JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. D12, PAGES 23,265-23,274, DECEMBER 20, 1993

Fractal Analysis of High-Resolution Rainfall Time Series

JONAS OLSSON, J ANUSZ NIEMCZYNOWICZ, AND RONNY BERNDTSSON

Department of Water Resources Engineering, Lund Institute of 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 but with different dimensions for different time aggregation periods. The time periods where changes in dimension occur can be related to average rainfall event durations and average dry period lengths. 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 applied to 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, which indicates that the rainfall process can be described by a multiplicative cascade process.

1. INTRODUCTION

The question whether the rainfall process 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., Lovejoy and Mandelbrot, 1985; Waymire, 1985; Kedem and Chiu, 1987; Zawadzki, 1987; Lovejoy and Schertzer, 1991].

Early studies of 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 statistical properties 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 fractal behavior) was presented by Lovefiry and Mandelbrot [1985]. However, it was soon realized that this approach was insufficient to describe crucial features of clouds and rainfall fields such as anisotropy and stratification. To overcome these limitations, a concept named generalized scale invariance was proposed by Lovejoy and Schertzer [1985], where a general scale-changing operation was included to obtain more 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, rather than 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 sealing behavior could be connected with a hypothesis

Copyright 1993 by the American Geophysical Union.

Paper number 93JD02658. 0148-0227/93/93 J D-02658 $05.00

concerning the governing physical process, that suggests multiplicative cascade processes to be responsible for the concentration of water and energy fluxes into successively smaller parts of the atmosphere. The basic idea is that a large-scale flux is successively broken up into smaller and smaller "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 applicability to atmospheric processes was tested in empirical analyses [e.g., Lovejoy and Schertzer, 1990, 1991; Gupta and Wa3,mire, 1990, 1993; Tessier eta!., 19931.

In spite of recent advances in scaling of atmospheric fields and multidimensional fractal techniques according to the above, very few studies from different geographical areas have been made to determine empirical fractal properties of meteorological observations. This is especially true for rain gage observations since it is extremely expensive and time consuming to observe rainlb.11 by gages over large space and timescales. Even so, most of the historical data collections have been made by gages and it would be highly beneficial if common properties, e.g., scaling parameters, can be found for data over different scales for these observations. It is also very important to establish relationships between scaling properties for gage, radar, and satellite observations. This can only be done by finding common empirical scaling properties using these kinds of observations.

In view of the above we have two main objectives in this paper. The first objective is to investigate the scaling properties of a set of rainfall time series. For this we employ three analysis methods to a high-quality data base comprising 1-min rainfall observations 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 versions of techniques for estimating the fractal dimension of sets (box and correlation dimension). We compare the results of each method. The second, equally important, aim of this investigation is to interpret the outcome of the methods in terms of physical characteristics and traditional descriptive statistics of the rainfall process and possible effects of the data collection system used.

23,265

23,266 OLdSSON ET AL.' FRACTAL ANALYSIS OF RAINFALL TIME SERIES

2. METHODOLOGY

When analyzing rainfall fields to detect temporal scaling properties, different approaches have been used. One approach has been to analyze probability distributions of rainfall intensity fluctuations and their variation with time separation [e.g., Lovejoy and Mandelbrot, 1985; Zawadzki, 1987]. Others have used box dimensions to characterize rainfall time series [Hubert and Carbonnel, 1989; Olsson et al., 1992]. Ladoy et al. [1991] used spectral analyses of time series to identify scaling behavior. Recently, rainfall time series have been investigated using methods developed within the framework of multiscaling, which aim to determine the scaling of probability distributions and moments [Hubert et al., 1993].

In this study we have used three related but conceptually different analysis methods to investigate the scaling behavior of rainfall time series. The first two methods are based on a

monodimensional fractal approach, thus assuming that the process can be described by a single fractal dimension. In general, the dimension may be interpreted as a characteristic "degree of irregularity" of the series, independent of scale and arising from underlying physical properties of the process. Thus the dimension of a raint•.11 time series would be a characteristic value of the

variability of the rainfall distribution at all scales within the series, originating from the rainfall-generating mechanisms.

Assuming that a process can be described by a single dimension D allows one to use the energy spectrum E(J) (where f is the frequency) of the observed variable X(t) for scaling investigations. The energy spectrum is scaling when it can be described by a power law relationship according to [e.g., Ladoy et al., 1991]

E(f) = f-• (1)

The exponent/5 and the dimension D are related according to

•3 = 2D+I (2)

In practical terms this means that fluctuations AX at small scales AthS, •J > 1, are related to those at large scales At by

AX(At/•J) d ZkX(At)/•jD (3)

where

ZkX(At) = X(tl)-X(to) At = tl-t 0

fiJ((At/tS) = X(t2)-X(to) t 2 = to+(tl-to)h5 (5)

The sign __a indicates equality in probability distributions. One way to estimate the dimension is by box counting. This

method is a straightforward and widely used way to investigate if a data set exhibits scale-invariant properties [Falconer, 1990]. The general procedure is to divide the space of observation into nonoverlapping segments (boxes) of characteristic size L and count the number of boxes N(L) needed to cover the data set. If the set is monodimensionally fraetal, then it can be characterized by the expression

N(L) o• L -Ds (6)

where D s is the box dimension. Thus plotting log[N(L)] as a function of log(L) will produce a straight line where the slope is an estimation of -D B. When applying the box-counting method to

a time series, the space of observation equals the total length of the series and the boxes represent time intervals.

Another monofractal measure of the irregularity by which a set is distributed over the available space is the correlation dimension [Hentschel and Procaccia, 1983]. The general idea is to investigate whether the average number of points within a region < C(R) > varies with the characteristic size of the region R as

< C(R) > o• R Dc (7)

where D c is the correlation dimension [e.g., Lovejoy et al., 1986]. For each point in the set the number of other points C(R) within distance R is counted, after which an average value < C(R) > may be calculated. As for the box-counting method, the validity of the expression is examined by plotting log[ < C(R) > ] as a function of log(R). When applied to a time series, R represents time periods. This method will be referred to below as the correlation technique.

Both of the above described monodimensional methods, the box- counting method and the correlation technique, are designed to be applied to sets of points. Thus when analyzing a time series, the data must first be transformed into a set of points along the time axis. In this study, the points in a set are defined as the minutes with a registered rainfall intensity exceeding an intensity threshold T. Using a zero threshold consequently here means including all minutes with a registered rainfall in the analysis.

The third method employed is the recently developed PDMS method [Lovejoy and Schertzer, 1990], which is based on a multidi•nensional fractal approach, i.e., that the variability of the distribution at different scales is connected through a dimension function instead of one single dimension. Contrary to the monodi•nensional methods, when using the PDMS method, the data are viewed as a measure rather than a set. The PDMS

method investigates whether the probability distribution related to dift•rent intensity levels is characterized by a scaling behavior. This may be expressed by the equation below ("the general characterization of multifractal fields," Lovejoy and Schertzer [1991]), which follows as a consequence from the mathematical description of a multiplicative cascade process. The relationship between average field intensity •x and scale ratio 3, is expressed as

Pr(•x > X •) = X -c(•) (8)

where Pr stands for probability and the exponent ?, the order of singularity, is related to the rarity of different intensity levels. Instead of a single dimension characterizing the process, a scale- invariant dimension function c(,y) that depends on the intensity level is assumed. Values of the function c(,y) are called codimensions, defined as the dimension in which the process is studied subtracted by the corresponding dimension of the process. The codimension is thus independent of the dimension used for the observations; e.g., the codimensions from spatial rainfall measurements are directly comparable to the ones obtained from time series measurements. Note that the codimension is inversely proportional to the actual dimension of the process.

In the present analysis the PDMS method, as described by Lavallie et al. [1991], was used in one dimension. Before performing the calculations, the measured time series must be normalized (nondimensionalized), which is done by dividing all values by the average of the total series. Then the series is divided into successively doubled time intervals (boxes). In general, the scale ratio X for each step is computed as the total

OLSSON ET AL.' FRACTAL ANALYSIS OF RAINFALL TIME SERmS 23,267

space of observation divided by the characteristic size of the boxes. Thus when analyzing a time series (one dimension, d= 1), X becomes equivalent to the number of boxes the series is divided into, N x. The average field intensity •x for each box is computed as the average value of the normalized intensity over the particular box interval. The probability Pr(• x > X *) is then approximated by the ratio NX(?)/N X, where NX(?) denotes the number of boxes with an average field intensity •x exceeding X *. Thus X * is specifying a threshold value for the average normalized intensity. This procedure is performed with boxes ranging from the resolution of the series (X equals the length of the series) up to the total length of the series (X equals 1). Finally, the probability, Pr(•x > X *) (i.e., NX('¾)/Nx), is plotted as a function of the scale ratio X in a double-logarithmic diagram, and if the series exhibit multiscaling as expressed by (8), the points fall on a straight line with a slope that is an approximation of-c(-y). To obtain the complete codimension function, the above method must be carried out for different values of the exponent ,y. The possible range of '),values in a certain analysis must be found by trial and error.

The codimensions should, according to Schertzer and Lovejoy [1987], belong to a "universality class" of codimension functions de fined by

c(•) : Cl( • + -• )"' (9) Cica, c•

where C 1 is the ½odimension of the mean process and is bounded between 0 and the dimension d of the observation set (i.e., for a

time series 0 < C 1 < 1). The parameter c•, 0 < c• < 2, is called the L•vy index and indicates how strongly the process differs from a monodimensional fractal behavior by specifying to which type of multiffactal process the probability distribution belongs. The parameter c•' is given by the relation 1/c• + 1/r•' = 1.

an area of approximately 30 km 2 [Niemczynowicz, 1984]. The average distance between the stations was 1.3 km. The rainfall intensity was simultaneously measured every minute by tipping- bucket gages with a resolution of 0.035 mm/min. The longest continuous measurement period was 2 years (1979-1980), and we used the six most complete time series from this period in the present study. Missing data were replaced by the values of the nearest functioning gage. For further details about the database and observation area see Niemczynowicz [1984, 1986a, b].

Figure 1 shows an example of properties of the time series used. The figure shows the autocorrelation for one of the 2-year time series. It is seen that a significant correlation exists up to about 60 min. After this the correlation decreases but does not

reach zero until after about 400-500 min. This delay in the autocorrelation function indicates a temporal persistence that may be related to fractal properties of the time series [e.g., Rodriguez- lturbe et al. , 1989].

Figure 2 shows the corresponding power spectral density function. As is evident from the figure, the time series displays a classical f-5/3 power law shape over a range of frequencies (/'=wave number [e.g., Crane, 1990]). This may be viewed as a scaling behavior in the range described by the power law relationship [Ladoy et al., 1991]. The classical Kolmogorovf -5/3 power law relationship is the spectrum predicted for the fluctuations of a passive scalar introduced into a turbulent fluid. These fluctuations with scaling spectra and hyperbolic intermittency are often assumed to be the result of a multiplicative cascade process, as mentioned above. The spectral density function breaks at a point equal to 40-50 min. This corresponds to the average rainfall event duration and will be shown below to be related to the fractal dimension.

4. MONODIMENSIONAL FRACTAL PROPERTIES

3. RAINFALL DATABASE

During 1979-1981 a detailed observation program of the areal and dynamic properties of short-term rainfall was performed in the city of Lund, Sweden, where 12 stations were set up to cover

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1 10 1 O0 1•000 10000 Time leg (m n) Fig. 1. Autocorrelation function for one of the 2-year series of 1-min rainfall (l-min time lags).

4.1. Box Dimensions

Figure 3 shows the result when applying the box-counting method to one of the 2-year time series using an intensity threshold T of 0 mm/min. The resulting curve may be satisfactorily approximated by three straight-lined segments, to which straight lines have been fitted by regression (denoted 1-3

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23,268 OLSSON ET AL.' FRACTAL ANALYSIS OF RAINFALL TIME SEPdES

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l_ (min) Fig. 3. The box-counting method applied to one of the 2-year series of 1-min rainfall using T = 0 mm/min. The straight segments 1-3 were fitted by linear regression.

in Figure 3). The first segment extends from a boxsize of 1 to 45 min with an estimated box dimension DB=0.82, the second segment from 45 min to 1 week with DB=0.37, and the third segment from 1 week to 2 years with DB= 1.00.

These results imply that the rainfall distribution in time is scaling but with different properties (box dimensions) for different ranges. A reasonable assumption is that the break points in the graph correspond to characteristic time periods when changes in the temporal structure of the rainfall distribution occur. A similar interpretation may be valid regarding the spatial structure since the observed time series at a point reflects the three-dimensional spatial structure ot' the rainfall field.

As mentioned above, segment 1 in Figure 3 corresponds to the boxsize interval 1-45 min. This time interval encompasses single rainlhll events in the time domain or separate clusters of rain cells in the space dolnain. The relatively steep slope of segment I (D B is comparably large) indicates a dense structure with rainy minutes closely clustered together. At a time increment of 45 min the graph bends (as observed above for the power spectral density function), which suggests a transition to a new structure. Assuming 45 min to equal the typical duration of an individual rainl•tll event in Lund implies that when the boxsize exceeds 45 min, individual rainfall events will in most cases be enclosed by the same box. Hence the slope of the graph no longer reflects the distribution of individual rainy minutes but rather represents the distribution of single rainfall events. The milder slope of segment 2 indicates that rainfall events are more scattered over time, as

compared to the rainy minutes within the time increments 1-45 min. The interval of the time increments corresponding to segment 2 encompasses 45 rain to about 1 week. For all time increments above 1 week (1 week to 2 years) the slope of the curve is -1. This implies saturation of the process, which in our case means that rainfall is always occurring for time increments exceeding 1 week. The location of the second break (1 week) thus indicates that the rainfall in southern Sweden is relatively evenly distributed throughout the year, with only slight differences between seasons. On the analogy of relating the first break in the

curve to the average rainfall event duration, the second break may be related to the typical length of a dry period between rainfall events.

When T is increased above 0 mm/min (0.1, 0.2, and 0.3 mm/min), some significant changes in the appearance of the box- counting graph occur, as seen in Figure 4. The most notable change is the tendency of the slopes (i.e., box dimensions) to decrease with increasing threshold. This behavior indicates a multiscaling behavior, where the most intense regions are characterized by the lowest dimension.

Table 1 summarizes the box dimensions obtained from

analyzing series observed at six different stations. Since the times of the break points are somewhat arbitrarily estimated (depending on which points on the graph one chooses to include in each segment), the time interval limits for each segment are given as average values only. However, the differences between stations are generally small. It is seen that average values of box dimensions are large in relation to standard deviations, which means that the estimated dimensions are robust and independent of gage location on the scale in question.

Apart from the decrease in dimension, other alterations occur when the threshold is increased (Figure 4). The left part of the curves (segments 1 and 2) is displaced downward and the break points are shifted in a regular way. To understand this behavior, it is important to emphasize that different intensity thresholds reveal different properties of the rainfall process. When using a zero threshold, the rainfall process in total, as represented in the series, is analyzed simultaneously with no separation between the background, low-intensity rain resulting from large-scale fields and the high-intensity rain resulting from smaller spatial units embedded in the fields. When using a higher threshold, the background parts are "cut off" and only the properties of the high-intensity occurrences are represented in the resulting graph. With this in mind, the above mentioned changes in the graphs can be explained from a physical viewpoint. The larger the threshold used, the more sparse is the remaining temporal structure, i.e., fewer number of points are included in the analysis. Because of

OL,$SON ET AL.' F•CTAL ANALYSIS OF RAINFALL TIME SERIES 23,269

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this the curves are lowered and smoothed out, which makes the

first break point (from the left) less pronounced. The changes in the time increments where the graphs break agree with the expected decrease in raintMl event duration and increase in dry period length, when studying only the high-intensity events.

To verit•y the assumptions regarding the connection between the break points on the graphs and the traditional descriptive rainfall time series statistics, analyses of duration frequencies were performed tbr the minute series. To distinguish between rainfall events and dry periods, a limit of 10 min was used as a minimum value to define a dry period. For the threshold values previously used, frequency distributions of rainfall event duration and dry period length were derived. The average values of these distributions are marked with arrows in Figure 4. The first arrow (from let't to right) in each graph shows the average rainfall event duration and the second arrow shows the average dry period length. The average rainfall event duration is almost perfectly matching the first break on the graphs. The average dry period length appears similarly related to the second break on the graphs (except for slight discrepancy at the lowest thresholds). These observations give good support for the proposed hypothesis regarding the breaks on the graphs.

TABLE 1. Box Dimensions for the Six Rainfall Time Series

Threshold

(mm/min) Mean Sd. Min Max Time interval

0.0 0.84 0.08 0.80 1.00 1 min < L < 45 min

0.1 0.53 0.02 0.49 0.55 1 min < L < 25 min

0.2 0.50 0.04 0.47 0.54 1 rain < L < 15 rain

0.3 0.46 0.03 0.42 0.50 1 rain < L < 10 min

0.0 0.36 0.01 0.35 0.38 45 rain < L < 1 week

0.1 0.22 0.02 0.19 0.24 25 rain < L < 2 weeks

0.2 0.14 0.03 0.11 0.20 15 rain < L < 3 weeks

0.3 0.12 0.02 0.10 0.16 10 rain < L < 1 month

4.2. Correlation Dimensions

Figure 5 shows the result when applying the correlation technique to one of the 2-year time series (the same as was used in Figure 3) using an intensity threshold T of 0 mm/min. The main part of the resulting curve may be properly described by two straight-lined segments, to which straight lines have been fitted by regression (denoted 1-2 in Figure 5). However, the first (from the left) part of the curve, corresponding to time increments up to about 3 hours, does not show a straight-lined behavior, possibly with the exception of the part at the extreme left (R is about 1-3 min). The first straight segment extends from 3 hours up to 1 week with an estimated correlation dimension of Dc=0.36 and the second from 1 week to 2 years with Dc=0.84.

One important reason for estimating the correlation dimension is to confirm the results from the box-counting method. The two methods are similar when employed in one dimension, but whereas the box-counting method only determines whether rainfall occurs at some point within a time interval, the correlation technique is also sensitive to the number of occurrences and thus produces more accurate dimension estimates.

When the two graphs obtained with respective analysis method using a zero threshold are compared (Figures 3 and 5), some interesting differences emerge. For time increments up to about 1 hour the box-counting graph indicates a scaling behavior, whereas the correlation graph does not. Thus the correlation technique implies that the rainfall distribution at timescales corresponding to event durations would not be characterized by scaling. This discrepancy may be related to limitations of the tipping-bucket gages to correctly represent the small-scale rainfall variability. Rainfall intensities lower than the gage resolution (0.035 mm/min) are in the series represented by 0.035-mm registrations separated by 0-mm registrations. This artificial representation will affect the scaling behavior at low thresholds and small time increments. When using the somewhat rough box-

23,270 OLSSON ET AL.' FRACrAL ANALYSIS OF RAINFALL TIME SERIES

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counting method, this limitation does not seriously aftbct the result, but when using the more sensitive correlation technique, the result is influenced. The resulting graph (Figure 5) seems to approach a state of saturation in the limit of small time intervals, an observation that was verified by calculating the slope using the points at Rvalues of 1, 2, and 3 min only (De= 1.0). Thus the first (from the left) part of the correlation curve should be regarded as a transition from the state of saturation to segment 1. This segment agrees with the second segment of the box-counting graph, both regarding the estimated dimension (0.37 and 0.36, respectively) and the upper limit (about 1 week), which supports the previous conclusions concerning the relation between this break point and the average dry period length. The clear break at

1 week is probably also related to the size of large-scale synoptic areas [e.g., Berndtsson and Niemczynowicz, 1988]. At a boxsize of I week the box-counting method becomes saturated, but the correlation technique is able to reveal yet another straight-lined interval (segment 2, Figure 5), extending from 1 week to 2 years. The steeper slope (Dc=0.84) indicates a denser structure, as compared to the previous segment of the same graph. A possible interpretation is that this segment reflects the relatively uniform distribution of rainfall in time during a year in southern Sweden, with only slight differences between seasons. A close examination of the part of the curve corresponding to Rvalues larger than 6 months implies a deviation to a slope of approximately 1. This indicates that the annual dependence in the series leads to a

OLSSON ET AL.: FRACTAL ANALYSIS OF RAINFALL T•ME SERIES 23,271

saturation of the correlation technique at R =6 months. However, since the deviation is not significant, this part has been included in segment 2.

As seen in Figure 6, using a T larger than 0 mm/min leads to a behavior of the graphs similar to that of the box-counting method. The graphs are lowered, the slopes decrease, and the break point around 1 week changes position similarly. It must be emphasized that the graphs obtained from thresholds above 0.1 mm/min exhibit a more irregular appearance as compared to the others. This is a result of the small number of points that remains when rainfall less than 0.2 and 0.3 mm/min, respectively, is removed, a limitation that affects the correlation technique more than the box-counting method since the former is more sensitive to the number of points used. Thus for most of the six time series, straight-lined segments could not be satisfactorily fitted to the graphs corresponding to a T of 0.2 and 0.3 mm/min.

Table 2 summarizes the correlation dimensions obtained from

analyzing series observed at six different stations. As for the box dimensions, the intervals for each segment are given as average values. As expected, the standard deviations of the estimated correlation dimensions are very low for T=0 mm/min but significantly higher for T=0.1 mm/min depending on the sensitivity of the method regarding the number of points used in the analysis.

To conclude, the results obtained from applying two different methods, the box-counting method and the correlation technique, to 2-year series of 1-min rainfall intensity measurements show that the rainfall distribution in time, as represented in the series,

TABLE 2. Correlation Dimensions for the Six Rainfall Time Series

Threshold

(mm/min) Mean Sd. Min Max Time interval

0.0 0.36 0.004 0.36 0.37 3 hours < R < 1 week

0.1 0.21 0.05 0.14 0.27 1 hour < R < 2 weeks

0.0 0.85 0.01 0.84 0.87 1 week < R < 2 years 0.1 0.74 0.06 0.63 0.79 2 weeks < R < 2 years

is characterized by a scaling behavior. This behavior, however, displays different properties for different time intervals and different intensity levels and is thus not possible to describe by a single fractal dimension.

5. M 11LTIDIMENSIONAL FRACTAL PROPERTIFfS

The results presented in section 4 show that a single dimension is insufficient to describe the scaling properties of the series rainfall distribution in time. First, the dimension is clearly related to the intensity level: the higher intensity of the field the lower the dimension. Second, the physical mechanisms of rainfall generation impose limits on where the scaling symmetry, expressed in terms of a single dimension, is broken. This implies that a multidimensional fractal approach may be more suitable to characterize the rainfall time series, an assumption that may be verified by using the PDMS method.

As previously mentioned, the PDMS method is used to estimate the codimension function c('y). In the present analysis the possible range of 'yvalues turned out to be rather limited, 0.22 <'y < 0.55. The lower limit -y=0.22 is related to the difficulties at low rainlM1 intensities caused by the "discrete" appearance of rainfall time series obtained by tipping-bucket gages (see section 4.2), which led to saturation of the procedure for smaller 'yvalues. The corresponding thresholds became so small that 1 min with rain in a box was enough to make the average field intensity larger than the threshold. Thus to have thresholds of the same size as the

average field intensities, 'y had to be at least 0.22. This limitation can only be overcome by using data of higher resolution. The upper limit ,y=0.55 was determined by the requirement to have a sufficient number of points for the estimation of c('y). This limitation is thus related to the maximum rainfall intensity in the studied time series.

Figure 7 shows the result of PDMS applied to the 1-min rainfall intensity data. The curves were obtained as the average from the six series. Note that since the number of boxes at the smallest

Xvalues (below 64) were too few to allow for reliable estimations

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.

'-..• .

nth week d•y hour mot i 11 I I I III111 I I I 11t111 I I I III111 1 1 111111

A

lO lOO 1000 10000 10()()(-)(-) 1000000 100000()

Lambda Fig. 7. The probability distribution/multiple scaling method applied to the 2-year series of 1-min rainfall using values of ? from 0.25 to 0.55. The curves were obtained as the average from six gages. The straight lines were fitted by linear regression.

23,272 OhSSON ET AL.' FRACTAL ANALYSIS OF RAINFALL TIME SERIES

of the probabilities, these points have been omitted in the diagram. In the Mnterval of 2000 to 1,000,000 (corresponding to boxsizes between 1 min and 8 hours), the curves may be well described by straight lines, which have been fitted to the points by regression. The deviations from the straight lines at 3,--2000 may have several reasons. In general, the larger the interval the series is averaged over, the fewer the number of boxes and thus the higher the uncertainty in the estimated probability. To improve the estimations, more series would be needed. Another possible reason is that the deviations are descended from properties of the time series. The influence of the numerous zerovalues may reach some critical level when averaging the process over a time interval of 8 hours. For boxsizes below 8 hours the average intensities are indeed reflecting the values of the measured intensities, but above 8 hours the zerovalues may prevail to such an extent that the difference in average intensity between boxes normally is negligible, except for some boxes containing extreme events. Another feature of the graphs is the tendency to successively shorten in the direction of decreasing X, when increasing the 7value. The "missing part" refers to boxsizes where the probability is 0, i.e., all the averaged box intensities were lower than the threshold X 'r. The higher the 7 used, the smaller the boxsize where the average intensity for the first time exceeds the threshold value.

From the straight-lined parts of the curves in Figure 7 the average values of c(7 ) were estimated as the slopes. Table 3 summarizes the codimensions obtained from analyzing series observed at six different stations. The standard deviations are

similar to those of the previous methods, which shows that also

TABLE 3. Codimensions for the Six Rainfall Time Series

Mean sd. Min Max

0.3 0.31 0.016 0.28 0.33

0.4 0.58 0.021 0.55 0.61

0.5 0.69 0.027 0.65 0.72

the codimensions are robust and independent of gage location at the scale in question.

Figure 8 shows the resulting average codimensions (the point values) obtained from Figure 7. According to Lavallge et al. [1991], when analyzing n samples of maximum scale ratio kma x from a process, the maximum codimension possible to evaluate is given by Cmax(q• ) =d + d s, where d s = logn/logXma x. In our case, Cmax('y) = 1.13 (d= 1, n=6, )Xma x-- 1048576). As seen in Figure 8, this value is an accurate upper limit for the obtained codimensions.

To fit a function to the codimcnsions, a graphical method was used, as outlined by Lovejoy and Schertzer [1990]. This fitting procedure, however, is associated with large uncertainties. This is partly because the two parameters C 1 and c• are highly correlated, partly because of the limited range of q•values. The graphical method involves estimating c(0) together with c(q•) and q•values connected with a tangential line of slope 1. Since in this case the q•values were in the range 0.22 to 0.55, the value c(0) could only be somewhat arbitrarily estimated, and the slope-1 line is a tangent to the curve at a point outside of this range. However, by this way, approximate values of the parameters were obtained, after which leastsquares regression were used to improve the fit. The best fit was found using C 1 •0.15, c•= 1.8, and c•'--2.25, and the corresponding curve is shown in Figure 8 (the solid curve). The agreement between the fitted function and the calculated values is satisfactory, although the latter exhibit departures from the monotonically increasing curve. This behavior can be explained by the "discrete" character of rainfall time series obtained by tipping-bucket gages, which makes the averages over various time periods to have some preferred sizes. Thus a small change in q• may lead to an unexpectedly large increase in c(•) since the resulting threshold happened to exceed some common average intensity and vice versa. Consequently, the calculated codimensions would, in agreement with Figure 8, show an approximately stepwise increasing behavior.

Depending on the highly indirect and uncertain estimation, the avalue 1.8 should be regarded as very approximate. However, it

1.2

0,8 -

0,6 -

0,4 -

0,2 -

Fitted c:odlmen,slon function

Qodimer•$ion$ (obtained from Figure

Fig. 8. Codimensions for the rainfall time series, obtained as the slopes of the straight lines in Figure 7, and the fitted ½odimension function.

o o.2 o,a 0,4 o,5 0,6

Gamma

OL.•SON ET AL.: FRACTAL ANALYSIS OF RAINFALL TIME SERIES 23,273

seems as a should be in the range 1 < a <2 to get a c('y) that properly fats the curvature described by the obtained codimensions. An a in this range implies a L6vy process with unbounded singularities [Tessier et al., 1993], a "hard" multifractal process [Schertzer and Lovejoy, 1992] characterized by the existence of highly extreme but very rare values. This is contrary to other studies, where rainfall time series were found to exhibit a "so•t" multifractal behavior [Hubert et al., 1993]. However, we do not stress the obtained avalue but confine

ourselves to pointing out that the criteria for multiscaling are reasonably fulfilled for the studied rainfall time series.

6. SUMMARY AND CONCLUSIONS

resolution of the data collection system to any scale needed for the hydrological problem in question would be feasible, a prospect of tremendous practical importance. However, before developing methods for such practical applications, the validity of multiscaling to describe the rainfall process must be further established. This involves both the collection of rainfall data

covering extremely large intervals in both time and space scales and the development and application of further refined analysis methods.

Acknowledgments. We gratefully acknowledge the reviewers for providing helpful criticism and the Swedish Natural Science Research Council for supporting this research.

A database consisting of six 2-year series of 1-min rainfall intensity observations was analyzed to obtain information about possible scale-invariant properties of rainfall time series. The possibility to characterize the series by a monodimensional fractal approach was investigated by applying two analysis methods to the series. First, the box-counting method was employed to estimate the box dimension of the series. The resulting graphs indicated scale invariance but with different box dimensions for

different time intervals. The limits of the intervals were related

to average rainfall event duration and average dry period length for the time series. When studying the parts of the observed rainfall time series with higher intensity by using intensity thresholds, the estimated dimensions decreased. Second, the correlation technique was employed to estimate the correlation dimension of the series. The results were consistent with the

results from the box-counting method, thus indicating that a monodimensional fractal approach is insufficient for characterizing the variability of the rainfall distribution in time. Thc dimension dependence on intensity level suggested that a multidimensional fractal approach, where the process is characterized by a dimension function instead of one single dimension, would be more suitable. To test this hypothesis, the probability distribution/multiple scaling method was employed to estimate the codimension function. The results support the hypothesis of a multiscaling behavior of the rainfall time series, at least within a significant part of the studied timescales.

The indications of multiscaling are encouraging since they suggest that the rainfall-producing mechanisms can be characterized by a multiplicative cascade process. Models based on this conceptually simple approach have been shown to effectively imitate fully developed turbulence [e.g., Meneveau and Sreenivasan, 1987], and it may also have a great potential in modeling the complex structure of rainfall distribution in time. Typical features of the rainfall process, which have created severe problems for rainfall modelers, may be naturally accounted for in this approach. At large scales, a multiplicative cascade will automatically result in the cellular structure of rainfall fields that has been observed in empirical studies. Also, when decreasing the scale, the variability of rainfall will increase. The extreme variability in time and space observed in high-resolution measurements, with intensity peaks many times the average, is a direct consequence of a multiplicative cascade process, where the total flux is concentrated on successively smaller parts of the available space.

In practice, a multiscaling approach may solve some of the problems encountered when performing hydrological calculations involving rainfall variability. The existence of scale-invariant properties of the rainfall process indicates that scale shit•s are theoretically possible. Thus a shit't from the scale defined by the

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(Received May 3, 1993; revised September 13, 1993;

accepted September 16, 1993.)