New Directions in Oceanographic Time Series AnalysisThe main point The tools we use to look at data matter. Download this talk, and Matlab toolbox JLAB, from . Lilly, Olhede, Sykulski,

New Directions in Oceanographic Time Series Analysis

J. M. Lilly1,S. C. Olhede2, A. M. Sykulski2, S. Elipot3, S. N. Waterman4

1NorthWest Research Associates, 2University College London,3University of Miami, 4University of British Columbia

February 26, 2014

Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis

The main point

The tools we use to look at data matter.

Download this talk, and Matlab toolbox JLAB, fromwww.jmlilly.net.


www.jmlilly.net

A surface drifter trajectory

The box shows a region where the drifter is trapped in an eddy.


The first tool: your eyes

Oscillatory motions due to an eddy are seen in the record center.


Rotary spectra using the periodogram

The periodogram is the squared Fourier transform. Red line is f .The rotary spectrum is the spectrum of z(t) ≡ u(t) + iv(t).


Rotary spectral using the multitaper method

Low-frequency peak of the cyclonic eddy is clearly apparent.Mysterious super-inertial peak at negative frequencies.


Rotary wavelet transform

Strong frequency-shifting of inertial oscillations is revealed.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis

Separation of eddy currents from residual

Wavelet transform is used as a basis for extracting the eddycomponent. This local best fit is called wavelet ridge analysis.


Historical dataset of eddy-resolving subsurface floats

1471 different instruments, 700,000 data points.


A preliminary eddy atlas

Apparent eddies in Rossby number band Ro = 1/64 to Ro = 1.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis


Moral: Our data is limited by the limitations of our methods.


What is time series analysis for?

The goal of time series analysis is (i) to extract as much usefulinformation as possible from the data, while at the same time, (ii)avoiding mis-interpretation of artifacts and spurious features.


Trajectories from a model

From a simulation by E. Danioux.



The periodogram is the squared Fourier transform. Red line is f .


Rotary spectra using the adaptive multitaper method

The periodogram (green line) is dominated by variance andblurring. Periodogram slope and apparent peaks are false.


Schematic of aliasing for rotary spectra

For rotary spectra, aliasing is not folding. It’s wrapping.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis

Summary of fundamentals

Do not neglect the fundamentals. We need to know what can gowrong — or we will waste time by interpreting spurious features.

Aliasing (wrapping)

Spectral blurring (broadband bias)

Variance (false peaks)

Important!

Any ”spectrum” created from data is not the true spectrum. It isan estimate. That estimate is a joint function of the data and theestimation method.


Summary of fundamentals

“More lives have been lost looking at the raw periodogram than byany other action involving time series!” —J. W. Tukey

Recommendations

Use the multitaper method of Thomson (1982).

Read: Park et al. (1987), ”Multitaper spectral analysis ofhigh-frequency seismograms”, Journal of Geophysical Research.

Use routines mspec and sleptap from JLAB, available atwww.jmlilly.net


www.jmlilly.net

Wavelets

An exceedingly brief summary of some useful recent results.

What is a wavelet? It is an intermediate basis betweendelta-functions in time and sinusoids. It lets you describe your dataas being composed of localized oscillations. But...

There has been a lot of confusion about how to choose the rightwavelet: Cauchy / Klauder, Derivative of Gaussian, Shannon,Bessel, Morlet, ... which we have sorted out.

All of these are special cases of a much broader family, thegeneralized Morse wavelets, Lilly and Olhede (2009,2013).

Recommendations

Generally speaking, you only need one wavelet, the Airy waveletΨ(ω) = ωβe−ω

3, see Lilly and Olhede (2013).

Use routines wavetrans and morsewave from JLAB, available atwww.jmlilly.net.


www.jmlilly.net

A foundation for time-varying analysis

Seismogram and float trajectory:

Definitely not sinusoidal

Inhabit multiple dimensions

Contaminated by ’noise’

=⇒ Modulated multivariate oscillations

Eddies, internal wave packets,seismic waves, ENSO, ...


Time-varying analysis

There exists an extremely powerful method for analyzing andinterpreting time-varying properties of quasi-periodic orquasi-oscillatory signals.

“The theory of instantaneous moments” –or–“The analytic signal method”

Foundation papers: Gabor (1946), Boashash (1992), Cohen(1995), Vakman and Vainshtein (1977), Picinbono (1997)

Extension to 2D (e.g. horizontal velocity currents): Lilly andOlhede (2009a)

Extension to 3D (e.g. internal wave packets): Lilly (2010)

Extension to N-D (e.g. climate fluctuations): Lilly and Olhede(2012)

Relationship to physical quantities (e.g. angular momentum,Kelvin circulation): Lilly (2012)


Instantaneous frequency and bandwidth

Represent a signal x(t) as an oscillation with changing amplitudeand phase:

x(t) = a(t) cosφ(t)

For a given x(t), a(t) and φ(t) are uniquely defined via theanalytic signal

x+(t) ≡ x(t) + iHx(t) ≡ a(t)e iφ(t)

where x(t) = <x+(t). This defines the canonical pair a(t), φ(t).

If you wish to recover ao and ωot for x(t) = ao cos(ωot), there isno other alternative involving a linear filter (Vakman).

References

Gabor (1946), Vakman and Vainshtein (1977), Picinbono (1997)


Instantaneous frequency and bandwidth

For a univariate signal

x(t) = a(t) cosφ(t)

the instantaneous frequency and instantaneous bandwidth are

ω(t) ≡ d

dtφ(t)

υ(t) ≡ 1

a(t)

d

dta(t).

These fundamental quantities decompose the first two moments ofthe spectrum of x+(t) across time, relating time variation tofrequency-domain structure.

References

Gabor (1946), Boashash (1992), Cohen (1995),Gabor (1946)


Geometry of modulated oscillations in two dimensions

How can you depart from purely oscillatory (sinusoidal) behavior?

Amplitude modulation, distortion, and precession.These three signals have identical average spectra (!)Also by changing frequency with fixed geometry.


Geometry of modulated oscillations in three dimensions


Wavelet Ridge Analysis

A wavelet is not a foundation. It is a lens. The foundation is theanalytic signal, and wavelets give us a way to extract an estimateof an analytic signal from a noisy time series.

Wavelet ridge analysis is a powerful, flexible, and rigorous methodfor extracting quasi-periodic signals of unknown frequency fromindividual time series or arrays of time series.

Potential applications: eddies, internal wave packets, ENSOsignals, seismic waves...

Start here

Lilly, J. M., and S. C. Olhede (2009). Wavelet ridge estimation ofjointly modulated multivariate oscillations.

Use routines wavetrans and ridgewalk from JLAB, available atwww.jmlilly.net.


www.jmlilly.net

Rotary wavelet transform

Strong frequency-shifting of inertial oscillations is revealed.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis

Separation of eddy currents from residual

We also understand how to quantify the errors involved in thisquasi-periodic signal extraction process.


Deep connection to vortex dynamics

Red = one-point estimate | Lilly, Olhede, and Early (2014), in prep.


Covariance ellipse geometry and eddy forcing

Eddy vorticity flux convergence Q is related to covariance ellipses.The are only four ways that covariance ellipses can generate eddyvorticity flux divergence. Joint work with S. Waterman.


Covariance ellipse geometry and eddy forcing

For more information...

This evening’s Poster Session 058: Mesoscale ocean processes andtheir representation in earth system models

Waterman and Lilly: Geometric ingredients of eddy-mean flowfeedbacks, and a time-varying extension


Stochastic modeling

Trajectories from a QG model. How can these be described as astochastic process? Joint work with A. Sykulski, S. Olhede,E. Danioux, and J. Early.


Stochastic modeling


See Sykulski et al. (2013a,b), submitted manuscripts, athttp://www.ucl.ac.uk/statistics/people/adamsykulski.

Friday’s Session 071: Frontiers of oceanographic data and methods

09:30 Sykulski, Lilly, Olhede, Danioux, and Early: Stochasticmodels for Lagrangian data


http://www.ucl.ac.uk/statistics/people/adamsykulski

Covariance analysis using the analytic signal

SVD analysis of vector wind + SST using the analytic signal leadsto SST amplitude and phase, wind ellipses, and wind phase.=⇒ Eastward propagation of wind signals.


Covariance analysis using the analytic signal


Elipot (2013), “On Singular Value Decomposition of analyticcovariance matrices of univariate and bivariate variables, withapplication to El Nino Southern Oscillation”, submitted.

Talk to Shane.


The methods paradox

The tools we use to look at data matter... but there is a challenge.

The methods paradox

The need for powerful and trustworthy data analysis methodsincreases as datasets become larger and more complex, but

There are intrinsic barriers that prevent the appropriatemethods from reaching those who need them.


The methods paradox

What are these barriers?

Because of the increasing size of fields, those working withdata (e.g. oceanographers) and those working on methods fortreating data (e.g. statisticians and time series analysts) arefrequently unable to find each other.

Therefore, those on the data side very frequently are forced to’reinvent the wheel’, while meanwhile,...

Those on the methods side do not have enough information todevelop the right methods for the most pressing problems.

Both sides create their own jargon, and cite primarily withintheir own literature, reinforcing the barriers.

Based on my experience in three different areas: time seriesanalysis, mapping methods, and covariance analysis.


The methods paradox

Proposed solutions to the methods paradox:

We need long-term, sustained interactions between theoceanographic community and methods communities such astime series and statistics.

We need to embrace a spirit of collaboration for time seriesmethods and other data analysis expertise—rather than only’do it yourself’ or ’black box’ approaches.

We need to prioritize the creation of accessible tutorialliterature in modern data analysis methods that is tailored tothe needs of our community.

Just as a class of specialists in numerical modeling is integralto oceanographic research, we also need to train a new classof specialists in data analysis theory and methods.

The end. Thanks!


Documents

New Directions in Oceanographic Time Series AnalysisThe main point The tools we use to look at data matter. Download this talk, and Matlab toolbox JLAB, from . Lilly, Olhede, Sykulski,