laboratory in oceanography: data and methods rotary spectra · 2020-04-22 · as such, in general,...
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Sundermeyer
MAR 550
Spring 2020 1
Laboratory in Oceanography:
Data and Methods
MAR550, Spring 2020
Miles A. Sundermeyer
Rotary Spectra
Sundermeyer
MAR 550
Spring 2020 2
Rotary Spectra
Rotary Spectra decompose vector time series (e.g., current or wind data) into
clockwise and counter-clockwise components.
Suppose we have u and v components of velocity:
These can be written in complex form as:
)sin()cos()(
)sin()cos()(
tDtCtv
tBtAtu
)sin()()cos()(
)sin()cos()sin()cos(
tiDBtiCA
tDtCitBtA
ivuR
Sundermeyer
MAR 550
Spring 2020 3
Rotary Spectra
Now write R as a sum of clockwise and counter-clockwise rotating components
as follows:
(Note: eit = cos(t) + i sin(t) rotates counter-clockwise in the complex plane,
and e-it = cos(t) – i sin(t) rotates clockwise.)
Comparing this to the final expression on the previous slide, we had:
Equating the coefficients of the cosine and sine parts, we find:
)sin()()cos()(
)sin()cos()sin()cos(
tiRRtRR
titRtitR
eReRR titi
)sin()()cos()( tiDBtiCAR
)(2
1
)(2
1
BCiDAR
BCiDAR
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Rotary Spectra
The magnitudes of the rotary components follow as:
Note: since the CW and CCW components are rotating at the same frequency but
in opposite directions there will be times when they are additive (pointing in the
same direction) and times when they are opposing (pointing in opposite direction)
and tend to cancel each other out.
These additive and opposing times define the major and minor axes of an ellipse:
major axis = (R++ R-)
minor axis = (R+- R-)
21
22
21
22
2
1
2
1
BCDAR
BCDAR
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Rotary Spectra
While the orientation and phase of the ellipse is:
2
1 :phase
2
1 :norientatio
DA
BC
DA
BC
1
1
tan
tan
where:
Important note: This example is for a single frequency velocity time series, but
in general our data records will be made up of a broad spectrum of frequencies.
As such, in general, the rotary amplitudes and corresponding A, B, C, and D
will also vary with signal frequency, as will the major and minor axes,
orientation and phase. Also, signals must be de-meaned and de-trended
before computing spectra.
Sundermeyer
MAR 550
Spring 2020 6
Rotary Spectra Examples:
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MAR 550
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Rotary Spectra Examples:
Sundermeyer
MAR 550
Spring 2020 8
Rotary Spectra Examples:
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MAR 550
Spring 2020 9
Rotary Spectra Examples:
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MAR 550
Spring 2020 10
Rotary Spectra Example:
(fro
m E
me
ry a
nd
Th
om
pso
n, 2
00
4)
Inertial + tidal
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MAR 550
Spring 2020 11
Rotary Spectra
Advantages of rotary spectra over cartesian spectra:
1. Rotary spectra and associate coherence analysis is independent of the
coordinate system (i.e., invariant to rotation).
2. The results are suitable for motions that may be hightly non-rectilinear.
Sundermeyer
MAR 550
Spring 2020 12
Rotary Spectra
Suppose now that we have two time series, ocean current, and wind ...
Autospectrum: The autospectrum for each time series is:
e.g., Scc (f ≥ 0) is the power spectral density of the counter-clockwise
component of the current time-series, where the area under this curve versus
frequency will equal the combined variance of two cartesian coordinate current
velocity components (i.e., Parseval’s theorem)
0,)]([
0,)]([2
2
ffA
ffAS
c
c
cc
0,)]([
0,)]([2
2
ffA
ffAS
w
w
ww
Sundermeyer
MAR 550
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Rotary Spectra
Inner cross-spectrum: The inner cross-spectrum of two time series compares
the joint energy of the two time series for the rotary components rotating in the
same direction (e.g. the clockwise component of one vector to the clockwise
component of the other vector):
with * denoting the complex conjugate and <> representing an ensemble
average.
)()()( * fWfWfS wccw
0,)()(
0,)()()]([
)]([
fefAfA
fefAfA
wc
wc
i
wc
i
wc
Sundermeyer
MAR 550
Spring 2020 14
Rotary Spectra
Inner coherence squared: The inner coherence-squared between the wind and
current time series at frequency (f) is calculated according to:
The coherence ranges from 0 to 1, and represents the similarity (or variability)
of the two time series to each other. A value near unity indicates a high degree
of correlation, while a coherence near zero indicates a negligible correlation.
Using a 95% confidence interval, a limiting value, or level to which coherence-
squared values occur by chance is given by:
where DOF represents the degrees of freedom contained in the time-series.
0,/)sin()cos(
0,/)sin()cos(
2222
2222
fAAAAAA
fAAAAAAC
wcwwwcwcwc
wcwcwcwcwc
cw
)]2/(2[5.01 DOFtsignifican
cwC
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Rotary Spectra
Inner phase: The inner phase for the cross spectrum and coherence measures
the phase lead of the rotary component of the one time-series with respect to
the other time-series. It can be calculated according to the following equation:
0,)cos(/)sin(
0,)cos(/)sin()tan(
22
22
fAAAA
fAAAA
wcwcwcwc
wcwcwcwc
cw
Sundermeyer
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Rotary Spectra
The following are similarly defined for the rotary components rotating in the
opposite direction (e.g. the clockwise component of one vector to the
anticlockwise component of the other vector)
• Outer cross-spectrum
• Outer coherence squared
• Outer phase
Sundermeyer
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Spring 2020 17
Additional Topics – Rotary Spectra
Key Points:
• Rotary spectra decompose complex time series into CW and CCW rotating
components.
• Complex data could be wind, currents, T & S, etc.
• Can be used to analyze wind, waves or currents and/or to isolate inertial
motions, tidal motions, and certain classes of waves.
• Rotary spectra are invariant under coordinate rotation.
• Cross spectra and coherence squared can be used to assess phase-lagged
coherence between different data series.
References:
• Mooers, C. N. K., 1973. A technique for the cross spectrum analysis of pairs
of complex-valued time series, with emphasis on properties of polarized
components and rotational invariants. DSR, 1973, Vol. 20, 1129-1141
• Emery and Thompson, Data Analysis Methods in Physical Oceanography,
2nd Edition, 2004, Elsevier.
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