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with
g
(
H
)
=−
H
and constant values of
h
=
1. The steady-state pdf of
H
(
D'Odorico
and Fagherazzi
,
2003
)is
=
C
(
H
+
b
)
−
a
−
1
(1
−
H
)
a
−
1
p
(
H
)
,
(4.18)
b
)
b
λ/
a
. We can obtain the deterministic counterpart of this
process by replacing the stochastic rate of erosion with its mean value
where
C
=
(
λ/
a
)(1
+
H
. Thus the
underlying deterministic process has only one stable state and no unstable state within
the [0, 1] interval [Fig.
4.15
(c), dashed curve with
λ
1], and the stochastic dynamics
have a U-shaped distribution [Fig.
4.15
(d)], suggesting that there is a statistically un-
stable state between 0 and 1; 0 and 1 are the stable states of the stochastic process. This
result is noise induced and resembles the case of noise-induced bistability discussed
in the previous two sections. The Poisson noise is able to convert a deterministic
system with only one stable state into a stochastic one with two preferential states and
an intermediate unstable state. This effect is due to the term
α
=
g
(
H
) in the equation,
λα
i.e., to the multiplicative character of noise (Eq.
2.69
).
4.3.4 Stochastic soil-moisture dynamics
Stochastic differential equations driven by white shot noise have been used in recent
years to model the soil-water balance (
Rodriguez-Iturbe et al.
,
1999b
;
Laio et al.
,
2001
), providing a theoretical framework to investigate the effect of climate, soils, and
vegetation on soil-moisture dynamics in the root zone. The probability distributions
of the soil-water content determined with this model have been used in a number
of ecohydrological applications, including the study of water-stress conditions in
vegetation, the modeling of land-atmosphere interactions, and the analysis of the
hydrologic controls on photosynthesis and nutrient cycling (see
Rodriguez-Iturbe and
Porporato
,
2005
). In this subsection we use this model to investigate the effect of
soil-moisture-precipitation feedbacks on the dynamics of soil moisture. Following
Laio et al.
(
2001
), we consider the water balance of a surface soil layer of depth
Z
r
(Fig.
4.16
). The only input to the soil-water balance is due to precipitation, which is
modeled as a stochastic process. We use a Poisson process of rainfall occurrences at
rate
(average storm frequency), with each storm having an exponentially distributed
random depth
h
with mean storm depth
λ
. The output is due to evapotranspiration and
leakage, which are modeled as deterministic functions of soil moisture (Fig.
4.17
). We
express soil-moisture dynamics through a stochastic differential equation expressing
the soil-water balance at a point (
Laio et al.
,
2001
)as
nZ
r
d
s
(
t
)
d
t
α
=
ϕ
[
s
(
t
)
,
t
]
−
χ
[
s
(
t
)]
,
(4.19)
where
s
is soil moisture,
t
is time,
n
is the soil porosity,
Z
r
is the active soil depth (i.e.,
the root zone, which is active in the exchange of water with the overlying atmosphere),
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