiance of the probability distributions of peak flows under translation on this indexing set defines statistical homogeneity in a network. It implies that floods can be indexed by the network magnitude, or equivalently the drainage area, which serves as a scale parameter. This definition generalizes to homogeneity of flows in a geographical region containing several networks which are not necessarily the subnetworks of a single network. The widely used quanfile regression method of the United States Geological Survey (USGS) provides one simple criterion to approximately designate homogeneous geographic regions. It is argued that the redefinition of homogeneity via the constancy of the coefficient of variation (CV) of floods implied by the index flood assumption is ad hoe. Invafiance of the probability distributions of peak flows under scale change is Used to develop the simple scaling and the multiscaling theories of regional floods in terms of their quamiles. The simple scaling theory predicts a constant CV and log-log lineafity between flood quantiles and drainage areas such that the slopes in these equations do not change with the probability of exceedance. Multiscaling theory of floods is developed to exhibit differences in floods between small and large basins. This theory shows that the CV for small basins increases, and for large basins it decreases, as area increases. Moreover, the quamiles do not obey log-log lineafity with respect to drainage areas. However, for large basins an approximate log-log lineafity between quantiles and drainage areas is shown to hold. The slopes in these equations decrease as p decreases; i.e., larger floods have smaller slopes than smaller floods. This approximation provides a theoretical interpretation of the results of the empirical quantile regression method in homogeneous regions where simple scaling or the index flood assumption does not hold. Recent results on physical interpretations of the scaling theories are also summarized here. A simple nonlinear method is developed to estimate the parameters in the multiscaling theory. This method and other features of the theory are illustrated using flood data from central Appalachia in the United States.

Abstract. A filter for separating base flow from streamflow in a river basin is derived from a mass balance equation for base flow from a hillside. Unlike other existing filters, this filter is physically based both during streamflow recession and rise, and each of the parameters in it has a well-defined physical meaning. Unlike other approaches to base flow separation, the filter is applied without any calibration or constraints. Yet, using two criteria as measures of filter performance, it performs as well as or better than two existing filters that require calibration. Using the same two criteria, weaknesses in the filter are identified and attributed mainly to a ratio of two parameters that are related to groundwater recharge and overland flow. This ratio is estimated using precipitation and streamflow data, and its estimate is conjectured to be inaccurate because of the poor spatial resolution of the precipitation data. Moreover, this ratio is assumed to be constant in time, yet, on physical grounds, it should be treated as a time-dependent quantity. IntroductionBase flow time series are needed to understand the spatial and temporal variability of runoff processes in river basins, to extrapolate runoff processes to ungauged locations [Gupta and Waymire, 1998], and, in practice, to manage water quantity and quality. However, there is no direct way to continuously measure base flow throughout a basin or processes that affect base flow such as overland flow, evapotranspiration, interflow, and groundwater recharge. Consequently, many approaches have been developed to estimate or separate base flow from streamflow continuously in time [e.g., see Wittenberg, 1999;Chapman, 1999;Birtles, 1978]. None of these approaches are physically based under all streamflow conditions and consist of only a few parameters. As a result, they rely heavily on calibration which masks the physics behind base flow estimation particularly during times when streamflow is rising. In this paper, a mathematical filter for continuous base flow separation is developed that is physically based under all streamflow conditions. It consists of four well-defined physical parameters, three of which can be estimated using both rainfall and streamflow data. Our filter is physically based both during streamflow recession and rise, and its physical basis requires that it be applied once and only forward in time to produce an estimate of base flow. By contrast, application of some other filters to streamflow data appears arbitrary [e.g., Nathan and McMahon, 1990; Chapman, 1991]. Unlike other filters, our filter is not founded on the assumption that base flow and overland flow are the low-and high-frequency components of streamflow, respectively [Spongberg, 2000]. Thus it accounts for the possibility that low-frequency variations exist in overland flow because of those in precipitation. The filter consists of four physical constants that must be evaluated before using it. Three of these are estimated from streamflow and precipitation data, and the filter...