JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. D12, PAGES 23,265-23,274, DECEMBER 20, 1993 Fractal Analysisof High-Resolution Rainfall Time Series JONAS OLSSON, JANUSZ NIEMCZYNOWICZ, AND RONNY BERNDTSSON Department of WaterResources Engineering,Lund Instituteof Technology, Lund University, Lund, Sweden Two-year series of 1-min rainfall intensities observed by rain gages at six different points are analyzed to obtain inibrmation about the fractal behavior of the rainfall distribution in time. First, the rainfall time series are investigated using a monodimensional fractal approach (simple scaling)by calculating the box and correlation dimensions, respectively. The results indicate scaling butwith different dimensions for different time aggregation periods. The time periods where changes in dimension occurcan be related to average rainfall eventdurations andaverage dry periodlengths. Also, the dimension is shown to be a decreasing function of the rainfall intensity level. This suggests a multidimensional fractal behavior(multiscaling), and to test this hypothesis, the probability distribution/multiple scaling method was appliedto the time series. The results confirm that the investigated rainfall time series display a multidimensional fractal behavior, at least within a significant part of the studied timescales, whichindicates that the rainfallprocess can be described by a multiplicative cascade process. 1. INTRODUCTION The question whether therainfallprocess exhibits properties that are independent of scale, i.e., the existence of scale invariance or scaling, and the physical reasons for this remain a widely discussed topic among hydrologists, meteorologists, and mathematicians [e.g., Lovejoyand Mandelbrot, 1985; Waymire, 1985; Kedem and Chiu, 1987; Zawadzki, 1987; Lovejoy and Schertzer, 1991]. Early studiesof scaling in atmospheric processes as well as many later studies were based on pure geometrical resemblance between difl•rent scales, i.e., self-similarity [e.g., Lovejoy, 1982]. The motivation for these studies was to calculate the fractal dimension of the process, i.e., a scaling parameter constituting a direct link between statisticalproperties of the process at all scales. The value of the fractal dimension was obtained by analyzing different geometrical and statistical properties of clouds and rainfall fields in time and space. A rainfall model based on self-similarity using a single fractal dimension (monodimensional fractalbehavior)was presented by Lovefiry and Mandelbrot[1985]. However, it was soonrealized that this approach was insufficient to describe crucial features of clouds andrainfall fieldssuch asanisotropy and stratification. To overcome theselimitations, a concept namedgeneralized scale invariance wasproposed by Lovejoy and Schertzer [1985], where a general scale-changing operation was included to obtainmore realistic models. By analyses of rainfall data using further refined methods, it was concluded that atmospheric processes exhibit a much more complex structure than previously assumed, where statistical properties at different scales are related through different intensity-dependent dimensions, ratherthan a single dimension [e.g., Schertzer and Lovejoy, 1985; Lovejoy et al., 1987]. Schertzer and Lovejoy [1987] showed that this behavior, called multiple scaling or multiscaling,may be interpreted as the outcome of a so-called multiplicative cascade process. In view of this the sealingbehaviorcould be connected with a hypothesis Copyright 1993 by the AmericanGeophysical Union. Papernumber 93JD02658. 0148-0227/93/93 JD-02658 $05.00 concerning the governing physical process, that suggests multiplicative cascade processes to be responsible for the concentration of waterandenergy fluxes intosuccessively smaller parts of the atmosphere. The basic ideais that a large-scale flux is successively broken up into smaller andsmaller "cascades" (or "eddies"), each receiving an amount of the total flux specified by a multiplicative parameter. As a result, the process is characterized by an infinite hierarchy of intensity-dependent dimensions. The notion of multiscaling was later refined by further theoretical developments, and its applicabilityto atmospheric processes was tested in empirical analyses[e.g., Lovejoy and Schertzer, 1990, 1991; Gupta and Wa3,mire, 1990, 1993; Tessier eta!., 19931. In spiteof recentadvances in scaling of atmospheric fields and multidimensional fractaltechniques according to the above, very few studies from different geographical areas have been madeto determine empirical fractal properties of meteorological observations. This is especiallytrue for rain gage observations since it is extremely expensive and time consuming to observe rainlb.11 by gages over large space andtimescales. Even so, most of the historicaldata collections have been made by gages and it would be highly beneficialif common properties, e.g., scaling parameters, canbe foundfor dataover different scales for these observations. It is also very important to establish relationships between scaling properties for gage, radar, and satellite observations. This can onlybe done by finding common empirical scaling properties using these kindsof observations. In view of the above we havetwo mainobjectives in thispaper. The first objective is to investigate the scaling properties of a set of rainfalltime series. For thiswe employ threeanalysis methods to a high-quality database comprising 1-minrainfallobservations with a length of 2 years, each observed at six different points. One of the methods used, probability distribution/multiple scaling (PDMS), is a recently developed technique to identify multiscaling properties of, for example, geophysical processes. The two other methods are refined versionsof techniques for estimating the fractal dimension of sets (box and correlation dimension). We compare the results of each method. The second, equallyimportant, aim of this investigation is to interpret the outcome of the methods in termsof physical characteristics and traditional descriptive statistics of therainfall process and possible effectsof the data collectionsystem used. 23,265