9.5 Spectral and Statistical Analysis of Time Series
333
Fourier analysis of finite length records results in inherent side lobes in the
spectral domain. The large side lobes allow leakage of energy at frequencies
well separated from the main lobe. To reduce the leakage problem, it is common practice to introduce a time window that tapers the time-history data,
to eliminate the discontinuities at the beginning and end of the records being
analyzed. There are numerous such windows in current use. For example, the
Hanning data window has the form:
{
~ [1- cos (~)] = 1- cos 2 (~)
v(t) =
2
L6.t
L6.t
o
otherwise.
for 0:::; t :s: L6.t,
(9.61)
If the total number of points, N, cannot be made arbitrarily large to get a
more accurate spectrum estimation, then a reasonable procedure is to overlap
the segments by one half of their length. The number of overlapping segments
now becomes:
2N
Kl = - - 1 = 2K - 1
L
'
in which K is the number of non-overlapped segments.
The final estimate of the one-sided spectral density S(w) becomes:
27rk
in which Wk = L6.t'
(9.62)
(9.63)
Spectra of Vector Series. Vector type time series such as currents and wind
are recorded as speed and direction by rotor-type meters and as orthogonal
components by acoustic and electromagnetic meters. Therefore, to calculate
the spectra of current and wind, the data has to be resolved into orthogonal
components related to the Earth-referenced Cartesian coordinate system with
two orthogonal horizontal components and a vertical component. In the open
sea, the horizontal velocity vector is usually resolved into eastward (u) and
northward (v) component, whereas in the coastal region the v axis is oriented
approximately parallel to the coastline and u axis is directed normally to the
coastline.
When investigating currents over abrupt topography or other forms of narrowband oscillatory flows, it is useful to separate the velocity vector into clockwise and counterclockwise rotating circular components. Resulting spectra are
known as rotary spectra (Gonella, 1972; Mooers, 1973).
333
Fourier analysis of finite length records results in inherent side lobes in the
spectral domain. The large side lobes allow leakage of energy at frequencies
well separated from the main lobe. To reduce the leakage problem, it is common practice to introduce a time window that tapers the time-history data,
to eliminate the discontinuities at the beginning and end of the records being
analyzed. There are numerous such windows in current use. For example, the
Hanning data window has the form:
{
~ [1- cos (~)] = 1- cos 2 (~)
v(t) =
2
L6.t
L6.t
o
otherwise.
for 0:::; t :s: L6.t,
(9.61)
If the total number of points, N, cannot be made arbitrarily large to get a
more accurate spectrum estimation, then a reasonable procedure is to overlap
the segments by one half of their length. The number of overlapping segments
now becomes:
2N
Kl = - - 1 = 2K - 1
L
'
in which K is the number of non-overlapped segments.
The final estimate of the one-sided spectral density S(w) becomes:
27rk
in which Wk = L6.t'
(9.62)
(9.63)
Spectra of Vector Series. Vector type time series such as currents and wind
are recorded as speed and direction by rotor-type meters and as orthogonal
components by acoustic and electromagnetic meters. Therefore, to calculate
the spectra of current and wind, the data has to be resolved into orthogonal
components related to the Earth-referenced Cartesian coordinate system with
two orthogonal horizontal components and a vertical component. In the open
sea, the horizontal velocity vector is usually resolved into eastward (u) and
northward (v) component, whereas in the coastal region the v axis is oriented
approximately parallel to the coastline and u axis is directed normally to the
coastline.
When investigating currents over abrupt topography or other forms of narrowband oscillatory flows, it is useful to separate the velocity vector into clockwise and counterclockwise rotating circular components. Resulting spectra are
known as rotary spectra (Gonella, 1972; Mooers, 1973).