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NCAR Technical Notes NCAR/TN-557+STR

Quantifying the Spatial Structure of Tropical Cyclone Imagery

Trey McNeelyAnn B. LeeDorit HammerlingKimberly Wood

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National Center for Atmospheric Research

P. O. Box 3000

Boulder, Colorado 80307-3000

NCAR/TN-557+STR

NCAR Technical Note

______________________________________________

2019-June

Quantifying the Spatial Structure of Tropical Cyclone

Imagery

Trey McNeely

Department of Statistics & Data Science, Carnegie Mellon University, Pittsburgh, PA Ann B. Lee Department of Statistics & Data Science, Carnegie Mellon University, Pittsburgh, PA Dorit Hammerling

Analytics and Integrative Machine Learning Group, National Center for Atmospheric Research, Boulder, CO Kimberly Wood Department of Geosciences, Mississippi State University, Mississippi State, MS

NCAR Laboratory

NCAR Division ______________________________________________________

NATIONAL CENTER FOR ATMOSPHERIC RESEARCH P. O. Box 3000

BOULDER, COLORADO 80307-3000 ISSN Print Edition 2153-2397

ISSN Electronic Edition 2153-2400

Quantifying the Spatial Structure of Tropical Cyclone Imagery

Trey McNeely ∗1, Ann B. Lee †1, Dorit Hammerling ‡2, and Kimberly Wood §3

1Carnegie Mellon University Department of Statistics & Data Science2National Center for Atmospheric Research

3Mississippi State University Department of Geosciences

Abstract

Tropical cyclones are highly organized, rotating storms which rank among the most costlynatural disasters in the United States. The processes which drive intensification and weakeningof such storms are still not fully understood, particularly when these intensity changes occur onshort time scales. The physical and environmental factors used in intensity prediction schemesoften do not consider the spatial structure of the storm and, taken alone, are inadequate fordescribing the evolution of storms during rapid intensity change events. Since tropical cyclonesgenerally form far from land-based observing networks, we often rely on satellite observations toassess these storms, particularly infrared observations that reveal cloud top temperatures as aproxy for strength of convection in these storms. In addition, advances in satellite instrumenta-tion has continued to improve the spatial and temporal resolution of these observations. To takeadvantage of this information, we develop a suite of features which quantify the spatial structureof convection within the storm via infrared brightness temperature data from the GeostationaryOperational Environmental Satellites. These features target the bulk morphology, core struc-ture, and overall organization of the storm to provide a rich, interpretable description of thespatial structure of convection within the storm. This quantification provides a foundation forapplying powerful but otherwise hard-to-interpret machine learning techniques to further ourunderstanding of the physical processes behind rapid intensity change in tropical cyclones andsupport future improvements in forecasting such changes.

Keywords: Tropical Cyclones, Image Analysis, Spatiotemporal, Cloud Top Temperature

∗imcneely@andrew.cmu.edu†annlee@andrew.cmu.edu‡dorith@ucar.edu§kimberly.wood@msstate.edu

1

Contents

1 Introduction 3

2 Data 4

3 Features to Quantify Spatial Structure 53.1 Level Sets for Bulk Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 Skew Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Shape Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.4 Coldest Cloud Tops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Radial Profiles for Core Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Image Centering and Eye Detection . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Eyewall Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Image Gradients for Axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 Deviation Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Deviation Angle Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Conclusion 14

2

1 Introduction

Tropical cyclones (TCs) are long-lived, powerful, and spatially extensive; these three factors placethem among the most damaging and costly natural disasters in the United States (Klotzbachet al., 2018). Over the past several decades, the forecasting of TC trajectories via statistical andphysical models has shown marked improvement. Meanwhile, intensity guidance has shown lessdrastic improvement; much of said improvement in intensity guidance is a side-effect of improvedtrajectory forecasting (DeMaria et al., 2014). Knowing how strong a storm will be at landfall is ascritical as knowing where or whether it will make landfall. This information informs allocation ofemergency response supplies and evacuation guidance.

Of particular note are periods of rapid intensity change. Forecast errors tend to be highestwhen storm intensities change most rapidly. These rapid changes in the intensity of a storm can beboth deadly and costly if not predicted. For example, if a storm unexpectedly intensifies, peoplemay be caught unprepared, or the window for orderly evacuation may have passed. Likewise, therapid weakening of storms can invalidate costly evacuations, leaving authorities appearing to have“cried wolf” and damaging public trust in future guidance.

In addition to humanitarian and fiscal considerations, prediction of rapid intensity changesremain a scientific puzzle. The TC intensification process resembles a Carnot heat engine by whichthe secondary (vertical) circulation converts heat energy into mechanical energy that strengthensthe horizontal circulation and thus the maximum winds, the subtleties of which are not fullyunderstood (e.g., the summary provided by Emanuel 2018). This problem has sparked researchinto cases of rapid intensification in particular. For example, the Rapid Intensification Index ofKaplan et al. (2010) attempts to model the probability of rapid intensification events using areaaverages such as the wind shear within 500 km of the storm center (SHRD) or the percentageof the region between 50 and 200 km covered by cloud tops with temperatures below −30◦C(PX30). These predictors are in turn drawn from the Statistical Hurricane Intensity PredictionScheme (SHIPS; DeMaria and Kaplan 1999) for general intensity guidance, which consist primarilyof averaged quantities over storm-centered rings and thus do not capture spatial structure in thefields from which they are computed. However, traditional sets of physical and environmentalpredictors, such as wind shear or average sea surface temperatures, appear inadequate to explainrapid intensity changes on their own given our current level of understanding.

In order to better understand the processes underlying rapid intensity change, we intend toquantify how the actual convective structure of the storm changes over time. Infrared brightnesstemperature imagery of TC cloud tops provides a proxy for convective strength, where low tem-peratures indicate stronger updrafts are likely generating deeper convection. The GeostationaryOperational Environmental Satellite 16 (GOES-16) provides a real-time stream of such images overthe North Atlantic basin, while previous iterations of GOES provide historical data stretching backseveral decades. The aforementioned SHIPS scheme and its accompanying Rapid IntensificationIndex begin to incorporate this information with the PX30 statistic mentioned above, but do notcapture spatial patterns in TC convection. We will develop a suite of characterizations of thestructure of cloud top temperatures in the infrared imagery from GOES. A rich characterizationof the convective structure within a TC will both provide insight into the evolution of the stormsduring such events, serving as a first step on the road to improved guidance.

An ideal, strong storm is nearly axisymmetric, with a warm eye surrounded by a ring of verycold cloud tops in the eyewall. Beyond the eyewall, cloud top temperature slowly rises up to theedge of the storm’s cirrus canopy, where the warm sea surface becomes visible. We can see thisapproximate structure in 2014’s Hurricane Edouard (Figure 1), which exhibits a well-defined, warmeye in the infrared, as well as a clear ring-shaped eyewall surrounding it.

3

Figure 1: Hurricane Edouard at 18 UTC 16 September 2014.

In this work, we seek to quantify deviations from this idealized structure as well as describethe location and concentration of regions of strong convection. We do so with the dual focus ofinterpretability and descriptiveness. Interpretable features can target physical qualities of knownor suspected physical importance, or they can simply quantify a qualitative description, such as astorm appearing “skewed” or “stretched.” The whole feature set must be descriptive, reducing animage (say, 200,000 pixel values) to a much shorter feature vector (say, 100 feature values) whileretaining most of the useful information in the image. Such a quantification will enable comparisonof the evolution of convective structure during typical TC intensity change with the evolution duringrapid intensity changes.

In this paper, we will first describe the data used in developing our spatial features in Section2. We will then develop a suite of tools for quantifying various aspects of the spatial structurein GOES imagery throughout Section 3. In Section 3.1, we will examine the bulk morphology ofthe storms. In Section 3.2, we will examine the core structures in the storm, such as the eye andeyewall. Finally, in Section 3.3, we will examine the global organization of the storm.

2 Data

High-resolution spatio-temporal observation of TCs is not necessarily common, particularly forover-water storms. Many TC observations come from land-based, airborne, or polar-orbiting mea-surement systems which are limited in temporal resolution or fail to cover the full basin. Asgeostationary platforms, the GOES-East/West satellites provide high-resolution imagery on a reg-ular basis (every 15 minutes with the newest satellites). The GOES data are available for bothover-water and land-threatening storms. The GridSat-GOES database provides historical, hourlyGOES Channel 4 (infrared band) brightness temperatures remapped to an even 0.04◦ grid over thewhole GOES domain in the Western Hemisphere, up to GOES-15 (Knapp and Wilkins, 2018).

To determine the location and strength of storms, we use NOAA’s HURDAT2 data set, areanalysis of TC best tracks (Landsea and Franklin, 2013). We use these data to: i) identifycandidate storms, limiting our data set to storms with intensities above 50 kt, where storm structureis more well-defined; ii) compute the 24-hour intensity change of the storm, used to identify periodsof rapid weakening; and iii) retrieve interpolated best-track centers of the TC in order to extractGOES imagery for a storm.

We restrict our sample to over-water (>250km from land) storms with maximum sustained

4

wind speeds in excess of 50 kt. We focus on the North Atlantic basin, where over-water weakeningis less frequent, and define a rapid weakening (RW) event as a 24-hour decay of 25 kt or more. Weextract the GOES imagery within 800 km of the interpolated best track center.

Between 1994 (the beginning of GOES-8) and 2016, there are 2,954 6-hour HURDAT2 entrieson storms meeting our criteria. Of these, only 385 reach or exceed the RW threshold. The 2,954entries comprise 204 different storms, 72 of which undergo at least one RW period. We have 14,475hourly GOES images of overwater storms. Sometimes images are fully or partially missing, or sufferfrom corruption, and we drop these for now. Without further investigation of recoverable levelsof missingness or corruption, we have GOES imagery corresponding to 1,627 of the 2,954 6-hourobservations.

3 Features to Quantify Spatial Structure

In this section, we lay out the suite of features for quantifying structure in the infrared imageryof TCs. We intend for these features to be rich enough to capture meaningful patterns in theimages without sacrificing our ability to interpret the features. We categorize these features bytheir conceptual foundations. In Section 3.1, we utilize level sets to characterize the morphologyof the cloud top temperatures in bulk. This includes quantification of qualitative descriptions suchas “skew” along with scientifically motivated features such as tracking the location of the coldestcloud tops within the storm. Section 3.2 develops radial profiles of mean cloud top temperaturesacross different radii in order to describe several core elements of TC anatomy, such as the eyeand eyewall. Finally, Section 3.3 implements the deviation angles introduced by Pineros et al.2008, targeting storm-wide symmetries. Descriptive and interpretable feature design will enableapplication of powerful machine learning methods with scientifically meaningful results.

3.1 Level Sets for Bulk Morphology

A sublevel set of a real-valued function is the subset of the function’s domain over which thatfunction takes values below a certain level; formally, for f : R2 7→ R,

L(f, c) = {(r, θ)|f(r, θ) ≤ c}. (1)

In this work, we will frequently work in polar coordinates in order to leverage the idealized struc-ture of TCs, which is symmetric about some center. Take T as the polar representation of theoriginal brightness temperature image, T (r, θ). Then, temperature level sets are given by selectinga temperature threshold c:

L(T, c) = {(r, θ)|T (r, θ) ≤ c}. (2)

We note the particularly important set L(T, 0◦). We denote this set as L?. This level set capturesthe region of the storm where convection is strong enough (and thus the cloud tops are high enough)that ice may begin to form in the clouds. The presence of ice is not measured or considered in thiswork, and the threshold serves primarily to exclude low, warm cloud tops.

We can also define the coverage of a temperature level set by the percentage of L? they cover.Let 0 < ε ≤ 1. Then, we choose the smallest threshold c such that∫

L(T,c)dA ≥ ε

∫L?dA, (3)

where dA is the differential area element. This gives the image region of lowest temperaturecontaining at least ε of the area of the baseline L?. The specification of a baseline set is critical

5

(a) Temperature level sets for Edouard,c ∈ {0◦,−20◦,−40◦,−60◦}.

(b) Coverage level sets for Edouard,ε ∈ {0.1, 0.25, 0.5, 1}.

Figure 2: The two types of sublevel sets for Edouard computed on a vector of temperatures c and coverages

for reducing edge effects (even with an image extending 800 km from the TC center). We computethese sets for different values of the coverage ε, and refer to the sets as coverage level sets at ε.Figure 2 shows temperature and coverage levels sets for Edouard.

The two types of sets serve different purposes. Examining fixed levels c in the temperature levelsets provides sets which are comparable in their minimum intensity; across all storms, the 0◦C levelsets are capable of forming ice in the cloud tops, while the −60◦C sets always have some minimumlevel of convection in the cloud tops. We can use this to estimate the size (via L?) of a storm orcoverage of strong convection (via L(T,−60◦)). Meanwhile, a fixed value of ε in the coverage setsresults in sets which are less comparable between storms, as the exact level adapts to the overallstrength of the storm as well as the size of the storm (as measured by L?). These sets are moreuseful in tracking features such as the location of strongest convection (via some low-ε set).

This document samples the levels sets over discrete sets of c and ε. This is for the purposesof input to linear or tree-based regression models. The level-sets, however, are defined on sets−∞ < c <∞ and 0 < ε ≤ 1. Then, the features computed on level sets can actually be continuousfunctions on those same domains. We have picked vectors which we believe to be a reasonablesampling of those domains.

For each of the (temperature or coverage) level sets, we compute a group of morphology featuressuited to its type. Size is computed on the temperature level sets. We track the location of thecoldest 5% of cloud tops via the coverage level sets. Finally, skew and shape asymmetry measuresare computed on both temperature level sets and coverage level sets. These morphology features,then, use the size, shape, and location of various level sets to characterize the bulk morphology ofthe storm. Table 1 summarizes associated morphology features for each type of level set.

Level Set TypeTemperature Coverage

Size, Shape/Skew Coldest Cloud Top Shape/Skewc ∈ {0◦,−20◦,−40◦,−60◦} ε = 0.05 ε ∈ {0.10, 0.25, 0.50}

Table 1: Summary of associated bulk features for each type of level set.

6

3.1.1 Size

We define the size of a storm at a given level as the area spanned by cloud tops below a certaintemperature. This captures the size of the storm at various levels of convection. For example,the Size(L?) could describe the overall storm size, while Size

(L(T,−60◦)

)describes the extent of

powerful convection within the storm. We do not compute the size of coverage sets, as they aretrivially ε-multiples of Size(L?).

Formally, define the size of a level set L as

Size(L) =

∫LdA. (4)

This is computed for the temperature levels sets (using coverage level sets would be trivial). Wereport this size in 1,000’s of km2. Note that size (and all morphology features) are computed onthe 800 km-radius stamp; the maximum coverage is thus about 2,000,000 km2. For Edouard, thesizes of the four temperature sets in Figure 2(a) are 591, 408, 232, and 9.3.

We note here that the size at lower levels is likely to be inflated by cloud tops beyond the stormitself, such as at the top of Figure 1(a); improvements to the level set features should begin witha screening rule to identify and discard such regions. For asymmetric sets, such as the left-skewedsets in this example, using this notion of storm size has clear benefits over approaches attemptingto assign a radius to the storm; the use of level sets gives a notion of size which is organizationagnostic.

3.1.2 Skew Asymmetry

We define the skew of a storm as an overloading or unbalancing of the cloud tops; when a storm isskewed, there is more cloud-mass to one side of the storm than the other. We quantify the skew ofa storm using both temperature and coverage level sets. The temperature threshold sets providea consistent level across storms, while the coverage sets adapt to the temperature profile of anindividual storm.

Consider again the level sets L(T, c). Then, we define the skew asymmetry at level c (or coverageε) as the vector (from polar origin) to the center of mass of a level set. Formally, let (x0, y0) be thecenter of the storm as defined by the best track. Then,

x =

∫∫L(T,c)

(x− x0)dxdy (5)

is the x-coordinate of the center of mass relative to the center. We repeat for the y-coordinate,then compute

Skew(L) =

(r̃, θ

)=

(1

r

√x2 + y2, arctan(y, x)

)(6)

r =

∫LrdA

/∫LdA (7)

where r̃ captures the magnitude of asymmetry (normalized by the average radius of the set, r)and θ captures the direction. In Figure 4(a), asymmetry is higher for the high-c (0◦C) coveragesets, where the storm has excess cloud tops to the left and top of the storm. In Figure 4(b), themagnitude of asymmetry is smaller, but still points towards the excess clouds to the left. Wesuccessfully capture the overloading-type asymmetry.

7

(a) At 0◦C, Edouard is skewed left. After nor-malizing for the average set radius of 371km, theskew of this set is 0.45.

(b) At −40◦C, Edouard is still skewed left. Afternormalizing for the average set radius of 224km,the skew of this set is 0.34.

Figure 3: The skew of two temperature level sets for Edouard. As the center of mass (red point) approachesthe average set radius (gold ring), the skew magnitude approaches 1.

3.1.3 Shape Asymmetry

Where the skew asymmetry measure (Skew(L)) is sensitive to overloading-type asymmetry, it doesnot capture ellipticity or other regular departures from the ideal nested-ring shape. We develop asecond method for capturing this shape asymmetry.

Let Sθ be the sector{

(r, φ) : r ≤ rmax, θ − δθ2 ≤ φ ≤ θ + δθ

2

}. Then, let

ρ(θ) =

∫L⋂SθdA∫

L dA

be the sector ratio. This is the fraction of a level set contained within the sector Sθ. As wesweep this sector about the storm, we get an angular profile of the level set, smoothed by δθ. Wedemonstrate this computation in Figure 4(a).

We use the Mean Absolute Deviation (MAD) from a fitted circle for this summary. Formally,fit a circle with the general form

ρ̂(θ) = r0 cos (θ + θ0) +√A2

0 − r20 sin2 (θ + θ0), (8)

to ρ(θ), where (r0, θ0) is the displacement of the circle center from the pole, and A0 is the radiusof the circle. Then, compute the MAD as

Shape(L) = 100

∫|ρ(θ)− ρ̂(θ)|dθ, (9)

where the factor of 100 brings the measure to an easily read magnitude. This value grows as thecold cloud tops become less uniformly distributed in angle about their center.

We can also compute an effective eccentricity of ρ. To do this, we simply chose the minimum-radius circle (concentric with ρ̂) which contains ρ and the maximum-radius circle contained by ρ.Formally, take the circle

R(θ,A) = r0 cos(θ + θ0) +√A2 − r2

0 sin2(θ + θ0), (10)

8

(a) Demonstration of the ρ-curve calculation forc = −40◦C. As we sweep Sθ (red) about thecenter of the storm, we compute the fraction (ρ,gold) of the set within Sθ.

(b) Fitted ρ̂(θ) (red) for the ρ(θ) (gold) in (a),resulting in Shape(L) = 0.65. The remaining cir-cles (black) give Rinner and Router for fixed cen-ters, resulting in εeff = 0.941.

(c) The black circles give Rinner and Router formoving centers, resulting in εeff = 0.837.

Figure 4: An example of computing the shape measures of a set for Edouard. Note in particular the high εeff

resulting from fixed centers in (b). Allowing the centers of the black circles to move (c) results in a bettermeasure of storm shape.

where r0 and θ0 come from the fit in Equation 9. Then,

Rinner = max{A > 0|R(θ,A) < ρ(θ) ∀ θ}, (11)

and, similarly,Router = min{A > 0|R(θ,A) > ρ(θ) ∀ θ}. (12)

We then define the effective eccentricity as

εeff =

√1−

R2inner

R2outer

, (13)

the eccentricity of an ellipse with minor axis Rinner and major axis Router. This can be very unstablefor more eccentric shapes. For a more stable measure of effective eccentricity, we lift the requirementthat the circles associated with Rinner and Router be concentric with ρ̂.

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3.1.4 Coldest Cloud Tops

We are also interested in the location of the coldest cloud tops in a storm. These represent theregions of strongest convection, and are thus the most energetic regions of the storm. Following thegroundwork laid in Equations 5 and 6, we can take an non-normalized skew vector for a coveragelevel set.

Formally, we apply Equation 5 to the 5% coverage level set(L(T, ε = 0.05)

)to obtain the center

of mass of the level set. Then,

Cold(L) =

(√x2 + y2, arctan(y, x)

)(14)

gives the location of the coldest cloud tops. This is similar in computation to the skew of a coverageset, but we do not normalize by r, the average radius of the set, as we did in Equation 6. We alsorecord the 5% temperature, the value of c in Equation 3 when ε = .05. These three numbers(magnitude and direction of vector to coldest set, maximum temperature of coldest set) provideinsight into the peak intensity within the storm.

3.2 Radial Profiles for Core Processes

In general, the core of a TC (the eye and eyewall) will be fairly axisymmetric. Axisymmetry hererefers to a rotational symmetry about a vertical axis through the eye of the TC; that is, a perfectlyaxisymmetric storm is invariant under rotations about its eye. The idealized TC organization,constructed of ring-shaped sets, is axisymmetric. The most severe deviations occur in the overcast.If perfect axisymmetry is assumed, then we can discard the rotational dimension entirely and reducethe storm to a radial profile.

Let T (r) represent the radial profile, defined as the average cloud top temperature at a givenradius:

T (r) =1

2π

∫ 2π

0T (r, θ)dθ. (15)

Then, in the idealized storm, we expect a general structure of high temperatures at low r (in theeye), low temperatures at intermediate values of r (in the eyewall), and high temperatures at highr (in the overcast and outside of the storm). We demonstrate this in Figure 5, but note that theeye and eyewall are not sharply defined. This is because we have selected the image center basedon the interpolated best track. We address a better centering of the image below.

3.2.1 Image Centering and Eye Detection

In Figure 5, we notice that the eye is poorly defined; the center of our image, drawn from an inter-polation of the best track, does not align with the center of the eye. Since we seek a characterizationof eye features, it stands to reason that we should align our coordinate system to maximize thedifference between the eye and the eyewall in our radial profile.

Before we can maximize this difference, we need to define a few key points along the profile.First, define the point of peak convection, which should locate the eyewall. Let

(rmin, Tmin) =

(arg min

rT (r), min

rT (r)

). (16)

Then, we can define the eye using a level set on T (r): let

E = {(r, θ) : r < rmin, T (r) ≥ Tmin + c}. (17)

10

Figure 5: Radial profile (T (r)) for Edouard. Near r = 0, the temperatures are high, as expected, but basedon Figure 1 we would expect these temperatures to exceed 0◦C. This discrepancy is a consequence of poorimage centering.

Figure 6: Radial profile of Edouard about the center which maximizes average eye temperature.

This captures the region of high temperature interior to the eyewall. In practice, we use a temper-ature difference of c = 25◦. Then, we can shift our coordinate center around near the center of theimage in order to maximize the average value of T (r) within E . This centers the coordinate systemon the eye and increases the contrast between eye and eyewall in T (r). This is demonstrated inFigure 6. If there is no center for which an eye is present, the best track center is used.

We describe the eye based on its size only. We assume a circular eye; we could assess eccentricityof the eye, but the radial profile does not provide a strong tool for doing so. Likewise, any asymmetryor disorder measures are applied to the eyewall rather than the eye.

Since the eye is assumed to be circular, we characterize it only by a radius. Formally,

Radius(E) = max{r : r ∈ E ∪ {0}}. (18)

If there is no eye visible in the infrared, the radius will be 0. Under this definition, the radius is thedistance from the center at which the average brightness temperature drops within 25◦C of Tmin.

3.2.2 Eyewall Features

Following once more the idealized structure of a mature TC, the eyewall of the storm is an annulusof low temperature where the strongest winds (which define intensity) are typically found. Theorganization and size of this region is likely to change over time during a rapid intensity change

11

event. We can again use a level set over the radial profile to define the eyewall. Formally, let

W = convr({(r, θ) : r < rmin, T (r) ≤ Tmin + c′}

), (19)

where convr(·) denotes the convex hull of the set of radii r (but not the angles θ). This operationforces the set to be a single annulus. In practice, we use c′ = 5◦C.

We do not need to record the minimum radial temperature Tmin, as we capture this in moredetail with the Cold(L) feature in Section 3.1.4. Instead, we describe the size and disorder of theset. The size is again simply described by the width in radius, in similar fashion to the radius ofthe eye. Formally,

Width(W) = max({r : r ∈ W})−min({r : r ∈ W}). (20)

This is the distance between the inner and outer edges of the eyewall.We describe the disorganization of the eyewall in two ways. The first of these imposes no

structural assumptions, simply taking the variance of the temperature across the eyewall. Formally,

Var(W) =

∫W

(T (r, θ)− T )2dA/

∫WdA, (21)

where T =∫W T (r, θ)dA/

∫W dA. Note that in spite of the definition of the eyewall, T (r, θ) is

not actually bounded; the original level set only bounds T (r) to a 5◦C range, but the convex hulloperation may include higher temperatures, and returning to the 2-dimensional temperature imagecertainly will. If the eyewall is fragmented or imbalanced (colder on one side than the other), thiswill increase the variance.

Our second measure of asymmetry places structural assumptions on the image gradients withinthe eyewall, using the deviation angle variance derived throughout Section 3.3. See Equation 25 inthe context of that section for a formal definition.

3.3 Image Gradients for Axisymmetry

With an accurate image center in hand, we implement a method of quantifying axisymmetryusing image gradients. Recall again the idealized TC. Under this structure, all gradients of thetemperature structure point either directly towards the eye (for points inside the peak convectionin the eyewall) or directly away from the eye (for points outside the peak of the eyewall). Thefoundation of image gradients provides a reasonable place to begin quantification of global stormdisorder.

For the remainder of this section, we describe mathematically the Deviation Angle Variancecomputation introduced by Pineros et al. 2008. For reasons of interpretability, we do not follow theprocedure for locating a central reference laid out therein, but use the identified eye location fromSection 3.2. We believe the eye to be the most intuitive reference, as it corresponds to the identifiedcenter in an ideal storm. As a result, however, we rely on the interpolated HURDAT2 best tracklocation in the absences of an eye. Practically, the center will typically only move by a few pixels asa result of our centering technique, so the difference in deviation angles is usually minimal beyondthe immediate neighborhood of the referenced center. A more computationally expensive approachwould involve calculating deviation angles for multiple centers, taking the lowest DAV center asthe storm location.

The foundation of the method relies on the direction of gradients of temperature images, denoted∇T . Before the gradients are computed, we smooth the infrared image with a two-dimensionalGaussian of standard deviation 50 km. Then, the gradients in the horizontal and vertical directionsare computed via a Sobel filter, from which the direction of ∇T can be computed.

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Figure 7: The deviation angle at a given point r is based on the direction of the image gradient ∇T (r)) atr relative a tangent vector v(r) of an origin-centered circle through r. In the figure above, the deviationangle ψ = |φ| − π/2, so that ψ = 0 when the gradient is aligned with the radial line, and ψ = ±π/2 when∇T (r) = ∓v(r), respectively.

3.3.1 Deviation Angles

The deviation angle compares the gradient angles of an infrared image to those expected of anidealized storm. At any point r = (x, y), consider a circle centered on the origin with radius|r|, and let v(r) be the (counter-clockwise) tangent vector at that point. Then, we define anintermediate angle, φ(r), to be angle representing a counter-clockwise rotation from v(r) to ∇T (r),the temperature gradient at r. This is illustrated in Figure 7. That is,

φ(r) = ∠(v(r),∇T (r)

), (22)

where the angle φ is largest and taking values ±π when the gradient ∇T (r) = −v(r), and smallestand taking the value 0 when ∇T (r) = v(r). When the gradient points towards or away from theimage center, φ takes values near π/2 and −π/2, respectively.

Recall that we are quantifying deviations from the idealized storm, wherein all gradients pointalong these radial lines. That is, we care only how far clockwise or counterclockwise from this radialline the gradient angle is. Thus, we define the deviation angle ψ as

ψ(r) = |φ(r)| − π/2 =

∣∣∣∣∠(v(r),∇T (r))∣∣∣∣− π/2. (23)

The deviation angles now take the value 0 when the gradient is pointed along the radial line, thevalue −π/2 when ∇T (r) = v(r), and the value π/2 when ∇T (r) = −v(r). The above gives us anintuitive notion of the gradient’s deviation from that of an idealized storm at any point r.

3.3.2 Deviation Angle Variance

A natural way to use the deviation angles is to look at the variance. We use a cumulative measure,looking at the variance of the deviation angles within a given radius. This function is defined as

DAV(r) = Var[ψ(r)

∣∣|r| ≤ r] (24)

We plot this function in Figure 8(c). While the DAV is typically defined over circles, we can alsoapply it to the set W. Formally,

DAV(W) = Var[ψ(r)∣∣r ∈ W]. (25)

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(a) IR brightness cloudtop temperatures forEdouard.

(b) Deviation angles for Edouard. The ring (gold)marks the radius of minimum DAV.

(c) DAV vs radius for Edouard. Each point on the line indicates the variance of ψwithin a circle of the given radius, where ψ is computed about the fixed center chosenin Section 3.2.1. The vertical line (gold) marks the radius of minimum DAV.

Figure 8: Deviation angles and DAV for Edouard. Notice the discontinuities (and corresponding DAV spikes)where the magnitude of the gradient is near zero (such as the center of the eye, the bottom of the eyewall).The minimum DAV radius appears to occur just outside of the eyewall, beyond which we would expectdisorganization to increase.

This provides an alternative quantification of disorder in the eyewall of the storm.Note that the magnitude of the gradient does not appear in the deviation angles or DAV

calculations. This allows the deviation angles to capture organization in the whole overcast of thestorm, where gradient magnitudes may be small, rather than letting high-gradient features likethe edge of the eye or outer edges of the overcast dominate the feature. However, it also meansthat regions of near-zero gradient may see the deviation angles changing between extreme values,inflating the variance. For example, deviation angles near the center of the eye or the top of theeyewall (local extrema) are often near π/2 or −π/2. We can see the eyewall clearly in Figure 8(c)near 50 km, where the DAV peaks as a result of this effect. A more stable approach ignores pixelswhere gradient magnitudes are below a certain threshold.

4 Conclusion

The features of Section 3 follow from three conceptual foundations: level sets, the radial profile, anddeviation angles. Each of these provides a rich characterization of a clear aspect of the convectivestructure of a TC. The various applications of relatively few conceptual tools makes the math behindthe suite of features explicable. Each feature can be linked back to a fundamental aspect of theconvective structure in the storm, answering scientific questions such as “How variable is convectionover the eyewall of the storm?” This quantification of spatial structure allows us to describe theconvection in an image of a storm, which will enable future work to describe the temporal evolution

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of convective structure as a storm evolves via application to GOES imagery at 1 hour or 15 minuteresolutions. A quantification of structural change over time will allow us to identify differences inthe evolution of rapidly changing storms, providing insight into the most poorly understood eventsand ultimately identifying predictors which can improve operational forecasts.

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