Autor: Ignacio Rodriguez Iturbe Fecha: No disponible

1. ECOHYDRO-GEOMORPHOLOGY OF Rl VER BASINS: A
CHALLENGE FOR THE NEXT DECAPE
Ignacio Rodriguez-Iturbe, Theodora Shelton Pitney Professor of Environmental Sciences and Professor of
Civil and Environmental Engineering.
Princeton University, Princeton, N.J.08544
Abstract
An overview is presented of a stochastic soil moisture model that describes the soil water balance via a
non-linear differential equation driven by a state dependent marked Poisson process. The analytical resuits
of the steady state probability density function and of the crossing properties provide a synthetic
representation of plant response to hydroclimatic forcing that describes quite well vegetation conditions in
water controlled ecosystems. These ecosystems are complex evolving structures whose characteristics and
dynamic properties depend on many interrelated iinks between climate, soil and vegetation. The analysis
demonstrate that, with careful simplifications, and the inclusion of stochastic components such dynamics
can be approached theoretically in a coherent manner.
The aboye framework is then used to examine the linkages between the drainage network and the patterns
of soil water balance components determined by the organization of vegetation, soils and climate m semi
arid river basins. Research durmg the last 10 years has conclusively shown a high degree of organization
and unitíing principies behind the structure of the drainage network and the 3-1) geometry of river basins.
This cohesion exists despite the infinite variety of shapes and forrns one observes in natural watersheds.
What has been relatively unexplored in a quantitative and general manner is the question whether or not the
interaction of soils, vegetation and climate aiso display a similar set of uniiying characteristics. The
presence of such co-organization is here undertaken. It is shown that that the drainage network acts as a
template for the organization of both vegetation and hydrological pattems which exhibit self-afline
characteristics in their distribution across river basiis.
1. Introduction
Water-controlled ecosysterns are cornplex evolving structures whose characteristics and dmarnic properties
depend on many interrelated links between climate, soil, and vegetation (Rodriguez-Iturbe, 2000; Porporato
and Rodriguez-Iturbe, 2002; Rodriguez-Iturbe and Porporato, 2004). On one hand, climate and soil control
vegetation dynarnics, on the other hand vegetation exerts important control on the entire water balance and
is responsible for many feed-backs to the atmosphere.
There are two characteristics which make especially daunting the quantitative analysis of the problern: (1)
the very large number of different processes and phenomena which make up the dynamics, and (2) the
extremely large degree of variability iii time and space that the phenomena present. The first of the aboye
characteristics obviously calis for simplif,'ing assumptions in the modeling scheme while still preserving
the rnost important features of the dynamics, while the second one calis for a stochastic description of sorne
of the processes controlling the overall dynarnics (especially precipitation).
The dynamics of the soil-plant-atmosphere system is interpreted at the daily time scale. Moreover, for the
sake of simplicity, oniy cases with negligible seasonal and interannual rainfall components are considered.
The growing season is thus assumed to be statisticaiiy homogeneous: the effects of the initial soil moisture
condition are considered to last for a relatively short time and the probability distribution of the soil water
content to settle iii a state independent of time. Possible extensions to unsteady conditions are discussed in
Rodriguez-Iturbe and Porporato (2004).

2. 2. Soil Water Balance at a Point
Soil moisture is the key variable synthesizing the interrelationship between climate, soil, and vegetation.
Its dynamics is described by the soil water balance, i.e. the mass conservation of soil water as a fiinction of
time. We will consider the water balance vertically-averaged over the root zone, focusing on the most
important components as sketched in Figure 1. The state variable regulating the water balance is the
relative soil moisture, s, which represents the fraction of pore volume containing water.
!NPLT: RAINI AL!.
(intcrnhittcnt-
Fi 5tOCI135tiC)
Evapo- (
transprTat ion
rrouhfaII
Runoff
Mazar
Effective porosity. n
FI!1ive porosilv, n
Lcakagc
Figure 1: Schematic representation of the various mechanisms of the soil water balance with emphasis on
the role of different ffinctional vegetation types. After Laio et al. (2001 a).
Under the simp1iQ'ing assumption that the lateral contributions can be neglected (i.e., negligible
topographic effects over the area under consideration), the vertically-averaged soil moisture balance
at a point may be expressed as
2

3. rZ
-í1
where t is time, ' L' ''
(.L»
1. '
¿
.1 is the rate of infiltration from rainíail, n is the soil porosity and
{ • mC of soil moisture losses from the root zone. The terms oil the r.h.s. of Equation (1)
represent water fluxes, i.e., volumes of water per unit area of ground and per unit of time (e.g., mm/d).
The infiltration from rainíall,J, [s(t). t] is the stochastic component of the balance. It represents the part
of rainfali that actually reaches the soil coluinn, i.e.,
= R(t) - I(f,:
'Ii
(2)
where R(t) is the rainfali rate, 1(t) is the amount of rainfail lost through canopy interception, and Q[s(t); t] is
the rate of runoff.
The water Iosses from the soil are from two different mechanisms,
= E±Y —L[s(•):.
being E[s(t)] and L[s(t)] the rates of evapotranspiration and leakage, respectively.
2.1. Rainfail Modeling
At small spatial scales, where the contribution of local soil-moisttire recycling to rainfali is negligible, the
rainíali input can be treated as an external random forcing, mdependent of the soil moisture state. Since
both the occurrence and amount of rainfali can be considered to be stochastic, the occurrence of rainíali is
idealized as a series of point events in continuous time, ansing according to a Poisson process of rate
and each carrymg a random ainount of rainfail extracted from a given distribution. The temporal structure
within each rain event is ignored and the marked Poisson process representing precipitation is physically
interpreted at a daily time scale, where the pulses of rainfali correspond to daily precipitation assumed to be
concentrated at an instant in time.
The depth of rainfail events is assunied to be an independent random variable ti, descnbed by an
exponential probability density function
1
fH(1l)=°. forh>O,
a
where la is the mean depth of rainfail events. Iii the following we will often refer to the value of the mean
rainfali depth normalized by the active soil depth, i.e.,
3

4. la
- flZr
Both the Poisson process and the exponential distribution are of common use in simplified modeis of
rainfail at daily time scale. Canopy interception is incorporated in the stochastic model by simply assuming
that, depending on the kind of vegetation, a given amount of water can be potentially intercepted from each
rainfali event (Rodriguez-Iturbe et al., 1999). The rainfail process is thus transformed into a new marked-
Poisson process, called a censored process, where the frequency of rainfail events is now
=
fHh:ldh =
and the depths have the same distribution as before.
2.2. Infiltration and Runoff
It is assumed that, when the soil has enough available storage to accommodate the totality of the incoming
rainfail event, the increment in water storage is equal to the rainfali depth of the event; whenever the
rainfail depth exceeds the available storage, the excess is converted in surface runoff. Since it depends on
both rainfali and soil moisture content, infiltration from rainfali results to be a stochastic, state-dependent
component, whose magnitude and temporal occurrence are controlled by the entire soil moisture dynamics.
The probability distribution of the infiltration component may be easily written in terms of the exponential
rainfall-depth distribution (6) and the soil moisture state s. Referring to its dimensionless counterpart y (i.e.,
the mfiltrated depth of water normalized by nZr) one can write
4- -- ' — 1 +
F1—5
"
for 0,<1—s.
where is defined in (5). Equation (7) is thus the probability distribution of having a jump in soil moisture
equal toy, starting from a level s. The mass at (1 -s) represents the probability that a storm will produce
saturation when the soil has moisture s. This sets the upper bound of the process at s = 1, making the soil
moisture balance evolution a bounded shot noise process.
2.3. Evapotranspiration and Leakage
The term E[s(t)] in Equation (3) represents the sum of the losses resulting from plant transpiration and
evaporation from the soil. Although these are governed by different mechanisms, we will consider them
together. When soil moisture is high, the evapotranspiration rate depends mainly on the type of plant and
climatic conditions (e.g., leafarea index, wind speed, air temperature and humidity, etc.). As long as soil
moisture content is sufficient to permit the normal course of the plant physiological processes,
evapotranspiration is assumed to occur at a maximum rate E, which is independent of s. When soil
moisture content falis below a given point s, which depends on both vegetation and soil characteristics,
plant transpiration (at the daily time scale) is reduced by stomatal closure to prevent internal water losses
4

5. and soil water availability becomes a key factor m determining the actual evapotranspiration rate.
Transpiration and root water uptake continue at a reduced rate until soil moisture reaches the so-called
wilting point s. Below wilting point, s soil water is fiirther depleted only by evaporation at a very low
rate up to the so-called hygroscopic point, Sh.
From the aboye arguments, daily evapotranspiration losses are assumed to happen at a constant rate E m ,
for s < s < 1, and then to Iinearly decrease with s, from E. to a value E w at s. Below s, only evaporation
from the soil is present and the loss rate is assumed to decrease linearly from E w to zero at Sh. Leakage or
deep mfiltration is modeled as starting from zero at the so-called fleld capacity sk and then increasing
exponentially to Ks at saturation. The total soil water losses are showns in Figure 2.
kis) (cmJd)
1.
0.
O Sp1 SW S' 1
Figure 2: Soil water losses (evapotranspiration and leakage), (s), as function of relative soil moisture for
typical climate, soil, and vegetation characteristics in semiarici ecosystems. Aíter Laio et al. (200 la).
3. Probabilistic Evolution of the Soil Moisture Process
The probability density function of soil moisture, p(s; t), can be derived from the Chapman-Kolmogorov
forward equation for the process under analysis (Rodriguez-Iturbe et al., 1999; see also Cox and Miller,
1965; Cox and Isham, 1986)
[p(, *)p(s)] - )''p(s. t + A , p( u, t) fy(s - u.
(8)
The complete solution of Equation (8) presents serious mathematical difficulties. Only formal solutions in
terms of Laplace transforms have been obtained for simple cases when the process is not bounded at s = 1
5

6. (Cox and lsham, 1986 and references therein). Attention is focused here on the steady-state case. Under
such conditions, the bound at 1 acts as a reflecting boundary. Accordingly, the solutions for the bounded
and unbounded case only differ by an arbitrary constant ofiritegration. The general form of the steady-state
solution can be shown to be (Rodriguez-Iturbe et al., 1999)
du
p(:= for 3h<:s<l,
p
(9)
where C is the normalization constant. The lirnits of the integral in the exponential terrn of Equation (9)
must be chosen so to assure the continuity of p(s) at the end points of the four different components of the
loss function (Cox and Isham, 1986; Rodriguez-Iturbe et al., 1999).
Figure 3 shows sorne examples of the soil moisture pdf. The two different types of soil are loamy sand and
loam with two different values of active soil depth. These are chosen in order to ernphasize the role of soil
in the soil moisture dynamics. The role of clirnate is studicd only in relation to changes in thc frequency of
storrn events keeping fixed the mean rainfail depth a and the maximum evapotranspiration rate E,.
A coarser soil texture corresponds to a consistent shift of the pdftoward drier conditions, which in the most
extreme case can reach a difference of 0.2 in the location of the mode. The shape of the pdf also undergoes
marked changes, with the broadest pdfs for shallower soils.
6halow SI Deep Sail
E
o
5
o
P5I a:
10
0.2 0.4 0.6 0.8
ci
8
6
i s
Pi SI e
la
8
10
8
6 J (4
2
5
0.2 0.4 0.6 0.8 1
8
6
Piii
16
8
2
S
Figure 3: Examples of pdfs of relative soil moisture for different type of soil, soil depth, and mean rainfail
rate. Continuous unes refer to loamy sand, dashed unes to loam. Left column corresponds to Z = 30 cm,
right colunui to 90 cm. Top, center, and bottom graphs have a mean rainfall rate of 0.1, 0.2, and 0.5 d'
respectively. Common parameters to all graphs are = 1:5 cm, = O cm, E. = 0:01 cmld, and E. =
0:45 cmld. Afler Lalo et al. (200 la).
rol

7. Crossing Properties and Mean First Passage Times of Soil Moisture
Dynamics
The frequency and duration of excursions of the soil moisture process aboye and below sorne leveis
directly related to the physiological dynarnics of plants are of crucial importance for ecohydrology. The
analysis of the crossing properties is also important to gain insights on the transient dynamics of soil
moisture, either at the beginning of the growing season, to evaluate the time to reach the stress leveis after
the soil winter storage is depleted, or after a drought, to estimate the time to recover from a situation of
intense water stress.
The derivation of exact expressions for the mean first passage times (MFPT's) for systems driven by white
shot noise, as the soil rnoisture dynamics described before, has received considerable attention, both
because of its analytical tractability and because of the large number of its possible applications. Laio et al.
(200 Ib), present simple interpretable results for the MFPT's of stochastic processes driven by white shot
noise derived for cases relevant to soil moisture dynamics.
As an example in Figure 4, T is studied as a function of the frequency of the rainfali events A and of the
mean rainfail depth C , iii such a way that the product Ct', remains constant for different values of the
maximum evapotranspiration rate. This is to compare environments with the same total rainíail during a
growing season, but with differences in the timing and average amount of the precipitation events. In case
of high maximum transpiration rates, plants may experience longer periods of stress either where the
rainfali evenis are very rare but intense or where the events are veiy frequent and light. From a physical
viewpoint, this is due to the relevant water losses by leakage, runoff or canopy interception, indicating
possible optimal conditions for vegetation.
Figure 4 points out the existence of an optimum ratio between A and
(cm)
1,5 1 075 0,6 0.5
80
70
60
. 50
-. 40
30
20
E- 4.5
Emax4 1
0,1 0,2 0,3 0.4 0,5 0,6
A (d 1 )
Figure 4: Mean duration of excursions below s, T,* , as a function of the frequency of the rainfali events
when the total rainfail during the growing season is kept fixed at 650 mm for different values of the
maximum evapotranspiration rate. The root depth is Zr = 60 cm, the soil is a loam. After Laio et al.
(200 ib).
Plant Water Stress
The reduction of soil moisture content during droughts lowers the plant water potential and leads to a
decrease in transpiration. This in turn causes a reduction of celi turgor and relative water content in plants
7

8. which bnngs about a sequence of damages of increasing seriousness. Using the links between soil moisture
and plant conditions, the mean crossing properties of soil moisture can be used to define an index of plant
water stress which combines fue intensity, duration, and frequency of periods of soil water deflcit.
Porporato et al. (2001) use the points of incipient and complete stomatal closure as indicators of the starting
and Ihe maximum point of water stress. They define a "static' stress that is zero when soil moisture is
aboye the level of incipient reduction of transpiration, s*, and grows nonhinearly to a maximum value equal
to one when soil moisture is at the level of complete stomatal closure (wilting), s, i.e.
for E - -
where q is a measure of the nonlinearity of the effects of soil moisture deficit on plant conditions. The
mean water stress given that s < s (indicated as ) can be calculated from the soil moisture pdf
A more complete measure of water stress which combines the previously defined "static' stress, with
the mean duration and frequency of water stress through the variables andwas proposed by
Porporato et al. (2001). This, referred to as "dynamic" water stress or mean total dynamic
stress during the growing season, 5, was defined as
(r?
if 7T
=
k -
1 otherwise.
The role of the parameters k and r is discussed in Porporato et al. (2001). The dynamic water stress well
describes plant conditions and has been used with success in different ecohydrological applications (e.g.
Rodriguez-Iturbe and Porporato 2004).
6. Ecohydro-geomorphology
Recent years have seen dramatic advances in the quantitative description of the geomorphologic structure
of river basins (Rodriguez-Iturbe and Rinaldo, 1997). The interconnected system of hilislopes and the
channel network has been shown to posses a profound order that manifests itself in a number of
probabilistic features whose basic characteristics remain unchanged regardless of scale, geology, or
climate. Despite the deep symmetry of structural organization in geomorphologic properties, the
convergence of the biological and geophysical study of river basins is a remaining frontier in hydrological
science. In particular, there exists a need to understand the interrelationship among biological, geophysical
and geochemical approaches to the study of the earth system. In this regard, soil moisture is a crucial link
between hydrological and biogeophysical processes through its controlling influence on transpiration,
runoffgeneration, carbon assimilation and nutrient absorption by plants. Therefore, efforts to integrate the
biological and geophysical aspects of river basins will require a focus on the interactive manner by which
patterns of climate, vegetation and geomorphology are coupled in landscape patterns and dynamics (Ridolfi
et al., 2003).
The beliefthat ecological processes are evident ¡u vegetation patterns has often been used to investigate the
relationships between the spatial structure of vegetation and the nature of competition, disturbance, and
resource heterogeneity across a range of ecosystems. Even still, many outstanding issues in plant ecology
rl]

9. are directly related to an incomplete understanding of the dynamics and persistence of spatial patterns
(Levin, 1992). These include (1) the reiationship between competition11ciiitation, spatiaI pattern, and the
persistence of biodiversity; (2) the relative importance of biotic and abiotic fctors in structuring vegetation
communities; (3) the role of both current and former plant patterns in determining the spatial distribution of
resource availability; and (4) the time and space scales over which various disturbances affect spatial
pattems and the consequences of spatial disturbances on iong-term stability of vegetation cornmunities.
Understanding how vegetation patterns are expressed within landscapes organized around river networks is
a central challenge that integrates each of the four issues usted aboye (Bridge and Johnson, 2000). Here we
focus on the manner by which drainage networks act as a tempiate for the organization of ecohydrological
interactions that determines vegetation patterns within landscapes. We link observed patterns in vegetation
organization with the hydrological dynarnics operating within the basin. The principies of such
organization have important consequences regarding the impact of land cover change on hydrological
dynamics in river basins, as well as the geornorphoiogical evolution of landscapes under varying climate
and vegetation regimes.
Given the deep coherence in geomorphologicai structure across different basins and the strong interactions
between climate, soils, and vegetation in determining hydrological dynamics, a number of questions arise:
Is the known geomorphological organization connected to the spatiai characteristics of vegetation and soil
of a river basin? If so, where does it manifest itself? Specifically, does vegetation present a spatial structure
in river basins which may be described by sorne features of a statistical character linked to the existing
geomorphologic order? Do the patterns of soil moisture and the components of the hydroiogic balance
resulting frorn the interaction of climate, soil and vegetation contain a stili unknown spatiai organization?
We propose a framework of analysis to study these questions using the channei network as a ternplate
around which the possibie organization of vegetation, soils and the components of the hydroiogic balance
takes place. The signatures of this organization will be, by necessity, of a statistical character. Thus, our
hope is that an infinite variety of spatial patterns in the aboye variables will nevertheless exhibit sorne
fttndamental thernes that rernain unchanged in a probabiiistic sense throughout different spatiai scaies and
different ciimatic and geologic conditions. Because our present approach focuses on water controlled
ecosystems, where soil moisture is a critical controlling resource, we will atternpt this goal through the
study of a semiarid river network, the Rio Salado basin in New Mexico.
7. The Rio Salado nver basin
Our aim is to address the questions posed aboye by appiying a frarnework to investigate the coupied
geomorphological and ecohydrological patterns rnanifested within a typical semiarid drainage network. To
this end, we analyze the Upper Rio Salado basin, iocated near the Sevilleta Long-term Ecological Research
(LTER) site in central New Mexico (Figure 5a). Our anaiysis considers the upper portion of the Rio Salado
basin, and three subbasins contained within. The basin covers an area of 464 km2, and its eievation ranges
from 1985 rneters aboye sea level (a.s.l.) to 2880 m a.s.l. The stream network (Figure 5b) is derived from
30-meter resolution USGS digital elevation rnodeis available frorn the searnless data distribution system
In this study, the basin organization is represented using the geornorphoiogical area fiinction W(x), which is
a well described indicator of the fractal morphoiogy contained within river networks (Marani et al., 1991;
Rinaldo et al. 1993). The geomorphologicai arce flmction is reiated to the width function, which measures
the number of stream iinks at a given distance frorn the outlet measured through the network. The arce
fhnction is found by dividing the basin into eiementary areas and determining the distance of each area
from the outiet as rneasured through the network flow path. In this way, the area function rnaps the two
dimensional structure of the basin into a one dimensional support. Thus, at each distance, the value of W(x)
is determined according to

10. YA
YA Lxr
where A denotes those elementary areas that are located at the same distance x from the outlet and ±- A.
is the total sum of A for all x (i.e. the total area of the basin). Here x is measured through the stream
network in constant intervais of Ax and takes values of O <x 1 through normalization by the maximum
upstream distance from the outlet. The resulting function may be interpreted as the probability distribution
of area within the basin, such that
The statistical structure of W(x) may be characterized through its spectral density fiinction, denoted by
Marani et al. (1994) show that Sw(f) exhibits power-law scaling for many basins of different
10

11. New Mexico, USA
(A)
TER
(B) ¡
Upper Rio Salado Basin j
Rio Salado Sub-Basins
E2sr
6an4 W
11
SfreamOvder
III
0 2.5 5 10 t'JIçnt,i
Figure 5 - Location of the Upper Rio Salado basin near Sevilleta LTER in New Mexico, USA (A). The
lower panel (B) depicts the overail basin river network, as well as, the location of the three subbasins used
in the study.
11

12. '
geomorphological characteristics, such that
S1 O(
' with the value of fi found to vary between 1.7
and 1.9. Functions which exhibit this behavior are said to be selí-afline, and belong to the general class of
processes known as fractional Brownian motion. The presence of seIí-affinity implies statistical scale
invariance, so that the process remains statistically unchanged when proper scaling factors are applied to
each axis, e.g., distance from the outlet and area distribution. Therefore, for the self-affine
geomorphological area function W(x), we can state that
W(x + )x) - W(x) yh'[W(x + x) -
(14)
where d means equality in the statistical sense, H is the Hurst exponent, which is related tofi according to
H = (fi —1)1 - , y is an arbitraiy rescaling factor along the distance axis, and y' is the self-affine scaling
factor along the area distribution axis. The values of H and fi indicate the persistence of a function, such
that when > 1/2 aucl fi >
2 we say that the fractional Brownian motion exhibits persistence, and
when .H <1/2 aud fi < 2 the process is said to display antipersistence. Ordinary Brownian motion
is a special case of fractional Brownian motion that occurs when FT = 112 and /1 = 2
Ml of the area functions for the Rio Salado basin and subbasins display the scaling relationship
5(f) °c f (with values offi in the range between 1.8 and 1.9. Having established the self-affine
organization of the basin network through the analysis of the geomorphological area function, we now
focus on the distribution of soils and vegetation within the Rio Salado basin.
The pattern of soil texture within the Rio Salado basin is taken from the USDA STATSGO soil database
(USDA, 1994). The basin is defined by the distribution of three different soil textures - loam, sandy loam,
and silty loam. In the upper part of the basin, loam and silty loam are most common, while sandy loam is
fornid in the lower portion of the basin particularly along the channel network. Specffic values of saturated
hydraulic conductivity K, porosity n, fleid capacity '1' 9 , and the hygroscopic point 'Ph are related to soil
texture using relationships defined in Clapp ¿md Hornberger (1978).
The distribution of vegetation composition (Figure 6a) is taken from the USGS 28.5 meter National Land
Cover Dataset (NLCD) based on Landsat imageiy (Vogelmann et al., 2001), which we resample at 30
meter resolution usmg a nearest neighbor techuique. The composition of vegetation cover is analyzed here
in terms of different plant functional types, which vaiy in their structure ¿md use of water. Thus the cover
vanes between grassland (25.4%), shrubland (28%), ¿md forest (45.7% of the total basin area). In the Rio
Salado area, forests are composed of open or moderately closed woodlands of pinyon pine (Pinus edulis)
¿md stands of one-seed juniper (Juniperus monosperma). Shrublands within the Rio Salado basin are

13. Landcover
(A)
9nhiIet
- -
Grcwirig Season Slandard Deviabon
RelaVve SdI Moislure
SM
(B) . -.
-
- Sit
'4
Grawing Season Average
Rabve SI Mosture
::
- ES
Figure 6
Figure 6— Spatial patterns of the vegetation (A) and soil (B) distributions within the Rio Salado basin, as
well as the standard deviation (C) and mean (D) of the steady-state relative soil moisture during the
growing season.
13

14. dom inated by the evergreen creosote bush (Larrea tridentata),which is a widespread and characteristic
evergreen shrub of the Chihuahuan and Sonoran deserts. The most common species within the grasslands
are those with Great Basin affinities such as galleta (Hilaría jamesii), and blue gramma (Bouteloua
gracilis). The area fi.mction concept defined aboye is used to analyze the patterns of vegetation and soil
distribution within the Rio Salado basin. For any arbitrary quantityf distributed within the basin, we define
the fiinction W(x), which is analogous to the area function presented aboye and describes the distribution of
the quantityf at a distance x measured through the network,
x
(15)
Through this definition of lV(x) we use the land cover data to determine the vegetation area function for the
various land cover components (tree, shrub, and grass). In the case of the vegetation area fiinction, the
number of elementaiy arcas containing a given land cover type at each distance is divided by the total
number of elementary arcas containing the assigned land cover type. The resulting vegetation area
functions for tices, shrubs and grasses (W ree(X), WShrub(X), and W rass(X) respectively) are given for the main
Rio Salado basin (Basin 1) in Figure 7. These vegetation functions exhibit markedly different
characteristics for each of the land cover types, but comparisons between the total basin and three subbasins
demonstrates consistency in these differences. As in the case of the geomorphological area function, the
power spectra of the vegetation area functions exhibit clear power law scaling (Figure 7). The spectra of
shrub and grass vegetation distribution have smaller siopes than the spectra of area distribution, indicating
an enrichment of high frequency variation in thc distribution of these types of vegetation compared to the
distribution of tice vegetation. The analysis is now further extended to investigate the spatial pattem of the
steady-state mean relative soil moisture. With the objective of linking relative soil moisture patterns with
the geomorphological organization present iii river basins, we introduce the soil moisture profile of the
basin, < >(x)• This function describes the changing values of mean relative soil moisture throughout
the basin, and is defined as the average value of the steady-state mean relative soil moisture of aH the points
located at the same distance x from the basin outlet measured through the network. Thus,
s >
<s >(x)= (16)
where N is the number of elementaiy pixeis at distance x measured through the drainage network, and
<s > is mean value of the steady-state probability distribution of relative soil moisture at each of these N
locations. As in the use of the geomorphological area function, these functions collapse the two
dimensional spatial structure of the soil moisture into a one-dimensional function that describes its
variability through the basin network. The loss of information regarding soil moisture pattern due to the
degradation of the 2-dimensional field into a single linear transect is balanced by the coherence of
associating the soil moisture pattern to the structure of the drainage network. The soil moisture profile and
corresponding power spectrum that result from this analysis as applied to basin 1 are reported in Figure S.
The power-law spectral density function indicates that <r >(X ) is a seIf-affine process despite Ihe
extended 2-dimensional patterns of mean soil moisture imposed by our coarse speciíication of soil texture.
14

15. 3 -
AA)
1.5
0.5
1
7(B)
04
1,04
.4
10l io 2 i •'
10
102
10
- -1.60
4)
4
10
104 -
•;4
... .;,
lo:
-1.70
4,
'n- L
0.2 0.4 0.6 0.8
1
3.5
2.5
1.5
CO04O
T
lo_• 10_ 10•• 10 10
x f
Figure 7
Figure 7— Normalized vegetation area functions and Iheir power spectra density functions for each of the
three vegetation types —trees (A), shrubs (B) and grasses (C) in the main basin of the upper Rio Salado.
15

16. 0.37
0.36 (A)
0.35
0.34
0.33
I 0.32
0.31
0.3
0.29
0.28
A fl
1J.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
100
1 02 (B)
1 -1.68
1
1 08
10,3 i0 2 10
f
Figure 8
100 101
Figure 8 - Soil moisture profile s and associated power spectra for the Upper Rio Salado basin
8. Water Stress and Vegetation Patterns
The dynamic water stress described in Section 5 is based on the assumption that the transient dynamic of
soil moisture associated with the initial condition (so) is not a significant factor in determining the overali
seasonal dynamics of the temporal evolution of soil moisture. However, in the case that the initial condition
of soil moisture is very high due to near-saturated conditions at the start of the growing season, it is
necessary to detennine the role of initial conditions m determining the overail stress conditions experienced
by the vegetation during the subsequent growing season. For any initial condition sO aboye the steady-state
16

17. mean soil moisture <s>,it is possible to determine the mean first passage time Lo (in days) of the
stochastic process between s0 and <s> (Laio et al. 200 ib). The value of T (50)
can then be used to
rescale the dynaimc water stress experienced by the vegetalion. This reformulation of the dynamic water
stress reflects only the stress experienced by the vegetation during the portion of the growing season that is
not influenced by the transient dynamics associated wilh the initial condition, when water is readily
available. The determination of T
foliows the presentation of Laio et al. (200 ib), and it is used to
scale the dynaniic water stress 0'according to the following relationship (Rodriguez-Iturbe et aL, 200 ib)
0. = ç0. - :i:.: . (3)
(17)
This simple modified dynamic water stress e' allows for an effective synthesis of the interaction between
plants, soils, and climate systems with important transient responses thaI occur at the start of the growing
season. In this case, we have assigned initial conditions of SSfc in the portion of the Rio Salado basin thaI
lies aboye 2350 meters. We examine the resulting effects of this initial condition on the modffied dynamic
water stress experienced by the vegetation within the basin.
Figure 9 exhibits the differences in dynamic water stress experienced by each of the vegetation types for a
silty loam soil in the presence and absence of initially wet conditions. For the case without muja!
condilions (Figure 9a), the water stress of the tree vegetation is consistently al or near one, indicating the
lack of suitability of the Rio Salado basin for tree vegetation when considering only the steady-state soil
moisture distribution that arises from the growing season climate. The inclusion of an initial condition such
that fue soil moisture at the beginning of the growing season is equal to field capacity al elevations aboye
2350 meters predicts a dramatic reduction in tree vegetation dynamic water stress at locations aboye 2350
meters with little effect of initial conditions on fue water stress experienced by shrub ¿md grass vegetalion
The reduction in water stress for woodland vegetation in the upper portion of the Rio Salado basin is m
agreement with fue general observation that signiflcant winter snow accumulation ¿md high soil moisture
values during the subsequent spring snowmelt are common in semiarid Pinyon-Juniper woodlands (Wilcox,
1994). Therefore, we suggest thaI the presence of both transient ¿md steady-state dynainics must be
considered when contemplating the distribution of vegetation patterns within highly seasonal semiarid
ecosystems. The modified dynamic water stress profile based on the consideration of w nter snow
accumulation is presented in Figure 10. This modified proflie represents the average )' for all pixeis
located al fue same distance from the outlet measured throughout the network. As the distance from the
outlet increases, the average modifled dynamic water stress decreases. These resulta indicate the strong
effect thaI the initial conditions have on the dynaniuc water stress, particularly in the tree vegetation, which
can mosl effectively exploit the large store of soil moisture that exists al the start of the growing season due
lo spnng snowmelt. The shailower rooting depth of shrub ¿md grass vegetation reduces the effect of the
snowmelt processes by limiting the anlount of additional soil moisture available to these vegetation types at
the start of the growing season.
17

18. 1 .
- Tree
Shrub
-'.- Grass
ci)
4.,
0.5
o
O
j
r
o
w4..
4..
l)
4..
U)
. 0.
u
E
2100 2200 2300 2400 2500 2600 2700 2800
EIevaton (m)
- Tree
(3) —1--- Shrub
—a.— Grass
2100 2200 2300 2400 2500 2600 2700 2800
Elevation (m)
Figure 9
Figure 9 - Dynamic water stress (A) and modified dynamic water stress (B) for tree, shrub, and grass
vegetation across the Rio Salado elevation gradient. The modified dynamic water stress (B) is derived for
an initial condition ofs=s51 at the start of the growing season. In both cases, stress is calculated for the case
of silt loam soil only.
18

19. 1
(A)
0.8
0.6
I
0.4
0.2
0I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
100
N..
(8)
-2
10
-1.55
u,
10
1
10-a l
-3 -2 -1 0
10 10 10 10 10
f
Figure 10
Figure 10 - Modified dynamic stress profile and associated power spectra for the Upper Rio Salado basin.
19

20. Given the importance of water stress in determining the distribution of vegetation within semiarid river
basins, the derivation of a characteristic dynamic water stress profile may provide insight into the extent to
which the current organization of vegetation is optimized with respect to dynamic water stress. This is
investigated through a comparison of the existing vegetation pattern to two alternative hypothetical
vegetation distributions. The flrst is the distribution of vegetation that arises from a random assignment of
vegetation type at each location within the basin, under the constraint that the proportions of overali land
cover composition are preserved. The second hypothetical pattern is the one arising from the specification
of the vegetation type that exhibits the lowest dynamic water stress at each location within the basin. From
the sole point of view of water stress we could say that the second pattern represents and optimal or ideal
distribution of vegetation. Figure 11 portrays the basin patterns resulting from each of these two alternative
specifications of land cover, as well as the actual pattern observed within the Rio Salado basin. A visual
comparison of these three patterns (actual, random, and ideal) suggests that the actual pattern of vegetation
distribution contains elements of both the highly organized large-scale ideal pattern, as well the
characteristic small-scale variation associated with the random pattern. The modified dynamic stress profile
based on each of the two hypothetical distributions (random and ideal), as well as the actual dynamic water
stress profile are presented in Figure 12. Our results suggest that the current vegetation pattern is
configured such that it is well-constrained by these two extremes of vegetation organization, so that the
basin tends to experience an intermediate level of water stress that is neither random nor ideal. The
existence of a distribution of water stress that is globally bounded by the random and ideal vegetation
distribution may allow for the development of dynamic modeling approaches for predicting the distribution
of vegetation pattern in river basins under conditions of changing climatic and edaphic regimes. Moreover,
it is likely that the vegetation patterns in water-controlled ecosystems tend to approach an optimal
configuration in terms of water stress but are subject to important and decisive random contingencies of an
altogether different character. Conceptually, this is not different than the notion of feasible optimality at
work in the organization of the drainage network (Rodriguez-Iturbe and Rinaldo, 1997).
Acknowledgements
Major parts ofthis paper are taken from Caylor at al. (2004) and Porporato et al. (2004). This research was
supported through National Science Foundation grants through the National Center for Earth-Surface
Dynamics (EAR-0120914) and Biocomplexity (DEB-0083566)
20

21. -
ShrubI'u1
OrassIid
Figure 11
Figure 11 - Actual pattern of vegetation in the Rio Salado basin (A) compared to two hypothetical patterns.
Panel B depicts a vegetation pattern resulting from a random assignment of vegetation type that preserves
the overail proportion of each type. Panel C shows the pattern of vegetation type that corresponds to the
minimuni modified dynamic water stress at each location.
21

22. 0.9
0.8
0.7
0.6
e.
0.4
0.3
0.2
0.1
LP
4•i lii.1
.JI
Random
- Actual
Ideal
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 12
Figure 12 - Comparison of the actual modified dynaniic stress profile (-) to random (-) and ideal (-)
distributions of vegetation within the Rio Salado basin.
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