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A Lagrangian Method for Extracting Eddy Boundaries in the Red Sea and the Gulf of Aden

TL;DR: This work visualize the eddies as tubes in space-time to enable the analysis of their movement and deformation over several weeks and proposes the following model: Inside an eddy, particles rotate around a common core and thereby remain at a constant distance under a certain parametrization.
Abstract: Mesoscale ocean eddies play a major role for both the intermixing of water and the transport of biological mass. This makes the identification and tracking of their shape, location and deformation over time highly important for a number of applications. While eddies maintain a roughly circular shape in the free ocean, the narrow basins of the Red Sea and Gulf of Aden lead to the formation of irregular eddy shapes that existing methods struggle to identify. We propose the following model: Inside an eddy, particles rotate around a common core and thereby remain at a constant distance under a certain parametrization. The transition to the more unpredictable flow on the outside can thus be identified as the eddy boundary. We apply this algorithm on a realistic simulation of the Red Sea circulation, where we are able to identify the shape of irregular eddies robustly and more coherently than previous methods. We visualize the eddies as tubes in space-time to enable the analysis of their movement and deformation over several weeks.

Summary (3 min read)

1 INTRODUCTION

  • Ocean eddies are bowl-shaped bodies of water that rotate around a vertical axis.
  • Most notably, they transport waters of a certain properties, like temperature and trace concentrations.
  • This causes the shape to deviate considerably from a circle and restricts the lifetime of Red Sea eddies to about two weeks [28].
  • To overcome these obstacles in this specific application, the authors present an algorithm for a reliable extraction and tracking.
  • This provides a clear boundary between particles that rotate jointly around a core line and those that diverge.

1.1 Eddy Properties

  • At first glance, most eddies are easily made visible by the sea level anomaly (SLA), i.e., the variation of the water surface height from the mean measured over several decades.
  • Depending on their rotation, eddies will appear as domes or bowls.
  • Another approach is to quantify the transport of mass directly, since an eddy is supposed to contain a relatively stable mass with little exchange across its boundary.
  • This also means that water inside an eddy remains in circular rotation.
  • Finally, eddies feature high vorticity, which quantifies rotating movement, and are usually surrounded by a ring of high shear where interacting with their surroundings.

1.2 Red Sea and Gulf of Aden

  • The Red Sea is a long and narrow basin that only connects to the open ocean through the shallow strait of Bab el Mandeb in the south.
  • Directly behind this strait, water intermixes within the very narrow Gulf of Aden, which is part of the Arabian Sea and thus the Indian Ocean.
  • Due to this shape, most eddies cover half or more of its width, providing rapid transport of organisms and nutrients between the coasts [25, 28].
  • Understanding the statistical properties and variability of eddies in the Red Sea will not only serve to improve local ocean forecasts but would also lead to greater understanding of the important physicalbiological and air-sea interactions in its basin [18].
  • The first step is a stable extraction of the eddies’ area and their evolution over time.

1.3 Dataset

  • The authors analyze the output of a general circulation model that has been recently implemented and validated for the Red Sea and Gulf of Aden [26, 27].
  • The authors have run the simulation on (0.04◦)2 sized grid cells and a temporal resolution of 6 hours.
  • Different initial conditions generated 10 similar instances.
  • With the SLA field, the authors identify the locations of three eddies.
  • Other extrema are not regarded as their lifetime is shorter than one week.

Local Eddy Detection

  • The simplest approach to identify eddies is by thresholding the sea level anomaly (SLA).
  • A framework to analyze the uncertain SLA field of the Red Sea has been implemented by Höllt et al. [11, 12].
  • Regions fulfilling fOW <=−0.2×σOW are classified as eddies, with σOW the standard deviation of fOW .
  • Note that the computation then is no longer local.
  • While the Okubo-Weiss criterion is abundantly used, it can be shown that it is not apt to define vortex area and should thus be apploed with care [2, 4].

Eulerian Topology-Based Eddy Detection

  • A number of algorithms have been proposed to detect such eddies and trace them across time slices [22, 24].
  • When no closed stream lines are present, nearly closed stream lines are extracted instead.
  • Here, the winding angle method from Saderjon and Post [20] is frequently used.
  • In case of eddies touching boundaries, as is often the case in the Red Sea, the criterion can be relaxed to accept smaller overall angles [28].
  • Another feature of 2D steady topology, Eulerian bifurcations, have been regarded to detect eddy formation [8].

Lagrangian Topology-Based Eddy Detection

  • Eddies trap mass, allowing no or little intermixing with their surroundings.
  • They can thus be treated as topological regions even in unsteady flows, with their boundaries describing Lagrangian coherent structures (LCS).
  • In FTLE fields, hyperbolic LCSs appear as ridge lines under forward and backward integration.
  • To their knowledge FTLE has not been utilized to extract the exact boundaries of eddies, but rather to investigate their splitting from other structures or their transportation.
  • This approach finds elliptical structures that deform uniformly over time.

Transport-Based Eddy Detection

  • One final set of algorithms are transport-based methods.
  • These regions, called finite-time coherent sets, have been computed to extract extremely long-lived eddies [6].

3 ANGULAR DISTANCE METRIC

  • Eddies in the Red Sea, the Gulf of Aden and other constricted seas show some special behavior.
  • Their coast is an important part of the eddy creation and often part of their boundary.
  • On the other hand, the water body loses kinetic energy when interacting with the basin walls and other shearing structures induced by them.
  • Of the aforementioned methods, many do not apply well to this setting.
  • Local methods do not utilize the context of the flow field, and thus rely on the Jacobian derivative, decomposed into shear and rotation components.

Motivation

  • The authors base their method for eddy shape extraction on a simple observation: Inside an eddy, path lines starting close together will move very similarly, revolving around a common core.
  • They can not deviate far from each other, as they are constricted to the inside of the eddy boundary.
  • To this end, the authors utilize an FTLE-like approach, making a few alterations to the original algorithm: Path lines are re-parametrized to polar coordinates, i.e. angle and radius.
  • This excludes shear strain induced by particles traveling very close geometrically, but at differing angular velocities.
  • The authors sum up the distances along the path lines instead of analyzing the end positions only.

Our Approach

  • Under a rotating flow, path lines seeded close together will only keep their distance constant if the flow exhibits a constant angular velocity.
  • This introduces a high error value contrary to the overall similar path.
  • (2) α steers the maximal integration time.

4 EDDY BOUNDARY DETECTION

  • The resulting field, shown for example in Figure 2d, has a clear area of low error inside the eddy and a sharp edge to high error values outside.
  • Due to the very clear edge, the result is stable under a wide range of contour parameterizations.
  • The area of the eddy is by far best defined in the Ec field, while E, Ec and the FTLE ridges show very similar structures.
  • This includes limitations like the reliance on long integration times and the high computational cost, while retaining advantages like Galilean invariance.
  • All together, their algorithm executes as follows: Identify the Core Lines:.

Boundaries

  • The authors executed their algorithm on two simulations of the dataset described in subsection 1.3.
  • The results are shown in Figure 5: In the Gulf of Aden, the authors can observe a very regular behavior, indicated by straight tubes dispatching from the coast line.
  • Around the stable eddy, spiraling patterns occur.
  • To analyze the pertinence of the extracted boundaries, the authors compare their results to those of the Okubo-Weiss criterion, the standard in the geophysics community.
  • In the last row, the boundaries from both methods are overlaid.

Coherence

  • To quantify this property, the authors measure the time each particle spends within the eddy boundary.
  • This quantity is summarized in a single coherence value: the average time a particle from within the eddy remains in it.
  • The authors compute the coherence for their method, FTLE and Okubo-Weiss.
  • While their method and the Okubo-Weiss criterion output an area directly, a comparison to the FTLE output is difficult.
  • This confirms that the method is not suitable for their application.

Summary

  • Eddies in the Red Sea are difficult to extract for a number of reasons.
  • It is based on the radial distance between path lines around a common core.
  • To visually encode the extracted eddies, the authors connected their boundaries to tubes over time in a compact visualization, depicting their deformation, movement and lifetime.
  • The domain expert co-authoring this paper has confirmed these findings and the overall validity of the result.
  • Especially unexpected was the boundary movement of the eastern eddy in the Gulf of Aden:.

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Content maybe subject to copyright    Report

A Lagrangian Method for Extracting Eddy
Boundaries in the Red Sea and the Gulf of Aden
Item Type Conference Paper
Authors Friederici, Anke; Toye, Habib; Hoteit, Ibrahim; Weinkauf, Tino;
Theisel, Holger; Hadwiger, Markus
Citation Friederici, A., Mahamadou Kele, H. T., Hoteit, I., Weinkauf, T.,
Theisel, H., & Hadwiger, M. (2018). A Lagrangian Method for
Extracting Eddy Boundaries in the Red Sea and the Gulf of
Aden. 2018 IEEE Scientific Visualization Conference (SciVis).
doi:10.1109/scivis.2018.8823600
Eprint version Post-print
DOI 10.1109/scivis.2018.8823600
Publisher Institute of Electrical and Electronics Engineers (IEEE)
Rights © 2019 IEEE. Personal use of this material is permitted.
Permission from IEEE must be obtained for all other uses, in any
current or future media, including reprinting/republishing this
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Download date 09/08/2022 22:04:15
Link to Item http://hdl.handle.net/10754/656836

A Lagrangian Method for Extracting Eddy Boundaries
in the Red Sea and the Gulf of Aden
Anke Friederici
Habib Toye Mahamadou Kele
Ibrahim Hoteit
Tino Weinkauf
Holger Theisel
Markus Hadwiger
ABSTRACT
Mesoscale ocean eddies play a major role for both the intermixing of
water and the transport of biological mass. This makes the identifica-
tion and tracking of their shape, location and deformation over time
highly important for a number of applications. While eddies maintain a
roughly circular shape in the free ocean, the narrow basins of the Red
Sea and Gulf of Aden lead to the formation of irregular eddy shapes
that existing methods struggle to identify. We propose the following
model: Inside an eddy, particles rotate around a common core and
thereby remain at a constant distance under a certain parametrization.
The transition to the more unpredictable flow on the outside can thus be
identified as the eddy boundary. We apply this algorithm on a realistic
simulation of the Red Sea circulation, where we are able to identify the
shape of irregular eddies robustly and more coherently than previous
methods. We visualize the eddies as tubes in space-time to enable the
analysis of their movement and deformation over several weeks.
Index Terms:
Flow visualization—Mesoscale eddies—Red Sea——
Gulf of Aden—Vortex area—Lagrangian Boundaries
1 INTRODUCTIO N
Ocean eddies are bowl-shaped bodies of water that rotate around a
vertical axis. They contribute to the energy conversions between large-
scale circulation and mesoscale dynamics, and regulate the bio-geo-
chemical ocean’s global cycle [21]. Most notably, they transport waters
of a certain properties, like temperature and trace concentrations.
To support an analysis of eddy properties, a robust detection of their
area is required. In the open ocean, their horizontal shape is circular, and
the structure may remain stable for months. Several algorithms exist to
identify and track them. The standard in the oceanography community
is the local Okubo-Weiss criterion [16, 23]. Another good indicator of
eddy boundaries are Lagrangian Coherent Structures (LCS).
Eddies in the Red Sea region differ from their long-lived counterparts.
The narrowness of the Red Sea exposes them to high strain. This causes
the shape to deviate considerably from a circle and restricts the lifetime
of Red Sea eddies to about two weeks [28]. These characteristics pose
significant challenges to the above mentioned extraction algorithms.
To overcome these obstacles in this specific application, we present
an algorithm for a reliable extraction and tracking. Given a vortex
core line in the center of a Red Sea eddy, we analyze the behavior
of particles circulating around it. Our goal is to distinguish between
particles remaining near the vortex core over longer integration times,
and particles drifting away from it. To this end, we re-parameterize the
corresponding path lines based on their angular position, which allows
for a meaningful comparison of lines with different angular speeds.
Then, we integrate over the distance they build up to their neighbors
KTH Royal Institute of Technology: {ankef, weinkauf}@kth.se
King Abdullah University of Science and Technology (KAUST):
{habib.toyemahamadoukele, ibrahim.hoteit, markus.hadwiger}@kaust.edu.sa
University of Magdeburg: theisel@ovgu.de
over time. This provides a clear boundary between particles that rotate
jointly around a core line and those that diverge. We apply our method
on two velocity time series generated by a simulation of the Red Sea.
-20cm
+20cm
Sea Level Anomaly
Red Sea
Bab el Mandeb
Gulf of Aden
AFRICA
ARABIA
0.2δ
-0.12
0.12
Shear Stress
Red Sea Eddy
Stable Eddy
Gulf Eddy
Figure 1: The Red Sea dataset. Geographical structures are annotated.
a) Velocity field presented by a LIC, overlaid by color coded SLA. The
three eddies appear as extremal areas. Several other vortical areas with
slight surface level deviations disappear within less than a week.
b) Shear field. Areas of high shear overlap with eddy SLA regions.
1.1 Eddy Properties
The presence of an eddy can be observed in several ocean properties. At
first glance, most eddies are easily made visible by the sea level anomaly
(SLA), i.e., the variation of the water surface height from the mean
measured over several decades. Depending on their rotation, eddies will
appear as domes or bowls. As eddies transport water, the distribution
of temperature, salinity and other tracers can be an indication of eddies,
too. Another approach is to quantify the transport of mass directly,
since an eddy is supposed to contain a relatively stable mass with little
exchange across its boundary. This also means that water inside an
eddy remains in circular rotation. Finally, eddies feature high vorticity,
which quantifies rotating movement, and are usually surrounded by a
ring of high shear where interacting with their surroundings.
1.2 Red Sea and Gulf of Aden
The Red Sea is a long and narrow basin that only connects to the open
ocean through the shallow strait of Bab el Mandeb in the south. Directly
behind this strait, water intermixes within the very narrow Gulf of Aden,
which is part of the Arabian Sea and thus the Indian Ocean. Figure 1
provides a geographical overview of the region. Due to this shape, most
eddies cover half or more of its width, providing rapid transport of
organisms and nutrients between the coasts [25,28].
Understanding the statistical properties and variability of eddies in
the Red Sea will not only serve to improve local ocean forecasts but
would also lead to greater understanding of the important physical-
biological and air-sea interactions in its basin [18]. The first step is a
stable extraction of the eddies’ area and their evolution over time.
1.3 Dataset
We analyze the output of a general circulation model that has been
recently implemented and validated for the Red Sea and Gulf of Aden
[26, 27]. The dataset is shown in Figure 1. We have run the simulation
on
(0.04
)
2
sized grid cells and a temporal resolution of 6 hours. The
output dataset contains
500 × 500
cells over 32 days, totaling
128
time
steps. Different initial conditions generated 10 similar instances.
With the SLA field, we identify the locations of three eddies. Other
extrema are not regarded as their lifetime is shorter than one week.

low high
Shear Strain / Separation
(a) Shear Strain (b) Forward FTLE (c) Euclidean path line distance (d) Core distance error E
c
Figure 2: Comparison of distance-based methods. Integration times are set to 40 time steps in each case, the core position is marked.
a) Shear strain values
S
shear
. b) Forward FTLE. The ridges mark fragments of the eddy boundary, but are difficult to extract. c) Euclidean distance
between path lines over time. d) Our method: Error of distance over angle (
E
c
). The clear cut borders are similar to the FTLE ridges but considerably
better suited for a shape extraction.
2 RELATED WORK
Local Eddy Detection
The simplest approach to identify eddies is by thresholding the sea level
anomaly (SLA). A framework to analyze the uncertain SLA field of
the Red Sea has been implemented by Höllt et al. [11, 12]. While this
method is useful to obtain a general idea of position and size, no exact
boundary can be extracted with this alone.
Treating an eddy as a general vortex allows to apply standard vortex
extractors. Early works applied a thresholding of vorticity alone [10], or
set it in relation to the strain of the field [14]. Both vorticity
, strain
S
and shear strain
s
shear
can be computed directly from the local Jacobian
J
. The most popular local method based on strain and vorticity in the
geophysics community is the
R
2
criterion, also known as the Okubo-
Weiss criterion [16,23]. Locally,
f
OW
(x, y) =
k
S
k
2
k
k
2
is evaluated.
Regions fulfilling
f
OW
<= 0.2 × σ
OW
are classified as eddies, with
σ
OW
the standard deviation of
f
OW
. This threshold is commonly found
in the meteorological community after Isern-Fontanet et al. [13]. An
extension of the Okubo-Weiss criterion to unsteady fields was proposed
by Mezie
´
c et al. [15], using the gradient of the flow map instead of
J
directly. Note that the computation then is no longer local. While the
Okubo-Weiss criterion is abundantly used, it can be shown that it is not
apt to define vortex area and should thus be apploed with care [2, 4].
Eulerian Topology-Based Eddy Detection
Within a 2D steady flow, the boundary of an eddy is expected to be
a circular stream line, closed or nearly closed [19]. A number of
algorithms have been proposed to detect such eddies and trace them
across time slices [22, 24]. The algorithm by Petz et al. [17] utilizes
lines whose tangent is not necessarily parallel to the underlying field,
but incident at a constant angle. When no closed stream lines are
present, nearly closed stream lines are extracted instead. Here, the
winding angle method from Saderjon and Post [20] is frequently used.
In case of eddies touching boundaries, as is often the case in the Red
Sea, the criterion can be relaxed to accept smaller overall angles [28].
Another feature of 2D steady topology, Eulerian bifurcations, have
been regarded to detect eddy formation [8]. They mark the formation
of a saddle point, detaching the eddy as a new topological region.
Lagrangian Topology-Based Eddy Detection
Eddies trap mass, allowing no or little intermixing with their surround-
ings. They can thus be treated as topological regions even in unsteady
flows, with their boundaries describing Lagrangian coherent structures
(LCS). Several works approach LCS eddy boundaries using the finite-
time Lyapunov exponent (FTLE) [1,3, 9]. In FTLE fields, hyperbolic
LCSs appear as ridge lines under forward and backward integration. An
example of the forward FTLE field computed around an eddy within
the Gulf of Aden is shown in Figure 2b. However, to our knowledge
FTLE has not been utilized to extract the exact boundaries of eddies,
but rather to investigate their splitting from other structures or their
transportation. The extraction of ridges as geometry is non-trivial, and
an eddy is seldom fully enclosed by FTLE structures.
Another method by Haller and Beron-Vera utilizes elliptical LCS [7].
This approach finds elliptical structures that deform uniformly over
time. Beron-Vera et al. [4] extract least stretching closed shear lines,
detecting especially long-lived eddies.
Transport-Based Eddy Detection
One final set of algorithms are transport-based methods. When inte-
grating particles within an eddy, they are expected to remain inside
a connected region, as in Figure 3. These regions, called finite-time
coherent sets, have been computed to extract extremely long-lived
eddies [6].
0
10π
Cumulative Angle Relative to Core Line
Time t
Eddy Boundary
Figure 3: Example of 19 path lines moving around an eddy core. Time
is displayed as height, while the angle around the common core is
color-coded. All particles up to a certain distance to the core stay in a
synchronous circular motion, fulfilling several turns. Further apart, paths
terminate early to the right or hit the coast to the left, ending in darker
colors. The path at the boundary is marked blue.
3 ANGULAR DISTANCE METRIC
Eddies in the Red Sea, the Gulf of Aden and other constricted seas show
some special behavior. Their coast is an important part of the eddy
creation and often part of their boundary. On the other hand, the water
body loses kinetic energy when interacting with the basin walls and
other shearing structures induced by them. This results in smaller sizes,
very irregular boundary shapes, and thus overall less stable structure.
Of the aforementioned methods, many do not apply well to this
setting. Local methods do not utilize the context of the flow field,
and thus rely on the Jacobian derivative, decomposed into shear and
rotation components. In the Red Sea, shear and vorticity behave more
erratically. The shear field of our data is illustrated in Figure 1.

Motivation
We base our method for eddy shape extraction on a simple observation:
Inside an eddy, path lines starting close together will move very simi-
larly, revolving around a common core. They can not deviate far from
each other, as they are constricted to the inside of the eddy boundary.
Outside, anything may happen in the flow. They may travel around
the eddy for a certain time, but will eventually move away. An ex-
ample of this is shown in Figure 3. Moving outwards from an eddy
center point, we seed a small number of representative path lines. It
is noteworthy that all particles staying in circular motion for the first
full turn continue to do so. Thus, they are inside the eddy, all other
lines are classified as outside. The eddy boundary can be defined as a
point where a relatively low distance of path lines abruptly changes to a
higher deviation. Another important factor visible in the example is the
different speed of particles. The outmost path line within the eddy is far
more elongated in time than the inner ones. It is therefor not sufficient
to compare the absolute position of particles to each other. Instead, we
will work in polar coordinates relative to the core line.
Several examples of distance based metrics are shown in Figure 2
on our dataset. The shear strain in Figure 2a captures distances for
infinitely small integration times. The Langrangian version, the forward
FTLE field, is shown in Figure 2b.
We want to use distance to find a point where a relatively stable
distance between path lines changes into a more chaotic, generally
expanding distance. To this end, we utilize an FTLE-like approach,
making a few alterations to the original algorithm:
Path lines are re-parametrized to polar coordinates, i.e. angle and
radius. This excludes shear strain induced by particles traveling
very close geometrically, but at differing angular velocities.
We sum up the distances along the path lines instead of analyzing
the end positions only. This gives the advantage of including high
variations that occurred along their way. In free flow, separation
is asymptotically exponential, compared to an almost constant
distance within the eddy. Thus, the error measure shows a rather
extreme jump across the eddy boundary.
The deviation is measured in radial direction only, instead of
looking at the maximal spread.
Time t
(a) Two path lines over τ
Angle ϕ
(b) Two path lines over ϕ
Figure 4: Two path lines in different flows around a core line (black).
a) Translation over time and a change in angular velocity strongly affect
the euclidean distance between particles. b) Mapping the lines to angle
in relation to the core line and subtracting the translation renders the
particle distance constant again.
Our Approach
Given a 2D time-dependent velocity field
v(x,t)
, we consider path lines
within the flow field φ(x,t, τ) obeying the differential equation
φ(x,t, τ)
τ
= v(φ(x,t, τ),t + τ) and φ(x, t, 0) = x. (1)
We can compare the deviation
d
between two path lines from seeds
x
1
and
x
2
at time
t +τ
by simply computing their Euclidean distance.
We call the average of the distance along the line, seen in Figure 2c,
path line distance:
E(x
1
, x
2
,t, τ) =
1
k
τ
k
R
s
0
k
φ(x
2
,t, s) φ(x
1
,t, τ)
k
ds.
In order to minimize the impact of shear flow, we apply an angle-
based re-parametrization. Under a rotating flow, path lines seeded
close together will only keep their distance constant if the flow exhibits
a constant angular velocity. For a realistic radial profile of velocity
strength, the lines will diverge (see Figure 4a, 3). This introduces a
high error value contrary to the overall similar path.
We re-parametrize a point on a path line to an angle
ϕ
c
and radius
r
c
around a moving center point
c(t)
as in Figure 4b. Segments that move
backwards are cut off, allowing us to define the inverse function of
ϕ
c
as
τ
c
(x,t, ϕ
c
(x,t, τ)) = τ
. In this reference frame, the error function is
e
c
(x
1
, x
2
,t, α) =
1
α
Z
α
0
k
r
c
(x
1
,t, τ
c
(x
1
,t, β )) r
c
(x
2
,t, τ
c
(x
2
,t, β ))
k
dβ .
(2)
α
steers the maximal integration time. As we can see for example in
Figure 4a, particles commonly diverge from their circular motion after
no more than one full turn. Vice versa, if a path line keeps a low error
distance for at least one turn, it is very likely to keep doing so. This
means that an α > 2π is a sufficient integration bound.
4 EDDY BOUNDARY DETECTION
We define our error measure
E
c
(x,t, β ) =
1
ε
e
c
(x, x + ε
x c(t)
k
x c(t)
k
, t, β) (3)
i.e., comparing particles a small
ε
apart from each other. The resulting
field, shown for example in Figure 2d, has a clear area of low error
inside the eddy and a sharp edge to high error values outside. This
edge marks the boundary clearly, which is an important advantage over
FTLE: Ridge extraction is an unsolved problem, which becomes even
harder when seeking a closed boundary for area computations.
To extract a closed boundary, we utilize an active contour algorithm
with balloon forces [5]. This basically means inflating a small circle
around the core that tends to stop at edges, while balancing this with
a maximal curvature stress. Due to the very clear edge, the result is
stable under a wide range of contour parameterizations.
In Figure 2, we computed FTLE, path line distance
E
and radial error
E
c
around one eddy in the Gulf of Aden, all limited to the maximum
integration length of one third of the data set. The area of the eddy is
by far best defined in the
E
c
field, while
E
,
E
c
and the FTLE ridges
show very similar structures. All measures would profit from longer
integration times, which is unfortunately not feasible in this case. In
general, our method shares most properties of FTLE. This includes
limitations like the reliance on long integration times and the high
computational cost, while retaining advantages like Galilean invariance.
All together, our algorithm executes as follows:
Identify the Core Lines:
For each eddy, a core is traced. Several
core extractors were tested, but the result was barely influenced as long
as the line remains well inside the eddy (see supplemental material).
For simplicity, we use a stream line core for all further computations.
Integrate Path Lines around the Core:
We integrate
n
path lines
on a number of outgoing radii and evaluate the error between consecu-
tive seed points, resulting in n 1 values of E
c
.
Compute Error Measure:
In relation to the selected core, the path
lines are mapped to angle and radius. Up to a maximum angle
α
,
differences are integrated to obtain E
c
.
Threshold Error:
The boundaries of the eddy lay on the ring where
the error rapidly increases. We have found the respective edges to
be very sharp: While inner values commonly lay between 0 and 1,
meaning no or negative separation over time, the edges often reach
values above 10. Also, gradients are especially high there, meaning the
threshold value can safely be chosen anywhere between 3 and 8.
Extract Boundary:
We grow an active contour from the center. It
terminates at the edges in the thresholded error field and bridges gaps.
Connect Results of all Time Slices:
The process is executed on
each time slice, generating a set of closed curves. By connecting them,
we gain a tubular structure in space time. We again display time in
height, making the overall movement and transformation of the eddies
visible, see Figure 5. The distance to the respective core line is encoded
into the color of the tube, allowing to more easily identify structures.

5 RESULTS
Boundaries
We executed our algorithm on two simulations of the dataset described
in subsection 1.3. The results are shown in Figure 5: In the Gulf of
Aden, we can observe a very regular behavior, indicated by straight
tubes dispatching from the coast line. Around the stable eddy, spiraling
patterns occur. due to small cusps that slowly rotate over time.
In the simulations, we observed zero to two eddies within the Red
Sea. Other vortical structures appear, but are too short-lived to be
considered as eddies. In the first dataset displayed, a single eddy
transforms from an elongated shape to a mostly circular one. It decays
after 10 days, when highly separating ridge-like structures form in the
eddy area as the structure diffuses. With neither method were we able
to define a boundary from then on.
low
high
Distance to Core Line
Time t
(a) Boundary tubes in simula-
tion 0
Time t
(b) Boundary tubes in simula-
tion 3
Figure 5: Boundaries for all time steps, connected to tubes in time. The
distance to the core line is color-coded on the surface. Grey lines indicate
the positions in time where a boundary was computed.
To analyze the pertinence of the extracted boundaries, we compare
our results to those of the Okubo-Weiss criterion, the standard in the
geophysics community. The results are shown in Figure 6.
In the
E
c
field, values above 3 were cut for the boundary extraction.
The Okubo-Weiss field is thresholded at 0.2 the standard deviation.
In the last row, the boundaries from both methods are overlaid. For
the stable eddy at the left, the areas mostly agree. Over time, the
extracted boundaries are deforming only slowly, showing more rotation
than deformation in the Gulf of Aden, as can be observed in Figure 5
and the accompanying video.
0
3
Ours, E
c
0.2δ
Okubo-Weiss
Both
t = 0 t = 20 t = 40 t = 60
Figure 6: Boundary lines of eddies in the Gulf of Aden
First row: Error graph E
c
and extracted boundaries at 4 points in time.
Second row: Okubo-Weiss field with respective boundaries.
Third row: Both results overlaid, E
c
in black, Okubo-Weiss in blue.
0% 100%
Time Inside Boundary
(a) Our algorithm
Coherence 80.65%
(b) Forward FTLE
Coherence 74.03%
(c) Okubo-Weiss
Coherence 23.26%
Figure 7: Coherence of the eddy tracks. The initial boundary is shown in
black. We display the time that each path line leaves the boundary.
Coherence
For an estimation of the boundary quality, we compare the coherence
of the extracted areas, as a defining quality of eddies is their transport
of a coherent water mass over a long time. To quantify this property,
we measure the time each particle spends within the eddy boundary.
In a completely stable case, each particle within the eddy will remain
inside indefinitely. This quantity is summarized in a single coherence
value: the average time a particle from within the eddy remains in it.
We compute the coherence for our method, FTLE and Okubo-Weiss.
While our method and the Okubo-Weiss criterion output an area directly,
a comparison to the FTLE output is difficult. We generated the FTLE
area by adapting our boundaries to coincide with the ridges, since the
difference between both tracks is rather small.
The time that each particle remains within the boundary is displayed
in Figure 7, with the overall average noted below. As expected, FTLE
and our method perform very similarly. In contrast, the coherence
achieved by the local Okubo-Weiss criterion is rather low, as expected.
This confirms that the method is not suitable for our application.
6 CONCLUSION
Summary
Eddies in the Red Sea are difficult to extract for a number of reasons.
We have developed a new Lagrangian method to determine the bound-
ary of eddies in this environment and applied it successfully. It is based
on the radial distance between path lines around a common core.
To visually encode the extracted eddies, we connected their bound-
aries to tubes over time in a compact visualization, depicting their
deformation, movement and lifetime. Applying it on two simulation
instances of the Red Sea, we have proven that we find sensible ar-
eas with both regular and highly irregular shapes. The domain expert
co-authoring this paper has confirmed these findings and the overall va-
lidity of the result. Especially unexpected was the boundary movement
of the eastern eddy in the Gulf of Aden: While it seems to separate
from the coast early on, we have found its boundary to lay at the coast
for 18 days before loosing the connection.
Within our application, we were able to extract eddy tracks that are
considerably more coherent than those computed using the Okubo-
Weiss measure. While the resulting boundaries are very close to the
FTLE ridges, our algorithm allows for an easy and automatic shape
extraction that FTLE can not provide.
Future Work
While several works investigate eddy boundaries, their vertical shape
and mass is an open but interesting question.
By defining the boundary through future particle behavior, we see
structures from the eddy dispersal while the eddy is still coherent. To
mitigate this, one could consider integrating particles not only forward
in time, but also backward, since the mass remains relatively stable.
Even though our method targets these irregular eddies, it should work
well on datasets of more stable eddies and can then be further compared
to previous methods. Vice versa, other bays and straits such as the Gulf
of Mexico are known to frequently form eddies from coastal shearing.
We thus expect our algorithm to perform well in these regions.
ACKNOWLEDGMENTS
This work was supported in part by funding from King Abdullah Uni-
versity of Science and Technology (KAUST).

Citations
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Abstract: This study was supported by the Office of Sponsored Research (OSR) at King Abdullah University of Science and Technology (KAUST) under the “Virtual Red Sea Initiative” (award #REP/1/3268-01-01) We want to acknowledge Dr Hari Prasad Dasari for his help and contribution in organizing the interviews with domain experts We also like to thank KAUST Visualization Core Lab for their help and support

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Proceedings ArticleDOI
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TL;DR: Geospatial analysis is crucial for addressing many of the world's most pressing challenges and there is immense value in improving and expanding the visualization techniques used to communicate geospatial data as discussed by the authors.
Abstract: Geospatial analysis is crucial for addressing many of the world’s most pressing challenges Given this, there is immense value in improving and expanding the visualization techniques used to communicate geospatial data In this work, we explore this important intersection – between geospatial analytics and visualization – by examining a set of recent IEEE VIS Conference papers (a selection from 2017-2019) to assess the inclusion of geospatial data and geospatial analyses within these papers After removing the papers with no geospatial data, we organize the remaining literature into geospatial data domain categories and provide insight into how these categories relate to VIS Conference paper types We also contextualize our results by investigating the use of geospatial terms in IEEE Visualization publications over the last 30 years Our work provides an understanding of the quantity and role of geospatial subject matter in recent IEEE VIS publications and supplies a foundation for future meta-analytical work around geospatial analytics and geovisualization that may shed light on opportunities for innovation

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DOI
01 Jan 2021
TL;DR: It is shown that the extraction produces stable and coherent geometries even for highly irregular eddies in the Red Sea, which indicates that the methods will be considered for newly simulated, even larger data sets.
Abstract: Oceanic eddies, which are highly mass-coherent vortices traveling through the earth’s waters, are of special interest for their mixing properties. Therefore, large-scale ensemble simulations are performed to approximate their possible evolution. Analyzing their development and transport behavior requires a stable extraction of both their shape and properties of water masses within. We present a framework for extracting the time series of full 3D eddy geometries based on an winding angle criterion. Our analysis tools enables users to explore the results in-depth by linking extracted volumes to extensive statistics collected across several ensemble members. The methods are showcased on an ensemble simulation of the Red Sea. We show that our extraction produces stable and coherent geometries even for highly irregular eddies in the Red Sea. These capabilities are utilized to evaluate the stability of our method with respect to variations of user-defined parameters. Feedback gathered from domain experts was very positive and indicates that our methods will be considered for newly simulated, even larger data sets.

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A lagrangian method for extracting eddy boundaries in the red sea and the gulf of aden" ?

The authors propose the following model: Inside an eddy, particles rotate around a common core and thereby remain at a constant distance under a certain parametrization.