Imperfect scaling of time and space–time rainfall

Daniele Venezianoa,*, Pierluigi Furcolob, Vito Iacobellisc

aDepartment of Civil and Environmental Engineering, MIT, Cambridge, MA 02139, USAbDipartimento di Ingegneria Civile, Universita degli Studi di Salerno, Fisciano (SA), Italy

cDipartimento di Ingegneria delle Acque e di Chimica, Politecnico di Bari, Bari, Italy

Received 5 February 2004; revised 28 September 2004; accepted 8 February 2005

Abstract

Scale invariance is the most fertile concept to be introduced in stochastic rainfall modeling in 15 years. In particular, a form

of scale invariance called multifractality has been exploited to construct parsimonious representations of rainfall in time and

space and address fundamental problems of hydrology such as rainfall extremes, downscaling, and forecasting. However,

several authors have observed that rainfall is scale invariant only in approximation and within limited ranges. Here, we make a

systematic analysis of the deviations of time and space–time rainfall from multifractality. We use a flexible multiplicative

cascade model, which produces multifractality as a special case while allowing deviations from scale invariance to occur. By

fitting the model to rainfall records from different climates and over land or ocean, we find significant and consistent departures

from multifractality in both the alternation of wet and dry conditions and the fluctuations of precipitation intensity when it rains.

The fractal dimension of the rain support increases with increasing rain rate and the (multiplicative) fluctuations are larger at

smaller scales and for lighter rainfall. A plausible explanation of these departures from scaling is that the rate of water vapor

condensation in the atmosphere is a multifractal process in three space dimensions plus time, but multifractality is destroyed

when the condensation rate is integrated to produce rainfall intensity at fixed altitudes.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Rainfall models; Scale invariance; Multifractal processes

1. Introduction

During the past two decades, stochastic models of

rainfall have increasingly exploited the property of

multifractal scale invariance. This property states that

rainfall fields are statistically invariant under a group

of transformations that involve contraction of

0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2005.02.044

* Corresponding author. Tel.: C1 617 253 7199; fax: C1 617 253

6044.

E-mail address: venezian@mit.edu (D. Veneziano).

the support and multiplication of the field by a non-

negative random factor (Gupta and Waymire, 1990;

Veneziano, 1999). Rainfall models based on multi-

fractal scale invariance have been proposed by

Schertzer and Lovejoy (1987), Gupta and Waymire

(1990, 1993), Tessier et al. (1993), Over and Gupta

(1994, 1996), Svensson et al. (1996), Perica and

Foufoula-Georgiou (1996), Menabde et al. (1997),

Olsson and Berndtsson (1998), Harris et al. (1998),

Schmitt et al. (1998), and Deidda et al. (1999), among

others.

Journal of Hydrology 322 (2006) 105–119

www.elsevier.com/locate/jhydrol

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119106

Multifractal models have several advantages over

conventional representations of rainfall. One is that

they are simpler and involve a smaller number of

parameters: once the fluctuations at a given scale are

understood, those at other scales are deduced from

scale invariance and need not be independently

specified. Other advantages come from the fact that

multifractal fields have a simple probabilistic struc-

ture (they are the product of statistically identical

fluctuations at equally spaced log-scales). One can use

this construct to deduce the marginal distribution of

rainfall intensity and the behavior of rainfall extremes

from minimal information on the component fluctu-

ations (Benjoudi et al., 1997, 1999; Menabde et al.,

1999; Veneziano, 2002; Veneziano and Furcolo,

2002, 2003), construct downscaling methods (Harris

et al., 1998; Olsson and Berndtsson, 1998; Venugopal

et al., 1999; Deidda et al., 1999; Deidda, 2000), and

devise forecasting procedures (Marsan et al., 1996).

It is generally thought that rainfall inherits its

scaling properties from atmospheric turbulence (on

the multifractality of turbulence, see for example

Frish, 1985; Frish and Parisi, 1985; Meneveau and

Sreenevasan, 1987), but the detailed transfer of

multifractality from turbulence to rainfall is not

clear. Complications arise from the fact that water

in the atmosphere is not a passive tracer, water vapor

condensation is affected by moisture and temperature,

and raindrops follow trajectories different from those

of gaseous particles. Hence, rainfall may violate scale

invariance also if atmospheric turbulence is perfectly

multifractal. This is confirmed by spectral analyses of

rainfall time series (Fraedrich and Larnder, 1993;

Olsson et al., 1993; Olsson, 1995) and from the

distribution of the duration of wet and dry periods

(Schmitt et al., 1998).

Departures from multifractality are expected also

in space–time rainfall, although on this issue there is

less empirical evidence and consensus. Even the type

of scale invariance in space–time rainfall is unclear.

Some studies (Over and Gupta, 1994, 1996) suggest

that there is scaling in space but not in time. Others

(Marsan et al., 1996; Venugopal et al., 1999; Deidda

et al., 1999, 2002) find scaling in both time and space

but conclude differently on whether rainfall remains

statistically invariant under isotropic or anisotropic

contraction of space and time. Still other studies

suggest invariance under more complicated

transformations that include anisotropic contraction

of space and time, as well as rotation (Lovejoy and

Schertzer, 1995).

To better understand how rainfall could be non-

scaling one may notice that, in the simple case of

isotropic multifractality, rainfall intensity R(x,y,t) is

the product of independent and identically distributed

oscillations Wj(x,y,t) at different resolutions jZ1,2,..

The oscillations satisfy the scaling relation

Wjðx; y; tÞZdWðrjox; r

joy; r

jotÞ (1)

where roO1 is a contraction factor, W(x,y,t) is a non-

negative random field with mean value 1, and ¼d

denotes equality in distribution. Key features of this

construction are that fluctuations at different log-

scales combine in a multiplicative way (multiplicative

property) and are statistically identical, after scaling

of the support (id property). The multiplicative

property is generally supported by data (Veneziano

et al., 1996; Carsteanu and Foufoula-Georgiou, 1996;

Menabde et al., 1997), but deviations from the id

property have been found in the form of dependencies

of Wj on scale j (Veneziano et al., 1996; Menabde

et al., 1997) and on covariates such as large-scale

rainfall intensity (Over and Gupta, 1996). In the latter

study, Over and Gupta fitted a so-called beta-

lognormal cascade to each frame of the GATE-1

and GATE-2 (GARP Atlantic Tropical Experiment,

Phases 1 and 2) radar sequences and examined how

the multifractal parameters depend on the mean

rainfall intensity �R over the frame. In the beta-

lognormal model, the generator W has a non-zero

probability mass at WZ0 and [WjWO0] has lognor-

mal distribution such that E[W]Z1. Scaling depends

on two parameters, Cbeta and CLN, which, respect-

ively, control the sparseness (wet/dry alternation) and

intensity fluctuations of rainfall. Over and Gupta

found that Cbeta depends strongly on �R, increasing for

decreasing average rainfall intensity. The dependence

of CLN on �R was found to be more modest, with a

maximum of CLN at intermediate values of �R. This

type of analysis has since been used by other

researchers (Ferraris et al., 2003; Deidda, 2000),

largely confirming the results of Over and Gupta

(1996).

One should be cautioned that conditioning on �Rproduces biased results, with a dependence of Cbeta on

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 107

�R similar to the one observed empirically (when �R is

low, precipitation tends to be sparse, with a low-

dimensional fractal support and a high Cbeta coeffi-

cient). Additional bias in the same direction is caused

by the fact that, below the radar sensitivity threshold,

rainfall intensity is reported as zero. The artificial

zeros increase the sparseness of rainfall and hence

increase the estimate of Cbeta in radar images with low

rainfall intensity �R. Due to these biases, it is not clear

whether the observed dependence of W on large-scale

rainfall intensity �R is real or is an artifact of the

method of analysis.

Here, we make a detailed investigation of lack of

multifractality in rainfall, extending the scope of the

Over and Gupta (1996) analysis while correcting for

the biases. Sections 2 and 3 describe our rainfall

model and analysis method and Section 4 shows

applications to several rainfall data sets. We find that

deviations from the id property are significant and

systematically present in all records. Section 5

summarizes our findings and suggests a physical

mechanism that may be responsible for the observed

lack of scale invariance.

2. Rainfall model

Our rainfall model differs from that of Over and

Gupta (1996) in three respects. First, Over and Gupta

consider the generators Wj at all cascade levels j to be

independent and identically distributed. They further

allow the common distribution to vary with the

average rainfall intensity �R over the radar frame (or

more precisely over the 256 km!256 km region used

in the analysis). However, one may argue that the

scale of the radar image has no special physical or

statistical significance. A more plausible assumption,

which we make here, is that the generator Wj from

level jK1 to level j depends on the ‘bare’ rainfall

intensity Rb;jK1ZR0

Qk!j Wk in the host cascade tile

at level jK1. Notice that Rb,jK1 is the intensity when

the cascade construction is terminated at level jK1,

whereas Over and Gupta’s �R is a ‘dressed’ intensity,

obtained by averaging the completely developed

cascade. (We use a subscript b for bare intensities

and no subscript for dressed intensities.)

Second, we assume like Marsan et al. (1996),

Deidda et al. (1999) and others that rainfall is

approximately multifractal in both space and

time not just space. Hence, our cascade tiles are

‘cubes’ in space–time, whose size depends on the

cascade level j.

Third, we allow the generator Wj to depend not

only on Rb,jK1, but also on the cascade level j. As was

noted in Section 1, temporal rainfall series show

evidence of this type of dependence (Veneziano et al.,

1996; Menabde et al., 1997).

To complete the model, one needs to specify the

distribution of the cascade generators Wj. In the case

of multifractal rainfall, when all the Wj are distributed

like W, the distribution of W has a non-zero

probability mass P0 at WZ0. This probability mass

is associated with the so-called ‘beta’ component of

the process and controls the fractal dimension of the

wet set; see below. For positive W, the distribution of

(log WjWO0) is often taken to be Levy stable, with

skewness coefficient bZK1 and stability index

0!a%2 (on Levy stable distributions, see for

example Samorodnitsky and Taqqu, 1994). This is a

special case (no fractional integration) of the ‘uni-

versal’ multifractal model of Schertzer and Lovejoy

(1987). For aZ2, the stable distribution becomes

normal. Hence, we refer to the above distribution of W

as ‘beta-logstable’ or ‘beta-lognormal’, depending on

whether a%2 or aZ2. This includes most multi-

fractal models of rainfall proposed in the past.

It is typical in multifractal analysis to work not

directly with the distribution of W but with the

moment-scaling function

KðqÞZ log2 E½Wq� (2)

where the base of the log is the multiplicity of the

cascade (or more in general, the scale-change factor to

which W refers). Here, we use a change-scale factor of

2, but any other choice would produce very similar

results. For a beta-logstable distribution of W, K(q) is

given by (Schertzer and Lovejoy, 1987)

KðqÞZCbetaðqK1ÞCCls

aK1ðqa KqÞ (3)

where Cbeta, Cls and a are parameters, which we

collect into a parameter vector qZ ½Cbeta; Cls; a�.

The relationships between Cbeta, the probability P0,

and the fractal dimension Dwet of the wet set is

CbetaZKlog 2ð1KP0ÞZdKDwet, where d is the

Euclidean dimension of the rain support (dZ1 for

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119108

temporal rainfall, dZ3 for space–time rainfall). In the

beta-lognormal case (aZ2), Cls is related to the

variance s2 of ðlog2 WjWO0Þ as ClsZs2/2.

In our model, we assume that the cascade generator

W has a beta-logstable distribution as above, but we

allow q to vary with scale j and the bare rainfall

intensity Rb,jK1. The main objective of our analysis is

to determine whether such dependence indeed exists.

3. Method of analysis

For this purpose, it would be ideal to observe the

bare densities Rb,j and obtain sample values of W as

WZRb,j/Rb,jK1 (it is implicit in this and similar

expressions that the cascade cell at level j is part of the

cascade cell at level jK1). Then one could use the

samples to infer the distribution of W for different j

and Rb,jK1. But unfortunately, the bare densities Rb,j

are not observable.

In multifractal analysis, a standard way to

circumvent this problem is to estimate the function

K(q) from the way the dressed moments E½Rqj � vary

with scale j (this is why K(q) is called the moment-

scaling function). When the distribution of W varies

with j and Rb,jK1, one would be tempted to estimate

Kðqjj;RjK1ÞZ log2ðE½Wqðj;Rb;jK1Þ�Þ as

Kðqjj;RjK1ÞZ log2ðE½Rqj jRjK1�ÞK log2ðR

qjK1Þ (4)

where E½Rqj jRjK1� is the empirical qth moment of

RjjRjK1 and the rainfall intensity RjK1 is discretized

into small intervals. However, when the distribution

of W varies with j and Rb,jK1, the connection of

K(qjRb,jK1) with the dressed moments is lost and

Kðqjj;RjK1Þ, which is based on a moment ratio, is a

biased estimator of K(qjj,Rb,jK1). As we shall say

later, another important source of bias is the fact that

very low rainfall intensities cannot be accurately

recorded and are set to zero.

In spite of being biased, Kðqjj;RjK1Þ conveys

important information on K(qjj,Rb,jK1). In our

approach, we use Kðqjj;RjK1Þ and adjust the distri-

bution of W(j,Rb,jK1) such that rainfall simulations

using this distribution closely reproduce the Kðqjj;RjK1Þ

function from actual data. By so doing, the bias of K is

implicitly accounted for. More specifically, the pro-

cedure consists of two stages, each with several steps:

Stage 1: rainfall data analysis

1.1

From the rainfall record, calculate the average

rainfall intensities Rj inside space–time boxes at

different cascade levels j;

1.2

Define (j, RjK1) classes by discretizing RjK1;

1.3

For each (j, RjK1) class, calculate KðqÞ using Eq.

(4). Fit to KðqÞ a function of the type in Eq. (3),

with parameters qZ ½Cbeta; Cls; a�;

1.4

Plot Cbeta, Cls and a against RjK1 for different j;

Stage 2: inference of qðj;Rb;jK1Þ

2.1

Start by assuming qðj;Rb;jK1ÞZ qðj;RjK1Þ from

Stage 1;

2.2

Simulate multiplicative cascades of size compar-

able to the actual data set, censor the simulations

to represent the low-intensity cutoff of the rainfall

measurements, and use the procedure of Stage 1

to calculate qð j;RjK1Þ;

2.3

If the calculated parameters qðj;RjK1Þ differ from

those of Stage 1, modify qðj;Rb;jK1Þ and return to

Step 2.2. Iterate until a satisfactory agreement is

reached.

In Steps 1.3 and 2.2, the parameters {Cbeta, Cls, a}

can be obtained using different criteria. A simple

method is to find Cbeta as the negative of the Y intercept

of K, calculate Cls as the empirical slope of K at

qZ1K Cbeta, and find a through least squares fitting of

KðqÞ over a range of moment orders (in all applications

we have used the range 0!q!2, sampled at constant

increments). Somewhat different estimation criteria

were used by Over and Gupta (1996) to infer Cbeta and

Cln in their beta-lognormal model.

Unfortunately, no analytical relation exists

between q and q. Hence, in Step 2.3 one must adjust

q in a semi-judgmental way, using sensitivity runs and

the results from previous iteration steps.

4. Numerical results

To investigate the possible departure of rainfall

from multifractality, we have analyzed the following

data sets:

1.

The radar sequences GATE-1 and GATE-2 from the

tropical North Atlantic Global Atmospheric

Tab

Cha

Dat

Gat

Gat

Bol

Flo

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 109

Research Program (GARP) Atlantic Tropical Exper-

iment (GATE). These sequences have a 4 km!4 km resolution within a 200 km radius and a

temporal resolution of 15 min from June 28 to July

15 1974 (GATE-1) and from July 28 to August 15

(GATE-2); see Hudlow and Patterson (1979).

2.

A sequence of measurements from the doppler radar

at S. Pietro Capofiume near Bologna, Italy. The

original measurements have a 1 km spatial resol-

ution and a temporal resolution of 15 min. We have

used a sequence of 128 frames on a grid of 64!64 pixels, after aggregation of the data at 2 km.

3.

The rain gauge record from the Osservatorio

Ximeniano in Florence, Italy (Becchi and Castelli,

1989), which has a temporal resolution of 5 min

and covers the 24-year period from 1962 to 1985;

see Veneziano and Iacobellis (2002) for an

analysis of this data set.

This selection was made to maximize the diversity

of rainfall measurements (by rain gauge or radar),

space dimension (time or space–time), and climatic

conditions (tropical North Atlantic versus mid-

latitude land). In addition, we have isolated an

intense-rainfall sequence from the GATE-1 data

(‘Sequence 1’), to verify that results are robust

relative to sub-sampling and the elimination of dry

periods. Table 1 collects basic information on the

different data sets and Fig. 1 shows the temporal

evolution of rainfall intensity. The blackened high-

intensity segment of GATE-1 corresponds to

Sequence 1.

The functions Cbetaðj;RjK1Þ, Clsðj;RjK1Þ, and aðj;

RjK1Þ estimated from the five data sets are plotted in

Fig. 2. This is the end product of Stage 1 of the

analysis. Rather than the resolution index j, Fig. 2

shows the length L (km) or duration T (min) of the

region considered. Since L and T are proportional to

2Kj, smaller values of L or T correspond to higher

le 1

racteristics of the data sets used in the numerical analysis

a set Temporal resolution Spatial resolution Instrume

e-1 15 min 4 km!4 km Radar

e-2 15 min 4 km!4 km Radar

ogna 15 min 1 km!1 km Doppler

rence 5 min – Tipping

values of j. For the space–time data sets, T varies

proportionally to L with TZ15 min for LZ4 km.

Consider first the results for GATE-1. When it does

not depend on RjK1 and L, the coefficient Cbeta

characterizes the fractal dimension Dwet of the rain

support, which for space–time rainfall is given by

DwetZ3KCbeta. Hence, Cbeta is a measure of sparse-

ness of the rainfall support. In the case of GATE-1,

Cbeta has a strong dependence on both L and RjK1;

therefore, this interpretation is valid at most locally

over a small range of scales and rainfall intensities. As

RjK1 or L decrease , Cbeta increases rapidly from 0 to

almost 3 (the Euclidean space–time dimension 3 is the

maximum possible value for Cbeta), meaning that the

rain support is more compact at larger scales and for

more intense rainfall.

The decrease of Cbeta with increasing RjK1 is due to

a combination of factors. First, regions of more

intense precipitation are generally associated with a

more compact rainfall support. Second, below

0.03 mm/h the rainfall intensity in the GATE data

sets is generally reported as zero, artificially increas-

ing the sparseness of the rainfall support. Finally, low

RjK1 values often occur across the dry/wet boundary,

where the lacunarity of the rainfall support is highest

and the positive rainfall intensities are just above the

threshold of radar detection.

The increase of Cbeta with decreasing region size is

due at least in part to the radar sensitivity threshold of

0.03 mm/h: since the spatial resolution is 4 km, for

RjK1 below that threshold and LZ8 km there must be

some dry L/2Z4 km sub-cells (and the number of

such dry cells must increase with decreasing RjK1),

whereas this is not necessarily true at larger scales.

Also, the parameter Cls depends on both RjK1 and

L. Dependence on RjK1 is non-monotonic, with a

maximum that increases for larger L. However, the

low values of Cls for small rainfall intensities are

artificial and poorly related to the corresponding

nt Time period Spatial extension

June 28–July 15 1974 Disc with 200 km radius

July 28–August 15 1974 Disc with 200 km radius

radar 1995–2004 Disc with 250 km radius

bucket 1962–1985 –

Fig. 1. Time series of rainfall intensity for the data sets used in the numerical analysis. For the radar sequences, the value plotted is the average

intensity over the spatial region analyzed. The blackened subset of GATE-1 corresponds to Sequence 1. For Florence data, one of the 24 years of

data is shown.

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119110

parameter Cls: they are due to the fact that Cbeta is

large and the sum CtotZ CbetaC Cls cannot exceed 3,

the Euclidean dimension of the support. For high

rainfall intensities this bias is not present and Cls is

close to 0.25 at all scales.

Finally, the index of stability a has values between

about 1 and 1.8 and is higher for smaller L and higher

RjK1. One could examine in detail the reasons for this

peculiar behavior (including the increase for very low

rainfall intensities, which is due to the radar detection

threshold), but as we shall see the same behavior is not

shared by the index a, which is the parameter we are

ultimately interested in.

A striking feature of Fig. 2 is the similarity of the

statistics produced by different data sets. The results for

GATE-1 and GATE-2 are virtually undistinguishable

and are robust relative to sub-sampling. In addition to

using Sequence 1 (a segment of GATE-1 with high

rainfall intensity), we have repeated the analysis for

subsets of GATE-1 obtained by partitioning the time

axis into four equal intervals, as well as for the subsets

when RjK1 is increasing or decreasing over time. The

latter analysis was made to detect any effect of rainfall

build-up and decay on the multifractal parameters. In

all cases, the results were virtually identical to those for

the entire GATE-1 record.

Also, the results from the Bologna radar sequence

are very similar (this sequence is much shorter than

either GATE-1 or GATE-2 and the range of scales

that could be investigated is consequently narrower).

Finally, qualitative similarity is observed with the

statistics from the Florence record, considering that

for temporal series the Euclidean dimension of the

observation space is 1 and one must have

CbetaC Cls%1.

Fig. 3 shows results when the same procedure is

applied to simulations from five different multi-

plicative cascades in space–time:

1.

A lognormal multifractal cascade. The cascade is

uncensored, i.e. it has no lower detection threshold

(LN/UNC model);

Fig. 2. Empirical beta-logstable parameters for the GATE-1, GATE-2, Sequence 1 (a subset of GATE-1), Bologna, and Florence data sets.

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 111

Fig. 3. Beta-logstable parameters of simulated data sets from the following cascade models: uncensored LN (LN/UNC), uncensored beta-LN

(BETA-LN/UNC), censored LN (LN/CEN), censored beta-LN (BETA-LN/CEN), and best-fitting cascade with variable parameters (BETA-

LN/VAR/CEN).

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119112

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 113

2.

Fig

at 4

An uncensored beta-lognormal multifractal cas-

cade (BETA-LN/UNC model);

3.

Same as LN/UNC with censoring (LN/CEN

model). Censoring consists of setting to zero the

dressed intensities at 4 km resolution that are

below 0.01 mm/h and randomly setting to zero the

cells with intensity between 0.01 and 0.03 mm/h,

with a probability that decreases as rainfall

intensity increases. This is done to reflect the

sharp decrease in the empirical probability density

function of rainfall intensity at the 4 km scale in

the GATE records; see Fig. 4 for GATE-1. Similar

rapid decreases below 0.03 mm/h are present in the

other radar data sets;

4.

A similarly censored version of BETA-LN/UNC

(BETA-LN/CEN model);

5.

A censored beta-lognormal model like Model 4, in

which the parameters Cbeta and Cls vary optimally

with L and Rb,jK1 (BETA-LN/VAR/CEN model).

The parameters are shown as solid lines in Fig. 5.

They were obtained by trial and error as described

in Section 2 to best reproduce the {Cbeta, Cls, a}

statistics of Sequence 1 in Fig. 2. In generating

dressed measures, the laws in Table 2 have been

assumed to hold also at scales smaller than

4 km/15 min, with c(L)Zc(4 km) in the expression

of Cls.

The parameters used in the model simulations are

summarized in Table 2; they were generally selected

. 4. Probability density function of rainfall intensity for GATE-1

km/15 min resolution.

to reproduce overall characteristics of the GATE data

sets.

This model selection emphasizes differences in the

lacunarity of the rain support (no lacunarity in the

LN/UNC model, fractal lacunarity independent of

rainfall intensity in the BETA-LN/UNC model, non-

fractal lacunarity due to censoring at small scales and

for low rainfall intensities in the LN/CEN, BETA-

LN/CEN and BETA-LN/VAR/CEN models), and the

effect of fixed versus variable distribution of the

generator (Models 1–4 versus the BETA-LN/VAR/

CEN model). In all cases, we have set aZ2, thus

assuming that W has beta-lognormal distribution. The

main reason for setting aZ2 is that the beta-

lognormal simulations produce statistics q very close

to the empirical ones, including values of a much

smaller than 2. Another reason is that, while certainly

possible, values of a below 2 have a small domain of

attraction (in the sense of the central limit theorem)

and therefore are regarded as less likely to occur in

nature. For a discussion of this last issue, see

Veneziano and Furcolo (2003).

All cascade models are fully developed, meaning

that appropriate ‘dressing factors’ are applied to

account for fluctuations of rainfall intensity below the

scale of the observations. The parameters used in the

model simulations are summarized in Table 2; they

were generally selected to reproduce overall charac-

teristics of the GATE data sets.

The results from the uncensored models LN/UNC

and BETA-LN/UNC in Fig. 3 are very different from

those of radar rainfall data in Fig. 2. In particular, in

LN/UNC it rains everywhere, with unrealistic par-

ameters Cbetah0 and ClsZconstant. The addition of

a beta component (model BETA-LN/UNC) makes

Cbeta positive, but again the behavior of Cbeta and Cls

is very different from that of GATE. The introduction

of censoring in the LN/CEN and BETA-LN/CEN

models increases Cbeta for low RjK1, but this effect is

confined to rainfall intensities below the censoring

threshold and the dependence on L observed in GATE

is not reproduced. Another problem with the censored

beta-lognormal model is that Cbeta remains non-zero

also at high rainfall intensities. By contrast, physical

rainfall has a compact support in regions of high

intensity. One concludes that multifractal models

(models with identical distribution of the generators

W) are inadequate.

Fig. 5. Assumed generator parameters {Cbeta, Cls, aZ2} (solid lines) and estimated partition coefficient parameters {Cbeta, Cls, a} (dashed lines)

for best-fitting variable-parameter BETA-LN/VAR/CEN model.

Table 2

Parameters used in model simulations

Model Cbeta Cls a Censoring

LN CbZ0 ClsZ0.261 2 None

LN censored CbZ0 ClsZ0.261 2 0.01–0.03 mm

Beta-LN CbZ0.2 ClsZ0.4 2 None

Beta-LN censored CbZ0.2 ClsZ0.4 2 0.01–0.03 mm

Best fitting CbðR;LÞZK0:2K1:33 LogðLÞK

0:93 LogðRÞ

ClsðR; LÞZmax

0:15; 0:15KcðLÞ Log R10

� �� �c(L) as in

Fig. 5

2 0.01–0.03 mm

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119114

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 115

A far better match with the GATE results is

obtained by allowing the parameters Cbeta and Cls to

vary with L and Rb,jK1, as is done in the BETA-

LN/VAR/CEN model (hereafter called also ‘best-

fitting’ model). In this case, the empirical values of

{Cbeta, Cls, a} are very well reproduced, both

qualitatively and quantitatively, providing further

indication that space–time rainfall does not behave

like a multifractal process.

It is interesting to compare the parameters {Cbeta,

Cls, aZ2} of the variable generators W with the

parameters {Cbeta, Cls, a} obtained from the simu-

lations. This is done in Fig. 5, where the dashed lines

are the simulation estimates and the solid lines are the

true parameters. While Cbeta and Cls are close to Cbeta

and Cls at high rainfall intensities, Cls is much lower

than Cls at low rainfall intensities. The good

correspondence for high rainfall intensities is due to

the fact that, for high RjK1, the model behaves like a

lognormal multifractal cascade with a low co-dimen-

sion coefficient Cln. In this case, the function KðqÞ is

close to K(q), at least in the range 0!q%2 used to

find q. By contrast, when RjK1 is low the rainfall field

is highly lacunar (also due to censoring). Lacunarity

reduces the value of Cls below Cln. For example, in

Fig. 6. Comparison of marginal distributions for S

the extreme case when 7 out of the 8 sub-cells at level

j are dry, Cls is necessarily zero.

As previously noted, there is also a marked

difference between aZ2 and a, showing that direct

inference of the stability index a is severely biased.

The BETA-LN/VAR/CEN model performs much

better than multifractal models also in reproducing the

marginal distribution of rainfall at fine scales. A

comparison with the empirical distribution at

4 km/15 min resolution for RO0.1 mm/h is shown

in Fig. 6. A straight line corresponds to a lognormal

distribution (except for a short transient at low values

due to the condition RO0.1 mm/h). Clearly, neither a

lognormal nor a beta-lognormal multifractal cascade

produce acceptable fits, whereas the model with

variable parameters is very close to the Sequence 1

data. Fig. 7 shows a similar comparison for the

correlation functions in space (along the X and Y

coordinate directions) and time. In this case, the

correlation functions for Sequence 1 are intermediate

between those of the beta-lognormal and variable-

parameter models, whereas the lognormal cascade de-

correlates too fast.

Finally, we have conducted analyses of the type in

Over and Gupta (1996) on both the GATE

equence 1 and three models with censoring.

Fig. 7. Comparison of spatial and temporal correlation functions for Sequence 1 and three models with censoring.

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119116

and simulated sequences. Recall that in this type of

analysis one estimates Cbeta and Cls separately for

each spatial frame and then one examines the

dependence of these parameters on the average

rainfall intensity �R over the frame. Results for

Sequence 1, the best-fitting (variable parameter)

model and the censored beta-lognormal model are

compared in Fig. 8. Again, the variable-parameter

model reproduces quite well the empirical statistics,

whereas the fix-parameter multifractal alternative

shows little dependence on �R. It is also interesting

that the Cbeta parameter from the best-fitting model

generally follows the behavior assumed in Fig. 5,

whereas the parameter Cls does not and remains very

low (as for Sequence 1). This means that the Over–

Gupta analysis, which conditions on �R rather than j

and RjK1, produces parameters that may be very

different from the actual ones.

One may note that in some cases, the estimates of

Cls in Fig. 8 are negative. In the multifractal case, the

function K(q) from which these estimates are obtained

is concave and ClsR0. However, for non-multifractal

processes this condition of concavity needs not hold

and this is why in some cases we obtain negative

values of Cls. This tends to occur when Cbeta is large,

hence for low rainfall intensities �R. While the values

from Sequence 1 are generally positive, estimates

from low-intensity portions of the GATE records

produce negative values.

5. Critical evaluation and conclusions

We have used a flexible multiplicative cascade

model to investigate departures of time and space–

time rainfall from scale invariance. The model

possesses multifractal scale invariance under certain

conditions (when the generators W at different

cascade levels j are independent and identically

distributed). We have assumed that W has beta-

logstable distribution, which includes most of the

scaling models proposed in the past for rainfall, such

as the beta model of Gupta and Waymire (1990) and

Over and Gupta (1994), the beta-lognormal model of

Over and Gupta (1996), and the conservative

universal models of Schertzer and Lovejoy (1987)

Fig. 8. Over and Gupta statistics for Sequence 1, the best-fitting model (BETA-LN/VAR/CEN), and the censored beta-lognormal model

(BETA-LN/CEN).

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 117

and Marsan et al. (1996). The main novelty of the

model is that the distribution of W is allowed to

depend on the cascade level j and the bare rainfall

intensity at the immediately coarser scale, Rb,jK1.

Fitting the model to various data sets has revealed

significant departures from multifractality, as the

distribution of W has been found to depend on both

Rb,jK1 and j. These dependencies reflect the following

features of physical rainfall:

1.

Let Uj be a cascade tile at resolution level j, with

sub-tiles UjC1 at level jC1. Denote by Rbj the bare

rainfall intensity in Uj and by Pdry,jC1 the

probability that the generic sub-tile UjC1 is dry.

In the multifractal case, Pdry,jC1 is independent of

both j and Rbj, whereas in rainfall Pdry,jC1 is higher

for larger j and smaller Rbj. Hence, rainfall displays

more lacunarity in smaller regions and for lower

rainfall intensities;

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119118

2.

Consider now the region of space–time where it

rains. In multifractal rainfall, the fluctuations of

log-rainfall intensity at equally spaced log-scales

are statistically identical (this corresponds to the

generator W having the same distribution for all j).

In physical rainfall, the fluctuations at smaller

scales and in higher-intensity regions are statisti-

cally smaller. Hence, rainfall intensity is relatively

smooth at small scales and during heavy

downpours.

These deviations from multifractality are consist-

ently observed in temporal rain gauge records and

space–time radar sequences, and in different climates;

hence, they appear to be general properties of rainfall.

The above results are robust relative to the assumed

distribution of the generator W (here taken to be of the

beta-logstable type).

What is the implication of these findings on the

stochastic modeling of rainfall? Multiplicative cas-

cades with variable parameters are used here as a

means to probe the presence or absence of scale

invariance, not because they are the preferred models

of rainfall. Indeed, having found that a cascade

representation requires variable and dependent gen-

erators and hence extensive parameterization and

complex inference procedures, one might argue that

one should abandon scaling approaches to rainfall and

return to classical stochastic models. This is indeed

one possibility.

Another possibility, which we find more attractive,

is to seek new paradigms that include scale invariance

in ways different from the past. Specifically, if scale

invariance in rainfall is inherited from atmospheric

turbulence, then one must revisit the relation between

these two phenomena and see whether something

fundamental is left out in current scaling represen-

tations of rainfall. Atmospheric turbulence takes place

in 3 spatial dimensions plus time, i.e. in (3C1)-space.

This is true also for the condensation of water vapor.

By contrast, what one observes as precipitation is the

flow of condensed water through a constant-altitude

plane. One may reasonably expect that the rate of

water vapor condensation is multifractal largely due

to its link to atmospheric turbulence. However, the fall

of condensed particles and their observation as

constant-altitude rainfall destroy any scale invariance

that might be present in the condensation rate.

A simple way to represent the relation between

condensation rate in (3C1)-space and constant-

altitude rain rate in (2C1)-space is to view the latter

as a projection (an integral) of the former along a

space–time direction that depends on the direction and

speed of falling raindrops. This line of inquiry will be

the subject of a separate communication. Suffices here

to say that ‘integrated’ multifractal models of this type

indeed explain the deviations of rainfall from multi-

fractality that have been observed in the present study.

Hence, it appears that the simplicity of multifractal

scale invariance can be recovered at the expense

of augmenting the space dimension from (2C1) to

(3C1), starting with a model of the water vapor

condensation rate rather than rainfall itself.

Acknowledgements

This research was supported by the National

Science Foundation under Grant EAR-0228835.

Additional support was provided by the Italian

National Research Council through the GNDCI-

RAM program (Gruppo Nazionale per la Difesa

dalle Catastrofi Idrogeologiche—Ricerca Applicata

in Meteoidrologia). The authors are grateful to

Servizio Meteorologico Regionale of Regione Emilia

Romagna for providing data from the meteorological

radar of San Pietro Capofiume, Bologna and to an

anonimous reviewer for his/her useful comments.

References

Becchi, I., Castelli, F., 1989. Lettura e archiviazione digitali di

registrazioni pluviografiche ad alta risoluzione temporale.

Technical Report 1/89, Department of Civil Engineering,

University of Florence, Italy.

Benjoudi, H., Hubert, P., Schertzer, D., Lovejoy, S., 1997.

Interpretation multifractale des courbes intensite’ -duree-

frequance des precipitations. Geosciences de Surface (Hydro-

logie)/Surface Geosciences (Hydrology) 325, 323–336.

Benjoudi, H., Hubert, P., Schertzer, D., Lovejoy, S., 1999.

Multifractal explanation of rainfall intensity–duration–fre-

quency curves, Proceedings, EGS, 24th General Assembly,

The Hague, The Netherlands, April 19–23.

Carsteanu, A., Foufloula-Georgiou, E., 1996. Assessing dependence

among weights in a multiplicative cascade model of temporal

rainfall. Journal of Geophysical Research 101 (D21),

26,363–26,370.

D. Veneziano et al. / Journal of Hydrology 322 (2006) 105–119 119

Deidda, R., 2000. Rainfall downscaling in a space–time multifractal

framework. Water Resources Research 36 (7), 1779–1794.

Deidda, R., Benzi, R., Siccardi, F., 1999. Multifractal modeling of

anomalous scaling laws in rainfall. Water Resources Research

35 (6), 1853–1867.

Deidda, R., Badas, M.G., Piga, E., 2002. Space–time scaling

behavior of GATE and TOGA-COARE rainfall. EGS XXVII

General Assembly, Nice, France, April.

Ferraris, L., Gabellani, S., Parodi, U., Rebora, N., Von

Hardenberg, J., Provenzale, A., 2003. Revisiting multifractality

in rainfall fields. Journal of Hydrometeorology 4, 544–551.

Fraedrich, K., Larnder, C., 1993. Scaling regimes of composite

rainfall time series. Tellus 45A, 289–298.

Frish, U., 1985. Fully developed turbulence and intermittency. In:

Ghil, M., Benzi, R., Parisi, G. (Eds.), Turbulence and

Predictability in Geophysical Fluid Dynamics. Elsevier, New

York, pp. 71–84.

Frish, U., Parisi, G., 1985. On the singularity structure of fully

developed turbulence. In: Ghil, M., Benzi, R., Parisi, G. (Eds.),

Turbulence and Predictability in Geophysical Fluid Dynamics.

Elsevier, New York, pp. 84–88.

Gupta, V.K., Waymire, E., 1990. Multiscaling properties of spatial

rainfall and river flow distributions. Journal of Geophysical

Research 95 (D3), 1999–2009.

Gupta, V.K., Waymire, E.C., 1993. A statistical analysis of

mesoscale rainfall as a random cascade. Journal of Applied

Meteorology 32 (2), 251–267.

Harris, D., Menabde, M., Seed, A., Austin, G., 1998. Breakdown

coefficients and scaling properties of rain fields. Nonlinear

Processes in Geophysics 5, 93–104.

Hudlow, M.D., Patterson, V.L., 1979. Gate Radar Rainfall Atlas,

NOAA Special Report. US Department of Commerce,

Washington, DC.

Lovejoy, S., Schertzer, D., 1995. Multifractals and rain. In:

Kundzewicz, A.W. (Ed.), Uncertainty Concepts in Hydrology

and Hydrological Modelling. Cambridge Press, Cambridge,

pp. 62–103.

Marsan, D., Schertzer, D., Lovejoy, S., 1996. Causal space–time

multifractal processes: predictability and forecasting of rain

fields. Journal of Geophysical Research 101 (D21),

26333–26346.

Menabde, M., Harris, D., Seed, A., Austin, G., Stow, D., 1997.

Multiscaling properties of rainfall and bounded random

cascades. Water Resources Research 33 (12), 2823–2830.

Menabde, M., Seed, A., Pegram, G., 1999. A simple scaling

model for extreme rainfall. Water Resources Research 35 (1),

335–339.

Meneveau, C., Sreenevasan, K.R., 1987. The multifractal spectrum

of the dissipation field in turbulent flow. Nuclear Physics B,

Proceedings Supplements 2, 49–76.

Olsson, J., 1995. Limits and characteristics of the multifractal

behavior of a high-resolution rainfall time series. Nonlinear

Processes in Geophysics 2, 23–29.

Olsson, J., Berndtsson, R., 1998. Temporal rainfall disaggregation

based on scaling properties. Water Science Technology 37 (11),

73–79.

Olsson, J., Niemczynowicz, J., Berndtsson, R., 1993. Fractal

analysis of high-resolution rainfall time series. Journal of

Geophysical Research 98 (D12), 23265–23274.

Over, T.M., Gupta, V.K., 1994. Statistical analysis of masoscale

rainfall: dependence of a random cascade generator on large-

scale forcing. Journal of Applied Meteorology 33, 1526–1542.

Over, T.M., Gupta, V.K., 1996. A space–time theory of mesoscale

rainfall using random cascades. Journal of Geophysical

Research 101 (D21), 26319–26331.

Perica, S., Foufoula-Georgiou, E., 1996. Model for multiscale

disaggregation of spatial rainfall based on coupling meteor-

ological and scaling descriptions. Journal of Geophysical

Research 101 (D21), 26347–26361.

Samorodnitsky, G., Taqqu, M.S., 1994. Stable Non-Gaussian

Random Processes. Chapman & Hall, New York.

Schertzer, D., Lovejoy, S., 1987. Physical modeling and analysis of

rain and clouds by anisotropic scaling of multiplicative

processes. Journal of Geophysical Research D 92, 9693–9714.

Schmitt, F., Vannitsem, S., Barbosa, A., 1998. Modeling of rainfall

time series using two-state renewal processes and multifractals.

Journal of Geophysical Research 103 (D18), 23181–23193.

Svensson, C., Olsson, J., Berndtsson, R., 1996. Multifractal

properties of daily rainfall in two different climates. Water

Resources Research 32 (8), 2463–2472.

Tessier, Y., Lovejoy, S., Schertzer, D., 1993. Universal multi-

fractals: theory and observations for rain and clouds. Journal of

Applied Meteorology 32 (2), 223–250.

Veneziano, D., 1999. Basic properties and characterization of

stochastically self-similar processes in Rd. Fractals 7, 59–78.

Veneziano, D., 2002. Large deviations of multifractal measures.

Fractals 10 (1), 117–129.

Veneziano, D., Furcolo, P., 2002. Multifractality of rainfall and

scaling of the IDF curves. Water Resources Research 38 (12),

42.

Veneziano, D., Furcolo, P., 2003. Marginal distribution of

stationary multifractal measures and their haar wavelet

coefficients. Fractals 11 (3), 253–270.

Veneziano, D., Iacobellis, V., 2002. Multiscaling pulse representation

of temporal rainfall. Water Resources Research 38 (8), 13.

Veneziano, D., Bras, R.L., Niemann, J.D., 1996. Nonlinearity and

self-similarity of rainfall in time and a stochastic model. Journal

of Geophysical Research 101 (D21), 26371–26392.

Venugopal, V., Foufoula-Georgiou, E., Sapozhnikov, V., 1999.

Evidence of dynamic scaling in space–time rainfall. Journal of

Geophysical Research 104 (D24), 31599–31610.