Environmental Engineering Reference
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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 )
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
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
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,
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
(average storm frequency), with each storm having an exponentially distributed
random depth h with mean storm depth
. 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
= ϕ
[ s ( t )
t ]
[ s ( t )]
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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