50$, anticyclonic Loop Current Eddies dominate
To do: compare with eddies in numerical models of the Gulf, detected using this method and also standard Eulerian methods.
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# Thank you!
This talk is available at
[http://www.jmlilly.net/talks.html](http://www.jmlilly.net/talks.html)
Acknowledgment
This research has been funded by the Mexican National Council for Science and Technology - Mexican Ministry of Energy - Hydrocarbon Fund, project 201441. This is a contribution of the Gulf of Mexico Research Consortium (CIGoM). We acknowledge PEMEX’s specific request to the Hydrocarbon Fund to address the environmental effects of oil spills in the Gulf of Mexico.
.footnote[ P.S. Like the way this presentation looks? Check out [{Liminal}](https://github.com/jonathanlilly/liminal).]
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## Estimating Eddy Properties from Individual Trajectories
The method to accomplish this draws from signal analysis.
An AM/FM signal sends information through the amplitude or frequency modulation of a carrier wave.
The trajectory of a particle trapped in a coherent eddy is a modulated oscillation in two dimensions—a stereo signal.
No special deployment needed, high data density not needed.
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##A Model for Lagrangian Trajectories
A model for Lagrangian trajectories $z(t)=x(t) + \mathrm{i} y(t)$ with eddies:
$$z(t)=x(t) + \mathrm{i} y(t) =z_\epsilon(t) + \boxed{z_\star(t)}.$$
Turbulent background $z\_\epsilon(t)$ (stochastic) plus $z\_\star(t)$ (oscillatory).
The oscillatory signal $z\_\star(t)$ is modeled as a *modulated ellipse*:
$$z\_\star(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos \phi(t)+\mathrm{i}b(t)\sin \phi(t)\right].$$
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##Lagrangian Estimation of Eddy Properties
This problem has five parts.
2. From $\widehat{z}\_\star(t)$, estimate $a(t)$, $b(t)$, $\theta(t)$, and $\phi(t)$ that generated it.
2. Address errors in $\widehat{z}\_\star(t)$ due to modulation, a.k.a. bias error.
1. Address “false positives” due to inertial oscillations.
3. Address “false positives” of $\widehat{z}\_\star(t)$ due to the background $z_\epsilon(t)$.
1. Estimated the eddy signal $\widehat{z}\_\star(t)$ given a noisy time series $z(t)$.
Solutions:
1. The analytic signal of .cite[Gabor (1946)], see .cite[Lilly and Olhede (2009)].
2. The internal bias estimate of .cite[Lilly and Olhede (2012)].
1. Easy; specify $\omega/f>-1/2$, as shown shortly.
3. A significance test based on a stochastic model, introduced here.
4. Multivariate wavelet ridge analysis of .cite[Lilly and Olhede (2012)].
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## Inferring Ellipse Properties
$$z\_\star(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos \phi(t)+\mathrm{i}b(t)\sin \phi(t)\right]$$
Given $z\_\star(t)$, we wish to infer $a(t)$, $b(t)$, $\theta(t)$, and $\phi(t)$.
This is a generalization of, given $x\_\star(t)=a(t)\cos \phi(t)$, infer $a(t)$ and $\phi(t)$, solved in a landmark paper by .cite[Gabor (1946)].
The solution is to form the analytic signal $x\_\star^+(t)$, by zeroing negative frequencies and doubling positive frequencies.
Then $x\_\star(t)=\Re \left\\\{x\_\star^+(t)\right\\\}$, and $\widehat{a}(t)\mathrm{e}^{\mathrm{i}\widehat{\phi}(t)}\equiv x\_\star^+(t)$ gives sensible estimates.
This inverse problem has no other sensible solution .cite[(Vakman 1996)], i.e. giving the right answer for a pure sinusoid.
Generalizing to two variables, we form $\mathbf{z}(t)=[x(t)\,\,y(t)]^T$ and take its analytic part $\mathbf{z}_+(t)$ .cite[(Lilly and Olhede, 2009)]. This defines $\widehat{a}(t)$, $\widehat{b}(t)$, $\widehat{\theta}(t)$, and $\widehat{\phi}(t)$ which are again the only sensible estimates.
When noise is present, we can't apply this method directly. The analytic signal is implicitly employed in wavelet ridge analysis.
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## Example of Wavelet Ridge Analysis
An elliptical signal is estimated by $\widehat{\mathbf{z}}(t)\equiv \mathbf{w}\left(t,\widehat{s}(t)\right)$ where $\widehat{s}(t)$ are “ridges”, maxima curves of the transform modulus $\|\mathbf{w}(t,s)\|$.
Constrained on physical grounds to end at $|w\_+(t,s)|=|w\_-(t,s)|$.
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## A Bias Estimate
The leading-order term in the bias error is .cite[(Lilly and Olhede, 2012)]
$$\frac{\|\widehat{\mathbf{z}}_+(t)-\mathbf{z}_+(t)\|}{\left\|\mathbf{z}_+(t)\right\|}
\approx \varsigma(t)\equiv \frac{1}{2} \frac{P}{\omega^2(t)} \frac{\left\|\mathbf{c}(t)\right\|}{\left\|\mathbf{z}_+(t)\right\|}$$
where $\mathbf{z}(t)=[x(t)\,\,\,y(t)]^T$, $\mathbf{z}_+(t)$ is its analytic part, and the parameter $P$ controls the wavelet time/frequency tradeoff.
The second-order departure of $\mathbf{z}_+(t)$ from a local oscillation at the frequency $\omega(t)$ is described by
$$
\mathbf{c}(t) = \frac{\mathrm{d}^2}{\mathrm{d}t^2}\,\mathbf{z}_+(t)-\mathrm{i}2\omega(t) \frac{\mathrm{d}}{\mathrm{d}t}\,\mathbf{z}_+(t)-\omega^2(t)\mathbf{z}_+(t)
$$
where the joint instantaneous frequency is the natural measure of time-varying frequency in the bivariate case .cite[(Lilly and Olhede, 2009)]
$$\omega(t)\equiv\frac{\Im\left\{\mathbf{z}_+^H(t)\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{z}_+(t)\right\}}{\|\mathbf{z}_+(t)\|^ 2}.$$
This states that error is proportional to the second-order deviation of the signal from a pure sinusoid over the time scale of the wavelet.
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## Bias Near Ridge Edges
Left is histogram of all points on edge distance $D$ vs. circularity $\zeta$.
Right is mean value of estimated bias $\varsigma$ over this plane.
Low circularities and high errors occur near edges.
Suggests trimming $D<1/2$ to remove wavelet spin-up effects.
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## Removing Inertial Oscillations
Inertial oscillations generally occur at $\omega=-f$, but may be shifted by background vorticity to $\omega=-f-\zeta/2$ (Kunze, 1985).
The strongest possible nonlinear anticyclonic eddy is $\zeta/f=-1$ (e.g. Thomas, 2007), implying $\omega/f=-1/2$ for a circular eddy.
This most nonlinear anticyclone frequency-shifts inertial oscillations from $\omega=-f$ to $\omega=-f-\zeta/2$ or $\omega/f=-1/2$.
Thus, no eddies should occur below $\omega/f=-1/2$, and no inertial oscillations should occur above $\omega/f=-1/2$.
There is no comparable stability boundary on the cyclonic side.
Thus we keep only ridges with ridge-averaged Rossby numbers $$Ro\_\star\equiv\omega\_\star/f\_\star>-1/2.$$
Next we turn to examining false positives emerging from the stochastic background.
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##Validation Using a Model of an Eddy
An eddy at 24°N tracked by 512×256=131,072 particles, by J. Early.
The ellipses represent a time-varying best fit to local oscillatory variability in two dimensions. Color is estimated enclosed vorticity.
See the [{2018 Ocean Sciences talk}](http://www.jmlilly.net/talks/lilly18-os.html) for more details.
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##Estimated vs. Actual Enclosed Vorticity
Log10 histogram of estimated vs. true enclosed vorticity.