Space–time multifractality of remotely sensed...

$: Space–time multifractality of remotely sensed …unica2.unica.it/rdeidda/papers/2006_JH_deibadpig.pdftwo different scenarios for applying multifractal theory to the analysis and$
Space–time multifractality of remotely sensed rainfall fields

Roberto Deidda*, Maria Grazia Badas, Enrico Piga

Dipartimento di Ingegneria del Territorio, Universita di Cagliari, Cagliari 09123, Italy

Accepted 8 February 2005

Abstract

A methodology aimed at characterizing the scaling properties of precipitation fields in space and time is revised and applied

to remotely sensed rainfall data retrieved during two oceanic campaigns (GATE and TOGA-COARE), and a land campaign

(TRMM-LBA). The presence of spatial heterogeneity induced by orography is investigated on data retrieved over land

(TRMM-LBA): the performed analyses show that the orographic induced heterogeneity seems to be negligible for the examined

data. Moreover, the scaling properties observed on rainfall over land are compared with those detected on ocean rainfall for

several space–time events. Results of a multifractal analysis enable a common calibration of the STRAIN space–time rainfall

downscaling cascade model for the three datasets. Generated synthetic fields preserve the observed rainfall space–time

variability.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Rainfall downscaling; Rainfall intensity; Fractals and multifractals; Flood risk

1. Introduction

Characterization of scaling properties of precipi-

tation is essential for assessing the hydrological risk in

small basins when numerical weather prediction

(NWP) models are used to provide rainfall forecast

over large scales of space and time. The concept

‘large scale’ is used here to refer to meteorological

model resolution in comparison with the rainfall field

resolution required in space and time for streamflow

forecasting in small catchments. Indeed, some

operational NWP models presently offer, with a one

to two-day lead time, rainfall field predictions

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

doi:10.1016/j.jhydrol.2005.02.036

* Corresponding author. Tel.: C39 070675 5324; fax: C39

070675 5310.

E-mail address: [email protected] (R. Deidda).

resolved down (or likely, if clustered-up) to scales

on the order of thousands of square kilometres in

space and several hours in time. On the hydrological

side, there is a need to forecast watershed streamflows

of a few hundred square kilometres or less, charac-

terized by a concentration time of a few hours or less.

At these smaller space–time scales, and specifically in

very small catchments and in urban areas, rainfall

intensity presents larger fluctuations, and therefore

higher values, than at the scale of meteorological

models. Thus the reliability of forecasts provided by

meteorological models depends also on the efficiency

of downscaling models, able to reproduce the sub-grid

rain rate fluctuations unresolved by NWP models.

Knowledge of scaling properties of rainfall, required

for the development of multifractal downscaling

models, can be achieved by the analysis of rainfall

Journal of Hydrology 322 (2006) 2–13

www.elsevier.com/locate/jhydrol

http://www.elsevier.com/locate/jhydrol

Fig. 1. A schematization of a downscaling problem: a rainfall

measure (volume or mean intensity) is given over a region L!L!T

(e.g. the grid resolution of a NWP model), we want to determine the

probability distribution of rainfall intensity over any smaller region

l0!l0!t0.

R. Deidda et al. / Journal of Hydrology 322 (2006) 2–13 3

fields retrieved from remote sensors capable of

providing a fine spatial and temporal resolution.

Moreover, the feasibility to tune the model parameters

on the basis of NWP large scale variables, such as

large-scale rain rate as suggested by Over and Gupta

(1994) and Deidda (2000) or CAPE (Convective

Available Potential Energy) as argued by Perica and

Foufoula-Georgiou (1996), represents another crucial

aspect when applying rainfall downscaling models in

a forecast chain.

Until now, researchers have primarily proposed

two different scenarios for applying multifractal

theory to the analysis and numerical simulation of

space–time rainfall fields. In the first scenario, scale–

invariance in space–time rainfall is analysed in a

self-similar framework, where a scale independent

velocity parameter U is simply used to relate

statistical properties at space scale l to those at time

scales tZl/U. In the second scenario, self-affinity

between space and time is assumed to hold in rainfall

fields, where Generalized Scale Invariance (G.S.I. see

Lovejoy and Schertzer (1985); Schertzer and Lovejoy

(1985) for details) can be investigated by a scale

change operator based on a scale dependent velocity

parameter UlflH. The scaling anisotropy exponent

H characterizes the degree of scale anisotropy

between statistical properties in space and those in

time. Regarding the choice between self-similar or

self-affine framework for space–time rainfall analysis

and simulation, Lovejoy and Schertzer (1991);

Tessier et al. (1993); Marsan et al. (1996) advocated

a theoretical value of the anisotropy exponent HZ1/3

for space–time rainfall due to turbulence advection,

while empirical results presented in Marsan et al.

(1996); Venugopal et al. (1999), Deidda (2000);

Deidda et al. (2004) seem to highlight the opportunity

to approximate space–time rainfall with self-similar

processes (HZ0). Following this simpler approach a

downscaling problem can be schematized as in Fig. 1:

L and T are the largest scales of space and time (e.g.

the resolution of a meteorological model) where the

rainfall volume (or the mean rainfall intensity) is

known or predicted, while l0 and t0 are smaller scales

where we want to downscale rainfall probability

distribution. Due to rainfall intermittency, it is

obvious that we can expect that the smaller the ratios

l/L and t/T, the larger the increase of rainfall intensity

fluctuation on the smaller scales.

Beside the choice of a self-similar or self-affine

scenario, the influence of geographical location or the

presence of ocean or land at its boundaries may play

an important role in rainfall scaling properties. When

‘land’ precipitation is considered we should also

investigate the possible presence of spatial hetero-

geneity induced by orography. Actually, for land

precipitation the probability distribution function of

rainfall intensity i(x,y,t) may not actually be the same

in each point of the space (x,y), because of orographic

influence. If any kind of spatial heterogeneity is

detected, it should be accounted for in the structure of

rainfall generators. This question has been already

addressed by Jothityangkoon et al. (2000) and

Pahirana and Herath (2002), Purdy et al. (2001),

who analyzed rainfall fields in different regions

identifying the presence of spatial heterogeneity.

In this paper we analyse rainfall fields retrieved

during the TRMM-LBA (Tropical Rainfall Measure-

ment Mission—Large Scale Biosphere Atmosphere

Experiment) land campaign with the aim to investi-

gate the orographic influence and to compare the

scaling behaviour with previous findings on oceanic

precipitation fields extracted from GATE (GARP,

Global Atmospheric Research Program, Atlantic

Tropical Experiment) and TOGA-COARE (Tropical

Ocean Global Atmosphere Coupled Ocean-Atmos-

phere Response Experiment) datasets. Extensive

information on the GATE, TOGA-COARE,

R. Deidda et al. / Journal of Hydrology 322 (2006) 2–134

and TRMM-LBA campaigns can be found in Hudlow

and Patterson (1979); Kruger et al. (1999); Carey et al.

(2000), respectively.

2. A framework for multifractal analysis

of space–time rainfall fields

Let i(x,y,t) be rainfall intensity in the point

(x,y)2[0,L]2 at time t2[0,TZL/U]. We can introduce

a l-partition of the space–time domain L!L!T,

where the side lengths of each grid-box are l in space

and tZl/U in time, and U is a scale invariant velocity

parameter. The total number of grid-boxes in the

l-partition is N(l)2N(t)Z(L/l)3, being N(t)ZT/tZN(l). The total measure m (i.e. the rainfall volume) on

each grid-box is:

mijkðlÞZ

ðxiCl

xi

dx

ðyjCl

yj

dy

ðtkCl=U

tk

dt iðx; y; tÞ (1)

where xiZ(iK1)l, yjZ(jK1)l and tkZ(kK1)l/U

identify the spatial position and the initial time of

each grid-box in the l-partition.

We can then introduce the following q-order

structure functions:

SqðlÞZ h½mijkðlÞ�qiZ

1

NðlÞ3

XNðlÞiZ1

XNðlÞjZ1

XNðlÞkZ1

½mijkðlÞ�q

(2)

where !$O is an ensemble average operator.

Rainfall fields are said to be scale-invariant if there

exists at least a range of scales where the following

‘scale-invariance’ law holds for any fixed moment q

SqðrLÞyrzðqÞSqðLÞ (3)

where r is a scaling factor. More simply the existence

of scale invariance ranges is often investigated with

the following power law:

SqðlÞflzðqÞ (4)

If the exponents z(q) characterizing the scaling

laws (3) and (4) are a linear function of q, the scaling

is defined as simple and the measure m is monofractal,

otherwise the scaling is said to be anomalous and the

measure is multifractal.

Sometimes the scaling of the mean rainfall rate 3 is

investigated instead of the scaling of the rainfall

volume m:

3ijkðlÞZ1

l3

ðxiCl

xi

dx

ðyjCl

yj

dy

ðtkCl=U

tk

dt iðx; y; tÞ (5)

Correspondently, scale-invariance can be written

as follows:

h½3ijkðlÞ�qiflKðqÞ (6)

Since 3ijk(l)Zmijk(l)/l3 the following relationship

holds between the two scaling exponents K(q) and

z(q):

KðqÞZ zðqÞK3q (7)

Again, a non-linearity of the K(q) exponents with q

reveals a multifractal nature in the analyzed field.

In a downscaling problem, a rainfall volume (or the

mean rain rate) is known over an area L!L and is

cumulated in a time TZL/U. Our aim is to predict the

probability distribution P(m(l)) of the measure m(l)

over smaller sub-regions l!l!t where l!L and

t!T. Scale-invariance laws defined in Eqs. (3), (4) or

(6) should be investigated in a wide range of space

scales lnZLbKns and timescales tnZln=UZTbKn

t .

For space–time rainfall displaying self-similarity, the

branching number bs in space should be the same as

the one in time bt.

The framework of analysis defined above is drawn

for self-similar measures, but it can be easily

generalized to self-affine measures by the introduction

of a scale dependent velocity parameter UlflH in

Eq. (1) or in Eq. (5), accordingly to the G.S.I.

formalism (Lovejoy and Schertzer, 1985; Schertzer

and Lovejoy, 1985). Indeed, in the case of self-affinity

(or scale anisotropy), scale-invariance can be inves-

tigated under anisotropic space–time transformations:

x/x/bs, y/y/bs, t/t/bt, where the branching

number bs in space now differs from the one bt in

time. Introducing the G.S.I. formalism the branching

numbers are related by btZbs(1KH), while using the

‘dynamic scaling’ formalism (Kardar et al., 1986;

Czirok et al., 1993; Venugopal et al., 1999) the

previous relationship becomes btZbzs. Actually, the

degree of scale anisotropy is characterized by

the ‘scaling anisotropy exponent’ H in the G.S.I.

formalism, or by the ‘dynamic scaling exponent’

Fig. 2. Distribution of log zx for the 102 TOGA-COARE sequences.


z using the second approach. The relationship zZ1KH holds between the two exponents. As a particular

case, for self-similar fields we should find HZ0 (or

zZ1).

An estimate of these exponents can be obtained by

comparing the one-dimensional power spectra Ex(fx)

or Ey(fy) in space and Et(ft) in time. In a multifractal

field we would expect to observe the following power

laws of frequency ft and wave-numbers fx or fy

EtðftÞf fKstt ExðfxÞf fKsx

x EyðfyÞf fKsyy (8)

where exponents sx, sy and st are constant in the

scaling ranges where Eqs. (4) and (6) hold. For a

multifractal measure we should also obtain:

EtðbtftÞ

EtðftÞh

ExðbsfxÞ

ExðfxÞh

EyðbsfyÞ

EyðfyÞ(9)

Comparing (8) and (9) we can estimate the scaling

anisotropy exponent as HxZ1Ksx/st or HyZ1Ksy/stwhen using G.S.I. formalism, and zxZsx/st or zyZsy/stwhen using dynamic scaling formalism. Obviously, in

cases of spatial isotropy, the estimates Hx and Hy

should be equal, as well as zx and zy.

The scaling of power spectra can thus reveal and

detect the self-similar or self-affine nature of space–

time rainfall: in the latter case of self-affinity, the

Hs0 or zs1 exponents characterize the degree of

scale anisotropy. Analyzing the estimates of H and z

on several space–time sequences extracted from the

TOGA-COARE dataset, Deidda et al. (2004) argued

and showed that the mean of these exponents is a

biased estimator of the average anisotropy degree.

Consequently, in order to verify if the average

behaviour of analysed precipitation fields is self-

similar or self-affine, they suggested referring to log z,

which can be proved to be an unbiased anisotropy

estimator. As an example, Fig. 2 displays a histogram

of log z estimates on 102 sequences selected from the

TOGA-COARE dataset; they appear to be quite

symmetrically distributed around a mean value log

zZ0, meaning that self-similarity between space and

time can be assumed on the average.

3. Influence of orography in TRMM-LBA rainfall

The possible presence of spatial heterogeneity

induced by orography is investigated here on rainfall

fields retrieved by the S-band polarimetric (NCAR

S-POL) radar in Rondonia (Brazil) during TRMM-

LBA campaign.

The raw S-POL data were processed and validated

by the Radar Meteorology Group of the Colorado

State University: they were corrected for the presence

of clear-air echo, ground clutter, anomalous propa-

gation, second-trip echoes, partial beam blocking,

precipitation attenuation, and calibration biases by

means of polarimetric methods, while for gaseous

attenuation a simple range correction with a coeffi-

cient was adopted. Applying an optimal polarimetric

radar technique, maps of rain rate have been

calculated from observations of S-POL horizontal

reflectivity, differential reflectivity, and specific

differential phase every 10 min from 10 January to

28 February 1999. Monthly estimates were compared

to those coming from 33 tipping rain gauges available

within the radar area, the resulting standard error and

bias are typical for polarimetric radar-to-gauge

comparison and range from 14 to 20%. The data

analyzed in this paper were extracted from the product

of this validation procedure. Further details on the

processing and validation methodology are reported

in Carey et al. (2000).

Following the same approach used for GATE and

TOGA-COARE datasets (Deidda, 2000; Deidda et al.,

2004) a preliminary investigation on the degree of

self-affinity between space and time was conducted on

53 rainfall events extracted from TRMM-LBA

dataset. In particular it was found that also TRMM-

LBA fields can be assumed as self-similar on the

average, with a scale independent velocity parameter

UZ24 km/h relating coherent space and time scales.

Fig. 3. Orography of the area corresponding to selected sequences at

the resolution of 4 km!4 km.


The 53 independent events were extracted over a

square area of side LZ128 km, each one

collecting the most intense consecutive scans over

periods TZL/UZ5 h and 20 min in time. The finest

resolution of each sequence is l0Z4 km in space and

t0Z10 min in time. The spatial domain 128 km!128 km of the selected sequences is almost centered in

the radar area and external to the zone affected by

partial and occasionally complete blocking which was

identified during the validation process; selected

events are evenly spread over the analyzed period

(10 January–28 February 1999). Terrain elevation of

the radar area was extracted at 1 km resolution from

Fig. 4. Map of mean rainfall intensities over 4 km!4 km for the 53 TR

generated fields (right).

the web site of the GLOBE project (NOAA, National

Environmental Satellite, Data and Information Ser-

vice), and averaged on the same lattice (with 4 km!4 km resolution) of the selected rain fields (Fig. 3).

The comparison of Fig. 3 with the map of mean

rainfall intensity (time-averaged for each 4 km!4 km

grid cell on all the selected 53 TRMM-LBA

sequences), displayed in the left-hand plot of Fig. 4,

does not reveal any evident relationship between

rainfall intensity and terrain pattern.

The mean rainfall intensities displayed in the left-

hand plot of Fig. 4 are also plotted against the

corresponding elevations in the left-hand plot of

Fig. 5. In order to reduce the rain rate variability, the

same data were averaged on classes of 30 m of

elevation and are presented in the left-hand plot of

Fig. 6. In both Figs. 5 and 6, rain rates are spread

around a mean value of about 0.8 mm/h but there does

not seem to be any identifiable trend of rainfall

intensity relating to elevation.

In order to empirically investigate if data fluctu-

ations could be interpreted as just sample variability,

53 space–time rainfall sequences were generated with

the STRAIN model (described in Section 4 and in

greater detail in Deidda, 2000), whose parameters

were estimated on each of the TRMM-LBA selected

sequences. The right-hand plots of Figs. 4–6 are

obtained from the generated sequences and display

the same variables as the corresponding left-hand

plots discussed above. Although the model does not

MM-LBA observed sequences (left), and for the corresponding 53

Fig. 5. Mean rainfall intensities plotted against mean elevations for each cell 4 km!4 km computed on the 53 TRMM-LBA observed sequences

(left), and the 53 corresponding generations (right).


take into account the orography, since it generates

homogeneous and isotropic rainfall fields, graphs

referring to generated data (right-hand plots) are very

similar to those obtained from the observed sequences

(left-hand plots).

This comparison shows that, in this region,

orography does not seem to introduce significant

heterogeneity on examined rainfall sequences, and

supports the feasibility of performing a scale-

invariance analysis on TRMM-LBA sequences with-

out taking orography into account.

The opportunity of introducing a heterogeneous

component in the modeling of synthetic rainfall fields

Fig. 6. Mean values of rainfall intensity computed for classes of elevation

TRMM-LBA sequences (left), and the 53 generated sequences (right).

over land was investigated in previous works by

Jothityangkoon et al. (2000); Pahirana and Herath

(2002), Purdy et al. (2001) leading to different results

with respect to those obtained for TRMM-LBA

rainfall. Specifically Jothityangkoon et al. (2000);

Pahirana and Herath (2002) analyzed a 400 km!400 km area in southwestern Australia and a

128 km!128 km region centered in the Japanese

archipelago respectively, and multiplied a spatial

random cascade by a deterministic factor depending

on the spatial location in order to reproduce observed

spatial heterogeneity. Analyzing a transect along the

Southern Alps of New Zealand, Purdy et al. (2001)

plotted against corresponding mean elevations for the 53 observed


found a dependence of scaling parameters on

orography and rain features. We can identify mainly

two reasons accounting for the different outcome of

our analysis: a physical argument lies in the limited

orographic range (the maximum altitude of the

analyzed area is 500 m on the sea level) which does

not represent a significant obstacle for perturbations;

while a second explanation may be bound to the

relatively brief observation period, which may not

allow filtering the single events high space–time

variability from the possible spatial heterogeneity.

4. Multifractal properties of the analyzed

sequences

A multifractal analysis based on 187 space–time

rainfall sequences extracted from GATE, TOGA-

COARE and TRMM-LBA datasets is discussed with

the aim of investigating whether a common interpret-

ation of scaling behaviour of rainfall retrieved in

different geographical conditions can be applied.

The 102 sequences extracted from the TOGA-

COARE dataset are the same analyzed in Deidda et al.

(2004), and are defined on a domain with extension

LZ128 km in space and TZ5 h and 20 min in time,

while the finest resolution is l0Z4 km in space and

t0Z10 min in time, accordingly to a velocity

parameter UZ24 km/h. From the GATE dataset 32

new sequences were selected on a different domain

with respect to those analyzed in Deidda (2000) in

order to match a large scale dimension closer to the

one of the sequences extracted from the other datasets.

Specifically, these sequences are defined on a domain

with extension LZ128 km in space and TZ8 h in

time, while the finest resolution is l0Z4 km in space

and t0Z15 min in time, accordingly to a velocity

parameter UZ16 km/h. The 53 sequences extracted

from TRMM-LBA campaign were already described

in the previous section.

Scale-invariance under self-similar transform-

ations was investigated on the sequences belonging

to the three datasets. Structure functions Sq(l), as

defined in Eq. (2), were computed for different

moments q. As an example, the third order structure

functions S3(l) computed for the 25 highest intensity

sequences of each dataset are plotted in Fig. 7 in log–

log plane. The picture shows a very good scaling for

the whole range of examined scales and for all the

analysed fields. Sample points follow a linear trend

revealing that scale-invariance holds: the slopes of the

corresponding best fit linear regressions, also plotted

in the same Fig. 7, are sample estimates of multifractal

exponents. As an example, in Fig. 8 sample multi-

fractal exponents z(q) estimated on two sequences are

plotted for some moments q, the multifractality of

these fields is apparent from the non-linear behaviour

of exponents z(q).

Sample estimates of multifractal exponents z(q) can

be used to calibrate multifractal downscaling models,

for which the theoretical expectation of z(q) is usually

expressed as a function of generator parameters and q.

Here the behaviour of the multifractal exponents z(q)

is interpreted by the STRAIN cascade model (Deidda,

2000), which is based on a log-Poisson generator hZby, where y is a Poisson distributed variable with mean

c (Dubrulle, 1994; She and Leveque, 1994; She and

Waymire, 1995, Deidda et al. (1999)). The choice of a

log-Poisson generator has the following advantages:

(i) having the property of being a infinitive divisible

distribution, satisfying conditions of moments’ con-

vergence and consistency for model parameters

belonging to sample ranges (Ossiander and Waymire,

2000; Deidda et al., 2004); (ii) being a very simple

probability distribution to introduce in a multiplicative

cascade; and (iii) requiring the introduction and

calibration of only two parameters, namely b and c.

The theoretical expectation of multifractal exponents

for this model results:

zðqÞZ 3qCc

ln 2½ð1KbqÞKqð1KbÞ� (10)

The two model parameters b and c were estimated

by a least square difference method between the set of

sample multifractal exponents z(q) of each sequence

and the theoretical expectations (10), for qZ1–6,

using weights suggested in Deidda (2000). Since for

the three datasets the b parameter has shown a limited

variability around a mean value close to eK1, the best

fit procedure was applied again to estimate the c

parameter fixing bZeK1 in Eq. (10), allowing the

comparison among c parameter estimates obtained for

the three datasets.

For sequences extracted from the TRMM-LBA

data the dependence of the model parameters from

meteorological condition was also investigated.

Fig. 7. Structure functions S3(l) for the 25 most intense sequences of GATE, TOGA-COARE and TRMM-LBA, plotted in a log-log plane. Structure functions have arbitrary units to

allow displaying of several sequences on the same graph.

R.Deid

daet

al./JournalofHydrology322(2006)2–13

9

Fig. 8. Sample multifractal exponents z(q) estimated for two sequences. Lines represent theoretical expectation of multifractal exponents

accordingly to Eq. (10).


Analyzing TRMM-LBA data Carey et al. (2000)

identified two meteorological regimes occurring

during the wet season in Rondonia, namely westerly

and easterly regimes, different in thermodynamic,

kinematic, and mycrophysics. As a consequence

rainfall occurring in the two conditions have different

characteristics concerning vertical structure, horizon-

tal organization, amount of stratiform precipitation,

frequency of occurrence of intense rain rates; 27 of the

53 selected sequences belong to the westerly regime,

Fig. 9. Sample c estimates obtained for TRMM-LBA sequences belonging

plotted against mean field intensity together with the best fit line for all the s

plotted against mean field intensity, for the sequences extracted from TRM

while 25 occurred during the easterly regime and only

one falls between the two regimes. Although easterly

sequences have a greater mean intensity than westerly

ones, the trend of c parameter estimates with the large

scale rainfall intensity (left-hand plot of Fig. 9)

appears to be the same for the two groups, therefore

all the TRMM-LBA sequences were considered as a

whole.

The estimates of c parameter for all the analysed

data are plotted in Fig. 9 versus the large scale

to the easterly regime (circle) and to the westerly one (square) are

equences (left). Sample c estimates, and corresponding best fit lines,

M-LBA, TOGA-COARE and GATE datasets (right).


rainfall intensity I, using a different symbol for

each dataset. Despite a great dispersion of c

estimates it is apparent that there is a decreasing

trend of the sample c parameter with the sequence

mean rain intensity I, which was interpreted for

each dataset by equations in the following form,

plotted also in right-hand plot of Fig. 9:

cðIÞZ a expðKgIÞCcN (11)

Empirical analyses showed that the best fit line

(11) and in particular the asymptotic value cN is

very sensitive to the few c estimates corresponding

to the sequences with highest value of large scale

intensity I. For this reason, slight difference among

best fit lines do not seem to justify the choice of a

different relation between c and I for each dataset.

The best fit procedure was therefore applied on all

the data giving the following expression for the

mean of c values:

mcðIÞZ 0:8694 expðK0:7248IÞC0:9531 (12)

An analysis of the coefficient of variation of c

values (CVc) performed on classes of intensity did

not show any trend with large scale mean intensity

I, CVc was therefore considered as constant and

equal to its mean value 0.16. In Fig. 10, Eq. (12) is

plotted (solid line) together with the range c(I)Zmc(I)[1GCVc], represented with dashed lines, which

includes the bulk of the c estimates plotted again in

the same figure.

Fig. 10. Sample c estimates for the sequences of all the datasets,

corresponding best fit line mc(I), and the range mc(I)[1GCVc].

For each of the 187 examined sequences, a set of

100 synthetic space–time rainfall fields was generated

using the STRAIN model, with parameters bZeK1

and c given by Eq. (12) for the same large scale rain

rate I of the corresponding observed sequence.

The cumulative distribution functions (CDFs) of

rainfall intensities of each observed sequence at the

smallest space–time scales was compared with

the 90% confidence limits obtained by the corre-

sponding set of synthetic sequences. As an example,

Fig. 11 illustrates this kind of comparison for three

sequences. In each figure both the observed CDF and

the confidence limits are shown for the entire

distribution with a zoom on the probability axis

from 0.5 to 0.9, corresponding to the bulk of non-zero

rain in the examined sequences. Most of the observed

CDF, as well as the ones plotted in Fig. 11, stay within

the confidence limits meaning a correct reproduction

of rainfall at smaller scales of space and time, both for

the extreme values and for the core of the distribution.

Nevertheless, occasionally the STRAIN model seems

to fail the reproduction of observed CDF, as already

pointed out in Deidda et al. (2004), where it was

shown that this kind of failures can be attributed to the

simplicity of Eq. (12), relating the only free model

parameter to the large-scale rainfall intensity. Deidda

et al. (2004) proved also that applying the STRAIN

model with c sample parameters estimated on the

failing test cases, the STRAIN model was indeed able

to correctly reproduce the observed CDF. This would

suggest the opportunity to relate downscaling model

parameters also to other meteorological variables: the

evaluation of the advantages in using more predict-

able variables rather than only one, as in this work and

in the others already referred, can be the object of

further investigation activities.

5. Conclusions

Space–time precipitation fields retrieved by remote

sensors in different ocean and land campaigns were

analysed in order to characterize scale-invariance

properties and to investigate a possible common

scaling behaviour. It was considered advisable to

assess the influence of orography on precipitation data

retrieved during the TRMM-LBA land campaign.

Nevertheless, results of this analysis did not highlight

Fig

.1

1.C

um

ula

tiv

ed

istr

ibu

tio

nfu

nct

ions

of

rain

fall

rain

rate

ath

igh

erre

solu

tio

nar

eco

mp

ared

wit

hth

e9

0%

con

fid

ence

lim

its

esti

mat

edfr

om

syn

thet

icfi

elds

for

one

sequen

ceof

each

dat

aset

.T

he

com

par

iso

nis

sho

wed

for

all

the

dis

trib

uti

on

wit

ha

zoo

mo

nit

sco

re(F

(i)Z

0.5

–0

.9)

inth

esm

all

bo

x.


any evidence of relationship between rainfall intensity

and terrain pattern, and consequently spatial hom-

ogeneity was assumed to hold for the examined

sequences. It should be noted that the model has been

applied to rainfall occurring over a relatively flat

region, quite different results can be obtained analyz-

ing areas with steeper terrain. Furthermore the

analyzed data belong to a relatively short period and

a possible presence of spatial heterogeneity might

have been masked by rainfall random variability.

The rainfall fields analysed in the GATE, TOGA-

COARE, and TRMM-LBA campaigns were found to

be scale-invariant under self-similar transformations.

The observed scaling properties and the detected

multifractal behaviour allowed calibrating the

STRAIN model, in which a log-Poisson distribution

is used as cascade generator. A possible dependence

of the two model parameters on the large scale

intensity I has been examined. The b parameter did

not show any trend with intensity I, thus it was fixed to

the value bZeK1, while the c parameter remains the

only model parameter that requires being calibrated.

Since c estimates displayed a decreasing trend with

intensity I, a simple relationship, given in Eq. (12),

was determined to account for this dependence. This

simple relationship allows varying the only free

model parameter, and thus the scaling behavior as

well, accordingly to the large scale rainfall intensity I,

which is a quantity predicted by any NWP model.

Despite the simplicity of the assumed self-similar

framework and of the calibration relationship, the

proposed downscaling procedure allows a good

reproduction of the bulk of the observed rainfall

variability and sample scaling behavior.

Acknowledgements

We are grateful to Eric Smith, Steven Rutledge,

Walter Peterson and all the CSU Radar Meteorology

Group of the Colorado State University for making

TRMM-LBA SPOL radar data available. A special

thank to Robert Cifelli and Larry Carey for useful

discussion about data quality. Some studies presented

in this paper were partially supported by the Italian

Space Agency (ASI) and by the Italian Ministry of

Education, University and Research (MIUR).


References

Carey, L.D., Cifelli, R., Petersen, W.A., Rutledge, S.A., Preliminary

Report on TRMM-LBA Rainfall Estimation Using the S-Pol

Radar. Department of Atmospheric Science Paper No. 697,

Colorado State University, 2000.

Czirok, A., Somfai, E., Viscek, T., 1993. Experimental evidence for

self-affine roughening in a micromodel of geomorphological

evolution. Phys. Rev. Lett. 71 (23), 2154–2157.

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

framework. Water Resour. Res. 36, 1779–1794.

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

anomalous scaling laws in rainfall. Water Resour. Res. 35,

1853–1867.

Deidda, R., Badas, M.G., Piga, E., 2004. Space–time scaling in high-

intensity Tropical Ocean Global Atmosphere Coupled Ocean-

Atmosphere Response Experiment (TOGA-COARE) storms.

Water Resour. Res. 40 (2) (doi:10.1029/2003WR002574).

Dubrulle, B., 1994. Intermittency in fully developed turbulence:

Log-Poisson statistics and generalized scale-covariance. Phys.

Rev. Lett. 73, 959–962.

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

NOAA Special Report, US Department of Commer., Washing-

ton, DC.

Jothityangkoon, C., Sivapalan, M., Viney, N.R., 2000. Tests of a

space–time model of daily rainfall in southwestern Australia

based on nonhomogeneous random cascades. Water Resour.

Res. 36 (1), 267–284.

Kardar, M., Parisi, G., Zhang, Y., 1986. Dynamic scaling of

growing interfaces. Phys. Rev. Lett. 56, 889–892.

Kruger, A., Kucera, P.A., Krajewski, W.F., 1999. TOGA-COARE

shipborne radar-rainfall products. Water Resour. Res. 35 (8),

2597–2600.

Lovejoy, S., Schertzer, D., 1985. Generalized scale invariance and

fractal models of rain. Water Resour. Res. 21, 1233–1250.

Lovejoy, S., Schertzer, D., 1991. In: Schertzer, D., Lovejoy, S.

(Eds.), Multifractal analysis techniques and the rain and clouds

fields from 10K1 to 106 m, in Scaling, Fractals and Non-linear

Variability in Geophysics. Kluwer, Norwell, MA, pp. 111–144.

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

multi-fractal processes: predictability and forecasting of rain

fields. J. Geophys. Res. 101, 26,333–26,346.

Ossiander, M., Waymire, E.C., 2000. Statistical estimation for

multiplicative cascades. Ann. Stat. 28, 1533–1560.

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

rainfall: dependence of random cascade generator on large-scale

forcing. J. Appl. Meteorol. 33, 1526–1542.

Pahirana, A., Herath, S., 2002. Multifractal modelling and

simulation of rain fields exhibiting spatial heterogeneity.

Hydrol. Earth Sys. Sci. 6 (4), 695–708.

Perica, S., Foufoula-Georgiou, E., 1996. Linkage of scaling and

thermodynamic parameters of rainfall: results from midlatitude

meso-scale convective systems. J. Geophys. Res. 101,

7431–7448.

Purdy, J.C., Harris, D., Austin, G.L., Seed, A.W., Gray, W., 2001. A

case study of orographic rainfall processes incorporating

multiscaling characterization techniques. J. Geophys. Res. 106

(D8), 7837–7845.

Schertzer, D., Lovejoy, S., 1985. Generalized scale invariance in

turbulent phenomena. Phys. Chem. Hydrodyn. 6, 623–635.

She, Z.-S., Leveque, E., 1994. Universal scaling laws in fully

developed turbulence. Phys. Rev. Lett. 72, 336–339.

She, Z.-S., Waymire, E.C., 1995. Quantized energy cascade and

Log-Poisson statistics in fully developed turbulence. Phys. Rev.

Lett. 74, 262–265.

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

fractals: theory and observations for rain and clouds. J. Appl.

Meteorol. 32, 223–250.

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

Evidence of dynamic scaling in space–time rainfall. J. Geophys.

Res. 104, 31,599–31,610.

Space–time multifractality of remotely sensed...

Documents

Transcript of Space–time multifractality of remotely sensed...