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Laboratory in Oceanography:

Data and Methods

MAR550, Spring 2020

Miles A. Sundermeyer

Rotary Spectra

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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

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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.

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Rotary Spectra Examples:

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Rotary Spectra Examples:

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Rotary Spectra Examples:

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Rotary Spectra Examples:

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Rotary Spectra Example:

(fro

m E

me

ry a

nd

Th

om

pso

n, 2

00

4)

Inertial + tidal

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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.

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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

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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

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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

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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

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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.