Environmental Engineering Reference
In-Depth Information
2.3.2 White shot noise as a limiting case of the DMN
WSN can be obtained from DMN by taking the following parameter values:
1
=
α
k
1
,
2
=
0
,
k
1
→∞
,
k
2
=
λ.
(2.56)
It is easily verified that the average of WSN is recovered from Eq. (
2.7
)whenwe
use the values in (
2.56
). Analogously, the higher-order moments and cumulants are
found in the limiting case to diverge with rate
1
, consistent with results obtained
in the previous subsection. Also, autocorrelation function (
2.55
) can be obtained from
Eq. (
B2.1-5
) by taking the limit for
m
−
∞
lim
a
→
0
e
−|
t
/
a
|
/
τ
c
→
0 [note that
δ
(
t
)
=
(2
a
)].
ξ
sn
=
ξ
sn
−
λα
,is
considered. We obtain the zero-average shot noise from the DMN by taking the same
parameter values as in (
2.56
), except
In the following subsection, a zero-average shot-noise process,
2
=−
λα
.
2.3.3 Relevance of white shot noise in the biogeosciences
The infinitesimal duration of the spikes and the uncorrelation could suggest that
WSN is not well suited to realistically represent an external forcing in “real-world”
processes. However, this is not necessarily true. In fact, to determine whether WSN
is suitable for the modeling of a random driver, we should look at the ratio between
the duration of a single random event and the typical time scale of the deterministic
process forced by this random event. If this ratio is low, then the temporal structure
of the forcing in the course of each event has no influence on the overall dynamics,
and - at the time scale of the deterministic process - the external forcing can then be
modeled as a sequence of episodic instantaneous impulses.
For example, consider the plots shown in Fig.
2.10
. They report the same rainfall
sequence visualized at different time scales. When the rainfall is described at a 10-min
temporal resolution, several complex features of the rainfall events can be observed:
For example, the rainfall intensity is irregular and shows strong temporal gradients.
But as the process is observed at coarser time scales, the temporal resolution decreases
until the rainfall sequence appears as a sequence of spikes, whose height is equal to
the total rainfall in the corresponding event. In the study of a process driven by this
random forcing, the time scale of the process will obviously dictate the scale that has
to be adopted to model the noise. For example, if we are interested in the modeling
of the flood process in a relatively small basin (e.g., contributing area
100 km
2
),
we need to account for the hourly structure of precipitation; in contrast, if we are
investigating seasonal dynamics of vegetation, the daily time scale will be sufficient
and rainfall can be modeled as WSN (
Rodriguez-Iturbe et al.
,
1999b
).
The previous example clarifies that when the time scale of the deterministic pro-
cess is much greater than the duration of single random events, WSN is a suitable
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