Environmental Engineering Reference
In-Depth Information
The steady-state probability distribution of the state variable
ξ
dn
is then a discrete-
valued distribution that can assume only two values,
2
, with probability
P
1
and
P
2
, respectively. The steady-state moment-generating function (see, for example,
van Kampen
,
1992
, for a definition) is then
1
and
2
k
2
e
v
1
k
1
e
v
2
+
e
v
i
P
i
=
M
dn
(
v
)
=
,
(2.5)
k
1
+
k
2
i
=
1
and the corresponding cumulant-generating function (see
van Kampen
,
1992
,fora
definition) is
log[
k
2
e
v
1
k
1
e
v
2
]
K
dn
(
v
)
=
log[
M
dn
(
v
)]
=
+
−
log[
k
1
+
k
2
]
.
(2.6)
By definition of the cumulant-generating function, the steady-state cumulant of the
order of
m
is obtained as the
m
th derivative of
K
dn
(
v
) with respect to
v
, calculated in
v
=
0. Therefore the mean of the process,
κ
1dn
,is
v
=
0
=
d
K
dn
(
v
)
k
2
1
+
k
1
2
k
1
ξ
dn
=
κ
1dn
=
.
(2.7)
d
v
+
k
2
Because the DMN is used as a noise term in Eq. (
2.1
), it can be useful to consider
a zero-average process. If this is the case, using Eq. (
2.7
)wehave
1
k
2
+
2
k
1
=
τ
2
+
1
2
τ
1
=
0
.
(2.8)
In this case the (stationary) dichotomous Markov process is characterized by three
independent parameters. For example, we can choose (i) the two transition rates
k
1
and
k
2
(or the mean durations
τ
1
and
τ
2
) and (ii) assign the value of one of the states of
ξ
2
)byusingEq.(
2.8
). Unless explicitly
stated otherwise, in what follows we refer to the case of zero-mean [Eq. (
2.8
)] DMN.
The variance of the dichotomous process is
dn
, say
1
, and obtain the other value (i.e.,
v
=
0
=
d
2
K
dn
(
2
−
1
)
2
(
k
1
v
)
k
1
k
2
(
ξ
dn
−
κ
1dn
)
2
(
=
κ
2dn
=
=−
1
2
,
(2.9)
k
2
)
2
d
v
2
+
and the autocovariance function is
2
−
1
)
2
(
k
1
k
1
k
2
(
e
−|
t
−
t
|
(
k
1
+
k
2
)
2
e
−|
t
−
t
|
(
k
1
+
k
2
)
dn
(
t
)
ξ
dn
(
t
)
ξ
=
=−
,
(2.10)
1
k
2
)
2
+
as demonstrated in Box 2.1, Eq. (
B2.1-5
). The structure of the autocovariance function
shows that the dichotomous noise is a colored noise, i.e., it is autocorrelated. This
is an important characteristic that explains why this type of noise is commonly used
to mimic natural processes in the biogeosciences (see Subsection
2.2.2
). A typical
temporal scale of a correlated process is the integral scale
, defined as the ratio
between the area subtended by the autocovariace function (i.e., the integral of the
autocovariance function with respect to the lag) and the variance of the process. The
I
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