Environmental Engineering Reference
In-Depth Information
When the distribution
p
H
(
h
) of the jump sizes is exponential with mean
α
, we obtain
the following steady-state solution of Eq. (
B2.3-7
):
(
z
)
exp
(
z
)
d
z
1
z
α
−
λ
1
p
(
z
)
=
C
−
,
(B2.3-8)
ρ
ρ
z
where
C
is an integration constant. Backtransforming Eq. (
B2.3-8
)fromthe
z
to the
φ
domain, we recover Eq. (
2.69
). To this end, we recall that if
z
(
φ
) is a monotonic function
the derived distribution of
. Moreover, the expressions of
master equation (
B2.3-7
) and probability distribution (
B2.3-8
)of
z
determined in this
box are for the case of WSN,
φ
is
p
(
φ
)
=
p
Z
(
z
)
|
d
z
/
d
φ
|
ξ
sn
with mean
ξ
sn
=
αλ
, whereas (
2.69
) refers to the case
ξ
sn
. Thus, to compare with (
2.69
) the pdf of
of zero-mean WSN
φ
obtained as a derived
distribution of
p
Z
(
φ
)using(
B2.3-8
), we need to use the transformation
f
(
φ
)
→
f
(
φ
)
−
αλ
).
The results obtained in this box can be also generalized to the case of processes
driven by WSN with state-dependent rate
g
(
φ
). Following the same steps as in
(
B2.3-3
)-(
B2.3-7
) [with an additional Taylor expansion of
λ
(
φ
λ
(
φ
) about
φ
] and neglecting
the higher-order terms, we obtain the pdf of
φ
for the case of state-dependent WSN
(
Porporato and D'Odorico
,
2004
):
(
z
)
exp
(
z
)
d
z
(
z
)
1
z
α
−
λ
p
(
z
)
=
C
−
.
(B2.3-9)
ρ
ρ
z
The effect of the state dependency of the rate
λ
on the properties of the probability
distribution of
are discussed in Chapter 4; in Subsection
2.3.4.3
we show that the same
result can be obtained as a limit of the pdf of
φ
φ
in the case of state-dependent
dichotomous noise [Eq. (
2.44
)].
p
φ
3
2.5
2
1.5
1
0.5
2
φ
0.25
0.5
0.75
1
1.25
1.5
1.75
Figure 2.12. The pdf's for Examples 2.5 (continuous curve) and 2.6 (dotted curve)
with
α
=
0
.
5 and
λ
=
1.
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