Satellites and Basketballs

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.title[Observing Earth’s ocean currents with satellites and basketballs]
 
.author[Jonathan Lilly]
.institution[Planetary Science Institute, Tucson, Arizona]
 
.date[January 26, 2024]
 
.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)]

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##Multi-Scale Variability in Earth’s Oceans

Sea surface height slope magnitude from a global simulation by H. Simmons, University of Alaska Fairbanks.

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## High-Tech Observations of the Oceans

The Jason-class satellite altimeter measures the height of the ocean surface with an accuracy of a few mm (!!), producing global maps every 9.92 days for the past 32 years (!!).

These measurements give key insight into the ocean currents.

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## The Along-Track Dataset

Working with the original along-track data, you can leverage the `$\sim\!5$` km along-track spacing vs. `$\sim\!100$` km smoothing scale.

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## Low-Tech Observations of the Ocean

Surface drifter = basketball + sock + GPS.

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## Surface Drifter Basics
 
<img style="width:50%;" src="../figures/drifterschematic.gif">
 
Position communicated by satellite roughly every hour.

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## The Global Surface Drifter Dataset

Note: non-uniform sampling distribution, temporal variation,
        superposition of spatial scales

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## 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?

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## Coherent Eddies, A Feature of Interest

To make an eddy, stir your coffee  and multiply by about 10 million.

Very important features for the ocean circulation!

Here I discuss two new approaches to observing eddies that rely on techniques from signal analysis and statistics.

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##The GOAT of all Eddies

[{Voyager 1 time lapse by Björn Jónsson and Ian Regan}](https://lightsinthedark.com/2010/11/13/sweet-sixteen-jupiter-in-motion)

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## Eddy-Centered Analysis

Given an eddy center from a mapped product, improve eddy size, shape, and transport estimates with along-track composites.

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## Along-Track Event Detection

Alternatively, detect anomalies directly within alongtrack data.

Requires (i) a new detector and (ii) careful consideration of the 1D nature of the measurements to accurately infer eddy sizes, strengths, and populations.

This example, from the Labrador Sea, shows a long-lived anticyclonic anomaly is estimated to be twice and large and half as strong from the mapped product vs. the along-track data.

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## Basic Idea

The type of signal we expect for eddies observed by along-track data consists of short-duration, isolated ‘bursts’ or  ‘impulses’—that is, events which may be represented as a wavelet or the temporal integral of a wavelet.

This type of signal may also be appropriate for other data as well.

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.

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## Element Analysis

Element analysis is a general method for identifying isolated signal anomalies of the proposed form. It  consists of several steps:

1.  Choice of element
1.  Event detection
2.  Rejection of statistically insignificant events
3.  Rejection of non-isolated events

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)
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## Localized Wavelets

The method is suitable for events that themselves resemble wavelets, or the temporal integral of a wavelet (the `$\beta=0$` case).

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## 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.

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## 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.

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##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.

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##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.

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## Application to Along-Track Data

We're going to look at repeated observations from a single ground-track in the Labrador Sea.

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## Detection of Isolated Anomalies

A few dozen coefficients (center) capture all of the structure in this data; the residual (right) appears to be random noise.

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## The Global Surface Drifter Dataset

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###  Identifying Vortices from Particle Trajectories

Vortices (loops) are identified using only particles (dots).    
.cite[Lilly, Scott, and Olhede (2011)]

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##A Global Census

Identifying coherent eddies within the surface drifter dataset.

Here are the anticyclones (rotating oppositely from the direction of the Earth's rotation).

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##A Global Census

And here are the cyclones.

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##Elements of Lagrangian Vortex Extraction

This analysis 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)]

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## A Model for a Trajectory with Eddies

We conceptualize the motion of a particle trapped in an eddy as a time-varying ellipse.

<div>\[z_o(t)=x_o(t)+\mathrm{i}y_o(t)=\mathrm{e}^{\mathrm{i}\theta(t)}\left[a(t)\cos\phi(t) + \mathrm{i} b(t)\sin \phi(t)\right]\]</div>

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.

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##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). ]

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##Example of Vortex Extraction
<img style="width:100%" src="../figures/bandwidth_example.png">

The ridge is the curve made by tracing out the maximum modulus of the wavelet transform. 
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##Example of Vortex Extraction
<img style="width:100%" src="../figures/bandwidth_looper.png">
 
The transform value along this curve is an estimate of the oscillatory signal component, visualized as ellipses.
 
Ridge analysis separates modulated elliptical signal from low-frequency meandering and higher-frequency variability.

Physically the ellipses quantify the properties of oceanic vortices.

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##Extraction of Oscillatory Signals
 
<img style="width:100%;margin-top:-0.2em;margin-bottom:-0.2em" src="../figures/nonlineareddyallridges.png">

All trajectory segments that contain a quasi-oscillatory signal.

`\[z(t)=x(t) + \mathrm{i} y(t) =z_\epsilon(t) + \boxed{z_o(t)}\]` 
        
`$z_\epsilon(t) =$` background, `$z_o(t) =$` oscillatory (e.g. eddy)
        
At each moment, the oscillatory signal is represented by an ellipse.  
This ellipse is *assigned to* the particle by the analysis method.  
It is a material ring that the particle is inferred to belong to.

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##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.
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##Estimated vs. Actual Enclosed Vorticity

Log10 histogram of estimated vs. true enclosed vorticity.

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## The Gulf of Mexico Surface Currents

High resolution Gulf currents from drifters   
.cite[Lilly & Perez-Brunius (2021a)]

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##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)]

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## Case Study of a Loop Current Eddy

Evidence of three different propagation velocities: 
3.5 cm/s, 5 cm/s, stalled
 
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## $|Ro|\gt 1/6$ Anticyclones Colored by $Ro\_\star$

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##  $|Ro|\gt 1/6$ Cyclones Colored by $Ro\_\star$

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##  $|Ro|\gt 1/6$ Cyclones Colored by $L\_\star$

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## Comparison with a Numerical Model

Source of positive vorticity is inflow through the Yucatán Channel.

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##Conclusions

1.  Earth is a planet!
2.  Advances in data analysis, combining time series methodology and statistics, are helping us shed new light on coherent eddies, a key aspect of the ocean circulation.
3.  Maybe these methods have potential for other planets too.

Thanks!