class: center, middle
.title[Eddy Detection from Surface Drifters, with Application to the Gulf of Mexico] .author[Jonathan Lilly
1
] .coauthor[Jeffrey Early
2
, Sofia Olhede
3
, Paula Perez-Brunius
4
] .institution[
1
Planetary Science Institute,
2
NorthWest Research Associates,
3
École Polytechnique Fédérale de Lausanne (EPFL),
4
Centro de Investigacion Cientifica y de Educacion Superior de Ensenada (CICESE)] .date[January 27, 2023] .center[[{www.jmlilly.net}](http://www.jmlilly.net)]
.note[Created with [{Liminal}](https://github.com/jonathanlilly/liminal) using [{Remark.js}](http://remarkjs.com/) + [{Markdown}](https://github.com/adam-p/markdown-here/wiki/Markdown-Cheatsheet) + [{KaTeX}](https://katex.org)] --- class: center ##Eddies, Waves, and Turbulence
Sea surface height slope magnitude from a global simulation by H. Simmons, University of Washington. --- class: center ## The Global Surface Drifter Dataset
Note: non-uniform sampling distribution, temporal variation, superposition of spatial scales --- class: center ## Mean Surface Current Speed
Formed by binning in latitude and longitude, then averaging. Easy to compute maps of low-order statistics: mean, variance, etc. Clearly does not capture full richness of dataset. What else can be done? --- class: center ##Lagrangian Analysis of a Nonlinear Eddy
An eddy at 24°N tracked by 512×256=131,072 particles, by J. Early. Color is estimated enclosed vorticity in inferred ellipses. --- class: center ##Estimated vs. Actual Enclosed Vorticity
Log10 histogram of estimated vs. true enclosed vorticity. --- class: left ##Elements of Lagrangian Vortex Extraction This analyis method consists of four steps: 1. *Extracting* oscillatory velocity features using a technique called *wavelet ridge analysis*. .cite[Lilly, Scott, and Olhede (2011)] 2. *Thresholding* with a suitable error quantity to remove false positives. .cite[Lilly and Olhede (2012a)] 3. *Linking* ellipse properties to spatially-integrated properties using an extended version of Stokes' theorem. .cite[Lilly (2018)] 3. *Testing* statistical confidence through comparison with a null hypothesis. .cite[Lilly and Pérez-Brunius (2021b)] --- class: left ## A Model for a Trajectory with Eddies We conceptualize the motion of a particle trapped in an eddy as a
time-varying ellipse.
\[z_o(t)=x_o(t)+\mathrm{i}y_o(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos\phi(t) + \mathrm{i} b(t)\sin \phi(t)\right]\]
We construct a type of “filter” that can isolate such signals. This lets us approximately split a time series $z(t)=z\_o(t)+z\_\epsilon(t)$ into an oscillatory $z\_o(t)$ and stochastic $z\_\epsilon(t)$ portion. --- class: left ##Recovering Modulated Oscillations The wavelet transform is a tool for
recovering
or
estimating
the properties of modulated oscillation immersed in background noise. The idea is to project the time series onto an oscillatory test signal, or
wavelet
, to find a “best fit” frequencies at each moment. The fits are then chained together into a continuous curve called a
ridge.
For a vector-valued signal `$\mathbf{x}_o(t)=\begin{bmatrix}x_1(t) & x_2(t) & \cdots & x_N(t)\end{bmatrix}^T$` in noise `$\mathbf{x}_\epsilon(t)$`, `$\mathbf{x}(t)=\mathbf{x}_o(t)+\mathbf{x}_\epsilon(t)$`, define the wavelet transform `\[\mathbf{w}(t ,s) \equiv \int_{-\infty}^{\infty} \frac{1}{s} \psi^*\left(\frac{\tau-t}{s}\right)\,\mathbf{x}(\tau)\,\mathrm{d} \tau\]` (also a vector) and then find the
wavelet ridges
`$s(t)$` from `\[\frac{\partial}{\partial s}\, \left\|\mathbf{w}(t ,s)\right\| = 0,\quad\quad \frac{\partial^2}{\partial s^ 2}\, \left\|\mathbf{w}(t ,s)\right\| < 0.\]` The oscillation is estimated simply by `$\widehat{\mathbf{x}_o}(t)\equiv\Re\left\{\mathbf{w}(t,s(t))\right\}.$` .cite[See Delprat et al. (1992), Lilly and Olhede (2009, 2010, 2012a). ] --- class: left ## 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 $|w\_+(t,s)|=|w\_-(t,s)|$. --- class: center ## Extraction of Quasi-Oscillatory Signals
A detected oscillatory feature is called a “ridge.” $Ro\equiv \omega/f$ = Lagrangian frequency Rossby number = $\frac{1}{2}\zeta/f$ $L\_\star$ = ridge length in number of cycles completed Ridges are extracted in the band $1/64 \lt Ro \lt 2$ and with $L_\star\gt1$. Inertial oscillations occur at $Ro=-1$. Stability boundary at $Ro=-\frac{1}{2}$ or $\zeta=-f$. $Ro\_\star$ = time average of $Ro$ over an entire ridge --- class: center ### Identifying Vortices from Particle Trajectories
Vortices (loops) are identified using only particles (dots). .cite[Lilly, Scott, and Olhede (2011)] --- class: center ##Surface Drifter Data for the Gulf of Mexico
3770 trajectories, 28 years, 13 data sources, 4.5 M hourly points .cite[Lilly & Perez-Brunius (2021a)] --- class: center ## The Gulf Surface Currents
High resolution Gulf currents from drifters .cite[Lilly & Perez-Brunius (2021a)] --- class: center ## Ridge-Average Properties
Note: (i) cyclone / anticyclone asymmetry; (ii) gap near $Ro\_\star=-\frac{1}{2}$; (iii) $\approx 90$ % are inertial oscillations; (iv) possible false positives --- class: center ## A Noise Dataset & Null Hypothesis
To assess significance, a noise dataset is created. Isotropic velocity spectrum $S\_{\varepsilon\varepsilon}(\omega) \equiv \min\left\\\{\widehat S\_{++}(\omega),\widehat S\_{--}(\omega) \right\\\}$. Same variance, initial location, and mean velocity as original data. --- class: center ## A Significance Test
$\zeta\_\star$ = time-averaged circular polarization $\frac{2a|b|}{a^2+b^2}$ Color = ratio of survival or reverse cumulative density functions Gray = larger $\zeta\_\star $ or $L\_\star$ values 10$\times$ more frequent in data vs noise Black = 100$\times$ more frequent --- class: center ## Ridge Properties Before Editing
Remove inertial oscillations and less than 90% significant Trim edges to remove “spin-up” artifacts --- class: center ## Ridge Properties After Editing
Major asymmetries: (i) intense submesoscale cyclones; (ii) mesoscale cylcones; (iii) anticyclonic Loop Current Eddies --- class: center ## Case Study of a Loop Current Eddy
Evidence of three different propagation velocities: 3.5 cm/s, 5 cm/s, stalled --- class: center ## $|Ro|\gt 1/6$ Anticyclones Colored by $Ro\_\star$
--- class: center ## $|Ro|\gt 1/6$ Cyclones Colored by $Ro\_\star$
--- class: center ## $|Ro|\gt 1/6$ Cyclones Colored by $L\_\star$
--- class: center ## Origin of Strong Mesoscale Cyclones
Lagrangian ellipses combined with sea surface height from CMEMS. --- class: center ## Comparison with a Numerical Model
Source of positive vorticity is inflow through the Yucatán Channel. --- class: center ## Small Cyclone Ellipses, $R\_\star<10$ km
Focus on small cyclones, with 4x as many ellipses as before. --- class: center ## Small Cyclone Trajectories, $R\_\star<10$ km
Trajectories often cusp rather than loop. Larger-scale advection. --- class: left ##Conclusions An objective analysis of oscillatory featuress in drifter trajectories in the Gulf of Mexico reveals new features of the eddy field. An eddyless null hypothesis allows one to assess the statistical confidence that eddy ridges are not due to stochastic fluctuations. 1. Intense cyclones with $Ro\_\star>1/6$ throughout the Gulf, with no analogue on the anticyclonic side 2. Strong $10
50$, 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. --- class: center, middle # 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).] --- class: center, middle # Alongtrack Eddy Detection --- class: left ## Motivation The ridge analysis is intended for signals consisting of modulated oscillations in noise. Another type of signal that may occur is short-duration, isolated ‘bursts’ or ‘impulses’—that is, events which may be represented as a wavelet or the temporal integral of a wavelet. This suggests a model for a real, univariate signal of the form `\begin{equation}\label{signalmodel} x(t) = \sum_{n=1}^N \Re\left\{c_n \psi\left(\frac{t-t_n}{\rho_n}\right)\right\} +x_\epsilon(t) \end{equation}` consisting of superpositions of amplitude scaled, stretched, and phase-shifted versions of some basis signal `$\psi(t)$`, called the *element*. It is assumed that the different realizations of the element are sufficiently separated in time and frequency such that they do not interfere, in a way that will be made precise later. --- class: left ## Ingredients Element analysis consists of several steps: 1. Choice of element 1. Event detection 2. Rejection of statistically insignificant events 3. Rejection of non-isolated events Developed in .cite[Lilly, J. M. (2017). Element analysis: a wavelet-based method for analyzing time-localized events in noisy time series. Proceedings of the Royal Society of London, Series A, 473 (2200): 20160776, 1–28. [{10.7 Mb Pdf}](http://rspa.royalsocietypublishing.org/content/473/2200/20160776)] --- class: center ## Localized Wavelets
The method is suitable for events that themselves resemble wavelets, or the temporal integral of a wavelet (the `$\beta=0$` case). --- class: left ## Signal Model Our signal model, using the generalized Morse wavelets, is `\begin{equation}\label{morseelementmodel} x(t) = \sum_{n=1}^N \Re \left\{ c_n \psi_{\mu,\gamma}\left(\frac{t-t_n}{\rho_n}\right)\right\} +x_\epsilon(t) \end{equation}` where `$\mu$` is the most suitable value of the order or `$\beta$` parameter. The `$n$`th event is then characterized by coefficient `$c_n$`, time `$t_n$`, and scale `$\rho_n$`. We wish to *estimate* these unknown quantities. We then take the wavelet transform with an order `$\beta$` wavelet in the same `$\gamma$` family, leading to `\begin{equation}\label{transformofelementmodel} w_{\beta,\gamma}(\tau,s)=\frac{1}{2}\sum_{n=1}^N c_n\int_{-\infty}^{\infty} \frac{1}{s} \psi_{\beta,\gamma}^*\left(\frac{t-\tau}{s}\right)\psi_{\mu,\gamma}\left(\frac{t-t_n}{\rho_n}\right)\,d t+\varepsilon_{\beta,\gamma}(\tau,s). \end{equation}` Owing to the properties of the generalized Morse wavelets, the integral in the above equation has a closed-form expression in terms of a rescaled `$\psi_{\beta+\mu,\gamma}(t)$` wavelet. --- class: left ## Event Detection We then find *isolated maxima* of the transform, that is, time-scale `$(\tau,s)$` points at which `\begin{multline} \quad\quad\quad\quad\frac{\partial}{\partial \tau}\left|w_{\beta,\gamma}(\tau,s)\right| =\frac{\partial}{\partial s}\left|w_{\beta,\gamma}(\tau,s)\right|=0, \\\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\partial^2}{\partial \tau^2}\left|w_{\beta,\gamma}(\tau,s)\right|<0,\quad\frac{\partial^2}{\partial s^2}\left|w_{\beta,\gamma}(\tau,s)\right|<0.\label{maxconditions} \end{multline}` Because we have a simple expression for `$w_{\beta,\gamma}(\tau,s)$`, we can relate the event properties `$c_n$` `$t_n$`, and `$\rho_n$` to the value of `$w_{\beta,\gamma}(\tau,s)$` at a maxima. This lets us work backwards from the maxima points to estimates of the event properties. --- class: center ##An Example of Events in White Noise
The events are based on the `$\psi_{2,2}(t)$` wavelet in this case. Grey dots are all maxima, black are significant and isolated. --- class: left ##Assessing Statistical Significance This is the key step for large-scale applications. The obvious approach is to simulate many noise realizations suitable for the time series of interest, and then apply the method. This works great for one time series but not for 20,000. Fortunately, there is a workaround. It turns out that all we need to know is the statistics of a 5-vector, containing this point, the next point, the previous point, the point above, and the point below. Actually, given the assumption of power-law Gaussian noise, having a spectrum of the form `$A^2/\omega^{2\alpha}$`, the covariance statistics of this 5-vector are known analytically. This means that all one needs to do is simulation realizations of this 5-vector in order to determine the probability that a point at a given scale will be a maximum of a certain magnitude, thus establishing statistical significance. --- class: center ##Event Distributions for the Example
Color = 5-vector realizations, gray dots = all detected events Gray squares = values of true events Black circles = estimated values of significant events. Lines are significance curves for event detection rates. --- class: left ##Assessing Isolation This is done through the identification of *regions of influence* around isolated maxima, related to but distinct from concentration regions. Analytic expressions for these regions of influence may be found. An event is rejected if it is within the region of influence (subject to a threshold) of another, larger-amplitude event. This turns out to be mostly important for rejecting artifacts due to closely-spaced events. --- class: center ## A Real-World Application
Measurement tracks for a sea surface height satellite. One of the main windows into the ocean circulation. --- class: center ## A Real-World Application
From an short altimeter track segment crossing the Labrador Sea. --- class: left # Conclusions We discussed three techniques for studying coherent vortices and their impacts on Earth: 1. Wavelet ridge analysis for studying oscillatory features 2. Stochastic modeling 3. Element analysis for the detection of localized events These methods offer promising avenues for research into large-scale vortex and turbulence phenomenon on Earth, and possibly other planets as well! --- class: center, middle # Thank you!