Environmental Engineering Reference
In-Depth Information
generate persistent time series. We will consider as
measures of persistence the autocorrelation function
(Section 3.5.1) and spectral analysis (Section 3.5.3).
As models that generate persistence, we consider the
autoregressive (AR) model for short-range persistence
(Section 3.5.3) and then spectral filtering of white noises
with their resultant fractional noises and walks for
long-range persistence (Section 3.5.4). In Section 3.6,
we discuss how complex time series can be generated
using physical models. We then discuss and summarize
the context of this chapter in Section 3.7. To aid the
reader, a table of variables used is given in Table 3.1, and
acronyms/abbreviations in Table 3.2.
3.2 Examples of environmental
time series
For illustration of various concepts that will be discussed
in this chapter, five contrasting examples of environ-
mentally related time series are given in Figure 3.1. In
Figure 3.1a, the mean monthly concentrations of CO
2
Table 3.1
Variables.
Symbol
Description
Section
Equation
introduced
introduced
β
Power-law exponent of the power-spectral density, i.e. the strength of the long-range
persistence.
3.5.3
3.13
δ
Sampling interval (including units of time) in discrete time series,
x
n
.
3.3
x
Range over which
x
is evaluated for a pdf (the 'bin' size).
3.3
ε
Random values in a white noise time series.
3.4
3.4
φ
Prescribed coefficient for auto regressive (AR) process.
3.5.2
3.7
η
White noise term in space and time.
3.6
3.17
π
Irrational number pi.
3.3
3.2
σ
ε
Standard deviation of the white noise sequence,
ε
.
3.4
3.5
σ
n
Standard deviation of the time series after a specified number
n
values,
x
n
,
n
3.5.4
3.15
=
1, 2, 3,
...
,
N
.
σ
x
Standard deviation of all
N
values in the time series
x
n
,
n
=
1, 2, 3,
...
,
N
.
3.3
3.1
τ
Time lag.
3.5.1
3.6
τ
τ
C
(
)
Autocorrelation function as a function of time lag,
.
3.5.1
3.6
c
v
Coefficient of variation (parameter of the log-normal distribution).
3.4
f
(
x
)
(Noncumulative) probability distribution function (pdf).
3.3
3.2
F
(
x
)
Cumulative distribution function (cdf).
3.3
3.3
f
m
Frequency,
f
m
=
m
/
(
N
δ
), where
m
=
1, 2, 3,
...
,
N
/
2.
3.5.3
h
Function for one-dimensional topography.
3.6
3.17
Ha
Hausdorff measure.
3.5.4
3.15
i
Square root of
−
1.
3.5.3
3.11
M
W
Earthquake moment magnitude.
3.2
N
Number of data points in a given series.
3.3
3.1
S
m
Estimator of the power spectral density.
3.5.3
3.12
t
Time.
3.3
T
Length of time series,
T
=
N
δ
.
3.3
x
(
t
)
Continuous time series as a function of
t
.
3.3
x
n
Discrete time series,
x
1
,
x
2
,
...
,
x
N
.
3.3
3.1
x
Mean value of the time series,
x
1
,
x
2
,
...
,
x
N
.
3.3
3.1
X
m
Fourier transform
X
1
,
X
2
,
...
,
X
N
of the time series,
x
1
,
x
2
,
...
,
x
N
.
X
m
=
a
+
b
i
,aandb
are coefficients,
i
is the square root of
3.5.3
3.11
−
1.
+
b
2
)
0
.
5
.
Modulus of
X
m
,
|
X
m
|=
(a
2
|
X
m
|
3.5.3
3.12
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