Multifractal scaling behavior of long-term records of daily runoff time series in 32 subwatersheds covering a wide range of sizes was examined. These subwatersheds were associated with four agricultural watersheds with different climates and topography. The empirical moment scaling curves obtained using the trace moment method showed that the runoff time series exhibited a multifractal behavior, which was valid over a time scale range from one day to about three years. The multifractal scaling of the runoff time series was well described by the Universal Multifractal Model. The spectral analysis (β

INTRODUCTION

The concepts of fractal scaling have proven to be useful in describing scale invariance in the statistical distribution of observations in space or time of some natural processes (Mandelbrot, 1983; Olsson et al., 1992; Prokoph, 1999). In the early analyses, a single scaling exponent was used to describe the behavior of the moments of the statistical distribution at different time scales. This approach was in effect an extension of concepts associated with scaling of geometrical sets using single scaling exponents and can be regarded as the monofractal approach to describe fractal behavior. However, recent studies have indicated that multiple scaling exponents were needed to describe statistical scaling behavior in many natural time series. Analyses based on multiple scaling exponents can be regarded as the multifractal approach. Indeed it is now acknowledged that multifractal analyses provide a more general description of natural time series, and monofractals are indeed special cases (meaning that the multiple scaling exponents are the same) for continuous or discrete multifractals (Lovejoy et al., 1987; Lovejoy and Schertzer, 1990; Sivakumar, 2001).

A phenomenological model frequently used to mimic complex systems exhibiting multifractal behavior is the multiplicative cascade, which was initially used to describe observed turbulent fluid flows (Schertzer and Lovejoy, 1987). Multiplicative cascade models have become increasingly popular in modeling hydrological time series (e.g., rainfall) in recent years (Harris et al, 1996; Over and Gupta, 1996; Olsson and Niemczynowicz, 1996; Schmitt et al., 1998; de Lima and Grasman, 1999; Olsson et al., 1999). These models successfully reproduced the structures and patterns observed in real hydrological processes and their statistical properties. Examination of a long term hydrological record over time would reveal fewer large events interspersed among abundant small events; or spatially, clusters of high rainfall intensity occur in smaller areas that are embedded within clusters of lower intensity in a large mesoscale area (Gupta and Waymire, 1990).

The multiplicative cascade model generates fields with infinite hierarchies and associated dimensions. It is therefore naturally adapted to reproduce singularities (events or occurrences) of extreme orders and to generically exhibit asymptotic decays of such extreme events. This property is very useful in rainstorm or flood analysis. Flood magnitude can be related to network size or equivalently to drainage area, which serves as a fractal scaling variable. However, a critical drainage area exists at which the scaling behavior changes (Gupta et al., 1994).

Many reports exist on the application of multifractal concepts to rainfall time series (Lovejoy et al., 1987; Lovejoy and Schertzer, 1990; Hubert et al., 1993; Deidda et al., 1999; de Lima and Grasman, 1999). However, no extensive body of knowledge as yet exists on application of the multifractal approach to investigate scaling behavior of runoff processes in watersheds. There are no such applications to runoff in agricultural watersheds. Rainfall is the principal input for streamflow generation, and it is not unreasonable to assume that time series of rainfall generated runoff over a given area would inherit the multifractal nature of rainfall patterns over that area. On the other hand, runoff is influenced by other geomorphological factors that would tend to "smooth" the fluctuations of the input rain series (Tessier et al., 1996). Although there is little literature on multifractal analysis of river flow/runoff data compared to rainfall data, there are indications that increasing attention is being directed to investigation of the multifractal patterns of streamflow data (Gupta et al., 1994, Tessier et al, 1996, Pandey et al, 1998; Labat et al, 2002). Tessier et al (1996) applied multifractal analysis to streamflow data from 30 small basins in France; the results indicated that the time series of stream runoff were well modeled by multifractals. Other studies on river flows supported this conclusion (Pandey et al., 1998).

Geophysical and geographical systems are considered to be nonlinear dynamic systems characterized by extreme spatial and temporal variability spanning wide ranges of scales. An important property of multifractals is that their extremes are power law functions of their space time resolutions, and therefore these functions can be readily used to disaggregate the properties of river runoff time series from coarser to finer time scales. It was found that runoff generated by different mechanisms had different scaling patterns. Floods mostly generated by snowmelt runoff manifest themselves as monofractal series, while floods mostly produced by rainfall are multifractal (Gupta and Dawdy, 1995). However, Labat et al. (2002) found that basins of different internal structures had similar multifractal parameters when studying the scaling properties of karstic watersheds.