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Advances in Water Resources 23 (2000) 349±357

Space-time self-organization of mesoscale rainfall and soil moisture
Paolo D'Odorico a,*, Ignacio Rodrõguez-Iturbe a,b
a

Water Resources and Environmental Engineering Program, Princeton University, Princeton, NJ 08544, USA
b
Princeton Environmental Institute, Princeton University, Princeton, NJ 08544, USA
Received 3 January 1999; received in revised form 20 June 1999; accepted 3 July 1999

Abstract
The e€ects of mesoscale circulations induced by soil moisture heterogeneities are studied to assess the impact that the coupling of
the land surface and the planetary boundary layer has on the rainfall and soil moisture dynamics at di€erent scales. Our goal is to
single out the most important physical mechanisms which a€ect the behaviour of the land-atmosphere system at the mesoscale
leading to a dynamical evolution which shows the same features at di€erent scales both in space and time. In particular we aim to
understand if any hypothesis of self-organization in the system can ®nd its rationale in the mesoscale soil-atmosphere coupling. The
main mechanisms of interaction have been simulated through a cellular automata model which incorporates in a soil water balance
both the local and the large scale phenomena. Fractal features are shown to emerge in the simulated hydrological ®elds as an e€ect
of the land-atmosphere coupling and this suggests the possible occurrence of self-organization. Some comparisons with real data
tend to support such a hypothesis. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Land-atmosphere interactions; Self-organization; Precipitation; Soil moisture

1. Introduction
This paper deals with an extended version of a model
for the mesoscale land-atmosphere dynamics suggested
by Rodrõguez-Iturbe et al. [1], who investigated if the
emergence of scale invariance in the space-time structure
of rainfall and soil moisture can result from local interactions between the land and the atmosphere. The
cellular automata used by Rodrõguez-Iturbe et al. [1],
concentrated on the most basic aspects of the feedback
between soil and atmosphere treating them in an extremely schematic manner without emphasizing a realistic description of the hydrological phenomena
involved. Our aim is here to study the emergence of scale
invariance using a more realistic and physically motivated representation of the main physical phenomena
involved in the dynamics.
Convective clouds and rain occurring in a spatial
scale of several hundred kilometers play a crucial role in
the dynamics of the earth's atmosphere a€ecting its energy, momentum and moisture balances. Despite the
e€orts spent in the analysis of these phenomena, some

*

Corresponding author.
E-mail address: dodorico@princeton.edu (P. D'Odorico)

basic questions on the mesoscale dynamics still remain
unresolved. Simulations by General Circulation Models
(GCM's) cannot help the detailed understanding of
mesoscale processes, because their resolution is larger
than the scales at which convective clouds usually
originate [2]. Mesoscale circulations are thus subgrid
phenomena that are commonly parametrized within a
GCM. This procedure could undoubtedly bene®t from a
deeper understanding of the driving physical mechanisms.
The above problem is further complicated by the
dependence of the rain and cloud space-time patterns on
the landscape structure: sea-land breezes and other landatmosphere interactions due to soil heterogeneities are
among the main examples of how the earth surface may
a€ect the climate of a region [3]. Here we study landatmosphere interactions locally driven by landscape
discontinuities. Such dynamics are believed to play a
crucial role in the initiation and evolution of mesoscale

convective processes [1,4]. Some recent analyses of data
from Sahel [5,6] tend to support such hypothesis showing how the coupling between the land surface and the
planetary boundary layer (PBL) occurs through the heat
and moisture ¯uxes with soil-atmosphere interactions
which have been recently detected [6] at scales as low as
20 km.

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 3 0 - 5

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Heterogeneities in the landscape structure (e.g., in soil
moisture, surface roughness, albedo, vegetal land cover,
stomatal conductance, etc.) tend to enhance convective
phenomena [2,3], a€ecting the Bowen ratio and inducing
atmospheric circulations which may cause precipitations. Among the previous variables, soil moisture plays
a predominant role because of its key in¯uence on the

partitioning of energy into sensible and latent heat ¯uxes
at the ground surface. Thus, soil moisture is commonly
used as a representative variable to describe the land
surface heterogeneity [2,5,7].
In wet ground regions the input of solar energy is
partly used for evapotranspiration and returned to the
atmosphere through latent heat ¯ux whereas in the dry
areas it is dissipated through sensible heat ¯ux and long
wave radiation. The dry soils reach higher temperatures
yielding larger ¯uxes of sensible heat and thermal radiation to the atmosphere. Thus, as a result of the spatial
variability of the soil moisture content the atmosphere
develops local gradients in temperature and pressure
that enhance mesoscale circulations from the wet
(colder) areas to the dry (warmer) regions. These
ground-induced circulations transfer humid air toward
the dry areas and the convergence of these motions is
conducive to a vertical transport of heat and moisture to
the upper planetary boundary layer which may lead to
local precipitation on the dry regions and the release in
the atmosphere of (latent) heat of condensation [8].

The above dynamics is sustained by solar radiation
over areas with strong spatial gradients in soil moisture.
It occurs mainly in summer time when the radiation is
stronger and its scale of evolution is of the order of one
day. Thus it is during the morning when the input of the
radiative energy, needed for the mesoscale circulations,
takes place. This is followed by the formation of clouds,
the possible occurrence of precipitations and the dissipation of the horizontal gradients of pressure and temperature throughout the early evening [2].
The role of land cover heterogeneities in the development of mesoscale circulations has also been extensively investigated [3,9,10]. In particular the numerical
simulations of Segal et al. [8] point out the links between
vegetation cover and thermally induced convective motions at the mesoscale. Irrigation of vegetated areas in
the semiarid environments of the American Southwest,
the basin of Lake Chad, etc. has also been observed [8]
to be responsible for important soil thermal contrasts
leading to mesoscale circulations.
The spatial extent of the land heterogeneities needed
for the development of mesoscale circulations has been a
matter of extensive debate but the model studies of Chen
and Avissar [4] and Lynn et al. [11], as well as the recent
observational evidence reported by Taylor and Lebel [6]

suggest that soil moisture gradients even at scales as small
as 20 km may under certain conditions lead to the occurrence of rainfall patterns of di€erent characteristics.

External atmospheric forcings acting at larger scales
activate the system. They are responsible for precipitation which ¯uctuates in spatial coverage and intensity
and which in turn may initiate local circulations and the
spontaneous internal activity of the system itself.
The dynamics includes a large number of feedback
mechanisms which are part of the mass and energy
balances between the soil and the atmosphere. When
they occur, mesoscale induced precipitations locally increase the soil moisture, provide water for the evapotranspiration process, a€ect the soil albedo in bare
grounds [12] and provide a new scenario of spatially
heterogeneous soil moisture gradients which may enhance further mesoscale convections. The existence of
such a strongly non linear and possibly self-sustaining
dynamics may be an appealing physical mechanism for
the explanation of numerous features of temporal persistence and spatial scale invariance observed in the
space-time patterns of the hydrological ®elds [13]. Taylor and Lebel [6] argue that the in¯uence of vegetation
on the spatial variability of moisture content and temperature of the PBL can be itself related to the selfsustained patterns in soil moisture and rainfall which
may persist for several weeks. This itself induces spatial
patterns in the vegetation growth and in the transpiration rates with a positive feedback to the above dynamics.

Extensive data analyses in many di€erent regions
have detected the existence of long range correlations in
the time series of rainfall and of other hydrologic variables [13±15]. In a spatial context the size of the clouds
and the areas of connected regions (e.g., clusters) having
daily rainfall above a certain threshold are probabilistically distributed according to power laws in many
di€erent climatic regions [1,16]. Power laws valid
through an extended range of scales have been shown to
exist also in the relationship between perimeter and area
for rain and clouds [17,18]. Although such scaling
properties are commonly reproduced by algorithms in
the generation of rainfall ®elds [19±21] or through particular cumulus parametrizations in atmospheric models
[22], a general physical understanding of the responsible
dynamical processes which, acting in quite di€erent situations, lead to basically similar statistical signatures,
has yet to be achieved.
Recent evidences of di€erent physical character support the hypothesis that open, dissipative, non-linear
systems with many degrees of freedom tend to display
critical self-organization with a fractal growth in their
global space-time evolution [23]. This results from the
local dynamics and from the co-operative activity
among the components of the system. Numerous studies

have investigated the role of local interactions in the
global behaviour of forest ®res [24], earthquakes dynamics [25], drainage networks [26], species mutations
[27] and in many other contexts. In all these di€erent

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

phenomena the emergence of fractal features is shown to
occur both in space and time through very simple
models capturing the main driving mechanisms. Structures of all sizes and durations were shown to occur in
all cases displaying power law probability distributions
without characteristic scales.
We attempt to ®nd a possible mechanism which,
based on local dynamics and non-linear interactions,
could provide an explanation for the temporal and
spatial fractal signatures frequently found in rainfall and
soil moisture. The mechanism focuses on mesoscale
processes and is driven by soil moisture gradients responsible for local dynamics and possible convective
rain. The latter will in turn a€ect the spatial patterns of
soil moisture and the global evolution of the system.

2. The model
A cellular automaton operating on a square lattice of
256  256 pixels simulates the space-time evolution of
the water balance in a region subject to large scale atmospheric external forcings. The cellular automata
attempts a simplistic representation of local soil-atmosphere interactions, evapotranspiration, deep in®ltration
and runo€ at every pixel. The water balance is carried
out daily on a pixel scale of the order of 20 km  20 km.
We study the space-time evolution of soil moisture at
each pixel, x…x1 ; x2 †, through the water balance equation:
gZr

dS
ˆ Rp ‡ Rc ÿ E ÿ I ÿ D;
dt

…1†

where S is the relative soil moisture, g the porosity, Zr
the depth of the soil active in the moisture exchange
with the atmosphere and in the storage of water volumes

at the daily time scale; Rp the rain from the large scale
external forcings; Rc the rain resulting from the mesoscale circulations driven by local spatial gradients in soil
moisture; E the water exchange from ground and vegetation to the atmosphere by evapotranspiration; I the
loss to deep in®ltration and D the runo€. The hydrologic
representation of the terms in Eq. (1) is done in this
paper with considerable more physical realism than the
more schematic modeling of Rodrõguez-Iturbe et al. [1].
In what follows we give more details about the
modeling of the hydrological processes, as well as of the
soil properties.

351

considered a function of the saturated hydraulic conductivity through the relationship: hgi ˆ 0:375ÿ
0:0108 log KS [KS ˆ cm=s]. This is only an approximate
relation obtained by ®tting the data given in Clapp and
Hornberger [29] and used with the sole purpose that g
and KS are not generated as independent random ®elds.
The minimum water content that can be drained by
gravity, fc …x†, is known as ®eld capacity and is here

expressed as a function of the porosity (e.g., from data
in [30]), as fc …x† ˆ 1:25g…x† ÿ 0:2625.
The dependence of the unsaturated hydraulic conductivity, K…S†, on the soil moisture is assumed to follow a power law [31]
K…S† ˆ KS S c ;

…2†

where the exponent c is a function of the soil properties.
The analysis of data from several soils shows that c increases as KS decreases and this dependence is expressed
here as c ˆ aKSÿb with the parameters a ˆ 5:76 cm/s and
b ˆ 0:148 estimated by ®tting the data reported in [29].
Again as in the case of the relationship between KS and
hgi all the approximate equations relating di€erent
variables are used only to generate soil properties which
are not independent from each other. They are not
meant to be assigned any physical interpretation.
2.2. Evapotranspiration
The rate of evapotranspiration, E, is modeled as a
function of the soil moisture as shown in Fig. 1, in the
hypothesis that, because of the high evaporative demand
(summer time), the soil water content is the main limiting factor for evapotranspiration. This type of dependence is commonly assumed in hydrology [32], with Emax
representing the maximum evapotranspiration which in
general depends on both the climate and the vegetation
and is here assumed to be constant throughout the

2.1. The soil properties
The saturated hydraulic conductivity, KS …x†, is
modeled as a lognormally distributed random ®eld [28]
with mean hKS i ˆ 1000 mm/day and coecient of variation, CV ˆ 0.1. The porosity at any point, g…x†, is then
simulated as a random variable uniformly distributed
around a mean value, hgi, (i.e., g ˆ hgi  0:1) which is

Fig. 1. Rate of evapotranspiration as a function of soil moisture.

352

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

region. For relative soil moistures below a critical value
S  , we consider E as a linear function of S while, for S
larger than the S  , we assume E as a constant and equal
to Emax . S  is in general a function of the soil and the
vegetation but we assume ± as is frequently done ± that
S  is approximated by the ®eld capacity of the soil, fc …x†,
at each speci®c pixel. It is important to remark that this
is not conceptually correct since the soil moisture at
which vegetation decreases its transpiration below the
maximum possible value for a given climatic condition is
di€erent and lower than the ®eld capacity of the soil.
Our assumption has no implications for the purposes of
this paper and it is motivated only to avoid the generation of another ®eld of S  values based on the type of
assumed vegetation.
2.3. Deep in®ltration and surface runo€
The rate of losses through deep leakage is a function
of the soil water content and can be expressed as
I…x; t† ˆ KS S c :

…3†

When the relative soil moisture is above the ®eld capacity, S  , the values of K…S† allow drainage of the soil
by gravity while for lower values of S the deep leakage is
negligible and almost all losses of soil moisture are due
to evapotranspiration.
Runo€ occurs in a pixel whenever its relative soil
moisture calculated through Eq. (1) exceeds the unit
value.
2.4. Rainfall
Rainfall in the model occurs according to two different mechanisms operating at di€erent scales. These
two mechanisms attempt to represent the e€ect of (a)
large scale synoptic events and (b) mesoscale precipitation induced by local land-atmosphere interactions.
Synoptic rainfall takes place in the region with
probability, P, of occurrence in any given day. Its spatial
structure at the daily time scale is modeled by a Poisson
process in space [33] and consists of a set of rectangular
storm areas which may overlap. The number, N, of
storms at any day is a random variable sampled from a
Poisson distribution with mean, hN i. The sides of each
rectangular storm area are independent random variables, l, exponentially distributed with mean hli. An
uniform depth of rain, dp , is assigned to every storm and
is randomly generated from an exponential distribution.
In those pixels where storms overlap the total depth of
rain, Rp , is given by the sum of the depths of the storms.
The rain generated by mesoscale circulations is assumed to occur over those dry areas which have a soil
moisture gradient with any wet neighbour exceeding a
certain threshold, n. A rain depth, dc is then generated
from an exponential distribution with mean hdc i. Thus

mesoscale convective rain is assumed to take place as a
consequence of landscape discontinuities in the soil
moisture which is considered as the most representative
property of the ground characteristics a€ecting, through
its spatial gradients, the mesoscale soil-atmosphere interaction. Simulations were also performed where, once
the condition on the local gradients of soil moisture is
satis®ed, rainfall is generated over the dry areas with a
certain probability of occurrence. No substantial changes were observed in the results.
Estimates of the values of the gradient in soil moisture needed in order to start and sustain mesoscale circulations are not readily available. Taylor et al., [5]
during the HAPEX experiment in the semiarid region of
southwest Niger, detected moisture gradients of 30±
50 mm/10 km in the top 150 cm of soil in days when selfsustained mesoscale circulations were observed. In terms
of relative soil moisture this would mean that the
threshold n can assume values as low as 0.10 although
one needs to stress that the generation of mesoscale
circulations and their sustainability depends on many
interrelated factors.

3. Space-time patterns of hydrological variables
Although the modeling of the phenomena involved
requires of a number of parameters, only two of them, P
and n, are needed to describe synthetically the relative
role played by the external forcings and the local dynamics in the water balance For a given P, low values of
n mean relatively high amounts of locally driven mesoscale rain, while high n's allow almost exclusively rainfall generated by synoptic events.
In what follows the di€erent sets of values for the
parameters n, P and hN i in the local and synoptic
rainfall models have been chosen in a way to have in the
region approximately the same average daily evapotranspiration (2±2.5 mm/day) and average soil moisture
( 0:30) and a daily rainfall in the range 2.5±3.5 mm/
day. The following parameters were used for the soil and
climate modeling: average soil depth, Zr ˆ 40 cm, average hydraulic conductivity, hKS i ˆ 100 cm/day and
maximum rate of evapotranspiration, Emax ˆ 4 mm/day.
The depth of the rainfall induced by mesoscale circulations is a random variable (exponentially distributed)
with mean hdc i ˆ 15 mm/storm, while the random depth
of the individual storms in the Poissonian model of
synoptic rain is exponentially distributed with mean
hdp i ˆ 10 mm/event. Every storm is a rectangle with
independent and identically distributed (exponential
distribution) random sides with mean hli ˆ 7 pixels. The
scheme for generating the synoptic rain, the link of
convective precipitation to local gradients of soil moisture and the dependence of the rate of evapotranspiration on the soil water content impose a spatial (and

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

353

temporal) variability on the daily values of rainfall and
evapotranspiration which is further enhanced by the
heterogeneities of the soil properties.
The following sections show how in the presence of
soil-atmosphere interactions the model displays fractal
properties both in the temporal evolution of the
hydrologic variables as well as in their spatial con®guration. The emergence of these properties tends to support the hypothesis of occurrence of self-organized
dynamics due to the coupling between land surface and
the atmosphere.
3.1. Analysis in time
We have studied the time series of average soil
moisture and evapotranspiration in the whole region, as
well as those of soil moisture at a point and the number
of pixels evapotranspirating at the maximum rate.
Power laws have been detected in the power spectra of
these series (Figs. 2 and 3) irrespectively of the values of
P and n. This implies that the time series of soil moisture
and evapotranspiration generated by the model tend to
show a lack of characteristic scales regardless of the
occurrence or not of important interactions between the
soil and the near-surface atmosphere. A di€erent behaviour has been found in the time series of average
rainfall. In fact, with high values of n, the amount of

Fig. 3. Power spectra of time series generated by the model: average
soil moisture, (A), and average evapotranspiration, (B). The parameter
values used are hN i ˆ 300, P ˆ 0:95 and n ˆ 0:90.

locally driven rain is negligible and the time series of the
average rain show (Fig. 4(a)) a ¯at power spectrum (i.e.,
uncorrelated signal) as we would expect for the Poisson
process here modelled for the synoptic rain. Decreasing
n, the crucial role of the local dynamics becomes
stronger inducing a di€erent structure in the power
spectra of rain which behaves as 1/f b noise (Fig. 4(b)).
Thus local land-atmosphere interaction could be responsible for the scaling behaviour of rain which has
been experimentally detected in real data [1,15].
Time series of the daily rain depths at a point were
also studied through the analysis of the rescaled range
[34] of the data sets yielded by the model. The rescaled
be the signal
range is de®ned as follows. Let fXi giˆ1;2;...;T P
k
(i.e., rain at a point) and let Yk ˆ iˆ1 …Xi ÿ X †
…i ˆ 1; 2; . . . ; k† be the cumulative deviations from the
mean. We de®ne as the range the di€erence between the
maximum and the minimum values of Y in the interval
‰0; T Š and we denote it by RT ˆ maxfYk gÿ
minfYk g …1 < k < T †. In order to compare values of RT
related to di€erent series and lags, T, we rescale RT with
the standard deviation
v
u T
u1 X
2
ST ˆ t
…Xi ÿ X † :
T iˆ1
Fig. 2. Power spectra of some time series generated by the model:
average soil moisture, (A), and average evapotranspiration, (B). The
parameter values used are hN i ˆ 40, P ˆ 0:05 and n ˆ 0:15.

Thus the rescaled range, RT , is a dimensionless variable
which scales with T as

354

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

Fig. 4. Spectral analyses for the time series of average daily rainfall
depth (mm), with hN i ˆ 300, P ˆ 0:95, n ˆ 0:90, (A), and hN i ˆ 40,
P ˆ 0:05, n ˆ 0:15, (B).

RT ˆ

RT
 TH;
ST

Fig. 5. Rescaled range for the time series of daily rainfall depth (mm)
at a point generated by the model. The parameter values used are
hN i ˆ 40, n ˆ 0:15, P ˆ 0:05, (A), and hN i ˆ 300, n ˆ 0:9, P ˆ 0:95,
(B).

…4†

where H is called the Hurst's exponent. For time series
with ®nite memory structure H ˆ 0:5. A value of
H > 0:5 is known as Hurst's phenomenon [35] and it
implies that the integral in ‰0; 1‰ of the correlation
function of Xi does not converge. Thus, the process has
an in®nite memory and is called persistent. We observe
(Fig. 5(a)) that when soil-atmosphere interactions are
present the local dynamics can induce persistence in the
series of daily rain depth at a point, the Hurst's exponents being larger than 0.5. This favourably resembles
the experimental ®ndings in long time series of rain in
di€erent parts of the world [14]. For high values of n the
contribution due to the mesoscale dynamics becomes
negligible and series of daily rainfall depth do not show
any persistence (Fig. 5(b)). This could suggest the hypothesis that the persistence commonly observed in
rainfall time series may ®nd a physical explanation in
the interaction between land and atmosphere. The same
analysis has been performed (Fig. 6) for the time series
of relative soil moisture at a point which yield values of
H  0:9 for all cases when locally driven rainfall becomes dominant leading to the same conclusion made
for the series of precipitation at a point. A similar
analysis was undertaken for some real soil moisture
data. Fig. 7 shows the range analysis for several time

series of daily relative soil moisture recorded at a depth
of 5 cm in di€erent points of the Central Plains in the
United States (data available on web site:
www.wcc.nrcs.usda.gov/smst/smst.html. Values of soil
moisture have been determined at the hourly time scale
through high frequency electrical measurements of the
capacitive and conductive properties of the soil). The
Hurst's exponent is found to fall in the range 0.94±0.98.
3.2. Spatial con®guration
The spatial structures of both the soil moisture ®eld
and the rainfall ®eld were studied through the probability distribution of the cluster sizes de®ned as the set of
connected pixels of the domain where soil moisture (or
rain) exceeds a certain threshold.
Fig. 8 shows the distribution of clusters in the soil
moisture ®elds for di€erent values of the parameters P
and n. Power laws P ‰A P aŠ  aÿc are found with exponents in the range c ˆ 0:6 ÿ 1:00. Unfortunately, the
lack of spatial data of soil water content at this resolution does not allow for a comparison of this result with
real ®elds of soil moisture. What can be concluded here
is that the occurrence of local land-atmosphere interaction does not seem to be crucial to the statistical

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

Fig. 6. Rescaled range for the time series of daily values at a point of
relative soil moisture generated by the model. The parameter values
used are hN i ˆ 40, n ˆ 0:15, P ˆ 0:05, (A), and hN i ˆ 300, n ˆ 0:9,
P ˆ 0:95, (B).

355

Fig. 8. Distribution of the sizes of the clusters in which soil moisture
exceeds a certain threshold (S P 0:3) for two di€erent simulations: (A)
hN i ˆ 40, n ˆ 0:15, P ˆ 0:05 and (B) hN i ˆ 300, n ˆ 0:90, P ˆ 0:95.
Areas, a, are in pixel units ( 20 km  20 km).

Fig. 7. Analysis of the rescaled range for the time series of daily values of relative soil moisture at a point recorded at a depth of 5 cm in di€erent
locations in the Central US Plains during the summers of 1995, 1996, 1997 and 1998 (June, 1st to September, 15th).

356

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

properties characterizing the spatial structure of soil
moisture ®elds.
A similar analysis has been performed on the ®elds of
daily rainfall occurrence where clusters have been de®ned as connected areas receiving rain. Power laws are
encountered in the distribution of cluster sizes but only
for low values of n (Fig. 9(a)), while for high n (i.e., with
weak local dynamics) this scaling behaviour is gradually
lost (Fig. 9(b)). Thus, land-atmosphere interactions may
be important in explaining the scaling properties observed on the spatial con®guration of rainfall ®elds
[1,16]. Fig. 10 shows a similar analysis carried out on
real data for a region of 1340 km  636 km in the
Central Plains of North America, where no major relevant orographic in¯uences are present. We observe that
the sizes of the clusters of rain have well de®ned powerlaw distributions with exponents very similar to those
found in the ®elds generated by the model for low values
of n.
The occurrence of local dynamics arising from landatmosphere interaction may thus allow the emergence of
the fractal properties in the space structure of rainfall
®elds which can be detected in real data.

Fig. 10. Distribution of the sizes of the clusters of rain detected in real
data. (A) August 12th, 1997 and (B) August 6th, 1997. The data are
provided by the Arkansas Red-River Forecasting Center (available on
web site http://info.abrfc/ noaa.gov/) in a grid of 335159 pixels of
4 km  4 km. Areas, a, are expressed in pixel units.

4. Conclusions

Fig. 9. Distribution of the sizes of the clusters of areas receiving rain in
daily images generated with: hN i ˆ 40, n ˆ 0:15, P ˆ 0:05, (A), and
hN i ˆ 300, n ˆ 0:90, P ˆ 0:95, (B). Areas, a, are in pixel units
( 20 km  20 km).

We have simulated the mesoscale hydrologic dynamics through a cellular automata model in which the
evolution of the system is studied estimating local water
balances in presence of interactions between the land
surface and the atmosphere induced by heterogeneities
in the surface soil moisture.
The simulations show how such interactions may lead
to scale invariance in the space-time evolution of the
system. When the local land-atmosphere interactions are
important in the cellular automata, we observe 1=f b noise
in the time series of the average daily rainfall depth and
the occurrence of the Hurst's phenomenon in the time
series of daily rainfall at a point. The analysis of images of
daily precipitation shows that the distribution of the areas experiencing rain are power laws where no characteristic sizes are found. All these fractal features in the
space-time evolution of rainfall ®elds disappear whenever
the model does not incorporate the local interactions.
The above results suggest that the soil-atmosphere
coupling can explain the emergence of the fractal
properties experimentally observed in the space-time

P. D'Odorico, I. Rodrõguez-Iturbe / Advances in Water Resources 23 (2000) 349±357

patterns of rainfall. The local and self-sustained interactions of the components of the system strongly a€ect
its dynamics on a wide range of scales and the emergence of scale invariance both in time and in space is
suggestive of self-organization.
Acknowledgements
This research was supported by grants from NASA
(N A G W - 4 1 7 1 , N A G W - 4 7 6 6 ) and NSF (E A R - 9 7 0 5 8 6 1 ). The
research leading to this paper was carried out when the
authors were at the Civil Engineering Department of
Texas A&M University.
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